Research ArticleRational Rock Physics for Improved Velocity Prediction andReservoir Properties Estimation for Granite Wash (Tight Sands)in Anadarko Basin Texas

Muhammad Z A Durrani1 Keith Willson2 Jingyi Chen1 Bryan Tapp1 and Jubran Akram3

1 Department of Geosciences University of Tulsa 800 South Tucker Drive Tulsa OK 74104 USA2Newfield Exploration Company 101 E 2nd Street Tulsa OK 74103 USA3Department of Geoscience University of Calgary 2500 University Dr NW Calgary AB Canada T2N 1N4

Correspondence should be addressed to Jingyi Chen jingyi-chenutulsaedu

Received 22 January 2014 Revised 27 June 2014 Accepted 9 July 2014 Published 26 August 2014

Academic Editor Rudolf A Treumann

Copyright copy 2014 Muhammad Z A Durrani et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Due to the complex nature deriving elastic properties from seismic data for the prolific GraniteWash reservoir (Pennsylvanian age)in the western Anadarko BasinWheeler County (Texas) is quite a challenge In this paper we used rock physics tool to describe thediagenesis and accurate estimation of seismic velocities of P and Swaves inGraniteWash reservoirHertz-Mindlin andCementation(Dvorkinrsquos) theories are applied to analyze the nature of the reservoir rocks (uncemented and cemented) In the implementation ofrock physics diagnostics three classical rock physics (empirical relations Kuster-Toksoz and Berryman) models are comparativelyanalyzed for velocity prediction taking into account the pore shape geometry An empirical (119881P-119881S) relationship is also generatedcalibratedwith core data for shearwave velocity prediction Finally we discussed the advantages of each rock physicsmodel in detailIn addition cross-plots of unconventional attributes help us in the clear separation of anomalous zone and lithologic properties ofsand and shale facies over conventional attributes

1 Introduction

For the last sixty years the Anadarko Basin has been a prolificsetting for oil and natural gas explorations in North Americalargely focused on two natural gas plays the Woodfordshale and the Granite Wash The exploration activity in theGranite Wash play has increased over the last decade TheGraniteWash play straddles the Oklahoma-Texas border andcovers more than six counties (Hemphill Roberts WheelerBeckham Roger Mills and Washita) (Figure 1) The play is aseries of stacked potential play zones each of which should betreated differently and requires a more geophysically focusedapproach to achieve success Due to the low porosity andpermeability cementation and the presence ofmicrofracturesmake each interval a slightly different nut to crack [1]

Desmoinesian age (Granite Wash) deltaic sedimentsreflect a long diagenetic history with extensive cementation

by quartz and calcite Postcementation secondary dissolu-tion and leaching of calcite cement and framework grains(eg feldspar rock fragments) producedrecovered currentporosity in the facies [2] The geological and depositionalfactors such as lithology fluid and porosity (LFP) directlyaffect the elastic properties of the rock and subsequently theseismic responses Because of the complex nature of Granitewash it has mystified geologists and petrophysicists as wellas geophysicists alike thus making geophysical experimentaland numerical studies extremely challenging

The solution to the problems requires rational rockphysics to tackle complex mineralogy and poor under-standing of diagenetic conditions to reexamine the currentunderstanding of the reservoir properties of Granite WashRock physics bridges the reservoir quantitative (eg P-and S-wave velocities impedances and elastic moduli) andreservoir qualitative (eg porosity clay content lithology

Hindawi Publishing CorporationInternational Journal of GeophysicsVolume 2014 Article ID 209351 15 pageshttpdxdoiorg1011552014209351

2 International Journal of Geophysics

HugotonembaymentOklahoma

Cim

arro

n A

rch

Deep Anadarko Basin

Amarillo upliftPaloDuroBasin

HollisHardemanBasin

Wichita mountains

Texas

Texas

Marietta

Ardmore

Arbuckle

mountains

Kansas

Kansas

Studyarea

Northern shelf

Nem

aha u

plift

A

N

0 100(km)

Oklahoma

Study area

Anadarko Basin

NStudy areat

koko

Figure 1 Location map of the study area Stiles Ranch field Wheeler County Anadarko Basin Texas (USA)

and saturation) properties In the past decade rock physicsfor tight gas reservoirs has been relatively neglected dueto their complex depositional and diagenetic histories [3]Therefore the ultimate goal here is to exploit a consis-tent methodology which can systematically integrate small-scale rock microstructure using the rock physic diagnosticapproach for GraniteWash reservoirs in order to describe thediagenesis of Granite Wash and accurately estimate seismic(P- and S-wave) velocities To analyze the nature of thereservoir rocks (uncemented and cemented) we apply tworock physics models (1) a rigorous contact model that is acombination of the Hertz-Mindlin [4] theory and modifiedHashin-Shtrikman [5] elastic bounds (2) and Dvorkinrsquos [6]cementation model based on cementation theory Further-more we estimate and comparatively analyze seismic (P- andS-wave) velocity predictions using effective medium theoriesand employ the optimal rock physics model to address theinfluence of pore shapes in granite wash sands from differentaspects based on actual well dataThree standard rock physics[7ndash9] models are comparatively analyzed and discussed as tohow pore shape influences the P- and S-wave velocities in theGranite Wash in the Anadarko Basin

An empirical relation is established to predict S-wavevelocitywhen there is only P-wave data available for quartziticsandstone that has not yet been defined and calibrated byGranite Wash core data Finally we applied Gassmannrsquosequations in order to predict velocity changes resulting fromdifferent pore-fluid saturations and identified anomalous(hydrocarbons) zones by composite cross-plots of rock prop-erties from well logs (Marmaton formation) data

2 Geologic Background

The term Granite Wash generally refers to the sedimentsderived predominantly from the weathering of granitic rocksand closely resembles the original material [13] The progres-sion of Granite Wash plays in the Anadarko Basin startedmore than 300 million years ago as a result of sediment ero-sion of the Amarillo-Wichita uplift [10] Large volumes of

Land areasHigh or moderately high relief

High or moderately high relief

Low relief

Low relief

Marginal areas

Shallow marinePrincipally carbonate deposition

Deep marine

Direction of sediment supplyMajor sourceMinor source

Principally terrigenous clastic deposition

Principally terrigenous clastic deposition

Study areaWheeler County

Figure 2 Middle Pennsylvanian (Desmoinesian) paleogeographyin Midcontinent area from [10] modified

clastic sediments were shed off the uplift and deposited to thenorth in the rapidly subsiding Anadarko Basin (Figure 2)

Varieties of depositional settings have been postulated forvarious reservoirs in the Granite Wash that include braided-stream alluvial-fan fan-delta turbidite and debris-flow en-vironments [14] The deposition in the Anadarko Basinoccurred in a series of alluvial fans and fan deltas that formed

International Journal of Geophysics 3

Proximal areaof lobe

Distal and lateralmargins of lobe

70 NG45 NG

25 NG5 NG

NESW

Proximal areaof lobe

Distal and lateralmargins of lobe

70 NG4445 NG

25 NG5 G NG

ENSWW

Figure 3 Depositional facies distribution of Granite Wash fromproximal to distal settings from [11] Data is courtesy of NewfieldExploration Company

rims around the structural highs that developed duringthe Pennsylvanian period [15] In proximal submarine fansettings (Texas) theWashes are thick-bedded (amalgamated)sheet sands poorly sorted and coarse-grained [11] GraniteWash thickness varies from 1500 to 3500 ft of interbeddedconglomerate and amalgamated sheet sandstones depositedin deeper water setting as turbidite andor debris-flowdeposits Distal submarine fan settings (Oklahoma) containvery thin-bedded sheet sands that have interbedded shaleswith decreased NG ratio well sorted silt to fine-grainedminor clay and abundant carbonate cements (Figure 3)

Rock composition ismainly influenced by deposition andunroofing of Siluro-Devonian and Mississippian carbonatesand discontinuity of beds which are often very thin laterallydiscontinuous and diagenetically altered (cemented) Fromproximal to distal settings the rock pore shape geometrychanges due to several geological processes including initialdeposition of fine to very fine grained sediments and thepresence of various types of dispersed shales and clays inthe pores [16] Figure 4 shows highly variable grain sizeand mineralogy with reduced porosity as a result of variableformation lithology substantial cementation and cuspateto triangularrectangular geometries tied to the geologicalcharacter of the sediment source areas in the immediatelyadjacent Amarillo-Wichita Mountains [11]

Figure 5(a) provides the generalized stratigraphic columnof the Anadarko Basin Thick deposits of Granite Wash aremade up of varying sizes of tight sands containing morethan a dozen underground formations stacked on top of theothers each holding vast quantities of hydrocarbons [12]Theproduction hasmainly come from the targetedDesmoinesian(middle Pennsylvanian) series reservoirs with additionalproduction from Atoka Missourian and Virgilian seriesunits (Figure 5(b)) The target depths in the study area rangefrom 12000 to 16500 ft Examination of more than 100 coreswas studied byCore Lab Reference [11] in proximal and distalfan settings from the Pennsylvanian Granite Wash sectionreveals a complex diagenetic history which is a function ofthe depositional environment and the thermal history of thebasin

3 Data Set

The principal data sources for rock physics diagnostics arewell logs and core measurements Log data includes P- and

S-wave velocities density gamma ray and resistivity Weincorporated the core study that has already been done byNewfield Exploration Company (Tulsa OK USA) whichcontains a series of laboratory experiments performed onthe Granite Wash core samples yielding their petrophysicalproperties that are the key for calibrating and improving therock physics modeling

4 Rock Physics Theory

Generally the velocities depend on the ratio rock modulito density The moduli and density effects generate velocityvariations which depend on the rock properties such asporosity fluid saturation and texture The seismic (P- and S-wave) velocities of an isotropic rockmaterial can be estimatedusing known rock moduli and density as follows

119881P = radic119870sat + (43) 120583sat

120588

119881S = radic120583sat120588sat

(1)

where 119870 is bulk modulus 120583 is shear modulus and 120588 is thebulk density

Generally P- and S-wave velocities are not consideredthe best fluid indicator due to the coupling effect betweenP- and S-waves through the shear modulus and bulk density[17] Rock physics trends appear more discretely in themodulus-porosity plane than in the velocity-porosity planeBulk modulus shows sensitivity to pore fluid (water) anddeformation produced by a seismic wave resulting in changeof pore volume Shear modulus usually does not producea pore-volume change and consequently is not affected bydifferent fluids The work space for rock physics analysis isthe rock physics plane that may be (a) velocity-porosity (b)impedance-porosity andor (c)modulus-porosity plane Pri-marily we prefer elastic moduli-porosity planes to correlatefluid effects change in bulk modulus in comparison withvelocity-porosity planes for diagnosing rock

We apply Dvorkinrsquos [6] rock physics diagnostics tech-nique to log data from the Granite Wash (Marmaton sec-tion) calibrated with core data from a well in Stile Ranchfield Wheeler County Texas USA (Figure 6) Techniqueis applied by using Hertz-Mindlin contact theory com-bined with theoretical elastic bounds (Hashin-Shtrikman [5]bounds) and cementation theory based mathematical model(see overview in [6 18]) to analyze the elastic signaturesof porosity reduction related to depositional sorting anddiagenesis (mechanical and chemical compaction) Firstpressure dependent Hertz-Mindlin contact theory is usedto estimate the dry moduli of the rock at critical porosity(040) end member Then we constructed moduli of thesolid mineral for zero porosity The moduli of the dry rockbetween high porosity (critical porosity) and zero porosityare estimated by Hashin-Shtrikman bounds which providetheoretical approximations of effective elastic moduli of amixture of grains and pores Hashin-Shtrikman bounds

4 International Journal of Geophysics

CaIQo

Fo

01mm

(a)

ChIAq

20 microns

(b)

Ab-o

Ab-o

Ab

III

10 microns

(c)

Figure 4 Petrography and mineralogy Granite Wash (Marmaton section) sandstones pore-filling constituents are (a) calcite fill pores andquartz overgrowths feldspar overgrowths (b) euhedral microquartz crystals and (c) authigenic albite overgrowths and single crystals from[11] Data is courtesy of Newfield Exploration Company

Systemsseries

Ochoan

Guadalupian

Leonardian

Wolfcampian

Virgilian

Missourian

Desmoinesian

Atokan

Morrowan

ChesterianMiss

Penn

sylv

ania

nPe

rmia

n

Elk city sandstoneDoxey shale

Cloud chief formationWhitehorse group

El Reno groupHennessey shale

Garber sandstoneWellington formation

Chase groupCouncil grove group

Admire groupWabaunsee group

Shawnee groupDouglas groupOchelata groupSkiatook group

Marmaton groupCherokee group

Atoka group

Morrow group

Springer formation

Chester group

Anadarko BasinSw Oklahoma

Gra

nite

was

h

(a)

Permian

Virgilian

Missourian

Desmionesian

Atokan

Morrowan

Sprin

geran

Granite wash

Pm

Pm

Cam

bro-

Ord

ovic

ian

Silu

ro D

evon

ian

Miss

issip

pian

Top Pennsylvanian

Cherokee group

Prec

ambr

ian

0

5000

10000

15 000

0

1000

2000

3000

4000

5000

OilGas

Mississ

ippian

0 5 10(mL)

0 5 10 15(km)

Subs

ea d

epth

(m)

N

Subs

ea d

epth

(m)

washCarbonate-

chert

Dolomitewash

S

Datum sea level

(b)

Figure 5 (a) Anadarko Basin stratigraphic column [12] and (b) stratigraphic dip cross-sectionmdashunroofing sequence [11]

International Journal of Geophysics 5

12300

12350

12400

12450

12500

12550

12600

126500 100 00 06 45 24 245 000 015

GR Sw Vp Vs Density Porosity

Dep

th (f

eet)

12368

12370

12372

12374

12376

12378

12380

12382

12395

12396

12397

12398

12399

12400

12401

12402

12403

12404

12405

12406

12407

12408

12409

Figure 6Well log curves from Stiles Ranch fieldWheeler County Anadarko Basin Texas (USA) From left to right GR (API) 119878119908119881P (Kms)

119881S (Kms) density (gccc) porosity and FMI logs with superposed core [11] data (maroon) Data is courtesy of Newfield ExplorationCompany

assume that each component and the rock are isotropic andelastic [18]

The behaviors of elastic properties due to cementation iscaptured by contact-cement model [6] The cement modelassumes that porosity decreases from the initial criticalporosity value due to the uniform deposition of cement layerson the surface of the grains This cement may be diageneticquartz calcite or reactive clay (such as illite) The diageneticcement dramatically increases the stiffness of the sand byreinforcing the grain contacts The mathematical model isbased on a rigorous contact-problem solution by Dvorkinet al [19 20] It is assumed that themicrostructure used in themodels quantitatively mimics the data The rock diagnosticwas conducted on the log scale and was confirmed with thecore data (thin sections)

Furthermore we employ effective medium [7 9] modelsin order to estimate and comparatively analyze seismic veloc-ity (or moduli) based on pore aspect ratio to characterize theappropriate pore-shape factor in the GraniteWash reservoirsThe generalized formulas of the Kuster-Toksoz [9] model areexpressed by Berryman [7] as follows

(119870119870119879

minus 119870119898)

(119870119898

+ (43) 120583119898)

(119870119870119879

+ (43) 120583119898)

=

119873

sum

119894=1

119909119894(119870119894minus 119870119898) 119875119898119894

(120583119870119879

minus 120583119898)

(120583119898

+ 120577119898)

(120583119870119879

+ 120577119898)

=

119873

sum

119894=1

119909119894(120583119894minus 120583119898) 119876119898119894

(2)

where the summation is over the different inclusions withvolume concentration 119909

119894and

120577 =120583

6

(9119870 + 8120583)

(119870 + 2120583) (3)

The coefficients 119875119898119894 and 119876

119898119894 describe the effect of aninclusion of material ldquo119894rdquo in a background medium 119898 inclu-sion coefficients119875 and119876 describe the elastic properties of thepore phase (Table 1) Two types of calculation algorithms forthe coefficients 119875 and 119876 are followed (1) The Kuster-Toksozmodel (spheroidal inclusions) with arbitrary pore aspect ratiois named 2DKT model and (2) Berrymanrsquos [7] 3D specialpores (penny-shaped crack) are called 3DKT model

To estimate the effect of the pore fluids on the elasticproperties Gassmannrsquos [21] fluid substitution approach isused to model different scenarios Gassmannrsquos equationsrelate the bulk modulus of a rock to its pore frame and fluidproperties The bulk modulus of a saturated rock is given bythe Gassmann theory

119870sat = 119870119889+

(1 minus 119870119889119870119898)2

(120601119870119891) + ((1 minus 120601) 119870

119898) minus (119870

1198891198702119898)

(4)

where119870sat the bulkmodulus of a rock is saturated with a fluidof bulk modulus119870

119891119870119889is the frame bulk modulus119870

119898is the

matrix (grain) bulk modulus and 120601 is porosity Consider

120583sat = 120583119889 (5)

The bulk density (120588sat) is obtained from mass balanceequation

120588sat = 120588119898

(1 minus 120601) + 120601120588119891 (6)

where120601 120588sat 120588119898 and 120588119891are respectively the porosity density

of saturated rocks the matrix and pore fluids The density ofthe solid (mineral) phase is 120588

119898= (1 minus 119862)120588

119902+ 119862120588119888

In this study we take assumptions that reservoir sand-stone mineral grains are mainly (a) cemented (b) spherical

6 International Journal of Geophysics

Table 1 Coefficients of 119875 and 119876 for some specific shapes The subscripts ldquo119898rdquo and ldquo119894rdquo refer to the background and inclusion materials [7]

Inclusion shape 119875119898119894

119876119898119894

Spheres119870119898

+ (43)120583119898

119870119894+ (43)120583

119898

120583119898

+ 120577119898

120583119894+ 120577119898

Needles119870119898

+ 120583119898

+ (13)120583119894

119870119894+ 120583119898

+ (13)120583119894

1

5(

4120583119898

120583119898

+ 120583119894

+ 2120583119898

+ 120574119898

120583119894+ 120574119898

+119870119894+ (43)120583

119898

119870119894+ 120583119898

+ (13)120583119894

)

Disks119870119898

+ (43)120583119894

119870119894+ (43)120583

119894

120583119898

+ 120577119894

120583119894+ 120577119894

Penny cracks119870119898

+ (43)120583119894

119870119894+ (43)120583

119894+ 120587120572120573

119898

1

5(1 +

8120583119898

4120583119894+ 120587120572(120583

119898+ 2120573119898)+ 2

119870119894+ (23)(120583

119894+ 120583119898)

119870119894+ (43)120583

119894+ 120587120572120573

119898

)

Table 2 The core values of moduli and densities of minerals andfluids used in this study [11]

Minerals and fluids Density(gcc)

Bulk modulus(GPa)

Shear modulus(Gpa)

Quartz 265 5580 38Clays 258 209 685Dolomite 287 8010 4500calcite 271 7550 3400Oil 07 162 mdashGas 012 0032 mdash

shape and (c) randomly packed From the core study thepercentage volume of the rock matrix is dominantly clay(15) and quartz (40) The basic parameters used for fluidsproperties in velocity predictionwere taken from core studies(Table 2) Analysis of log and core data from well A suggeststhat these assumptions are met for our reservoir

5 Data Analysis and Results

51 Rock Physics Diagnostics Application to the Granite WashA common practice in rock physics analysis is to generatecross-plots of elastic properties to study diagenetic trends andproperty dependencies of sands and shales The cross-plotsexhibit interdependence among the various rock propertiesWe particularly focused on a thick Marmaton Wash section(12325ndash12435 ft) of well number A containing various logcurves (Figure 6) Marmaton section is marked by extremelylow and fairly constant (about 375) gamma-ray readings andhigh velocity (about 46 kms) surrounded by shale packages

In spite of the congruency between different log curvesin Figure 6 various cross-plots of Marmaton sands do notproduce unique trends (Figure 7) In Figure 7(a) low andhigh ldquoGRrdquo sediments do not show huge variations in bulkdensity implying that both sand and shale arewell compactedand cemented Figure 7(b) shows the same as the case inGR-porosity plot and Figure 7(c) shows that presence ofclay with significantly lower impedance than the quartz itcements Figures 7(d) 7(e) and 7(f) modulusvelocity versusporosity cross-plots show 119881 shape overturned paths Thesuperimposed119881-shaped paths actually represent clay contentfrom clean sands to pure shale The sands overlapped with

the clean shales probably indicate the sand cluster includingboth clean and shaly sands [22] From moduli-velocity-porosity cross-plots we can assess the progress of diagenesisand compaction trends of sediments For soft or uncementedand stiff sediments the paths are fairly gentle with notmuch of velocity contrast between sands and shales Forcemented sediments (case under examination) increasingdepth quartz cementation provides a large contrast in velocityfrom shaly sands to the pure shales This drop might beassociated with the effect of proceeding from grain supportedto clay-supported sediments under relatively high effectivepressure [23 24] Thus a large part of the scatter observedin modulivelocity versus porosity plots is due to presence ofclay in the rocks

We construct the Hashin and Shtrikman [5] bounds(upper and lower) to conceptually explain and model thedata scatter of the Granite Wash from four different wells(Figure 8) The bounds provide an idea of the narrowestpossible range of elasticmoduli without a detailed descriptionof the microstructure of Granite Wash rocks For boundsonly volume fractions of the constituents quartz is 40 byvolume of solid fraction calcite is 7 by volume of solidfraction and their elastic moduli obtained from core dataare used without specifying the geometric details of theirarrangements The upper and lower bounds are determinedby exchanging the materials one that fills the pores andanother constructing the shell The upper bound is realizedwhen stiffer material forms the shell the lower bound isrealized when the stiffer material is physically present in thecore of the shell [18] The separation between upper andlower bounds shows how elastically different the constituentsare The elastic contrast between pure sand and the criticalconcentration mixture is relatively strong because of thevariable elastic contrast between quartz and clay grains Thisimplies that the effect of quartz and clay minerals may becritical for seismic interpretation of Granite Wash

Moduli-porosity data point superimposed with core datafall between the upper and lower bounds which indicates thatMarmaton sands are consolidated sandstone with cemen-tations Superimposed core data from Marmaton sectionconfirms the presence of quartz and calcite cement

52 Confirming the Diagnostic From Figure 4 we observethatmicrocrystalline quartz is found in close contact betweengrains so this quartz cement has a stiffening effect on

International Journal of Geophysics 7

Bulk

den

sity

27

26

25

24

2350 100 150 200

Gamma ray

(a)

Poro

sity

015

010

005

00050 100 150 200

Density

Gamma ray

(b)14

12

10

50 100 150 200

Density

Gamma ray

IP

(c)

Vp

(km

s)

50

45

40

35000 005 010 015 020

Porosity

Density

(d)

70

60

50

40

000 005 010 015 020

Porosity

K(G

Pa)

Vsh

(e)

70

60

50

40

000 005 010 015 020

Sw

Porosity

K(G

Pa)

(f)

Figure 7 Cross-plots of Well log data Marmaton section from Figure 6 (a) bulk density versus GR (b) total porosity versus GR color codedby density (c) P-wave impedance versus GR color coded by density (d) P-wave versus total porosity color coded by density (e) bulkmodulusversus total porosity color coded by shale volume (119881sh) and (f) bulk modulus versus total porosity color coded by water saturation (119878

119908)

the elastic properties of the Granite Wash Studies havesupported the fact that dissolution and local precipitationof quartz accompanied by compaction cause long termdeformation of the aggregate resulting in permeability andporosity reduction [25] Hence quartz precipitation betweenthe grains causes major porosity reduction

For diagnostic purposes we cross-plotted the elasticproperties from well number A (Marmaton section) invelocity-porosity and moduli-porosity planes superimpos-ing theoretical rock physics models (Figure 9) The cross-plots show that gradually increasing clay content with depthcorresponds with decreasing elastic value Based on labora-tory data log data thin section analysis and rock-physicsmodels the data is sorted into Dvorkin and Nurrsquos [6] follow-ing three schemes uncemented (or soft)WashesWasheswithsmall amounts of quartz cement at grain contacts (scheme 1)and Washes from the same formation but with clay cement

(scheme 2) The upper two theoretical (schemes 1 and 2)curves correspond to the pure quartz cases while lower curveis for pure clay or uncemented rock Scheme 1 represents allcement deposited precisely at grain contacts and the slopesteep Scheme 2 curve corresponds to the case where thecontact cement region is formed at the intersection of cementshells that are uniformly deposited on grain surfaces cementfills the space of the large pores and results in a flat slopeIt is quite possible that the variation in microcracks maybe the major reason why the slope of velocity-porosity andmodulus-porosity planes differs in different schemes

We noticed that the maximum data points fall on therock physics scheme 2 line representing uniformly depositedcement on grain surfaceThe thin sections samples from wellnumber A confirmed the rock diagnostic that with increasingdepth and pressure the decreasing porosity is a result ofcontact cement in the rock-physics cross-plots It is quite

8 International Journal of Geophysics

125

100

75

50

25

0000 025 050 075 100

Bulk

mod

ulus

(GPa

)

Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

50

40

30

20

10

0000 025 050 075 100

Shea

r mod

ulus

(GPa

)Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

Figure 8 Moduli versus porosity for different wells and core [11] data with superposition of Hashin and Shtrikman [5] bounds

6

5

4

3

2

00 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Vp

(km

s)

Porosity

Vp versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(a)

80

60

40

20

000 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Bulk

mod

ulus

(GPa

)

Porosity

K versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(b)

Figure 9 (a) P-wave velocity versus porosity and (b) modulus versus porosity (core [11] data included) color coded with shale volume (119881sh)

of the Marmaton section in well number A with rock physics models curves superimposed

International Journal of Geophysics 9

12368

12370

12372

465 470 475

P-wave velocity

465 470 475

P-wave velocity

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

260 265 270 275

S-wave velocity

260 265 270 275

S-wave velocity

Vp2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-09

2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-082DKT-09

Vs

3DKT-013DKT-023DKT-033DKT-043DKT-053DKT-063DKT-073DKT-08

Vp

3DKT-09

3DKT-04

3DKT-013DKT-023DKT-03

3DKT-053DKT-063DKT-073DKT-083DKT-09

Vs

Figure 10 P- and S-wave velocities prediction by employing 2DKT and 3DKT models using different aspect ratios (01ndash09) of well numberA

interesting that the well-log data with resolution of tens-of-centimeter reflects what we observe at the microscaleBy establishing a site-specific rock physics transform (unce-mented and cemented models) between the total porositymineralogy and elastic properties we conclude the rockphysics diagnostics here In the next section we analyze poreshape influence on seismic velocities in order to generatelocally calibrated 119881P-119881S relationship for Granite Wash

53 Analysis of Pore Shapes Influencing Seismic VelocitiesTheoretically it has been reckoned that the velocities in twostates (dry or saturated) are mainly controlled by the effectivemoduli (bulk and shear) which in turn depend on poreshape geometry mineralogy properties of solids the porefluid degree of consolidation and cementation [9 26 27]Thus elastic moduli (or velocities) are basically the functionof all these factors combined Especially in low porosityand permeability rocks elastic moduli of the rock min-eral matrix with spherical pores (high aspect ratio) often

dominate those of the bulk rock [9 26 27] Therefore it istheoretically reasonable to employ the elastic moduli in rocksto estimate seismic velocities Here two effective medium(2DKT and 3DKT) models are investigated to evaluate poreshape geometry using different pore aspect ratios (valueranging from 01 to 09) to predict P- and S-wave velocitiesfrom the well data We know that the pores with high aspectratio (spherical) tend to be stiffer than those with low aspectratio (oblate or penny shape pores) That is to say P- and S-wave velocities in the reservoir with spherical pores tend tobe higher than one with oblate pores or penny shapeThe factcan be easily understood because stiffer pores contribute tothe stiffness of the whole rock while oblate pores contributeto the compliance of thewhole rockHere actual well numberA is taken as an example containing dipole sonic logs toestimate P- and S-wave velocity From Figure 10 we can seeclearly that the increase in P- and S-wave velocities is sharperwhen the pore aspect ratio increases from 01 to 06 for both2DKT and 3DKT models However the increasing trend isnominal when the pore aspect ratio exceeds beyond 06 in

10 International Journal of Geophysics

12366

12368

12370

1237247 48

Vp (kms) Vs (kms)

Dep

th (f

eet)

12366

12368

12370

12372

Dep

th (f

eet)

VpVp-2DKTVp-3DKT

26 27

VsVs-2DKTVs-3DKT

Figure 11 A comparison between the predicted and measured P- and S-wave velocities by employing 2DKT and 3DKTmodels using criticalaspect ratio (06) of well number A

both models Significantly velocities predicted by the 3DKTmodel agreewell with thosemeasured compared to the 2DKTmodel (Figure 11)

It is due to the fact that the model adjusts pore aspectratio or the proportion of various 3D pores and effectivelymodels the geometries of various pore shapes which is thedominant factor influencing the velocities of the elastic-wave propagating in the clastic reservoirs [28] Due to thesubstantial amount of clay (15) contents that may containmicrocracks the dry rock moduli estimation by the Kuster-Toksoz (2DKT) model (assuming ellipsoidal) could not beappropriate The addition of small volumes of cracks to therock matrix significantly weakens the rock The predicted P-and S-wave velocities from 3DKT are more consistent withthe measured ones although with some deviation Thus weconsider aspect ratio 06 as the critical value of the pore aspectratio which is the turning point of the increase trend ofpredicted velocities for Marmaton Wash

Thus variations in the elastic moduli are more influencedby pore shape geometry than porosity lower than 10and have a relatively small impact on 119881P119881S ratios Theintegrated application of the 2DKT and 3DKTmodels helpedus properly determine the pore shape geometry of GraniteWash reservoir to estimate site specific 119881P-119881S relationship

54 119881119875-119881119878Relationship of Granite Wash The 119881P-119881S relation

is normally used for 119881S prediction where only P-wave data isavailable and is also very important forAVOanalysis and porefluids prediction For S-wave velocity estimation regard-less of the approach taken it is important to understand

the underlying lithology The rock composition providesfirst order control on dry frame 119881P119881S ratio especially forconsolidated rocks [29] We take advantage of rock physics(effective medium) modeling and derive the 119881P-119881S relation-ship and compare it with the widely used empirical rela-tionship ldquo119881P = 116 lowast 119881S + 136rdquo presented by Castagnaet al [8] for clastic rocks We employ Berrymanrsquos [7] (3DKT)model considering rock composition porosity and poreshape After calibration of the predicted velocities with coredata the 119881P-119881S relationship is generated through regressionanalysis of least-square linear fit (1198772 = 090) as a function of119881P (Kms) (Figure 12)

The derived 119881P-119881S relation by 3DKT for Granite Washprovides closer approximation from 2DKT and global empir-ical relation by Castagna et al [8] Figure 13 shows that theS-wave velocity derived from Castagnarsquos mudrock is overpredicted with known 119881P when comparatively analyzed to2DKT and 3DKT models predictions However there areapparent deviations between the measured and predicteddata by the locally calibrated empirical relation and theeffectivemediummodelsWe actually do not advocate the useof global empirical relations for S-wave velocity predictionfor the Granite Wash but emphasize on site specific builtrelation based on the data of the corresponding study areausing effective medium models Next we employed the bestmodel (3DKT model) to discuss the fluid substitution effectson P- and S-wave velocities Marmaton Wash

55 Fluid Substitution in Granite Wash For practical fluidsubstitution results in tight sands using Gassmannrsquos [21]

International Journal of Geophysics 11

35

30

25

20

1530 35 40 45 50 55 60

0000012500250003750050000625007500087501000

R2 = 90

Vs

(km

s)

Vp (kms)

Vs = 0599 lowast Vp minus 019102

0 4 5 50 55

0000012500250037505000625075087100000

R2 = 90

s = 0599 lowast VpVV minus 019102

GR

Figure 12 Compressional and shearwave velocities ofGraniteWashformation from sonic logs data color coded with gamma ray (GR)from two wells numbers A and B and calibrated with core [11]measurements (Maroon data)

12340

12360

12380

12400

12420

12440

12460

20 22 24 26 28 30 32 34

003500

01556

02763

03969

05175

06381

07588

08794

1000

Dep

th (f

eet)

S-wave velocity (kms)

Vs log

Vs-2DKTVs-3DKT

Vs-RVs-C

Vsh

Figure 13 Comparison of S-wave velocity predictions (color codedwith 119881sh) of well number A with effective medium (2DKT and3DKT) models locally generated relation (119881

119904-119877) and Castagna

global empirical relationship (119881119904-119862)

equations necessitates an appropriate technique which real-istically stiffens low porosity rock by fluctuating the mineralpore fluid or dry rock moduli Application of Gassmannrsquosequation is dependent upon reasonable estimation of thebulk modulus of the dry rock frame (119870

119889) Dry frame rock

bulk (119870119889) modulus can be derived either from (1) laboratory

measurement on core data (2) application of empirical rela-tionships or effective medium theory [30ndash33] or (3) directcalculation from log data by rewriting Gassmannrsquos equation(4) for frame rock bulk (119870

119889) [34] which often generates

negative (a physical impossibility) or unrealistically largevalues in low porosity reservoirs [35] However of the variousalternatives we take essentially a pragmatic approach tocondition the dry frame rockmodel used in fluid substitution

For reliable fluid substitution model in clastic reservoir weconditioned the calculation of dry frame (119870

119889) rock data

implicit in Gassmannrsquos model and apply Biot [36] relationused by Wang et al [37] as follows

119870119889= 119870119898

(1 minus 120573) (7)

where 120573 is sometimes called the Biot coefficient and it is oftencalculated as a function of porosity and the form used in thepaper is derived by Krief et al [38]

(1 minus 120573) = [1 minus 120601]119898(120601)

119898 (120601) =3

(1 minus 120601)

(8)

where 120601 is the formation porosityConsequently the saturated bulk modulus of the rock

(119870sat) for the in situ conditions is first calculated using (1)Equation (7) is combined with Gassmannrsquos equations (4)to generate a pseudo dry frame rock bulk (119870

119889) modulus

suitable for fluid substitution The input parameters for fluidsubstitutions are generally roughly estimated and thus resultin substantial errorsTherefore densities bulkmoduli veloc-ities and viscosities of various pore fluids (brine and gas)are used both from core measurements and estimation usingBatzle and Wang [39] relationships The homogeneous fluid(119870fl) uniformly distributed throughout the pore space hasbeen computed by harmonic average (Reuss) using data fromthewell logs assuming twofluids (water and gas) phases in thereservoir The fluids (119870fl 120588fl) properties occupying the porespace and the bulk modulus of the mineral matrix (119870matrix)are determined by formulations given by Smith et al [35] Tocalculate the bulk modulus of the mineral matrix (119870matrix)the information on the composition (mineral abundance) ofthe rock is used from core study data [11] Mineral matrix(119870matrix) is then calculated via application of Voigt-Reuss-Hill (VRH) averaging the mineral constituents The elasticconstants and densities of pore fluid properties obtained fromcoremeasurements are given in Table 2 Figure 14 an exampleof composite log example from Granite Wash (Marmaton)section demonstrates the significant fluid substitution differ-ences between the wet and the gas effect on the seismic P-wave (119881P) velocity log for well number A The observed fluideffects reach amaximum in the cleanest sandwith the highest(13) porosity and low density response Fluid substitutioneffects correspond more closely with expectations based onraw logs increasing our confidence in the results of AVOmodeling and seismic inversion carried out in phase II

56 Cross-Plotting Attribute Analysis Cross-plotting of elas-tic rock properties from well logs enables the simultane-ous and meaningful evaluation of two attributes that helpsin making straightforward interpretation of hydrocarbonsaturated sands in a particular reservoir For comparisonwe cross-plotted both conventional (P- and S-impedances)and unconventional (lambda mu and rho) attributes in thederived elastic parameter cross-plot space to isolate potentialanomalous zones from a background relationship of well

12 International Journal of Geophysics

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Vsh

(a)

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Sw

(b)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

00 01

Porosity

(c)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

24 26

Density

(d)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

42 48

Vp

GasBrine

(e)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

025 030

PR

(f)

Figure 14 A composite log example fromGraniteWash (Marmaton) section Plots from left to right show logs of (a)119881sh (b)Water saturation(119878119908) (c) porosity (d) density (e) compressional velocity (119881P) and (f) Poisson ratio (PR) Fluid substitution is carried out for brine (blue) and

gas (red) scenarios using dry rock data derived from logs

number A (Figure 15) Figure 15(a) shows where a conven-tional impedance IP versus IS analysis has limitations inclearly discriminating between all the various lithologies andgas sand zone The P- and S-impedances log curves (whichshare both rigidity and density) tend to track each other withdepth and never crossover only S-impedance contrast in theMarmaton sand (12400

1015840

) is clearly less distinguishable thanthe comparable P-impedance contrast By contrast the 120582120588

and 120583120588 attributes have similar value ranges that do cross over

with 120582120588 lt 120583120588 showing a clear wide separation for a hydro-carbon zone while tracking together to clearly indicate thethin tight shale breaks (Figure 15(b)) In Figure 15(c) onlythe gas zone is discernable from background cut-off on the119881P119881S curve which varies between 16 and 20 Figures 15(d)and 15(e) demonstrate the significant advantage of the 120582120588120583120588 cross-plot in isolating both lithologic properties (such assand shale and clastic facies) as well as the gas zone com-pared to IP and IS cross-plot P- and S-impedance data points

International Journal of Geophysics 13

12340

12360

12380

12400

12420

12440

12460

4 6 8 10 12 14

IPIS

Dep

th (f

eet)

Impedance

(a)

12340

12360

12380

12400

12420

12440

12460

10 20 30 40 50 60 70 80

Tight shale

Gas zone

Lambda-rhoMu-rho

Shale

Dep

th (f

eet)

Lambda-rho Mu-rho

(b)

12340

12360

12380

12400

12420

12440

12460

15 16 17 18 19 20 21

Dep

th (f

eet)

VpVs

(c)

80

75

70

65

60

55

50

458 9 10 11 12 13 14

P-impedance

Gas zone

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

S-im

peda

nce

(d)

60

55

50

45

40

35

30

25

0 10 20 30 40 50 60 70

Gas zone

Lambda-rho

Mu-

rho

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

(e)

25

20

15

10

05

00minus15 minus10 minus5 0 5 10 15 20

Shale

Sst line

Wet sand

Gas zone

Lambda-Mu (difference)

Lam

bda

Mu

(rat

io)

003500

01556

02763

03969

05175

06381

07588

08794

1000

Shale

Sst line

Wet sand

Gas zone

Vsh

(f)

Figure 15 Well log curves and cross-plots from Granite Wash (Marmaton sections 12325ndash12470) of well number A (a) P- and S-impedanceslog curves (b) Lambda-rho and Mu-rho log versus depth (c) 119881

119901119881119904ratio versus depth (d) P-impedance versus S-impedance cross-plot color

coded by shale volume (119881sh) (e) Lambda-rho versus Mu-rho cross-plot color coded by shale volume (119881sh) and (f) Lambda-Mu (difference)versus Lambda-rhoMu-rho (ratio) cross-plot color coded by shale volume (119881sh)

14 International Journal of Geophysics

show a close linear relationship with shale having lowestvalues while in comparison the lowest 120582120588 (incompressibilitytimes density) value has the best gas sand values along with 120583120588

(rigidity times density) values higher than shalesThe other morecompetent pure lithologies (silts cemented sands) could beobserved in the opposite upper right quadrant relative to theshales

Following the cross-plots analysis for the same log dataset Figure 15(f) shows an interesting interpretation templatefor the 120582120583 ratio versus the 120582 minus 120583 (difference) cross-plotThe anomalous (gas) zone is marked in a lower left red boxbeyond the (06 minus5) point separating it from the wet sandswhich tend to align in the upper right zone The variationof the slope beyond the origin (1 0) with increasing gassaturation (blue to green dotted lines) reveals an unexpectedeffect of the gas on the rigidity of the sand This is due tothe effective porosity and lower density gas effect on rigidityleading to a slight steeping of the ldquosandstone linerdquo slopeThus in this case unconventional attributes plots provide aclear separation of the anomalous hydrocarbon zones and thelithologic properties of the sand and shale facies which havebeen color coded with shale volume

6 Conclusion

With the above discussion and limitations the followingconclusions can be derived Rock physics diagnostic and thinsections analysis indicate the variations in elastic propertiesof Granite Wash which can be explained by diagenetic phaseof quartz (or silica) cementation CementedWashes have rel-atively high elastic moduli and can be modeled primarily bycontact-cement model of scheme 2 Velocity prediction fromeffective medium models shows that pore shape geometrysignificantly influences the P- and S-wave velocities in ourreservoirs as observed in the velocity data evaluated Core andpetrographic observations also confirm cementations withcomplex pore geometry containing variable concentrations ofhigh and low pores aspect ratio Hence Berrymanrsquos (3DKT)model is proven to be the best one to discuss the influence ofpore shape on seismic velocities and used for the generationof locally calibrated empirical (119881P-119881S) relationship that bestdescribes our data set Finally we applied fluid substitution onlog data for calculating reservoir elastic properties with dif-ferent pore fluid (brine and gas) for an improved petrophys-ical discrimination of rock properties using unconventionalLame parameters 120582 (pure incompressibility) and 120583 (rigidity)attributes over conventional 119881P 119881S attributes analysis Theresearch outcomes are intended to be rigorously used as afoundation for ongoing reservoir characterization study inphase II

Disclosure

This paper is developed from an oral and poster presentationat American Association of Petroleum Geologists (AAPG)Annual Convention amp Exhibition David L Lawrence Con-vention Center Pittsburgh PA USA May 19ndash22 2013

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank The University of Tulsa and NewfieldExploration Company for support of this research The firstauthor would like to thank Jeff Kane (Corporate Petro-physicist) for his scholastic discussions They also thankthe anonymous reviewers for the constructive comments onimproving our paper

References

[1] S Ingram I Paterniti B Rothkopf and C Stevenson ldquoGranitewash field study-buffalo wallow field Texas panhandlerdquo inProceedings of the SPE Eastern Regional Meeting SPE104546Canton Ohio USA October 2006

[2] M I Odigi ldquoDiagenesis and reservoir quality of Cretaceoussandstone of Nkporo Shale (Campanian) Southeastern Benuetrough Nigeriardquo Journal of Geology and Mining Research vol 3pp 265ndash280 2011

[3] F Ruiz and A Cheng ldquoA rock physics model for tight gas sandrdquoThe Leading Edge vol 29 no 12 pp 1484ndash1489 2010

[4] RDMindlin ldquoCompliance of elastic bodies in contactrdquo Journalof Applied Mechanics vol 16 pp 259ndash268 1949

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[6] J Dvorkin and A Nur ldquoElasticity of high-porosity sandstonestheory for twoNorth Sea data setsrdquoGeophysics vol 61 no 5 pp1363ndash1370 1996

[7] J G Berryman ldquoLong-wavelength propagation in compositeelastic media II Ellipsoidal Inclusionsrdquo The Journal of theAcoustical Society of America vol 68 no 6 pp 1820ndash1831 1980

[8] J P Castagna M L Batzle and R L Eastwood ldquoRelationshipsbetween compressional-wave and shear-wave velocities in clas-tic silicate rocksrdquo Geophysics vol 50 no 4 pp 571ndash581 1985

[9] G T Kuster and M N Toksoz ldquoVelocity and attenuation ofseismic waves in two-phase mediardquo Geophysics vol 39 no 5pp 587ndash618 1974

[10] G E Moore ldquoPennsylvanian paleogeography of the south-ern Mid-Continentrdquo in Pennsylvanian Sandstones of the Mid-Continent N J Hyne Ed pp 2ndash12 Tulsa Geological Society(TGS) Tulsa Okla USA 1979

[11] Core Lab ldquoGraniteWash geological Petrophysical and comple-tions study for improved formation evaluation and optimal wellcompletionrdquo Proprietary industry study 2005

[12] K S Johnson and B J Cardott ldquoGeologic framework andhydrocarbon source rocks of Oklahomardquo in Proceedings of theSource Rocks in the Southern Midcontinent K S JohnsonB andJ Cardott Eds pp 21ndash37 OGS Circular 1992

[13] N R Gelphman ldquoWest sentinel oil field Washita CountyOklahoma sedimentology of the ldquoGraniteWashrdquo and structuralgeologyrdquoOklahoma City Geological Society Shale Shaker vol 10pp 234ndash247 1960

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

2 International Journal of Geophysics

HugotonembaymentOklahoma

Cim

arro

n A

rch

Deep Anadarko Basin

Amarillo upliftPaloDuroBasin

HollisHardemanBasin

Wichita mountains

Texas

Texas

Marietta

Ardmore

Arbuckle

mountains

Kansas

Kansas

Studyarea

Northern shelf

Nem

aha u

plift

A

N

0 100(km)

Oklahoma

Study area

Anadarko Basin

NStudy areat

koko

Figure 1 Location map of the study area Stiles Ranch field Wheeler County Anadarko Basin Texas (USA)

and saturation) properties In the past decade rock physicsfor tight gas reservoirs has been relatively neglected dueto their complex depositional and diagenetic histories [3]Therefore the ultimate goal here is to exploit a consis-tent methodology which can systematically integrate small-scale rock microstructure using the rock physic diagnosticapproach for GraniteWash reservoirs in order to describe thediagenesis of Granite Wash and accurately estimate seismic(P- and S-wave) velocities To analyze the nature of thereservoir rocks (uncemented and cemented) we apply tworock physics models (1) a rigorous contact model that is acombination of the Hertz-Mindlin [4] theory and modifiedHashin-Shtrikman [5] elastic bounds (2) and Dvorkinrsquos [6]cementation model based on cementation theory Further-more we estimate and comparatively analyze seismic (P- andS-wave) velocity predictions using effective medium theoriesand employ the optimal rock physics model to address theinfluence of pore shapes in granite wash sands from differentaspects based on actual well dataThree standard rock physics[7ndash9] models are comparatively analyzed and discussed as tohow pore shape influences the P- and S-wave velocities in theGranite Wash in the Anadarko Basin

An empirical relation is established to predict S-wavevelocitywhen there is only P-wave data available for quartziticsandstone that has not yet been defined and calibrated byGranite Wash core data Finally we applied Gassmannrsquosequations in order to predict velocity changes resulting fromdifferent pore-fluid saturations and identified anomalous(hydrocarbons) zones by composite cross-plots of rock prop-erties from well logs (Marmaton formation) data

2 Geologic Background

The term Granite Wash generally refers to the sedimentsderived predominantly from the weathering of granitic rocksand closely resembles the original material [13] The progres-sion of Granite Wash plays in the Anadarko Basin startedmore than 300 million years ago as a result of sediment ero-sion of the Amarillo-Wichita uplift [10] Large volumes of

Land areasHigh or moderately high relief

High or moderately high relief

Low relief

Low relief

Marginal areas

Shallow marinePrincipally carbonate deposition

Deep marine

Direction of sediment supplyMajor sourceMinor source

Principally terrigenous clastic deposition

Principally terrigenous clastic deposition

Study areaWheeler County

Figure 2 Middle Pennsylvanian (Desmoinesian) paleogeographyin Midcontinent area from [10] modified

clastic sediments were shed off the uplift and deposited to thenorth in the rapidly subsiding Anadarko Basin (Figure 2)

Varieties of depositional settings have been postulated forvarious reservoirs in the Granite Wash that include braided-stream alluvial-fan fan-delta turbidite and debris-flow en-vironments [14] The deposition in the Anadarko Basinoccurred in a series of alluvial fans and fan deltas that formed

International Journal of Geophysics 3

Proximal areaof lobe

Distal and lateralmargins of lobe

70 NG45 NG

25 NG5 NG

NESW

Proximal areaof lobe

Distal and lateralmargins of lobe

70 NG4445 NG

25 NG5 G NG

ENSWW

Figure 3 Depositional facies distribution of Granite Wash fromproximal to distal settings from [11] Data is courtesy of NewfieldExploration Company

rims around the structural highs that developed duringthe Pennsylvanian period [15] In proximal submarine fansettings (Texas) theWashes are thick-bedded (amalgamated)sheet sands poorly sorted and coarse-grained [11] GraniteWash thickness varies from 1500 to 3500 ft of interbeddedconglomerate and amalgamated sheet sandstones depositedin deeper water setting as turbidite andor debris-flowdeposits Distal submarine fan settings (Oklahoma) containvery thin-bedded sheet sands that have interbedded shaleswith decreased NG ratio well sorted silt to fine-grainedminor clay and abundant carbonate cements (Figure 3)

Rock composition ismainly influenced by deposition andunroofing of Siluro-Devonian and Mississippian carbonatesand discontinuity of beds which are often very thin laterallydiscontinuous and diagenetically altered (cemented) Fromproximal to distal settings the rock pore shape geometrychanges due to several geological processes including initialdeposition of fine to very fine grained sediments and thepresence of various types of dispersed shales and clays inthe pores [16] Figure 4 shows highly variable grain sizeand mineralogy with reduced porosity as a result of variableformation lithology substantial cementation and cuspateto triangularrectangular geometries tied to the geologicalcharacter of the sediment source areas in the immediatelyadjacent Amarillo-Wichita Mountains [11]

Figure 5(a) provides the generalized stratigraphic columnof the Anadarko Basin Thick deposits of Granite Wash aremade up of varying sizes of tight sands containing morethan a dozen underground formations stacked on top of theothers each holding vast quantities of hydrocarbons [12]Theproduction hasmainly come from the targetedDesmoinesian(middle Pennsylvanian) series reservoirs with additionalproduction from Atoka Missourian and Virgilian seriesunits (Figure 5(b)) The target depths in the study area rangefrom 12000 to 16500 ft Examination of more than 100 coreswas studied byCore Lab Reference [11] in proximal and distalfan settings from the Pennsylvanian Granite Wash sectionreveals a complex diagenetic history which is a function ofthe depositional environment and the thermal history of thebasin

3 Data Set

The principal data sources for rock physics diagnostics arewell logs and core measurements Log data includes P- and

S-wave velocities density gamma ray and resistivity Weincorporated the core study that has already been done byNewfield Exploration Company (Tulsa OK USA) whichcontains a series of laboratory experiments performed onthe Granite Wash core samples yielding their petrophysicalproperties that are the key for calibrating and improving therock physics modeling

4 Rock Physics Theory

Generally the velocities depend on the ratio rock modulito density The moduli and density effects generate velocityvariations which depend on the rock properties such asporosity fluid saturation and texture The seismic (P- and S-wave) velocities of an isotropic rockmaterial can be estimatedusing known rock moduli and density as follows

119881P = radic119870sat + (43) 120583sat

120588

119881S = radic120583sat120588sat

(1)

where 119870 is bulk modulus 120583 is shear modulus and 120588 is thebulk density

Generally P- and S-wave velocities are not consideredthe best fluid indicator due to the coupling effect betweenP- and S-waves through the shear modulus and bulk density[17] Rock physics trends appear more discretely in themodulus-porosity plane than in the velocity-porosity planeBulk modulus shows sensitivity to pore fluid (water) anddeformation produced by a seismic wave resulting in changeof pore volume Shear modulus usually does not producea pore-volume change and consequently is not affected bydifferent fluids The work space for rock physics analysis isthe rock physics plane that may be (a) velocity-porosity (b)impedance-porosity andor (c)modulus-porosity plane Pri-marily we prefer elastic moduli-porosity planes to correlatefluid effects change in bulk modulus in comparison withvelocity-porosity planes for diagnosing rock

We apply Dvorkinrsquos [6] rock physics diagnostics tech-nique to log data from the Granite Wash (Marmaton sec-tion) calibrated with core data from a well in Stile Ranchfield Wheeler County Texas USA (Figure 6) Techniqueis applied by using Hertz-Mindlin contact theory com-bined with theoretical elastic bounds (Hashin-Shtrikman [5]bounds) and cementation theory based mathematical model(see overview in [6 18]) to analyze the elastic signaturesof porosity reduction related to depositional sorting anddiagenesis (mechanical and chemical compaction) Firstpressure dependent Hertz-Mindlin contact theory is usedto estimate the dry moduli of the rock at critical porosity(040) end member Then we constructed moduli of thesolid mineral for zero porosity The moduli of the dry rockbetween high porosity (critical porosity) and zero porosityare estimated by Hashin-Shtrikman bounds which providetheoretical approximations of effective elastic moduli of amixture of grains and pores Hashin-Shtrikman bounds

4 International Journal of Geophysics

CaIQo

Fo

01mm

(a)

ChIAq

20 microns

(b)

Ab-o

Ab-o

Ab

III

10 microns

(c)

Figure 4 Petrography and mineralogy Granite Wash (Marmaton section) sandstones pore-filling constituents are (a) calcite fill pores andquartz overgrowths feldspar overgrowths (b) euhedral microquartz crystals and (c) authigenic albite overgrowths and single crystals from[11] Data is courtesy of Newfield Exploration Company

Systemsseries

Ochoan

Guadalupian

Leonardian

Wolfcampian

Virgilian

Missourian

Desmoinesian

Atokan

Morrowan

ChesterianMiss

Penn

sylv

ania

nPe

rmia

n

Elk city sandstoneDoxey shale

Cloud chief formationWhitehorse group

El Reno groupHennessey shale

Garber sandstoneWellington formation

Chase groupCouncil grove group

Admire groupWabaunsee group

Shawnee groupDouglas groupOchelata groupSkiatook group

Marmaton groupCherokee group

Atoka group

Morrow group

Springer formation

Chester group

Anadarko BasinSw Oklahoma

Gra

nite

was

h

(a)

Permian

Virgilian

Missourian

Desmionesian

Atokan

Morrowan

Sprin

geran

Granite wash

Pm

Pm

Cam

bro-

Ord

ovic

ian

Silu

ro D

evon

ian

Miss

issip

pian

Top Pennsylvanian

Cherokee group

Prec

ambr

ian

0

5000

10000

15 000

0

1000

2000

3000

4000

5000

OilGas

Mississ

ippian

0 5 10(mL)

0 5 10 15(km)

Subs

ea d

epth

(m)

N

Subs

ea d

epth

(m)

washCarbonate-

chert

Dolomitewash

S

Datum sea level

(b)

Figure 5 (a) Anadarko Basin stratigraphic column [12] and (b) stratigraphic dip cross-sectionmdashunroofing sequence [11]

International Journal of Geophysics 5

12300

12350

12400

12450

12500

12550

12600

126500 100 00 06 45 24 245 000 015

GR Sw Vp Vs Density Porosity

Dep

th (f

eet)

12368

12370

12372

12374

12376

12378

12380

12382

12395

12396

12397

12398

12399

12400

12401

12402

12403

12404

12405

12406

12407

12408

12409

Figure 6Well log curves from Stiles Ranch fieldWheeler County Anadarko Basin Texas (USA) From left to right GR (API) 119878119908119881P (Kms)

119881S (Kms) density (gccc) porosity and FMI logs with superposed core [11] data (maroon) Data is courtesy of Newfield ExplorationCompany

assume that each component and the rock are isotropic andelastic [18]

The behaviors of elastic properties due to cementation iscaptured by contact-cement model [6] The cement modelassumes that porosity decreases from the initial criticalporosity value due to the uniform deposition of cement layerson the surface of the grains This cement may be diageneticquartz calcite or reactive clay (such as illite) The diageneticcement dramatically increases the stiffness of the sand byreinforcing the grain contacts The mathematical model isbased on a rigorous contact-problem solution by Dvorkinet al [19 20] It is assumed that themicrostructure used in themodels quantitatively mimics the data The rock diagnosticwas conducted on the log scale and was confirmed with thecore data (thin sections)

Furthermore we employ effective medium [7 9] modelsin order to estimate and comparatively analyze seismic veloc-ity (or moduli) based on pore aspect ratio to characterize theappropriate pore-shape factor in the GraniteWash reservoirsThe generalized formulas of the Kuster-Toksoz [9] model areexpressed by Berryman [7] as follows

(119870119870119879

minus 119870119898)

(119870119898

+ (43) 120583119898)

(119870119870119879

+ (43) 120583119898)

=

119873

sum

119894=1

119909119894(119870119894minus 119870119898) 119875119898119894

(120583119870119879

minus 120583119898)

(120583119898

+ 120577119898)

(120583119870119879

+ 120577119898)

=

119873

sum

119894=1

119909119894(120583119894minus 120583119898) 119876119898119894

(2)

where the summation is over the different inclusions withvolume concentration 119909

119894and

120577 =120583

6

(9119870 + 8120583)

(119870 + 2120583) (3)

The coefficients 119875119898119894 and 119876

119898119894 describe the effect of aninclusion of material ldquo119894rdquo in a background medium 119898 inclu-sion coefficients119875 and119876 describe the elastic properties of thepore phase (Table 1) Two types of calculation algorithms forthe coefficients 119875 and 119876 are followed (1) The Kuster-Toksozmodel (spheroidal inclusions) with arbitrary pore aspect ratiois named 2DKT model and (2) Berrymanrsquos [7] 3D specialpores (penny-shaped crack) are called 3DKT model

To estimate the effect of the pore fluids on the elasticproperties Gassmannrsquos [21] fluid substitution approach isused to model different scenarios Gassmannrsquos equationsrelate the bulk modulus of a rock to its pore frame and fluidproperties The bulk modulus of a saturated rock is given bythe Gassmann theory

119870sat = 119870119889+

(1 minus 119870119889119870119898)2

(120601119870119891) + ((1 minus 120601) 119870

119898) minus (119870

1198891198702119898)

(4)

where119870sat the bulkmodulus of a rock is saturated with a fluidof bulk modulus119870

119891119870119889is the frame bulk modulus119870

119898is the

matrix (grain) bulk modulus and 120601 is porosity Consider

120583sat = 120583119889 (5)

The bulk density (120588sat) is obtained from mass balanceequation

120588sat = 120588119898

(1 minus 120601) + 120601120588119891 (6)

where120601 120588sat 120588119898 and 120588119891are respectively the porosity density

of saturated rocks the matrix and pore fluids The density ofthe solid (mineral) phase is 120588

119898= (1 minus 119862)120588

119902+ 119862120588119888

In this study we take assumptions that reservoir sand-stone mineral grains are mainly (a) cemented (b) spherical

6 International Journal of Geophysics

Table 1 Coefficients of 119875 and 119876 for some specific shapes The subscripts ldquo119898rdquo and ldquo119894rdquo refer to the background and inclusion materials [7]

Inclusion shape 119875119898119894

119876119898119894

Spheres119870119898

+ (43)120583119898

119870119894+ (43)120583

119898

120583119898

+ 120577119898

120583119894+ 120577119898

Needles119870119898

+ 120583119898

+ (13)120583119894

119870119894+ 120583119898

+ (13)120583119894

1

5(

4120583119898

120583119898

+ 120583119894

+ 2120583119898

+ 120574119898

120583119894+ 120574119898

+119870119894+ (43)120583

119898

119870119894+ 120583119898

+ (13)120583119894

)

Disks119870119898

+ (43)120583119894

119870119894+ (43)120583

119894

120583119898

+ 120577119894

120583119894+ 120577119894

Penny cracks119870119898

+ (43)120583119894

119870119894+ (43)120583

119894+ 120587120572120573

119898

1

5(1 +

8120583119898

4120583119894+ 120587120572(120583

119898+ 2120573119898)+ 2

119870119894+ (23)(120583

119894+ 120583119898)

119870119894+ (43)120583

119894+ 120587120572120573

119898

)

Table 2 The core values of moduli and densities of minerals andfluids used in this study [11]

Minerals and fluids Density(gcc)

Bulk modulus(GPa)

Shear modulus(Gpa)

Quartz 265 5580 38Clays 258 209 685Dolomite 287 8010 4500calcite 271 7550 3400Oil 07 162 mdashGas 012 0032 mdash

shape and (c) randomly packed From the core study thepercentage volume of the rock matrix is dominantly clay(15) and quartz (40) The basic parameters used for fluidsproperties in velocity predictionwere taken from core studies(Table 2) Analysis of log and core data from well A suggeststhat these assumptions are met for our reservoir

5 Data Analysis and Results

51 Rock Physics Diagnostics Application to the Granite WashA common practice in rock physics analysis is to generatecross-plots of elastic properties to study diagenetic trends andproperty dependencies of sands and shales The cross-plotsexhibit interdependence among the various rock propertiesWe particularly focused on a thick Marmaton Wash section(12325ndash12435 ft) of well number A containing various logcurves (Figure 6) Marmaton section is marked by extremelylow and fairly constant (about 375) gamma-ray readings andhigh velocity (about 46 kms) surrounded by shale packages

In spite of the congruency between different log curvesin Figure 6 various cross-plots of Marmaton sands do notproduce unique trends (Figure 7) In Figure 7(a) low andhigh ldquoGRrdquo sediments do not show huge variations in bulkdensity implying that both sand and shale arewell compactedand cemented Figure 7(b) shows the same as the case inGR-porosity plot and Figure 7(c) shows that presence ofclay with significantly lower impedance than the quartz itcements Figures 7(d) 7(e) and 7(f) modulusvelocity versusporosity cross-plots show 119881 shape overturned paths Thesuperimposed119881-shaped paths actually represent clay contentfrom clean sands to pure shale The sands overlapped with

the clean shales probably indicate the sand cluster includingboth clean and shaly sands [22] From moduli-velocity-porosity cross-plots we can assess the progress of diagenesisand compaction trends of sediments For soft or uncementedand stiff sediments the paths are fairly gentle with notmuch of velocity contrast between sands and shales Forcemented sediments (case under examination) increasingdepth quartz cementation provides a large contrast in velocityfrom shaly sands to the pure shales This drop might beassociated with the effect of proceeding from grain supportedto clay-supported sediments under relatively high effectivepressure [23 24] Thus a large part of the scatter observedin modulivelocity versus porosity plots is due to presence ofclay in the rocks

We construct the Hashin and Shtrikman [5] bounds(upper and lower) to conceptually explain and model thedata scatter of the Granite Wash from four different wells(Figure 8) The bounds provide an idea of the narrowestpossible range of elasticmoduli without a detailed descriptionof the microstructure of Granite Wash rocks For boundsonly volume fractions of the constituents quartz is 40 byvolume of solid fraction calcite is 7 by volume of solidfraction and their elastic moduli obtained from core dataare used without specifying the geometric details of theirarrangements The upper and lower bounds are determinedby exchanging the materials one that fills the pores andanother constructing the shell The upper bound is realizedwhen stiffer material forms the shell the lower bound isrealized when the stiffer material is physically present in thecore of the shell [18] The separation between upper andlower bounds shows how elastically different the constituentsare The elastic contrast between pure sand and the criticalconcentration mixture is relatively strong because of thevariable elastic contrast between quartz and clay grains Thisimplies that the effect of quartz and clay minerals may becritical for seismic interpretation of Granite Wash

Moduli-porosity data point superimposed with core datafall between the upper and lower bounds which indicates thatMarmaton sands are consolidated sandstone with cemen-tations Superimposed core data from Marmaton sectionconfirms the presence of quartz and calcite cement

52 Confirming the Diagnostic From Figure 4 we observethatmicrocrystalline quartz is found in close contact betweengrains so this quartz cement has a stiffening effect on

International Journal of Geophysics 7

Bulk

den

sity

27

26

25

24

2350 100 150 200

Gamma ray

(a)

Poro

sity

015

010

005

00050 100 150 200

Density

Gamma ray

(b)14

12

10

50 100 150 200

Density

Gamma ray

IP

(c)

Vp

(km

s)

50

45

40

35000 005 010 015 020

Porosity

Density

(d)

70

60

50

40

000 005 010 015 020

Porosity

K(G

Pa)

Vsh

(e)

70

60

50

40

000 005 010 015 020

Sw

Porosity

K(G

Pa)

(f)

Figure 7 Cross-plots of Well log data Marmaton section from Figure 6 (a) bulk density versus GR (b) total porosity versus GR color codedby density (c) P-wave impedance versus GR color coded by density (d) P-wave versus total porosity color coded by density (e) bulkmodulusversus total porosity color coded by shale volume (119881sh) and (f) bulk modulus versus total porosity color coded by water saturation (119878

119908)

the elastic properties of the Granite Wash Studies havesupported the fact that dissolution and local precipitationof quartz accompanied by compaction cause long termdeformation of the aggregate resulting in permeability andporosity reduction [25] Hence quartz precipitation betweenthe grains causes major porosity reduction

For diagnostic purposes we cross-plotted the elasticproperties from well number A (Marmaton section) invelocity-porosity and moduli-porosity planes superimpos-ing theoretical rock physics models (Figure 9) The cross-plots show that gradually increasing clay content with depthcorresponds with decreasing elastic value Based on labora-tory data log data thin section analysis and rock-physicsmodels the data is sorted into Dvorkin and Nurrsquos [6] follow-ing three schemes uncemented (or soft)WashesWasheswithsmall amounts of quartz cement at grain contacts (scheme 1)and Washes from the same formation but with clay cement

(scheme 2) The upper two theoretical (schemes 1 and 2)curves correspond to the pure quartz cases while lower curveis for pure clay or uncemented rock Scheme 1 represents allcement deposited precisely at grain contacts and the slopesteep Scheme 2 curve corresponds to the case where thecontact cement region is formed at the intersection of cementshells that are uniformly deposited on grain surfaces cementfills the space of the large pores and results in a flat slopeIt is quite possible that the variation in microcracks maybe the major reason why the slope of velocity-porosity andmodulus-porosity planes differs in different schemes

We noticed that the maximum data points fall on therock physics scheme 2 line representing uniformly depositedcement on grain surfaceThe thin sections samples from wellnumber A confirmed the rock diagnostic that with increasingdepth and pressure the decreasing porosity is a result ofcontact cement in the rock-physics cross-plots It is quite

8 International Journal of Geophysics

125

100

75

50

25

0000 025 050 075 100

Bulk

mod

ulus

(GPa

)

Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

50

40

30

20

10

0000 025 050 075 100

Shea

r mod

ulus

(GPa

)Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

Figure 8 Moduli versus porosity for different wells and core [11] data with superposition of Hashin and Shtrikman [5] bounds

6

5

4

3

2

00 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Vp

(km

s)

Porosity

Vp versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(a)

80

60

40

20

000 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Bulk

mod

ulus

(GPa

)

Porosity

K versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(b)

Figure 9 (a) P-wave velocity versus porosity and (b) modulus versus porosity (core [11] data included) color coded with shale volume (119881sh)

of the Marmaton section in well number A with rock physics models curves superimposed

International Journal of Geophysics 9

12368

12370

12372

465 470 475

P-wave velocity

465 470 475

P-wave velocity

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

260 265 270 275

S-wave velocity

260 265 270 275

S-wave velocity

Vp2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-09

2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-082DKT-09

Vs

3DKT-013DKT-023DKT-033DKT-043DKT-053DKT-063DKT-073DKT-08

Vp

3DKT-09

3DKT-04

3DKT-013DKT-023DKT-03

3DKT-053DKT-063DKT-073DKT-083DKT-09

Vs

Figure 10 P- and S-wave velocities prediction by employing 2DKT and 3DKT models using different aspect ratios (01ndash09) of well numberA

interesting that the well-log data with resolution of tens-of-centimeter reflects what we observe at the microscaleBy establishing a site-specific rock physics transform (unce-mented and cemented models) between the total porositymineralogy and elastic properties we conclude the rockphysics diagnostics here In the next section we analyze poreshape influence on seismic velocities in order to generatelocally calibrated 119881P-119881S relationship for Granite Wash

53 Analysis of Pore Shapes Influencing Seismic VelocitiesTheoretically it has been reckoned that the velocities in twostates (dry or saturated) are mainly controlled by the effectivemoduli (bulk and shear) which in turn depend on poreshape geometry mineralogy properties of solids the porefluid degree of consolidation and cementation [9 26 27]Thus elastic moduli (or velocities) are basically the functionof all these factors combined Especially in low porosityand permeability rocks elastic moduli of the rock min-eral matrix with spherical pores (high aspect ratio) often

dominate those of the bulk rock [9 26 27] Therefore it istheoretically reasonable to employ the elastic moduli in rocksto estimate seismic velocities Here two effective medium(2DKT and 3DKT) models are investigated to evaluate poreshape geometry using different pore aspect ratios (valueranging from 01 to 09) to predict P- and S-wave velocitiesfrom the well data We know that the pores with high aspectratio (spherical) tend to be stiffer than those with low aspectratio (oblate or penny shape pores) That is to say P- and S-wave velocities in the reservoir with spherical pores tend tobe higher than one with oblate pores or penny shapeThe factcan be easily understood because stiffer pores contribute tothe stiffness of the whole rock while oblate pores contributeto the compliance of thewhole rockHere actual well numberA is taken as an example containing dipole sonic logs toestimate P- and S-wave velocity From Figure 10 we can seeclearly that the increase in P- and S-wave velocities is sharperwhen the pore aspect ratio increases from 01 to 06 for both2DKT and 3DKT models However the increasing trend isnominal when the pore aspect ratio exceeds beyond 06 in

10 International Journal of Geophysics

12366

12368

12370

1237247 48

Vp (kms) Vs (kms)

Dep

th (f

eet)

12366

12368

12370

12372

Dep

th (f

eet)

VpVp-2DKTVp-3DKT

26 27

VsVs-2DKTVs-3DKT

Figure 11 A comparison between the predicted and measured P- and S-wave velocities by employing 2DKT and 3DKTmodels using criticalaspect ratio (06) of well number A

both models Significantly velocities predicted by the 3DKTmodel agreewell with thosemeasured compared to the 2DKTmodel (Figure 11)

It is due to the fact that the model adjusts pore aspectratio or the proportion of various 3D pores and effectivelymodels the geometries of various pore shapes which is thedominant factor influencing the velocities of the elastic-wave propagating in the clastic reservoirs [28] Due to thesubstantial amount of clay (15) contents that may containmicrocracks the dry rock moduli estimation by the Kuster-Toksoz (2DKT) model (assuming ellipsoidal) could not beappropriate The addition of small volumes of cracks to therock matrix significantly weakens the rock The predicted P-and S-wave velocities from 3DKT are more consistent withthe measured ones although with some deviation Thus weconsider aspect ratio 06 as the critical value of the pore aspectratio which is the turning point of the increase trend ofpredicted velocities for Marmaton Wash

Thus variations in the elastic moduli are more influencedby pore shape geometry than porosity lower than 10and have a relatively small impact on 119881P119881S ratios Theintegrated application of the 2DKT and 3DKTmodels helpedus properly determine the pore shape geometry of GraniteWash reservoir to estimate site specific 119881P-119881S relationship

54 119881119875-119881119878Relationship of Granite Wash The 119881P-119881S relation

is normally used for 119881S prediction where only P-wave data isavailable and is also very important forAVOanalysis and porefluids prediction For S-wave velocity estimation regard-less of the approach taken it is important to understand

the underlying lithology The rock composition providesfirst order control on dry frame 119881P119881S ratio especially forconsolidated rocks [29] We take advantage of rock physics(effective medium) modeling and derive the 119881P-119881S relation-ship and compare it with the widely used empirical rela-tionship ldquo119881P = 116 lowast 119881S + 136rdquo presented by Castagnaet al [8] for clastic rocks We employ Berrymanrsquos [7] (3DKT)model considering rock composition porosity and poreshape After calibration of the predicted velocities with coredata the 119881P-119881S relationship is generated through regressionanalysis of least-square linear fit (1198772 = 090) as a function of119881P (Kms) (Figure 12)

The derived 119881P-119881S relation by 3DKT for Granite Washprovides closer approximation from 2DKT and global empir-ical relation by Castagna et al [8] Figure 13 shows that theS-wave velocity derived from Castagnarsquos mudrock is overpredicted with known 119881P when comparatively analyzed to2DKT and 3DKT models predictions However there areapparent deviations between the measured and predicteddata by the locally calibrated empirical relation and theeffectivemediummodelsWe actually do not advocate the useof global empirical relations for S-wave velocity predictionfor the Granite Wash but emphasize on site specific builtrelation based on the data of the corresponding study areausing effective medium models Next we employed the bestmodel (3DKT model) to discuss the fluid substitution effectson P- and S-wave velocities Marmaton Wash

55 Fluid Substitution in Granite Wash For practical fluidsubstitution results in tight sands using Gassmannrsquos [21]

International Journal of Geophysics 11

35

30

25

20

1530 35 40 45 50 55 60

0000012500250003750050000625007500087501000

R2 = 90

Vs

(km

s)

Vp (kms)

Vs = 0599 lowast Vp minus 019102

0 4 5 50 55

0000012500250037505000625075087100000

R2 = 90

s = 0599 lowast VpVV minus 019102

GR

Figure 12 Compressional and shearwave velocities ofGraniteWashformation from sonic logs data color coded with gamma ray (GR)from two wells numbers A and B and calibrated with core [11]measurements (Maroon data)

12340

12360

12380

12400

12420

12440

12460

20 22 24 26 28 30 32 34

003500

01556

02763

03969

05175

06381

07588

08794

1000

Dep

th (f

eet)

S-wave velocity (kms)

Vs log

Vs-2DKTVs-3DKT

Vs-RVs-C

Vsh

Figure 13 Comparison of S-wave velocity predictions (color codedwith 119881sh) of well number A with effective medium (2DKT and3DKT) models locally generated relation (119881

119904-119877) and Castagna

global empirical relationship (119881119904-119862)

equations necessitates an appropriate technique which real-istically stiffens low porosity rock by fluctuating the mineralpore fluid or dry rock moduli Application of Gassmannrsquosequation is dependent upon reasonable estimation of thebulk modulus of the dry rock frame (119870

119889) Dry frame rock

bulk (119870119889) modulus can be derived either from (1) laboratory

measurement on core data (2) application of empirical rela-tionships or effective medium theory [30ndash33] or (3) directcalculation from log data by rewriting Gassmannrsquos equation(4) for frame rock bulk (119870

119889) [34] which often generates

negative (a physical impossibility) or unrealistically largevalues in low porosity reservoirs [35] However of the variousalternatives we take essentially a pragmatic approach tocondition the dry frame rockmodel used in fluid substitution

For reliable fluid substitution model in clastic reservoir weconditioned the calculation of dry frame (119870

119889) rock data

implicit in Gassmannrsquos model and apply Biot [36] relationused by Wang et al [37] as follows

119870119889= 119870119898

(1 minus 120573) (7)

where 120573 is sometimes called the Biot coefficient and it is oftencalculated as a function of porosity and the form used in thepaper is derived by Krief et al [38]

(1 minus 120573) = [1 minus 120601]119898(120601)

119898 (120601) =3

(1 minus 120601)

(8)

where 120601 is the formation porosityConsequently the saturated bulk modulus of the rock

(119870sat) for the in situ conditions is first calculated using (1)Equation (7) is combined with Gassmannrsquos equations (4)to generate a pseudo dry frame rock bulk (119870

119889) modulus

suitable for fluid substitution The input parameters for fluidsubstitutions are generally roughly estimated and thus resultin substantial errorsTherefore densities bulkmoduli veloc-ities and viscosities of various pore fluids (brine and gas)are used both from core measurements and estimation usingBatzle and Wang [39] relationships The homogeneous fluid(119870fl) uniformly distributed throughout the pore space hasbeen computed by harmonic average (Reuss) using data fromthewell logs assuming twofluids (water and gas) phases in thereservoir The fluids (119870fl 120588fl) properties occupying the porespace and the bulk modulus of the mineral matrix (119870matrix)are determined by formulations given by Smith et al [35] Tocalculate the bulk modulus of the mineral matrix (119870matrix)the information on the composition (mineral abundance) ofthe rock is used from core study data [11] Mineral matrix(119870matrix) is then calculated via application of Voigt-Reuss-Hill (VRH) averaging the mineral constituents The elasticconstants and densities of pore fluid properties obtained fromcoremeasurements are given in Table 2 Figure 14 an exampleof composite log example from Granite Wash (Marmaton)section demonstrates the significant fluid substitution differ-ences between the wet and the gas effect on the seismic P-wave (119881P) velocity log for well number A The observed fluideffects reach amaximum in the cleanest sandwith the highest(13) porosity and low density response Fluid substitutioneffects correspond more closely with expectations based onraw logs increasing our confidence in the results of AVOmodeling and seismic inversion carried out in phase II

56 Cross-Plotting Attribute Analysis Cross-plotting of elas-tic rock properties from well logs enables the simultane-ous and meaningful evaluation of two attributes that helpsin making straightforward interpretation of hydrocarbonsaturated sands in a particular reservoir For comparisonwe cross-plotted both conventional (P- and S-impedances)and unconventional (lambda mu and rho) attributes in thederived elastic parameter cross-plot space to isolate potentialanomalous zones from a background relationship of well

12 International Journal of Geophysics

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Vsh

(a)

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Sw

(b)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

00 01

Porosity

(c)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

24 26

Density

(d)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

42 48

Vp

GasBrine

(e)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

025 030

PR

(f)

Figure 14 A composite log example fromGraniteWash (Marmaton) section Plots from left to right show logs of (a)119881sh (b)Water saturation(119878119908) (c) porosity (d) density (e) compressional velocity (119881P) and (f) Poisson ratio (PR) Fluid substitution is carried out for brine (blue) and

gas (red) scenarios using dry rock data derived from logs

number A (Figure 15) Figure 15(a) shows where a conven-tional impedance IP versus IS analysis has limitations inclearly discriminating between all the various lithologies andgas sand zone The P- and S-impedances log curves (whichshare both rigidity and density) tend to track each other withdepth and never crossover only S-impedance contrast in theMarmaton sand (12400

1015840

) is clearly less distinguishable thanthe comparable P-impedance contrast By contrast the 120582120588

and 120583120588 attributes have similar value ranges that do cross over

with 120582120588 lt 120583120588 showing a clear wide separation for a hydro-carbon zone while tracking together to clearly indicate thethin tight shale breaks (Figure 15(b)) In Figure 15(c) onlythe gas zone is discernable from background cut-off on the119881P119881S curve which varies between 16 and 20 Figures 15(d)and 15(e) demonstrate the significant advantage of the 120582120588120583120588 cross-plot in isolating both lithologic properties (such assand shale and clastic facies) as well as the gas zone com-pared to IP and IS cross-plot P- and S-impedance data points

International Journal of Geophysics 13

12340

12360

12380

12400

12420

12440

12460

4 6 8 10 12 14

IPIS

Dep

th (f

eet)

Impedance

(a)

12340

12360

12380

12400

12420

12440

12460

10 20 30 40 50 60 70 80

Tight shale

Gas zone

Lambda-rhoMu-rho

Shale

Dep

th (f

eet)

Lambda-rho Mu-rho

(b)

12340

12360

12380

12400

12420

12440

12460

15 16 17 18 19 20 21

Dep

th (f

eet)

VpVs

(c)

80

75

70

65

60

55

50

458 9 10 11 12 13 14

P-impedance

Gas zone

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

S-im

peda

nce

(d)

60

55

50

45

40

35

30

25

0 10 20 30 40 50 60 70

Gas zone

Lambda-rho

Mu-

rho

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

(e)

25

20

15

10

05

00minus15 minus10 minus5 0 5 10 15 20

Shale

Sst line

Wet sand

Gas zone

Lambda-Mu (difference)

Lam

bda

Mu

(rat

io)

003500

01556

02763

03969

05175

06381

07588

08794

1000

Shale

Sst line

Wet sand

Gas zone

Vsh

(f)

Figure 15 Well log curves and cross-plots from Granite Wash (Marmaton sections 12325ndash12470) of well number A (a) P- and S-impedanceslog curves (b) Lambda-rho and Mu-rho log versus depth (c) 119881

119901119881119904ratio versus depth (d) P-impedance versus S-impedance cross-plot color

coded by shale volume (119881sh) (e) Lambda-rho versus Mu-rho cross-plot color coded by shale volume (119881sh) and (f) Lambda-Mu (difference)versus Lambda-rhoMu-rho (ratio) cross-plot color coded by shale volume (119881sh)

14 International Journal of Geophysics

show a close linear relationship with shale having lowestvalues while in comparison the lowest 120582120588 (incompressibilitytimes density) value has the best gas sand values along with 120583120588

(rigidity times density) values higher than shalesThe other morecompetent pure lithologies (silts cemented sands) could beobserved in the opposite upper right quadrant relative to theshales

Following the cross-plots analysis for the same log dataset Figure 15(f) shows an interesting interpretation templatefor the 120582120583 ratio versus the 120582 minus 120583 (difference) cross-plotThe anomalous (gas) zone is marked in a lower left red boxbeyond the (06 minus5) point separating it from the wet sandswhich tend to align in the upper right zone The variationof the slope beyond the origin (1 0) with increasing gassaturation (blue to green dotted lines) reveals an unexpectedeffect of the gas on the rigidity of the sand This is due tothe effective porosity and lower density gas effect on rigidityleading to a slight steeping of the ldquosandstone linerdquo slopeThus in this case unconventional attributes plots provide aclear separation of the anomalous hydrocarbon zones and thelithologic properties of the sand and shale facies which havebeen color coded with shale volume

6 Conclusion

With the above discussion and limitations the followingconclusions can be derived Rock physics diagnostic and thinsections analysis indicate the variations in elastic propertiesof Granite Wash which can be explained by diagenetic phaseof quartz (or silica) cementation CementedWashes have rel-atively high elastic moduli and can be modeled primarily bycontact-cement model of scheme 2 Velocity prediction fromeffective medium models shows that pore shape geometrysignificantly influences the P- and S-wave velocities in ourreservoirs as observed in the velocity data evaluated Core andpetrographic observations also confirm cementations withcomplex pore geometry containing variable concentrations ofhigh and low pores aspect ratio Hence Berrymanrsquos (3DKT)model is proven to be the best one to discuss the influence ofpore shape on seismic velocities and used for the generationof locally calibrated empirical (119881P-119881S) relationship that bestdescribes our data set Finally we applied fluid substitution onlog data for calculating reservoir elastic properties with dif-ferent pore fluid (brine and gas) for an improved petrophys-ical discrimination of rock properties using unconventionalLame parameters 120582 (pure incompressibility) and 120583 (rigidity)attributes over conventional 119881P 119881S attributes analysis Theresearch outcomes are intended to be rigorously used as afoundation for ongoing reservoir characterization study inphase II

Disclosure

This paper is developed from an oral and poster presentationat American Association of Petroleum Geologists (AAPG)Annual Convention amp Exhibition David L Lawrence Con-vention Center Pittsburgh PA USA May 19ndash22 2013

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank The University of Tulsa and NewfieldExploration Company for support of this research The firstauthor would like to thank Jeff Kane (Corporate Petro-physicist) for his scholastic discussions They also thankthe anonymous reviewers for the constructive comments onimproving our paper

References

[1] S Ingram I Paterniti B Rothkopf and C Stevenson ldquoGranitewash field study-buffalo wallow field Texas panhandlerdquo inProceedings of the SPE Eastern Regional Meeting SPE104546Canton Ohio USA October 2006

[2] M I Odigi ldquoDiagenesis and reservoir quality of Cretaceoussandstone of Nkporo Shale (Campanian) Southeastern Benuetrough Nigeriardquo Journal of Geology and Mining Research vol 3pp 265ndash280 2011

[3] F Ruiz and A Cheng ldquoA rock physics model for tight gas sandrdquoThe Leading Edge vol 29 no 12 pp 1484ndash1489 2010

[4] RDMindlin ldquoCompliance of elastic bodies in contactrdquo Journalof Applied Mechanics vol 16 pp 259ndash268 1949

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[6] J Dvorkin and A Nur ldquoElasticity of high-porosity sandstonestheory for twoNorth Sea data setsrdquoGeophysics vol 61 no 5 pp1363ndash1370 1996

[7] J G Berryman ldquoLong-wavelength propagation in compositeelastic media II Ellipsoidal Inclusionsrdquo The Journal of theAcoustical Society of America vol 68 no 6 pp 1820ndash1831 1980

[8] J P Castagna M L Batzle and R L Eastwood ldquoRelationshipsbetween compressional-wave and shear-wave velocities in clas-tic silicate rocksrdquo Geophysics vol 50 no 4 pp 571ndash581 1985

[9] G T Kuster and M N Toksoz ldquoVelocity and attenuation ofseismic waves in two-phase mediardquo Geophysics vol 39 no 5pp 587ndash618 1974

[10] G E Moore ldquoPennsylvanian paleogeography of the south-ern Mid-Continentrdquo in Pennsylvanian Sandstones of the Mid-Continent N J Hyne Ed pp 2ndash12 Tulsa Geological Society(TGS) Tulsa Okla USA 1979

[11] Core Lab ldquoGraniteWash geological Petrophysical and comple-tions study for improved formation evaluation and optimal wellcompletionrdquo Proprietary industry study 2005

[12] K S Johnson and B J Cardott ldquoGeologic framework andhydrocarbon source rocks of Oklahomardquo in Proceedings of theSource Rocks in the Southern Midcontinent K S JohnsonB andJ Cardott Eds pp 21ndash37 OGS Circular 1992

[13] N R Gelphman ldquoWest sentinel oil field Washita CountyOklahoma sedimentology of the ldquoGraniteWashrdquo and structuralgeologyrdquoOklahoma City Geological Society Shale Shaker vol 10pp 234ndash247 1960

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

International Journal of Geophysics 3

Proximal areaof lobe

Distal and lateralmargins of lobe

70 NG45 NG

25 NG5 NG

NESW

Proximal areaof lobe

Distal and lateralmargins of lobe

70 NG4445 NG

25 NG5 G NG

ENSWW

Figure 3 Depositional facies distribution of Granite Wash fromproximal to distal settings from [11] Data is courtesy of NewfieldExploration Company

rims around the structural highs that developed duringthe Pennsylvanian period [15] In proximal submarine fansettings (Texas) theWashes are thick-bedded (amalgamated)sheet sands poorly sorted and coarse-grained [11] GraniteWash thickness varies from 1500 to 3500 ft of interbeddedconglomerate and amalgamated sheet sandstones depositedin deeper water setting as turbidite andor debris-flowdeposits Distal submarine fan settings (Oklahoma) containvery thin-bedded sheet sands that have interbedded shaleswith decreased NG ratio well sorted silt to fine-grainedminor clay and abundant carbonate cements (Figure 3)

Rock composition ismainly influenced by deposition andunroofing of Siluro-Devonian and Mississippian carbonatesand discontinuity of beds which are often very thin laterallydiscontinuous and diagenetically altered (cemented) Fromproximal to distal settings the rock pore shape geometrychanges due to several geological processes including initialdeposition of fine to very fine grained sediments and thepresence of various types of dispersed shales and clays inthe pores [16] Figure 4 shows highly variable grain sizeand mineralogy with reduced porosity as a result of variableformation lithology substantial cementation and cuspateto triangularrectangular geometries tied to the geologicalcharacter of the sediment source areas in the immediatelyadjacent Amarillo-Wichita Mountains [11]

Figure 5(a) provides the generalized stratigraphic columnof the Anadarko Basin Thick deposits of Granite Wash aremade up of varying sizes of tight sands containing morethan a dozen underground formations stacked on top of theothers each holding vast quantities of hydrocarbons [12]Theproduction hasmainly come from the targetedDesmoinesian(middle Pennsylvanian) series reservoirs with additionalproduction from Atoka Missourian and Virgilian seriesunits (Figure 5(b)) The target depths in the study area rangefrom 12000 to 16500 ft Examination of more than 100 coreswas studied byCore Lab Reference [11] in proximal and distalfan settings from the Pennsylvanian Granite Wash sectionreveals a complex diagenetic history which is a function ofthe depositional environment and the thermal history of thebasin

3 Data Set

The principal data sources for rock physics diagnostics arewell logs and core measurements Log data includes P- and

S-wave velocities density gamma ray and resistivity Weincorporated the core study that has already been done byNewfield Exploration Company (Tulsa OK USA) whichcontains a series of laboratory experiments performed onthe Granite Wash core samples yielding their petrophysicalproperties that are the key for calibrating and improving therock physics modeling

4 Rock Physics Theory

Generally the velocities depend on the ratio rock modulito density The moduli and density effects generate velocityvariations which depend on the rock properties such asporosity fluid saturation and texture The seismic (P- and S-wave) velocities of an isotropic rockmaterial can be estimatedusing known rock moduli and density as follows

119881P = radic119870sat + (43) 120583sat

120588

119881S = radic120583sat120588sat

(1)

where 119870 is bulk modulus 120583 is shear modulus and 120588 is thebulk density

Generally P- and S-wave velocities are not consideredthe best fluid indicator due to the coupling effect betweenP- and S-waves through the shear modulus and bulk density[17] Rock physics trends appear more discretely in themodulus-porosity plane than in the velocity-porosity planeBulk modulus shows sensitivity to pore fluid (water) anddeformation produced by a seismic wave resulting in changeof pore volume Shear modulus usually does not producea pore-volume change and consequently is not affected bydifferent fluids The work space for rock physics analysis isthe rock physics plane that may be (a) velocity-porosity (b)impedance-porosity andor (c)modulus-porosity plane Pri-marily we prefer elastic moduli-porosity planes to correlatefluid effects change in bulk modulus in comparison withvelocity-porosity planes for diagnosing rock

We apply Dvorkinrsquos [6] rock physics diagnostics tech-nique to log data from the Granite Wash (Marmaton sec-tion) calibrated with core data from a well in Stile Ranchfield Wheeler County Texas USA (Figure 6) Techniqueis applied by using Hertz-Mindlin contact theory com-bined with theoretical elastic bounds (Hashin-Shtrikman [5]bounds) and cementation theory based mathematical model(see overview in [6 18]) to analyze the elastic signaturesof porosity reduction related to depositional sorting anddiagenesis (mechanical and chemical compaction) Firstpressure dependent Hertz-Mindlin contact theory is usedto estimate the dry moduli of the rock at critical porosity(040) end member Then we constructed moduli of thesolid mineral for zero porosity The moduli of the dry rockbetween high porosity (critical porosity) and zero porosityare estimated by Hashin-Shtrikman bounds which providetheoretical approximations of effective elastic moduli of amixture of grains and pores Hashin-Shtrikman bounds

4 International Journal of Geophysics

CaIQo

Fo

01mm

(a)

ChIAq

20 microns

(b)

Ab-o

Ab-o

Ab

III

10 microns

(c)

Figure 4 Petrography and mineralogy Granite Wash (Marmaton section) sandstones pore-filling constituents are (a) calcite fill pores andquartz overgrowths feldspar overgrowths (b) euhedral microquartz crystals and (c) authigenic albite overgrowths and single crystals from[11] Data is courtesy of Newfield Exploration Company

Systemsseries

Ochoan

Guadalupian

Leonardian

Wolfcampian

Virgilian

Missourian

Desmoinesian

Atokan

Morrowan

ChesterianMiss

Penn

sylv

ania

nPe

rmia

n

Elk city sandstoneDoxey shale

Cloud chief formationWhitehorse group

El Reno groupHennessey shale

Garber sandstoneWellington formation

Chase groupCouncil grove group

Admire groupWabaunsee group

Shawnee groupDouglas groupOchelata groupSkiatook group

Marmaton groupCherokee group

Atoka group

Morrow group

Springer formation

Chester group

Anadarko BasinSw Oklahoma

Gra

nite

was

h

(a)

Permian

Virgilian

Missourian

Desmionesian

Atokan

Morrowan

Sprin

geran

Granite wash

Pm

Pm

Cam

bro-

Ord

ovic

ian

Silu

ro D

evon

ian

Miss

issip

pian

Top Pennsylvanian

Cherokee group

Prec

ambr

ian

0

5000

10000

15 000

0

1000

2000

3000

4000

5000

OilGas

Mississ

ippian

0 5 10(mL)

0 5 10 15(km)

Subs

ea d

epth

(m)

N

Subs

ea d

epth

(m)

washCarbonate-

chert

Dolomitewash

S

Datum sea level

(b)

Figure 5 (a) Anadarko Basin stratigraphic column [12] and (b) stratigraphic dip cross-sectionmdashunroofing sequence [11]

International Journal of Geophysics 5

12300

12350

12400

12450

12500

12550

12600

126500 100 00 06 45 24 245 000 015

GR Sw Vp Vs Density Porosity

Dep

th (f

eet)

12368

12370

12372

12374

12376

12378

12380

12382

12395

12396

12397

12398

12399

12400

12401

12402

12403

12404

12405

12406

12407

12408

12409

Figure 6Well log curves from Stiles Ranch fieldWheeler County Anadarko Basin Texas (USA) From left to right GR (API) 119878119908119881P (Kms)

119881S (Kms) density (gccc) porosity and FMI logs with superposed core [11] data (maroon) Data is courtesy of Newfield ExplorationCompany

assume that each component and the rock are isotropic andelastic [18]

The behaviors of elastic properties due to cementation iscaptured by contact-cement model [6] The cement modelassumes that porosity decreases from the initial criticalporosity value due to the uniform deposition of cement layerson the surface of the grains This cement may be diageneticquartz calcite or reactive clay (such as illite) The diageneticcement dramatically increases the stiffness of the sand byreinforcing the grain contacts The mathematical model isbased on a rigorous contact-problem solution by Dvorkinet al [19 20] It is assumed that themicrostructure used in themodels quantitatively mimics the data The rock diagnosticwas conducted on the log scale and was confirmed with thecore data (thin sections)

Furthermore we employ effective medium [7 9] modelsin order to estimate and comparatively analyze seismic veloc-ity (or moduli) based on pore aspect ratio to characterize theappropriate pore-shape factor in the GraniteWash reservoirsThe generalized formulas of the Kuster-Toksoz [9] model areexpressed by Berryman [7] as follows

(119870119870119879

minus 119870119898)

(119870119898

+ (43) 120583119898)

(119870119870119879

+ (43) 120583119898)

=

119873

sum

119894=1

119909119894(119870119894minus 119870119898) 119875119898119894

(120583119870119879

minus 120583119898)

(120583119898

+ 120577119898)

(120583119870119879

+ 120577119898)

=

119873

sum

119894=1

119909119894(120583119894minus 120583119898) 119876119898119894

(2)

where the summation is over the different inclusions withvolume concentration 119909

119894and

120577 =120583

6

(9119870 + 8120583)

(119870 + 2120583) (3)

The coefficients 119875119898119894 and 119876

119898119894 describe the effect of aninclusion of material ldquo119894rdquo in a background medium 119898 inclu-sion coefficients119875 and119876 describe the elastic properties of thepore phase (Table 1) Two types of calculation algorithms forthe coefficients 119875 and 119876 are followed (1) The Kuster-Toksozmodel (spheroidal inclusions) with arbitrary pore aspect ratiois named 2DKT model and (2) Berrymanrsquos [7] 3D specialpores (penny-shaped crack) are called 3DKT model

To estimate the effect of the pore fluids on the elasticproperties Gassmannrsquos [21] fluid substitution approach isused to model different scenarios Gassmannrsquos equationsrelate the bulk modulus of a rock to its pore frame and fluidproperties The bulk modulus of a saturated rock is given bythe Gassmann theory

119870sat = 119870119889+

(1 minus 119870119889119870119898)2

(120601119870119891) + ((1 minus 120601) 119870

119898) minus (119870

1198891198702119898)

(4)

where119870sat the bulkmodulus of a rock is saturated with a fluidof bulk modulus119870

119891119870119889is the frame bulk modulus119870

119898is the

matrix (grain) bulk modulus and 120601 is porosity Consider

120583sat = 120583119889 (5)

The bulk density (120588sat) is obtained from mass balanceequation

120588sat = 120588119898

(1 minus 120601) + 120601120588119891 (6)

where120601 120588sat 120588119898 and 120588119891are respectively the porosity density

of saturated rocks the matrix and pore fluids The density ofthe solid (mineral) phase is 120588

119898= (1 minus 119862)120588

119902+ 119862120588119888

In this study we take assumptions that reservoir sand-stone mineral grains are mainly (a) cemented (b) spherical

6 International Journal of Geophysics

Table 1 Coefficients of 119875 and 119876 for some specific shapes The subscripts ldquo119898rdquo and ldquo119894rdquo refer to the background and inclusion materials [7]

Inclusion shape 119875119898119894

119876119898119894

Spheres119870119898

+ (43)120583119898

119870119894+ (43)120583

119898

120583119898

+ 120577119898

120583119894+ 120577119898

Needles119870119898

+ 120583119898

+ (13)120583119894

119870119894+ 120583119898

+ (13)120583119894

1

5(

4120583119898

120583119898

+ 120583119894

+ 2120583119898

+ 120574119898

120583119894+ 120574119898

+119870119894+ (43)120583

119898

119870119894+ 120583119898

+ (13)120583119894

)

Disks119870119898

+ (43)120583119894

119870119894+ (43)120583

119894

120583119898

+ 120577119894

120583119894+ 120577119894

Penny cracks119870119898

+ (43)120583119894

119870119894+ (43)120583

119894+ 120587120572120573

119898

1

5(1 +

8120583119898

4120583119894+ 120587120572(120583

119898+ 2120573119898)+ 2

119870119894+ (23)(120583

119894+ 120583119898)

119870119894+ (43)120583

119894+ 120587120572120573

119898

)

Table 2 The core values of moduli and densities of minerals andfluids used in this study [11]

Minerals and fluids Density(gcc)

Bulk modulus(GPa)

Shear modulus(Gpa)

Quartz 265 5580 38Clays 258 209 685Dolomite 287 8010 4500calcite 271 7550 3400Oil 07 162 mdashGas 012 0032 mdash

shape and (c) randomly packed From the core study thepercentage volume of the rock matrix is dominantly clay(15) and quartz (40) The basic parameters used for fluidsproperties in velocity predictionwere taken from core studies(Table 2) Analysis of log and core data from well A suggeststhat these assumptions are met for our reservoir

5 Data Analysis and Results

51 Rock Physics Diagnostics Application to the Granite WashA common practice in rock physics analysis is to generatecross-plots of elastic properties to study diagenetic trends andproperty dependencies of sands and shales The cross-plotsexhibit interdependence among the various rock propertiesWe particularly focused on a thick Marmaton Wash section(12325ndash12435 ft) of well number A containing various logcurves (Figure 6) Marmaton section is marked by extremelylow and fairly constant (about 375) gamma-ray readings andhigh velocity (about 46 kms) surrounded by shale packages

In spite of the congruency between different log curvesin Figure 6 various cross-plots of Marmaton sands do notproduce unique trends (Figure 7) In Figure 7(a) low andhigh ldquoGRrdquo sediments do not show huge variations in bulkdensity implying that both sand and shale arewell compactedand cemented Figure 7(b) shows the same as the case inGR-porosity plot and Figure 7(c) shows that presence ofclay with significantly lower impedance than the quartz itcements Figures 7(d) 7(e) and 7(f) modulusvelocity versusporosity cross-plots show 119881 shape overturned paths Thesuperimposed119881-shaped paths actually represent clay contentfrom clean sands to pure shale The sands overlapped with

the clean shales probably indicate the sand cluster includingboth clean and shaly sands [22] From moduli-velocity-porosity cross-plots we can assess the progress of diagenesisand compaction trends of sediments For soft or uncementedand stiff sediments the paths are fairly gentle with notmuch of velocity contrast between sands and shales Forcemented sediments (case under examination) increasingdepth quartz cementation provides a large contrast in velocityfrom shaly sands to the pure shales This drop might beassociated with the effect of proceeding from grain supportedto clay-supported sediments under relatively high effectivepressure [23 24] Thus a large part of the scatter observedin modulivelocity versus porosity plots is due to presence ofclay in the rocks

We construct the Hashin and Shtrikman [5] bounds(upper and lower) to conceptually explain and model thedata scatter of the Granite Wash from four different wells(Figure 8) The bounds provide an idea of the narrowestpossible range of elasticmoduli without a detailed descriptionof the microstructure of Granite Wash rocks For boundsonly volume fractions of the constituents quartz is 40 byvolume of solid fraction calcite is 7 by volume of solidfraction and their elastic moduli obtained from core dataare used without specifying the geometric details of theirarrangements The upper and lower bounds are determinedby exchanging the materials one that fills the pores andanother constructing the shell The upper bound is realizedwhen stiffer material forms the shell the lower bound isrealized when the stiffer material is physically present in thecore of the shell [18] The separation between upper andlower bounds shows how elastically different the constituentsare The elastic contrast between pure sand and the criticalconcentration mixture is relatively strong because of thevariable elastic contrast between quartz and clay grains Thisimplies that the effect of quartz and clay minerals may becritical for seismic interpretation of Granite Wash

Moduli-porosity data point superimposed with core datafall between the upper and lower bounds which indicates thatMarmaton sands are consolidated sandstone with cemen-tations Superimposed core data from Marmaton sectionconfirms the presence of quartz and calcite cement

52 Confirming the Diagnostic From Figure 4 we observethatmicrocrystalline quartz is found in close contact betweengrains so this quartz cement has a stiffening effect on

International Journal of Geophysics 7

Bulk

den

sity

27

26

25

24

2350 100 150 200

Gamma ray

(a)

Poro

sity

015

010

005

00050 100 150 200

Density

Gamma ray

(b)14

12

10

50 100 150 200

Density

Gamma ray

IP

(c)

Vp

(km

s)

50

45

40

35000 005 010 015 020

Porosity

Density

(d)

70

60

50

40

000 005 010 015 020

Porosity

K(G

Pa)

Vsh

(e)

70

60

50

40

000 005 010 015 020

Sw

Porosity

K(G

Pa)

(f)

Figure 7 Cross-plots of Well log data Marmaton section from Figure 6 (a) bulk density versus GR (b) total porosity versus GR color codedby density (c) P-wave impedance versus GR color coded by density (d) P-wave versus total porosity color coded by density (e) bulkmodulusversus total porosity color coded by shale volume (119881sh) and (f) bulk modulus versus total porosity color coded by water saturation (119878

119908)

the elastic properties of the Granite Wash Studies havesupported the fact that dissolution and local precipitationof quartz accompanied by compaction cause long termdeformation of the aggregate resulting in permeability andporosity reduction [25] Hence quartz precipitation betweenthe grains causes major porosity reduction

For diagnostic purposes we cross-plotted the elasticproperties from well number A (Marmaton section) invelocity-porosity and moduli-porosity planes superimpos-ing theoretical rock physics models (Figure 9) The cross-plots show that gradually increasing clay content with depthcorresponds with decreasing elastic value Based on labora-tory data log data thin section analysis and rock-physicsmodels the data is sorted into Dvorkin and Nurrsquos [6] follow-ing three schemes uncemented (or soft)WashesWasheswithsmall amounts of quartz cement at grain contacts (scheme 1)and Washes from the same formation but with clay cement

(scheme 2) The upper two theoretical (schemes 1 and 2)curves correspond to the pure quartz cases while lower curveis for pure clay or uncemented rock Scheme 1 represents allcement deposited precisely at grain contacts and the slopesteep Scheme 2 curve corresponds to the case where thecontact cement region is formed at the intersection of cementshells that are uniformly deposited on grain surfaces cementfills the space of the large pores and results in a flat slopeIt is quite possible that the variation in microcracks maybe the major reason why the slope of velocity-porosity andmodulus-porosity planes differs in different schemes

We noticed that the maximum data points fall on therock physics scheme 2 line representing uniformly depositedcement on grain surfaceThe thin sections samples from wellnumber A confirmed the rock diagnostic that with increasingdepth and pressure the decreasing porosity is a result ofcontact cement in the rock-physics cross-plots It is quite

8 International Journal of Geophysics

125

100

75

50

25

0000 025 050 075 100

Bulk

mod

ulus

(GPa

)

Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

50

40

30

20

10

0000 025 050 075 100

Shea

r mod

ulus

(GPa

)Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

Figure 8 Moduli versus porosity for different wells and core [11] data with superposition of Hashin and Shtrikman [5] bounds

6

5

4

3

2

00 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Vp

(km

s)

Porosity

Vp versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(a)

80

60

40

20

000 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Bulk

mod

ulus

(GPa

)

Porosity

K versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(b)

Figure 9 (a) P-wave velocity versus porosity and (b) modulus versus porosity (core [11] data included) color coded with shale volume (119881sh)

of the Marmaton section in well number A with rock physics models curves superimposed

International Journal of Geophysics 9

12368

12370

12372

465 470 475

P-wave velocity

465 470 475

P-wave velocity

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

260 265 270 275

S-wave velocity

260 265 270 275

S-wave velocity

Vp2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-09

2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-082DKT-09

Vs

3DKT-013DKT-023DKT-033DKT-043DKT-053DKT-063DKT-073DKT-08

Vp

3DKT-09

3DKT-04

3DKT-013DKT-023DKT-03

3DKT-053DKT-063DKT-073DKT-083DKT-09

Vs

Figure 10 P- and S-wave velocities prediction by employing 2DKT and 3DKT models using different aspect ratios (01ndash09) of well numberA

interesting that the well-log data with resolution of tens-of-centimeter reflects what we observe at the microscaleBy establishing a site-specific rock physics transform (unce-mented and cemented models) between the total porositymineralogy and elastic properties we conclude the rockphysics diagnostics here In the next section we analyze poreshape influence on seismic velocities in order to generatelocally calibrated 119881P-119881S relationship for Granite Wash

53 Analysis of Pore Shapes Influencing Seismic VelocitiesTheoretically it has been reckoned that the velocities in twostates (dry or saturated) are mainly controlled by the effectivemoduli (bulk and shear) which in turn depend on poreshape geometry mineralogy properties of solids the porefluid degree of consolidation and cementation [9 26 27]Thus elastic moduli (or velocities) are basically the functionof all these factors combined Especially in low porosityand permeability rocks elastic moduli of the rock min-eral matrix with spherical pores (high aspect ratio) often

dominate those of the bulk rock [9 26 27] Therefore it istheoretically reasonable to employ the elastic moduli in rocksto estimate seismic velocities Here two effective medium(2DKT and 3DKT) models are investigated to evaluate poreshape geometry using different pore aspect ratios (valueranging from 01 to 09) to predict P- and S-wave velocitiesfrom the well data We know that the pores with high aspectratio (spherical) tend to be stiffer than those with low aspectratio (oblate or penny shape pores) That is to say P- and S-wave velocities in the reservoir with spherical pores tend tobe higher than one with oblate pores or penny shapeThe factcan be easily understood because stiffer pores contribute tothe stiffness of the whole rock while oblate pores contributeto the compliance of thewhole rockHere actual well numberA is taken as an example containing dipole sonic logs toestimate P- and S-wave velocity From Figure 10 we can seeclearly that the increase in P- and S-wave velocities is sharperwhen the pore aspect ratio increases from 01 to 06 for both2DKT and 3DKT models However the increasing trend isnominal when the pore aspect ratio exceeds beyond 06 in

10 International Journal of Geophysics

12366

12368

12370

1237247 48

Vp (kms) Vs (kms)

Dep

th (f

eet)

12366

12368

12370

12372

Dep

th (f

eet)

VpVp-2DKTVp-3DKT

26 27

VsVs-2DKTVs-3DKT

Figure 11 A comparison between the predicted and measured P- and S-wave velocities by employing 2DKT and 3DKTmodels using criticalaspect ratio (06) of well number A

both models Significantly velocities predicted by the 3DKTmodel agreewell with thosemeasured compared to the 2DKTmodel (Figure 11)

It is due to the fact that the model adjusts pore aspectratio or the proportion of various 3D pores and effectivelymodels the geometries of various pore shapes which is thedominant factor influencing the velocities of the elastic-wave propagating in the clastic reservoirs [28] Due to thesubstantial amount of clay (15) contents that may containmicrocracks the dry rock moduli estimation by the Kuster-Toksoz (2DKT) model (assuming ellipsoidal) could not beappropriate The addition of small volumes of cracks to therock matrix significantly weakens the rock The predicted P-and S-wave velocities from 3DKT are more consistent withthe measured ones although with some deviation Thus weconsider aspect ratio 06 as the critical value of the pore aspectratio which is the turning point of the increase trend ofpredicted velocities for Marmaton Wash

Thus variations in the elastic moduli are more influencedby pore shape geometry than porosity lower than 10and have a relatively small impact on 119881P119881S ratios Theintegrated application of the 2DKT and 3DKTmodels helpedus properly determine the pore shape geometry of GraniteWash reservoir to estimate site specific 119881P-119881S relationship

54 119881119875-119881119878Relationship of Granite Wash The 119881P-119881S relation

is normally used for 119881S prediction where only P-wave data isavailable and is also very important forAVOanalysis and porefluids prediction For S-wave velocity estimation regard-less of the approach taken it is important to understand

the underlying lithology The rock composition providesfirst order control on dry frame 119881P119881S ratio especially forconsolidated rocks [29] We take advantage of rock physics(effective medium) modeling and derive the 119881P-119881S relation-ship and compare it with the widely used empirical rela-tionship ldquo119881P = 116 lowast 119881S + 136rdquo presented by Castagnaet al [8] for clastic rocks We employ Berrymanrsquos [7] (3DKT)model considering rock composition porosity and poreshape After calibration of the predicted velocities with coredata the 119881P-119881S relationship is generated through regressionanalysis of least-square linear fit (1198772 = 090) as a function of119881P (Kms) (Figure 12)

The derived 119881P-119881S relation by 3DKT for Granite Washprovides closer approximation from 2DKT and global empir-ical relation by Castagna et al [8] Figure 13 shows that theS-wave velocity derived from Castagnarsquos mudrock is overpredicted with known 119881P when comparatively analyzed to2DKT and 3DKT models predictions However there areapparent deviations between the measured and predicteddata by the locally calibrated empirical relation and theeffectivemediummodelsWe actually do not advocate the useof global empirical relations for S-wave velocity predictionfor the Granite Wash but emphasize on site specific builtrelation based on the data of the corresponding study areausing effective medium models Next we employed the bestmodel (3DKT model) to discuss the fluid substitution effectson P- and S-wave velocities Marmaton Wash

55 Fluid Substitution in Granite Wash For practical fluidsubstitution results in tight sands using Gassmannrsquos [21]

International Journal of Geophysics 11

35

30

25

20

1530 35 40 45 50 55 60

0000012500250003750050000625007500087501000

R2 = 90

Vs

(km

s)

Vp (kms)

Vs = 0599 lowast Vp minus 019102

0 4 5 50 55

0000012500250037505000625075087100000

R2 = 90

s = 0599 lowast VpVV minus 019102

GR

Figure 12 Compressional and shearwave velocities ofGraniteWashformation from sonic logs data color coded with gamma ray (GR)from two wells numbers A and B and calibrated with core [11]measurements (Maroon data)

12340

12360

12380

12400

12420

12440

12460

20 22 24 26 28 30 32 34

003500

01556

02763

03969

05175

06381

07588

08794

1000

Dep

th (f

eet)

S-wave velocity (kms)

Vs log

Vs-2DKTVs-3DKT

Vs-RVs-C

Vsh

Figure 13 Comparison of S-wave velocity predictions (color codedwith 119881sh) of well number A with effective medium (2DKT and3DKT) models locally generated relation (119881

119904-119877) and Castagna

global empirical relationship (119881119904-119862)

equations necessitates an appropriate technique which real-istically stiffens low porosity rock by fluctuating the mineralpore fluid or dry rock moduli Application of Gassmannrsquosequation is dependent upon reasonable estimation of thebulk modulus of the dry rock frame (119870

119889) Dry frame rock

bulk (119870119889) modulus can be derived either from (1) laboratory

measurement on core data (2) application of empirical rela-tionships or effective medium theory [30ndash33] or (3) directcalculation from log data by rewriting Gassmannrsquos equation(4) for frame rock bulk (119870

119889) [34] which often generates

negative (a physical impossibility) or unrealistically largevalues in low porosity reservoirs [35] However of the variousalternatives we take essentially a pragmatic approach tocondition the dry frame rockmodel used in fluid substitution

For reliable fluid substitution model in clastic reservoir weconditioned the calculation of dry frame (119870

119889) rock data

implicit in Gassmannrsquos model and apply Biot [36] relationused by Wang et al [37] as follows

119870119889= 119870119898

(1 minus 120573) (7)

where 120573 is sometimes called the Biot coefficient and it is oftencalculated as a function of porosity and the form used in thepaper is derived by Krief et al [38]

(1 minus 120573) = [1 minus 120601]119898(120601)

119898 (120601) =3

(1 minus 120601)

(8)

where 120601 is the formation porosityConsequently the saturated bulk modulus of the rock

(119870sat) for the in situ conditions is first calculated using (1)Equation (7) is combined with Gassmannrsquos equations (4)to generate a pseudo dry frame rock bulk (119870

119889) modulus

suitable for fluid substitution The input parameters for fluidsubstitutions are generally roughly estimated and thus resultin substantial errorsTherefore densities bulkmoduli veloc-ities and viscosities of various pore fluids (brine and gas)are used both from core measurements and estimation usingBatzle and Wang [39] relationships The homogeneous fluid(119870fl) uniformly distributed throughout the pore space hasbeen computed by harmonic average (Reuss) using data fromthewell logs assuming twofluids (water and gas) phases in thereservoir The fluids (119870fl 120588fl) properties occupying the porespace and the bulk modulus of the mineral matrix (119870matrix)are determined by formulations given by Smith et al [35] Tocalculate the bulk modulus of the mineral matrix (119870matrix)the information on the composition (mineral abundance) ofthe rock is used from core study data [11] Mineral matrix(119870matrix) is then calculated via application of Voigt-Reuss-Hill (VRH) averaging the mineral constituents The elasticconstants and densities of pore fluid properties obtained fromcoremeasurements are given in Table 2 Figure 14 an exampleof composite log example from Granite Wash (Marmaton)section demonstrates the significant fluid substitution differ-ences between the wet and the gas effect on the seismic P-wave (119881P) velocity log for well number A The observed fluideffects reach amaximum in the cleanest sandwith the highest(13) porosity and low density response Fluid substitutioneffects correspond more closely with expectations based onraw logs increasing our confidence in the results of AVOmodeling and seismic inversion carried out in phase II

56 Cross-Plotting Attribute Analysis Cross-plotting of elas-tic rock properties from well logs enables the simultane-ous and meaningful evaluation of two attributes that helpsin making straightforward interpretation of hydrocarbonsaturated sands in a particular reservoir For comparisonwe cross-plotted both conventional (P- and S-impedances)and unconventional (lambda mu and rho) attributes in thederived elastic parameter cross-plot space to isolate potentialanomalous zones from a background relationship of well

12 International Journal of Geophysics

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Vsh

(a)

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Sw

(b)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

00 01

Porosity

(c)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

24 26

Density

(d)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

42 48

Vp

GasBrine

(e)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

025 030

PR

(f)

Figure 14 A composite log example fromGraniteWash (Marmaton) section Plots from left to right show logs of (a)119881sh (b)Water saturation(119878119908) (c) porosity (d) density (e) compressional velocity (119881P) and (f) Poisson ratio (PR) Fluid substitution is carried out for brine (blue) and

gas (red) scenarios using dry rock data derived from logs

number A (Figure 15) Figure 15(a) shows where a conven-tional impedance IP versus IS analysis has limitations inclearly discriminating between all the various lithologies andgas sand zone The P- and S-impedances log curves (whichshare both rigidity and density) tend to track each other withdepth and never crossover only S-impedance contrast in theMarmaton sand (12400

1015840

) is clearly less distinguishable thanthe comparable P-impedance contrast By contrast the 120582120588

and 120583120588 attributes have similar value ranges that do cross over

with 120582120588 lt 120583120588 showing a clear wide separation for a hydro-carbon zone while tracking together to clearly indicate thethin tight shale breaks (Figure 15(b)) In Figure 15(c) onlythe gas zone is discernable from background cut-off on the119881P119881S curve which varies between 16 and 20 Figures 15(d)and 15(e) demonstrate the significant advantage of the 120582120588120583120588 cross-plot in isolating both lithologic properties (such assand shale and clastic facies) as well as the gas zone com-pared to IP and IS cross-plot P- and S-impedance data points

International Journal of Geophysics 13

12340

12360

12380

12400

12420

12440

12460

4 6 8 10 12 14

IPIS

Dep

th (f

eet)

Impedance

(a)

12340

12360

12380

12400

12420

12440

12460

10 20 30 40 50 60 70 80

Tight shale

Gas zone

Lambda-rhoMu-rho

Shale

Dep

th (f

eet)

Lambda-rho Mu-rho

(b)

12340

12360

12380

12400

12420

12440

12460

15 16 17 18 19 20 21

Dep

th (f

eet)

VpVs

(c)

80

75

70

65

60

55

50

458 9 10 11 12 13 14

P-impedance

Gas zone

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

S-im

peda

nce

(d)

60

55

50

45

40

35

30

25

0 10 20 30 40 50 60 70

Gas zone

Lambda-rho

Mu-

rho

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

(e)

25

20

15

10

05

00minus15 minus10 minus5 0 5 10 15 20

Shale

Sst line

Wet sand

Gas zone

Lambda-Mu (difference)

Lam

bda

Mu

(rat

io)

003500

01556

02763

03969

05175

06381

07588

08794

1000

Shale

Sst line

Wet sand

Gas zone

Vsh

(f)

Figure 15 Well log curves and cross-plots from Granite Wash (Marmaton sections 12325ndash12470) of well number A (a) P- and S-impedanceslog curves (b) Lambda-rho and Mu-rho log versus depth (c) 119881

119901119881119904ratio versus depth (d) P-impedance versus S-impedance cross-plot color

coded by shale volume (119881sh) (e) Lambda-rho versus Mu-rho cross-plot color coded by shale volume (119881sh) and (f) Lambda-Mu (difference)versus Lambda-rhoMu-rho (ratio) cross-plot color coded by shale volume (119881sh)

14 International Journal of Geophysics

show a close linear relationship with shale having lowestvalues while in comparison the lowest 120582120588 (incompressibilitytimes density) value has the best gas sand values along with 120583120588

(rigidity times density) values higher than shalesThe other morecompetent pure lithologies (silts cemented sands) could beobserved in the opposite upper right quadrant relative to theshales

Following the cross-plots analysis for the same log dataset Figure 15(f) shows an interesting interpretation templatefor the 120582120583 ratio versus the 120582 minus 120583 (difference) cross-plotThe anomalous (gas) zone is marked in a lower left red boxbeyond the (06 minus5) point separating it from the wet sandswhich tend to align in the upper right zone The variationof the slope beyond the origin (1 0) with increasing gassaturation (blue to green dotted lines) reveals an unexpectedeffect of the gas on the rigidity of the sand This is due tothe effective porosity and lower density gas effect on rigidityleading to a slight steeping of the ldquosandstone linerdquo slopeThus in this case unconventional attributes plots provide aclear separation of the anomalous hydrocarbon zones and thelithologic properties of the sand and shale facies which havebeen color coded with shale volume

6 Conclusion

With the above discussion and limitations the followingconclusions can be derived Rock physics diagnostic and thinsections analysis indicate the variations in elastic propertiesof Granite Wash which can be explained by diagenetic phaseof quartz (or silica) cementation CementedWashes have rel-atively high elastic moduli and can be modeled primarily bycontact-cement model of scheme 2 Velocity prediction fromeffective medium models shows that pore shape geometrysignificantly influences the P- and S-wave velocities in ourreservoirs as observed in the velocity data evaluated Core andpetrographic observations also confirm cementations withcomplex pore geometry containing variable concentrations ofhigh and low pores aspect ratio Hence Berrymanrsquos (3DKT)model is proven to be the best one to discuss the influence ofpore shape on seismic velocities and used for the generationof locally calibrated empirical (119881P-119881S) relationship that bestdescribes our data set Finally we applied fluid substitution onlog data for calculating reservoir elastic properties with dif-ferent pore fluid (brine and gas) for an improved petrophys-ical discrimination of rock properties using unconventionalLame parameters 120582 (pure incompressibility) and 120583 (rigidity)attributes over conventional 119881P 119881S attributes analysis Theresearch outcomes are intended to be rigorously used as afoundation for ongoing reservoir characterization study inphase II

Disclosure

This paper is developed from an oral and poster presentationat American Association of Petroleum Geologists (AAPG)Annual Convention amp Exhibition David L Lawrence Con-vention Center Pittsburgh PA USA May 19ndash22 2013

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank The University of Tulsa and NewfieldExploration Company for support of this research The firstauthor would like to thank Jeff Kane (Corporate Petro-physicist) for his scholastic discussions They also thankthe anonymous reviewers for the constructive comments onimproving our paper

References

[1] S Ingram I Paterniti B Rothkopf and C Stevenson ldquoGranitewash field study-buffalo wallow field Texas panhandlerdquo inProceedings of the SPE Eastern Regional Meeting SPE104546Canton Ohio USA October 2006

[2] M I Odigi ldquoDiagenesis and reservoir quality of Cretaceoussandstone of Nkporo Shale (Campanian) Southeastern Benuetrough Nigeriardquo Journal of Geology and Mining Research vol 3pp 265ndash280 2011

[3] F Ruiz and A Cheng ldquoA rock physics model for tight gas sandrdquoThe Leading Edge vol 29 no 12 pp 1484ndash1489 2010

[4] RDMindlin ldquoCompliance of elastic bodies in contactrdquo Journalof Applied Mechanics vol 16 pp 259ndash268 1949

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[6] J Dvorkin and A Nur ldquoElasticity of high-porosity sandstonestheory for twoNorth Sea data setsrdquoGeophysics vol 61 no 5 pp1363ndash1370 1996

[7] J G Berryman ldquoLong-wavelength propagation in compositeelastic media II Ellipsoidal Inclusionsrdquo The Journal of theAcoustical Society of America vol 68 no 6 pp 1820ndash1831 1980

[8] J P Castagna M L Batzle and R L Eastwood ldquoRelationshipsbetween compressional-wave and shear-wave velocities in clas-tic silicate rocksrdquo Geophysics vol 50 no 4 pp 571ndash581 1985

[9] G T Kuster and M N Toksoz ldquoVelocity and attenuation ofseismic waves in two-phase mediardquo Geophysics vol 39 no 5pp 587ndash618 1974

[10] G E Moore ldquoPennsylvanian paleogeography of the south-ern Mid-Continentrdquo in Pennsylvanian Sandstones of the Mid-Continent N J Hyne Ed pp 2ndash12 Tulsa Geological Society(TGS) Tulsa Okla USA 1979

[11] Core Lab ldquoGraniteWash geological Petrophysical and comple-tions study for improved formation evaluation and optimal wellcompletionrdquo Proprietary industry study 2005

[12] K S Johnson and B J Cardott ldquoGeologic framework andhydrocarbon source rocks of Oklahomardquo in Proceedings of theSource Rocks in the Southern Midcontinent K S JohnsonB andJ Cardott Eds pp 21ndash37 OGS Circular 1992

[13] N R Gelphman ldquoWest sentinel oil field Washita CountyOklahoma sedimentology of the ldquoGraniteWashrdquo and structuralgeologyrdquoOklahoma City Geological Society Shale Shaker vol 10pp 234ndash247 1960

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

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Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

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Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

4 International Journal of Geophysics

CaIQo

Fo

01mm

(a)

ChIAq

20 microns

(b)

Ab-o

Ab-o

Ab

III

10 microns

(c)

Figure 4 Petrography and mineralogy Granite Wash (Marmaton section) sandstones pore-filling constituents are (a) calcite fill pores andquartz overgrowths feldspar overgrowths (b) euhedral microquartz crystals and (c) authigenic albite overgrowths and single crystals from[11] Data is courtesy of Newfield Exploration Company

Systemsseries

Ochoan

Guadalupian

Leonardian

Wolfcampian

Virgilian

Missourian

Desmoinesian

Atokan

Morrowan

ChesterianMiss

Penn

sylv

ania

nPe

rmia

n

Elk city sandstoneDoxey shale

Cloud chief formationWhitehorse group

El Reno groupHennessey shale

Garber sandstoneWellington formation

Chase groupCouncil grove group

Admire groupWabaunsee group

Shawnee groupDouglas groupOchelata groupSkiatook group

Marmaton groupCherokee group

Atoka group

Morrow group

Springer formation

Chester group

Anadarko BasinSw Oklahoma

Gra

nite

was

h

(a)

Permian

Virgilian

Missourian

Desmionesian

Atokan

Morrowan

Sprin

geran

Granite wash

Pm

Pm

Cam

bro-

Ord

ovic

ian

Silu

ro D

evon

ian

Miss

issip

pian

Top Pennsylvanian

Cherokee group

Prec

ambr

ian

0

5000

10000

15 000

0

1000

2000

3000

4000

5000

OilGas

Mississ

ippian

0 5 10(mL)

0 5 10 15(km)

Subs

ea d

epth

(m)

N

Subs

ea d

epth

(m)

washCarbonate-

chert

Dolomitewash

S

Datum sea level

(b)

Figure 5 (a) Anadarko Basin stratigraphic column [12] and (b) stratigraphic dip cross-sectionmdashunroofing sequence [11]

International Journal of Geophysics 5

12300

12350

12400

12450

12500

12550

12600

126500 100 00 06 45 24 245 000 015

GR Sw Vp Vs Density Porosity

Dep

th (f

eet)

12368

12370

12372

12374

12376

12378

12380

12382

12395

12396

12397

12398

12399

12400

12401

12402

12403

12404

12405

12406

12407

12408

12409

Figure 6Well log curves from Stiles Ranch fieldWheeler County Anadarko Basin Texas (USA) From left to right GR (API) 119878119908119881P (Kms)

119881S (Kms) density (gccc) porosity and FMI logs with superposed core [11] data (maroon) Data is courtesy of Newfield ExplorationCompany

assume that each component and the rock are isotropic andelastic [18]

The behaviors of elastic properties due to cementation iscaptured by contact-cement model [6] The cement modelassumes that porosity decreases from the initial criticalporosity value due to the uniform deposition of cement layerson the surface of the grains This cement may be diageneticquartz calcite or reactive clay (such as illite) The diageneticcement dramatically increases the stiffness of the sand byreinforcing the grain contacts The mathematical model isbased on a rigorous contact-problem solution by Dvorkinet al [19 20] It is assumed that themicrostructure used in themodels quantitatively mimics the data The rock diagnosticwas conducted on the log scale and was confirmed with thecore data (thin sections)

Furthermore we employ effective medium [7 9] modelsin order to estimate and comparatively analyze seismic veloc-ity (or moduli) based on pore aspect ratio to characterize theappropriate pore-shape factor in the GraniteWash reservoirsThe generalized formulas of the Kuster-Toksoz [9] model areexpressed by Berryman [7] as follows

(119870119870119879

minus 119870119898)

(119870119898

+ (43) 120583119898)

(119870119870119879

+ (43) 120583119898)

=

119873

sum

119894=1

119909119894(119870119894minus 119870119898) 119875119898119894

(120583119870119879

minus 120583119898)

(120583119898

+ 120577119898)

(120583119870119879

+ 120577119898)

=

119873

sum

119894=1

119909119894(120583119894minus 120583119898) 119876119898119894

(2)

where the summation is over the different inclusions withvolume concentration 119909

119894and

120577 =120583

6

(9119870 + 8120583)

(119870 + 2120583) (3)

The coefficients 119875119898119894 and 119876

119898119894 describe the effect of aninclusion of material ldquo119894rdquo in a background medium 119898 inclu-sion coefficients119875 and119876 describe the elastic properties of thepore phase (Table 1) Two types of calculation algorithms forthe coefficients 119875 and 119876 are followed (1) The Kuster-Toksozmodel (spheroidal inclusions) with arbitrary pore aspect ratiois named 2DKT model and (2) Berrymanrsquos [7] 3D specialpores (penny-shaped crack) are called 3DKT model

To estimate the effect of the pore fluids on the elasticproperties Gassmannrsquos [21] fluid substitution approach isused to model different scenarios Gassmannrsquos equationsrelate the bulk modulus of a rock to its pore frame and fluidproperties The bulk modulus of a saturated rock is given bythe Gassmann theory

119870sat = 119870119889+

(1 minus 119870119889119870119898)2

(120601119870119891) + ((1 minus 120601) 119870

119898) minus (119870

1198891198702119898)

(4)

where119870sat the bulkmodulus of a rock is saturated with a fluidof bulk modulus119870

119891119870119889is the frame bulk modulus119870

119898is the

matrix (grain) bulk modulus and 120601 is porosity Consider

120583sat = 120583119889 (5)

The bulk density (120588sat) is obtained from mass balanceequation

120588sat = 120588119898

(1 minus 120601) + 120601120588119891 (6)

where120601 120588sat 120588119898 and 120588119891are respectively the porosity density

of saturated rocks the matrix and pore fluids The density ofthe solid (mineral) phase is 120588

119898= (1 minus 119862)120588

119902+ 119862120588119888

In this study we take assumptions that reservoir sand-stone mineral grains are mainly (a) cemented (b) spherical

6 International Journal of Geophysics

Table 1 Coefficients of 119875 and 119876 for some specific shapes The subscripts ldquo119898rdquo and ldquo119894rdquo refer to the background and inclusion materials [7]

Inclusion shape 119875119898119894

119876119898119894

Spheres119870119898

+ (43)120583119898

119870119894+ (43)120583

119898

120583119898

+ 120577119898

120583119894+ 120577119898

Needles119870119898

+ 120583119898

+ (13)120583119894

119870119894+ 120583119898

+ (13)120583119894

1

5(

4120583119898

120583119898

+ 120583119894

+ 2120583119898

+ 120574119898

120583119894+ 120574119898

+119870119894+ (43)120583

119898

119870119894+ 120583119898

+ (13)120583119894

)

Disks119870119898

+ (43)120583119894

119870119894+ (43)120583

119894

120583119898

+ 120577119894

120583119894+ 120577119894

Penny cracks119870119898

+ (43)120583119894

119870119894+ (43)120583

119894+ 120587120572120573

119898

1

5(1 +

8120583119898

4120583119894+ 120587120572(120583

119898+ 2120573119898)+ 2

119870119894+ (23)(120583

119894+ 120583119898)

119870119894+ (43)120583

119894+ 120587120572120573

119898

)

Table 2 The core values of moduli and densities of minerals andfluids used in this study [11]

Minerals and fluids Density(gcc)

Bulk modulus(GPa)

Shear modulus(Gpa)

Quartz 265 5580 38Clays 258 209 685Dolomite 287 8010 4500calcite 271 7550 3400Oil 07 162 mdashGas 012 0032 mdash

shape and (c) randomly packed From the core study thepercentage volume of the rock matrix is dominantly clay(15) and quartz (40) The basic parameters used for fluidsproperties in velocity predictionwere taken from core studies(Table 2) Analysis of log and core data from well A suggeststhat these assumptions are met for our reservoir

5 Data Analysis and Results

51 Rock Physics Diagnostics Application to the Granite WashA common practice in rock physics analysis is to generatecross-plots of elastic properties to study diagenetic trends andproperty dependencies of sands and shales The cross-plotsexhibit interdependence among the various rock propertiesWe particularly focused on a thick Marmaton Wash section(12325ndash12435 ft) of well number A containing various logcurves (Figure 6) Marmaton section is marked by extremelylow and fairly constant (about 375) gamma-ray readings andhigh velocity (about 46 kms) surrounded by shale packages

In spite of the congruency between different log curvesin Figure 6 various cross-plots of Marmaton sands do notproduce unique trends (Figure 7) In Figure 7(a) low andhigh ldquoGRrdquo sediments do not show huge variations in bulkdensity implying that both sand and shale arewell compactedand cemented Figure 7(b) shows the same as the case inGR-porosity plot and Figure 7(c) shows that presence ofclay with significantly lower impedance than the quartz itcements Figures 7(d) 7(e) and 7(f) modulusvelocity versusporosity cross-plots show 119881 shape overturned paths Thesuperimposed119881-shaped paths actually represent clay contentfrom clean sands to pure shale The sands overlapped with

the clean shales probably indicate the sand cluster includingboth clean and shaly sands [22] From moduli-velocity-porosity cross-plots we can assess the progress of diagenesisand compaction trends of sediments For soft or uncementedand stiff sediments the paths are fairly gentle with notmuch of velocity contrast between sands and shales Forcemented sediments (case under examination) increasingdepth quartz cementation provides a large contrast in velocityfrom shaly sands to the pure shales This drop might beassociated with the effect of proceeding from grain supportedto clay-supported sediments under relatively high effectivepressure [23 24] Thus a large part of the scatter observedin modulivelocity versus porosity plots is due to presence ofclay in the rocks

We construct the Hashin and Shtrikman [5] bounds(upper and lower) to conceptually explain and model thedata scatter of the Granite Wash from four different wells(Figure 8) The bounds provide an idea of the narrowestpossible range of elasticmoduli without a detailed descriptionof the microstructure of Granite Wash rocks For boundsonly volume fractions of the constituents quartz is 40 byvolume of solid fraction calcite is 7 by volume of solidfraction and their elastic moduli obtained from core dataare used without specifying the geometric details of theirarrangements The upper and lower bounds are determinedby exchanging the materials one that fills the pores andanother constructing the shell The upper bound is realizedwhen stiffer material forms the shell the lower bound isrealized when the stiffer material is physically present in thecore of the shell [18] The separation between upper andlower bounds shows how elastically different the constituentsare The elastic contrast between pure sand and the criticalconcentration mixture is relatively strong because of thevariable elastic contrast between quartz and clay grains Thisimplies that the effect of quartz and clay minerals may becritical for seismic interpretation of Granite Wash

Moduli-porosity data point superimposed with core datafall between the upper and lower bounds which indicates thatMarmaton sands are consolidated sandstone with cemen-tations Superimposed core data from Marmaton sectionconfirms the presence of quartz and calcite cement

52 Confirming the Diagnostic From Figure 4 we observethatmicrocrystalline quartz is found in close contact betweengrains so this quartz cement has a stiffening effect on

International Journal of Geophysics 7

Bulk

den

sity

27

26

25

24

2350 100 150 200

Gamma ray

(a)

Poro

sity

015

010

005

00050 100 150 200

Density

Gamma ray

(b)14

12

10

50 100 150 200

Density

Gamma ray

IP

(c)

Vp

(km

s)

50

45

40

35000 005 010 015 020

Porosity

Density

(d)

70

60

50

40

000 005 010 015 020

Porosity

K(G

Pa)

Vsh

(e)

70

60

50

40

000 005 010 015 020

Sw

Porosity

K(G

Pa)

(f)

Figure 7 Cross-plots of Well log data Marmaton section from Figure 6 (a) bulk density versus GR (b) total porosity versus GR color codedby density (c) P-wave impedance versus GR color coded by density (d) P-wave versus total porosity color coded by density (e) bulkmodulusversus total porosity color coded by shale volume (119881sh) and (f) bulk modulus versus total porosity color coded by water saturation (119878

119908)

the elastic properties of the Granite Wash Studies havesupported the fact that dissolution and local precipitationof quartz accompanied by compaction cause long termdeformation of the aggregate resulting in permeability andporosity reduction [25] Hence quartz precipitation betweenthe grains causes major porosity reduction

For diagnostic purposes we cross-plotted the elasticproperties from well number A (Marmaton section) invelocity-porosity and moduli-porosity planes superimpos-ing theoretical rock physics models (Figure 9) The cross-plots show that gradually increasing clay content with depthcorresponds with decreasing elastic value Based on labora-tory data log data thin section analysis and rock-physicsmodels the data is sorted into Dvorkin and Nurrsquos [6] follow-ing three schemes uncemented (or soft)WashesWasheswithsmall amounts of quartz cement at grain contacts (scheme 1)and Washes from the same formation but with clay cement

(scheme 2) The upper two theoretical (schemes 1 and 2)curves correspond to the pure quartz cases while lower curveis for pure clay or uncemented rock Scheme 1 represents allcement deposited precisely at grain contacts and the slopesteep Scheme 2 curve corresponds to the case where thecontact cement region is formed at the intersection of cementshells that are uniformly deposited on grain surfaces cementfills the space of the large pores and results in a flat slopeIt is quite possible that the variation in microcracks maybe the major reason why the slope of velocity-porosity andmodulus-porosity planes differs in different schemes

We noticed that the maximum data points fall on therock physics scheme 2 line representing uniformly depositedcement on grain surfaceThe thin sections samples from wellnumber A confirmed the rock diagnostic that with increasingdepth and pressure the decreasing porosity is a result ofcontact cement in the rock-physics cross-plots It is quite

8 International Journal of Geophysics

125

100

75

50

25

0000 025 050 075 100

Bulk

mod

ulus

(GPa

)

Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

50

40

30

20

10

0000 025 050 075 100

Shea

r mod

ulus

(GPa

)Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

Figure 8 Moduli versus porosity for different wells and core [11] data with superposition of Hashin and Shtrikman [5] bounds

6

5

4

3

2

00 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Vp

(km

s)

Porosity

Vp versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(a)

80

60

40

20

000 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Bulk

mod

ulus

(GPa

)

Porosity

K versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(b)

Figure 9 (a) P-wave velocity versus porosity and (b) modulus versus porosity (core [11] data included) color coded with shale volume (119881sh)

of the Marmaton section in well number A with rock physics models curves superimposed

International Journal of Geophysics 9

12368

12370

12372

465 470 475

P-wave velocity

465 470 475

P-wave velocity

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

260 265 270 275

S-wave velocity

260 265 270 275

S-wave velocity

Vp2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-09

2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-082DKT-09

Vs

3DKT-013DKT-023DKT-033DKT-043DKT-053DKT-063DKT-073DKT-08

Vp

3DKT-09

3DKT-04

3DKT-013DKT-023DKT-03

3DKT-053DKT-063DKT-073DKT-083DKT-09

Vs

Figure 10 P- and S-wave velocities prediction by employing 2DKT and 3DKT models using different aspect ratios (01ndash09) of well numberA

interesting that the well-log data with resolution of tens-of-centimeter reflects what we observe at the microscaleBy establishing a site-specific rock physics transform (unce-mented and cemented models) between the total porositymineralogy and elastic properties we conclude the rockphysics diagnostics here In the next section we analyze poreshape influence on seismic velocities in order to generatelocally calibrated 119881P-119881S relationship for Granite Wash

53 Analysis of Pore Shapes Influencing Seismic VelocitiesTheoretically it has been reckoned that the velocities in twostates (dry or saturated) are mainly controlled by the effectivemoduli (bulk and shear) which in turn depend on poreshape geometry mineralogy properties of solids the porefluid degree of consolidation and cementation [9 26 27]Thus elastic moduli (or velocities) are basically the functionof all these factors combined Especially in low porosityand permeability rocks elastic moduli of the rock min-eral matrix with spherical pores (high aspect ratio) often

dominate those of the bulk rock [9 26 27] Therefore it istheoretically reasonable to employ the elastic moduli in rocksto estimate seismic velocities Here two effective medium(2DKT and 3DKT) models are investigated to evaluate poreshape geometry using different pore aspect ratios (valueranging from 01 to 09) to predict P- and S-wave velocitiesfrom the well data We know that the pores with high aspectratio (spherical) tend to be stiffer than those with low aspectratio (oblate or penny shape pores) That is to say P- and S-wave velocities in the reservoir with spherical pores tend tobe higher than one with oblate pores or penny shapeThe factcan be easily understood because stiffer pores contribute tothe stiffness of the whole rock while oblate pores contributeto the compliance of thewhole rockHere actual well numberA is taken as an example containing dipole sonic logs toestimate P- and S-wave velocity From Figure 10 we can seeclearly that the increase in P- and S-wave velocities is sharperwhen the pore aspect ratio increases from 01 to 06 for both2DKT and 3DKT models However the increasing trend isnominal when the pore aspect ratio exceeds beyond 06 in

10 International Journal of Geophysics

12366

12368

12370

1237247 48

Vp (kms) Vs (kms)

Dep

th (f

eet)

12366

12368

12370

12372

Dep

th (f

eet)

VpVp-2DKTVp-3DKT

26 27

VsVs-2DKTVs-3DKT

Figure 11 A comparison between the predicted and measured P- and S-wave velocities by employing 2DKT and 3DKTmodels using criticalaspect ratio (06) of well number A

both models Significantly velocities predicted by the 3DKTmodel agreewell with thosemeasured compared to the 2DKTmodel (Figure 11)

It is due to the fact that the model adjusts pore aspectratio or the proportion of various 3D pores and effectivelymodels the geometries of various pore shapes which is thedominant factor influencing the velocities of the elastic-wave propagating in the clastic reservoirs [28] Due to thesubstantial amount of clay (15) contents that may containmicrocracks the dry rock moduli estimation by the Kuster-Toksoz (2DKT) model (assuming ellipsoidal) could not beappropriate The addition of small volumes of cracks to therock matrix significantly weakens the rock The predicted P-and S-wave velocities from 3DKT are more consistent withthe measured ones although with some deviation Thus weconsider aspect ratio 06 as the critical value of the pore aspectratio which is the turning point of the increase trend ofpredicted velocities for Marmaton Wash

Thus variations in the elastic moduli are more influencedby pore shape geometry than porosity lower than 10and have a relatively small impact on 119881P119881S ratios Theintegrated application of the 2DKT and 3DKTmodels helpedus properly determine the pore shape geometry of GraniteWash reservoir to estimate site specific 119881P-119881S relationship

54 119881119875-119881119878Relationship of Granite Wash The 119881P-119881S relation

is normally used for 119881S prediction where only P-wave data isavailable and is also very important forAVOanalysis and porefluids prediction For S-wave velocity estimation regard-less of the approach taken it is important to understand

the underlying lithology The rock composition providesfirst order control on dry frame 119881P119881S ratio especially forconsolidated rocks [29] We take advantage of rock physics(effective medium) modeling and derive the 119881P-119881S relation-ship and compare it with the widely used empirical rela-tionship ldquo119881P = 116 lowast 119881S + 136rdquo presented by Castagnaet al [8] for clastic rocks We employ Berrymanrsquos [7] (3DKT)model considering rock composition porosity and poreshape After calibration of the predicted velocities with coredata the 119881P-119881S relationship is generated through regressionanalysis of least-square linear fit (1198772 = 090) as a function of119881P (Kms) (Figure 12)

The derived 119881P-119881S relation by 3DKT for Granite Washprovides closer approximation from 2DKT and global empir-ical relation by Castagna et al [8] Figure 13 shows that theS-wave velocity derived from Castagnarsquos mudrock is overpredicted with known 119881P when comparatively analyzed to2DKT and 3DKT models predictions However there areapparent deviations between the measured and predicteddata by the locally calibrated empirical relation and theeffectivemediummodelsWe actually do not advocate the useof global empirical relations for S-wave velocity predictionfor the Granite Wash but emphasize on site specific builtrelation based on the data of the corresponding study areausing effective medium models Next we employed the bestmodel (3DKT model) to discuss the fluid substitution effectson P- and S-wave velocities Marmaton Wash

55 Fluid Substitution in Granite Wash For practical fluidsubstitution results in tight sands using Gassmannrsquos [21]

International Journal of Geophysics 11

35

30

25

20

1530 35 40 45 50 55 60

0000012500250003750050000625007500087501000

R2 = 90

Vs

(km

s)

Vp (kms)

Vs = 0599 lowast Vp minus 019102

0 4 5 50 55

0000012500250037505000625075087100000

R2 = 90

s = 0599 lowast VpVV minus 019102

GR

Figure 12 Compressional and shearwave velocities ofGraniteWashformation from sonic logs data color coded with gamma ray (GR)from two wells numbers A and B and calibrated with core [11]measurements (Maroon data)

12340

12360

12380

12400

12420

12440

12460

20 22 24 26 28 30 32 34

003500

01556

02763

03969

05175

06381

07588

08794

1000

Dep

th (f

eet)

S-wave velocity (kms)

Vs log

Vs-2DKTVs-3DKT

Vs-RVs-C

Vsh

Figure 13 Comparison of S-wave velocity predictions (color codedwith 119881sh) of well number A with effective medium (2DKT and3DKT) models locally generated relation (119881

119904-119877) and Castagna

global empirical relationship (119881119904-119862)

equations necessitates an appropriate technique which real-istically stiffens low porosity rock by fluctuating the mineralpore fluid or dry rock moduli Application of Gassmannrsquosequation is dependent upon reasonable estimation of thebulk modulus of the dry rock frame (119870

119889) Dry frame rock

bulk (119870119889) modulus can be derived either from (1) laboratory

measurement on core data (2) application of empirical rela-tionships or effective medium theory [30ndash33] or (3) directcalculation from log data by rewriting Gassmannrsquos equation(4) for frame rock bulk (119870

119889) [34] which often generates

negative (a physical impossibility) or unrealistically largevalues in low porosity reservoirs [35] However of the variousalternatives we take essentially a pragmatic approach tocondition the dry frame rockmodel used in fluid substitution

For reliable fluid substitution model in clastic reservoir weconditioned the calculation of dry frame (119870

119889) rock data

implicit in Gassmannrsquos model and apply Biot [36] relationused by Wang et al [37] as follows

119870119889= 119870119898

(1 minus 120573) (7)

where 120573 is sometimes called the Biot coefficient and it is oftencalculated as a function of porosity and the form used in thepaper is derived by Krief et al [38]

(1 minus 120573) = [1 minus 120601]119898(120601)

119898 (120601) =3

(1 minus 120601)

(8)

where 120601 is the formation porosityConsequently the saturated bulk modulus of the rock

(119870sat) for the in situ conditions is first calculated using (1)Equation (7) is combined with Gassmannrsquos equations (4)to generate a pseudo dry frame rock bulk (119870

119889) modulus

suitable for fluid substitution The input parameters for fluidsubstitutions are generally roughly estimated and thus resultin substantial errorsTherefore densities bulkmoduli veloc-ities and viscosities of various pore fluids (brine and gas)are used both from core measurements and estimation usingBatzle and Wang [39] relationships The homogeneous fluid(119870fl) uniformly distributed throughout the pore space hasbeen computed by harmonic average (Reuss) using data fromthewell logs assuming twofluids (water and gas) phases in thereservoir The fluids (119870fl 120588fl) properties occupying the porespace and the bulk modulus of the mineral matrix (119870matrix)are determined by formulations given by Smith et al [35] Tocalculate the bulk modulus of the mineral matrix (119870matrix)the information on the composition (mineral abundance) ofthe rock is used from core study data [11] Mineral matrix(119870matrix) is then calculated via application of Voigt-Reuss-Hill (VRH) averaging the mineral constituents The elasticconstants and densities of pore fluid properties obtained fromcoremeasurements are given in Table 2 Figure 14 an exampleof composite log example from Granite Wash (Marmaton)section demonstrates the significant fluid substitution differ-ences between the wet and the gas effect on the seismic P-wave (119881P) velocity log for well number A The observed fluideffects reach amaximum in the cleanest sandwith the highest(13) porosity and low density response Fluid substitutioneffects correspond more closely with expectations based onraw logs increasing our confidence in the results of AVOmodeling and seismic inversion carried out in phase II

56 Cross-Plotting Attribute Analysis Cross-plotting of elas-tic rock properties from well logs enables the simultane-ous and meaningful evaluation of two attributes that helpsin making straightforward interpretation of hydrocarbonsaturated sands in a particular reservoir For comparisonwe cross-plotted both conventional (P- and S-impedances)and unconventional (lambda mu and rho) attributes in thederived elastic parameter cross-plot space to isolate potentialanomalous zones from a background relationship of well

12 International Journal of Geophysics

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Vsh

(a)

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Sw

(b)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

00 01

Porosity

(c)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

24 26

Density

(d)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

42 48

Vp

GasBrine

(e)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

025 030

PR

(f)

Figure 14 A composite log example fromGraniteWash (Marmaton) section Plots from left to right show logs of (a)119881sh (b)Water saturation(119878119908) (c) porosity (d) density (e) compressional velocity (119881P) and (f) Poisson ratio (PR) Fluid substitution is carried out for brine (blue) and

gas (red) scenarios using dry rock data derived from logs

number A (Figure 15) Figure 15(a) shows where a conven-tional impedance IP versus IS analysis has limitations inclearly discriminating between all the various lithologies andgas sand zone The P- and S-impedances log curves (whichshare both rigidity and density) tend to track each other withdepth and never crossover only S-impedance contrast in theMarmaton sand (12400

1015840

) is clearly less distinguishable thanthe comparable P-impedance contrast By contrast the 120582120588

and 120583120588 attributes have similar value ranges that do cross over

with 120582120588 lt 120583120588 showing a clear wide separation for a hydro-carbon zone while tracking together to clearly indicate thethin tight shale breaks (Figure 15(b)) In Figure 15(c) onlythe gas zone is discernable from background cut-off on the119881P119881S curve which varies between 16 and 20 Figures 15(d)and 15(e) demonstrate the significant advantage of the 120582120588120583120588 cross-plot in isolating both lithologic properties (such assand shale and clastic facies) as well as the gas zone com-pared to IP and IS cross-plot P- and S-impedance data points

International Journal of Geophysics 13

12340

12360

12380

12400

12420

12440

12460

4 6 8 10 12 14

IPIS

Dep

th (f

eet)

Impedance

(a)

12340

12360

12380

12400

12420

12440

12460

10 20 30 40 50 60 70 80

Tight shale

Gas zone

Lambda-rhoMu-rho

Shale

Dep

th (f

eet)

Lambda-rho Mu-rho

(b)

12340

12360

12380

12400

12420

12440

12460

15 16 17 18 19 20 21

Dep

th (f

eet)

VpVs

(c)

80

75

70

65

60

55

50

458 9 10 11 12 13 14

P-impedance

Gas zone

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

S-im

peda

nce

(d)

60

55

50

45

40

35

30

25

0 10 20 30 40 50 60 70

Gas zone

Lambda-rho

Mu-

rho

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

(e)

25

20

15

10

05

00minus15 minus10 minus5 0 5 10 15 20

Shale

Sst line

Wet sand

Gas zone

Lambda-Mu (difference)

Lam

bda

Mu

(rat

io)

003500

01556

02763

03969

05175

06381

07588

08794

1000

Shale

Sst line

Wet sand

Gas zone

Vsh

(f)

Figure 15 Well log curves and cross-plots from Granite Wash (Marmaton sections 12325ndash12470) of well number A (a) P- and S-impedanceslog curves (b) Lambda-rho and Mu-rho log versus depth (c) 119881

119901119881119904ratio versus depth (d) P-impedance versus S-impedance cross-plot color

coded by shale volume (119881sh) (e) Lambda-rho versus Mu-rho cross-plot color coded by shale volume (119881sh) and (f) Lambda-Mu (difference)versus Lambda-rhoMu-rho (ratio) cross-plot color coded by shale volume (119881sh)

14 International Journal of Geophysics

show a close linear relationship with shale having lowestvalues while in comparison the lowest 120582120588 (incompressibilitytimes density) value has the best gas sand values along with 120583120588

(rigidity times density) values higher than shalesThe other morecompetent pure lithologies (silts cemented sands) could beobserved in the opposite upper right quadrant relative to theshales

Following the cross-plots analysis for the same log dataset Figure 15(f) shows an interesting interpretation templatefor the 120582120583 ratio versus the 120582 minus 120583 (difference) cross-plotThe anomalous (gas) zone is marked in a lower left red boxbeyond the (06 minus5) point separating it from the wet sandswhich tend to align in the upper right zone The variationof the slope beyond the origin (1 0) with increasing gassaturation (blue to green dotted lines) reveals an unexpectedeffect of the gas on the rigidity of the sand This is due tothe effective porosity and lower density gas effect on rigidityleading to a slight steeping of the ldquosandstone linerdquo slopeThus in this case unconventional attributes plots provide aclear separation of the anomalous hydrocarbon zones and thelithologic properties of the sand and shale facies which havebeen color coded with shale volume

6 Conclusion

With the above discussion and limitations the followingconclusions can be derived Rock physics diagnostic and thinsections analysis indicate the variations in elastic propertiesof Granite Wash which can be explained by diagenetic phaseof quartz (or silica) cementation CementedWashes have rel-atively high elastic moduli and can be modeled primarily bycontact-cement model of scheme 2 Velocity prediction fromeffective medium models shows that pore shape geometrysignificantly influences the P- and S-wave velocities in ourreservoirs as observed in the velocity data evaluated Core andpetrographic observations also confirm cementations withcomplex pore geometry containing variable concentrations ofhigh and low pores aspect ratio Hence Berrymanrsquos (3DKT)model is proven to be the best one to discuss the influence ofpore shape on seismic velocities and used for the generationof locally calibrated empirical (119881P-119881S) relationship that bestdescribes our data set Finally we applied fluid substitution onlog data for calculating reservoir elastic properties with dif-ferent pore fluid (brine and gas) for an improved petrophys-ical discrimination of rock properties using unconventionalLame parameters 120582 (pure incompressibility) and 120583 (rigidity)attributes over conventional 119881P 119881S attributes analysis Theresearch outcomes are intended to be rigorously used as afoundation for ongoing reservoir characterization study inphase II

Disclosure

This paper is developed from an oral and poster presentationat American Association of Petroleum Geologists (AAPG)Annual Convention amp Exhibition David L Lawrence Con-vention Center Pittsburgh PA USA May 19ndash22 2013

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank The University of Tulsa and NewfieldExploration Company for support of this research The firstauthor would like to thank Jeff Kane (Corporate Petro-physicist) for his scholastic discussions They also thankthe anonymous reviewers for the constructive comments onimproving our paper

References

[1] S Ingram I Paterniti B Rothkopf and C Stevenson ldquoGranitewash field study-buffalo wallow field Texas panhandlerdquo inProceedings of the SPE Eastern Regional Meeting SPE104546Canton Ohio USA October 2006

[2] M I Odigi ldquoDiagenesis and reservoir quality of Cretaceoussandstone of Nkporo Shale (Campanian) Southeastern Benuetrough Nigeriardquo Journal of Geology and Mining Research vol 3pp 265ndash280 2011

[3] F Ruiz and A Cheng ldquoA rock physics model for tight gas sandrdquoThe Leading Edge vol 29 no 12 pp 1484ndash1489 2010

[4] RDMindlin ldquoCompliance of elastic bodies in contactrdquo Journalof Applied Mechanics vol 16 pp 259ndash268 1949

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[6] J Dvorkin and A Nur ldquoElasticity of high-porosity sandstonestheory for twoNorth Sea data setsrdquoGeophysics vol 61 no 5 pp1363ndash1370 1996

[7] J G Berryman ldquoLong-wavelength propagation in compositeelastic media II Ellipsoidal Inclusionsrdquo The Journal of theAcoustical Society of America vol 68 no 6 pp 1820ndash1831 1980

[8] J P Castagna M L Batzle and R L Eastwood ldquoRelationshipsbetween compressional-wave and shear-wave velocities in clas-tic silicate rocksrdquo Geophysics vol 50 no 4 pp 571ndash581 1985

[9] G T Kuster and M N Toksoz ldquoVelocity and attenuation ofseismic waves in two-phase mediardquo Geophysics vol 39 no 5pp 587ndash618 1974

[10] G E Moore ldquoPennsylvanian paleogeography of the south-ern Mid-Continentrdquo in Pennsylvanian Sandstones of the Mid-Continent N J Hyne Ed pp 2ndash12 Tulsa Geological Society(TGS) Tulsa Okla USA 1979

[11] Core Lab ldquoGraniteWash geological Petrophysical and comple-tions study for improved formation evaluation and optimal wellcompletionrdquo Proprietary industry study 2005

[12] K S Johnson and B J Cardott ldquoGeologic framework andhydrocarbon source rocks of Oklahomardquo in Proceedings of theSource Rocks in the Southern Midcontinent K S JohnsonB andJ Cardott Eds pp 21ndash37 OGS Circular 1992

[13] N R Gelphman ldquoWest sentinel oil field Washita CountyOklahoma sedimentology of the ldquoGraniteWashrdquo and structuralgeologyrdquoOklahoma City Geological Society Shale Shaker vol 10pp 234ndash247 1960

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

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Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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MineralogyInternational Journal of

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Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

International Journal of Geophysics 5

12300

12350

12400

12450

12500

12550

12600

126500 100 00 06 45 24 245 000 015

GR Sw Vp Vs Density Porosity

Dep

th (f

eet)

12368

12370

12372

12374

12376

12378

12380

12382

12395

12396

12397

12398

12399

12400

12401

12402

12403

12404

12405

12406

12407

12408

12409

Figure 6Well log curves from Stiles Ranch fieldWheeler County Anadarko Basin Texas (USA) From left to right GR (API) 119878119908119881P (Kms)

119881S (Kms) density (gccc) porosity and FMI logs with superposed core [11] data (maroon) Data is courtesy of Newfield ExplorationCompany

assume that each component and the rock are isotropic andelastic [18]

The behaviors of elastic properties due to cementation iscaptured by contact-cement model [6] The cement modelassumes that porosity decreases from the initial criticalporosity value due to the uniform deposition of cement layerson the surface of the grains This cement may be diageneticquartz calcite or reactive clay (such as illite) The diageneticcement dramatically increases the stiffness of the sand byreinforcing the grain contacts The mathematical model isbased on a rigorous contact-problem solution by Dvorkinet al [19 20] It is assumed that themicrostructure used in themodels quantitatively mimics the data The rock diagnosticwas conducted on the log scale and was confirmed with thecore data (thin sections)

Furthermore we employ effective medium [7 9] modelsin order to estimate and comparatively analyze seismic veloc-ity (or moduli) based on pore aspect ratio to characterize theappropriate pore-shape factor in the GraniteWash reservoirsThe generalized formulas of the Kuster-Toksoz [9] model areexpressed by Berryman [7] as follows

(119870119870119879

minus 119870119898)

(119870119898

+ (43) 120583119898)

(119870119870119879

+ (43) 120583119898)

=

119873

sum

119894=1

119909119894(119870119894minus 119870119898) 119875119898119894

(120583119870119879

minus 120583119898)

(120583119898

+ 120577119898)

(120583119870119879

+ 120577119898)

=

119873

sum

119894=1

119909119894(120583119894minus 120583119898) 119876119898119894

(2)

where the summation is over the different inclusions withvolume concentration 119909

119894and

120577 =120583

6

(9119870 + 8120583)

(119870 + 2120583) (3)

The coefficients 119875119898119894 and 119876

119898119894 describe the effect of aninclusion of material ldquo119894rdquo in a background medium 119898 inclu-sion coefficients119875 and119876 describe the elastic properties of thepore phase (Table 1) Two types of calculation algorithms forthe coefficients 119875 and 119876 are followed (1) The Kuster-Toksozmodel (spheroidal inclusions) with arbitrary pore aspect ratiois named 2DKT model and (2) Berrymanrsquos [7] 3D specialpores (penny-shaped crack) are called 3DKT model

To estimate the effect of the pore fluids on the elasticproperties Gassmannrsquos [21] fluid substitution approach isused to model different scenarios Gassmannrsquos equationsrelate the bulk modulus of a rock to its pore frame and fluidproperties The bulk modulus of a saturated rock is given bythe Gassmann theory

119870sat = 119870119889+

(1 minus 119870119889119870119898)2

(120601119870119891) + ((1 minus 120601) 119870

119898) minus (119870

1198891198702119898)

(4)

where119870sat the bulkmodulus of a rock is saturated with a fluidof bulk modulus119870

119891119870119889is the frame bulk modulus119870

119898is the

matrix (grain) bulk modulus and 120601 is porosity Consider

120583sat = 120583119889 (5)

The bulk density (120588sat) is obtained from mass balanceequation

120588sat = 120588119898

(1 minus 120601) + 120601120588119891 (6)

where120601 120588sat 120588119898 and 120588119891are respectively the porosity density

of saturated rocks the matrix and pore fluids The density ofthe solid (mineral) phase is 120588

119898= (1 minus 119862)120588

119902+ 119862120588119888

In this study we take assumptions that reservoir sand-stone mineral grains are mainly (a) cemented (b) spherical

6 International Journal of Geophysics

Table 1 Coefficients of 119875 and 119876 for some specific shapes The subscripts ldquo119898rdquo and ldquo119894rdquo refer to the background and inclusion materials [7]

Inclusion shape 119875119898119894

119876119898119894

Spheres119870119898

+ (43)120583119898

119870119894+ (43)120583

119898

120583119898

+ 120577119898

120583119894+ 120577119898

Needles119870119898

+ 120583119898

+ (13)120583119894

119870119894+ 120583119898

+ (13)120583119894

1

5(

4120583119898

120583119898

+ 120583119894

+ 2120583119898

+ 120574119898

120583119894+ 120574119898

+119870119894+ (43)120583

119898

119870119894+ 120583119898

+ (13)120583119894

)

Disks119870119898

+ (43)120583119894

119870119894+ (43)120583

119894

120583119898

+ 120577119894

120583119894+ 120577119894

Penny cracks119870119898

+ (43)120583119894

119870119894+ (43)120583

119894+ 120587120572120573

119898

1

5(1 +

8120583119898

4120583119894+ 120587120572(120583

119898+ 2120573119898)+ 2

119870119894+ (23)(120583

119894+ 120583119898)

119870119894+ (43)120583

119894+ 120587120572120573

119898

)

Table 2 The core values of moduli and densities of minerals andfluids used in this study [11]

Minerals and fluids Density(gcc)

Bulk modulus(GPa)

Shear modulus(Gpa)

Quartz 265 5580 38Clays 258 209 685Dolomite 287 8010 4500calcite 271 7550 3400Oil 07 162 mdashGas 012 0032 mdash

shape and (c) randomly packed From the core study thepercentage volume of the rock matrix is dominantly clay(15) and quartz (40) The basic parameters used for fluidsproperties in velocity predictionwere taken from core studies(Table 2) Analysis of log and core data from well A suggeststhat these assumptions are met for our reservoir

5 Data Analysis and Results

51 Rock Physics Diagnostics Application to the Granite WashA common practice in rock physics analysis is to generatecross-plots of elastic properties to study diagenetic trends andproperty dependencies of sands and shales The cross-plotsexhibit interdependence among the various rock propertiesWe particularly focused on a thick Marmaton Wash section(12325ndash12435 ft) of well number A containing various logcurves (Figure 6) Marmaton section is marked by extremelylow and fairly constant (about 375) gamma-ray readings andhigh velocity (about 46 kms) surrounded by shale packages

In spite of the congruency between different log curvesin Figure 6 various cross-plots of Marmaton sands do notproduce unique trends (Figure 7) In Figure 7(a) low andhigh ldquoGRrdquo sediments do not show huge variations in bulkdensity implying that both sand and shale arewell compactedand cemented Figure 7(b) shows the same as the case inGR-porosity plot and Figure 7(c) shows that presence ofclay with significantly lower impedance than the quartz itcements Figures 7(d) 7(e) and 7(f) modulusvelocity versusporosity cross-plots show 119881 shape overturned paths Thesuperimposed119881-shaped paths actually represent clay contentfrom clean sands to pure shale The sands overlapped with

the clean shales probably indicate the sand cluster includingboth clean and shaly sands [22] From moduli-velocity-porosity cross-plots we can assess the progress of diagenesisand compaction trends of sediments For soft or uncementedand stiff sediments the paths are fairly gentle with notmuch of velocity contrast between sands and shales Forcemented sediments (case under examination) increasingdepth quartz cementation provides a large contrast in velocityfrom shaly sands to the pure shales This drop might beassociated with the effect of proceeding from grain supportedto clay-supported sediments under relatively high effectivepressure [23 24] Thus a large part of the scatter observedin modulivelocity versus porosity plots is due to presence ofclay in the rocks

We construct the Hashin and Shtrikman [5] bounds(upper and lower) to conceptually explain and model thedata scatter of the Granite Wash from four different wells(Figure 8) The bounds provide an idea of the narrowestpossible range of elasticmoduli without a detailed descriptionof the microstructure of Granite Wash rocks For boundsonly volume fractions of the constituents quartz is 40 byvolume of solid fraction calcite is 7 by volume of solidfraction and their elastic moduli obtained from core dataare used without specifying the geometric details of theirarrangements The upper and lower bounds are determinedby exchanging the materials one that fills the pores andanother constructing the shell The upper bound is realizedwhen stiffer material forms the shell the lower bound isrealized when the stiffer material is physically present in thecore of the shell [18] The separation between upper andlower bounds shows how elastically different the constituentsare The elastic contrast between pure sand and the criticalconcentration mixture is relatively strong because of thevariable elastic contrast between quartz and clay grains Thisimplies that the effect of quartz and clay minerals may becritical for seismic interpretation of Granite Wash

Moduli-porosity data point superimposed with core datafall between the upper and lower bounds which indicates thatMarmaton sands are consolidated sandstone with cemen-tations Superimposed core data from Marmaton sectionconfirms the presence of quartz and calcite cement

52 Confirming the Diagnostic From Figure 4 we observethatmicrocrystalline quartz is found in close contact betweengrains so this quartz cement has a stiffening effect on

International Journal of Geophysics 7

Bulk

den

sity

27

26

25

24

2350 100 150 200

Gamma ray

(a)

Poro

sity

015

010

005

00050 100 150 200

Density

Gamma ray

(b)14

12

10

50 100 150 200

Density

Gamma ray

IP

(c)

Vp

(km

s)

50

45

40

35000 005 010 015 020

Porosity

Density

(d)

70

60

50

40

000 005 010 015 020

Porosity

K(G

Pa)

Vsh

(e)

70

60

50

40

000 005 010 015 020

Sw

Porosity

K(G

Pa)

(f)

Figure 7 Cross-plots of Well log data Marmaton section from Figure 6 (a) bulk density versus GR (b) total porosity versus GR color codedby density (c) P-wave impedance versus GR color coded by density (d) P-wave versus total porosity color coded by density (e) bulkmodulusversus total porosity color coded by shale volume (119881sh) and (f) bulk modulus versus total porosity color coded by water saturation (119878

119908)

the elastic properties of the Granite Wash Studies havesupported the fact that dissolution and local precipitationof quartz accompanied by compaction cause long termdeformation of the aggregate resulting in permeability andporosity reduction [25] Hence quartz precipitation betweenthe grains causes major porosity reduction

For diagnostic purposes we cross-plotted the elasticproperties from well number A (Marmaton section) invelocity-porosity and moduli-porosity planes superimpos-ing theoretical rock physics models (Figure 9) The cross-plots show that gradually increasing clay content with depthcorresponds with decreasing elastic value Based on labora-tory data log data thin section analysis and rock-physicsmodels the data is sorted into Dvorkin and Nurrsquos [6] follow-ing three schemes uncemented (or soft)WashesWasheswithsmall amounts of quartz cement at grain contacts (scheme 1)and Washes from the same formation but with clay cement

(scheme 2) The upper two theoretical (schemes 1 and 2)curves correspond to the pure quartz cases while lower curveis for pure clay or uncemented rock Scheme 1 represents allcement deposited precisely at grain contacts and the slopesteep Scheme 2 curve corresponds to the case where thecontact cement region is formed at the intersection of cementshells that are uniformly deposited on grain surfaces cementfills the space of the large pores and results in a flat slopeIt is quite possible that the variation in microcracks maybe the major reason why the slope of velocity-porosity andmodulus-porosity planes differs in different schemes

We noticed that the maximum data points fall on therock physics scheme 2 line representing uniformly depositedcement on grain surfaceThe thin sections samples from wellnumber A confirmed the rock diagnostic that with increasingdepth and pressure the decreasing porosity is a result ofcontact cement in the rock-physics cross-plots It is quite

8 International Journal of Geophysics

125

100

75

50

25

0000 025 050 075 100

Bulk

mod

ulus

(GPa

)

Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

50

40

30

20

10

0000 025 050 075 100

Shea

r mod

ulus

(GPa

)Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

Figure 8 Moduli versus porosity for different wells and core [11] data with superposition of Hashin and Shtrikman [5] bounds

6

5

4

3

2

00 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Vp

(km

s)

Porosity

Vp versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(a)

80

60

40

20

000 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Bulk

mod

ulus

(GPa

)

Porosity

K versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(b)

Figure 9 (a) P-wave velocity versus porosity and (b) modulus versus porosity (core [11] data included) color coded with shale volume (119881sh)

of the Marmaton section in well number A with rock physics models curves superimposed

International Journal of Geophysics 9

12368

12370

12372

465 470 475

P-wave velocity

465 470 475

P-wave velocity

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

260 265 270 275

S-wave velocity

260 265 270 275

S-wave velocity

Vp2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-09

2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-082DKT-09

Vs

3DKT-013DKT-023DKT-033DKT-043DKT-053DKT-063DKT-073DKT-08

Vp

3DKT-09

3DKT-04

3DKT-013DKT-023DKT-03

3DKT-053DKT-063DKT-073DKT-083DKT-09

Vs

Figure 10 P- and S-wave velocities prediction by employing 2DKT and 3DKT models using different aspect ratios (01ndash09) of well numberA

interesting that the well-log data with resolution of tens-of-centimeter reflects what we observe at the microscaleBy establishing a site-specific rock physics transform (unce-mented and cemented models) between the total porositymineralogy and elastic properties we conclude the rockphysics diagnostics here In the next section we analyze poreshape influence on seismic velocities in order to generatelocally calibrated 119881P-119881S relationship for Granite Wash

53 Analysis of Pore Shapes Influencing Seismic VelocitiesTheoretically it has been reckoned that the velocities in twostates (dry or saturated) are mainly controlled by the effectivemoduli (bulk and shear) which in turn depend on poreshape geometry mineralogy properties of solids the porefluid degree of consolidation and cementation [9 26 27]Thus elastic moduli (or velocities) are basically the functionof all these factors combined Especially in low porosityand permeability rocks elastic moduli of the rock min-eral matrix with spherical pores (high aspect ratio) often

dominate those of the bulk rock [9 26 27] Therefore it istheoretically reasonable to employ the elastic moduli in rocksto estimate seismic velocities Here two effective medium(2DKT and 3DKT) models are investigated to evaluate poreshape geometry using different pore aspect ratios (valueranging from 01 to 09) to predict P- and S-wave velocitiesfrom the well data We know that the pores with high aspectratio (spherical) tend to be stiffer than those with low aspectratio (oblate or penny shape pores) That is to say P- and S-wave velocities in the reservoir with spherical pores tend tobe higher than one with oblate pores or penny shapeThe factcan be easily understood because stiffer pores contribute tothe stiffness of the whole rock while oblate pores contributeto the compliance of thewhole rockHere actual well numberA is taken as an example containing dipole sonic logs toestimate P- and S-wave velocity From Figure 10 we can seeclearly that the increase in P- and S-wave velocities is sharperwhen the pore aspect ratio increases from 01 to 06 for both2DKT and 3DKT models However the increasing trend isnominal when the pore aspect ratio exceeds beyond 06 in

10 International Journal of Geophysics

12366

12368

12370

1237247 48

Vp (kms) Vs (kms)

Dep

th (f

eet)

12366

12368

12370

12372

Dep

th (f

eet)

VpVp-2DKTVp-3DKT

26 27

VsVs-2DKTVs-3DKT

Figure 11 A comparison between the predicted and measured P- and S-wave velocities by employing 2DKT and 3DKTmodels using criticalaspect ratio (06) of well number A

both models Significantly velocities predicted by the 3DKTmodel agreewell with thosemeasured compared to the 2DKTmodel (Figure 11)

It is due to the fact that the model adjusts pore aspectratio or the proportion of various 3D pores and effectivelymodels the geometries of various pore shapes which is thedominant factor influencing the velocities of the elastic-wave propagating in the clastic reservoirs [28] Due to thesubstantial amount of clay (15) contents that may containmicrocracks the dry rock moduli estimation by the Kuster-Toksoz (2DKT) model (assuming ellipsoidal) could not beappropriate The addition of small volumes of cracks to therock matrix significantly weakens the rock The predicted P-and S-wave velocities from 3DKT are more consistent withthe measured ones although with some deviation Thus weconsider aspect ratio 06 as the critical value of the pore aspectratio which is the turning point of the increase trend ofpredicted velocities for Marmaton Wash

Thus variations in the elastic moduli are more influencedby pore shape geometry than porosity lower than 10and have a relatively small impact on 119881P119881S ratios Theintegrated application of the 2DKT and 3DKTmodels helpedus properly determine the pore shape geometry of GraniteWash reservoir to estimate site specific 119881P-119881S relationship

54 119881119875-119881119878Relationship of Granite Wash The 119881P-119881S relation

is normally used for 119881S prediction where only P-wave data isavailable and is also very important forAVOanalysis and porefluids prediction For S-wave velocity estimation regard-less of the approach taken it is important to understand

the underlying lithology The rock composition providesfirst order control on dry frame 119881P119881S ratio especially forconsolidated rocks [29] We take advantage of rock physics(effective medium) modeling and derive the 119881P-119881S relation-ship and compare it with the widely used empirical rela-tionship ldquo119881P = 116 lowast 119881S + 136rdquo presented by Castagnaet al [8] for clastic rocks We employ Berrymanrsquos [7] (3DKT)model considering rock composition porosity and poreshape After calibration of the predicted velocities with coredata the 119881P-119881S relationship is generated through regressionanalysis of least-square linear fit (1198772 = 090) as a function of119881P (Kms) (Figure 12)

The derived 119881P-119881S relation by 3DKT for Granite Washprovides closer approximation from 2DKT and global empir-ical relation by Castagna et al [8] Figure 13 shows that theS-wave velocity derived from Castagnarsquos mudrock is overpredicted with known 119881P when comparatively analyzed to2DKT and 3DKT models predictions However there areapparent deviations between the measured and predicteddata by the locally calibrated empirical relation and theeffectivemediummodelsWe actually do not advocate the useof global empirical relations for S-wave velocity predictionfor the Granite Wash but emphasize on site specific builtrelation based on the data of the corresponding study areausing effective medium models Next we employed the bestmodel (3DKT model) to discuss the fluid substitution effectson P- and S-wave velocities Marmaton Wash

55 Fluid Substitution in Granite Wash For practical fluidsubstitution results in tight sands using Gassmannrsquos [21]

International Journal of Geophysics 11

35

30

25

20

1530 35 40 45 50 55 60

0000012500250003750050000625007500087501000

R2 = 90

Vs

(km

s)

Vp (kms)

Vs = 0599 lowast Vp minus 019102

0 4 5 50 55

0000012500250037505000625075087100000

R2 = 90

s = 0599 lowast VpVV minus 019102

GR

Figure 12 Compressional and shearwave velocities ofGraniteWashformation from sonic logs data color coded with gamma ray (GR)from two wells numbers A and B and calibrated with core [11]measurements (Maroon data)

12340

12360

12380

12400

12420

12440

12460

20 22 24 26 28 30 32 34

003500

01556

02763

03969

05175

06381

07588

08794

1000

Dep

th (f

eet)

S-wave velocity (kms)

Vs log

Vs-2DKTVs-3DKT

Vs-RVs-C

Vsh

Figure 13 Comparison of S-wave velocity predictions (color codedwith 119881sh) of well number A with effective medium (2DKT and3DKT) models locally generated relation (119881

119904-119877) and Castagna

global empirical relationship (119881119904-119862)

equations necessitates an appropriate technique which real-istically stiffens low porosity rock by fluctuating the mineralpore fluid or dry rock moduli Application of Gassmannrsquosequation is dependent upon reasonable estimation of thebulk modulus of the dry rock frame (119870

119889) Dry frame rock

bulk (119870119889) modulus can be derived either from (1) laboratory

measurement on core data (2) application of empirical rela-tionships or effective medium theory [30ndash33] or (3) directcalculation from log data by rewriting Gassmannrsquos equation(4) for frame rock bulk (119870

119889) [34] which often generates

negative (a physical impossibility) or unrealistically largevalues in low porosity reservoirs [35] However of the variousalternatives we take essentially a pragmatic approach tocondition the dry frame rockmodel used in fluid substitution

For reliable fluid substitution model in clastic reservoir weconditioned the calculation of dry frame (119870

119889) rock data

implicit in Gassmannrsquos model and apply Biot [36] relationused by Wang et al [37] as follows

119870119889= 119870119898

(1 minus 120573) (7)

where 120573 is sometimes called the Biot coefficient and it is oftencalculated as a function of porosity and the form used in thepaper is derived by Krief et al [38]

(1 minus 120573) = [1 minus 120601]119898(120601)

119898 (120601) =3

(1 minus 120601)

(8)

where 120601 is the formation porosityConsequently the saturated bulk modulus of the rock

(119870sat) for the in situ conditions is first calculated using (1)Equation (7) is combined with Gassmannrsquos equations (4)to generate a pseudo dry frame rock bulk (119870

119889) modulus

suitable for fluid substitution The input parameters for fluidsubstitutions are generally roughly estimated and thus resultin substantial errorsTherefore densities bulkmoduli veloc-ities and viscosities of various pore fluids (brine and gas)are used both from core measurements and estimation usingBatzle and Wang [39] relationships The homogeneous fluid(119870fl) uniformly distributed throughout the pore space hasbeen computed by harmonic average (Reuss) using data fromthewell logs assuming twofluids (water and gas) phases in thereservoir The fluids (119870fl 120588fl) properties occupying the porespace and the bulk modulus of the mineral matrix (119870matrix)are determined by formulations given by Smith et al [35] Tocalculate the bulk modulus of the mineral matrix (119870matrix)the information on the composition (mineral abundance) ofthe rock is used from core study data [11] Mineral matrix(119870matrix) is then calculated via application of Voigt-Reuss-Hill (VRH) averaging the mineral constituents The elasticconstants and densities of pore fluid properties obtained fromcoremeasurements are given in Table 2 Figure 14 an exampleof composite log example from Granite Wash (Marmaton)section demonstrates the significant fluid substitution differ-ences between the wet and the gas effect on the seismic P-wave (119881P) velocity log for well number A The observed fluideffects reach amaximum in the cleanest sandwith the highest(13) porosity and low density response Fluid substitutioneffects correspond more closely with expectations based onraw logs increasing our confidence in the results of AVOmodeling and seismic inversion carried out in phase II

56 Cross-Plotting Attribute Analysis Cross-plotting of elas-tic rock properties from well logs enables the simultane-ous and meaningful evaluation of two attributes that helpsin making straightforward interpretation of hydrocarbonsaturated sands in a particular reservoir For comparisonwe cross-plotted both conventional (P- and S-impedances)and unconventional (lambda mu and rho) attributes in thederived elastic parameter cross-plot space to isolate potentialanomalous zones from a background relationship of well

12 International Journal of Geophysics

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Vsh

(a)

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Sw

(b)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

00 01

Porosity

(c)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

24 26

Density

(d)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

42 48

Vp

GasBrine

(e)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

025 030

PR

(f)

Figure 14 A composite log example fromGraniteWash (Marmaton) section Plots from left to right show logs of (a)119881sh (b)Water saturation(119878119908) (c) porosity (d) density (e) compressional velocity (119881P) and (f) Poisson ratio (PR) Fluid substitution is carried out for brine (blue) and

gas (red) scenarios using dry rock data derived from logs

number A (Figure 15) Figure 15(a) shows where a conven-tional impedance IP versus IS analysis has limitations inclearly discriminating between all the various lithologies andgas sand zone The P- and S-impedances log curves (whichshare both rigidity and density) tend to track each other withdepth and never crossover only S-impedance contrast in theMarmaton sand (12400

1015840

) is clearly less distinguishable thanthe comparable P-impedance contrast By contrast the 120582120588

and 120583120588 attributes have similar value ranges that do cross over

with 120582120588 lt 120583120588 showing a clear wide separation for a hydro-carbon zone while tracking together to clearly indicate thethin tight shale breaks (Figure 15(b)) In Figure 15(c) onlythe gas zone is discernable from background cut-off on the119881P119881S curve which varies between 16 and 20 Figures 15(d)and 15(e) demonstrate the significant advantage of the 120582120588120583120588 cross-plot in isolating both lithologic properties (such assand shale and clastic facies) as well as the gas zone com-pared to IP and IS cross-plot P- and S-impedance data points

International Journal of Geophysics 13

12340

12360

12380

12400

12420

12440

12460

4 6 8 10 12 14

IPIS

Dep

th (f

eet)

Impedance

(a)

12340

12360

12380

12400

12420

12440

12460

10 20 30 40 50 60 70 80

Tight shale

Gas zone

Lambda-rhoMu-rho

Shale

Dep

th (f

eet)

Lambda-rho Mu-rho

(b)

12340

12360

12380

12400

12420

12440

12460

15 16 17 18 19 20 21

Dep

th (f

eet)

VpVs

(c)

80

75

70

65

60

55

50

458 9 10 11 12 13 14

P-impedance

Gas zone

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

S-im

peda

nce

(d)

60

55

50

45

40

35

30

25

0 10 20 30 40 50 60 70

Gas zone

Lambda-rho

Mu-

rho

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

(e)

25

20

15

10

05

00minus15 minus10 minus5 0 5 10 15 20

Shale

Sst line

Wet sand

Gas zone

Lambda-Mu (difference)

Lam

bda

Mu

(rat

io)

003500

01556

02763

03969

05175

06381

07588

08794

1000

Shale

Sst line

Wet sand

Gas zone

Vsh

(f)

Figure 15 Well log curves and cross-plots from Granite Wash (Marmaton sections 12325ndash12470) of well number A (a) P- and S-impedanceslog curves (b) Lambda-rho and Mu-rho log versus depth (c) 119881

119901119881119904ratio versus depth (d) P-impedance versus S-impedance cross-plot color

coded by shale volume (119881sh) (e) Lambda-rho versus Mu-rho cross-plot color coded by shale volume (119881sh) and (f) Lambda-Mu (difference)versus Lambda-rhoMu-rho (ratio) cross-plot color coded by shale volume (119881sh)

14 International Journal of Geophysics

show a close linear relationship with shale having lowestvalues while in comparison the lowest 120582120588 (incompressibilitytimes density) value has the best gas sand values along with 120583120588

(rigidity times density) values higher than shalesThe other morecompetent pure lithologies (silts cemented sands) could beobserved in the opposite upper right quadrant relative to theshales

Following the cross-plots analysis for the same log dataset Figure 15(f) shows an interesting interpretation templatefor the 120582120583 ratio versus the 120582 minus 120583 (difference) cross-plotThe anomalous (gas) zone is marked in a lower left red boxbeyond the (06 minus5) point separating it from the wet sandswhich tend to align in the upper right zone The variationof the slope beyond the origin (1 0) with increasing gassaturation (blue to green dotted lines) reveals an unexpectedeffect of the gas on the rigidity of the sand This is due tothe effective porosity and lower density gas effect on rigidityleading to a slight steeping of the ldquosandstone linerdquo slopeThus in this case unconventional attributes plots provide aclear separation of the anomalous hydrocarbon zones and thelithologic properties of the sand and shale facies which havebeen color coded with shale volume

6 Conclusion

With the above discussion and limitations the followingconclusions can be derived Rock physics diagnostic and thinsections analysis indicate the variations in elastic propertiesof Granite Wash which can be explained by diagenetic phaseof quartz (or silica) cementation CementedWashes have rel-atively high elastic moduli and can be modeled primarily bycontact-cement model of scheme 2 Velocity prediction fromeffective medium models shows that pore shape geometrysignificantly influences the P- and S-wave velocities in ourreservoirs as observed in the velocity data evaluated Core andpetrographic observations also confirm cementations withcomplex pore geometry containing variable concentrations ofhigh and low pores aspect ratio Hence Berrymanrsquos (3DKT)model is proven to be the best one to discuss the influence ofpore shape on seismic velocities and used for the generationof locally calibrated empirical (119881P-119881S) relationship that bestdescribes our data set Finally we applied fluid substitution onlog data for calculating reservoir elastic properties with dif-ferent pore fluid (brine and gas) for an improved petrophys-ical discrimination of rock properties using unconventionalLame parameters 120582 (pure incompressibility) and 120583 (rigidity)attributes over conventional 119881P 119881S attributes analysis Theresearch outcomes are intended to be rigorously used as afoundation for ongoing reservoir characterization study inphase II

Disclosure

This paper is developed from an oral and poster presentationat American Association of Petroleum Geologists (AAPG)Annual Convention amp Exhibition David L Lawrence Con-vention Center Pittsburgh PA USA May 19ndash22 2013

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank The University of Tulsa and NewfieldExploration Company for support of this research The firstauthor would like to thank Jeff Kane (Corporate Petro-physicist) for his scholastic discussions They also thankthe anonymous reviewers for the constructive comments onimproving our paper

References

[1] S Ingram I Paterniti B Rothkopf and C Stevenson ldquoGranitewash field study-buffalo wallow field Texas panhandlerdquo inProceedings of the SPE Eastern Regional Meeting SPE104546Canton Ohio USA October 2006

[2] M I Odigi ldquoDiagenesis and reservoir quality of Cretaceoussandstone of Nkporo Shale (Campanian) Southeastern Benuetrough Nigeriardquo Journal of Geology and Mining Research vol 3pp 265ndash280 2011

[3] F Ruiz and A Cheng ldquoA rock physics model for tight gas sandrdquoThe Leading Edge vol 29 no 12 pp 1484ndash1489 2010

[4] RDMindlin ldquoCompliance of elastic bodies in contactrdquo Journalof Applied Mechanics vol 16 pp 259ndash268 1949

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[6] J Dvorkin and A Nur ldquoElasticity of high-porosity sandstonestheory for twoNorth Sea data setsrdquoGeophysics vol 61 no 5 pp1363ndash1370 1996

[7] J G Berryman ldquoLong-wavelength propagation in compositeelastic media II Ellipsoidal Inclusionsrdquo The Journal of theAcoustical Society of America vol 68 no 6 pp 1820ndash1831 1980

[8] J P Castagna M L Batzle and R L Eastwood ldquoRelationshipsbetween compressional-wave and shear-wave velocities in clas-tic silicate rocksrdquo Geophysics vol 50 no 4 pp 571ndash581 1985

[9] G T Kuster and M N Toksoz ldquoVelocity and attenuation ofseismic waves in two-phase mediardquo Geophysics vol 39 no 5pp 587ndash618 1974

[10] G E Moore ldquoPennsylvanian paleogeography of the south-ern Mid-Continentrdquo in Pennsylvanian Sandstones of the Mid-Continent N J Hyne Ed pp 2ndash12 Tulsa Geological Society(TGS) Tulsa Okla USA 1979

[11] Core Lab ldquoGraniteWash geological Petrophysical and comple-tions study for improved formation evaluation and optimal wellcompletionrdquo Proprietary industry study 2005

[12] K S Johnson and B J Cardott ldquoGeologic framework andhydrocarbon source rocks of Oklahomardquo in Proceedings of theSource Rocks in the Southern Midcontinent K S JohnsonB andJ Cardott Eds pp 21ndash37 OGS Circular 1992

[13] N R Gelphman ldquoWest sentinel oil field Washita CountyOklahoma sedimentology of the ldquoGraniteWashrdquo and structuralgeologyrdquoOklahoma City Geological Society Shale Shaker vol 10pp 234ndash247 1960

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mining

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Journal of

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International Journal of

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OceanographyInternational Journal of

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GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

6 International Journal of Geophysics

Table 1 Coefficients of 119875 and 119876 for some specific shapes The subscripts ldquo119898rdquo and ldquo119894rdquo refer to the background and inclusion materials [7]

Inclusion shape 119875119898119894

119876119898119894

Spheres119870119898

+ (43)120583119898

119870119894+ (43)120583

119898

120583119898

+ 120577119898

120583119894+ 120577119898

Needles119870119898

+ 120583119898

+ (13)120583119894

119870119894+ 120583119898

+ (13)120583119894

1

5(

4120583119898

120583119898

+ 120583119894

+ 2120583119898

+ 120574119898

120583119894+ 120574119898

+119870119894+ (43)120583

119898

119870119894+ 120583119898

+ (13)120583119894

)

Disks119870119898

+ (43)120583119894

119870119894+ (43)120583

119894

120583119898

+ 120577119894

120583119894+ 120577119894

Penny cracks119870119898

+ (43)120583119894

119870119894+ (43)120583

119894+ 120587120572120573

119898

1

5(1 +

8120583119898

4120583119894+ 120587120572(120583

119898+ 2120573119898)+ 2

119870119894+ (23)(120583

119894+ 120583119898)

119870119894+ (43)120583

119894+ 120587120572120573

119898

)

Table 2 The core values of moduli and densities of minerals andfluids used in this study [11]

Minerals and fluids Density(gcc)

Bulk modulus(GPa)

Shear modulus(Gpa)

Quartz 265 5580 38Clays 258 209 685Dolomite 287 8010 4500calcite 271 7550 3400Oil 07 162 mdashGas 012 0032 mdash

shape and (c) randomly packed From the core study thepercentage volume of the rock matrix is dominantly clay(15) and quartz (40) The basic parameters used for fluidsproperties in velocity predictionwere taken from core studies(Table 2) Analysis of log and core data from well A suggeststhat these assumptions are met for our reservoir

5 Data Analysis and Results

51 Rock Physics Diagnostics Application to the Granite WashA common practice in rock physics analysis is to generatecross-plots of elastic properties to study diagenetic trends andproperty dependencies of sands and shales The cross-plotsexhibit interdependence among the various rock propertiesWe particularly focused on a thick Marmaton Wash section(12325ndash12435 ft) of well number A containing various logcurves (Figure 6) Marmaton section is marked by extremelylow and fairly constant (about 375) gamma-ray readings andhigh velocity (about 46 kms) surrounded by shale packages

In spite of the congruency between different log curvesin Figure 6 various cross-plots of Marmaton sands do notproduce unique trends (Figure 7) In Figure 7(a) low andhigh ldquoGRrdquo sediments do not show huge variations in bulkdensity implying that both sand and shale arewell compactedand cemented Figure 7(b) shows the same as the case inGR-porosity plot and Figure 7(c) shows that presence ofclay with significantly lower impedance than the quartz itcements Figures 7(d) 7(e) and 7(f) modulusvelocity versusporosity cross-plots show 119881 shape overturned paths Thesuperimposed119881-shaped paths actually represent clay contentfrom clean sands to pure shale The sands overlapped with

the clean shales probably indicate the sand cluster includingboth clean and shaly sands [22] From moduli-velocity-porosity cross-plots we can assess the progress of diagenesisand compaction trends of sediments For soft or uncementedand stiff sediments the paths are fairly gentle with notmuch of velocity contrast between sands and shales Forcemented sediments (case under examination) increasingdepth quartz cementation provides a large contrast in velocityfrom shaly sands to the pure shales This drop might beassociated with the effect of proceeding from grain supportedto clay-supported sediments under relatively high effectivepressure [23 24] Thus a large part of the scatter observedin modulivelocity versus porosity plots is due to presence ofclay in the rocks

We construct the Hashin and Shtrikman [5] bounds(upper and lower) to conceptually explain and model thedata scatter of the Granite Wash from four different wells(Figure 8) The bounds provide an idea of the narrowestpossible range of elasticmoduli without a detailed descriptionof the microstructure of Granite Wash rocks For boundsonly volume fractions of the constituents quartz is 40 byvolume of solid fraction calcite is 7 by volume of solidfraction and their elastic moduli obtained from core dataare used without specifying the geometric details of theirarrangements The upper and lower bounds are determinedby exchanging the materials one that fills the pores andanother constructing the shell The upper bound is realizedwhen stiffer material forms the shell the lower bound isrealized when the stiffer material is physically present in thecore of the shell [18] The separation between upper andlower bounds shows how elastically different the constituentsare The elastic contrast between pure sand and the criticalconcentration mixture is relatively strong because of thevariable elastic contrast between quartz and clay grains Thisimplies that the effect of quartz and clay minerals may becritical for seismic interpretation of Granite Wash

Moduli-porosity data point superimposed with core datafall between the upper and lower bounds which indicates thatMarmaton sands are consolidated sandstone with cemen-tations Superimposed core data from Marmaton sectionconfirms the presence of quartz and calcite cement

52 Confirming the Diagnostic From Figure 4 we observethatmicrocrystalline quartz is found in close contact betweengrains so this quartz cement has a stiffening effect on

International Journal of Geophysics 7

Bulk

den

sity

27

26

25

24

2350 100 150 200

Gamma ray

(a)

Poro

sity

015

010

005

00050 100 150 200

Density

Gamma ray

(b)14

12

10

50 100 150 200

Density

Gamma ray

IP

(c)

Vp

(km

s)

50

45

40

35000 005 010 015 020

Porosity

Density

(d)

70

60

50

40

000 005 010 015 020

Porosity

K(G

Pa)

Vsh

(e)

70

60

50

40

000 005 010 015 020

Sw

Porosity

K(G

Pa)

(f)

Figure 7 Cross-plots of Well log data Marmaton section from Figure 6 (a) bulk density versus GR (b) total porosity versus GR color codedby density (c) P-wave impedance versus GR color coded by density (d) P-wave versus total porosity color coded by density (e) bulkmodulusversus total porosity color coded by shale volume (119881sh) and (f) bulk modulus versus total porosity color coded by water saturation (119878

119908)

the elastic properties of the Granite Wash Studies havesupported the fact that dissolution and local precipitationof quartz accompanied by compaction cause long termdeformation of the aggregate resulting in permeability andporosity reduction [25] Hence quartz precipitation betweenthe grains causes major porosity reduction

For diagnostic purposes we cross-plotted the elasticproperties from well number A (Marmaton section) invelocity-porosity and moduli-porosity planes superimpos-ing theoretical rock physics models (Figure 9) The cross-plots show that gradually increasing clay content with depthcorresponds with decreasing elastic value Based on labora-tory data log data thin section analysis and rock-physicsmodels the data is sorted into Dvorkin and Nurrsquos [6] follow-ing three schemes uncemented (or soft)WashesWasheswithsmall amounts of quartz cement at grain contacts (scheme 1)and Washes from the same formation but with clay cement

(scheme 2) The upper two theoretical (schemes 1 and 2)curves correspond to the pure quartz cases while lower curveis for pure clay or uncemented rock Scheme 1 represents allcement deposited precisely at grain contacts and the slopesteep Scheme 2 curve corresponds to the case where thecontact cement region is formed at the intersection of cementshells that are uniformly deposited on grain surfaces cementfills the space of the large pores and results in a flat slopeIt is quite possible that the variation in microcracks maybe the major reason why the slope of velocity-porosity andmodulus-porosity planes differs in different schemes

We noticed that the maximum data points fall on therock physics scheme 2 line representing uniformly depositedcement on grain surfaceThe thin sections samples from wellnumber A confirmed the rock diagnostic that with increasingdepth and pressure the decreasing porosity is a result ofcontact cement in the rock-physics cross-plots It is quite

8 International Journal of Geophysics

125

100

75

50

25

0000 025 050 075 100

Bulk

mod

ulus

(GPa

)

Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

50

40

30

20

10

0000 025 050 075 100

Shea

r mod

ulus

(GPa

)Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

Figure 8 Moduli versus porosity for different wells and core [11] data with superposition of Hashin and Shtrikman [5] bounds

6

5

4

3

2

00 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Vp

(km

s)

Porosity

Vp versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(a)

80

60

40

20

000 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Bulk

mod

ulus

(GPa

)

Porosity

K versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(b)

Figure 9 (a) P-wave velocity versus porosity and (b) modulus versus porosity (core [11] data included) color coded with shale volume (119881sh)

of the Marmaton section in well number A with rock physics models curves superimposed

International Journal of Geophysics 9

12368

12370

12372

465 470 475

P-wave velocity

465 470 475

P-wave velocity

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

260 265 270 275

S-wave velocity

260 265 270 275

S-wave velocity

Vp2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-09

2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-082DKT-09

Vs

3DKT-013DKT-023DKT-033DKT-043DKT-053DKT-063DKT-073DKT-08

Vp

3DKT-09

3DKT-04

3DKT-013DKT-023DKT-03

3DKT-053DKT-063DKT-073DKT-083DKT-09

Vs

Figure 10 P- and S-wave velocities prediction by employing 2DKT and 3DKT models using different aspect ratios (01ndash09) of well numberA

interesting that the well-log data with resolution of tens-of-centimeter reflects what we observe at the microscaleBy establishing a site-specific rock physics transform (unce-mented and cemented models) between the total porositymineralogy and elastic properties we conclude the rockphysics diagnostics here In the next section we analyze poreshape influence on seismic velocities in order to generatelocally calibrated 119881P-119881S relationship for Granite Wash

53 Analysis of Pore Shapes Influencing Seismic VelocitiesTheoretically it has been reckoned that the velocities in twostates (dry or saturated) are mainly controlled by the effectivemoduli (bulk and shear) which in turn depend on poreshape geometry mineralogy properties of solids the porefluid degree of consolidation and cementation [9 26 27]Thus elastic moduli (or velocities) are basically the functionof all these factors combined Especially in low porosityand permeability rocks elastic moduli of the rock min-eral matrix with spherical pores (high aspect ratio) often

dominate those of the bulk rock [9 26 27] Therefore it istheoretically reasonable to employ the elastic moduli in rocksto estimate seismic velocities Here two effective medium(2DKT and 3DKT) models are investigated to evaluate poreshape geometry using different pore aspect ratios (valueranging from 01 to 09) to predict P- and S-wave velocitiesfrom the well data We know that the pores with high aspectratio (spherical) tend to be stiffer than those with low aspectratio (oblate or penny shape pores) That is to say P- and S-wave velocities in the reservoir with spherical pores tend tobe higher than one with oblate pores or penny shapeThe factcan be easily understood because stiffer pores contribute tothe stiffness of the whole rock while oblate pores contributeto the compliance of thewhole rockHere actual well numberA is taken as an example containing dipole sonic logs toestimate P- and S-wave velocity From Figure 10 we can seeclearly that the increase in P- and S-wave velocities is sharperwhen the pore aspect ratio increases from 01 to 06 for both2DKT and 3DKT models However the increasing trend isnominal when the pore aspect ratio exceeds beyond 06 in

10 International Journal of Geophysics

12366

12368

12370

1237247 48

Vp (kms) Vs (kms)

Dep

th (f

eet)

12366

12368

12370

12372

Dep

th (f

eet)

VpVp-2DKTVp-3DKT

26 27

VsVs-2DKTVs-3DKT

Figure 11 A comparison between the predicted and measured P- and S-wave velocities by employing 2DKT and 3DKTmodels using criticalaspect ratio (06) of well number A

both models Significantly velocities predicted by the 3DKTmodel agreewell with thosemeasured compared to the 2DKTmodel (Figure 11)

It is due to the fact that the model adjusts pore aspectratio or the proportion of various 3D pores and effectivelymodels the geometries of various pore shapes which is thedominant factor influencing the velocities of the elastic-wave propagating in the clastic reservoirs [28] Due to thesubstantial amount of clay (15) contents that may containmicrocracks the dry rock moduli estimation by the Kuster-Toksoz (2DKT) model (assuming ellipsoidal) could not beappropriate The addition of small volumes of cracks to therock matrix significantly weakens the rock The predicted P-and S-wave velocities from 3DKT are more consistent withthe measured ones although with some deviation Thus weconsider aspect ratio 06 as the critical value of the pore aspectratio which is the turning point of the increase trend ofpredicted velocities for Marmaton Wash

Thus variations in the elastic moduli are more influencedby pore shape geometry than porosity lower than 10and have a relatively small impact on 119881P119881S ratios Theintegrated application of the 2DKT and 3DKTmodels helpedus properly determine the pore shape geometry of GraniteWash reservoir to estimate site specific 119881P-119881S relationship

54 119881119875-119881119878Relationship of Granite Wash The 119881P-119881S relation

is normally used for 119881S prediction where only P-wave data isavailable and is also very important forAVOanalysis and porefluids prediction For S-wave velocity estimation regard-less of the approach taken it is important to understand

the underlying lithology The rock composition providesfirst order control on dry frame 119881P119881S ratio especially forconsolidated rocks [29] We take advantage of rock physics(effective medium) modeling and derive the 119881P-119881S relation-ship and compare it with the widely used empirical rela-tionship ldquo119881P = 116 lowast 119881S + 136rdquo presented by Castagnaet al [8] for clastic rocks We employ Berrymanrsquos [7] (3DKT)model considering rock composition porosity and poreshape After calibration of the predicted velocities with coredata the 119881P-119881S relationship is generated through regressionanalysis of least-square linear fit (1198772 = 090) as a function of119881P (Kms) (Figure 12)

The derived 119881P-119881S relation by 3DKT for Granite Washprovides closer approximation from 2DKT and global empir-ical relation by Castagna et al [8] Figure 13 shows that theS-wave velocity derived from Castagnarsquos mudrock is overpredicted with known 119881P when comparatively analyzed to2DKT and 3DKT models predictions However there areapparent deviations between the measured and predicteddata by the locally calibrated empirical relation and theeffectivemediummodelsWe actually do not advocate the useof global empirical relations for S-wave velocity predictionfor the Granite Wash but emphasize on site specific builtrelation based on the data of the corresponding study areausing effective medium models Next we employed the bestmodel (3DKT model) to discuss the fluid substitution effectson P- and S-wave velocities Marmaton Wash

55 Fluid Substitution in Granite Wash For practical fluidsubstitution results in tight sands using Gassmannrsquos [21]

International Journal of Geophysics 11

35

30

25

20

1530 35 40 45 50 55 60

0000012500250003750050000625007500087501000

R2 = 90

Vs

(km

s)

Vp (kms)

Vs = 0599 lowast Vp minus 019102

0 4 5 50 55

0000012500250037505000625075087100000

R2 = 90

s = 0599 lowast VpVV minus 019102

GR

Figure 12 Compressional and shearwave velocities ofGraniteWashformation from sonic logs data color coded with gamma ray (GR)from two wells numbers A and B and calibrated with core [11]measurements (Maroon data)

12340

12360

12380

12400

12420

12440

12460

20 22 24 26 28 30 32 34

003500

01556

02763

03969

05175

06381

07588

08794

1000

Dep

th (f

eet)

S-wave velocity (kms)

Vs log

Vs-2DKTVs-3DKT

Vs-RVs-C

Vsh

Figure 13 Comparison of S-wave velocity predictions (color codedwith 119881sh) of well number A with effective medium (2DKT and3DKT) models locally generated relation (119881

119904-119877) and Castagna

global empirical relationship (119881119904-119862)

equations necessitates an appropriate technique which real-istically stiffens low porosity rock by fluctuating the mineralpore fluid or dry rock moduli Application of Gassmannrsquosequation is dependent upon reasonable estimation of thebulk modulus of the dry rock frame (119870

119889) Dry frame rock

bulk (119870119889) modulus can be derived either from (1) laboratory

measurement on core data (2) application of empirical rela-tionships or effective medium theory [30ndash33] or (3) directcalculation from log data by rewriting Gassmannrsquos equation(4) for frame rock bulk (119870

119889) [34] which often generates

negative (a physical impossibility) or unrealistically largevalues in low porosity reservoirs [35] However of the variousalternatives we take essentially a pragmatic approach tocondition the dry frame rockmodel used in fluid substitution

For reliable fluid substitution model in clastic reservoir weconditioned the calculation of dry frame (119870

119889) rock data

implicit in Gassmannrsquos model and apply Biot [36] relationused by Wang et al [37] as follows

119870119889= 119870119898

(1 minus 120573) (7)

where 120573 is sometimes called the Biot coefficient and it is oftencalculated as a function of porosity and the form used in thepaper is derived by Krief et al [38]

(1 minus 120573) = [1 minus 120601]119898(120601)

119898 (120601) =3

(1 minus 120601)

(8)

where 120601 is the formation porosityConsequently the saturated bulk modulus of the rock

(119870sat) for the in situ conditions is first calculated using (1)Equation (7) is combined with Gassmannrsquos equations (4)to generate a pseudo dry frame rock bulk (119870

119889) modulus

suitable for fluid substitution The input parameters for fluidsubstitutions are generally roughly estimated and thus resultin substantial errorsTherefore densities bulkmoduli veloc-ities and viscosities of various pore fluids (brine and gas)are used both from core measurements and estimation usingBatzle and Wang [39] relationships The homogeneous fluid(119870fl) uniformly distributed throughout the pore space hasbeen computed by harmonic average (Reuss) using data fromthewell logs assuming twofluids (water and gas) phases in thereservoir The fluids (119870fl 120588fl) properties occupying the porespace and the bulk modulus of the mineral matrix (119870matrix)are determined by formulations given by Smith et al [35] Tocalculate the bulk modulus of the mineral matrix (119870matrix)the information on the composition (mineral abundance) ofthe rock is used from core study data [11] Mineral matrix(119870matrix) is then calculated via application of Voigt-Reuss-Hill (VRH) averaging the mineral constituents The elasticconstants and densities of pore fluid properties obtained fromcoremeasurements are given in Table 2 Figure 14 an exampleof composite log example from Granite Wash (Marmaton)section demonstrates the significant fluid substitution differ-ences between the wet and the gas effect on the seismic P-wave (119881P) velocity log for well number A The observed fluideffects reach amaximum in the cleanest sandwith the highest(13) porosity and low density response Fluid substitutioneffects correspond more closely with expectations based onraw logs increasing our confidence in the results of AVOmodeling and seismic inversion carried out in phase II

56 Cross-Plotting Attribute Analysis Cross-plotting of elas-tic rock properties from well logs enables the simultane-ous and meaningful evaluation of two attributes that helpsin making straightforward interpretation of hydrocarbonsaturated sands in a particular reservoir For comparisonwe cross-plotted both conventional (P- and S-impedances)and unconventional (lambda mu and rho) attributes in thederived elastic parameter cross-plot space to isolate potentialanomalous zones from a background relationship of well

12 International Journal of Geophysics

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Vsh

(a)

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Sw

(b)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

00 01

Porosity

(c)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

24 26

Density

(d)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

42 48

Vp

GasBrine

(e)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

025 030

PR

(f)

Figure 14 A composite log example fromGraniteWash (Marmaton) section Plots from left to right show logs of (a)119881sh (b)Water saturation(119878119908) (c) porosity (d) density (e) compressional velocity (119881P) and (f) Poisson ratio (PR) Fluid substitution is carried out for brine (blue) and

gas (red) scenarios using dry rock data derived from logs

number A (Figure 15) Figure 15(a) shows where a conven-tional impedance IP versus IS analysis has limitations inclearly discriminating between all the various lithologies andgas sand zone The P- and S-impedances log curves (whichshare both rigidity and density) tend to track each other withdepth and never crossover only S-impedance contrast in theMarmaton sand (12400

1015840

) is clearly less distinguishable thanthe comparable P-impedance contrast By contrast the 120582120588

and 120583120588 attributes have similar value ranges that do cross over

with 120582120588 lt 120583120588 showing a clear wide separation for a hydro-carbon zone while tracking together to clearly indicate thethin tight shale breaks (Figure 15(b)) In Figure 15(c) onlythe gas zone is discernable from background cut-off on the119881P119881S curve which varies between 16 and 20 Figures 15(d)and 15(e) demonstrate the significant advantage of the 120582120588120583120588 cross-plot in isolating both lithologic properties (such assand shale and clastic facies) as well as the gas zone com-pared to IP and IS cross-plot P- and S-impedance data points

International Journal of Geophysics 13

12340

12360

12380

12400

12420

12440

12460

4 6 8 10 12 14

IPIS

Dep

th (f

eet)

Impedance

(a)

12340

12360

12380

12400

12420

12440

12460

10 20 30 40 50 60 70 80

Tight shale

Gas zone

Lambda-rhoMu-rho

Shale

Dep

th (f

eet)

Lambda-rho Mu-rho

(b)

12340

12360

12380

12400

12420

12440

12460

15 16 17 18 19 20 21

Dep

th (f

eet)

VpVs

(c)

80

75

70

65

60

55

50

458 9 10 11 12 13 14

P-impedance

Gas zone

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

S-im

peda

nce

(d)

60

55

50

45

40

35

30

25

0 10 20 30 40 50 60 70

Gas zone

Lambda-rho

Mu-

rho

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

(e)

25

20

15

10

05

00minus15 minus10 minus5 0 5 10 15 20

Shale

Sst line

Wet sand

Gas zone

Lambda-Mu (difference)

Lam

bda

Mu

(rat

io)

003500

01556

02763

03969

05175

06381

07588

08794

1000

Shale

Sst line

Wet sand

Gas zone

Vsh

(f)

Figure 15 Well log curves and cross-plots from Granite Wash (Marmaton sections 12325ndash12470) of well number A (a) P- and S-impedanceslog curves (b) Lambda-rho and Mu-rho log versus depth (c) 119881

119901119881119904ratio versus depth (d) P-impedance versus S-impedance cross-plot color

coded by shale volume (119881sh) (e) Lambda-rho versus Mu-rho cross-plot color coded by shale volume (119881sh) and (f) Lambda-Mu (difference)versus Lambda-rhoMu-rho (ratio) cross-plot color coded by shale volume (119881sh)

14 International Journal of Geophysics

show a close linear relationship with shale having lowestvalues while in comparison the lowest 120582120588 (incompressibilitytimes density) value has the best gas sand values along with 120583120588

(rigidity times density) values higher than shalesThe other morecompetent pure lithologies (silts cemented sands) could beobserved in the opposite upper right quadrant relative to theshales

Following the cross-plots analysis for the same log dataset Figure 15(f) shows an interesting interpretation templatefor the 120582120583 ratio versus the 120582 minus 120583 (difference) cross-plotThe anomalous (gas) zone is marked in a lower left red boxbeyond the (06 minus5) point separating it from the wet sandswhich tend to align in the upper right zone The variationof the slope beyond the origin (1 0) with increasing gassaturation (blue to green dotted lines) reveals an unexpectedeffect of the gas on the rigidity of the sand This is due tothe effective porosity and lower density gas effect on rigidityleading to a slight steeping of the ldquosandstone linerdquo slopeThus in this case unconventional attributes plots provide aclear separation of the anomalous hydrocarbon zones and thelithologic properties of the sand and shale facies which havebeen color coded with shale volume

6 Conclusion

With the above discussion and limitations the followingconclusions can be derived Rock physics diagnostic and thinsections analysis indicate the variations in elastic propertiesof Granite Wash which can be explained by diagenetic phaseof quartz (or silica) cementation CementedWashes have rel-atively high elastic moduli and can be modeled primarily bycontact-cement model of scheme 2 Velocity prediction fromeffective medium models shows that pore shape geometrysignificantly influences the P- and S-wave velocities in ourreservoirs as observed in the velocity data evaluated Core andpetrographic observations also confirm cementations withcomplex pore geometry containing variable concentrations ofhigh and low pores aspect ratio Hence Berrymanrsquos (3DKT)model is proven to be the best one to discuss the influence ofpore shape on seismic velocities and used for the generationof locally calibrated empirical (119881P-119881S) relationship that bestdescribes our data set Finally we applied fluid substitution onlog data for calculating reservoir elastic properties with dif-ferent pore fluid (brine and gas) for an improved petrophys-ical discrimination of rock properties using unconventionalLame parameters 120582 (pure incompressibility) and 120583 (rigidity)attributes over conventional 119881P 119881S attributes analysis Theresearch outcomes are intended to be rigorously used as afoundation for ongoing reservoir characterization study inphase II

Disclosure

This paper is developed from an oral and poster presentationat American Association of Petroleum Geologists (AAPG)Annual Convention amp Exhibition David L Lawrence Con-vention Center Pittsburgh PA USA May 19ndash22 2013

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank The University of Tulsa and NewfieldExploration Company for support of this research The firstauthor would like to thank Jeff Kane (Corporate Petro-physicist) for his scholastic discussions They also thankthe anonymous reviewers for the constructive comments onimproving our paper

References

[1] S Ingram I Paterniti B Rothkopf and C Stevenson ldquoGranitewash field study-buffalo wallow field Texas panhandlerdquo inProceedings of the SPE Eastern Regional Meeting SPE104546Canton Ohio USA October 2006

[2] M I Odigi ldquoDiagenesis and reservoir quality of Cretaceoussandstone of Nkporo Shale (Campanian) Southeastern Benuetrough Nigeriardquo Journal of Geology and Mining Research vol 3pp 265ndash280 2011

[3] F Ruiz and A Cheng ldquoA rock physics model for tight gas sandrdquoThe Leading Edge vol 29 no 12 pp 1484ndash1489 2010

[4] RDMindlin ldquoCompliance of elastic bodies in contactrdquo Journalof Applied Mechanics vol 16 pp 259ndash268 1949

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[6] J Dvorkin and A Nur ldquoElasticity of high-porosity sandstonestheory for twoNorth Sea data setsrdquoGeophysics vol 61 no 5 pp1363ndash1370 1996

[7] J G Berryman ldquoLong-wavelength propagation in compositeelastic media II Ellipsoidal Inclusionsrdquo The Journal of theAcoustical Society of America vol 68 no 6 pp 1820ndash1831 1980

[8] J P Castagna M L Batzle and R L Eastwood ldquoRelationshipsbetween compressional-wave and shear-wave velocities in clas-tic silicate rocksrdquo Geophysics vol 50 no 4 pp 571ndash581 1985

[9] G T Kuster and M N Toksoz ldquoVelocity and attenuation ofseismic waves in two-phase mediardquo Geophysics vol 39 no 5pp 587ndash618 1974

[10] G E Moore ldquoPennsylvanian paleogeography of the south-ern Mid-Continentrdquo in Pennsylvanian Sandstones of the Mid-Continent N J Hyne Ed pp 2ndash12 Tulsa Geological Society(TGS) Tulsa Okla USA 1979

[11] Core Lab ldquoGraniteWash geological Petrophysical and comple-tions study for improved formation evaluation and optimal wellcompletionrdquo Proprietary industry study 2005

[12] K S Johnson and B J Cardott ldquoGeologic framework andhydrocarbon source rocks of Oklahomardquo in Proceedings of theSource Rocks in the Southern Midcontinent K S JohnsonB andJ Cardott Eds pp 21ndash37 OGS Circular 1992

[13] N R Gelphman ldquoWest sentinel oil field Washita CountyOklahoma sedimentology of the ldquoGraniteWashrdquo and structuralgeologyrdquoOklahoma City Geological Society Shale Shaker vol 10pp 234ndash247 1960

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

International Journal of Geophysics 7

Bulk

den

sity

27

26

25

24

2350 100 150 200

Gamma ray

(a)

Poro

sity

015

010

005

00050 100 150 200

Density

Gamma ray

(b)14

12

10

50 100 150 200

Density

Gamma ray

IP

(c)

Vp

(km

s)

50

45

40

35000 005 010 015 020

Porosity

Density

(d)

70

60

50

40

000 005 010 015 020

Porosity

K(G

Pa)

Vsh

(e)

70

60

50

40

000 005 010 015 020

Sw

Porosity

K(G

Pa)

(f)

Figure 7 Cross-plots of Well log data Marmaton section from Figure 6 (a) bulk density versus GR (b) total porosity versus GR color codedby density (c) P-wave impedance versus GR color coded by density (d) P-wave versus total porosity color coded by density (e) bulkmodulusversus total porosity color coded by shale volume (119881sh) and (f) bulk modulus versus total porosity color coded by water saturation (119878

119908)

the elastic properties of the Granite Wash Studies havesupported the fact that dissolution and local precipitationof quartz accompanied by compaction cause long termdeformation of the aggregate resulting in permeability andporosity reduction [25] Hence quartz precipitation betweenthe grains causes major porosity reduction

For diagnostic purposes we cross-plotted the elasticproperties from well number A (Marmaton section) invelocity-porosity and moduli-porosity planes superimpos-ing theoretical rock physics models (Figure 9) The cross-plots show that gradually increasing clay content with depthcorresponds with decreasing elastic value Based on labora-tory data log data thin section analysis and rock-physicsmodels the data is sorted into Dvorkin and Nurrsquos [6] follow-ing three schemes uncemented (or soft)WashesWasheswithsmall amounts of quartz cement at grain contacts (scheme 1)and Washes from the same formation but with clay cement

(scheme 2) The upper two theoretical (schemes 1 and 2)curves correspond to the pure quartz cases while lower curveis for pure clay or uncemented rock Scheme 1 represents allcement deposited precisely at grain contacts and the slopesteep Scheme 2 curve corresponds to the case where thecontact cement region is formed at the intersection of cementshells that are uniformly deposited on grain surfaces cementfills the space of the large pores and results in a flat slopeIt is quite possible that the variation in microcracks maybe the major reason why the slope of velocity-porosity andmodulus-porosity planes differs in different schemes

We noticed that the maximum data points fall on therock physics scheme 2 line representing uniformly depositedcement on grain surfaceThe thin sections samples from wellnumber A confirmed the rock diagnostic that with increasingdepth and pressure the decreasing porosity is a result ofcontact cement in the rock-physics cross-plots It is quite

8 International Journal of Geophysics

125

100

75

50

25

0000 025 050 075 100

Bulk

mod

ulus

(GPa

)

Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

50

40

30

20

10

0000 025 050 075 100

Shea

r mod

ulus

(GPa

)Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

Figure 8 Moduli versus porosity for different wells and core [11] data with superposition of Hashin and Shtrikman [5] bounds

6

5

4

3

2

00 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Vp

(km

s)

Porosity

Vp versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(a)

80

60

40

20

000 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Bulk

mod

ulus

(GPa

)

Porosity

K versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(b)

Figure 9 (a) P-wave velocity versus porosity and (b) modulus versus porosity (core [11] data included) color coded with shale volume (119881sh)

of the Marmaton section in well number A with rock physics models curves superimposed

International Journal of Geophysics 9

12368

12370

12372

465 470 475

P-wave velocity

465 470 475

P-wave velocity

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

260 265 270 275

S-wave velocity

260 265 270 275

S-wave velocity

Vp2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-09

2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-082DKT-09

Vs

3DKT-013DKT-023DKT-033DKT-043DKT-053DKT-063DKT-073DKT-08

Vp

3DKT-09

3DKT-04

3DKT-013DKT-023DKT-03

3DKT-053DKT-063DKT-073DKT-083DKT-09

Vs

Figure 10 P- and S-wave velocities prediction by employing 2DKT and 3DKT models using different aspect ratios (01ndash09) of well numberA

interesting that the well-log data with resolution of tens-of-centimeter reflects what we observe at the microscaleBy establishing a site-specific rock physics transform (unce-mented and cemented models) between the total porositymineralogy and elastic properties we conclude the rockphysics diagnostics here In the next section we analyze poreshape influence on seismic velocities in order to generatelocally calibrated 119881P-119881S relationship for Granite Wash

53 Analysis of Pore Shapes Influencing Seismic VelocitiesTheoretically it has been reckoned that the velocities in twostates (dry or saturated) are mainly controlled by the effectivemoduli (bulk and shear) which in turn depend on poreshape geometry mineralogy properties of solids the porefluid degree of consolidation and cementation [9 26 27]Thus elastic moduli (or velocities) are basically the functionof all these factors combined Especially in low porosityand permeability rocks elastic moduli of the rock min-eral matrix with spherical pores (high aspect ratio) often

dominate those of the bulk rock [9 26 27] Therefore it istheoretically reasonable to employ the elastic moduli in rocksto estimate seismic velocities Here two effective medium(2DKT and 3DKT) models are investigated to evaluate poreshape geometry using different pore aspect ratios (valueranging from 01 to 09) to predict P- and S-wave velocitiesfrom the well data We know that the pores with high aspectratio (spherical) tend to be stiffer than those with low aspectratio (oblate or penny shape pores) That is to say P- and S-wave velocities in the reservoir with spherical pores tend tobe higher than one with oblate pores or penny shapeThe factcan be easily understood because stiffer pores contribute tothe stiffness of the whole rock while oblate pores contributeto the compliance of thewhole rockHere actual well numberA is taken as an example containing dipole sonic logs toestimate P- and S-wave velocity From Figure 10 we can seeclearly that the increase in P- and S-wave velocities is sharperwhen the pore aspect ratio increases from 01 to 06 for both2DKT and 3DKT models However the increasing trend isnominal when the pore aspect ratio exceeds beyond 06 in

10 International Journal of Geophysics

12366

12368

12370

1237247 48

Vp (kms) Vs (kms)

Dep

th (f

eet)

12366

12368

12370

12372

Dep

th (f

eet)

VpVp-2DKTVp-3DKT

26 27

VsVs-2DKTVs-3DKT

Figure 11 A comparison between the predicted and measured P- and S-wave velocities by employing 2DKT and 3DKTmodels using criticalaspect ratio (06) of well number A

both models Significantly velocities predicted by the 3DKTmodel agreewell with thosemeasured compared to the 2DKTmodel (Figure 11)

It is due to the fact that the model adjusts pore aspectratio or the proportion of various 3D pores and effectivelymodels the geometries of various pore shapes which is thedominant factor influencing the velocities of the elastic-wave propagating in the clastic reservoirs [28] Due to thesubstantial amount of clay (15) contents that may containmicrocracks the dry rock moduli estimation by the Kuster-Toksoz (2DKT) model (assuming ellipsoidal) could not beappropriate The addition of small volumes of cracks to therock matrix significantly weakens the rock The predicted P-and S-wave velocities from 3DKT are more consistent withthe measured ones although with some deviation Thus weconsider aspect ratio 06 as the critical value of the pore aspectratio which is the turning point of the increase trend ofpredicted velocities for Marmaton Wash

Thus variations in the elastic moduli are more influencedby pore shape geometry than porosity lower than 10and have a relatively small impact on 119881P119881S ratios Theintegrated application of the 2DKT and 3DKTmodels helpedus properly determine the pore shape geometry of GraniteWash reservoir to estimate site specific 119881P-119881S relationship

54 119881119875-119881119878Relationship of Granite Wash The 119881P-119881S relation

is normally used for 119881S prediction where only P-wave data isavailable and is also very important forAVOanalysis and porefluids prediction For S-wave velocity estimation regard-less of the approach taken it is important to understand

the underlying lithology The rock composition providesfirst order control on dry frame 119881P119881S ratio especially forconsolidated rocks [29] We take advantage of rock physics(effective medium) modeling and derive the 119881P-119881S relation-ship and compare it with the widely used empirical rela-tionship ldquo119881P = 116 lowast 119881S + 136rdquo presented by Castagnaet al [8] for clastic rocks We employ Berrymanrsquos [7] (3DKT)model considering rock composition porosity and poreshape After calibration of the predicted velocities with coredata the 119881P-119881S relationship is generated through regressionanalysis of least-square linear fit (1198772 = 090) as a function of119881P (Kms) (Figure 12)

The derived 119881P-119881S relation by 3DKT for Granite Washprovides closer approximation from 2DKT and global empir-ical relation by Castagna et al [8] Figure 13 shows that theS-wave velocity derived from Castagnarsquos mudrock is overpredicted with known 119881P when comparatively analyzed to2DKT and 3DKT models predictions However there areapparent deviations between the measured and predicteddata by the locally calibrated empirical relation and theeffectivemediummodelsWe actually do not advocate the useof global empirical relations for S-wave velocity predictionfor the Granite Wash but emphasize on site specific builtrelation based on the data of the corresponding study areausing effective medium models Next we employed the bestmodel (3DKT model) to discuss the fluid substitution effectson P- and S-wave velocities Marmaton Wash

55 Fluid Substitution in Granite Wash For practical fluidsubstitution results in tight sands using Gassmannrsquos [21]

International Journal of Geophysics 11

35

30

25

20

1530 35 40 45 50 55 60

0000012500250003750050000625007500087501000

R2 = 90

Vs

(km

s)

Vp (kms)

Vs = 0599 lowast Vp minus 019102

0 4 5 50 55

0000012500250037505000625075087100000

R2 = 90

s = 0599 lowast VpVV minus 019102

GR

Figure 12 Compressional and shearwave velocities ofGraniteWashformation from sonic logs data color coded with gamma ray (GR)from two wells numbers A and B and calibrated with core [11]measurements (Maroon data)

12340

12360

12380

12400

12420

12440

12460

20 22 24 26 28 30 32 34

003500

01556

02763

03969

05175

06381

07588

08794

1000

Dep

th (f

eet)

S-wave velocity (kms)

Vs log

Vs-2DKTVs-3DKT

Vs-RVs-C

Vsh

Figure 13 Comparison of S-wave velocity predictions (color codedwith 119881sh) of well number A with effective medium (2DKT and3DKT) models locally generated relation (119881

119904-119877) and Castagna

global empirical relationship (119881119904-119862)

equations necessitates an appropriate technique which real-istically stiffens low porosity rock by fluctuating the mineralpore fluid or dry rock moduli Application of Gassmannrsquosequation is dependent upon reasonable estimation of thebulk modulus of the dry rock frame (119870

119889) Dry frame rock

bulk (119870119889) modulus can be derived either from (1) laboratory

measurement on core data (2) application of empirical rela-tionships or effective medium theory [30ndash33] or (3) directcalculation from log data by rewriting Gassmannrsquos equation(4) for frame rock bulk (119870

119889) [34] which often generates

negative (a physical impossibility) or unrealistically largevalues in low porosity reservoirs [35] However of the variousalternatives we take essentially a pragmatic approach tocondition the dry frame rockmodel used in fluid substitution

For reliable fluid substitution model in clastic reservoir weconditioned the calculation of dry frame (119870

119889) rock data

implicit in Gassmannrsquos model and apply Biot [36] relationused by Wang et al [37] as follows

119870119889= 119870119898

(1 minus 120573) (7)

where 120573 is sometimes called the Biot coefficient and it is oftencalculated as a function of porosity and the form used in thepaper is derived by Krief et al [38]

(1 minus 120573) = [1 minus 120601]119898(120601)

119898 (120601) =3

(1 minus 120601)

(8)

where 120601 is the formation porosityConsequently the saturated bulk modulus of the rock

(119870sat) for the in situ conditions is first calculated using (1)Equation (7) is combined with Gassmannrsquos equations (4)to generate a pseudo dry frame rock bulk (119870

119889) modulus

suitable for fluid substitution The input parameters for fluidsubstitutions are generally roughly estimated and thus resultin substantial errorsTherefore densities bulkmoduli veloc-ities and viscosities of various pore fluids (brine and gas)are used both from core measurements and estimation usingBatzle and Wang [39] relationships The homogeneous fluid(119870fl) uniformly distributed throughout the pore space hasbeen computed by harmonic average (Reuss) using data fromthewell logs assuming twofluids (water and gas) phases in thereservoir The fluids (119870fl 120588fl) properties occupying the porespace and the bulk modulus of the mineral matrix (119870matrix)are determined by formulations given by Smith et al [35] Tocalculate the bulk modulus of the mineral matrix (119870matrix)the information on the composition (mineral abundance) ofthe rock is used from core study data [11] Mineral matrix(119870matrix) is then calculated via application of Voigt-Reuss-Hill (VRH) averaging the mineral constituents The elasticconstants and densities of pore fluid properties obtained fromcoremeasurements are given in Table 2 Figure 14 an exampleof composite log example from Granite Wash (Marmaton)section demonstrates the significant fluid substitution differ-ences between the wet and the gas effect on the seismic P-wave (119881P) velocity log for well number A The observed fluideffects reach amaximum in the cleanest sandwith the highest(13) porosity and low density response Fluid substitutioneffects correspond more closely with expectations based onraw logs increasing our confidence in the results of AVOmodeling and seismic inversion carried out in phase II

56 Cross-Plotting Attribute Analysis Cross-plotting of elas-tic rock properties from well logs enables the simultane-ous and meaningful evaluation of two attributes that helpsin making straightforward interpretation of hydrocarbonsaturated sands in a particular reservoir For comparisonwe cross-plotted both conventional (P- and S-impedances)and unconventional (lambda mu and rho) attributes in thederived elastic parameter cross-plot space to isolate potentialanomalous zones from a background relationship of well

12 International Journal of Geophysics

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Vsh

(a)

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Sw

(b)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

00 01

Porosity

(c)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

24 26

Density

(d)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

42 48

Vp

GasBrine

(e)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

025 030

PR

(f)

Figure 14 A composite log example fromGraniteWash (Marmaton) section Plots from left to right show logs of (a)119881sh (b)Water saturation(119878119908) (c) porosity (d) density (e) compressional velocity (119881P) and (f) Poisson ratio (PR) Fluid substitution is carried out for brine (blue) and

gas (red) scenarios using dry rock data derived from logs

number A (Figure 15) Figure 15(a) shows where a conven-tional impedance IP versus IS analysis has limitations inclearly discriminating between all the various lithologies andgas sand zone The P- and S-impedances log curves (whichshare both rigidity and density) tend to track each other withdepth and never crossover only S-impedance contrast in theMarmaton sand (12400

1015840

) is clearly less distinguishable thanthe comparable P-impedance contrast By contrast the 120582120588

and 120583120588 attributes have similar value ranges that do cross over

with 120582120588 lt 120583120588 showing a clear wide separation for a hydro-carbon zone while tracking together to clearly indicate thethin tight shale breaks (Figure 15(b)) In Figure 15(c) onlythe gas zone is discernable from background cut-off on the119881P119881S curve which varies between 16 and 20 Figures 15(d)and 15(e) demonstrate the significant advantage of the 120582120588120583120588 cross-plot in isolating both lithologic properties (such assand shale and clastic facies) as well as the gas zone com-pared to IP and IS cross-plot P- and S-impedance data points

International Journal of Geophysics 13

12340

12360

12380

12400

12420

12440

12460

4 6 8 10 12 14

IPIS

Dep

th (f

eet)

Impedance

(a)

12340

12360

12380

12400

12420

12440

12460

10 20 30 40 50 60 70 80

Tight shale

Gas zone

Lambda-rhoMu-rho

Shale

Dep

th (f

eet)

Lambda-rho Mu-rho

(b)

12340

12360

12380

12400

12420

12440

12460

15 16 17 18 19 20 21

Dep

th (f

eet)

VpVs

(c)

80

75

70

65

60

55

50

458 9 10 11 12 13 14

P-impedance

Gas zone

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

S-im

peda

nce

(d)

60

55

50

45

40

35

30

25

0 10 20 30 40 50 60 70

Gas zone

Lambda-rho

Mu-

rho

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

(e)

25

20

15

10

05

00minus15 minus10 minus5 0 5 10 15 20

Shale

Sst line

Wet sand

Gas zone

Lambda-Mu (difference)

Lam

bda

Mu

(rat

io)

003500

01556

02763

03969

05175

06381

07588

08794

1000

Shale

Sst line

Wet sand

Gas zone

Vsh

(f)

Figure 15 Well log curves and cross-plots from Granite Wash (Marmaton sections 12325ndash12470) of well number A (a) P- and S-impedanceslog curves (b) Lambda-rho and Mu-rho log versus depth (c) 119881

119901119881119904ratio versus depth (d) P-impedance versus S-impedance cross-plot color

coded by shale volume (119881sh) (e) Lambda-rho versus Mu-rho cross-plot color coded by shale volume (119881sh) and (f) Lambda-Mu (difference)versus Lambda-rhoMu-rho (ratio) cross-plot color coded by shale volume (119881sh)

14 International Journal of Geophysics

show a close linear relationship with shale having lowestvalues while in comparison the lowest 120582120588 (incompressibilitytimes density) value has the best gas sand values along with 120583120588

(rigidity times density) values higher than shalesThe other morecompetent pure lithologies (silts cemented sands) could beobserved in the opposite upper right quadrant relative to theshales

Following the cross-plots analysis for the same log dataset Figure 15(f) shows an interesting interpretation templatefor the 120582120583 ratio versus the 120582 minus 120583 (difference) cross-plotThe anomalous (gas) zone is marked in a lower left red boxbeyond the (06 minus5) point separating it from the wet sandswhich tend to align in the upper right zone The variationof the slope beyond the origin (1 0) with increasing gassaturation (blue to green dotted lines) reveals an unexpectedeffect of the gas on the rigidity of the sand This is due tothe effective porosity and lower density gas effect on rigidityleading to a slight steeping of the ldquosandstone linerdquo slopeThus in this case unconventional attributes plots provide aclear separation of the anomalous hydrocarbon zones and thelithologic properties of the sand and shale facies which havebeen color coded with shale volume

6 Conclusion

With the above discussion and limitations the followingconclusions can be derived Rock physics diagnostic and thinsections analysis indicate the variations in elastic propertiesof Granite Wash which can be explained by diagenetic phaseof quartz (or silica) cementation CementedWashes have rel-atively high elastic moduli and can be modeled primarily bycontact-cement model of scheme 2 Velocity prediction fromeffective medium models shows that pore shape geometrysignificantly influences the P- and S-wave velocities in ourreservoirs as observed in the velocity data evaluated Core andpetrographic observations also confirm cementations withcomplex pore geometry containing variable concentrations ofhigh and low pores aspect ratio Hence Berrymanrsquos (3DKT)model is proven to be the best one to discuss the influence ofpore shape on seismic velocities and used for the generationof locally calibrated empirical (119881P-119881S) relationship that bestdescribes our data set Finally we applied fluid substitution onlog data for calculating reservoir elastic properties with dif-ferent pore fluid (brine and gas) for an improved petrophys-ical discrimination of rock properties using unconventionalLame parameters 120582 (pure incompressibility) and 120583 (rigidity)attributes over conventional 119881P 119881S attributes analysis Theresearch outcomes are intended to be rigorously used as afoundation for ongoing reservoir characterization study inphase II

Disclosure

This paper is developed from an oral and poster presentationat American Association of Petroleum Geologists (AAPG)Annual Convention amp Exhibition David L Lawrence Con-vention Center Pittsburgh PA USA May 19ndash22 2013

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank The University of Tulsa and NewfieldExploration Company for support of this research The firstauthor would like to thank Jeff Kane (Corporate Petro-physicist) for his scholastic discussions They also thankthe anonymous reviewers for the constructive comments onimproving our paper

References

[1] S Ingram I Paterniti B Rothkopf and C Stevenson ldquoGranitewash field study-buffalo wallow field Texas panhandlerdquo inProceedings of the SPE Eastern Regional Meeting SPE104546Canton Ohio USA October 2006

[2] M I Odigi ldquoDiagenesis and reservoir quality of Cretaceoussandstone of Nkporo Shale (Campanian) Southeastern Benuetrough Nigeriardquo Journal of Geology and Mining Research vol 3pp 265ndash280 2011

[3] F Ruiz and A Cheng ldquoA rock physics model for tight gas sandrdquoThe Leading Edge vol 29 no 12 pp 1484ndash1489 2010

[4] RDMindlin ldquoCompliance of elastic bodies in contactrdquo Journalof Applied Mechanics vol 16 pp 259ndash268 1949

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[6] J Dvorkin and A Nur ldquoElasticity of high-porosity sandstonestheory for twoNorth Sea data setsrdquoGeophysics vol 61 no 5 pp1363ndash1370 1996

[7] J G Berryman ldquoLong-wavelength propagation in compositeelastic media II Ellipsoidal Inclusionsrdquo The Journal of theAcoustical Society of America vol 68 no 6 pp 1820ndash1831 1980

[8] J P Castagna M L Batzle and R L Eastwood ldquoRelationshipsbetween compressional-wave and shear-wave velocities in clas-tic silicate rocksrdquo Geophysics vol 50 no 4 pp 571ndash581 1985

[9] G T Kuster and M N Toksoz ldquoVelocity and attenuation ofseismic waves in two-phase mediardquo Geophysics vol 39 no 5pp 587ndash618 1974

[10] G E Moore ldquoPennsylvanian paleogeography of the south-ern Mid-Continentrdquo in Pennsylvanian Sandstones of the Mid-Continent N J Hyne Ed pp 2ndash12 Tulsa Geological Society(TGS) Tulsa Okla USA 1979

[11] Core Lab ldquoGraniteWash geological Petrophysical and comple-tions study for improved formation evaluation and optimal wellcompletionrdquo Proprietary industry study 2005

[12] K S Johnson and B J Cardott ldquoGeologic framework andhydrocarbon source rocks of Oklahomardquo in Proceedings of theSource Rocks in the Southern Midcontinent K S JohnsonB andJ Cardott Eds pp 21ndash37 OGS Circular 1992

[13] N R Gelphman ldquoWest sentinel oil field Washita CountyOklahoma sedimentology of the ldquoGraniteWashrdquo and structuralgeologyrdquoOklahoma City Geological Society Shale Shaker vol 10pp 234ndash247 1960

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

8 International Journal of Geophysics

125

100

75

50

25

0000 025 050 075 100

Bulk

mod

ulus

(GPa

)

Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

50

40

30

20

10

0000 025 050 075 100

Shea

r mod

ulus

(GPa

)Porosity

Well AWell BWell CWell D

Core dataHS-UBHS-LB

Figure 8 Moduli versus porosity for different wells and core [11] data with superposition of Hashin and Shtrikman [5] bounds

6

5

4

3

2

00 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Vp

(km

s)

Porosity

Vp versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(a)

80

60

40

20

000 01 02 03 04

003500

01556

02763

03969

05175

06381

07588

08794

1000

Bulk

mod

ulus

(GPa

)

Porosity

K versus porosityCora data Uncement

Vsh

Cement 2

Cement 1

(b)

Figure 9 (a) P-wave velocity versus porosity and (b) modulus versus porosity (core [11] data included) color coded with shale volume (119881sh)

of the Marmaton section in well number A with rock physics models curves superimposed

International Journal of Geophysics 9

12368

12370

12372

465 470 475

P-wave velocity

465 470 475

P-wave velocity

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

260 265 270 275

S-wave velocity

260 265 270 275

S-wave velocity

Vp2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-09

2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-082DKT-09

Vs

3DKT-013DKT-023DKT-033DKT-043DKT-053DKT-063DKT-073DKT-08

Vp

3DKT-09

3DKT-04

3DKT-013DKT-023DKT-03

3DKT-053DKT-063DKT-073DKT-083DKT-09

Vs

Figure 10 P- and S-wave velocities prediction by employing 2DKT and 3DKT models using different aspect ratios (01ndash09) of well numberA

interesting that the well-log data with resolution of tens-of-centimeter reflects what we observe at the microscaleBy establishing a site-specific rock physics transform (unce-mented and cemented models) between the total porositymineralogy and elastic properties we conclude the rockphysics diagnostics here In the next section we analyze poreshape influence on seismic velocities in order to generatelocally calibrated 119881P-119881S relationship for Granite Wash

53 Analysis of Pore Shapes Influencing Seismic VelocitiesTheoretically it has been reckoned that the velocities in twostates (dry or saturated) are mainly controlled by the effectivemoduli (bulk and shear) which in turn depend on poreshape geometry mineralogy properties of solids the porefluid degree of consolidation and cementation [9 26 27]Thus elastic moduli (or velocities) are basically the functionof all these factors combined Especially in low porosityand permeability rocks elastic moduli of the rock min-eral matrix with spherical pores (high aspect ratio) often

dominate those of the bulk rock [9 26 27] Therefore it istheoretically reasonable to employ the elastic moduli in rocksto estimate seismic velocities Here two effective medium(2DKT and 3DKT) models are investigated to evaluate poreshape geometry using different pore aspect ratios (valueranging from 01 to 09) to predict P- and S-wave velocitiesfrom the well data We know that the pores with high aspectratio (spherical) tend to be stiffer than those with low aspectratio (oblate or penny shape pores) That is to say P- and S-wave velocities in the reservoir with spherical pores tend tobe higher than one with oblate pores or penny shapeThe factcan be easily understood because stiffer pores contribute tothe stiffness of the whole rock while oblate pores contributeto the compliance of thewhole rockHere actual well numberA is taken as an example containing dipole sonic logs toestimate P- and S-wave velocity From Figure 10 we can seeclearly that the increase in P- and S-wave velocities is sharperwhen the pore aspect ratio increases from 01 to 06 for both2DKT and 3DKT models However the increasing trend isnominal when the pore aspect ratio exceeds beyond 06 in

10 International Journal of Geophysics

12366

12368

12370

1237247 48

Vp (kms) Vs (kms)

Dep

th (f

eet)

12366

12368

12370

12372

Dep

th (f

eet)

VpVp-2DKTVp-3DKT

26 27

VsVs-2DKTVs-3DKT

Figure 11 A comparison between the predicted and measured P- and S-wave velocities by employing 2DKT and 3DKTmodels using criticalaspect ratio (06) of well number A

both models Significantly velocities predicted by the 3DKTmodel agreewell with thosemeasured compared to the 2DKTmodel (Figure 11)

It is due to the fact that the model adjusts pore aspectratio or the proportion of various 3D pores and effectivelymodels the geometries of various pore shapes which is thedominant factor influencing the velocities of the elastic-wave propagating in the clastic reservoirs [28] Due to thesubstantial amount of clay (15) contents that may containmicrocracks the dry rock moduli estimation by the Kuster-Toksoz (2DKT) model (assuming ellipsoidal) could not beappropriate The addition of small volumes of cracks to therock matrix significantly weakens the rock The predicted P-and S-wave velocities from 3DKT are more consistent withthe measured ones although with some deviation Thus weconsider aspect ratio 06 as the critical value of the pore aspectratio which is the turning point of the increase trend ofpredicted velocities for Marmaton Wash

Thus variations in the elastic moduli are more influencedby pore shape geometry than porosity lower than 10and have a relatively small impact on 119881P119881S ratios Theintegrated application of the 2DKT and 3DKTmodels helpedus properly determine the pore shape geometry of GraniteWash reservoir to estimate site specific 119881P-119881S relationship

54 119881119875-119881119878Relationship of Granite Wash The 119881P-119881S relation

is normally used for 119881S prediction where only P-wave data isavailable and is also very important forAVOanalysis and porefluids prediction For S-wave velocity estimation regard-less of the approach taken it is important to understand

the underlying lithology The rock composition providesfirst order control on dry frame 119881P119881S ratio especially forconsolidated rocks [29] We take advantage of rock physics(effective medium) modeling and derive the 119881P-119881S relation-ship and compare it with the widely used empirical rela-tionship ldquo119881P = 116 lowast 119881S + 136rdquo presented by Castagnaet al [8] for clastic rocks We employ Berrymanrsquos [7] (3DKT)model considering rock composition porosity and poreshape After calibration of the predicted velocities with coredata the 119881P-119881S relationship is generated through regressionanalysis of least-square linear fit (1198772 = 090) as a function of119881P (Kms) (Figure 12)

The derived 119881P-119881S relation by 3DKT for Granite Washprovides closer approximation from 2DKT and global empir-ical relation by Castagna et al [8] Figure 13 shows that theS-wave velocity derived from Castagnarsquos mudrock is overpredicted with known 119881P when comparatively analyzed to2DKT and 3DKT models predictions However there areapparent deviations between the measured and predicteddata by the locally calibrated empirical relation and theeffectivemediummodelsWe actually do not advocate the useof global empirical relations for S-wave velocity predictionfor the Granite Wash but emphasize on site specific builtrelation based on the data of the corresponding study areausing effective medium models Next we employed the bestmodel (3DKT model) to discuss the fluid substitution effectson P- and S-wave velocities Marmaton Wash

55 Fluid Substitution in Granite Wash For practical fluidsubstitution results in tight sands using Gassmannrsquos [21]

International Journal of Geophysics 11

35

30

25

20

1530 35 40 45 50 55 60

0000012500250003750050000625007500087501000

R2 = 90

Vs

(km

s)

Vp (kms)

Vs = 0599 lowast Vp minus 019102

0 4 5 50 55

0000012500250037505000625075087100000

R2 = 90

s = 0599 lowast VpVV minus 019102

GR

Figure 12 Compressional and shearwave velocities ofGraniteWashformation from sonic logs data color coded with gamma ray (GR)from two wells numbers A and B and calibrated with core [11]measurements (Maroon data)

12340

12360

12380

12400

12420

12440

12460

20 22 24 26 28 30 32 34

003500

01556

02763

03969

05175

06381

07588

08794

1000

Dep

th (f

eet)

S-wave velocity (kms)

Vs log

Vs-2DKTVs-3DKT

Vs-RVs-C

Vsh

Figure 13 Comparison of S-wave velocity predictions (color codedwith 119881sh) of well number A with effective medium (2DKT and3DKT) models locally generated relation (119881

119904-119877) and Castagna

global empirical relationship (119881119904-119862)

equations necessitates an appropriate technique which real-istically stiffens low porosity rock by fluctuating the mineralpore fluid or dry rock moduli Application of Gassmannrsquosequation is dependent upon reasonable estimation of thebulk modulus of the dry rock frame (119870

119889) Dry frame rock

bulk (119870119889) modulus can be derived either from (1) laboratory

measurement on core data (2) application of empirical rela-tionships or effective medium theory [30ndash33] or (3) directcalculation from log data by rewriting Gassmannrsquos equation(4) for frame rock bulk (119870

119889) [34] which often generates

negative (a physical impossibility) or unrealistically largevalues in low porosity reservoirs [35] However of the variousalternatives we take essentially a pragmatic approach tocondition the dry frame rockmodel used in fluid substitution

For reliable fluid substitution model in clastic reservoir weconditioned the calculation of dry frame (119870

119889) rock data

implicit in Gassmannrsquos model and apply Biot [36] relationused by Wang et al [37] as follows

119870119889= 119870119898

(1 minus 120573) (7)

where 120573 is sometimes called the Biot coefficient and it is oftencalculated as a function of porosity and the form used in thepaper is derived by Krief et al [38]

(1 minus 120573) = [1 minus 120601]119898(120601)

119898 (120601) =3

(1 minus 120601)

(8)

where 120601 is the formation porosityConsequently the saturated bulk modulus of the rock

(119870sat) for the in situ conditions is first calculated using (1)Equation (7) is combined with Gassmannrsquos equations (4)to generate a pseudo dry frame rock bulk (119870

119889) modulus

suitable for fluid substitution The input parameters for fluidsubstitutions are generally roughly estimated and thus resultin substantial errorsTherefore densities bulkmoduli veloc-ities and viscosities of various pore fluids (brine and gas)are used both from core measurements and estimation usingBatzle and Wang [39] relationships The homogeneous fluid(119870fl) uniformly distributed throughout the pore space hasbeen computed by harmonic average (Reuss) using data fromthewell logs assuming twofluids (water and gas) phases in thereservoir The fluids (119870fl 120588fl) properties occupying the porespace and the bulk modulus of the mineral matrix (119870matrix)are determined by formulations given by Smith et al [35] Tocalculate the bulk modulus of the mineral matrix (119870matrix)the information on the composition (mineral abundance) ofthe rock is used from core study data [11] Mineral matrix(119870matrix) is then calculated via application of Voigt-Reuss-Hill (VRH) averaging the mineral constituents The elasticconstants and densities of pore fluid properties obtained fromcoremeasurements are given in Table 2 Figure 14 an exampleof composite log example from Granite Wash (Marmaton)section demonstrates the significant fluid substitution differ-ences between the wet and the gas effect on the seismic P-wave (119881P) velocity log for well number A The observed fluideffects reach amaximum in the cleanest sandwith the highest(13) porosity and low density response Fluid substitutioneffects correspond more closely with expectations based onraw logs increasing our confidence in the results of AVOmodeling and seismic inversion carried out in phase II

56 Cross-Plotting Attribute Analysis Cross-plotting of elas-tic rock properties from well logs enables the simultane-ous and meaningful evaluation of two attributes that helpsin making straightforward interpretation of hydrocarbonsaturated sands in a particular reservoir For comparisonwe cross-plotted both conventional (P- and S-impedances)and unconventional (lambda mu and rho) attributes in thederived elastic parameter cross-plot space to isolate potentialanomalous zones from a background relationship of well

12 International Journal of Geophysics

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Vsh

(a)

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Sw

(b)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

00 01

Porosity

(c)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

24 26

Density

(d)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

42 48

Vp

GasBrine

(e)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

025 030

PR

(f)

Figure 14 A composite log example fromGraniteWash (Marmaton) section Plots from left to right show logs of (a)119881sh (b)Water saturation(119878119908) (c) porosity (d) density (e) compressional velocity (119881P) and (f) Poisson ratio (PR) Fluid substitution is carried out for brine (blue) and

gas (red) scenarios using dry rock data derived from logs

number A (Figure 15) Figure 15(a) shows where a conven-tional impedance IP versus IS analysis has limitations inclearly discriminating between all the various lithologies andgas sand zone The P- and S-impedances log curves (whichshare both rigidity and density) tend to track each other withdepth and never crossover only S-impedance contrast in theMarmaton sand (12400

1015840

) is clearly less distinguishable thanthe comparable P-impedance contrast By contrast the 120582120588

and 120583120588 attributes have similar value ranges that do cross over

with 120582120588 lt 120583120588 showing a clear wide separation for a hydro-carbon zone while tracking together to clearly indicate thethin tight shale breaks (Figure 15(b)) In Figure 15(c) onlythe gas zone is discernable from background cut-off on the119881P119881S curve which varies between 16 and 20 Figures 15(d)and 15(e) demonstrate the significant advantage of the 120582120588120583120588 cross-plot in isolating both lithologic properties (such assand shale and clastic facies) as well as the gas zone com-pared to IP and IS cross-plot P- and S-impedance data points

International Journal of Geophysics 13

12340

12360

12380

12400

12420

12440

12460

4 6 8 10 12 14

IPIS

Dep

th (f

eet)

Impedance

(a)

12340

12360

12380

12400

12420

12440

12460

10 20 30 40 50 60 70 80

Tight shale

Gas zone

Lambda-rhoMu-rho

Shale

Dep

th (f

eet)

Lambda-rho Mu-rho

(b)

12340

12360

12380

12400

12420

12440

12460

15 16 17 18 19 20 21

Dep

th (f

eet)

VpVs

(c)

80

75

70

65

60

55

50

458 9 10 11 12 13 14

P-impedance

Gas zone

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

S-im

peda

nce

(d)

60

55

50

45

40

35

30

25

0 10 20 30 40 50 60 70

Gas zone

Lambda-rho

Mu-

rho

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

(e)

25

20

15

10

05

00minus15 minus10 minus5 0 5 10 15 20

Shale

Sst line

Wet sand

Gas zone

Lambda-Mu (difference)

Lam

bda

Mu

(rat

io)

003500

01556

02763

03969

05175

06381

07588

08794

1000

Shale

Sst line

Wet sand

Gas zone

Vsh

(f)

Figure 15 Well log curves and cross-plots from Granite Wash (Marmaton sections 12325ndash12470) of well number A (a) P- and S-impedanceslog curves (b) Lambda-rho and Mu-rho log versus depth (c) 119881

119901119881119904ratio versus depth (d) P-impedance versus S-impedance cross-plot color

coded by shale volume (119881sh) (e) Lambda-rho versus Mu-rho cross-plot color coded by shale volume (119881sh) and (f) Lambda-Mu (difference)versus Lambda-rhoMu-rho (ratio) cross-plot color coded by shale volume (119881sh)

14 International Journal of Geophysics

show a close linear relationship with shale having lowestvalues while in comparison the lowest 120582120588 (incompressibilitytimes density) value has the best gas sand values along with 120583120588

(rigidity times density) values higher than shalesThe other morecompetent pure lithologies (silts cemented sands) could beobserved in the opposite upper right quadrant relative to theshales

Following the cross-plots analysis for the same log dataset Figure 15(f) shows an interesting interpretation templatefor the 120582120583 ratio versus the 120582 minus 120583 (difference) cross-plotThe anomalous (gas) zone is marked in a lower left red boxbeyond the (06 minus5) point separating it from the wet sandswhich tend to align in the upper right zone The variationof the slope beyond the origin (1 0) with increasing gassaturation (blue to green dotted lines) reveals an unexpectedeffect of the gas on the rigidity of the sand This is due tothe effective porosity and lower density gas effect on rigidityleading to a slight steeping of the ldquosandstone linerdquo slopeThus in this case unconventional attributes plots provide aclear separation of the anomalous hydrocarbon zones and thelithologic properties of the sand and shale facies which havebeen color coded with shale volume

6 Conclusion

With the above discussion and limitations the followingconclusions can be derived Rock physics diagnostic and thinsections analysis indicate the variations in elastic propertiesof Granite Wash which can be explained by diagenetic phaseof quartz (or silica) cementation CementedWashes have rel-atively high elastic moduli and can be modeled primarily bycontact-cement model of scheme 2 Velocity prediction fromeffective medium models shows that pore shape geometrysignificantly influences the P- and S-wave velocities in ourreservoirs as observed in the velocity data evaluated Core andpetrographic observations also confirm cementations withcomplex pore geometry containing variable concentrations ofhigh and low pores aspect ratio Hence Berrymanrsquos (3DKT)model is proven to be the best one to discuss the influence ofpore shape on seismic velocities and used for the generationof locally calibrated empirical (119881P-119881S) relationship that bestdescribes our data set Finally we applied fluid substitution onlog data for calculating reservoir elastic properties with dif-ferent pore fluid (brine and gas) for an improved petrophys-ical discrimination of rock properties using unconventionalLame parameters 120582 (pure incompressibility) and 120583 (rigidity)attributes over conventional 119881P 119881S attributes analysis Theresearch outcomes are intended to be rigorously used as afoundation for ongoing reservoir characterization study inphase II

Disclosure

This paper is developed from an oral and poster presentationat American Association of Petroleum Geologists (AAPG)Annual Convention amp Exhibition David L Lawrence Con-vention Center Pittsburgh PA USA May 19ndash22 2013

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank The University of Tulsa and NewfieldExploration Company for support of this research The firstauthor would like to thank Jeff Kane (Corporate Petro-physicist) for his scholastic discussions They also thankthe anonymous reviewers for the constructive comments onimproving our paper

References

[1] S Ingram I Paterniti B Rothkopf and C Stevenson ldquoGranitewash field study-buffalo wallow field Texas panhandlerdquo inProceedings of the SPE Eastern Regional Meeting SPE104546Canton Ohio USA October 2006

[2] M I Odigi ldquoDiagenesis and reservoir quality of Cretaceoussandstone of Nkporo Shale (Campanian) Southeastern Benuetrough Nigeriardquo Journal of Geology and Mining Research vol 3pp 265ndash280 2011

[3] F Ruiz and A Cheng ldquoA rock physics model for tight gas sandrdquoThe Leading Edge vol 29 no 12 pp 1484ndash1489 2010

[4] RDMindlin ldquoCompliance of elastic bodies in contactrdquo Journalof Applied Mechanics vol 16 pp 259ndash268 1949

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[6] J Dvorkin and A Nur ldquoElasticity of high-porosity sandstonestheory for twoNorth Sea data setsrdquoGeophysics vol 61 no 5 pp1363ndash1370 1996

[7] J G Berryman ldquoLong-wavelength propagation in compositeelastic media II Ellipsoidal Inclusionsrdquo The Journal of theAcoustical Society of America vol 68 no 6 pp 1820ndash1831 1980

[8] J P Castagna M L Batzle and R L Eastwood ldquoRelationshipsbetween compressional-wave and shear-wave velocities in clas-tic silicate rocksrdquo Geophysics vol 50 no 4 pp 571ndash581 1985

[9] G T Kuster and M N Toksoz ldquoVelocity and attenuation ofseismic waves in two-phase mediardquo Geophysics vol 39 no 5pp 587ndash618 1974

[10] G E Moore ldquoPennsylvanian paleogeography of the south-ern Mid-Continentrdquo in Pennsylvanian Sandstones of the Mid-Continent N J Hyne Ed pp 2ndash12 Tulsa Geological Society(TGS) Tulsa Okla USA 1979

[11] Core Lab ldquoGraniteWash geological Petrophysical and comple-tions study for improved formation evaluation and optimal wellcompletionrdquo Proprietary industry study 2005

[12] K S Johnson and B J Cardott ldquoGeologic framework andhydrocarbon source rocks of Oklahomardquo in Proceedings of theSource Rocks in the Southern Midcontinent K S JohnsonB andJ Cardott Eds pp 21ndash37 OGS Circular 1992

[13] N R Gelphman ldquoWest sentinel oil field Washita CountyOklahoma sedimentology of the ldquoGraniteWashrdquo and structuralgeologyrdquoOklahoma City Geological Society Shale Shaker vol 10pp 234ndash247 1960

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

International Journal of Geophysics 9

12368

12370

12372

465 470 475

P-wave velocity

465 470 475

P-wave velocity

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

12368

12370

12372

Dep

th (f

eet)

260 265 270 275

S-wave velocity

260 265 270 275

S-wave velocity

Vp2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-09

2DKT-012DKT-022DKT-032DKT-042DKT-052DKT-062DKT-072DKT-082DKT-09

Vs

3DKT-013DKT-023DKT-033DKT-043DKT-053DKT-063DKT-073DKT-08

Vp

3DKT-09

3DKT-04

3DKT-013DKT-023DKT-03

3DKT-053DKT-063DKT-073DKT-083DKT-09

Vs

Figure 10 P- and S-wave velocities prediction by employing 2DKT and 3DKT models using different aspect ratios (01ndash09) of well numberA

interesting that the well-log data with resolution of tens-of-centimeter reflects what we observe at the microscaleBy establishing a site-specific rock physics transform (unce-mented and cemented models) between the total porositymineralogy and elastic properties we conclude the rockphysics diagnostics here In the next section we analyze poreshape influence on seismic velocities in order to generatelocally calibrated 119881P-119881S relationship for Granite Wash

53 Analysis of Pore Shapes Influencing Seismic VelocitiesTheoretically it has been reckoned that the velocities in twostates (dry or saturated) are mainly controlled by the effectivemoduli (bulk and shear) which in turn depend on poreshape geometry mineralogy properties of solids the porefluid degree of consolidation and cementation [9 26 27]Thus elastic moduli (or velocities) are basically the functionof all these factors combined Especially in low porosityand permeability rocks elastic moduli of the rock min-eral matrix with spherical pores (high aspect ratio) often

dominate those of the bulk rock [9 26 27] Therefore it istheoretically reasonable to employ the elastic moduli in rocksto estimate seismic velocities Here two effective medium(2DKT and 3DKT) models are investigated to evaluate poreshape geometry using different pore aspect ratios (valueranging from 01 to 09) to predict P- and S-wave velocitiesfrom the well data We know that the pores with high aspectratio (spherical) tend to be stiffer than those with low aspectratio (oblate or penny shape pores) That is to say P- and S-wave velocities in the reservoir with spherical pores tend tobe higher than one with oblate pores or penny shapeThe factcan be easily understood because stiffer pores contribute tothe stiffness of the whole rock while oblate pores contributeto the compliance of thewhole rockHere actual well numberA is taken as an example containing dipole sonic logs toestimate P- and S-wave velocity From Figure 10 we can seeclearly that the increase in P- and S-wave velocities is sharperwhen the pore aspect ratio increases from 01 to 06 for both2DKT and 3DKT models However the increasing trend isnominal when the pore aspect ratio exceeds beyond 06 in

10 International Journal of Geophysics

12366

12368

12370

1237247 48

Vp (kms) Vs (kms)

Dep

th (f

eet)

12366

12368

12370

12372

Dep

th (f

eet)

VpVp-2DKTVp-3DKT

26 27

VsVs-2DKTVs-3DKT

Figure 11 A comparison between the predicted and measured P- and S-wave velocities by employing 2DKT and 3DKTmodels using criticalaspect ratio (06) of well number A

both models Significantly velocities predicted by the 3DKTmodel agreewell with thosemeasured compared to the 2DKTmodel (Figure 11)

It is due to the fact that the model adjusts pore aspectratio or the proportion of various 3D pores and effectivelymodels the geometries of various pore shapes which is thedominant factor influencing the velocities of the elastic-wave propagating in the clastic reservoirs [28] Due to thesubstantial amount of clay (15) contents that may containmicrocracks the dry rock moduli estimation by the Kuster-Toksoz (2DKT) model (assuming ellipsoidal) could not beappropriate The addition of small volumes of cracks to therock matrix significantly weakens the rock The predicted P-and S-wave velocities from 3DKT are more consistent withthe measured ones although with some deviation Thus weconsider aspect ratio 06 as the critical value of the pore aspectratio which is the turning point of the increase trend ofpredicted velocities for Marmaton Wash

Thus variations in the elastic moduli are more influencedby pore shape geometry than porosity lower than 10and have a relatively small impact on 119881P119881S ratios Theintegrated application of the 2DKT and 3DKTmodels helpedus properly determine the pore shape geometry of GraniteWash reservoir to estimate site specific 119881P-119881S relationship

54 119881119875-119881119878Relationship of Granite Wash The 119881P-119881S relation

is normally used for 119881S prediction where only P-wave data isavailable and is also very important forAVOanalysis and porefluids prediction For S-wave velocity estimation regard-less of the approach taken it is important to understand

the underlying lithology The rock composition providesfirst order control on dry frame 119881P119881S ratio especially forconsolidated rocks [29] We take advantage of rock physics(effective medium) modeling and derive the 119881P-119881S relation-ship and compare it with the widely used empirical rela-tionship ldquo119881P = 116 lowast 119881S + 136rdquo presented by Castagnaet al [8] for clastic rocks We employ Berrymanrsquos [7] (3DKT)model considering rock composition porosity and poreshape After calibration of the predicted velocities with coredata the 119881P-119881S relationship is generated through regressionanalysis of least-square linear fit (1198772 = 090) as a function of119881P (Kms) (Figure 12)

The derived 119881P-119881S relation by 3DKT for Granite Washprovides closer approximation from 2DKT and global empir-ical relation by Castagna et al [8] Figure 13 shows that theS-wave velocity derived from Castagnarsquos mudrock is overpredicted with known 119881P when comparatively analyzed to2DKT and 3DKT models predictions However there areapparent deviations between the measured and predicteddata by the locally calibrated empirical relation and theeffectivemediummodelsWe actually do not advocate the useof global empirical relations for S-wave velocity predictionfor the Granite Wash but emphasize on site specific builtrelation based on the data of the corresponding study areausing effective medium models Next we employed the bestmodel (3DKT model) to discuss the fluid substitution effectson P- and S-wave velocities Marmaton Wash

55 Fluid Substitution in Granite Wash For practical fluidsubstitution results in tight sands using Gassmannrsquos [21]

International Journal of Geophysics 11

35

30

25

20

1530 35 40 45 50 55 60

0000012500250003750050000625007500087501000

R2 = 90

Vs

(km

s)

Vp (kms)

Vs = 0599 lowast Vp minus 019102

0 4 5 50 55

0000012500250037505000625075087100000

R2 = 90

s = 0599 lowast VpVV minus 019102

GR

Figure 12 Compressional and shearwave velocities ofGraniteWashformation from sonic logs data color coded with gamma ray (GR)from two wells numbers A and B and calibrated with core [11]measurements (Maroon data)

12340

12360

12380

12400

12420

12440

12460

20 22 24 26 28 30 32 34

003500

01556

02763

03969

05175

06381

07588

08794

1000

Dep

th (f

eet)

S-wave velocity (kms)

Vs log

Vs-2DKTVs-3DKT

Vs-RVs-C

Vsh

Figure 13 Comparison of S-wave velocity predictions (color codedwith 119881sh) of well number A with effective medium (2DKT and3DKT) models locally generated relation (119881

119904-119877) and Castagna

global empirical relationship (119881119904-119862)

equations necessitates an appropriate technique which real-istically stiffens low porosity rock by fluctuating the mineralpore fluid or dry rock moduli Application of Gassmannrsquosequation is dependent upon reasonable estimation of thebulk modulus of the dry rock frame (119870

119889) Dry frame rock

bulk (119870119889) modulus can be derived either from (1) laboratory

measurement on core data (2) application of empirical rela-tionships or effective medium theory [30ndash33] or (3) directcalculation from log data by rewriting Gassmannrsquos equation(4) for frame rock bulk (119870

119889) [34] which often generates

negative (a physical impossibility) or unrealistically largevalues in low porosity reservoirs [35] However of the variousalternatives we take essentially a pragmatic approach tocondition the dry frame rockmodel used in fluid substitution

For reliable fluid substitution model in clastic reservoir weconditioned the calculation of dry frame (119870

119889) rock data

implicit in Gassmannrsquos model and apply Biot [36] relationused by Wang et al [37] as follows

119870119889= 119870119898

(1 minus 120573) (7)

where 120573 is sometimes called the Biot coefficient and it is oftencalculated as a function of porosity and the form used in thepaper is derived by Krief et al [38]

(1 minus 120573) = [1 minus 120601]119898(120601)

119898 (120601) =3

(1 minus 120601)

(8)

where 120601 is the formation porosityConsequently the saturated bulk modulus of the rock

(119870sat) for the in situ conditions is first calculated using (1)Equation (7) is combined with Gassmannrsquos equations (4)to generate a pseudo dry frame rock bulk (119870

119889) modulus

suitable for fluid substitution The input parameters for fluidsubstitutions are generally roughly estimated and thus resultin substantial errorsTherefore densities bulkmoduli veloc-ities and viscosities of various pore fluids (brine and gas)are used both from core measurements and estimation usingBatzle and Wang [39] relationships The homogeneous fluid(119870fl) uniformly distributed throughout the pore space hasbeen computed by harmonic average (Reuss) using data fromthewell logs assuming twofluids (water and gas) phases in thereservoir The fluids (119870fl 120588fl) properties occupying the porespace and the bulk modulus of the mineral matrix (119870matrix)are determined by formulations given by Smith et al [35] Tocalculate the bulk modulus of the mineral matrix (119870matrix)the information on the composition (mineral abundance) ofthe rock is used from core study data [11] Mineral matrix(119870matrix) is then calculated via application of Voigt-Reuss-Hill (VRH) averaging the mineral constituents The elasticconstants and densities of pore fluid properties obtained fromcoremeasurements are given in Table 2 Figure 14 an exampleof composite log example from Granite Wash (Marmaton)section demonstrates the significant fluid substitution differ-ences between the wet and the gas effect on the seismic P-wave (119881P) velocity log for well number A The observed fluideffects reach amaximum in the cleanest sandwith the highest(13) porosity and low density response Fluid substitutioneffects correspond more closely with expectations based onraw logs increasing our confidence in the results of AVOmodeling and seismic inversion carried out in phase II

56 Cross-Plotting Attribute Analysis Cross-plotting of elas-tic rock properties from well logs enables the simultane-ous and meaningful evaluation of two attributes that helpsin making straightforward interpretation of hydrocarbonsaturated sands in a particular reservoir For comparisonwe cross-plotted both conventional (P- and S-impedances)and unconventional (lambda mu and rho) attributes in thederived elastic parameter cross-plot space to isolate potentialanomalous zones from a background relationship of well

12 International Journal of Geophysics

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Vsh

(a)

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Sw

(b)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

00 01

Porosity

(c)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

24 26

Density

(d)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

42 48

Vp

GasBrine

(e)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

025 030

PR

(f)

Figure 14 A composite log example fromGraniteWash (Marmaton) section Plots from left to right show logs of (a)119881sh (b)Water saturation(119878119908) (c) porosity (d) density (e) compressional velocity (119881P) and (f) Poisson ratio (PR) Fluid substitution is carried out for brine (blue) and

gas (red) scenarios using dry rock data derived from logs

number A (Figure 15) Figure 15(a) shows where a conven-tional impedance IP versus IS analysis has limitations inclearly discriminating between all the various lithologies andgas sand zone The P- and S-impedances log curves (whichshare both rigidity and density) tend to track each other withdepth and never crossover only S-impedance contrast in theMarmaton sand (12400

1015840

) is clearly less distinguishable thanthe comparable P-impedance contrast By contrast the 120582120588

and 120583120588 attributes have similar value ranges that do cross over

with 120582120588 lt 120583120588 showing a clear wide separation for a hydro-carbon zone while tracking together to clearly indicate thethin tight shale breaks (Figure 15(b)) In Figure 15(c) onlythe gas zone is discernable from background cut-off on the119881P119881S curve which varies between 16 and 20 Figures 15(d)and 15(e) demonstrate the significant advantage of the 120582120588120583120588 cross-plot in isolating both lithologic properties (such assand shale and clastic facies) as well as the gas zone com-pared to IP and IS cross-plot P- and S-impedance data points

International Journal of Geophysics 13

12340

12360

12380

12400

12420

12440

12460

4 6 8 10 12 14

IPIS

Dep

th (f

eet)

Impedance

(a)

12340

12360

12380

12400

12420

12440

12460

10 20 30 40 50 60 70 80

Tight shale

Gas zone

Lambda-rhoMu-rho

Shale

Dep

th (f

eet)

Lambda-rho Mu-rho

(b)

12340

12360

12380

12400

12420

12440

12460

15 16 17 18 19 20 21

Dep

th (f

eet)

VpVs

(c)

80

75

70

65

60

55

50

458 9 10 11 12 13 14

P-impedance

Gas zone

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

S-im

peda

nce

(d)

60

55

50

45

40

35

30

25

0 10 20 30 40 50 60 70

Gas zone

Lambda-rho

Mu-

rho

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

(e)

25

20

15

10

05

00minus15 minus10 minus5 0 5 10 15 20

Shale

Sst line

Wet sand

Gas zone

Lambda-Mu (difference)

Lam

bda

Mu

(rat

io)

003500

01556

02763

03969

05175

06381

07588

08794

1000

Shale

Sst line

Wet sand

Gas zone

Vsh

(f)

Figure 15 Well log curves and cross-plots from Granite Wash (Marmaton sections 12325ndash12470) of well number A (a) P- and S-impedanceslog curves (b) Lambda-rho and Mu-rho log versus depth (c) 119881

119901119881119904ratio versus depth (d) P-impedance versus S-impedance cross-plot color

coded by shale volume (119881sh) (e) Lambda-rho versus Mu-rho cross-plot color coded by shale volume (119881sh) and (f) Lambda-Mu (difference)versus Lambda-rhoMu-rho (ratio) cross-plot color coded by shale volume (119881sh)

14 International Journal of Geophysics

show a close linear relationship with shale having lowestvalues while in comparison the lowest 120582120588 (incompressibilitytimes density) value has the best gas sand values along with 120583120588

(rigidity times density) values higher than shalesThe other morecompetent pure lithologies (silts cemented sands) could beobserved in the opposite upper right quadrant relative to theshales

Following the cross-plots analysis for the same log dataset Figure 15(f) shows an interesting interpretation templatefor the 120582120583 ratio versus the 120582 minus 120583 (difference) cross-plotThe anomalous (gas) zone is marked in a lower left red boxbeyond the (06 minus5) point separating it from the wet sandswhich tend to align in the upper right zone The variationof the slope beyond the origin (1 0) with increasing gassaturation (blue to green dotted lines) reveals an unexpectedeffect of the gas on the rigidity of the sand This is due tothe effective porosity and lower density gas effect on rigidityleading to a slight steeping of the ldquosandstone linerdquo slopeThus in this case unconventional attributes plots provide aclear separation of the anomalous hydrocarbon zones and thelithologic properties of the sand and shale facies which havebeen color coded with shale volume

6 Conclusion

With the above discussion and limitations the followingconclusions can be derived Rock physics diagnostic and thinsections analysis indicate the variations in elastic propertiesof Granite Wash which can be explained by diagenetic phaseof quartz (or silica) cementation CementedWashes have rel-atively high elastic moduli and can be modeled primarily bycontact-cement model of scheme 2 Velocity prediction fromeffective medium models shows that pore shape geometrysignificantly influences the P- and S-wave velocities in ourreservoirs as observed in the velocity data evaluated Core andpetrographic observations also confirm cementations withcomplex pore geometry containing variable concentrations ofhigh and low pores aspect ratio Hence Berrymanrsquos (3DKT)model is proven to be the best one to discuss the influence ofpore shape on seismic velocities and used for the generationof locally calibrated empirical (119881P-119881S) relationship that bestdescribes our data set Finally we applied fluid substitution onlog data for calculating reservoir elastic properties with dif-ferent pore fluid (brine and gas) for an improved petrophys-ical discrimination of rock properties using unconventionalLame parameters 120582 (pure incompressibility) and 120583 (rigidity)attributes over conventional 119881P 119881S attributes analysis Theresearch outcomes are intended to be rigorously used as afoundation for ongoing reservoir characterization study inphase II

Disclosure

This paper is developed from an oral and poster presentationat American Association of Petroleum Geologists (AAPG)Annual Convention amp Exhibition David L Lawrence Con-vention Center Pittsburgh PA USA May 19ndash22 2013

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank The University of Tulsa and NewfieldExploration Company for support of this research The firstauthor would like to thank Jeff Kane (Corporate Petro-physicist) for his scholastic discussions They also thankthe anonymous reviewers for the constructive comments onimproving our paper

References

[1] S Ingram I Paterniti B Rothkopf and C Stevenson ldquoGranitewash field study-buffalo wallow field Texas panhandlerdquo inProceedings of the SPE Eastern Regional Meeting SPE104546Canton Ohio USA October 2006

[2] M I Odigi ldquoDiagenesis and reservoir quality of Cretaceoussandstone of Nkporo Shale (Campanian) Southeastern Benuetrough Nigeriardquo Journal of Geology and Mining Research vol 3pp 265ndash280 2011

[3] F Ruiz and A Cheng ldquoA rock physics model for tight gas sandrdquoThe Leading Edge vol 29 no 12 pp 1484ndash1489 2010

[4] RDMindlin ldquoCompliance of elastic bodies in contactrdquo Journalof Applied Mechanics vol 16 pp 259ndash268 1949

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[6] J Dvorkin and A Nur ldquoElasticity of high-porosity sandstonestheory for twoNorth Sea data setsrdquoGeophysics vol 61 no 5 pp1363ndash1370 1996

[7] J G Berryman ldquoLong-wavelength propagation in compositeelastic media II Ellipsoidal Inclusionsrdquo The Journal of theAcoustical Society of America vol 68 no 6 pp 1820ndash1831 1980

[8] J P Castagna M L Batzle and R L Eastwood ldquoRelationshipsbetween compressional-wave and shear-wave velocities in clas-tic silicate rocksrdquo Geophysics vol 50 no 4 pp 571ndash581 1985

[9] G T Kuster and M N Toksoz ldquoVelocity and attenuation ofseismic waves in two-phase mediardquo Geophysics vol 39 no 5pp 587ndash618 1974

[10] G E Moore ldquoPennsylvanian paleogeography of the south-ern Mid-Continentrdquo in Pennsylvanian Sandstones of the Mid-Continent N J Hyne Ed pp 2ndash12 Tulsa Geological Society(TGS) Tulsa Okla USA 1979

[11] Core Lab ldquoGraniteWash geological Petrophysical and comple-tions study for improved formation evaluation and optimal wellcompletionrdquo Proprietary industry study 2005

[12] K S Johnson and B J Cardott ldquoGeologic framework andhydrocarbon source rocks of Oklahomardquo in Proceedings of theSource Rocks in the Southern Midcontinent K S JohnsonB andJ Cardott Eds pp 21ndash37 OGS Circular 1992

[13] N R Gelphman ldquoWest sentinel oil field Washita CountyOklahoma sedimentology of the ldquoGraniteWashrdquo and structuralgeologyrdquoOklahoma City Geological Society Shale Shaker vol 10pp 234ndash247 1960

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

10 International Journal of Geophysics

12366

12368

12370

1237247 48

Vp (kms) Vs (kms)

Dep

th (f

eet)

12366

12368

12370

12372

Dep

th (f

eet)

VpVp-2DKTVp-3DKT

26 27

VsVs-2DKTVs-3DKT

Figure 11 A comparison between the predicted and measured P- and S-wave velocities by employing 2DKT and 3DKTmodels using criticalaspect ratio (06) of well number A

both models Significantly velocities predicted by the 3DKTmodel agreewell with thosemeasured compared to the 2DKTmodel (Figure 11)

It is due to the fact that the model adjusts pore aspectratio or the proportion of various 3D pores and effectivelymodels the geometries of various pore shapes which is thedominant factor influencing the velocities of the elastic-wave propagating in the clastic reservoirs [28] Due to thesubstantial amount of clay (15) contents that may containmicrocracks the dry rock moduli estimation by the Kuster-Toksoz (2DKT) model (assuming ellipsoidal) could not beappropriate The addition of small volumes of cracks to therock matrix significantly weakens the rock The predicted P-and S-wave velocities from 3DKT are more consistent withthe measured ones although with some deviation Thus weconsider aspect ratio 06 as the critical value of the pore aspectratio which is the turning point of the increase trend ofpredicted velocities for Marmaton Wash

Thus variations in the elastic moduli are more influencedby pore shape geometry than porosity lower than 10and have a relatively small impact on 119881P119881S ratios Theintegrated application of the 2DKT and 3DKTmodels helpedus properly determine the pore shape geometry of GraniteWash reservoir to estimate site specific 119881P-119881S relationship

54 119881119875-119881119878Relationship of Granite Wash The 119881P-119881S relation

is normally used for 119881S prediction where only P-wave data isavailable and is also very important forAVOanalysis and porefluids prediction For S-wave velocity estimation regard-less of the approach taken it is important to understand

the underlying lithology The rock composition providesfirst order control on dry frame 119881P119881S ratio especially forconsolidated rocks [29] We take advantage of rock physics(effective medium) modeling and derive the 119881P-119881S relation-ship and compare it with the widely used empirical rela-tionship ldquo119881P = 116 lowast 119881S + 136rdquo presented by Castagnaet al [8] for clastic rocks We employ Berrymanrsquos [7] (3DKT)model considering rock composition porosity and poreshape After calibration of the predicted velocities with coredata the 119881P-119881S relationship is generated through regressionanalysis of least-square linear fit (1198772 = 090) as a function of119881P (Kms) (Figure 12)

The derived 119881P-119881S relation by 3DKT for Granite Washprovides closer approximation from 2DKT and global empir-ical relation by Castagna et al [8] Figure 13 shows that theS-wave velocity derived from Castagnarsquos mudrock is overpredicted with known 119881P when comparatively analyzed to2DKT and 3DKT models predictions However there areapparent deviations between the measured and predicteddata by the locally calibrated empirical relation and theeffectivemediummodelsWe actually do not advocate the useof global empirical relations for S-wave velocity predictionfor the Granite Wash but emphasize on site specific builtrelation based on the data of the corresponding study areausing effective medium models Next we employed the bestmodel (3DKT model) to discuss the fluid substitution effectson P- and S-wave velocities Marmaton Wash

55 Fluid Substitution in Granite Wash For practical fluidsubstitution results in tight sands using Gassmannrsquos [21]

International Journal of Geophysics 11

35

30

25

20

1530 35 40 45 50 55 60

0000012500250003750050000625007500087501000

R2 = 90

Vs

(km

s)

Vp (kms)

Vs = 0599 lowast Vp minus 019102

0 4 5 50 55

0000012500250037505000625075087100000

R2 = 90

s = 0599 lowast VpVV minus 019102

GR

Figure 12 Compressional and shearwave velocities ofGraniteWashformation from sonic logs data color coded with gamma ray (GR)from two wells numbers A and B and calibrated with core [11]measurements (Maroon data)

12340

12360

12380

12400

12420

12440

12460

20 22 24 26 28 30 32 34

003500

01556

02763

03969

05175

06381

07588

08794

1000

Dep

th (f

eet)

S-wave velocity (kms)

Vs log

Vs-2DKTVs-3DKT

Vs-RVs-C

Vsh

Figure 13 Comparison of S-wave velocity predictions (color codedwith 119881sh) of well number A with effective medium (2DKT and3DKT) models locally generated relation (119881

119904-119877) and Castagna

global empirical relationship (119881119904-119862)

equations necessitates an appropriate technique which real-istically stiffens low porosity rock by fluctuating the mineralpore fluid or dry rock moduli Application of Gassmannrsquosequation is dependent upon reasonable estimation of thebulk modulus of the dry rock frame (119870

119889) Dry frame rock

bulk (119870119889) modulus can be derived either from (1) laboratory

measurement on core data (2) application of empirical rela-tionships or effective medium theory [30ndash33] or (3) directcalculation from log data by rewriting Gassmannrsquos equation(4) for frame rock bulk (119870

119889) [34] which often generates

negative (a physical impossibility) or unrealistically largevalues in low porosity reservoirs [35] However of the variousalternatives we take essentially a pragmatic approach tocondition the dry frame rockmodel used in fluid substitution

For reliable fluid substitution model in clastic reservoir weconditioned the calculation of dry frame (119870

119889) rock data

implicit in Gassmannrsquos model and apply Biot [36] relationused by Wang et al [37] as follows

119870119889= 119870119898

(1 minus 120573) (7)

where 120573 is sometimes called the Biot coefficient and it is oftencalculated as a function of porosity and the form used in thepaper is derived by Krief et al [38]

(1 minus 120573) = [1 minus 120601]119898(120601)

119898 (120601) =3

(1 minus 120601)

(8)

where 120601 is the formation porosityConsequently the saturated bulk modulus of the rock

(119870sat) for the in situ conditions is first calculated using (1)Equation (7) is combined with Gassmannrsquos equations (4)to generate a pseudo dry frame rock bulk (119870

119889) modulus

suitable for fluid substitution The input parameters for fluidsubstitutions are generally roughly estimated and thus resultin substantial errorsTherefore densities bulkmoduli veloc-ities and viscosities of various pore fluids (brine and gas)are used both from core measurements and estimation usingBatzle and Wang [39] relationships The homogeneous fluid(119870fl) uniformly distributed throughout the pore space hasbeen computed by harmonic average (Reuss) using data fromthewell logs assuming twofluids (water and gas) phases in thereservoir The fluids (119870fl 120588fl) properties occupying the porespace and the bulk modulus of the mineral matrix (119870matrix)are determined by formulations given by Smith et al [35] Tocalculate the bulk modulus of the mineral matrix (119870matrix)the information on the composition (mineral abundance) ofthe rock is used from core study data [11] Mineral matrix(119870matrix) is then calculated via application of Voigt-Reuss-Hill (VRH) averaging the mineral constituents The elasticconstants and densities of pore fluid properties obtained fromcoremeasurements are given in Table 2 Figure 14 an exampleof composite log example from Granite Wash (Marmaton)section demonstrates the significant fluid substitution differ-ences between the wet and the gas effect on the seismic P-wave (119881P) velocity log for well number A The observed fluideffects reach amaximum in the cleanest sandwith the highest(13) porosity and low density response Fluid substitutioneffects correspond more closely with expectations based onraw logs increasing our confidence in the results of AVOmodeling and seismic inversion carried out in phase II

56 Cross-Plotting Attribute Analysis Cross-plotting of elas-tic rock properties from well logs enables the simultane-ous and meaningful evaluation of two attributes that helpsin making straightforward interpretation of hydrocarbonsaturated sands in a particular reservoir For comparisonwe cross-plotted both conventional (P- and S-impedances)and unconventional (lambda mu and rho) attributes in thederived elastic parameter cross-plot space to isolate potentialanomalous zones from a background relationship of well

12 International Journal of Geophysics

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Vsh

(a)

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Sw

(b)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

00 01

Porosity

(c)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

24 26

Density

(d)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

42 48

Vp

GasBrine

(e)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

025 030

PR

(f)

Figure 14 A composite log example fromGraniteWash (Marmaton) section Plots from left to right show logs of (a)119881sh (b)Water saturation(119878119908) (c) porosity (d) density (e) compressional velocity (119881P) and (f) Poisson ratio (PR) Fluid substitution is carried out for brine (blue) and

gas (red) scenarios using dry rock data derived from logs

number A (Figure 15) Figure 15(a) shows where a conven-tional impedance IP versus IS analysis has limitations inclearly discriminating between all the various lithologies andgas sand zone The P- and S-impedances log curves (whichshare both rigidity and density) tend to track each other withdepth and never crossover only S-impedance contrast in theMarmaton sand (12400

1015840

) is clearly less distinguishable thanthe comparable P-impedance contrast By contrast the 120582120588

and 120583120588 attributes have similar value ranges that do cross over

with 120582120588 lt 120583120588 showing a clear wide separation for a hydro-carbon zone while tracking together to clearly indicate thethin tight shale breaks (Figure 15(b)) In Figure 15(c) onlythe gas zone is discernable from background cut-off on the119881P119881S curve which varies between 16 and 20 Figures 15(d)and 15(e) demonstrate the significant advantage of the 120582120588120583120588 cross-plot in isolating both lithologic properties (such assand shale and clastic facies) as well as the gas zone com-pared to IP and IS cross-plot P- and S-impedance data points

International Journal of Geophysics 13

12340

12360

12380

12400

12420

12440

12460

4 6 8 10 12 14

IPIS

Dep

th (f

eet)

Impedance

(a)

12340

12360

12380

12400

12420

12440

12460

10 20 30 40 50 60 70 80

Tight shale

Gas zone

Lambda-rhoMu-rho

Shale

Dep

th (f

eet)

Lambda-rho Mu-rho

(b)

12340

12360

12380

12400

12420

12440

12460

15 16 17 18 19 20 21

Dep

th (f

eet)

VpVs

(c)

80

75

70

65

60

55

50

458 9 10 11 12 13 14

P-impedance

Gas zone

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

S-im

peda

nce

(d)

60

55

50

45

40

35

30

25

0 10 20 30 40 50 60 70

Gas zone

Lambda-rho

Mu-

rho

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

(e)

25

20

15

10

05

00minus15 minus10 minus5 0 5 10 15 20

Shale

Sst line

Wet sand

Gas zone

Lambda-Mu (difference)

Lam

bda

Mu

(rat

io)

003500

01556

02763

03969

05175

06381

07588

08794

1000

Shale

Sst line

Wet sand

Gas zone

Vsh

(f)

Figure 15 Well log curves and cross-plots from Granite Wash (Marmaton sections 12325ndash12470) of well number A (a) P- and S-impedanceslog curves (b) Lambda-rho and Mu-rho log versus depth (c) 119881

119901119881119904ratio versus depth (d) P-impedance versus S-impedance cross-plot color

coded by shale volume (119881sh) (e) Lambda-rho versus Mu-rho cross-plot color coded by shale volume (119881sh) and (f) Lambda-Mu (difference)versus Lambda-rhoMu-rho (ratio) cross-plot color coded by shale volume (119881sh)

14 International Journal of Geophysics

show a close linear relationship with shale having lowestvalues while in comparison the lowest 120582120588 (incompressibilitytimes density) value has the best gas sand values along with 120583120588

(rigidity times density) values higher than shalesThe other morecompetent pure lithologies (silts cemented sands) could beobserved in the opposite upper right quadrant relative to theshales

Following the cross-plots analysis for the same log dataset Figure 15(f) shows an interesting interpretation templatefor the 120582120583 ratio versus the 120582 minus 120583 (difference) cross-plotThe anomalous (gas) zone is marked in a lower left red boxbeyond the (06 minus5) point separating it from the wet sandswhich tend to align in the upper right zone The variationof the slope beyond the origin (1 0) with increasing gassaturation (blue to green dotted lines) reveals an unexpectedeffect of the gas on the rigidity of the sand This is due tothe effective porosity and lower density gas effect on rigidityleading to a slight steeping of the ldquosandstone linerdquo slopeThus in this case unconventional attributes plots provide aclear separation of the anomalous hydrocarbon zones and thelithologic properties of the sand and shale facies which havebeen color coded with shale volume

6 Conclusion

With the above discussion and limitations the followingconclusions can be derived Rock physics diagnostic and thinsections analysis indicate the variations in elastic propertiesof Granite Wash which can be explained by diagenetic phaseof quartz (or silica) cementation CementedWashes have rel-atively high elastic moduli and can be modeled primarily bycontact-cement model of scheme 2 Velocity prediction fromeffective medium models shows that pore shape geometrysignificantly influences the P- and S-wave velocities in ourreservoirs as observed in the velocity data evaluated Core andpetrographic observations also confirm cementations withcomplex pore geometry containing variable concentrations ofhigh and low pores aspect ratio Hence Berrymanrsquos (3DKT)model is proven to be the best one to discuss the influence ofpore shape on seismic velocities and used for the generationof locally calibrated empirical (119881P-119881S) relationship that bestdescribes our data set Finally we applied fluid substitution onlog data for calculating reservoir elastic properties with dif-ferent pore fluid (brine and gas) for an improved petrophys-ical discrimination of rock properties using unconventionalLame parameters 120582 (pure incompressibility) and 120583 (rigidity)attributes over conventional 119881P 119881S attributes analysis Theresearch outcomes are intended to be rigorously used as afoundation for ongoing reservoir characterization study inphase II

Disclosure

This paper is developed from an oral and poster presentationat American Association of Petroleum Geologists (AAPG)Annual Convention amp Exhibition David L Lawrence Con-vention Center Pittsburgh PA USA May 19ndash22 2013

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank The University of Tulsa and NewfieldExploration Company for support of this research The firstauthor would like to thank Jeff Kane (Corporate Petro-physicist) for his scholastic discussions They also thankthe anonymous reviewers for the constructive comments onimproving our paper

References

[1] S Ingram I Paterniti B Rothkopf and C Stevenson ldquoGranitewash field study-buffalo wallow field Texas panhandlerdquo inProceedings of the SPE Eastern Regional Meeting SPE104546Canton Ohio USA October 2006

[2] M I Odigi ldquoDiagenesis and reservoir quality of Cretaceoussandstone of Nkporo Shale (Campanian) Southeastern Benuetrough Nigeriardquo Journal of Geology and Mining Research vol 3pp 265ndash280 2011

[3] F Ruiz and A Cheng ldquoA rock physics model for tight gas sandrdquoThe Leading Edge vol 29 no 12 pp 1484ndash1489 2010

[4] RDMindlin ldquoCompliance of elastic bodies in contactrdquo Journalof Applied Mechanics vol 16 pp 259ndash268 1949

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[6] J Dvorkin and A Nur ldquoElasticity of high-porosity sandstonestheory for twoNorth Sea data setsrdquoGeophysics vol 61 no 5 pp1363ndash1370 1996

[7] J G Berryman ldquoLong-wavelength propagation in compositeelastic media II Ellipsoidal Inclusionsrdquo The Journal of theAcoustical Society of America vol 68 no 6 pp 1820ndash1831 1980

[8] J P Castagna M L Batzle and R L Eastwood ldquoRelationshipsbetween compressional-wave and shear-wave velocities in clas-tic silicate rocksrdquo Geophysics vol 50 no 4 pp 571ndash581 1985

[9] G T Kuster and M N Toksoz ldquoVelocity and attenuation ofseismic waves in two-phase mediardquo Geophysics vol 39 no 5pp 587ndash618 1974

[10] G E Moore ldquoPennsylvanian paleogeography of the south-ern Mid-Continentrdquo in Pennsylvanian Sandstones of the Mid-Continent N J Hyne Ed pp 2ndash12 Tulsa Geological Society(TGS) Tulsa Okla USA 1979

[11] Core Lab ldquoGraniteWash geological Petrophysical and comple-tions study for improved formation evaluation and optimal wellcompletionrdquo Proprietary industry study 2005

[12] K S Johnson and B J Cardott ldquoGeologic framework andhydrocarbon source rocks of Oklahomardquo in Proceedings of theSource Rocks in the Southern Midcontinent K S JohnsonB andJ Cardott Eds pp 21ndash37 OGS Circular 1992

[13] N R Gelphman ldquoWest sentinel oil field Washita CountyOklahoma sedimentology of the ldquoGraniteWashrdquo and structuralgeologyrdquoOklahoma City Geological Society Shale Shaker vol 10pp 234ndash247 1960

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

International Journal of Geophysics 11

35

30

25

20

1530 35 40 45 50 55 60

0000012500250003750050000625007500087501000

R2 = 90

Vs

(km

s)

Vp (kms)

Vs = 0599 lowast Vp minus 019102

0 4 5 50 55

0000012500250037505000625075087100000

R2 = 90

s = 0599 lowast VpVV minus 019102

GR

Figure 12 Compressional and shearwave velocities ofGraniteWashformation from sonic logs data color coded with gamma ray (GR)from two wells numbers A and B and calibrated with core [11]measurements (Maroon data)

12340

12360

12380

12400

12420

12440

12460

20 22 24 26 28 30 32 34

003500

01556

02763

03969

05175

06381

07588

08794

1000

Dep

th (f

eet)

S-wave velocity (kms)

Vs log

Vs-2DKTVs-3DKT

Vs-RVs-C

Vsh

Figure 13 Comparison of S-wave velocity predictions (color codedwith 119881sh) of well number A with effective medium (2DKT and3DKT) models locally generated relation (119881

119904-119877) and Castagna

global empirical relationship (119881119904-119862)

equations necessitates an appropriate technique which real-istically stiffens low porosity rock by fluctuating the mineralpore fluid or dry rock moduli Application of Gassmannrsquosequation is dependent upon reasonable estimation of thebulk modulus of the dry rock frame (119870

119889) Dry frame rock

bulk (119870119889) modulus can be derived either from (1) laboratory

measurement on core data (2) application of empirical rela-tionships or effective medium theory [30ndash33] or (3) directcalculation from log data by rewriting Gassmannrsquos equation(4) for frame rock bulk (119870

119889) [34] which often generates

negative (a physical impossibility) or unrealistically largevalues in low porosity reservoirs [35] However of the variousalternatives we take essentially a pragmatic approach tocondition the dry frame rockmodel used in fluid substitution

For reliable fluid substitution model in clastic reservoir weconditioned the calculation of dry frame (119870

119889) rock data

implicit in Gassmannrsquos model and apply Biot [36] relationused by Wang et al [37] as follows

119870119889= 119870119898

(1 minus 120573) (7)

where 120573 is sometimes called the Biot coefficient and it is oftencalculated as a function of porosity and the form used in thepaper is derived by Krief et al [38]

(1 minus 120573) = [1 minus 120601]119898(120601)

119898 (120601) =3

(1 minus 120601)

(8)

where 120601 is the formation porosityConsequently the saturated bulk modulus of the rock

(119870sat) for the in situ conditions is first calculated using (1)Equation (7) is combined with Gassmannrsquos equations (4)to generate a pseudo dry frame rock bulk (119870

119889) modulus

suitable for fluid substitution The input parameters for fluidsubstitutions are generally roughly estimated and thus resultin substantial errorsTherefore densities bulkmoduli veloc-ities and viscosities of various pore fluids (brine and gas)are used both from core measurements and estimation usingBatzle and Wang [39] relationships The homogeneous fluid(119870fl) uniformly distributed throughout the pore space hasbeen computed by harmonic average (Reuss) using data fromthewell logs assuming twofluids (water and gas) phases in thereservoir The fluids (119870fl 120588fl) properties occupying the porespace and the bulk modulus of the mineral matrix (119870matrix)are determined by formulations given by Smith et al [35] Tocalculate the bulk modulus of the mineral matrix (119870matrix)the information on the composition (mineral abundance) ofthe rock is used from core study data [11] Mineral matrix(119870matrix) is then calculated via application of Voigt-Reuss-Hill (VRH) averaging the mineral constituents The elasticconstants and densities of pore fluid properties obtained fromcoremeasurements are given in Table 2 Figure 14 an exampleof composite log example from Granite Wash (Marmaton)section demonstrates the significant fluid substitution differ-ences between the wet and the gas effect on the seismic P-wave (119881P) velocity log for well number A The observed fluideffects reach amaximum in the cleanest sandwith the highest(13) porosity and low density response Fluid substitutioneffects correspond more closely with expectations based onraw logs increasing our confidence in the results of AVOmodeling and seismic inversion carried out in phase II

56 Cross-Plotting Attribute Analysis Cross-plotting of elas-tic rock properties from well logs enables the simultane-ous and meaningful evaluation of two attributes that helpsin making straightforward interpretation of hydrocarbonsaturated sands in a particular reservoir For comparisonwe cross-plotted both conventional (P- and S-impedances)and unconventional (lambda mu and rho) attributes in thederived elastic parameter cross-plot space to isolate potentialanomalous zones from a background relationship of well

12 International Journal of Geophysics

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Vsh

(a)

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Sw

(b)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

00 01

Porosity

(c)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

24 26

Density

(d)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

42 48

Vp

GasBrine

(e)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

025 030

PR

(f)

Figure 14 A composite log example fromGraniteWash (Marmaton) section Plots from left to right show logs of (a)119881sh (b)Water saturation(119878119908) (c) porosity (d) density (e) compressional velocity (119881P) and (f) Poisson ratio (PR) Fluid substitution is carried out for brine (blue) and

gas (red) scenarios using dry rock data derived from logs

number A (Figure 15) Figure 15(a) shows where a conven-tional impedance IP versus IS analysis has limitations inclearly discriminating between all the various lithologies andgas sand zone The P- and S-impedances log curves (whichshare both rigidity and density) tend to track each other withdepth and never crossover only S-impedance contrast in theMarmaton sand (12400

1015840

) is clearly less distinguishable thanthe comparable P-impedance contrast By contrast the 120582120588

and 120583120588 attributes have similar value ranges that do cross over

with 120582120588 lt 120583120588 showing a clear wide separation for a hydro-carbon zone while tracking together to clearly indicate thethin tight shale breaks (Figure 15(b)) In Figure 15(c) onlythe gas zone is discernable from background cut-off on the119881P119881S curve which varies between 16 and 20 Figures 15(d)and 15(e) demonstrate the significant advantage of the 120582120588120583120588 cross-plot in isolating both lithologic properties (such assand shale and clastic facies) as well as the gas zone com-pared to IP and IS cross-plot P- and S-impedance data points

International Journal of Geophysics 13

12340

12360

12380

12400

12420

12440

12460

4 6 8 10 12 14

IPIS

Dep

th (f

eet)

Impedance

(a)

12340

12360

12380

12400

12420

12440

12460

10 20 30 40 50 60 70 80

Tight shale

Gas zone

Lambda-rhoMu-rho

Shale

Dep

th (f

eet)

Lambda-rho Mu-rho

(b)

12340

12360

12380

12400

12420

12440

12460

15 16 17 18 19 20 21

Dep

th (f

eet)

VpVs

(c)

80

75

70

65

60

55

50

458 9 10 11 12 13 14

P-impedance

Gas zone

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

S-im

peda

nce

(d)

60

55

50

45

40

35

30

25

0 10 20 30 40 50 60 70

Gas zone

Lambda-rho

Mu-

rho

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

(e)

25

20

15

10

05

00minus15 minus10 minus5 0 5 10 15 20

Shale

Sst line

Wet sand

Gas zone

Lambda-Mu (difference)

Lam

bda

Mu

(rat

io)

003500

01556

02763

03969

05175

06381

07588

08794

1000

Shale

Sst line

Wet sand

Gas zone

Vsh

(f)

Figure 15 Well log curves and cross-plots from Granite Wash (Marmaton sections 12325ndash12470) of well number A (a) P- and S-impedanceslog curves (b) Lambda-rho and Mu-rho log versus depth (c) 119881

119901119881119904ratio versus depth (d) P-impedance versus S-impedance cross-plot color

coded by shale volume (119881sh) (e) Lambda-rho versus Mu-rho cross-plot color coded by shale volume (119881sh) and (f) Lambda-Mu (difference)versus Lambda-rhoMu-rho (ratio) cross-plot color coded by shale volume (119881sh)

14 International Journal of Geophysics

show a close linear relationship with shale having lowestvalues while in comparison the lowest 120582120588 (incompressibilitytimes density) value has the best gas sand values along with 120583120588

(rigidity times density) values higher than shalesThe other morecompetent pure lithologies (silts cemented sands) could beobserved in the opposite upper right quadrant relative to theshales

Following the cross-plots analysis for the same log dataset Figure 15(f) shows an interesting interpretation templatefor the 120582120583 ratio versus the 120582 minus 120583 (difference) cross-plotThe anomalous (gas) zone is marked in a lower left red boxbeyond the (06 minus5) point separating it from the wet sandswhich tend to align in the upper right zone The variationof the slope beyond the origin (1 0) with increasing gassaturation (blue to green dotted lines) reveals an unexpectedeffect of the gas on the rigidity of the sand This is due tothe effective porosity and lower density gas effect on rigidityleading to a slight steeping of the ldquosandstone linerdquo slopeThus in this case unconventional attributes plots provide aclear separation of the anomalous hydrocarbon zones and thelithologic properties of the sand and shale facies which havebeen color coded with shale volume

6 Conclusion

With the above discussion and limitations the followingconclusions can be derived Rock physics diagnostic and thinsections analysis indicate the variations in elastic propertiesof Granite Wash which can be explained by diagenetic phaseof quartz (or silica) cementation CementedWashes have rel-atively high elastic moduli and can be modeled primarily bycontact-cement model of scheme 2 Velocity prediction fromeffective medium models shows that pore shape geometrysignificantly influences the P- and S-wave velocities in ourreservoirs as observed in the velocity data evaluated Core andpetrographic observations also confirm cementations withcomplex pore geometry containing variable concentrations ofhigh and low pores aspect ratio Hence Berrymanrsquos (3DKT)model is proven to be the best one to discuss the influence ofpore shape on seismic velocities and used for the generationof locally calibrated empirical (119881P-119881S) relationship that bestdescribes our data set Finally we applied fluid substitution onlog data for calculating reservoir elastic properties with dif-ferent pore fluid (brine and gas) for an improved petrophys-ical discrimination of rock properties using unconventionalLame parameters 120582 (pure incompressibility) and 120583 (rigidity)attributes over conventional 119881P 119881S attributes analysis Theresearch outcomes are intended to be rigorously used as afoundation for ongoing reservoir characterization study inphase II

Disclosure

This paper is developed from an oral and poster presentationat American Association of Petroleum Geologists (AAPG)Annual Convention amp Exhibition David L Lawrence Con-vention Center Pittsburgh PA USA May 19ndash22 2013

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank The University of Tulsa and NewfieldExploration Company for support of this research The firstauthor would like to thank Jeff Kane (Corporate Petro-physicist) for his scholastic discussions They also thankthe anonymous reviewers for the constructive comments onimproving our paper

References

[1] S Ingram I Paterniti B Rothkopf and C Stevenson ldquoGranitewash field study-buffalo wallow field Texas panhandlerdquo inProceedings of the SPE Eastern Regional Meeting SPE104546Canton Ohio USA October 2006

[2] M I Odigi ldquoDiagenesis and reservoir quality of Cretaceoussandstone of Nkporo Shale (Campanian) Southeastern Benuetrough Nigeriardquo Journal of Geology and Mining Research vol 3pp 265ndash280 2011

[3] F Ruiz and A Cheng ldquoA rock physics model for tight gas sandrdquoThe Leading Edge vol 29 no 12 pp 1484ndash1489 2010

[4] RDMindlin ldquoCompliance of elastic bodies in contactrdquo Journalof Applied Mechanics vol 16 pp 259ndash268 1949

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[6] J Dvorkin and A Nur ldquoElasticity of high-porosity sandstonestheory for twoNorth Sea data setsrdquoGeophysics vol 61 no 5 pp1363ndash1370 1996

[7] J G Berryman ldquoLong-wavelength propagation in compositeelastic media II Ellipsoidal Inclusionsrdquo The Journal of theAcoustical Society of America vol 68 no 6 pp 1820ndash1831 1980

[8] J P Castagna M L Batzle and R L Eastwood ldquoRelationshipsbetween compressional-wave and shear-wave velocities in clas-tic silicate rocksrdquo Geophysics vol 50 no 4 pp 571ndash581 1985

[9] G T Kuster and M N Toksoz ldquoVelocity and attenuation ofseismic waves in two-phase mediardquo Geophysics vol 39 no 5pp 587ndash618 1974

[10] G E Moore ldquoPennsylvanian paleogeography of the south-ern Mid-Continentrdquo in Pennsylvanian Sandstones of the Mid-Continent N J Hyne Ed pp 2ndash12 Tulsa Geological Society(TGS) Tulsa Okla USA 1979

[11] Core Lab ldquoGraniteWash geological Petrophysical and comple-tions study for improved formation evaluation and optimal wellcompletionrdquo Proprietary industry study 2005

[12] K S Johnson and B J Cardott ldquoGeologic framework andhydrocarbon source rocks of Oklahomardquo in Proceedings of theSource Rocks in the Southern Midcontinent K S JohnsonB andJ Cardott Eds pp 21ndash37 OGS Circular 1992

[13] N R Gelphman ldquoWest sentinel oil field Washita CountyOklahoma sedimentology of the ldquoGraniteWashrdquo and structuralgeologyrdquoOklahoma City Geological Society Shale Shaker vol 10pp 234ndash247 1960

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

12 International Journal of Geophysics

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Vsh

(a)

12340

12360

12380

12400

12420

12440

12460

00 05

Dep

th (f

eet)

Sw

(b)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

00 01

Porosity

(c)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

24 26

Density

(d)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

42 48

Vp

GasBrine

(e)

12340

12360

12380

12400

12420

12440

12460

Dep

th (f

eet)

025 030

PR

(f)

Figure 14 A composite log example fromGraniteWash (Marmaton) section Plots from left to right show logs of (a)119881sh (b)Water saturation(119878119908) (c) porosity (d) density (e) compressional velocity (119881P) and (f) Poisson ratio (PR) Fluid substitution is carried out for brine (blue) and

gas (red) scenarios using dry rock data derived from logs

number A (Figure 15) Figure 15(a) shows where a conven-tional impedance IP versus IS analysis has limitations inclearly discriminating between all the various lithologies andgas sand zone The P- and S-impedances log curves (whichshare both rigidity and density) tend to track each other withdepth and never crossover only S-impedance contrast in theMarmaton sand (12400

1015840

) is clearly less distinguishable thanthe comparable P-impedance contrast By contrast the 120582120588

and 120583120588 attributes have similar value ranges that do cross over

with 120582120588 lt 120583120588 showing a clear wide separation for a hydro-carbon zone while tracking together to clearly indicate thethin tight shale breaks (Figure 15(b)) In Figure 15(c) onlythe gas zone is discernable from background cut-off on the119881P119881S curve which varies between 16 and 20 Figures 15(d)and 15(e) demonstrate the significant advantage of the 120582120588120583120588 cross-plot in isolating both lithologic properties (such assand shale and clastic facies) as well as the gas zone com-pared to IP and IS cross-plot P- and S-impedance data points

International Journal of Geophysics 13

12340

12360

12380

12400

12420

12440

12460

4 6 8 10 12 14

IPIS

Dep

th (f

eet)

Impedance

(a)

12340

12360

12380

12400

12420

12440

12460

10 20 30 40 50 60 70 80

Tight shale

Gas zone

Lambda-rhoMu-rho

Shale

Dep

th (f

eet)

Lambda-rho Mu-rho

(b)

12340

12360

12380

12400

12420

12440

12460

15 16 17 18 19 20 21

Dep

th (f

eet)

VpVs

(c)

80

75

70

65

60

55

50

458 9 10 11 12 13 14

P-impedance

Gas zone

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

S-im

peda

nce

(d)

60

55

50

45

40

35

30

25

0 10 20 30 40 50 60 70

Gas zone

Lambda-rho

Mu-

rho

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

(e)

25

20

15

10

05

00minus15 minus10 minus5 0 5 10 15 20

Shale

Sst line

Wet sand

Gas zone

Lambda-Mu (difference)

Lam

bda

Mu

(rat

io)

003500

01556

02763

03969

05175

06381

07588

08794

1000

Shale

Sst line

Wet sand

Gas zone

Vsh

(f)

Figure 15 Well log curves and cross-plots from Granite Wash (Marmaton sections 12325ndash12470) of well number A (a) P- and S-impedanceslog curves (b) Lambda-rho and Mu-rho log versus depth (c) 119881

119901119881119904ratio versus depth (d) P-impedance versus S-impedance cross-plot color

coded by shale volume (119881sh) (e) Lambda-rho versus Mu-rho cross-plot color coded by shale volume (119881sh) and (f) Lambda-Mu (difference)versus Lambda-rhoMu-rho (ratio) cross-plot color coded by shale volume (119881sh)

14 International Journal of Geophysics

show a close linear relationship with shale having lowestvalues while in comparison the lowest 120582120588 (incompressibilitytimes density) value has the best gas sand values along with 120583120588

(rigidity times density) values higher than shalesThe other morecompetent pure lithologies (silts cemented sands) could beobserved in the opposite upper right quadrant relative to theshales

Following the cross-plots analysis for the same log dataset Figure 15(f) shows an interesting interpretation templatefor the 120582120583 ratio versus the 120582 minus 120583 (difference) cross-plotThe anomalous (gas) zone is marked in a lower left red boxbeyond the (06 minus5) point separating it from the wet sandswhich tend to align in the upper right zone The variationof the slope beyond the origin (1 0) with increasing gassaturation (blue to green dotted lines) reveals an unexpectedeffect of the gas on the rigidity of the sand This is due tothe effective porosity and lower density gas effect on rigidityleading to a slight steeping of the ldquosandstone linerdquo slopeThus in this case unconventional attributes plots provide aclear separation of the anomalous hydrocarbon zones and thelithologic properties of the sand and shale facies which havebeen color coded with shale volume

6 Conclusion

With the above discussion and limitations the followingconclusions can be derived Rock physics diagnostic and thinsections analysis indicate the variations in elastic propertiesof Granite Wash which can be explained by diagenetic phaseof quartz (or silica) cementation CementedWashes have rel-atively high elastic moduli and can be modeled primarily bycontact-cement model of scheme 2 Velocity prediction fromeffective medium models shows that pore shape geometrysignificantly influences the P- and S-wave velocities in ourreservoirs as observed in the velocity data evaluated Core andpetrographic observations also confirm cementations withcomplex pore geometry containing variable concentrations ofhigh and low pores aspect ratio Hence Berrymanrsquos (3DKT)model is proven to be the best one to discuss the influence ofpore shape on seismic velocities and used for the generationof locally calibrated empirical (119881P-119881S) relationship that bestdescribes our data set Finally we applied fluid substitution onlog data for calculating reservoir elastic properties with dif-ferent pore fluid (brine and gas) for an improved petrophys-ical discrimination of rock properties using unconventionalLame parameters 120582 (pure incompressibility) and 120583 (rigidity)attributes over conventional 119881P 119881S attributes analysis Theresearch outcomes are intended to be rigorously used as afoundation for ongoing reservoir characterization study inphase II

Disclosure

This paper is developed from an oral and poster presentationat American Association of Petroleum Geologists (AAPG)Annual Convention amp Exhibition David L Lawrence Con-vention Center Pittsburgh PA USA May 19ndash22 2013

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank The University of Tulsa and NewfieldExploration Company for support of this research The firstauthor would like to thank Jeff Kane (Corporate Petro-physicist) for his scholastic discussions They also thankthe anonymous reviewers for the constructive comments onimproving our paper

References

[1] S Ingram I Paterniti B Rothkopf and C Stevenson ldquoGranitewash field study-buffalo wallow field Texas panhandlerdquo inProceedings of the SPE Eastern Regional Meeting SPE104546Canton Ohio USA October 2006

[2] M I Odigi ldquoDiagenesis and reservoir quality of Cretaceoussandstone of Nkporo Shale (Campanian) Southeastern Benuetrough Nigeriardquo Journal of Geology and Mining Research vol 3pp 265ndash280 2011

[3] F Ruiz and A Cheng ldquoA rock physics model for tight gas sandrdquoThe Leading Edge vol 29 no 12 pp 1484ndash1489 2010

[4] RDMindlin ldquoCompliance of elastic bodies in contactrdquo Journalof Applied Mechanics vol 16 pp 259ndash268 1949

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[6] J Dvorkin and A Nur ldquoElasticity of high-porosity sandstonestheory for twoNorth Sea data setsrdquoGeophysics vol 61 no 5 pp1363ndash1370 1996

[7] J G Berryman ldquoLong-wavelength propagation in compositeelastic media II Ellipsoidal Inclusionsrdquo The Journal of theAcoustical Society of America vol 68 no 6 pp 1820ndash1831 1980

[8] J P Castagna M L Batzle and R L Eastwood ldquoRelationshipsbetween compressional-wave and shear-wave velocities in clas-tic silicate rocksrdquo Geophysics vol 50 no 4 pp 571ndash581 1985

[9] G T Kuster and M N Toksoz ldquoVelocity and attenuation ofseismic waves in two-phase mediardquo Geophysics vol 39 no 5pp 587ndash618 1974

[10] G E Moore ldquoPennsylvanian paleogeography of the south-ern Mid-Continentrdquo in Pennsylvanian Sandstones of the Mid-Continent N J Hyne Ed pp 2ndash12 Tulsa Geological Society(TGS) Tulsa Okla USA 1979

[11] Core Lab ldquoGraniteWash geological Petrophysical and comple-tions study for improved formation evaluation and optimal wellcompletionrdquo Proprietary industry study 2005

[12] K S Johnson and B J Cardott ldquoGeologic framework andhydrocarbon source rocks of Oklahomardquo in Proceedings of theSource Rocks in the Southern Midcontinent K S JohnsonB andJ Cardott Eds pp 21ndash37 OGS Circular 1992

[13] N R Gelphman ldquoWest sentinel oil field Washita CountyOklahoma sedimentology of the ldquoGraniteWashrdquo and structuralgeologyrdquoOklahoma City Geological Society Shale Shaker vol 10pp 234ndash247 1960

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

International Journal of Geophysics 13

12340

12360

12380

12400

12420

12440

12460

4 6 8 10 12 14

IPIS

Dep

th (f

eet)

Impedance

(a)

12340

12360

12380

12400

12420

12440

12460

10 20 30 40 50 60 70 80

Tight shale

Gas zone

Lambda-rhoMu-rho

Shale

Dep

th (f

eet)

Lambda-rho Mu-rho

(b)

12340

12360

12380

12400

12420

12440

12460

15 16 17 18 19 20 21

Dep

th (f

eet)

VpVs

(c)

80

75

70

65

60

55

50

458 9 10 11 12 13 14

P-impedance

Gas zone

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

S-im

peda

nce

(d)

60

55

50

45

40

35

30

25

0 10 20 30 40 50 60 70

Gas zone

Lambda-rho

Mu-

rho

003500

01556

02763

03969

05175

06381

07588

08794

1000

Gas zone

Vsh

(e)

25

20

15

10

05

00minus15 minus10 minus5 0 5 10 15 20

Shale

Sst line

Wet sand

Gas zone

Lambda-Mu (difference)

Lam

bda

Mu

(rat

io)

003500

01556

02763

03969

05175

06381

07588

08794

1000

Shale

Sst line

Wet sand

Gas zone

Vsh

(f)

Figure 15 Well log curves and cross-plots from Granite Wash (Marmaton sections 12325ndash12470) of well number A (a) P- and S-impedanceslog curves (b) Lambda-rho and Mu-rho log versus depth (c) 119881

119901119881119904ratio versus depth (d) P-impedance versus S-impedance cross-plot color

coded by shale volume (119881sh) (e) Lambda-rho versus Mu-rho cross-plot color coded by shale volume (119881sh) and (f) Lambda-Mu (difference)versus Lambda-rhoMu-rho (ratio) cross-plot color coded by shale volume (119881sh)

14 International Journal of Geophysics

show a close linear relationship with shale having lowestvalues while in comparison the lowest 120582120588 (incompressibilitytimes density) value has the best gas sand values along with 120583120588

(rigidity times density) values higher than shalesThe other morecompetent pure lithologies (silts cemented sands) could beobserved in the opposite upper right quadrant relative to theshales

Following the cross-plots analysis for the same log dataset Figure 15(f) shows an interesting interpretation templatefor the 120582120583 ratio versus the 120582 minus 120583 (difference) cross-plotThe anomalous (gas) zone is marked in a lower left red boxbeyond the (06 minus5) point separating it from the wet sandswhich tend to align in the upper right zone The variationof the slope beyond the origin (1 0) with increasing gassaturation (blue to green dotted lines) reveals an unexpectedeffect of the gas on the rigidity of the sand This is due tothe effective porosity and lower density gas effect on rigidityleading to a slight steeping of the ldquosandstone linerdquo slopeThus in this case unconventional attributes plots provide aclear separation of the anomalous hydrocarbon zones and thelithologic properties of the sand and shale facies which havebeen color coded with shale volume

6 Conclusion

With the above discussion and limitations the followingconclusions can be derived Rock physics diagnostic and thinsections analysis indicate the variations in elastic propertiesof Granite Wash which can be explained by diagenetic phaseof quartz (or silica) cementation CementedWashes have rel-atively high elastic moduli and can be modeled primarily bycontact-cement model of scheme 2 Velocity prediction fromeffective medium models shows that pore shape geometrysignificantly influences the P- and S-wave velocities in ourreservoirs as observed in the velocity data evaluated Core andpetrographic observations also confirm cementations withcomplex pore geometry containing variable concentrations ofhigh and low pores aspect ratio Hence Berrymanrsquos (3DKT)model is proven to be the best one to discuss the influence ofpore shape on seismic velocities and used for the generationof locally calibrated empirical (119881P-119881S) relationship that bestdescribes our data set Finally we applied fluid substitution onlog data for calculating reservoir elastic properties with dif-ferent pore fluid (brine and gas) for an improved petrophys-ical discrimination of rock properties using unconventionalLame parameters 120582 (pure incompressibility) and 120583 (rigidity)attributes over conventional 119881P 119881S attributes analysis Theresearch outcomes are intended to be rigorously used as afoundation for ongoing reservoir characterization study inphase II

Disclosure

This paper is developed from an oral and poster presentationat American Association of Petroleum Geologists (AAPG)Annual Convention amp Exhibition David L Lawrence Con-vention Center Pittsburgh PA USA May 19ndash22 2013

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank The University of Tulsa and NewfieldExploration Company for support of this research The firstauthor would like to thank Jeff Kane (Corporate Petro-physicist) for his scholastic discussions They also thankthe anonymous reviewers for the constructive comments onimproving our paper

References

[1] S Ingram I Paterniti B Rothkopf and C Stevenson ldquoGranitewash field study-buffalo wallow field Texas panhandlerdquo inProceedings of the SPE Eastern Regional Meeting SPE104546Canton Ohio USA October 2006

[2] M I Odigi ldquoDiagenesis and reservoir quality of Cretaceoussandstone of Nkporo Shale (Campanian) Southeastern Benuetrough Nigeriardquo Journal of Geology and Mining Research vol 3pp 265ndash280 2011

[3] F Ruiz and A Cheng ldquoA rock physics model for tight gas sandrdquoThe Leading Edge vol 29 no 12 pp 1484ndash1489 2010

[4] RDMindlin ldquoCompliance of elastic bodies in contactrdquo Journalof Applied Mechanics vol 16 pp 259ndash268 1949

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[6] J Dvorkin and A Nur ldquoElasticity of high-porosity sandstonestheory for twoNorth Sea data setsrdquoGeophysics vol 61 no 5 pp1363ndash1370 1996

[7] J G Berryman ldquoLong-wavelength propagation in compositeelastic media II Ellipsoidal Inclusionsrdquo The Journal of theAcoustical Society of America vol 68 no 6 pp 1820ndash1831 1980

[8] J P Castagna M L Batzle and R L Eastwood ldquoRelationshipsbetween compressional-wave and shear-wave velocities in clas-tic silicate rocksrdquo Geophysics vol 50 no 4 pp 571ndash581 1985

[9] G T Kuster and M N Toksoz ldquoVelocity and attenuation ofseismic waves in two-phase mediardquo Geophysics vol 39 no 5pp 587ndash618 1974

[10] G E Moore ldquoPennsylvanian paleogeography of the south-ern Mid-Continentrdquo in Pennsylvanian Sandstones of the Mid-Continent N J Hyne Ed pp 2ndash12 Tulsa Geological Society(TGS) Tulsa Okla USA 1979

[11] Core Lab ldquoGraniteWash geological Petrophysical and comple-tions study for improved formation evaluation and optimal wellcompletionrdquo Proprietary industry study 2005

[12] K S Johnson and B J Cardott ldquoGeologic framework andhydrocarbon source rocks of Oklahomardquo in Proceedings of theSource Rocks in the Southern Midcontinent K S JohnsonB andJ Cardott Eds pp 21ndash37 OGS Circular 1992

[13] N R Gelphman ldquoWest sentinel oil field Washita CountyOklahoma sedimentology of the ldquoGraniteWashrdquo and structuralgeologyrdquoOklahoma City Geological Society Shale Shaker vol 10pp 234ndash247 1960

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

14 International Journal of Geophysics

show a close linear relationship with shale having lowestvalues while in comparison the lowest 120582120588 (incompressibilitytimes density) value has the best gas sand values along with 120583120588

(rigidity times density) values higher than shalesThe other morecompetent pure lithologies (silts cemented sands) could beobserved in the opposite upper right quadrant relative to theshales

Following the cross-plots analysis for the same log dataset Figure 15(f) shows an interesting interpretation templatefor the 120582120583 ratio versus the 120582 minus 120583 (difference) cross-plotThe anomalous (gas) zone is marked in a lower left red boxbeyond the (06 minus5) point separating it from the wet sandswhich tend to align in the upper right zone The variationof the slope beyond the origin (1 0) with increasing gassaturation (blue to green dotted lines) reveals an unexpectedeffect of the gas on the rigidity of the sand This is due tothe effective porosity and lower density gas effect on rigidityleading to a slight steeping of the ldquosandstone linerdquo slopeThus in this case unconventional attributes plots provide aclear separation of the anomalous hydrocarbon zones and thelithologic properties of the sand and shale facies which havebeen color coded with shale volume

6 Conclusion

With the above discussion and limitations the followingconclusions can be derived Rock physics diagnostic and thinsections analysis indicate the variations in elastic propertiesof Granite Wash which can be explained by diagenetic phaseof quartz (or silica) cementation CementedWashes have rel-atively high elastic moduli and can be modeled primarily bycontact-cement model of scheme 2 Velocity prediction fromeffective medium models shows that pore shape geometrysignificantly influences the P- and S-wave velocities in ourreservoirs as observed in the velocity data evaluated Core andpetrographic observations also confirm cementations withcomplex pore geometry containing variable concentrations ofhigh and low pores aspect ratio Hence Berrymanrsquos (3DKT)model is proven to be the best one to discuss the influence ofpore shape on seismic velocities and used for the generationof locally calibrated empirical (119881P-119881S) relationship that bestdescribes our data set Finally we applied fluid substitution onlog data for calculating reservoir elastic properties with dif-ferent pore fluid (brine and gas) for an improved petrophys-ical discrimination of rock properties using unconventionalLame parameters 120582 (pure incompressibility) and 120583 (rigidity)attributes over conventional 119881P 119881S attributes analysis Theresearch outcomes are intended to be rigorously used as afoundation for ongoing reservoir characterization study inphase II

Disclosure

This paper is developed from an oral and poster presentationat American Association of Petroleum Geologists (AAPG)Annual Convention amp Exhibition David L Lawrence Con-vention Center Pittsburgh PA USA May 19ndash22 2013

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank The University of Tulsa and NewfieldExploration Company for support of this research The firstauthor would like to thank Jeff Kane (Corporate Petro-physicist) for his scholastic discussions They also thankthe anonymous reviewers for the constructive comments onimproving our paper

References

[1] S Ingram I Paterniti B Rothkopf and C Stevenson ldquoGranitewash field study-buffalo wallow field Texas panhandlerdquo inProceedings of the SPE Eastern Regional Meeting SPE104546Canton Ohio USA October 2006

[2] M I Odigi ldquoDiagenesis and reservoir quality of Cretaceoussandstone of Nkporo Shale (Campanian) Southeastern Benuetrough Nigeriardquo Journal of Geology and Mining Research vol 3pp 265ndash280 2011

[3] F Ruiz and A Cheng ldquoA rock physics model for tight gas sandrdquoThe Leading Edge vol 29 no 12 pp 1484ndash1489 2010

[4] RDMindlin ldquoCompliance of elastic bodies in contactrdquo Journalof Applied Mechanics vol 16 pp 259ndash268 1949

[5] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[6] J Dvorkin and A Nur ldquoElasticity of high-porosity sandstonestheory for twoNorth Sea data setsrdquoGeophysics vol 61 no 5 pp1363ndash1370 1996

[7] J G Berryman ldquoLong-wavelength propagation in compositeelastic media II Ellipsoidal Inclusionsrdquo The Journal of theAcoustical Society of America vol 68 no 6 pp 1820ndash1831 1980

[8] J P Castagna M L Batzle and R L Eastwood ldquoRelationshipsbetween compressional-wave and shear-wave velocities in clas-tic silicate rocksrdquo Geophysics vol 50 no 4 pp 571ndash581 1985

[9] G T Kuster and M N Toksoz ldquoVelocity and attenuation ofseismic waves in two-phase mediardquo Geophysics vol 39 no 5pp 587ndash618 1974

[10] G E Moore ldquoPennsylvanian paleogeography of the south-ern Mid-Continentrdquo in Pennsylvanian Sandstones of the Mid-Continent N J Hyne Ed pp 2ndash12 Tulsa Geological Society(TGS) Tulsa Okla USA 1979

[11] Core Lab ldquoGraniteWash geological Petrophysical and comple-tions study for improved formation evaluation and optimal wellcompletionrdquo Proprietary industry study 2005

[12] K S Johnson and B J Cardott ldquoGeologic framework andhydrocarbon source rocks of Oklahomardquo in Proceedings of theSource Rocks in the Southern Midcontinent K S JohnsonB andJ Cardott Eds pp 21ndash37 OGS Circular 1992

[13] N R Gelphman ldquoWest sentinel oil field Washita CountyOklahoma sedimentology of the ldquoGraniteWashrdquo and structuralgeologyrdquoOklahoma City Geological Society Shale Shaker vol 10pp 234ndash247 1960

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

International Journal of Geophysics 15

[14] L R Brown Jr ldquoDeltaic sandstone facies of the midcontinentrdquoin Pennsylvanian Sandstones of the Midcontinent vol 1 pp 35ndash63 Tulsa Geological Society (TGS) Special Publication TulsaOkla USA 1979

[15] S P Dutton ldquoPennsylvanian fan-delta and carbonate deposi-tion Mobeetie Field Texas PanhandlerdquoThe American Associa-tion of Petroleum Geologists Bulletin vol 66 no 4 pp 389ndash4071982

[16] J W Neasham ldquoApplications of scanning electron microscopyto the characteristics of hydrocarbon-bearing rocksrdquo ScanningElectron Microscopy vol 1 pp 101ndash108 1977

[17] D Han and M L Batzle ldquoGassmannrsquos equation and fluid-sat-uration effects on seismic velocitiesrdquo Geophysics vol 69 no 2pp 398ndash405 2004

[18] G Mavko T Mukerji and J Dvorkin The Rock Physics Hand-book Cambridge University Press New York NY USA 2009

[19] J Dvorkin G Mavko and A Nur ldquoThe effect of cementationon the elastic properties of granular materialrdquo Mechanics ofMaterials vol 12 no 3-4 pp 207ndash217 1991

[20] J Dvorkin A Nur and H Yin ldquoEffective properties of cement-ed granular materialsrdquoMechanics of Materials vol 18 no 4 pp351ndash366 1994

[21] F Gassmann Uber die elastizitatporosermedien Veirteljahrs-schrift der NaturforschendenGesellschaft in Zurich vol 96 1951English translation httpsepwwwstanfordedusepberry-manPSgassmannpdf

[22] P Avseth T Mukerji and G Mavko Quantitative SeismicInterpretation Applying Rock Physics Tools to Reduce Interpreta-tion Risk Cambridge University Press Cambridge Mass USA2005

[23] S L Herron M M Herron and R A Plumb ldquoIdentificationof clay-supported and framework-supported domains fromgeochemical and geophysical well log datardquo inProceedings of theSPE Annual Technical Conference and Exhibition vol 24726 pp667ndash680 October 1992

[24] D Marion Acoustical mechanical and transport propertiesof sediments and granular materials [PhD thesis] StanfordUniversity 1990

[25] D Fairbairn ldquoSynthetic quartziterdquo American Mineralogist vol35 pp 735ndash748 1950

[26] D H Han A Nur and D Morgan ldquoEffects of porosity and claycontent onwave velocities in sandstonesrdquoGeophysics vol 51 no11 pp 2093ndash2107 1986

[27] F Ruiz and J Dvorkin ldquoPredicting elasticity in nonclastic rockswith a differential effective mediummodelrdquo Geophysics vol 75no 1 pp E41ndashE53 2010

[28] H Wang S Z Sun H Yang H Gao Y Xiao and H Hu ldquoTheinfluence of pore structure on P- amp S-wave velocities in complexcarbonate reservoirs with secondary storage spacerdquo PetroleumScience vol 8 no 4 pp 394ndash405 2011

[29] T M Smith ldquoPractical seismic petrophysics the effective use oflog data for seismic analysisrdquoThe Leading Edge vol 30 no 10pp 1128ndash1141 2011

[30] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[31] W Murphy and A Reischer ldquoModulus decomposition of com-pressional and shear velocities in sand bodiesrdquo Geophysics vol58 no 2 pp 227ndash239 1993

[32] J W Spencer Jr M E Cates and D D Thompson ldquoFramemoduli of unconsolidated sands and sandstonesrdquo Geophysicsvol 59 no 9 pp 1352ndash1361 1994

[33] Z Wang ldquoFundamentals of seismic rock physicsrdquo Geophysicsvol 66 no 2 pp 398ndash412 2001

[34] X Zhu and G A McMechan ldquoDirect estimation of the bulkmodulus of the frame in fluid saturated elastic medium by Biottheoryrdquo in Proceedings of the 60th Annual InternationalMeetingSociety of Exploration Geophysicist (SEGrsquo90) pp 787ndash790 1990

[35] T M Smith C H Sondergeld and C S Rai ldquoGassmann fluidsubstitutions a tutorialrdquoGeophysics vol 68 no 2 pp 430ndash4402003

[36] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoThe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956

[37] H Wang S Z Sun Y Li and X Li ldquoEvaluation and improve-ment of velocity-prediction models and fluid substitution forargilliferous clastic reservoirs rockrdquo in Proceedings of the Inter-national Exposition and Annual Meeting Society of ExplorationGeophysicists (SEG) Houston Tex USA 2009

[38] M Krief J Garat J Stellingwerff and J Ventre ldquoA petrophysicalinterpretation using the velocities of P and S waves (full-waveform sonic)rdquoThe Log Analyst vol 31 pp 355ndash369 1990

[39] M Batzle and Z Wang ldquoSeismic properties of pore fluidsrdquoGeophysics vol 57 no 11 pp 1396ndash1408 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in