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GEOPHYSICS, VOL. 72, NO. 6 �NOVEMBER-DECEMBER 2007�; P. A75–A79, 3 FIGS.10.1190/1.2772400

eneralization of Gassmann equations for porous mediaaturated with a solid material

adim Ciz1 and Serge A. Shapiro1

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ABSTRACT

Gassmann equations predict effective elastic properties ofan isotropic homogeneous bulk rock frame filled with a fluid.This theory has been generalized for an anisotropic porousframe by Brown and Korringa’s equations. Here, we developa new model for effective elastic properties of porous rocks— a generalization of Brown and Korringa’s and Gassmannequations for a solid infill of the pore space. We derive theelastic tensor of a solid-saturated porous rock consideringsmall deformations of the rock skeleton and the pore infillmaterial upon loading them with the confining and pore-space stresses. In the case of isotropic material, the solutionreduces to two generalized Gassmann equations for the bulkand shear moduli. The applicability of the new model is test-ed by independent numerical simulations performed on themicroscale by finite-difference and finite-element methods.The results show very good agreement between the new theo-ry and the numerical simulations. The generalized Gass-mann model introduces a new heuristic parameter, character-izing the elastic properties of average deformation of thepore-filling solid material. In many cases, these elastic modu-li can be substituted by the elastic parameters of the infillgrain material. They can also represent a proper viscoelasticmodel of the pore-filling material. Knowledge of the effec-tive elastic properties for such a situation is required, for ex-ample, when predicting seismic velocities in some heavy oilreservoirs, where a highly viscous material fills the pores.The classical Gassmann fluid substitution is inapplicable fora configuration in which the fluid behaves as a quasi-solid.

INTRODUCTION

Physical properties of porous rocks, such as seismic velocity, de-end on elastic properties of the porous frame and the pore-space

Manuscript received by the Editor 1 June 2007; revised manuscript receive1Freie Universität Berlin, Fachrichtung Geophysik, Berlin, Germany. E-m2007 Society of Exploration Geophysicists.All rights reserved.

A75

lling material. Gassmann equation �Gassmann, 1951� commonly ispplied to predict the bulk modulus of rocks saturated with differentuids. This equation assumes that all pores are interconnected and

hat the pore pressure is in equilibrium in the pore space. The porousrame is macroscopically and microscopically homogeneous andsotropic. Brown and Korringa �1975� generalize Gassmann equa-ions for inhomogeneous anisotropic material of the porous frame.

In this paper, we further extend this theory to the case when theore-filling material is an anisotropic elastic solid. In the isotropicase, i.e., in the case equivalent to Gassmann equation but with aossibility of solid pore-space infill, the effective bulk modulus is es-imated from the mineral and dry bulk moduli of the porous frame,orosity, and a new parameter: bulk modulus of the pore space filledy a solid material. The saturated shear modulus differs significantlyrom the dry shear modulus. This is one of the main advantages ofur approach in comparison with Gassmann, and Brown and Korrin-a’s equations, which apply only when the pore-filling material is auid. The developed model has its applicability for computing effec-

ive elastic properties of heavy oil reservoirs when the highly vis-ous �and possibly non-Newtonian� liquid saturating the pores actss a solid.

STRESS/STRAIN THEORY

We consider a porous rock of porosity �. It is possible that an elas-ic solid fills up the pore space. Let � be the external surface of theorous rock. It cuts and seals the pores �jacketed sample�. The porepace is assumed to be interconnected, representing Biot’s mediumBiot, 1962�. We also define the surface of the pore space � . The sur-ace � coincides with the surface � where it cuts the pores. Theirormals are opposite in these points, i.e., nj� � �nj. The tractionomponent � i at any given point x of a surface � is given by

� i � � ijc nj�x� , �1�

here nj�x� are the components of the outward normal of � �see Fig-re 1 in Shapiro and Kaselow, 2005� and � ij

c is the uniform confining

2007; published online 5 September 2007.geophysik.fu-berlin.de; shapiro@geophysik.fu-berlin.de.

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A76 Ciz and Shapiro

tress. We assume application of small uniform changes in both con-ning stress � ij

c and pore stress � ijf .

The pore stress term now represents the stress field in the solid fill-ng the pore space. When the pore-space saturating material is a flu-d, the pore stress reduces to the pore pressure. Because of the load,oints of the external surface � are displaced by ui�x� to their finalosition �see Figure 2 in Shapiro and Kaselow, 2005�. This displace-ent is assumed to be very small in comparison to the size of the

ock volume under consideration.According to Brown and Korringa �1975� and Shapiro and Ka-

elow �2005�, we can describe the deformation of a rock sample byymmetric tensors representing the deformation of the rock sample,

� ij � ��

1

2�uinj � ujni�d2x , �2�

nd the deformation of the pore space,

� ij � ��

1

2�uinj� � ujni��d

2x . �3�

For a continuous elastic body replacing the porous matrix and theontinuous pore-filling elastic material, Gauss’s theorem applies:

� ij � �V

1

2�� jui � � iuj�d3x , �4�

� ij � �V�

1

2�� jui � � iuj�d3x . �5�

he integrands in equations 4 and 5 are the strain tensors. The quan-ity � ij/V represents a volume-averaged strain of the bulk volume,nd the quantity � ij/V� denotes a volume-averaged strain of the poreolume. Here, V is the volume of the porous body and V� is the vol-me of all its connected pores.

Three fundamental compliances of an anisotropic porous bodyan be introduced:

Sijkldry �

1

V� �� ij

�� kld �

� f, �6�

Sijklgr �

1

V� �� ij

�� klf �

� d, �7�

nd

Sijkl� � �

1

V�� �� ij

�� klf �

� d, �8�

here � kld � � kl

c � � klf is the differential stress, and where indices

ry, gr, and � are related to the dry porous frame, the grain materialf the frame, and the pore space of the dry porous frame, respective-y. These quantities in expressions 6–8 represent the tensorial gener-lization of Brown and Korringa’s �1975� expressions 4a–4c.

The fourth compliance tensor,

Sijkl� � �1

V� �� ij

�� kld �

� f, �9�

s not independent because of the reciprocity theorem �Shapiro andaselow, 2005�:

Sijkl� � Sijkldry � Sijkl

gr . �10�

The fifth tensor is required to describe the compliance of the porepace filled by a solid material. We define it heuristically in the fol-owing way:

Sijklif � �

1

V�� �� ij

�� klf �

con

, �11�

here the index if �infill� is related to the body of the pore-space infillnd where con is constant infill mass. This generalized �in the sensehat the infill can be solid� compliance tensor Sijkl

if is related to the vol-me-averaged strain of the pore space and therefore differs from theompliance tensor of the grain material of the pore infill Sijkl

ifgr �indexfgr denotes the pore-infill grain material�. Later we explain how tostimate Sijkl

if . The effective compliance tensor of the composite po-ous rock with a solid infill Sijkl

* , which is the subject of our consider-tion, is defined as follows:

Sijkl* �

1

V� �� ij

�� klc �

con

. �12�

urther, we evaluate the changes of � ij and of � ij by applying � � d

nd by keeping the pore stress � f constant and the effect of applying� ij

f from inside and outside while leaving � d constant:

� � ij � � �� ij

�� kld �

� f� � kl

d � � �� ij

�� klf �

� d� � kl

f �13�

nd

� � ij � � �� ij

�� kld �

� f� � kl

d � � �� ij

�� klf �

� d� � kl

f . �14�

hese two expressions serve for the derivation of the effective com-liance tensor Sijkl

* .

GENERALIZED BROWN AND KORRINGA’SEQUATIONS FOR SOLID INFILL OF THE

PORE SPACE

The definition of the effective compliance tensor in expression 12,he definitions in expressions 6–8, and the resulting change in thealue of the frame deformation tensor � ij given by expression 13ield

� � ij

V� Sijkl

* � � klc � Sijkl

dry�� � klc �� � kl

f � � Sijklgr � � kl

f . �15�

he requirement of the conservation of mass of a pore-filling materi-l upon loading it with the pore stress � � kl

f , using expressions 3, 9,nd 8, gives

�� � ij

V�

� Sijklif � � kl

f �1

�Sijkl� �� � kl

c �� � klf � � Sijkl

� � � klf ,

�16�

here � � V� /V denotes porosity. To obtain the effective compli-nce tensor S* , we eliminate � � c and � � f from equations 15 and

ijkl kl kl

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Gassmann equations for solid saturation A77

6 and use equation 10. The result yields the generalized anisotropicrown and Korringa’s �1975� equation for a solid filling the pore

pace:

Sijkl* � Sijkl

dry � �Sijkldry � Sijkl

gr ����Sif � S��

� �Sdry � Sgr��mnqp�1 �Smnqp

dry � Smnqpgr � . �17�

quation 17 is the main result of this paper. �Note the tensorial naturef this equation.�

ISOTROPIC GENERALIZED BROWN ANDKORRINGA’S EQUATIONS FOR A SOLID INFILL

OF THE PORE SPACE

In the case of isotropic materials, the compliance tensor S can bexpressed in terms of bulk and shear moduli K and � �Mavko et al.,998�. We substitute individual compliances Sijkl

dry, Sijklgr , Sijkl

� , and Sijklif

nto expression 17. Solving the system of equations 17 for the isotro-ic case, we obtain the solid-saturated bulk and shear moduli:

sat*�1 � Kdry

�1��Kdry

�1 � Kgr�1�2

��Kif�1 � K�

�1� � �Kdry�1 � Kgr

�1��18�

nd

sat*�1 � �dry

�1���dry

�1 � �gr�1�2

���if�1 � ��

�1� � ��dry�1 � �gr

�1�, �19�

here Ksat* and �sat

* are solid saturated bulk and shear moduli, Kdry anddry denote drained bulk and shear moduli of the porous frame, Kgr

nd �gr represent bulk and shear moduli of the grain material of therame, K� and �� are bulk and shear moduli related to the pore spacef the frame, and Kif and �if are the newly defined bulk and shearoduli related to the solid body of the pore infill. Equations 18 and

9 represent the isotropic Gassmann equations for a solid-saturatedorous rock. �Note that the equations for the solid-saturated bulk andhear moduli are of the same form.�

If the porous frame material is homogeneous, it follows �Brownnd Korringa, 1975� K� � Kgr and �� � �gr. If the pore infill is auid ��if � 0�, equations 18 and 19 reduce to Brown and Korringa’squations �1975�. In the case of a single mineral in the porous frameK� � Kgr� and a fluid in the pore space, equations 18 and 19 reduceo the standard equations of Gassmann �1951�.

COMPARISON WITH KNOWN THEORIES

When all constituents of an elastic composite have the same shearodulus �, Hill �1963� shows that the effective bulk modulus Keff is

iven by an exact formula:

1

Keff �4

3�

�1 � �

Kgr �4

3�

��

Kifgr �4

3�

, �20�

here Kifgr represents the grain bulk modulus of the pore-fillingaterial.In this case, �gr � �ifgr � �if, and equation 19 reduces to �sat

*

�gr. For this case, K� � Kgr also. In equation 18, we still have annknown heuristic parameter, Kif. This special case of identicalhear moduli allows us to analyze this new parameter. We use equa-ion 20 to calculate K .

if

Both equations 18 and 20 express the effective bulk moduli ofhe mixture of matrix and pore materials. Thus, Ksat

* � Keff and webtain

if � ��1 �Kdry

Kgr�2

Keff � Kdry�

�1 �Kdry

Kgr� � �

Kgr

�1

. �21�

To analyze equation 21, we keep constant elastic parameters of theorous frame and vary the grain bulk modulus of the saturating solidn the pore space. The porous frame has the following parameters:ulk and shear moduli of grains is Kgr � 36.7 GPa and �gr

22 GPa, porosity � � 0.22, and drained bulk and shear modulif the frame is Kdry � 10 GPa and �dry � 7.6 GPa. The grain shearoduli of the porous frame and the pore-filling material are equal:

ifgr � �gr � 22 GPa. �Note that �ifgr represents the shear modulusf grains of the pore infill, whereas �if denotes shear modulus relatedo the volume-averaged shear deformation of the pore-filling solidody.�

Analysis of equation 21 shows that for the low contrast in bulkoduli of the frame material and of the pore-filling material, the un-

nown modulus Kif can be approximated very well by the bulk mod-lus Kifgr �i.e., bulk modulus of the infill material�. This result greatlyimplifies the use of equations 18 and 19. For the contrast in elasticoduli up to 20%, the unknown parameters Kif and �if can be almost

xactly substituted with the bulk and shear moduli of the pore-fillingaterial, Kifgr and �ifgr. This approximation is still applicable for the

ontrast in the elastic moduli in the range from 20% to 40%.

COMPARISON WITH NUMERICAL SIMULATIONS

To check the correctness of expressions 18 and 19, in Figures 1nd 2 we compare the derived analytical model with the numericalimulations. For the comparison, we use two numerical approaches:nite-element and finite-difference �FD� modeling.

0 5 10 15 20 25

25

20

15

10

5

0

µifgr

(GPa)

µ* sat

New model

FD simulations

FEM simulations

HS average

HS upper bound

HS lower bound

(GP

a)

igure 1. Effective shear modulus �sat* in dependence of the shear

odulus of the pore-filling solid �ifgr. The results for the new modelre obtained from equation 19, assuming �if � �ifgr, �gr � 22 GPa,nd �dry � 7.6 GPa. FD � finite difference; FEM � finite-ele-ent modeling; HS � Hashin-Shtrikman.

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A78 Ciz and Shapiro

inite-element modeling

The finite-element �FE� modeling algorithm was developed origi-ally by Garboczi and Day �1995� and adopted for the study of po-ous rocks by Arns et al. �2002�. The algorithm uses a formulation ofhe static linear elastic equations and finds the steady-state solutiony minimizing the strain energy of the system. We implement this al-orithm here to compute the effective elastic constants of the isotro-ic porous model saturated in the pore space with another elastic sol-d material. The numerical simulations provide the static effectiveulk and shear moduli of such a model. The tests are performed onhe Gaussian random field models �GRF5 and GRF1� created andnalyzed by Saenger et al. �2005�.

inite-difference modeling

The elastic version of the rotated staggered-grid FD algorithm,eveloped by Saenger et al. �2000�, has been extended to viscoelas-icity �Saenger et al., 2005�. This algorithm is proven to be effectiven simulations of wave propagation in porous media on the micro-cale. In this study, we perform several simulations on the artificialock sample GRF5 to obtain the effective P- and S-wave velocities.his enables us to derive the dynamic effective bulk and shear mod-li of the analyzed piece of porous rock saturated with elastic solidaterial. Of course, we expect a coincidence of the static and dy-

amic moduli for the case of low enough frequencies of propagationaves.

esults

The comparisons of both numerical methods with the new modelre shown in Figures 1 and 2. These figures illustrate the effectiveolid-saturated shear moduli dependent on the shear moduli of theore-filling material �ifgr. For completeness, we also plot theashin-Shtrikman bounds and their average value �Hashin and Sh-

rikman, 1963�. The porous matrix model is represented by the

10−3

10−2

10−1

100

0

5

10

15

20

25

µifgr

(GPa)

µ* sat

New model

FEM simulations

HS average

HS upper

HS lower

(GP

a)

igure 2. Effective shear modulus �sat* dependent of the shear modu-

us of the pore-filling solid �ifgr. The results for the new model areomputed using �if � �ifgr. For the low-porosity medium and theigh contrast in shear moduli of a porous matrix and a pore-spacelling material, the new model performs better than the Hashin-Sh-

rikman �1963� average.

RF5 �in Figure 1� and GRF1 �in Figure 2� models with the samerame parameters given after equation 21.

The results in Figure 1 show the six numerical simulations per-ormed with the following bulk and shear moduli �Kifgr,�ifgr�: �36.7,2�, �25, 20�, �20, 15�, �13.34, 10�, �2.25, 0�, and �0,0� GPa, wherehe last one represents the drained porous sample. The numericalimulations are in very good agreement with the theoretical results.he theoretical effective shear moduli are obtained using equation9, where the unknown shear parameter �if is taken to be equal to itsrain counterpart �ifgr. These results confirm the conclusions statedn the previous section. The low contrast in the elastic parameters ofhe elastic solid matrix and the pore-filling material enables us to car-y out the substitutions Kif � Kifgr and �if � �ifgr. When the pore in-ll is a fluid, the new model reduces to the classical Gassmann equa-

ion, as shown earlier. This situation confirms the numerical simula-ion for the point �2.25,0� GPa.

Figure 2 shows an example where the Hashin-Shtrikman averageeviates from the effective shear modulus obtained by numericalimulations. This is the case of rock �model GRF1� with low porositynd high contrast in the frame and the pore-filling materials. The pa-ameters of the model are � � 0.0342, bulk and shear moduli of therame grains are Kgr � 36.7 GPa and �gr � 22 GPa, and drainedoduli of the porous frame are Kdry � 29 GPa and �dry � 18.7Pa. The results in Figure 2 show four numerical simulations per-

ormed with the following bulk and shear moduli �Kifgr,�ifgr�: �2.2,.001�, �2.2, 0.01�, �2.2, 0.1�, and �2.2,1� GPa. This example rep-esents the situation when the developed model performs betterhan the Hashin-Shtrikman average. Such a situation can be ob-erved in the case of fractured �low-porosity� rocks filled witheavy oils.

VISCOELASTIC EXTENSION

We heuristically extend the elastic equations 17–19 for the vis-oelastic material filling the pore space. For this, complex bulk andhear moduli Kif and �if are introduced into equations 18 and 19. Theumerical code implements the viscoelastic infill as a generalizedaxwell body. The S-wave velocity in the rock is then given by

Vs ���sat*

, �22�

here �sat* is obtained from equation 19 using the viscoelastic exten-

ion of �if �Maxwell fluid model�:

�if�� ���

�i��

�� 1

. �23�

ere, �� is the real shear modulus of the infill medium at high fre-uencies, � is the dynamic shear viscosity of the same medium, and� �1 � ��gr � � f is the overall density.In Figure 3, we show the results of numerical experiments for the

-wave transmission through porous rock filled with a viscous fluidor varying viscosity from 1 to 107 kg/ms. We use the viscoelasticxtension of the finite-difference code developed by Saenger et al.2005�. The porous rock is represented by the GRF5 model. The P-nd S-wave velocities and the density of the porous frame grain ma-erial are Vp � 5100 m/s, Vs � 2944 m/s, and gr � 2540 kg/m3,espectively; � � 0.22; drained bulk and shear moduli are K

dry

�

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Gassmann equations for solid saturation A79

10 GPa and �dry � 7.6 GPa; and fluid density is f � 1000kg/m3.Apressure seismic source radiates a Ricker wavelet with theominant frequency fdom � 80 kHz.

Figure 3 compares the new viscoelastic Gassmann equation forhe shear modulus and the results of our numerical simulations. Theew model in equation 22 is in very good agreement with the numer-cal results. For comparison, we also plot the elastic solution giveny the classical Gassmann equation, Vs � ��dry/, which is frequen-y and viscosity independent. Thus, the new model developed here,long with its viscoelastic extension, is important for modeling seis-ic responses of rocks containing heavy oil in the pore space.

CONCLUSIONS

The main result of this paper is a new analytical model of effectivelastic properties of porous rock. This model extends Brown and Ko-ringa’s anisotropic version of Gassmann equations to the situationith an elastic solid filling the pore space. The pore space filled with

lastic material is characterized by a new tensor of elastic compli-

100 102 104 106 1081800

2000

2200

2400

2600

2800

3000

3200

3400

Viscosity (kg/ms)

Vs(m/s)

Classical GassmannNew modelNumerics

igure 3. The S-wave velocity Vs in dependence of the fluid viscosi-y. The classical Gassmann model computes S-wave velocity fromrained bulk modulus, whereas the new model implements the newiscoelastic Gassmann equation for saturated shear modulus �equa-ion 22�.

nces. This tensor is related to the pore-space volume-averaged

train. It can be approximated well by a tensor of infill compliances.or an isotropic material, our formalism reduces to two equationsescribing effective saturated bulk and shear moduli of the solid-sat-rated porous rock.

The numerical simulations support the validity of this model. Theodel is particularly suited for computing effective properties of

ock saturated with heavy oil. For very heavy oils, the viscosity isigh and the material behaves like a quasi-solid. The classical Gas-mann equation is inapplicable if the pore-filling material is a solidr a liquid whose shear modulus has a finite component. Our modelttempts to overcome these limitations and is extendable for the vis-oelastic pore-space infill.

ACKNOWLEDGMENTS

This work was supported by sponsors of the PHASE universityonsortium. The authors thank E. H. Saenger for providing the fi-ite-difference numerical code and C. Arns and M. Knackstedt forroviding the finite-element modeling numerical code. The authorslso thank B. Gurevich for stimulating discussions and J. M. Car-ione and two anonymous reviewers for useful comments.

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rns, C., M. Knackstedt, W. Pinczewski, and E. Garboczi, 2002, Computa-tion of linear elastic properties from microtomographic images: Method-ology and agreement between theory and experiment: Geophysics, 67,1396–1405.

iot, M. A., 1962, Mechanics of deformation and acoustic propagation in po-rous media: Journal ofApplied Physics, 33, 1482–1498.

rown, R. J. S., and J. Korringa, 1975, On the dependence of the elastic prop-erties of a porous rock on the compressibility of the pore fluid: Geophysics,40, 608–616.

arboczi, E. J., and A. R. Day, 1995, An algorithm for calculating the effec-tive linear elastic properties of heterogeneous materials: Three dimension-al results for composites with equal phase Poisson ratios: Journal of theMechanics and Physics of Solids, 43, 1349–1362.

assmann, F., 1951, Über die Elastizität poröser Medien: Vierteljahrsschriftder Naturforschende Gesellschaft, 96, 1–23.

ashin, Z., and S. Shtrikman, 1963, A variational approach to the elastic be-havior of multiphase materials: Journal of the Mechanics and Physics ofSolids, 11, 127–140.

ill, R., 1963, Elastic properties of reinforced solids, Some theoretical prin-ciples: Journal of the Mechanics and Physics of Solids, 11, 357–372.avko, G., T. Mukerji, and J. Dvorkin, 1998, Rock physics handbook: Cam-bridge University Press.

aenger, E. H., N. Gold, and S. A. Shapiro, 2000, Modeling the propagationof elastic waves using a modified finite-difference grid: Wave Motion, 31,77–92.

aenger, E. H., S. A. Shapiro, and Y. Keehm, 2005, Seismic effects of viscousBiot-coupling, finite difference simulations on micro-scale: GeophysicalResearch Letters, 32, L14310.

hapiro, S. A., and A. Kaselow, 2005, Porosity and elastic anisotropy ofrocks under tectonic stress and pore-pressure changes: Geophysics, 70, no.

5, N27–N38.