May01 Rock Physics

42 CSEG Recorder May, 2001

What is Rock Physics?

Rock Physics describes a reservoir rock by physical propertiessuch as porosity, rigidity, compressibility; properties that will affecthow seismic waves physically travel through the rocks. The RockPhysicist seeks to establish relations between these material prop-erties and the observed seismic response, and to develop a predic-tive theory so that these properties may be detected seismically.

Establishing relationships between seismic expression andphysical rock properties therefore requires 1) knowledge about theelastic properties of the pore fluid and rock frame, and 2) modelsfor rock-fluid interactions. This is the domain of Rock Physics.

Rock Physics / Petrophysics: What’s the difference?

Stated very simply, (and therefore with apologies to RockPhysicists and to Petrophysicists):

Continued on Page 43

FEATURE ARTICLE

ROCK PHYSICS FOR THE REST OF US –AN INFORMAL DISCUSSION

By Jan Dewar - Scott Pickford, a Core Laboratories Company, Calgary

Rock Physics... Petrophysics...

Rock Physics uses sonic logs, density logs, and also dipole(shear velocity) logs if available.

Rock Physics aims to establish P-wave velocity (Vp), S-wavevelocity (Vs), density, and their relationships to elastic moduli κ (bulk modulus) and µ (rigidity Modulus), porosity,pore fluid, temperature, pressure, etc. for given lithologiesand fluid types.

Rock Physics talks about velocities and elastic parameters,because these are what link physical rock properties to seismicexpressions.

Rock Physics may use information provided by thePetrophysicist, such as shale volume, saturation levels, andporosity in establishing relations between rock properties orin performing fluid substitution analyses.

Rock Physics is the interest of Geophysicists (and maybePhysicists).

Petrophysics uses all kinds of logs, core data and productiondata; and integrates all pertinent information.

Petrophysics aims at obtaining the physical properties suchas porosity, saturation and permeability, which are related toproduction parameters.

Petrophysics is generally less concerned with seismic, andmore concerned with using wellbore measurements to contribute to reservoir description.

Petrophysics can provide things like porosity, saturation, per-meability, net pay, fluid contacts, shale volume, and reservoirzonation.

Petrophysics is the interest of Petroleum Engineers, Well LogAnalysts, Core Analysts, Geologists and Geophysicists.

Why do we need Rock Physics?

Accurate relations between rock properties and seismicattributes can to put “flesh on the bones” of a seismic interpretation.That is, Rock Physics allows the interpreter to put “rock propertiestogether with seismic horizons.” (Peeters). Information aboutporosity, pore-fill, and lithology becomes available to augment theseismic interpretation.

What can Rock Physics studies contribute?

Typical Rock Physics studies will answer questions such as:

• can porosity and saturation be obtained from (seismic) inter-val velocity?

• what are the velocity-porosity relations in various lithologies?• is seismic AVO response sensitive to gas saturation levels in

a particular play?

■ how sensitive? ■ what increments of gas saturation can be detected

seismically?• how robust would AVO seismic response be in situations of

varying thickness?• how robust would AVO seismic response be in situations of

varying porosity?• can an optimal processing strategy be determined for process-

ing the seismic data for AVO analysis?• what seismic characteristics or attributes may be useful to dis-

tinguish gas-sand from shale?• what is the influence of gas saturation on velocities, density

and reflectivity in a particular play?• what is the influence of clay content?• what is an appropriate mudrock line value for further AVO

analysis of a particular play?

May, 2001 CSEG Recorder 43

Rock Physics in Practice: some examples

Example 1: Cross-plotting well log data

A typical task in a Rock Physics study is to calculate AVOattributes from the well logs and cross-plot various attributes fromselected geologic units. These cross-plots are used to:

• understand how the rock properties are related • determine the resolution of rock properties in various litholo-

gies in the area• determine the sensitivity of various attributes to fluid effects • contribute velocity constraints such as mudrock line values to

AVO analysis of seismic data. • contribute to interpretation of attribute sections: which

attribute(s) are best for describing a given reservoir?

The following example is from a well log from western Canada.Data points from various depth intervals are cross-plotted in vari-ous attribute spaces. Note how data from various depth intervals(lithologies) cluster. Note also how the clusters are separated moreeasily in some attribute spaces than in others.


FEATURE ARTICLE Cont’d

ROCK PHYSICS FOR THE REST OF US – AN INFORMAL DISCUSSIONContinued from Page 42

Figure 1: Crossplots of parameters calculated from wireline data. Suchcrossplots indicate which attributes will be helpful to discriminate gassands in a particular play.


Example 2: Crossplotting seismic data

This example illustrates how crossplots can be used to see whichattributes may show clear separation for the zone of interest. In thiscase, the zone of interest is an oil-bearing sand.

On the following page are crossplots from AVO Attribute sec-tions derived from pre-stack seismic data. The highlighted datapoints are taken from data at well locations A, B, C, and D, over thezone of interest (as interpreted on the AVO Attribute time sections).Note differences in separation of clusters in the Ip versus Is (P-

Impedance versus S-Impedance) crossplots as compared to theLambda*Rho versus Mu*Rho crossplots. This set of crossplots illus-trates how, at least in this study area, good sand may be distin-guished from coal, shale, wet sand, and regional wet sand bycrossplotting AVO attributes or elastic rock properties, and that var-ious attributes are available for crossplotting.

One could then use this knowledge of crossplot cluster patternsat known wells to further investigate cluster patterns at variousother well locations, and at potential locations beyond.




Example 3: S- wave velocity prediction

Shear wave velocities are required for most AVO analyses.However, the S-wave velocities are not commonly logged, andoften must be predicted from P-wave sonic logs. This is anothertypical task of the Rock Physics study.

In this example from northern Mexico, the P-wave velocitymodel was built by calibrating the interval velocities obtained fromNMO analysis of the seismic data with the sonic velocities. Themud-rock line, estimated from the well’s P- and S- velocities, andshown in the figure below, was used to scale and shift the P-wavevelocities to obtain the S-wave velocity field.

Figure 3: Vp vs Vs Graph. The S-wave velocity function was obtainedfrom the mud-rock line estimated from the well’s dipole sonic log. Thegreen points, which correspond to the gas sand were not included in thelinear fit.


Example 4: Fluid substitutions

Another practical application of Rock Physics is to perform fluidsubstitutions in well log data, to assess ‘what if’ possibilities. Forexample, the gas-charged zone of a well log can be edited to simu-late the wet well case. Fluid substitutions should not be consideredto be trivial matters because fluid type plays an influence on manyproperties. For instance, one cannot simply lower the P-velocity tosubstitute gas for brine, the density will also be affected when gasis substituted for brine. The main point here is that fluid substitu-tion studies cannot be approached casually.

There are two main modeling algorithms: Zoeppritz and Elasticwave equation. The Zoeppritz equations modeling uses ray tracingand approximates the source signal as a plane wave propagatingthrough our earth model. The more computationally intensiveElastic Wave Equation modeling provides a more sophisticatedresult by using the full wave equation to propagate the sphericalwavefront through the depth model. It accounts for peg-leg multi-ples, surface multiples, absorption, transmission losses, and con-verted waves.

The synthetic pre-stack gather can then be processed in the sameway as one would process any real pre-stack gathers. This includesextracting AVO attributes and Lame’s parameters from modeledsynthetic gathers in just the same way as one extracts these fromreal pre-stack data. In this way one can predict from the modelthose attributes, if any, which will be useful AVO hydrocarbonindicators for actual seismic data.

As mentioned earlier, one can vary reservoir properties such asthe fluid type, porosity, thickness, to study the resulting seismicAVO response and investigate possible non-unique physical causesof the observed seismic response. We can also study the sensitivityof the seismic to acquisition parameters, signal to noise ratio, band-width, and processing parameters.




Figure 4: In this example from a carbonate play, the effects of porosityand fluid content have been investigated. This particular example,shows that Lambda*Rho is decreased by gas, and the gas effect is greaterat low porosities. Mu*Rho may be increased or decreased, but is gener-ally relatively unchanged by the presence of gas.

In addition to fluid substitutions, sensitivity to thickness of arock layer may be examined by perturbing the well logs to alter thethickness, or even to remove the layer. Attributes can be calculatedfrom the modified wells to simulate seismic response to the variousthickness increments.

Similarly, porosity, saturation, or other physical propertiescould be incrementally changed by editing the logs, then calculateVp, Vs, density, and elastic moduli relationships to investigate seis-mic response. This is illustrated in the next example.

Example 5: 1.5D pre-stack seismic modeling

Reservoir rocks have physical material properties. Pre-stackseismic modeling means calculating the synthetic pre-stack seismicgather that would result from these rock properties. Pre-stack seis-mic modeling essentially presents similar information as doescross-plotting, but since synthetic seismic wiggle traces are output,one has the added benefit of being able to view seismic expressionfor the modeled earth properties, not just well logs or crossplots.

Synthetic seismic gathers are generated by calculating how thewavefield would pass through an earth model. The earth model isfirst constructed by using sonic, density, and shear wave logs toform a description of the subsurface depth.

Figure 5: Pre-Stack Synthetic gather. The depth model is a well fromwestern Canada; the well data is shown here in two-way time.

Example 6: 2D pre-stack modeling

Using horizon based interpolation, several well logs can be usedto construct a 2D geologic earth model and generate an interpolat-ed ‘2D Line’ of synthetic pre-stack gathers. One can stack this ‘line’of modeled pre-stack gathers to study the stack response. One canalso extract pre-stack AVO attributes and assess the relative impor-tance of attributes before investing time and effort in an AVO anal-ysis of actual field seismic data.

Digging Deeper: Where does all this come from?

Two approaches: Theoretical and Empirical

Equations that attempt to describe the relationships betweenseismic velocities and lithology, porosity, pore fluid, etc. are eithertheoretical or empirical. Theoretical relationships start with underlyingphysical principles andattempt to propose a univer-sal relationship (at least forthe assumptions theymay be forced to make).One of the amazingthings about many theo-retical relationships inPhysics is how well theycan work. But they allbreak down at somepoint or in some particu-lar circumstances whenassumptions, sometimeshidden, are violated.

Empirical relationshipsare derived fromexperiment. Physical properties of a suite of rock samples are mea-sured, analysed, graphed, and a mathematical function (often a lin-ear equation of the form y = mx + b) is fit to the data points.Sometimes a linear fit is natural (the data points are linear for themost part, but in some manner fall away from a linear trend, oftennear the end points). Sometimes measured data points fit to non-linear equations. Empirical relations most often work very well forthe data they were derived from, but can be difficult to comparefrom one research project to the next. With empirical relations onemust be careful about ascribing physical meaning to what areessentially generic mathematical formulae.


Zhijing Wang summarizes the tension between theoretical andempirical approaches this way:

Most direct measurements are carried out either in the laborato-ry or inside a borehole, whereas most theoretical calculations arebased on the Gassmann equation (Gassmann, 1951) because of itssimplicity and ease of use. Direct laboratory measurements are car-ried out uncontrolled, simulated reservoir environments and pro-vide accurate effects of pore fluids on seismic properties. Directborehole measurements, however, are often affected by uncontrol-lable factors such as stress concentration, hole washout, mud inva-sion/filtration, and saturation conditions. In both laboratory andborehole measurements, the wave frequencies are higher than seis-mic frequencies.

Theoretical calculations such as those using the Gassmann equa-tion require input parameters that have to be directly measured inthe laboratory. In fact, some theories require input parameters thatare often hard to obtain with a reasonable precision. On the otherhand, some theories, particularly the Gassmann equation, are basedon frequencies comparable to seismic wave frequencies. Therefore,a direct comparison of laboratory results with theoretical calcula-tions often involves the dispersion problem, where dispersionmeans that seismic velocities are functions of the wave frequency.(Wang, 2000, p 8)

The crux of all this is that there are a great number of relation-ships between seismic velocities (and constituent elastic properties)and rock parameters that are all valid to some degree but not validalways, and many that do not illuminate the physical principlesinvolved. The trick is to try to gain a fundamental understanding sothat, on a practical basis, the different relationships can be evaluat-ed for their applicability to solving specific problems.

Gassman equations: an overview

Gassman’s (1951) equations provide a way to calculate the bulkmodulus of a fluid-saturated porous medium using the knownbulk moduli of the solid matrix, of the frame, and of the pore fluid.For a rock, the solid matrix consists of the rock-forming minerals,the frame refers to the rock sample with empty pores (dry rock),and the pore fluid can be a gas, oil, water, or a mixture. The equa-tions also express that the shear modulus is not affected by fluidsaturation.

Assumptions in Gassman’s equations

“The rock-fluid system is so complicated that virtually all thetheories for such a system have to make major assumptions to sim-plify the mathematics. “ (Wang 2000, pp 9,10)

So what can be said about the uses and limitations ofGassmann’s equation?

The Gassmann equation works reasonably well for rocks withinterconnected high aspect-ratio pores such as unconsolidated




Figure 6: 2D Modeled (Lambda*Rho - Mu*Rho) Difference Section.The pre-stack modeling is based on well log data and rock physics anal-yses.


clean sands and sandstones at high effective pressures. (“worksreasonably well” means little difference exists between theGassmann-calculated and laboratory-measured seismic velocities).

Carbonate rocks have strong elastic frames and very differentpore systems compared to siliciclastic rocks. Because many poresin carbonate rocks are not well connected, the Gassmann equationis in general inadequate for carbonate rocks.

For rocks with flat pores, cracks, or fractures, and rocks saturat-ed with high-viscosity fluids, Assumption 5 cannot be satisfied, sothe Gassmann-calculated Vp is always less than the measured Vp.This is because such pore shapes do not allow room for the fluid toequilibriate in the half-wavelength time period required by theequation.

At high pressures, the flat pores, cracks, or fractures are closed,so then the Gassmann-calculated velocities agree better with themeasured values.

In unconsolidated or poorly consolidated sands, the measuredshear modulus of the dry frame usually has high uncertaintybecause of high shear wave attenuation. A 10% uncertainty in dryframe shear modulus yields a 2% uncertainty in the Gassmann-cal-culated Vp.

Biot

Gassman’s equations are not adequate to calculate frame mod-uli in the high frequency range of laboratory data. Biot’s (1956) the-ory includes the entire frequency range up to the point where thegrain scattering becomes important and the rocks can no longer beconsidered homogeneous. Gassmann’s equations are the low fre-quency limit of Biot’s more general relationships. Note that for thecase of perfect coupling, Biot’s equations reduce to the zero-fre-quency case (Gassmann’s Assumption #5 is met).




1. The rock (both the matrix and the frame) is macroscopically homogeneous and isotropic.

2. All the pores are interconnected or communicating.

3. The pores are filled with a frictionless fluid (liquid, gas, ormixture)

4. The rock-fluid system under study is closed (undrained).

5. When the rock is excited by a wave, the relative motionbetween the fluid and the solid rock is negligibly small compared to the motion of the whole saturated rock itself.

6. The pore fluid does not interact with the solid in a way thatwould soften or harden the frame.

The basic assumptions in the Gassmann equation.(The devil is in the details)

What does this mean?

This common assumption ensures that the wavelength is longcompared to the grain and pore sizes. Most rock can generallymeet this assumption for seismic (20-200 Hz) to laboratory fre-quencies (100 kHz - 1 MHz).

This implies that porosity and permeability are high. The rea-son behind this assumption is to ensure full equilibrium of thepore fluid flow induced by the passing wave can be attainedwithin the time frame of half a wave period. For seismicwaves,only unconsolidated sands can approximately meet thisassumption because of the finite wavelength.

The viscosity of the saturating fluid is zero. The purpose of thisassumption again is to ensure full equilibrium of the pore fluidflow. In reality, because all fluids have finite viscosities and all waves have finite wavelengths, most calculations using the Gassmann equation will violate thisassumption.

For a lab rock sample, this means that the rock-fluid system issealed so that no fluid can flow in or out of the rock’s surface.For a reservoir rock, the volume v which is under study mustbe part of a much larger volume V, and be located far enoughfrom the surface of V that the passing seismic wave does notcause any apprreciable flow through the surface of v.

This key assumption is the essence of the Gassmann equation. Itrequires that wavelength be infinity (or the frequency be zero).It is also perhaps the reason why the measured bulk modulusor velocity are usually higher than those calculated by theGassmann equation. This is because at high frequencies, relative motion between the solid matrix and pore fluid willoccur so that 6the waves are dispersive.

In reality, the pore fluid will interact with the rock’s solidmatrix to change the surface energy. When a rock is saturatedby a fluid, the fluid may either soften or harden the matrix.


Further Refinements

Much of the ongoing research work in Rock Physics is toapproach physical reality more closely. For example, the Biot equa-tions say that the fluid must participate in the solid’s motion by vis-cous friction and inertial coupling, but it is known that fluid alsosquirts out of pores when deformed by a passing seismic wave.Traditionally, the Biot mechanism has been treated macroscopically,and the squirt-flow mechanism at the individual pore level. Workby Dvorak and Nur (1992) offers a model which treats both mecha-nisms as coupled processes and relates Vp and attenuation tomacroscopic parameters: the Biot poroelastic constants, porosity,permeability, fluid compressibility and viscosity, and a new micro-scopic-scale parameter - a fundamental and measurable character-istic squirt-flow length. Such local flow models are representative ofcurrent work to extend Gassmann’s description to include morerealistic portrayals of rocks, and to relative poroelastic behavior tomacroscopic measurable parameters such as permeability, porosity,saturation, pore-fluid compressibility, density, and viscosity).Please see ‘For Further Reading’ for some excellent papers in thisregard.

Issues in Rock Physics

The challenge of scale

The scales at which Geophysics and Petrophysics work are verydifferent. Logs and cores give resolution less than 0.3 metres, whileseismic resolution is often no better than 15 metres. This may beexpressed in terms of the frequency ranges used:

• The range of seismic frequencies is typically considered to be20 - 200 Hz (and more realistically 10 - 80 Hz)

• Logging frequencies are around 10 kHz

• Laboratory frequencies have traditionally been 100 kHz - 1MHz. Also note that in the lab, it has been nearly impossibleto carry out wave propagation measurements at low (seismicrealm) frequencies as the sample length is required to be atleast half a wavelength. For a rock with 4000m/s Vp, a 50Hzwave would require a 40m sample. However, recent promis-ing advances in Rock Mechanics have succeeded in velocitymeasurements in the lab at the 100Hz range, which is verypromising indeed (Lewis Lacy, Director of Geomechanics,CoreLab Rock Mechanics Lab, personal communication).

The challenge of focus

Advances in computing power mean that efforts are turningfrom the traditional seismic task of getting an accurate image ofsubsurface structures to extracting more and more rock propertyinformation from the seismic wavelet. Computing power also ben-efits Reservoir Modeling, and as more complex dynamic simula-tions become possible, real integration of seismic and reservoircharacterization gets closer. Establishing accurate relationsbetween rock properties and acoustic parameters is becoming moreimperative. “If the trend in computering power continues, a 3-D

earth model with geological, petrophysical, and geophysical data inall grid-blocks will be available soon. This ‘unified model’ wouldhave the resolution of cores near the well bore; of logs in most otherplaces; and will be used for both static and dynamic modeling.”(Peeters)

The challenge of calibration

Seismic sections that show rock and fluid properties will bemore meaningful if they can be calibrated with forward modelingfrom accurate rock properties measured at seismic frequencies.There is a huge difference between a 50 Hz seismic wavlet (approx-imately 40 m resolution) and a 10 kHz sonic wireline tool (20 cm).Which high frequency results can be extrapolated to the seismicrealm?

The lack of measured shear velocities

Dipole logs are not commonly performed, and estimates of Vscarry uncertainty, particularly in unconsolidated sediments.

The promise of the prize

Just a few of the problems facing the geoscientist are mentionedabove.There are plenty of issues and shortcomings with RockPhysics, as with any discipline handling complex physical process-es. Nonetheless, the Rock Physics effort is progressing in a validdirection. The achievement of rock properties from seismic data,integrated with wireline, petrophysical, and geologic knowledge isa goal that is becoming more attainable. Rock physics draws togeth-er the disciplines of Geophysics and Petrophysics, bringing thepossibility of a unified 3-dimensional earth model within reach. R




Biography

Jan Dewar graduated from theUniversity of Alberta in 1981 with aB.Sc. in Physics. Jan is currently work-ing with Scott Pickford in Calgary,with a special enthusiasm for commu-nicating technical concepts includingAVO, Inversion, Modeling, VSP andTransfer Filter processing, RockPhysics, and just about anything else

that can be puzzling to the average [email protected]


Acknowledgments

I am indebted to my colleagues Yongyi Li, Alvaro Chaveste,and Michael Burianyk for their assistance.

References & Further Reading

Biot, 1956, Theory of propagation of elastic waves in a fluid saturatedporous solid, 1. Low frequency range, J. Acoust. Soc. Am., 28, 179-191.

Castagna, J., 1993, AVO Analysis - Tutorialand Review, Chapter one of Offset-dependentreflectivity - theory and practice of AVO analy-sis, Investigations in Geophysics No. 8, SEG.

Castagna, J.,Han, D., & Batzle, M.L., 1995,Issues in rock physics and implications for DHIinterpretation, The Leading Edge, August1995.

Dvorkin, J., & Nur,A., 1993, Dynamic poroe-lasticity: A unified model with the squirt and theBiot mechanisms, Geophysics 58, 524-533.

Dvorkin, J., Nolen_Hoeksems, R. & Nur, A.,1994, The squirt-flow mechanism: Macroscopicdescription, Geophysics, 59, 428-438.

Gassmann,F., 1951, Elastic waves through apacking of spheres: Geophysics, 16, 673-685.

Kelder, O., & Smeulders, D.M.J., 1997,Observation of the Biot slow compressional wavein water-saturated Nivelsteiner sandstone,Geophysics, 62, p 1794.

Peeters, Ir. M., Physical Reservoir Models:From Pictures to Properties, an address pre-sented at the commencement of the BakerHughes Distinguished Chair ofPetrophysics and Borehole Geophysics,Colorado School of Mines,

http://www.geophysics.mines.edu/max/Resmod.html

Wang, Z., 2000, The Gassmann equation revisited: Comparing labo-ratory data with Gassmann’s predictions, Seismic and AcousticVelocities in Reser voir Rocks, Vol. 3, Recent Developments, SEGReprint Series, pp 1-23.



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Whether you are looking for the natural fractures or variationsof fluid properties from your producing fields,CGG’s 4D/4C Processing is the answer.

Radial / transverse energyratio time slice

Fast S1 axisdirectionconfirmedon panoramicreceiver stack.

Determine the naturalfracture orientation fromthe shear wave analysisof 4C seismic data. Inthis example, S1 wavesindicate a primaryfracture orientation of104 degrees from North.

-57100.0

-57200.0

-57400.0

-57500.0

570000.0

-57300.0 284°

104°

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Time-Lapse multi-component seismic analysis provides a direct image ofthe evolution of the fluid properties from the producing reservoir.

Initial Survey Time-Lapse Survey

Amplitude difference mapover the reservoir interval

Increase in amplitude in the Time-Lapse dataindicates gas coming out of solution. Also notea separate compartment which appears to beconnected to the main reservoir.

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May01 Rock Physics