GEOPHYSICS, VOL. 64, NO. 5 (SEPTEMBER-OCTOBER 1999); P. 1627–1629

Tutorial

Origin of Gassmann’s equations

James G. Berryman∗

INTRODUCTION

Gassmann’s relations are receiving more attention as seismicdata are increasingly used for reservoir monitoring. Correctinterpretation of underground fluid migration from seismicdata requires a quantitative understanding of the relationshipsamong the velocity data and fluid properties in the form offluid substitution formulas, and these formulas are very com-monly based on Gassmann’s equations. Nevertheless, confu-sion persists about the basic assumptions and the derivationof Gassmann’s (1951) well-known equation in poroelasticityrelating dry or drained bulk elastic constants to those for fluid-saturated and undrained conditions. It is frequently stated, forexample, but quite incorrect to say that Gassmann assumesthe shear modulus is constant (i.e., mechanically independentof the presence of the saturating fluid). This note clarifiesthe situation by presenting an unusually brief derivation ofGassmann’s relations that emphasizes the true origin of theconstant shear modulus result, while also clarifying the roleplayed by the shear modulus in the derivation of the betterunderstood result for the bulk modulus.

DERIVATION FOR ISOTROPIC POROUS MEDIA

I now present a very concise, but nevertheless complete,derivation of Gassmann’s famous results. For the sake of sim-plicity, the analysis that follows is limited to isotropic systems,but it can be generalized with little difficulty to anisotropic sys-tems (Gassmann, 1951; Brown and Korringa, 1975; Berryman,1998). Gassmann’s (1951) equations relate the bulk and shearmoduli of a saturated isotropic porous, monomineralic mediumto the bulk and shear moduli of the same medium in the drainedcase and shows furthermore that the shear modulus must bemechanically independent of the presence of the fluid. An im-portant implicit assumption is that there is no chemical inter-action between porous rock and fluid that affects the moduli;if such effects are present, we assume the medium is drained(rather than dry) but otherwise neglect chemical effects for this

Manuscript received by the Editor November 16, 1998; revised manuscript received May 28, 1999.∗Lawrence Livermore National Laboratory, Earth and Environmental Sciences Directorate, P.O. Box 808 L-200, Livermore, California 94551-9900.E-mail: berryman1@llnl.gov.c© 1999 Society of Exploration Geophysicists. All rights reserved.

argument. Gassmann’s paper is concerned with the quasi-static(low-frequency) analysis of the elastic moduli and that is whatwe emphasize here also. Generalization to higher frequencyeffects and complications arising in wave propagation due tofrequency dispersion are well beyond the scope of what wepresent.

In contrast to simple elasticity with stress tensorσi j and straintensor ei j , the presence of a saturating porefluid in porous me-dia introduces the possibility of an additional control field andan additional type of strain variable. The pressure pf in thefluid is the new field parameter that can be controlled. Allow-ing sufficient time (equivalent to a low-frequency assumption)for global pressure equilibration will permit us to consider pf

to be a constant throughout the percolating (connected) porefluid, while restricting the analysis to quasi-static processes. Thechange ζ in the amount of fluid mass contained in the poresis the new type of strain variable, measuring how much of theoriginal fluid in the pores is squeezed out during the compres-sion of the pore volume while including the effects of compres-sion or expansion of the pore fluid itself due to changes in pf .It is most convenient to write the resulting equations in termsof compliances si j rather than stiffnesses ci j , so for an isotropicporous medium (chosen only for the sake of its simplicity) thebasic equation to be considered takes the form:

e11

e22

e33

−ζ

=

s11 s12 s12 −βs12 s11 s12 −βs12 s12 s11 −β−β −β −β γ

σ11

σ22

σ33

−pf

. (1)

The constants β and γ appearing in the matrix on the righthand side will be defined later. It is important to write theequations this way rather than using the inverse relation interms of the stiffnesses, because the compliances si j appearingin equation (1) are simply and directly related to the drainedconstants λdr and µdr (the Lame parameters for the isotropic

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1628 Berryman

porous medium in the drained case) in the same way they arerelated in normal elasticity (the matrix si j is just the inverseof the matrix ci j ), whereas the individual stiffnesses c∗i j (the∗ superscript indicates the constants for the saturated case)obtained by inverting equation (1) must contain coupling termsthrough the parameters β and γ that depend on the porousmedium and fluid compliances. Using the standard relationsfor the isotropic moduli, I find that

s11 = 1Edr= λdr + µdr

µdr (3λdr + 2µdr )= 1

9Kdr+ 1

3µdr(2)

and

s12 = − νdr

Edr= 1

9Kdr− 1

6µdr, (3)

where the drained Young’s modulus Edr is defined in terms ofthe drained bulk modulus Kdr and shear modulus µdr by thesecond equality of equation (2) and the drained Poisson’s ratiois determined by

νdr = λdr

2(λdr + µdr ). (4)

The fundamental results of interest (Gassmann’s equations)are found by considering the saturated (and undrained) casesuch that

ζ ≡ 0, (5)

which [making use of equation (1)] implies that the pore pres-sure must respond to external applied stresses according to

pf = −βγ

(σ11 + σ22 + σ33). (6)

Equation (6) is often called the “pore-pressure buildup” equa-tion (Skempton, 1954). Then, using this result to eliminate bothζ and pf from equation (1), I obtaine11

e22

e33

=s∗11 s∗12 s∗12

s∗12 s∗11 s∗12

s∗12 s∗12 s∗11

σ11

σ22

σ33

=

s11 s12 s12

s12 s11 s12

s12 s12 s11

− β2

γ

1 1 1

1 1 1

1 1 1

σ11

σ22

σ33

,(7)

where s∗i j is the desired compliance including the effects of thetrapped fluid, while si j is the compliance in the absence of thefluid. Since for elastic isotropy there are only two independentcoefficients (s11 and s12), I find that equation (7) reduces to oneexpression for the diagonal compliance:

s∗11 = s11 − β2

γ, (8)

and another for the off-diagonal compliance:

s∗12 = s12 − β2

γ. (9)

If K ∗ and µ∗ are, respectively, the undrained bulk and shearmoduli, then equations (2) and (3) together with equations (8)and (9) imply that

19K ∗+ 1

3µ∗= 1

9Kdr+ 1

3µdr− β

2

γ, (10)

and

19K ∗− 1

6µ∗= 1

9Kdr− 1

6µdr− β

2

γ. (11)

Subtracting equation (11) from (10) shows immediately that1/2µ∗ = 1/2µdr or, equivalently, that

µ∗ = µdr . (12)

Thus, the first result of Gassmann is that, for purely mechan-ical effects, the shear modulus for the case with trapped fluid(undrained) is the same as that for the case with no fluid(drained). Then, substituting equation (12) back into eitherequation (10) or equation (11) gives one form of the resultcommonly known as Gassmann’s equation for the bulk modu-lus:

1K ∗= 1

Kdr− 9β2

γ. (13)

I want to emphasize that the analysis presented shows clearlythat equation (12) is a definite result of this analysis, not anassumption. In fact, we must have equation (12) in order forequation (13) to hold, and furthermore, if equation (13) holds,then so must equation (12). Thus, monitoring any changes inshear modulus with changes of fluid content (say, through shearvelocity measurements) provides a test of both Gassmann’sassumptions and results.

To obtain one of the more common forms of Gassmann’sresult for the bulk modulus, first note that

3β = 1Kdr− 1

Kg≡ α

Kdr, (14)

where Kg is the grain modulus of the solid constituent presentand α is the Biot-Willis parameter (Biot and Willis, 1957). Fur-thermore, the parameter γ is related through equation (6) toSkempton’s (1954) pore-pressure buildup coefficient B, so that

3βγ= B. (15)

Substituting these results into equation (13) gives

K ∗ = Kdr

1− αB, (16)

which is another form (Carroll, 1980) of Gassmann’s standardresult for the bulk modulus.

ACKNOWLEDGMENTS

I thank Michael L. Batzle and Patricia A. Berge for helpfulcomments on the manuscript. This work was performed underthe auspices of the U. S. Department of Energy by the LawrenceLivermore National Laboratory under contract No. W-7405-ENG-48 and supported specifically by the Geosciences Re-search Program of the DOE Office of Energy Research within

Origin of Gassmann’s Equations 1629

the Office of Basic Energy Sciences, Division of Engineeringand Geosciences.

REFERENCES

Berryman, J. G., 1998, Transversely isotropic poroelasticity arisingfrom thin isotropic layers, in Golden, K. M., Grimmett, G. R. , James,R. D., Milton, G. W., and Sen, P. N., Eds., Mathematics of multiscalematerials: Springer-Verlag, 37–50.

Biot, M. A., and Willis, D. G., 1957, The elastic coefficients of the theory

of consolidation: J. App. Mech., 24, 594–601.Brown, R. J. S., and Korringa, J., 1975, On the dependence of the elastic

properties of a porous rock on the compressibility of the pore fluid:Geophysics, 40, 608–616.

Carroll, M. M., 1980, Mechanical response of fluid-saturated porousmaterials, in Rimrott, F. P. J., and Tabarrok, B., Eds., Theoretical andApplied Mechanics: North-Holland Publ. Co., 251–262.

Gassmann, F., 1951, Uber die Elastizitat poroser Medien: Veirteljahrss-chrift der Naturforschenden Gesellschaft in Zurich, 96, 1–23.

Skempton, A. W., 1954, The pore-pressure coefficients A and B:Geotechnique, 4, 143–147.