Seismic scale saturation relations in turbidite reservoirs ... · all performance of a waterflood and potential encroach-ment on producing wells. In this context, there is a need

Geophysical Prospecting, 2008, 56, 693–714 doi:10.1111/j.1365-2478.2008.00712.x

Seismic scale saturation relations in turbidite reservoirs undergoingwaterflood

Colin MacBeth∗ and Karl StephenInstitute of Petroleum Engineering, Heriot-Watt University, Edinburgh EH14 4AS, UK

Received June 2007, revision accepted February 2008

ABSTRACTEstimates of the effective fluid modulus from seismic cannot be directly converted tothe true pore-volume weighted mean saturation Sw determined from fluid flow prin-ciples by using the saturation laws currently in use. One of the reasons is that seismicwaves sample the reservoir geology and production induced saturation heterogene-ity in a different way from the fluids. This mismatch prevents accurate quantitativeevaluation of saturation changes from 4D seismic analysis. To tackle this problem, areservoir-related saturation law is developed for a turbidite reservoir – this geologybeing chosen because the architecture for a single sand package can be modelled as astack of horizontal beds. An effective medium and perturbation theory are applied tothe determination of the seismic properties of this model. This calculation provides arelationship that connects the true saturation Sw to the effective fluid modulus fromseismic via statistical measures of the vertical spread of the porosity and saturationvariations in the reservoir. These statistics can be extracted from the simulation modeland if known, enable the new saturation law to deliver a significant improvement inaccuracy when estimating Sw compared to other well-known laws. The relationshipthat has been developed also captures the effect of inter-bedded shales and can there-fore be used to estimate true saturation in regions of the reservoir with moderate tolow net-to-gross, provided the fraction of the shale component is known. In practice,the final choice of saturation law depends upon the reservoir information available,the assumptions that can be tolerated and the accuracy required in any particularreservoir characterization study.

I N T R O D U C T I O N

Dynamic reservoir management requires accurate monitor-ing of saturation values as production proceeds. Thus, under-standing the lateral variation of saturation in the inter-wellvolumes is particularly important for evaluating the over-all performance of a waterflood and potential encroach-ment on producing wells. In this context, there is a needto reliably determine the position and nature of the water-front for the purposes of optimization of the oil displace-ment process and the associated well injectivity, to detect

∗E-mail: [email protected]

movement along preferential pathways and hence bypassedoil, or to assess overall sweep efficiency and reservoir connec-tivity. Ideally, an accurate estimation of the individual fluidsaturations and their gradients in each flow unit of the reser-voir would provide a means of differentiating situations suchas a breakthrough along a thin layer, gravity override or finger-ing and channelling, which might ultimately be detrimental toreservoir performance if left undetected. An emerging tool thatcan complement production logging tools and saturation logsfor this purpose is 4D seismic imaging. Here, there is particu-lar value to be gained from the aerial coverage offered by theseismic, albeit at a reduced vertical resolution and added non-uniqueness when compared to the more direct measurementsfrom well data. As an aid to this process, techniques have

C© 2008 European Association of Geoscientists & Engineers 693

694 C. MacBeth and K. Stephen

recently been developed to obtain saturations and pressuresfrom 4D seismic attributes that can be used to estimate satu-ration independently of the pressure change overprint: see forexample, Landrø (2001), Tura and Lumley (1999) and Ribeiroand MacBeth (2006).

Methods like those referenced above rely upon establish-ing the correct petroelastic model linking the fluid saturationsand pressures to the seismic attributes interpreted on the 4Dseismic data. This mathematical model is based on certain spe-cific assumptions about the physics of interaction between theelastic waves and the fluids and rocks in the reservoir. Onerecognized challenge in the use of seismic techniques that relysolely on this model is the way in which the seismic respondsto the reservoir’s fluid saturation. This is because the relation-ship that links the seismic properties to the individual fluidconstituents in the reservoir is known to be ill-defined andinconsistent with flow principles. Thus, it is known that dif-ferent sub-seismic scale heterogeneous fluid saturation condi-tions can give distinctly different effective seismic saturationlaws. Also, a significant problem is that the seismic estimateof saturation does not relate directly to the true pore-volumeweighted average saturation used in fluid flow calculations.Therefore, if the products from 4D seismic analysis are to beused in a precise quantitative manner for dynamic reservoircharacterization, the seismic saturation law must be revised togive a truer measure of reservoir saturation in the presence ofreservoir heterogeneity.

The work here addresses the topic described above by devel-oping a suitable equation for a specific type of heterogeneousreservoir. Firstly, an introduction to existing saturation laws isgiven. Next, these laws are compared with our proposed newlaw developed specifically for a turbidite reservoir. A reservoirsimulation is used to generate the data from which the correctsaturation and the effective fluid modulus are obtained forthese comparisons. The learned outcome from this research isthat the statistics of the internal geological architecture andsaturation variability provide an essential ingredient in ob-taining an accurate saturation law, information which has notbeen utilized to constrain this problem in the past.

H O W W E L L C A N W E C U R R E N T LYE S T I M AT E R E S E RV O I R S AT U R AT I O NS TAT E F R O M S E I S M I C ?

Changes in reservoir saturation affect the bulk modulus anddensity of the rock, which in turn influence the density (ρ), P-wave velocity (VP ) and S-wave velocity (VS ) and hence, impactthe overall seismic response. As it is assumed for the purposes

of this current work that neither the geological architecturenor the petrophysical components induce anisotropy or atten-uation in the seismic waves, the shear modulus μ is thereforeunaffected by saturation and we need only be concerned withthe density and bulk modulus changes. The effective densityat the seismic scale, ρeff , is given by:

ρeff = 〈ρdry〉 + 〈φ(Soρo + Swρw)〉= ρdry + φ((1 − Sw)ρo + Swρw) (1)

where the angular brackets and bars refer to a spatial averagetaken over an arbitrarily located elemental volume or voxel(�V) in the reservoir, typically 10 to 20 m in overall size andof similar vertical dimension to a sand package. The averagedquantities are therefore influenced by the sub-seismic scalereservoir properties. Here, So and Sw are the respective fluidsaturations for oil and water and ρo and ρw the correspond-ing fluid densities. ρdry is the mean density of the porous rockframe without the fluids, φ the corresponding mean effectiveporosity and Sw the desired pore-volume weighted saturation.At the seismic scale, the effective bulk modulus for a saturatedreservoir rock κ

effsat and the corresponding effective fluid modu-

lus are usually implicitly assumed to be related via the formulasuggested by Geertsma and Smit (1961):

κeffsat =

⟨κdry + γ 2

φ

κ f+ (γ−φ)

κm

⟩≈ κ

effdry + γ 2

φ

κefff

+ (γ−φ)κm

, (2)

where κeffdry is the effective bulk modulus of the rock without its

fluid constituents, κm the effective bulk modulus of the min-erals (in the grains, clays and cements) from which the solidframe is built and γ = (1 − κdry/κm). This equation capturesin a single relationship the detail of the low-frequency poroe-lastic interactions between the fluid and solid phases of therock upon excitation by seismic waves, the fundamental ba-sis of which originated from Gassmann (1951). It is typicallyapplied without question at the seismic scale for which theright-hand side of the equation is considered to be an equalityrather than an approximation. The single effective fluid bulkmodulus, κ

efff , in equation (2) gives the contribution of the

fluid mixture in the pore space, which is usually calculatedusing an algebraic combination of the individual fluid bulkmoduli for oil and water and their corresponding saturations.Our study here focuses on the specific uncertainty due to fluidand geological heterogeneity and its impact on the extractionof the desired saturation Sw from κ

efff . Thus, we do not con-

sider effects of the variability in κm and κdry introduced byclays and cements or other mineral components on the de-sired saturation result. Thus, the other averaged quantities in

C© 2008 European Association of Geoscientists & Engineers, Geophysical Prospecting, 56, 693–714

Seismic scale saturation relations in turbidite reservoirs 695

equation (2) are assumed to be relatively well understood andcalculable for the purposes of this current study.

Saturation heterogeneity and geological homogeneity

Improved calculation of κefff requires knowledge of the fluid

saturation distributions in the reservoir at scales below �V.Such sub-seismic saturation heterogeneity can exist prior toproduction but is more prevalent during dynamic fluid move-ment when employing improved oil recovery, or when drawingdown the reservoir during production. Figures 1 and 2 showsome possible reservoir scenarios that can lead to a sub-seismicsaturation distribution. This unevenly distributed fluid mix-ture is a problem, as it leads to a spatial variation in the fluidpressure induced by the incident seismic wave. Thus, predom-inantly oil-saturated rock laminae will experience differentlyinduced pressures from water-filled or gas-filled laminae. Thisis because the dominant sensitivity is from the fluids due totheir large relative compressibilities. Thus, the induced pres-sure field is less affected by variability in the rock porosity,pore-space compressibility and the mineral rock frame. Thepressure variations violate one of the key assumptions under-lying the theory of Gassmann (1951), which requires that thefluid in the pore space must be at a constant, stable pressure.

Figure 1 Saturation heterogeneity at the: (a) pore scale due to viscous-dominated (fast) water displacement of oil; (b) core-scale with residual oilbypassed in pores and groups of pores; (c) individual bed scale where trapping of residual oil in laminae is visible (after Pickup and Hern 2002).

This conflict can be resolved if the pore volume for the regionover which the average in equation (2) is taken is completelyconnected hydraulically and freely permits fluid pressure com-munication into and out of adjoining pore spaces. The pressuregradients can now be successfully eradicated during one cycleof the seismic wave. If the conditions are appropriate for this tooccur, then the fluids act in consort and the single effective fluidmodulus κ

efff is obtained by a harmonic average (Domenico

1974):

1

κefff

= 1 − Sw

κo+ Sw

κw. (3)

where the saturation is defined for the whole connected regionand may therefore be taken to be Sw. Equation (3) is identicalto the Wood (1949) equation that has been used in the pastto calculate the speed of sound in a mixture of fluids broughttogether in an open container which, due to the iso-pressureconstraint, is derived by volumetrically weighting the sum ofthe individual compressibilities. Although this law is clearlyideal and unlikely to be realized in practice, the resultant κ

efff

versus Sw relation is drawn as curve L in Fig. 3 for referencepurposes. Here, the saturations are defined between the usuallimits Sw = Swc and Sw = 1 − Sor , where Swc and Sor are theconnate water and residual oil saturations, respectively. The



Figure 2 A selection of possible sub-seismic saturation scenarios induced by reservoir production processes. (a) Layer-cake reservoirs with highlateral continuity and poor vertical sweep due to compartments separated by impermeable intercalations. This gives an uneven sweep and earlywater breakthrough – after Clark (1964); (b) interconnected sand bodies in which there are dead ends and gravity traps – after Weber (1999);(c) effect of rate of capillary displacement of oil from tight sand lenses for different production rates – an inappropriate balance of productionrate and capillary forces can create isolated zones of bypassed oil. (d) Unstable displacement of oil by a greater mobility water; (e) dissolved gasdrive reservoir, gas collecting up-structure – free gas pulled into well at high oil producing rates; (f) viscous fingering in essentially homogeneousrocks with little detectable permeability variation – a high mobility (less viscous) fluid displacing a less mobile fluid.



Figure 3 Estimates of the relationship between measured (seismic scale) fluid bulk modulus κefff and the true pore-volume weighted mean water

saturation Sw for a two-phase (oil-water) system under waterflood (increasing Sw). L and U – the reference curves given by equation (3) andthe non-physical end-member of equation (5). P1 – the saturation trajectory for displacement of oil in a smooth manner through homogeneousgeology as in equation (7); P1’ – saturation trajectory when bypassed oil is left behind as in equation (11). P2 – saturation law proposed in thispaper in equation (14) to take into account fluctuations in porosity and saturation state and their cross-correlation. It is drawn by assumingφ = 0.23, β = 0.35, σφ = 0.02, σ S = 0.13 and σφS = 0.02 in equation (A28).

figure is drawn with κw = 3 GPa and κo = 1.2 GPa, these be-ing extreme values drawn from a range of turbidite reservoirsknown to the authors and representing the stiffest brine andlightest oil.

If there is excellent pressure communication throughouteach volume �V of rock, then equation (3) remains valid de-spite the fluid constituents being distributed unevenly due toproduction (such as in the examples of Figs 1 and 2). However,pressure equilibrium is only established if the fluid pressure hassufficient time to propagate over �V during one seismic os-cillation. Generally, pressure equilibration cannot be expectedand instead the local distance over which equation (3) can bevalidly applied must be considered instead. This is defined byD, the distance travelled by the pressure waves in one seismicoscillation (Knight, Dvorkin and Nur 1998). The exact mag-nitude of this quantity is dependent on the fluid viscosities,fluid bulk moduli, the local permeability of the rock (hencethe pore throats, clays and cements) and the frequency of theseismic waves. For typical seismic frequencies, D lies inthe sub-bed region between 0.3 and 1 m (Kirstetter et al. 2006)such that �V � D3 and therefore equation (3) is only validover a volume smaller than the seismic scale voxel. In practice,the presence of shale barriers or baffles, or low-permeability

rock containing clays and cements (see the examples in Fig. 2)may also limit applicability to even smaller volumes. In conclu-sion, it is not possible to use equation (3) as a way of accuratelyestimating Sw and additional knowledge of how the small-scale saturation and geology heterogeneity distributes appearsnecessary.

Saturation heterogeneity and weak geological heterogeneity

In reservoir situations such as a layer-cake reservoir separatedby shale barriers or impermeable intercalations (Fig. 2a) orinter-connected sand bodies in which there are dead ends andgravity traps (Fig. 2b) it is appropriate to apply equation (3)separately to each connected region or layer of thickness ≤D

and then to use some appropriate effective medium theory todetermine the overall seismic properties for the volume �V.This can be calculated exactly, for example, by assuming theshear modulus μ to be invariant across the volume of inter-est. The Hill (1963) average can then be used to calculate theeffective saturated bulk modulus κ

effsat for the rock volume by

solving:

1

κeffsat + 4

3 μeff=

⟨1

κsat + 43 μ

⟩(4)



for κefff . Here, the individual rock laminae or thin beds have

bulk moduli κ(i)sat given by equations (2) and (3) and are

weighted according to their volumetric proportions. This par-ticular result is independent of the shape, size and distribu-tion of each discrete, isolated region of rock. In theory, theequation suggests that thin horizontal fluid streaks in lami-nated sands separated by a clay smear or shale (Fig. 2a) couldhave similar seismic signatures to discrete volumetric regions(Fig. 2c). This is valid provided the variations in μ are small.The solution of equation (4) for κ

efff can be given approxi-

mately by the arithmetic average (Domenico 1974):

κefff ≈

N∑i=1

wiκ f i , (5)

where κ fi is the effective fluid modulus for the ith reservoir sub-volume/layer and is given by equation (3) and wi is the volumefraction occupied by that layer. Equation (5) provides a way toevaluate the effective seismic properties of many thin (≤D inthickness) isolated beds with different values of connate wa-ter saturation and residual oil. Equation (5) is attractive as itpermits a direct comparison (arithmetic versus harmonic aver-ages) with the saturation model of equation (3). The approxi-mation in equation (5) always slightly overestimates the exactresult from equation (4) but is believed to work well when therock is stiff and well consolidated (Mavko and Mukerji 1998).Both equations (4) and (5) predict higher bulk moduli than inequation (3). This is because pressure equilibration betweencompartments is prevented, this effectively stiffening the over-all pore volume when seismic waves are incident. Again, it isused here only for reference and serves no other purpose. Atrajectory U is drawn in Fig. 3 defined by the highly artificial(mathematical) condition of completely isolated single phasefluid compartments (hence N = 2, w1 = 1 − Sw, w2 = Sw,κ f1 = κo and κ f2 = κw), assuming again that Sw represents Sw.Values for κw and κo used above to calculate L are again usedto calculate U.

Equation (5) can also be used to compute the generalizedseismic-scale saturation trajectory for oil displacement by wa-ter, defined by the evolution of the reservoir’s pore-volumeweighted average saturation Sw. Consider the elemental vol-ume �V to be located in the oil leg of a reservoir, which is thenswept by the passage of a waterfront from the injector – eitherlateral or basal sweep, ideally with a sharp oil-water contactand hence, piston-like displacement. Prior to production thereis connate water saturation Swc and oil in place So = 1 − Swc.Assuming firstly that the geology and saturation is homoge-nous in �V, then the pre-production condition κ ′

o is given

by:

1κ ′

o

= Swc

κw+ 1 − Swc

κo. (6)

If the waterfront progresses through the reservoir slowlyand efficiently enough for only minimal pore-scale residual oilSor to be left behind (see Fig. 1a), then saturation heterogeneityis kept to a minimum. After the displacement process, theeffective fluid bulk modulus κ ′

w is now given by:

1κ ′

w

= 1 − Sor

κw+ Sor

κo. (7)

At some intermediate time during the displacement pro-cess, the reservoir consists in two saturation states (swept andunswept) and evolves between the end points defined by equa-tions (6) and (7). The fraction FV of �V where the water hasbeen is defined by the two end-point saturations Sw = Swc (Ain Fig. 3) and Sw = 1 − Sorw (B in Fig. 3):

FV = Sw − Swc

1 − Sorw − Swc. (8)

where Sw grows with production time. In this definition, thewaterfront can be an arbitrary shape. By partitioning �V ac-cording to each saturated region and then summing accord-ing to equation (5), the effective fluid bulk modulus can beobtained. The final generalized trajectory for this case canthus be written (Zhang and Bentley 2000; Kirstetter et al.

2006):

κefff = (1 − FV)κ ′

o + FVκ ′w. (9)

This defines the linear trajectory P1 in Fig. 3 when plottedagainst Sw. Values of Swc = 0.25 and Sor = 0.15 are used hereto calculate P1.

As an alternate scenario, if oil displacement proceedsquickly and bypassed oil at scales larger than the pore scaleis left behind (Fig. 1b), the bypassed oil must be includedas an additional saturation state. The initial saturation isthe same but the final effective saturation must be revisedto Sw = (1 − α)(1 − Sor ) + αSwc(B’ in Fig. 3), where α isthe fraction of the volume that now contains the bypassedoil. The corresponding saturation trajectory P1’ should nowbe written as in equation (9) but with κ ′

w replaced by κ ′′w

where:

κ ′′w = ακ ′

o + (1 − α)κ ′w. (10)

To draw this trajectory in Fig. 3, α is arbitrarily chosen tobe 0.22 (for clarity of illustration only). Note that the volumefraction α is not generally known in practice and therefore



must be estimated. P1’ is also appropriate for oil trapped byseals or very low permeability (see Fig. 2).

In practice, bed and intra-bed scale fluctuations will pro-vide an additional contribution to the fluid heterogeneity in�V. Thus, for example, it is known that irreducible or connatewater saturation varies with rock properties and hence, litho-facies (Morrow and Melrose 1991). Indeed, Swc can typicallyvary from about 0.70 down to 0.05, depending on the natureof the reservoirs rocks. By contrast, different rock types havea variation of only a few per cent in the minimum Sor andit is approximately constant with permeability and rock type(Hamon 2003) – typically less than 0.40 (depending mainlyon wettability variation). Thus, the preproduction saturationis probably non-uniform, necessitating the use of equation (5)here too. Indeed, the exact start and end points and the waythe reservoir evolves over time depend on the initial saturationdistribution and the fine-scale geology in �V. Generalizationof the saturation law equations to the case of heterogeneousgeology is the topic of the next section, where a new approx-imate saturation law is developed. As our treatment is likelyto be reservoir-specific, we choose to focus only on turbiditereservoirs. It will be seen in the next section that for the caseof turbidite reservoirs, statistics derived from the simulationmodel can be used to provide a new saturation law that en-hances the accuracy of Sw estimated from the seismic κ

efff for

layered reservoirs.

E S T I M AT I O N O F R E S E RV O I R S AT U R AT I O NI N T H E P R E S E N C E O F G E O L O G I C A LH E T E R O G E N E I T Y

A specific working model for turbidite reservoirs

It is our belief that an understanding of geological hetero-geneity and the associated fluid distributions obtained fromreservoir simulation studies might provide the informationnecessary to accurately define seismic scale saturation esti-mates that can improve upon those offered by equations (6),(9) or (10). As such a treatment is likely to be reservoir-specificand may therefore be hard to generalize, in this particulararticle we choose to focus only on turbidite reservoirs. Thisreservoir type is selected because its geological characteristicsmake it more accessible to our analysis than most others. Itshould be emphasized however, that although individual chan-nel sand bodies (single genetic sedimentary units or architec-tural elements) do appear relatively simple in their description,turbidite reservoirs possess very complex inter-connectivitiesand fluid flow behaviour (Dromgoole et al. 2000; MacBeth,

Stephen and McInally 2005). The model of Stephen, Clarkand Gardiner (2001) is suitable for the majority of tur-bidite reservoirs, which are sand-rich and sheet-like in nature(Fig. 4a). Such reservoirs are usually the result of more dis-tal deposition in basin floor lobes but can include channelcomplexes with wide in-fill (see, for example, Leach et al.

1999 and Navarre et al. 2002). Sand bodies are composedof individual sand beds typically 20 cm to 1 m thick (bedscan be as thin as a few centimetres but only exceptionally asthick as 10s of metres – Hurst, Cronin and Hartley 2000).These are separated by vertical permeability barriers and baf-fles that arise from argillaceous beds, abandoned channel fills,mud-clast conglomerate layers, pebbly mudstones and diage-netic concretions. Thin argillaceous beds and shale barriersare deposited between turbidity currents but successive eventsmay scour them to produce amalgamated sand units. Turbiditereservoirs may therefore be visualized as horizontal layers ofsheet-like sandstone, separated vertically by discrete shales orsub-horizontal erosive surfaces.

Shale continuity and distribution are important for deter-mining vertical flow. A range of size of erosive holes exists,which in turn determines the extent of the vertical connectiv-ity. In the sand beds, porosity fluctuations are seldom morethan a few per cent (for example, 20 per cent ± 2 per cent(Chapter 20, Barwis, McPherson and Studlick 1990)), by con-trast however, permeability typically varies by much more.From the flow simulation perspective a useful working modelfor the internal architecture of each channel sand body is there-fore a series of thin, homogeneous and horizontal sand beds,isotropic to flow and sandwiched between shale eroded todiffering degrees (Fig. 4b). Thus, for the purposes of this par-ticular model, the contribution of variable bed thickness dueto say, variation in amalgamation ratio or coarsening or finingupwards of the sedimentary fabrics due to intra-bed hetero-geneity is assumed negligible. Saturation heterogeneity in thismodel relates only to the vertical direction, whilst horizon-tally within the channel sand body the geological propertiesare much more uniform. From the seismic perspective, an iden-tical working model to MacBeth et al. (2005) can be used, inwhich each seismic elemental volume �V represents a sandpackage thickness, with the fluid distribution in each individ-ual bed at the sub-package scale being equilibrated becausethe bed thickness is approximately D. This model is valid ir-respective of whether the channel sand beds are amalgamatedor not, although the shales will contribute to the average rockframe modulus and also influence the overall saturation dis-tribution.



Figure 4 (a) Photograph of sand-rich outcrop unit from Gres d’Annot, SE France. Internal fill of this large-scale sediment conduit is sheet-like.Lengths of discontinuous shales are evident between beds and amalgamation surfaces have also been mapped (courtesy of A. Gardiner, HWU).(b) Model used to represent different degrees of amalgamation in this work. Shales are in blue, sands in red. (c) Schematic of the impact of eachmodel on fluid flow between opposite sides of the model.

A new seismic-scale saturation law for turbidite reservoirs

We take each channel sand package to have a vertical thicknessroughly equal to a fifth of a seismic wavelength (≈25 m maxi-mum total thickness compared to a wavelength of 100 m) andit can thus be treated as a single homogeneous rock mass �V

with an equivalent set of elastic properties κeffsat, μeff , ρeff de-

fined by effective medium theory. The objective is to determinean accurate saturation law that relates the κ

efff obtained from

seismic using the seismic-scale Gassmann law in equation (2)for this equivalent rock mass, to the pore-volume weighted av-erage Sw used in the study of fluid flow. The starting point hereis the evaluation of the average saturated and dry frame mod-uli in equation (2). For the saturated rock, κ

effsat is obtained by

firstly applying Gassmann’s equation to each bed and then thewell-known theory of Backus (1962) to the resultant fine lay-ered channel model. For the dry frame κ

effdry, Backus (1962) is

again applied to the model but in the absence of the fluids. TheBackus averaging process is exact for normal incidence and ahorizontally layered medium. The relevant effective mediumsolution for vertical P-wave propagation is identical to that

given in equation (4):

1

κeff + 43 μeff

=⟨

1

κ + 43 μ

⟩(11)

but with the addition of:

1μeff

=⟨

1μ

⟩(12)

where the averages are weighted by the thickness normalizedby the total thickness for each of the N sand beds and shale lay-ers in the channel body. These equations are valid in this casefor both κ

effsat and κ

effdry. Our first goal is to determine the effective

fluid bulk modulus κefff that could be used in the seismic scale

Gassmann’s equation in equation (2) to saturate the effectiverock in its dry state and convert the Backus-averaged quan-tity κ

effdry to the Backus-averaged quantity κ

effsat. To solve for κ

efff ,

bed-scale fluctuations in porosity δφ and saturation δSw are de-fined such that the saturation Sw and porosity φ for each bedcan be written as Sw = Sw + δSw and φ = φ + δφ, where thehat represents a mean of the symmetrically distributed func-tion (these means are close to the desired quantities as Sw ≈ Sw



and φ ≈ φ). The perturbation terms δSw and δφ are distributedacross the reservoir and vary from bed to bed. These quantitiesare related to variances or covariances of their distributionsaccording to:

σ 2φ = 1

N

N∑i=1

(δφ)2, (13a)

σ 2φS = 1

N

N∑i=1

δφδSw, (13b)

and σ 2S = 1

N

N∑i=1

(δSw)2. (13c)

The desired estimate of κefff is obtained through application

of Backus (1962) and perturbation theory, yielding an adaptedharmonic average:

1

κefff

=(

Sw

κw+ 1 − Sw

κo

) (1 + aσ 2

φ + bκ f σ2φS + cκ2

f σ2S

)−1(14)

which is accurate to the second order in the saturation andporosity fluctuations (see Appendix). Here, κ f is the fluid bulkmodulus obtained using the harmonic average and is thereforea function of Sw. Equation (14) provides a transformation forthe observed fluid bulk modulus κ

efff in terms of Sw. The con-

stants a, b and c depend on the mean porosity φ and mate-rial and fluid constants, details of which can be found in theAppendix. The second bracketed term in equation (14) maytherefore be viewed as a correction to equation (3) to accountfor the geological and saturation fluctuations. Additionally,the inversion of the equation gives estimates of Sw in terms ofthe seismic-scale estimate of κ

efff obtained by the application

of equation (2).Equation (14) provides the desired saturation law that links

the seismic scale fluid bulk modulus κefff to the actual Sw

and therefore, it defines the new generalized saturation lawP2, illustrated in Fig. 3. The law requires statistics derivedfrom vertical distribution of porosity (geology) and satura-tion (ideally from simulation studies) as input. It predicts thatκ

efff should lie above the reference law L and close to P1 or

P1’. This backs up the conclusions of Sengupta and Mavko(2003), who showed that seismic velocities upscaled fromsmall-scale simulations follow the L law more closely than thelaw defined by U. With increasing reservoir saturation, thenew law then predicts a progressive divergence of κ

efff from

these laws. For turbidite reservoirs the porosity term σφ istypically small (≈0.02), while simulation runs (see the nextsection) have shown the saturation-related standard deviation

σ S to be much larger (≈0.13).The cross-term standard devia-tion σφS remains quite small when the porosity and saturationvariations are independent (≈0.02). In overall terms, it there-fore appears that saturation fluctuations tend to dominate.However, the new law also contains coefficients multiplyingthe variances and covariances that are large enough to renderthem significant. In turbidite reservoirs saturation heterogene-ity prior to production is generally quite small, due to the lackof significant porosity variation and the main saturation het-erogeneity is due to production-related fluid flow and hence,permeability. During production, as the waterfront progressesthrough the reservoir, it is anticipated that at any specific lo-cation in the reservoir the spread in saturation, σ S, will at firstgrow over time before reaching a roughly constant value (seeFigs 5b, d and f). Thus, the exact saturation law defined by P2

is likely to vary not only across the field but also with produc-tion time. Therefore, to obtain accurate Sw values it requiresan understanding of the expected spread of saturation valueswith production, which in turn requires calibration from nu-merical fluid flow simulation.

Test of the new saturation law using a numerical f lowsimulation study

To evaluate the accuracy of the new saturation law in equa-tion (14) against the alternative laws proposed above, tests areperformed based on a number of numerical flow simulations.For this purpose, the turbidite model of the previous section isused and flow through a single channel sand body is simulated.For the flow simulation, each bed in the channel sand bodyis taken to be laterally continuous and extensive and to haveuniform permeability. The shale inter-bedding is representedinitially by transmissibility multipliers between cell faces to re-strict vertical flow. The key assumption here is that the shalesare too thin to be represented by individual model cells. In thenext section, models are considered in which finite thickness isgiven to the shales. In both cases, shale erosion is modelled viaglobal statistics that assign a density and size distribution toscours that are then randomly inserted into the shales (Stephenet al. 2001). Lateral displacement of oil by a waterflood in anoil-water system is considered.

The accuracy of the saturation laws in equations (4), (9) and(10) (labelled as L, P1 and P2 in Fig. 3) is evaluated by usingeach law to invert the seismic scale κ

efff back to an estimate

of saturation. This is carried out at each production time stepfor a vertical column of cells in the centre of the model (seeFigs 5a–f). κ

efff is calculated using exact numerical compu-

tation by applying Gassmann’s equation to each cell, then



Figure 5 Instantaneous oil saturation for the 2D model with continuous shales and the corresponding histogram of water saturation for a rangeof times during production induced by a laterally-driven waterflood. The column highlighted at the centre of (a), (c) and (e) indicates the cellsused to gather the saturation statistics in (b), (d) and (f) respectively and also in the subsequent analysis.

Backus’s (1962) theory (11) to the column of cells and fi-nally performing a reverse Gassmann calculation using theupscaled relation in equation (2). The latter is obtained bycalculating Msat and Mdry (where Msat = κsat + 4

3 μdry andMdry = κdry + 4

3 μdry) using the Backus (1962) theory and also φ

as the volumetric average. For comparison purposes, the pore-volume weighted average saturation Sw is calculated for thesame column of cells from the simulation output and knownmodel pore volume. In this process it is assumed that κw andκo are known constants. It is also assumed that there is novariation in the mineral modulus due to clays and cements.

Results based on the simulation study

Initially, we work with a 2D cross-section of 100 × 95 cells,the cell size being 2 m horizontally and 5/19 m vertically (sim-

ilar to D), giving a total model size of 200 m × 25 m. In thefirst set of tests, the porosity is set to have a mean of 22.6per cent, net-to-gross to be unity and horizontal permeabil-ity (kh) assigned according to a normal distribution with amean of 1000 mD and coefficient of variation 0.5. The ini-tial oil saturation is obtained by evaluating 1 − Swc, whereSwc is determined from a laboratory-based empirical relation-ship connected directly to permeability (Morrow and Melrose1991):

Swc = 0.600 − 0.165 log10(kh). (15)

where kh is in milli-Darcies. A variety of shale erosion scenar-ios are considered, from none to full by generating a numberof stochastic realizations of partial erosion. Four models arefinally chosen: continuous shales, no shales and hence, fully



Figure 6 Histograms of saturation distributions as a function of production time for the four 2D models analysed. These statistics correspondto the column defined in Figs 5(a), (c) and (e). Histograms are drawn for 30-day time steps, ranging from 0 to 300 days.

amalgamated sands and two other models with partial ero-sion.

Figure 5 shows the spatial variation in oil saturation andcorresponding Sw distribution for the different time snapshotsduring a 300-day period of (initially uniform) waterflood. Thisdemonstrates the evolution of the water saturation distribu-tion during production. The histograms in Fig. 6 define thisvariation in more detail for the central column of cells forthe four different models and at a full range of time steps.Here, as expected, it can be seen that the inter-bed permeabil-ity variation and transmissibility have a strong control on thevertical saturation heterogeneity. Finally, Fig. 7 gives the av-erage saturation and its standard deviation evaluated over thecells in the selected vertical column. Saturation values increasefrom approximately 0.15 (Swc) to 0.70 (1 − Sor ) over time(Fig. 7a). Also, the saturation does not vary significantly atpre-production times (σ S is only 0.05) but does become morebroadly distributed as the waterflood propagates. On someoccasions during production, the saturation distribution pos-sesses dual peaks and also exhibits some skewness (see Fig. 6).The spread σ S in saturation grows to a maximum of around

0.18 before decreasing slightly (Fig. 7b). The results of esti-mating average saturation using each of the three laws L, P1and P2 and for the four different simulation models, are givenin Figs 8(a), (b) and (c). In percentage terms, the L curve esti-mates give the largest error (3 to 17 per cent with a mean of10 per cent), the P1 estimates (with α set to 0) provide a slightlybetter estimate (2 to 16 per cent with a mean of 9 per cent),whilst the P2 estimates reduce the error still further (2 to3 per cent with a mean of 2.5 per cent). Interestingly, for cer-tain periods during the time evolution the error estimates arerelatively constant in time (apart from a small transition area),suggesting that errors might possibly cancel out if differencesare taken between closely spaced surveys.

Next, the porosity variations in the model are increased sig-nificantly, whilst keeping other parameters unchanged. Theporosity is linked to the permeability variations via a logarith-mic relationship (Pickup and Sorbie 1996):

φ = 0.226 + d log10(kh) (16)

where φ is in absolute percentage units and kh is in milli-Darcies. The parameter d is varied from 0.025 to 0.100 in



TIme (days)

0 50 100 150 200 250 300 350

Sta

ndard

devia

tion

0.00

0.05

0.10

0.15

0.20

TIme (days)

0 50 100 150 200 250 300 350

Avera

ge s

atu

ration

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Continuous shales

Model 1

Model 3

No shales (a)

(b)

orS−1

wcS

Figure 7 Saturation statistics for waterflooding of the 2D modelswith a range of shale erosion characteristics. Results are for the col-umn of cells highlighted in Figs 5(a), (c) and (e) and as a functionof production time. (a) Average saturation defined as the arithmeticmean of the results from the simulation (same as the pore-volumeweighted average in this case); (b) standard deviation of the saturationdistribution.

steps of 0.025. This defines a range of porosity fluctuationsfrom those normally expected in turbidites with d < 0.05 tothose beyond – which are used to test the limitations of the the-ory for other possible depositional systems. The models withvariable porosity have slightly more pore volume. Comparingthe saturations at the 10th time step (300 days) to the uniformporosity case, the saturation distributions are almost identical.The resultant saturation estimates are compared to the trueaverage saturation in Figs 9(a), (b) and (c) for the continuousshale model and Figs 10(a), (b) and (c) for a partially erodedshale model. Here, it is observed that provided the porosityfluctuations are small (as in the turbidite case) then the P2estimates still provide an accurate estimate of Sw (errors are

less than a few per cent for d < 0.05), however the errors doappear to scale strongly with the porosity fluctuations.

Results for 3D simulations

3D simulations are also performed to provide a better under-standing of how the results might translate into practice. A3D model is built with 50 × 50 × 95 cells, giving total di-mensions of 200 m × 200 m × 25 m. The flow is controlledalong the face of the model to ensure that it is line driven andthat it more closely resembles flow in the inter-well region. Inthe 3D model the holes are now elliptical instead of being ofinfinite extent transverse to flow, however this difference doesnot alter the flow behaviour when small or large degrees oferosion are considered. The anisotropy of the hole size is setby the aspect ratio 2:1 (twice as long and parallel to flow).Permeability and porosity are defined as in the 2D modelling.Figures 11(a) and (b) show examples of the saturations in themodel midway through production for the case of continu-ous shale and partially eroded shales. Clearly, the continuousshales model does not require a 3D simulation if the injectionand production are uniform across the left and right faces, re-spectively. Figures 12(a), (b), (c) and (d) compare the resultsfor the 2D and 3D modelling, where again a central columnof cells has been chosen for the analysis. The 3D model withcontinuous shales has cells that are four times larger horizon-tally than the 2D model. This cell size difference produces aslight saturation difference between the models due to numer-ical dispersion and will affect the saturation distributions andhence, contributes to the slight differences of the saturationestimates. However, the variation of the shales in the third di-mension also adjusts the saturation distributions and appearsto cancel the numerical dispersion effects. Both of these effectsare quite small. These results demonstrate that for both mod-els there is little difference in the 2D and 3D case and thatconclusions derived for the 2D case may readily be applicableto 3D.

Results from varying net-to-gross

The shales are initially assumed to be of insignificant thicknessand hence, are replaced by transmissibility multipliers. Here,they are now considered to be of cell thickness (varying be-tween half and two cells thick), such that net-to-gross is nolonger unity but varies across and down through the model.For turbidite reservoirs in practice, shales are deposited be-tween turbidite events with a different depositional processfrom the sands, resulting in differences in thickness and grain



time (days)

0 60 120 180 240

perc

enta

ge e

rror

-20

-10

0

10

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30

time (days)

0 60 120 180 240

perc

enta

ge e

rror

-20

-10

0

10

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30

continuous shales

model 1

model 2

no shales

L curve P1 curve(b)(a)

time (days)

0 60 120 180 240

perc

enta

ge e

rror

-20

-10

0

10

20

30(c) P2 curve

Figure 8 Percentage error between the actual pore-volume weighted average saturation and the estimates of saturation obtained by invertingthe laws defined by (a) equation (6), (b) equation (9) and (c) equation (16) in the main text. Four models are considered, from continuous shalesto partial erosion and complete amalgamation.

size. Shale layers may be relatively thicker at the edges of chan-nels and lobes but also in situations where turbidity events areless frequent. Consideration of net-to-gross can be importantwhen trying to evaluate turbidite fields for production andsubsequent development, as 4D seismic detection of satura-tion changes in the main channel fairway and the field marginsmust be consistent and accurate (McInally et al. 2003).

Using the analysis above, saturation estimates can also beextracted from a sand package containing thick shales. Thisassumes an appropriate seismic estimation scheme is used tocorrect for the reservoir’s net-to-gross (see Appendix). The ex-act details of this procedure are beyond the scope of this paper.However, thick shales also have an effect on the flow simula-tion, which in turn will change the spread of saturation valuesand hence, the results of Figs 8, 9 and 10. The results of re-running the flow simulation using the same type of modelsas in the previous case but now changing net-to-gross, areshown in Figs 13, 14 and 15. Here it is observed that there isa similarity with the thin shale case, although there are slightdifferences (a few per cent) due to the different saturation val-ues and spreads in the models. This suggests that the current

analysis is sufficient for both low to moderate NTG regionsof the reservoir.

D I S C U S S I O N

Seismic-scale saturation estimates based on the laws defined byL and U in Fig. 1 are in common use today amongst geophysi-cists who regard these as extreme, end-member states embrac-ing reality. They are useful as they can be applied without theneed for reservoir information but do require strict underlyingassumptions to be honoured: L requires homogeneous geologyor pressure equilibration in the rock volume �V imaged by theseismic; U is no more than a mathematical limit and does notrepresent any known physical saturation distribution. For arealistic reservoir with heterogeneous geology and saturation,such laws do not accurately predict pore-volume weighted av-erage saturation Sw, as it has been shown that the errors canreach up to 30 per cent in this case. By contrast, saturationestimates obtained via the P1 (or P1’) laws have only recentlybeen suggested and are therefore used less often. However,provided geological heterogeneity is not strong, it appears that



(a)

(c)

(b)

time (days)

0 60 120 180 240

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ge e

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0

10

20

30

time (days)

0 60 120 180 240

perc

enta

ge e

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0

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30 L curve

P2 curve

time (days)

0 60 120 180 240

perc

enta

ge e

rror

-20

-10

0

10

20

30 P1 curve

d = 0.025

d = 0.05

d=0.075

d=0.1

Figure 9 Error in saturation estimates as in Fig. 8 but for five separate continuous shale models with increasing amounts of porosity fluctuationas defined by equation (18).

time (days)

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time (days)

0 60 120 180 240

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0

10

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(c)

P2 curve

L curve(a)

d = 0.025

d = 0.05

d=0.075

d=0.1

time (days)

0 60 120 180 240

perc

enta

ge e

rror

-20

-10

0

10

20

30

(b)

P1 curve

Figure 10 Error in saturation estimates as in Fig. 8 but for one particular partial erosion model realization with various amounts of porosityfluctuation as defined by equation (18).



Figure 11 Oil saturation after 150 days of injection through the 3Dmodels. (a) Continuous shales model; (b) partially eroded shale model.

these can potentially be used to estimate Sw, using seismic to ahigher accuracy with errors less than 20 per cent. These lawsare attractive, as they require as input information on the aver-age connate water saturation and residual oil for the reservoir,which should be readily available for any particular reservoir(although for P1’ the assignment of the more uncertain quan-tity α relating to the amount of bypassed oil is necessary).Finally, the P2 law estimates Sw with the smallest error, thisbeing a few per cent for modest porosity variations and up to10 per cent for more extreme vertical heterogeneity. However,to achieve this increase in accuracy, the law requires as inputmaps specifying the vertical spread in saturation and porosityand their covariance across the reservoir at the time/s of theseismic survey/s, information that can be extracted from thesimulation and geological model. In this case, seismic inver-sion for Sw must therefore be constrained by the statistics ofthe fluid flow simulation. In practice, the appropriate choiceof saturation law for any given situation will depend on thebalance between the level of accuracy required for the satura-tion estimates, which in turn depends on its ultimate use (forexample, as part of a seismic history matching project or fordynamic reservoir management for well placement), togetherwith the availability of the necessary reservoir data.

Figure 12 Comparison of the performance of the L, P1 and P2 esti-mates for the 2D and 3D simulation results. (a) Models with contin-uous shales; (b) models with thin, partially eroded shales.

For the application of these results to quantitative 4D seis-mic interpretation, an important finding is that for the lawsoutlined here, particularly P2, the saturation errors are mostlyconstant over large ranges of production time. This finding islinked to the way in which the saturation field evolves in thereservoir. It suggests that taking differences in saturation mea-surements between successive monitor surveys will cancel outthe majority of the error in the Sw estimate, particularly if thesesurveys are frequently shot. This conclusion is valid providedthe change in the spread of the vertical saturation distribu-tion over time is not significantly greater than the change inthe average saturation value being detected. Indeed, for theapplication of either the U, L, P1 or P1’ laws, it appears thattaking differences between the monitor and the baseline pre-production survey leads to large inaccuracies, whereas takingdifferences between frequently acquired monitor surveys givesmore accurate results.

The work in this paper has made a number of simplifying as-sumptions. One assumption is to neglect the influence of fluidpressure. Pressure change provides a complicating factor, as itcauses the fluid bulk moduli, densities and the rock matrix to



time (days)

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-20

-10

0

10

20

30

(c)

P2 curve

L curve(a) (b)

P1 curve

continuous shales

model 1

model 2

no shales

Figure 13 Comparison of the performance of the L, P1 and P2 estimates for four models ranging from continuous shales to partial erosion andcomplete amalgamation. Comparable to the models of Fig. 8 but with a bulk net-to-gross of 0.85.

time (days)

0 60 120 180 240

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(c)

P2 curve

(a)

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30 P1 curve

time (days)

0 60 120 180 240

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rror

-20

-10

0

10

20

30(b)

L curve

NTG = 0

NTG = 0.075

NTG = 0.15

NTG = 0.3

Figure 14 Comparison of the performance of the L, P1 and P2 estimates for the model of continuous (thick) shales. In this case net-to-gross(NTG) is varied.



(c)

(a) (b)

time (days)

0 60 120 180 240

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0

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0 60 120 180 240

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rcen

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or

-20

-10

0

10

20

30 evruc 1Pevruc L

P2 curve

NTG = 0

NTG = 0.075

NTG = 0.15

NTG = 0.3

Figure 15 Comparison of the performance of the L, P1 and P2 estimates for the one particular realization of the partial erosion model for thickshales. In this case net-to-gross (NTG) is varied.

vary, which may in turn affect some of the results shown here.However, as the reservoir interval considered in the presentstudy is thin and the pressure variations small, ignoring pres-sure effects is taken to be a reasonable assumption in the cur-rent case, although it remains a subject for future research.Another simplifying assumption is that the predominant errorsources are concentrated in the fluid modulus and porosityterms. However, it is known that additional error must alsoarise due to the variability of cements, clay minerals and thedegree of consolidation of the sands. It is anticipated that suchvariations may lead to a departure of the rock frame from thelinear porosity law of (A16) by only a few per cent, due to thenarrow range of porosity in the turbidite sands. The error con-tributions due to variable mineral modulus could potentiallybe much larger. The accuracy of the proposed saturation lawalso varies slightly with fluid contrast, so that the fluctuationof oil API, oil solution gas ratio or brine salinity will controlits ultimate accuracy. Tests for a range of North Sea turbiditefields show that this dependence on fluid contrast yields er-rors of only a few per cent. The cumulative effect of theseerrors on the accuracy of each of the saturation laws must beconsidered and it is not yet known how this will affect ourassessment of the accuracy of each. A further neglected factor

is the effect of tuning on the thin turbidite sand package (seefor example, McInally et al. 2003). Variation in sand thicknesscan produce amplitude changes that may be misinterpreted aschanges in the fluid properties and thus, must be compensated.There is therefore a requirement for a de-tuning stage prior tothe extraction of the fluid bulk modulus from seismic. Onefinal assumption in the development of the P2 saturation lawis that it is only strictly valid for vertically incidence P-waves.It should therefore be applicable only for near-offset seismicdata, while a generalization to larger offsets requires the in-corporation of seismic anisotropy and the introduction of theanisotropic Gassmann equation into the solution. A furtherrefinement to the current method would include the consid-eration of intra-bed variations to account for gradations ofshaley sand or sandy shale in individual beds.

C O N C L U S I O N S

Estimates of fluid-bulk moduli extracted from seismic do notcorrespond to the true (pore-volume weighted) average valueof saturation used for flow simulation purposes. The outputfrom 4D seismic interpretation is therefore not directly inter-pretable for reservoir engineering purposes. The reason for this



discrepancy is that the seismic waves sample the reservoir’s ge-ological and saturation heterogeneity differently from the fluidflow. This paper develops a solution to this problem specifi-cally for turbidite reservoirs, by using an effective mediumand perturbation theory. This leads to a new saturation lawthat depends on the vertical spread of saturation and porosityat each location in the reservoir as a function of productiontime. When comparing saturation estimates from alternativelaws to this new law, it is more accurate and does not violatethe underlying physics. However, the proposed law does needto be calibrated using statistics derived from the simulationand output or wireline logs and therefore requires more effortto integrate seismic and engineering disciplines. The resultsfrom this work have particular impact in those reservoirs forwhich saturation plays a dominant role – such as the Nelsonfield (McInally et al. 2003). It provides a correction factor formethods that estimate pressure and saturation simultaneouslyfrom 4D seismic data and may help towards an accurate sim-ulator to seismic modelling studies.

A C K N O W L E D G E M E N T S

This work was sponsored by the Edinburgh Time-LapseProject, Phase III and is published with approval from its spon-sors: BP, BG, Chevron, ConocoPhillips, ExxonMobil, IkonScience, Shell, Statoil, Total, Maersk, Marathon Oil, Norsk-Hydro, Petrobras, Norsar, Landmark and Woodside. The au-thors thank Andy Gardiner of IPE HWU for constructivediscussions on aspects of reservoir geology, particularly withreference to the work of the Genetic Units Project. Schlum-berger Geoquest is thanked for use of the Eclipse simulator.

R E F E R E N C E S

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Clark N.J. 1964. Elements of Petroleum Reservoirs. Society ofPetroleum Engineers.

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A P P E N D I X

Derivation of a geology controlled saturation law forturbidite reservoirs

The elastic properties of each channel sand package com-posing a turbidite channel complex can be modelled satis-factorily, from the flow, geology and elastic wave perspec-tive, as a stack of thin (<1 m thick) horizontal plane layers(beds). It is known from geological and petrophysical studies(Gardiner, personal communication) that the porosity fluctu-ations between individual beds in this stack are not large andthat the overall thickness, T, of the channel sand package isstill small compared to the seismic wavelength, λ, such thatλ/T < 5. It therefore follows that for the purposes of seismicwave propagation, the vertical heterogeneity arising from theinternal thin layering in such a reservoir can be adequatelyreplaced by a single saturated homogeneous layer represent-ing the entire channel sand package, with no loss of accuracy(see for example Folstad and Schoenberg 1992). The elasticmoduli κ

effsat and μeff of this effective layer can be calculated

by applying the theory of Backus (1962), who developed gen-eral averaging formulae for isotropic (or vertically transverseisotropic) layers. His replacement solution for wave propaga-tion normal to the beds is:

1

κeffsat + 4

3 μeff=

⟨1

κsat + 43 μ

⟩(A1)

and

1μeff

=⟨

1μ

⟩, (A2)

where the angular brackets represent a weighted arithmeticaverage over the fine-scale layers, each with individual moduli

κ(i)sat and μ(i) (i = 1 to N) and the weighting corresponds to

layer thickness normalized by the total channel sand packagethickness. The saturated bulk modulus κ

(i)sat for every single thin

bed, i, in the stack is obtained by applying Gassmann (1951)and the small-scale law equation (3) for the saturation of asingle homogeneous sand facies, with:

κ(i)sat = κ

(i)dry +

(1 − κ

(i)dry/κm

)2

φ(i)

(Swκw

+ (1−Sw)κo

)+

(1−κ

(i)dry/κm−φ(i)

)κm

, (A3)

where κdry is the dry frame modulus, κm the mineral grainmodulus and κw and κo the bulk moduli for water and oil re-spectively. Each individual thin homogeneous bed is saturatedby both water and oil with saturations S(i)

w and 1 − S(i)w respec-

tively. After this, the Backus (1962) theory is applied to obtainκ

effsat via equation (A1) and κ

effdry by using:

1

κeffdry + 4

3 μeff=

⟨w

κdry + 43 μ

⟩. (A4)

The objective is now to work backwards and isolate anexpression for the effective fluid bulk modulus κ

efff for the

composite stack of layers that should be inserted into a seismicscale Gassmann-type equation in equation (2), which saturatesthe effective dry rock modulus κ

effdry to produce the saturated

rock modulus κeffsat:

κeffsat = κ

effdry +

(1 − κ

effdry

/κm

)2

φ

κefff

+(

1−κeffdry

/κm−φ

)κm

(A5)

where φ is the mean layer porosity. The solution for κeffsat can

be achieved by considering equations (A1), (A2) and (A5),for two distinct geological scenarios that lead to channel sandpackages with shales of negligible thickness, which act purelyas transmissibility barriers and thicker shales whose propertiescontribute to the elasticity of the whole composite.

Thick sands and very thin shale beds

We first consider a model of the channel sand package inwhich the beds are separated from each other by thin im-permeable shales, which carry negligible volume or thickness.Here, the individual beds in the package have variable porosityand also water saturation – the latter being larger due to theoil-water displacement process itself. It is this variability thatin turn gives rise to variations in both the dry frame and sat-urated moduli in equations (A5) and (A1). To solve for κ

efff in



equation (A4) in terms of the small-scale fluctuations in poros-ity δφ and the larger-scale fluctuations in saturation δSw, thewater saturation, Sw and porosity, φ are firstly re-written asSw = Sw + δSw and φ = φ + δφ, where the bar represents themean of the symmetrically distributed function. The fluctu-ation terms δSw and δφ are considered to be randomly dis-tributed and varying from bed to bed and are related to thestandard deviations of the vertical distributions according to:

σφ =√√√√ 1

N

N∑i=1

(δφ)2 (A6a)

σφS =√√√√ 1

N

N∑i=1

δφδSw (A6b)

σS =√√√√ 1

N

N∑i=1

(δSw)2 (A6c)

We start by focusing on the evaluation of the saturation termin equation (A4), which can be written:

κeffsat − κ

effdry =

⟨1

Msat

⟩−1

−⟨

1Mdry

⟩−1

(A7)

where Msat = κsat + 43 μdry and Mdry = κdry + 4

3 μdry. We writethe saturated modulus Msat as the sum of the mean dry framemodulus Mdry, the fluctuations in the dry frame modulus δMdry

and a term, G:

Msat = Mdry + δMdry + G (A8)

where G is obtained from the difference in saturated anddry bulk modulus evaluated via equation (A4). Setting x =δMdry+G

Mdry, we can write:

⟨1

Msat

⟩−1

= Mdry

⟨1

1 + x

⟩−1

≈ Mdry(1 + 〈x〉 − 〈x2〉 + 〈x〉2+, . . .)(A9)

assuming x to be small compared to 1 and dropping termshigher than second order. This gives:

⟨1

Msat

⟩−1

≈ Mdry

⎛⎝1 +

⟨δMdry + G

Mdry

⟩−

⟨(δMdry + G

Mdry

)2⟩

+⟨

δMdry + G

Mdry

⟩2⎞⎠, (A10)

or⟨1

Msat

⟩−1

≈ Mdry

+〈G〉 +〈G〉2 − 〈G2〉 − 2〈GδMdry〉 −

⟨δM2

dry

⟩Mdry

.(A11)

Similarly

⟨1

Mdry

⟩−1

≈ Mdry −⟨δM2

dry

⟩Mdry

, (A12)

and thus⟨1

Msat

⟩−1

−⟨

1Mdry

⟩−1

= 〈G〉 + 〈G〉2 − 〈G2〉 − 2〈GδMdry〉Mdry

.

(A13)

Note that we assume that Mdry is a linear function of poros-ity (see below) so that 〈δMdry〉 = 0. Furthermore, if we writeG = G + δG, where G is the saturation term on the right-handside of equation (A4) and its corresponding fluctuations areδG, where 〈δG〉 = 0 necessarily, then:⟨

1Msat

⟩−1

−⟨

1Mdry

⟩−1

= G + 〈δG〉

+ 〈δG〉2 − 〈δG2〉 − 2〈δGδMdry〉Mdry

, (A14)

or⟨1

Msat

⟩−1

−⟨

1Mdry

⟩−1

= G +⟨δG

(1 − 2δMdry

Mdry

)⟩

+ 〈δG〉2 − 〈δG2〉Mdry

. (A15)

It is now assumed that over the narrow porosity range of therocks under consideration (with a variation of no more than afew per cent) that the dry frame moduli can be approximatedby the linear porosity laws:

κ(i)dry = κm

(1 − βφ(i)

), (A16a)

and

μ(i)dry = μm

(1 − βφ(i)

), (A16b)

where β is a constant. Such relations arise, for example, fromthe critical porosity model (Nur et al. 1995, where β = 1/φc

and φc is the critical porosity) – the validity of this assumptionis discussed in Chapter 7 of Mavko, Mukerji and Dvorkin(1998). These then give:

δMdry = −(

κm + 43

μm

)βδφ, (A17)



and

Mdry =(

κm + 43

μm

)(1 − βφ). (A18)

Thus, equation (A15) can now be written as:⟨1

Msat

⟩−1

−⟨

1Mdry

⟩−1

= G

+⟨δG

(1 + 2

(κm + 4

3 μm)βδφ(

κm + 43 μm

)(1 − βφ)

) ⟩+ 〈δG〉2 − 〈δG2〉

M.

(A19)

Finally, to second order in the bed fluctuations and by drop-ping the last ratio in equation (A19):⟨

1Msat

⟩−1

−⟨

1Mdry

⟩−1

= G +⟨δG

(1 + 2βδφ

(1 − βφ)

)⟩. (A20)

The calculation for the perturbation in the fluid sat-uration term δG can be obtained via Gassmann’s equa-tion in equation (A4). Expanding this equation in terms ofthe porosity and saturation fluctuations gives:

G + δG = β2(φ + δφ)2

(φ+δφ)κ f

(1 − a f κ f δSw) − (φ+δφ)(β−1)κm

(A21)

or

G + δG = β2(φ + δφ)1

κ f(1 − a f κ f δSw) − (β−1)

κm

(A22)

where a f = ( 1κ0

− 1κw

) (an arbitrary definition) and implicit in

this form is κ f = ( Swκw

+ 1−Swκo

)−1. Assuming that the 1κm

term inthe denominator is small (as κm � κ f ) and expanding 1/(1 −a f κ f δSw), dropping the third order terms, gives:

G + δG = β2κ f (φ + δφ)(1 + a f κ f δSw + a2

f κ2f δS2

w

)(A23)

and thus

δG = β2κ f φ(a f κ f δSw + a2

f κ2f δS2

w

)+ β2κ f δφ(1 + a f κ f δSw).

(A24)

Upon insertion of equation (A24) into equation (A20) andsubsequent re-arrangement, an expression for the saturationterm is provided:⟨

1Msat

⟩−1

−⟨

1Mdry

⟩−1

= G + β2κ f

⟨2β

(1 − βφ)δφ2 + a2

f κ2f φδS2

w

+(

a f κ f + 2βa f κ f φ

(1 − βφ)

)δφδSw

⟩(A25)

which can then be re-written as the final solution to secondorder:⟨

1Msat

⟩−1

−⟨

1Mdry

⟩−1

= G + β2κ f

×[

2β

(1 − βφ)σ 2

φ + a f κ f

(1 + φ

2β

(1 − βφ)

)σ 2

φS + φa2f κ

2f σ

2S

]

(A26)

From equation (A26) it is predicted that for completely in-dependent, random porosity and saturation fluctuations, thenσ S dominates, whilst σφ is small and the covariance term σφS

is also small. However, if the porosity and saturation fluctu-ations are correlated, then σφS also becomes prominent. Theaccuracy of this approximation has been tested numerically.For the purposes of this test, random porosity and saturationfluctuations are generated across the layer stack by consider-ing each parameter to be a normally distributed random vari-able with a specific preset standard deviation. A comparison ismade between the left-hand side of equation (A26), evaluatedexactly by numerical calculation using Backus averaging andthe approximate analytic expression on the right-hand sideof equation (A26). The agreement (not shown) appears to bebetter than 1 per cent error, indicating that equation (A26) isan acceptable approximation.

Equation (A26) can now be used to determine the desiredseismic estimate of fluid bulk modulus κ f (defined as κ

efff in the

main text) in terms of the true fluid bulk modulus κ f , assumingthis has been obtained by some appropriate inversion processusing the Gassmann equation applied to the seismic voxel �V.To obtain the final relationship, the left-hand side of equa-tion (A25) is replaced by one of the Gassmann approxima-tions proposed by Han and Batzle (2004), which is valid togood accuracy for porosities above 15 per cent with less than3 per cent error and is thus appropriate for the particulargeology under consideration here. This now gives:

β2κ f φ = β2κ f φ + β2κ f

×[

2β

(1 − βφ)σ 2

φ + a f κ f

(1 + φ

2β

(1 − βφ)

)σ 2

φS + φa2f κ

2f σ

2S

](A27)

or

κ f = κ f[1 + aσ 2

φ + bκ f σ2φS + cκ2

f σ2S

], (A28)

where a = 2β

φ(1−βφ), b = ( a f

φ)(1 + 2βφ

1−βφ) and c = a2

f .

For completeness, the desired solution for the mean fluidbulk modulus κ f , in terms of the measured fluid bulk modulusκ f and the individual porosity and saturation fluctuations, can



also be obtained by considering the ansatz κf = κ f (1 + � f ),which upon substitution into equation (A28) gives:

1 = (1 + � f ) + 1 + � f

φ

×[

2β

(1 − βφ)σ 2

φ + a f (1 + � f )κ f

(1 + φ

2β

(1 − βφ)

)σ 2

φS

+ φa2f

(1 + 2� f + �2

f

)κ2

f σ2S

]. (A29)

To first order in �f this then gives:

� f =−

[2β

φ(1−βφ)σ 2

φ + a f

φκ f

(1 + φ

2β

(1−βφ)

)σ 2

φS + a2f κ

2f σ

2S

][1 + 2β

φ(1−βφ)σ 2

φ + 2 a f

φκ f

(1 + φ

2β

(1−βφ)

)σ 2

φS + 3a2f κ

2f σ

2S

] ,

(A30)

and hence,

κ f = κ f

×⎛⎝ 1 + a f

φκ f

(1 + φ

2β

(1−βφ)

)σ 2

φS + 2a2f κ

2f σ

2S

1 + 2β

φ(1−βφ)σ 2

φ + 2 a f

φκ f

(1 + φ

2β

(1−βφ)

)σ 2

φS + 3a2f κ

2f σ

2S

⎞⎠ .

(A31)

This is the final desired saturation solution for calculatingthe mean bulk modulus from the seismic, from which the meansaturation can be obtained. The saturation obtained in thisway is close to the pore-volume weighted mean saturationobtained by consideration of fluid flow principles. The ex-pression in the brackets of equation (A31) defines the datarequired to align the different sampling regimes of the seis-mic and engineering measurements after the distortion by theBackus averaging for the seismic. It can therefore be used as acorrection factor if we have prior knowledge of the porosityand saturation fluctuations and their inter-relationship.

Thick shales and thinner sand beds

Another valid model for turbidites gives the shales a non-negligible thickness when compared to the sands. As shale

is eroded by the sand deposition, the shale beds are usuallymuch thinner, however at the channel margins, different ero-sion mechanisms prevail, which prevent this process. If shalesare thick, they cannot be assumed to have a negligible influ-ence on the elastic properties of the channel sand package asin the previous section and their properties must also be in-cluded in the calculation. This can be achieved by consideringthe following:(a) as the shales are impermeable and have low porosity, δG

is zero for shales because they do not (in general) change theirsaturation;(b) for the reasons given in (a), G is an average over the sandbeds only;(c) it may be assumed that the dry frame moduli for theshales are approximately constant throughout the chan-nel sand package, as the shale porosity and its associ-ated fluctuations are negligible. Shales in turbidites arisefrom the same depositional process as the sands, with thesame energy levels and sediment source. They might there-fore differ slightly from fluvial or shallow marine shales.However, variations in the properties of the shales are as-sumed to be second order for the purposes of the currentwork.

The equations in the previous section must be revised fora sand-thick shale sequence so that the angular brackets de-noting the averaged quantities over the reservoir interval mustnow be defined as averages over only the sands. The final sat-uration equations are therefore identical to the thin shale case,however the quantities involved refer only to the sand compo-nent of the reservoir. This presupposes that the contributiondue to the shale has been exactly taken out of the seismic re-sponse using, for example, the Backus averaging formulae ofequations (A1) and (A2). Thus, assuming the NTG and elasticproperties of the shales to be known it is possible to extracttheir effect from the overall measured elastic modulus fromseismic. For the current study this procedure is taken to beexact, however for application to observed seismic data therewill be an error involved in this step.


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Seismic scale saturation relations in turbidite reservoirs ... · all performance of a waterflood and potential encroach-ment on producing wells. In this context, there is a need