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GEOPHYSICS, VOL. 69, NO. 2 (MARCH-APRIL 2004); P. 398405, 8 FIGS., 2 TABLES.10.1190/1.1707059

Gassmanns equation and fluid-saturation effects on seismic velocities

De-hua Han and Michael L. Batzle

ABSTRACT

Gassmanns (1951) equations commonly are used topredict velocity changes resulting from different pore-

fluid saturations. However, the input parameters are of-ten crudely estimated, and the resulting estimates offluid effects can be unrealistic. In rocks, parameters suchas porosity, density, and velocity are not independent,and values must be kept consistent and constrained.Otherwise, estimating fluid substitution can result insubstantial errors. We recast the Gassmanns relationsin terms of a porosity-dependent normalized modu-lus Kn and the fluid sensitivity in terms of a simpli-fied gain function G . General Voigt-Reuss bounds andcritical porosity limits constrain the equations and pro-vide upper and lower bounds of the fluid-saturation ef-fect on bulk modulus. The D functions are simplifiedmodulus-porosity relations that are based on empiricalporosity-velocity trends. These functions are applicable

to fluid-substitution calculations and add important con-straints on the results. More importantly, the simplifiedGassmanns relationsprovidebetter physicalinsight intothe significance of each parameter. The estimated mod-uliremain physical,the calculationsare more stable, andthe results are more realistic.

INTRODUCTION

With improved resolution and cost efficiency, seismic tech-nologies havegained a central position in reservoir delineationand monitoring. Rock physics is an essential link connectingseismic data to the presence of in situ hydrocarbons and toreservoir characteristics. Modeling the effects of fluid on rockvelocity and density is a basic method used to ascertain the in-

fluence of porefluids on seismic data.Gassmanns (1951) equa-tions are the relations most widely used to calculate seismic-velocity changes resulting from different fluid saturations inreservoirs. These equations predominate in the analysis of di-rect hydrocarbon indicators (DHI), such as amplitude bright

Manuscript received by the Editor September 19, 2001; revised manuscript received July 26, 2003.University of Houston, Department of Geoscience, 312 Science & Research Building, Houston, Texas 77204-5007. E-mail: [email protected] School of Mines, Department of Geophysics, Golden, Colorado 80401-1887. E-mail: [email protected] 2004 Society of Exploration Geophysicists. All rights reserved.

spots, amplitude variation with offset (AVO), and time-lapsereservoir monitoring.

Despite the popularity of Gassmanns equations and theirincorporation in most software packages for seismic reservoirinterpretation, important aspects of these equations have notbeen thoroughly examined. Many of the basic assumptions areinvalid for common reservoir rocks and fluids. Previous effortsto understand the operation and application of Gassmannsequations (Han, 1992; Mavko and Mukerji, 1995, Mavko et al.,1998; Sengupta and Mavko, 1999) have focused mostly on in-dividual parameter effects.

Recently, Nolen-Hoeksema (2000) made a detailed effortto quantify changes in the pore-space modulus in response tochanges in fluid modulus. He introduced an effective fluid co-efficient by differentiating the bulk modulus of the fluid-filledpore spaceKpore. He included two ratios to control the effec-tive fluid coefficient: the ratio of fluid modulus to solid grainmodulus andthe ratio of theBiot coefficient (1941) to porosity.However, the physical meaning of the effective fluidcoefficientwas not clarified. The fluid effect was in conjunction with other

rock parameters. TheVoigt and Reuss models wereintroducedto test the fluid effect. We need to extend his analysis to deriveboth the mechanical bounds for porous media and the magni-tude of the fluid effect.

Because the full implications of parameter interactions inGassmanns equation are not well understood, in general prac-tice, no constraints are placed on input parameters and there isno quality control of the results. In particular, problems arisein automated analysis in which results are usually taken atface value. In this paper, we briefly list the assumptions forGassmanns equation. We then derive mechanical bounds forthe input parameters that provide stricter constraints on cal-culated results. The specific physical properties controlling thefluid-saturation effect are then better understood.

Effect of fluid saturation on seismic properties

The seismic response of reservoirs is directly controlled bycompressional (P-wave) and shear (S-wave) velocitiesVp andVsrespectively along with densities. Figure 1a shows measured

398


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Gassmanns E & Fluid Effects on V 399

dry and water-saturated P- and S-wave velocities of sandstoneas a function of differential pressure. With water saturation,P-wave velocity increases slightly, whereas S-wave velocity de-creases slightly. However, neither P- nor S-wave velocity is thebest indicator of any fluid saturation effect because of the cou-pling between P- and S-waves through the shear modulus and

bulk density:

Vp =

K+ 4/3

=

M

,

Vs =

, (1)

where K is bulk modulus, is shear modulus, Mis the com-pressional modulus, and is the bulk density. In contrast, if weplot bulk andshear modulias functionsof pressure(Figure1b),the water-saturation effect becomes evident: (1) the bulk mod-ulus increases about 50%, whereas (2) shear modulus remainsconstant.

Clearly, bulk modulus is more sensitive to water saturation.The bulk-volume deformation produced by a passing seismic

wave results in a pore-volume changeand causesa pressure in-crease in pore fluid (water). This pressure increase stiffens therock frame and causes an increase in bulk modulus. Shear de-formation, however, does not produce a pore-volume change,and consequently different fluids do not affect shear modulus.Therefore, any fluid-saturation effect should correlate mainlyto a change in bulk modulus.

Gassmanns equation

Gassmanns equations provide a simple model for estimat-ing the fluid-saturation effect on bulk modulus. We prefer the

Figure 1. (a) Measured P- and S-wave velocities on a sandstone sample at dry and water-saturated states, as a function of pressure.(b) Bulk and shear moduli at dry and water-saturated states, as a function of pressure.

following Gassmanns formulation because of its clearphysicalmeaning:

Ks = Kd+ Kd

Kd=K0(1 Kd/K0)

2

1

Kd/

K0+

K0/

Kf

, (2)

s = d, (3)

whereK0,Kf ,Kd,Ks , arethe bulk moduliof themineral grain,fluid, dry rock, and saturated rock frame, respectively; isporosity; ands anddare the saturated and dry-rock shearmoduli. Kdis an increment of bulk modulus as a result offluid saturation of dry rock. These equations indicate that fluidin pores will affect bulk modulus but not shear modulus, whichis consistent with our earlier discussion. As Berryman (1999)pointed out, a shear modulus that is independent of fluid sat-uration results directly from the assumptions used to deriveGassmanns equation.

Numerous assumptions are involved in the derivation andapplication of Gassmanns equation:

1. the porous material is isotropic, elastic, monomineralic,and homogeneous;

2. the pore space is well connected and in pressure equilib-rium (zero-frequency limit);

3. the medium is a closed system with no pore-fluid move-ment across boundaries;

4. there is no chemical interaction between fluids and rockframe (shear modulus remains constant).

Many of these assumptions may not be valid for hydrocar-bon reservoirs and depend on rock and fluid properties andthe in-situ conditions. For example, most rocks are anisotropicto some degree, invalidating assumption (1). The work ofBrown and Korringa (1975) provides an explicit form for an


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400 Han and Batzle

anisotropic fluid substitution. Also, in seismic applications, itis normally assumed that Gassmanns equation works best forseismic data at frequencies less than 100 Hz (Mavko et al.,1998). Recently published laboratory data (Batzle et al., 2001)show that acoustic waves may be dispersive in rocks within thetypical seismic band,invalidatingassumption (2).In such cases,

seismic frequencies may still be too high for the application ofGassmanns equation. Pore pressures may not have enoughtime to reach equilibrium, and the rock remains unrelaxed oronly partially relaxed. A Gassmann-type calculation providesan estimate of relaxed velocity at zero frequency, which is alower bound of the fluid-saturation effect. Other similar vio-lations of the assumptions often lead to the misapplication ofGassmanns equation. We will discuss this issue in detail in aseparate paper.

SIMPLIFIED GASSMANNS EQUATION

Gassmanns formulation is straightforward, and the simpleinput parameters typically can be directly measured or as-sumed. This simplicity is a primary reason for its wide applica-tion in geophysical techniques. However,derivation of the rockand fluid input parameters frequently remains ambiguous. Asa consequence, we frequently have littlecontrol over thevalid-ity of fluid-substitution calculations. We have made efforts toclarify how the fluid-saturation effects are controlled by rockparameters (Han, 1992). A simple graphic construction devel-oped by Mavko and Mukerji (1995) introduces an interceptporosityR for a dry rock that is based on bulk porosity, dry-bulk modulus, and solid-frame modulus. However, the physi-cal meaning of the intercept porosity is not clear. Additionally,multiple-parameter effects on Gassmanns calculation remainambiguous. As a consequence, for application of Gassmannscalculation, generallywe havenot beenable either to constraininput parameters or to obtain quality control on the results. Inthis section, we regroup Gassmanns equation with combinedrock parameters.Under certain conditions, we can simplify this

equation further and clearly define the controlling parametersfor fluid-saturation effects.

The primary measure of a rocks velocity sensitivity to fluidsaturation is its normalized modulusKn , the ratio of dry bulkmodulus to that of the mineral:

Kn = Kd/K0. (4)

This function can be complicated and depends on rock texture(porosity, clay content, pore geometry, grain size, grain con-tact, cementation, mineral composition, and so on) and reser-voir conditions (pressure and temperature). This Kn can bedetermined either empirically or theoretically. For relativelyclean sandstone at high differential pressure (>20 MPa), thecomplex dependence of Kn(x,y,z, . . .) can be simplified as a

function of porosity:

Kn(x,y,z, . . .) = Kn(). (5)

From equation (2), the bulk-modulus increment Kdis thenequal to

Kd=K0 [1 Kn()]

2

1 Kn()+ K0/Kf(6)

where (1 Kn()) is the Biot parameter (Biot, 1941). TheBiot parameter is a relative measure of the difference be-tween the mineral and dry-frame moduli. Furthermore, if weapply the Voigt bound forKn , and because usually K0 Kf , itis reasonable to assume that

0 1 Kn() K0/Kf (7)

for sedimentary rocks with high porosities ( >15%). There-fore, the fluid-saturation effect of the Gassmann equation canbe simplified as

Ks = Kd+ Kd Kd+ G() Kf , (8)

whereG () is the simplified gain function defined as

G() =[1 Kn()]

2

=

2

. (9)

Equation (8)is a simplifiedform of Gassmanns equation, withclear physical meaning: fluid-saturation effects on the bulkmodulus are proportional to a simplified gain function G()and the fluid modulus Kf . The G () in turn depends directlyon dry-rock properties: the normalized modulus and porosity.In general, G () is independent of fluid properties (ignoringinteractions between rock frame and pore fluid). Equation (9)also shows that the normalized modulus or the Biot parametermust be compatible with porosity. Otherwise,G() can be un-stable, particularly at small porosities. We need to know boththesimplified gain function of thedry rock frame andthe pore-fluid modulus to evaluate the fluid-saturation effect on seismicproperties.

Figure 2 shows that for sandstone samples (Han, 1986) athighdifferentialpressure(>20MPa), theKsof water-saturatedsands calculated using the simplified form is overestimated by3% for porosities greater than 15%. These errors will decreasesignificantly with a low fluid modulus (gas and light-oil satura-

tion). For low-porosity sands with high clay content, the sim-plified Gassmannsequation substantiallyoverestimates water-saturation effects.

This simplified Gassmanns equation has a clear physicalmeaning with systematic errors. This formulation can guide usin applying Gassmanns equation and in assessing the validityof such calculations.

Figure2. ThesimplifiedGassmannsformulationoverestimatesthe water-saturation effect on bulk modulus of sandstone sam-ples (1986) at a differential pressure greater than 20 MPa. Forporous sands (porosity>15%), the error is around 3%.


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CONSTRAINTS ON GASSMANNS EQUATION

The assumptions contained in Gassmanns equation do notconstrain basic rockparameters. In equations(2), thereare fiveparameters, and usually the only applied constraint is that theparameters are physically meaningful (>0). When one applies

Gassmanns equation, one generally handles input parametersas being completely independent. Values for K0 and Kf areestimated or assumed.Ksor Kdare calculated fromVpand Vs ,and density either comes from log data or is estimated alongwith. Incompatible or mismatched dataoften generatewrongor even unphysical results, such as a negative modulus. Unre-alistic results can be hidden when one is performing a fluidmodulus substitution without examiningKd. In reality, onlyK0and Kfare completely independent. Ks , Kd, and are actu-ally closely correlated. Bounds on Kdas a function of , forexample, constrain the bounds ofKs .

The Voigt bound

Assuming the porous medium is a Voigt material, which is ahigh bound forK

d,

Kd= K0 (1 ). (10)

Substituting equation (10) into Gassmanns equation (2) gives

Kdmin = Kf, (11)

and

Ks = Kd+ Kdmin = K0 (1 )+ Kf . (12)

Because this Voigt bound is the stiffest upper limit, the fluid-saturation effect on bulk modulus here (Kdmin) will be aminimum (Figure 3). This is the first constraint derived fromthe Gassmanns equation: The minimum of bulk modulus in-crement resulting from fluid saturation is proportional to the

porosity of the rock(the simplifiedgain functionG()=)andthe modulus of the pore fluid.We know that fluid modulus is a positive value, butKf K0.

Therefore, from equations (10) and (11),

K0 Ks Kd. (13)

The Reuss bound

The low modulus bound for porous media is the Reussbound:

1

KR=

(1 )

K0+

Kf, (14)

KR =K0 Kf

(1 ) Kf+ K0. (15)

For completely empty(dry)rocks, thefluid modulusKf isequalto zero. Thus, both the Reuss bound and the normalized mod-ulus (KnR ) for a dry rock in this case equals zero (for nonzeroporosity):

Kn R() = Kd/K0 = 0. (16)

Substituting equation (16) into the Gassmanns equation (2),we find the fluid-saturation effect on bulk modulus when the

frame is at this lower bound:

Kdmax =K0

1 + K0/Kf= KR. (17)

Note that the modulus incrementKfrom dry to fluid satura-tion is equal to the Reuss bound (equation 15):

Ks = Kd+ Kdmax = KR. (18)

Again, Gassmanns equation is consistent with the dry andfluid-saturated Reuss bounds. The Kdmax is the maximumfluid-saturation effect predicted from Gassmanns equation(Figure 3). Physically, with the weakest frame, fluids have amaximum effect. It is interesting that for the Voigt bound, thefluid-saturationeffect on the modulus increaseswith increasingporosity. This is opposite of the fluid-saturation effect on themodulus for the Reuss bound, which decreases with increas-ing porosity. At a porosity of 100%, both the Voigt and Reussboundsin theGassmanns calculation show that thefluid effecton modulus equals the fluid modulus.

In many reported applications, these general bound values

have often been ignored. For example, aKscalculated directlyfrom log data (Vp, Vs , and bulk density) may be lower thanthe Reuss bound. This results in a negative value forKd. Suchbounds of the fluid effect on bulk modulus provide constraintsfor the inputand output parameters of Gassmanns calculation.

Critical porosity

Reservoir rocks in general are far from the Voigt and Reussbounds, as Figure 4 shows forsandstones. Dolomite with vuggypores may approach the Voigt bound, and highly fracturedrocks may approach the Reuss bound. However, there is agreat difference between these idealized bounds and most realrocks: The bounds shown in Figure 4 do not limit most ob-

served, naturally occurring porosity. Thevast majority of rockshave an upper limit to their porosity, usually termed criti-cal porosity, c (Yin, 1992; Nur et al., 1995). At this high-porosity limit, we reach the threshold of grain contacts (Han,1986). Thisc modifies the Voigt model (Figure 3) to providetighter constraints for dry and fluid-saturated bulk moduli forsands. This triangle physically correlates with both the Voigtand Reuss bounds. The fluid-saturation effect on modulus isconsistent with the Voigt triangle: It increase with increasingporosity andis limited by theReussbound at thecriticalporos-ity. The modified Voigt triangle provides a linear formulationand a graphic procedure for Gassmanns calculation: the fluidsaturation effect on bulk modulus is proportional to normal-ized porosity and the maximum fluid saturation effect on bulkmodulus (Reuss bound) at the critical porosity (Figure 3):

Kd= /c KRc. (19)

This is consistent with the earlier work done by Mavko andMukerji (1995), with a slightly different physical formulation.

For typical sandstones,the critical porosityc isaround 40%.Thus, we can also generate a simplified numerical formula ofthe normalized modulusKnfor the modified Voigt model:

Kn() = 1 /c = 1 2.5 . (20)


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402 Han and Batzle

Using this in Gassmanns equation (6) yields the fluid-saturation effect

Kd=6.25 K0

1.5+ K0/Kf


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high pressures (>20 MPa), K0 for sandstone samples rangesfrom 33 to 39 MPa. K0 is not a constant and can increase bymorethan 10%with increasingeffective pressure. Clearly, min-eral modulus K0 can vary across a wide range, depending onmineral composition, distribution, and in situ conditions.

Figure 5. Normalized modulus based on Voigt and Reussbounds and empirical D functions for different rocks (SeeTable 1).

Figure 6. Comparison of the Gassmann and simplifiedGassmann calculation for brine-saturation effect (brine modu-lusKf= 2.8 GPa) forthe modified Voigt bound,withc = 40%and D = 2 function model.

Table 1. Compiled empirical relations and relativeD-por. models for different rocks.

Kd= (1 A + B 2 C 3) K0

V-emp. relation Kd= (1 D )2 K0

Rock type Pe = 40 MPa A B C D

Dry shaly sandstone Vp = 5.41 6.35 3.053 3.070 1.016 1.450Vs = 3.57 4.57 (K0 = 32.5 GPa)

Dry clean sandstone Vp = 5.97 7.85 3.206 3.349 1.143 1.523Vs = 4.03 5.85 (K0 = 37.0 GPa)

Silicate clastic Vp = 5.81 9.42 3.283 3.284 1.014 1.584(Castagna et al., 1985) Vs = 3.89 7.07 (K0 = 36.0 GPa)Dry vuggy limestone Vp = 6.47 5.84 2.815 2.639 0.824 1.340

Vs = 3.39 3.03 (K0 = 71.9 GPa)Dry limestone Vp = 6.19 9.80 4.244 5.820 2.605 1.970

Vs = 3.20 4.90 (K0 = 66.8 GPa)Dry dolomite Vp = 6.78 9.80 3.578 4.020 1.358 1.705

Vs = 3.72 5.20 (K0 = 94.4 GPa)

*Kdunits areV-emp. relation.

Figure 7 shows the influence of K0 on Gassmanns calcu-lation. This case uses a dry bulk modulus calculated with amineral modulus of 40 GPa and D = 2 in the D function. Thewater-saturation effect was calculated for three mineral mod-uli, of 65, 40, and 32 GPa, and a water modulus of 2.8 GPa.

Figure 7. Different mineral-frame bulk-modulus effects on cal-culated bulk modulus with water saturation (brine modulus:Kf= 2.8 GPa) and D = 2 function model.

Figure 8. Fluid-saturation effect on calculated bulk modulus,with typical water, oil, and gas saturation and D= 2 functionmodel.


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404 Han and Batzle

Results show that for the sameKdand Kf , the bulk-modulusincrementKdresulting from fluid saturation increases withincreasing mineral modulus K0. Errors resulting from uncer-tainty ofK0can be significant for low-porosity rocks.

Because of a lack of measurements on mineral moduli, of-ten we must use measured velocity-porosity-clay-content rela-

tionships for shaly sandstone (Han, 1986) to estimate the min-eral modulus. Assuming zero porosity and a grain density of2.65 g/cm3, we can derive mineral bulk and shear moduli frommeasured P- and S-wave velocities. The results are shown inTable 2.

For relatively clean sandstones (with clay content of a fewpercent), the mineral bulk modulus ( K0) typically is 39 GPa,which is a stable value for differential pressures higher than20 MPa. Mineral shear modulus (0) is around 33 GPa, whichis significantlylessthan the44 GPafor a pure-quartz aggregate.Shear modulus is more sensitive to differential pressure and toclay content. For shaly sandstones, the mineral bulk modulusdecreases about 1.7 GPa per 10% increment of clay content.Derived mineral bulk modulus canbe used forthe Gassmannscalculation, if there are no directly measured data or reliable

models for calculation.Lithology detection is often a goal of seismic interpretation.

In modeling the seismic response, we often face the challengeof how to separate the fluids influence from the lithology ef-fect. To estimate this effect, we need to perform lithologysubstitution by using different values for the simplified gainfunction in Gassmanns equation:

K21() [G2() G1()] Kf . (27)

We should incorporate this factor into the fluid-substitutionscheme.However, this process is not commonly performedandis still poorly understood.

FLUID MODULUS AND FLUID-SATURATION EFFECTS

The fluid modulus is another key independent parameter in

Gassmanns equation. Because identification of fluid types isoften the primary goal of a seismic program, fluid propertiesare of critical significance. However, fluid properties are oftenoversimplified in seismic applications. Although complex, fluidpropertiesare systematic. Oil propertiesdepend on density (orAPI gravity), gas-oil ratio (GOR), gas gravity, pressure, andtemperature conditions. Under different conditions, the fluidphaseand its seismic propertiescan varydramatically (HanandBatzle, 2000a, b; Batzle andWang, 1992). As we mentionedpre-viously, Gassmanns prediction is approximately proportionalto the fluid bulk modulus Kf , with different gain functions.Thus, the stiffer the pore fluid, the higher the bulk modulus.

Table 2. Grain bulk and shear modulus for shaly sands; min-eral modulus derived fromempiricalvelocitiesrelation of shaly

sandstones. Cis the fractional clay content.

C= 0 C= 0.1 C= 0.2

Pd K0 0 K0 0 K0 0(MPa) (GPa) (GPa) (GPa) (GPa) (GPa) (GPa)

40 39.03 32.83 37.27 29.40 35.51 26.1630 39.08 31.91 37.26 28.56 35.44 25.4020 39.27 30.45 37.30 27.29 35.35 24.3010 38.74 26.46 36.72 25.73 34.72 22.94

Fluid substitution is a primary application of Gassmannsequation. With a change of fluid saturation from fluid 1 to fluid2, the bulk modulus increment is equal to

K21() G() (Kf2 Kf1), (28)

where Kf1 and Kf2 are the moduli of fluids 1 and 2, respec-tively, and K21 represents the change in increment causedby substituting fluid 2 for fluid 1. Equation (28) uses the factthat the simplified gain function G() of the dry-rock frameremains constant as the fluid modulus changes (this may notbe true for some real rocks). The fluid-substitution effect onbulk modulus is simply proportional to the difference in fluidbulk moduli. This form of fluid substitution is similar to thatderived by Mavko et al. (1998):

Ks1

K0 Ks1

Kf1

(K0 Kf1)=

Ks2

K0 Ks2

Kf2

(K0 Kf2)=

Kd

K0 Kd. (29)

Note that the fluid-substitution effect calculatedthrough equa-

tion (29) is based on dry-frame properties. Deriving the drymodulus often helps us to examine validity of input parame-ters and output results of Gassmanns calculation.

If we know the simplifiedgain function fora rock formation,we can estimate the fluid-substitution effect without knowingshear modulus:

2V2

p2 1V2

p1 + G() (Kf2 Kf1), (30)

where 1, 2, Vp1 and Vp2 are the densities and velocities ofrock saturated, respectively, with fluid 1 and fluid 2. Bothequations (28) and (30) are direct results from simplifiedGassmanns equation (equation 8). Mavko et al. (1995) havesuggested a similar method.

In Figure 8, we show the typical fluid-modulus effect on the

saturated bulk modulusKs . Even at a modest porosity of 15%,changes canbe substantial.At in situ conditions,porefluidsareoften multiphase mixtures. A dynamic fluid modulus may alsodepend on fluid mobility, fluid distribution, rock compressibil-ity, and seismic wavelength.

Another approach is to use the P-wave modulus (M) to re-place bulk modulus in Gassmanns equation. This works rea-sonably well for sandstones. The validity of this simplificationresults from the approximate equivalence of the ratio of dry-frame bulk and shear modulus (Kd/d) to the ratio of mineral(quartz) bulk and shear modulus (K0/0).

ESTIMATING DRY BULK MODULUS

To estimate saturation effects on rocks practically, we needto obtain the dry bulk modulus. From Gassmanns equation,

we deriveKd fromfluid-saturatedKsandfluid-saturationeffectKs :

Kd= Ks Ks = Ks K0 (1 Ks/K0)

2

K0/Kf+ Ks/K0 1 ,

(31)

where the fluid-saturation effectKs is based on measuredfluid-saturated modulus Ks and results in a slightly differentformulation forKd in equation (2). If we reformulateKs


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with the Reuss boundKRof the rock with saturated fluid,

Ks =KR (K0 Ks)

2

K20 + Ks KR 2 K0 KR KR. (32)

Again, the Reuss bound provides the maximum fluid-saturation effect.

CONCLUSIONS

Gassmanns equations are used widely to calculate fluid-substitution effects.Unfortunately,t he underlying assumptionsare often violated, and the validity of the resulting calculationsis unknown. Several factors canbe incorporatedinto theanaly-sis to makethe results morephysically meaningful and reliable.

1) A simplified form of Gassmanns equation clarifies thephysical control of fluid-saturation effects on rock bulkmodulus, through thesimplified gain function of dryrock(dependent on the normalized dry bulk modulus andporosity) and fluid modulus. Rock parameters, such asnormalized dry bulk modulus and porosity, are stronglycorrelated, which effectively reduces the number of free

parameters.2) The dry and fluid-saturated Voigt-Reuss bounds of bulk

modulus provide physical limitations on Gassmannsequation. The minimum increment of bulk modulus re-sulting from fluid saturation, consistent with the Voigtbound,is proportionalto porosity andfluid modulus. Themaximum increment of bulk modulus with changing sat-uration, consistent with the Reuss bound, is equivalentto the Reuss bound of fluid-saturated rock itself. This isbecause the Reuss bound of dry rock is zero.

3) The normalized bulk modulus can be a complicatedfunction of rock textures and in situ conditions, whichmay lead to wide variations in the results of apply-ing Gassmanns equation. However, simplified modulus-porosity trends can be incorporated, resulting in simple

polynomial dependenceon porosity in sandstones. Fluid-substitution effects are then a straightforwardfunction ofporosity and the difference in fluid moduli.

4) The calculated dry-frame modulus or mineral moduluscan be in substantial error if rock properties are not con-sistent or assumptions are violated.

Although these relationships have been applied for manyyears, considerable basic research still needs to be done onmany of the controlling factors. Fundamental components,such as the mineral moduli (particularly for clays), are rarelymeasured. The exact character of the pore-volume modulus(Brownand Korringa, 1975)is unknown, andthe validity of re-placing it withthe mineral modulus is ambiguous. Theinfluenceofmixedfluidphasesandfluidmobilityisalsonotincorporated.Additionally, theory applies strictlyto thelow-frequencyrange,

thereby permitting no frequency dependence. With increasingapplication of seismic data to extracting and predicting reser-voir and fluid properties, we will need more constrained andtested forms of Gassmanns relations and other porous-mediatheories.

ACKNOWLEDGMENTS

We thank the industry sponsors of Phase III Fluid Consor-tium for guidance and financial support. We also thank theeditor and reviewers for their comments and suggestions.

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