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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 92 (2016) 28–54

http://d0022-50

n CorrE-m

journal homepage: www.elsevier.com/locate/jmps

Dynamic bulk and shear moduli due to grain-scale local fluidflow in fluid-saturated cracked poroelastic rocks: Theoreticalmodel

Yongjia Song a, Hengshan Hu a,n, John W. Rudnicki b

a Department of Astronautics and Mechanics, Harbin Institute of Technology, P. O. Box 344, Harbin 150001, PR Chinab Department of Civil and Environmental Engineering and Department of Mechanical Engineering, Northwestern University, Evanston, IL60208, U.S.A

a r t i c l e i n f o

Article history:Received 18 September 2015Received in revised form24 February 2016Accepted 22 March 2016Available online 28 March 2016

Keywords:Porous rocksEshelby transformSquirt flowWave attenuationMori–Tanaka Scheme

x.doi.org/10.1016/j.jmps.2016.03.01996/& 2016 Elsevier Ltd. All rights reserved.

esponding author.ail address: [email protected] (H. Hu).

a b s t r a c t

Grain-scale local fluid flow is an important loss mechanism for attenuating waves incracked fluid-saturated poroelastic rocks. In this study, a dynamic elastic modulus modelis developed to quantify local flow effect on wave attenuation and velocity dispersion inporous isotropic rocks. The Eshelby transform technique, inclusion-based effective med-ium model (the Mori–Tanaka scheme), fluid dynamics and mass conservation principleare combined to analyze pore-fluid pressure relaxation and its influences on overall elasticproperties. The derivation gives fully analytic, frequency-dependent effective bulk andshear moduli of a fluid-saturated porous rock. It is shown that the derived bulk and shearmoduli rigorously satisfy the Biot-Gassmann relationship of poroelasticity in the low-frequency limit, while they are consistent with isolated-pore effective medium theory inthe high-frequency limit. In particular, a simplified model is proposed to quantify thesquirt-flow dispersion for frequencies lower than stiff-pore relaxation frequency. Themain advantage of the proposed model over previous models is its ability to predict thedispersion due to squirt flow between pores and cracks with distributed aspect ratio in-stead of flow in a simply conceptual double-porosity structure. Independent input para-meters include pore aspect ratio distribution, fluid bulk modulus and viscosity, and bulkand shear moduli of the solid grain. Physical assumptions made in this model include(1) pores are inter-connected and (2) crack thickness is smaller than the viscous skindepth. This study is restricted to linear elastic, well-consolidated granular rocks.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

The presence of microscale cracks can have a large effect on the effective elastic properties of a solid even if the fractionalvolume occupied by the cracks is extremely small. The state of fluid saturation of the cracks has a corresponding large effect.When a cracked rock is compressed by passing waves, the fluid pressure response in the compliant/soft cracks will begreater than that in the stiffer pores. The induced pressure gradient creates grain-scale local flow from crack to adjacentpores. The resulting fluid flow is called “squirt flow (SF)” (e.g., Mavko and Nur, 1975) which also occurs in shear waves. In

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Nomenclature

ϕ total porosityϕc total crack porosityϕp total stiff-pore porosityϕi,ϕc

i, ϕpi porosity of the ith pore, crack, stiff pore

γ i aspect ratio of the ith crackγp aspect ratio of the stiff poresv volume of a rock sample

ϕv volume of the total pore spacevc

i , vpi volume of the ith crack, stiff pore

vp volume of the total stiff poresL0, K0, G0 elastic modulus tensor, bulk and shear moduli

of solid grainν0 Poisson's ratio of the solid grainLf , Kf elastic modulus tensor, bulk modulus of pore

fluidρf , η density, viscosity of pore fluidLd, Kd, Gd elastic modulus tensor, bulk and shear moduli

of dry sampleL , K , G effective elastic modulus tensor, bulk and

shear modulif , ω frequency, circular frequencySi, Sc

i , Spi Eshelby tensor of the ith pore, crack, stiff pore

σi, σci , σp

i averaged fluid stress tensor in the ith pore,crack, stiff pore

eci , ep

i strain tensor of the ith crack, stiff poreecf

i , epfi fluid strain tensor in the ith crack, stiff pore

qci, qp

i volumetric flow out of the ith crack, stiff poreqp averaged volumetric flow over stiff pores

ϕe averaged pore strain over all pore spaceef averaged fluid strain over all pore space

∞e externally applied strainσf averaged pore fluid stress tensor over all pore

spacepc

i , ppi averaged fluid pressure in the ith crack, stiff

porep̃c

i deformation-induced fluid pressure in the ithcrack

pp averaged fluid pressure over stiff porespf averaged fluid pressure over all poresr , θ , z cylindrical coordinatesvr , θv , vz fluid velocity components in a crackuf fluid particle displacement vector in a crack

(= ˜)k kr , kz wavenumber components in a crackfc

i relaxation function of the ith crackNp number of stiff poresSc upper and lower surfaces of a crackn normal vector directed outward from crack

fluidh, R thickness, radius of a crackL effective distance between stiff poresS effective flow cross section area of stiff poresδ dimensionless constant for stiff poresQ c

i , Q pi transform tensor relating ∞e to qc

i , qpi

Dci , Dp

i , C transform tensor relating ∞e to σci , σp

i , σf

T̃ci T̃p

i , ˜ϕT transform tensor relating ∞e to eci , ep

i , ϕeTc

i , Tpi transform tensor relating ∞e to ec

i , epi for iso-

lated pores

Y. Song et al. / J. Mech. Phys. Solids 92 (2016) 28–54 29

turn, the SF attenuates waves and changes effective elastic properties of rocks (e.g., David et al., 2013). Previous studies showthat the SF appears to be an important physical mechanism in attenuating waves at sonic and ultrasonic frequencies (Mavkoand Nur, 1975, 1979; Jones, 1986). Compared with Biot's global flow mechanism (Biot, 1956a, b), in which viscous-fluidmotion is induced by the wavelength-scale fluid pressure gradient, the SF is a consequence of pore microstructural geo-metry and can produce much greater phase velocity dispersion and wave attenuation. The SF can give rise to significanthydrocarbon signature in the acoustic measurement (Tang et al., 2012) and affect the borehole signals (Markova et al., 2013).Over the years, many efforts have been made to develop unified models of global flow and squirt flow for dynamics ofmacroscopically homogeneous porous rocks having grain-scale cracks. Those include works by Dvorkin and Nur (1993),Jakobsen and Chapman (2009), Gurevich et al. (2010), Tang (2011) and Tang et al. (2012). We restrict this discussion tosituations in which the Biot global flow mechanism is neglected.

In the low-frequency limit, the pore fluid pressure is in equilibrium. The elastic properties of rocks are described byGassmann's equations (Gassmann, 1951), in which undrained (or relaxed) bulk modulus are calculated from drained (or dry)bulk modulus, solid grain bulk modulus, fluid bulk modulus and porosity, while undrained shear modulus is identical todrained shear modulus. Experimental data presented by Thomsen (1985) suggests that the Gassmann's equations are indeedsatisfied for a wide range of different rock types. Pride and Berryman (1998) related variables controlled and measured inelastostatic laboratory experiments to the appropriate variables of poroelastic theory. With increasing frequency, inducedfluid pressure in cracks stiffens rocks, resulting in frequency-dependent velocities.

In the high-frequency limit (in which wavelength is required to be still larger than porous representative volume element),the fluid pressure does not have enough time to equilibrate between pores. Then no fluid mass communication occurs so thatall pores behave like isolated inclusions. It is more appropriate to estimate the unrelaxed effective elastic moduli using ef-fective medium theory. Recently, David and Zimmerman (2012) developed a procedure to extract the pore aspect ratio dis-tribution from the dry velocities. Their results showed that for ultrasonic velocity measurements, the predictions of saturatedvelocities using Mori–Tanaka (MT) scheme (Mori and Tanaka, 1973; Benveniste, 1987) and differential effective medium(LeRavalec and Guéguen, 1996) matched well the experimental data for a good number of sandstone data sets.

On the other hand, for a certain porous sample, its pore structure cannot change with frequency. The SF dispersion issolely due to pore-microstructure-induced fluid pressure relaxation. Because of this, the effective medium schemes are alsowidely used to estimate the static undrained elastic moduli (e.g., Berryman, 1981; Xu and White, 1995; LeRavalec and

Y. Song et al. / J. Mech. Phys. Solids 92 (2016) 28–5430

Guéguen, 1996; Suvorov and Selvadurai, 2011; Xu, 1998). The procedure is as follows: estimating the dry moduli at first andthen substituting the dry moduli into Gassmann's formula to obtain the undrained elastic moduli.

Dvorkin et al. (1995) incorporated the SF effect into solid grains. In their model, solid grains and soft pores are assumedto make up a modified solid phase, while stiff pores are assumed to occupy the main pore space in which fluid pressure isconstant during squirt-flow mass communication. The modified solid is modeled as an isotropic Biot, poroelastic materialbut in which only lateral flow is allowed. Then the bulk modulus of the modified solid is related to the frequency-dependentfluid pressure response via an axisymmetrical flow model. In their model, both of the bulk and shear moduli are dispersive,and the low-frequency limits are identical to those predicted by Gassmann's formula. However, Pride et al. (2004) showedthat Dvorkin et al. (1995) made so many approximations that the error introduced is often as large as the dispersion beingmodeled.

Pride et al. (2004) generalized the double-porosity framework to analyze the bulk modulus dispersion due to SF. In theirmodel, soft pores and solid grains are treated as a Biot, poroelastic medium while stiff pore space is treated as a full fluidspace. The advantage of this method is that the medium can be treated as poroelastic on the subpore scale. However, themodel does not provide prediction for shear dispersion. In fact, Laboratory measurements (e.g., Mavko and Jizba, 1991;Adam and Otheim; 2013) show that the shear dispersion is as large as bulk dispersion. Furthermore, extending the double-porosity framework to multiple crack aspect ratios will greatly increase the number of stiffness coefficients in the con-stitutive equations.

Gurevich et al. (2010) incorporated the SF effect into the solid frame. They considered a so-called modified rock frame inwhich only cracks are filled with the fluid, whereas stiff pores are empty. They assumed that (1) the fully saturated bulkmodulus at intermediate frequencies could be calculated from a modified Gassmann's equation in which the drained bulkmodulus is replaced by the modified frame bulk modulus, and (2) the saturated shear modulus is identical to the modifiedframe shear modulus. By introducing the SF relaxation effect into the modified frame, they obtain a dispersion relationwithin full frequency range. In their model, the derived bulk and shear moduli are also consistent with the Gassmann'sequations in the low-frequency limit. The shear dispersion is simply proportional to bulk dispersion via a constant scalecoefficient. While the pressure-dependent moduli predicted by their model have experimental support, the concept ofmodified frame moduli, however, is a merely logic consideration, corresponding to no real scenarios for fully saturatedrocks. Moreover, the model is also restricted to the conceptual double-porosity structure.

The SF models stated above are essentially developed upon poroelastic method: stiff pore response is assumed to bedescribed by a Biot poroelastic medium, occupying the main pore volume, whereas the SF is treated as an additional effectbeing incorporated into such a Biot medium to obtain the overall elastic properties.

On the other hand, effective medium theory provides alternative method to quantify SF dispersion. O’Connell and Bu-diansky (1977) were the first to use effective medium method to study SF dispersion and obtained dynamic elastic moduli.They considered a special fluid-saturated porous solid in which the pore space consists of cracks only. When the rock wassubjected to bulk compression, the same pressure would be induced in every crack, and thus no bulk dispersion occurred.Under shear stress the situation is different. The shear stress would in general induce different pressures in cracks ofdifferent orientations, and then the pressure gradient created flow.

Chapman et al. (2002) also applied the effective medium approach to study the grain-scale local flow on effective elasticmoduli. In their model, stiff pores are modeled as spherical pores, while cracks are modeled as spheroidal pores with smallaspect ratio. The model is consistent with Gassmann's formula in the low-frequency limit, and satisfies the isolated-poreeffective elastic scheme in the high-frequency limit. In the model, the characteristic frequency of dispersion is predicted tobe linearly proportional to fluid mobility and crack aspect ratio and is likely to be much higher than ultrasonic frequencies(∼MHz). Thus, their model cannot be used to interpret the observable attenuation at logging and ultrasonic frequencies.Furthermore, the derived dynamic elastic moduli are based on Eshelby's dilute-inclusion scheme, which is valid only forsmall porosity.

In the recent decade, Jakobsen and his collaborators published a series of papers (Jakobsen et al., 2003; Jakobsen, 2004;Jakobsen and Chapman, 2009; Ali and Jakobsen, 2014) to quantify the SF dispersion by using T-matrix approach. Theirmodel allows for non-dilute pores and introduces an independent constant, i.e. relaxation time, to quantify the char-acteristic frequency of the SF dispersion. But, they do not provide a method to obtain the key parameter.

In all existing works, certain restrictive assumptions have been made in developing the models. Although these modelshave provided important insight into the understanding of certain phenomena in wave attenuation and velocity dispersioncaused by SF, their applicability to more general cases is limited due to the assumptions made.

We argue that an ideal SF model should be rigorously consistent with Gassmann's equations in the low-frequency limit,and satisfy isolated-pore effective medium schemes in the high-frequency limits. Equally importantly, such a model couldalso be used to invert pore microstructures, in particular, the pore aspect ratio distribution. More importantly, the modelshould contain the least physical assumptions.

In this paper, we propose a dynamic elastic modulus model of squirt-flow attenuation that uses pore-pressure dependentMT scheme of Song et al. (submitted to Geophysics) in the conjunction with the Eshelby transformation technique and fluiddynamics. The resulting model is consistent with Gassmann and isolated-pore MT equations in the low- and high-frequencylimits, respectively. The model gives a fully analytical expression for frequency-dependent bulk and shear moduli which fillthe missing link between the low-frequency formula and high-frequency estimates. We show that the model can be ex-tended to cover a range of different crack aspect ratios without complicating its mathematical form. This allows us to treat


more realistic pore spaces. Although several approximations (such as pore geometry and reference material choice) areinevitably introduced to the effective medium scheme (MT scheme), the physical assumptions made in the model includethat: (1) the pore space is interconnected and (2) the crack thickness is smaller than viscous skin depth. The first assumptionis merely due to that the isolated pores cannot cause dispersion in the long wavelength situation. The second assumption isa common treatment for the squirt mechanism (Tang, 2011; Tang et al., 2012).

The paper is organized as follows. First, we develop a theoretical model of dynamic elastic moduli for the SF mechanism.Eshelby transform and fluid dynamics are combined to analyze the pore-fluid pressure relaxation. Transform relationsrelating induced fluid pressure, pore deformation and local flow to externally applied loadings are derived. Effective elasticmoduli are obtained via a pore-pressure dependent effective medium scheme. We then analyze the low- and high-fre-quency behaviors of this model. Finally, we provide numerical examples to illustrate the prediction of the model.

2. Fluid-pressure-dependent effective elastic moduli

Although Biot theory is widely used to describe the linear elastic properties of porous rocks, the theory pertains tohomogenized pores, neglecting the effects of pore microstructure. Because SF is a consequence of the effect of pore mi-crostructure, it is natural to use the effective medium method to quantify SF dispersion. Among the many effective mediumtheories, Mori–Tanaka scheme is an explicit scheme and also valid for non-dilute-pore rocks. We start with the formulationof pore-pressure dependent Mori–Tanaka scheme. According to Song et al. (submitted to Geophysics), the pore-pressure-dependent effective elastic modulus tensor L for a porous rock is given by

( ) ( )( ) ( )∑ ϕ ϕ ϕσ σ σ− − + = − −( )

ϕ− − − −

⎪ ⎪

⎪ ⎪⎧⎨⎩

⎡⎣⎢⎢

⎤⎦⎥⎥

⎫⎬⎭

L L L I S S L L L L e ,1

d di

i i ii f f

1 10

10

10

where Ld is the dry or drained elastic modulus tensor

( )∑ϕ ϕ ϕ= ( − ) ( − ) + −( )

−−⎡

⎣⎢⎢

⎤⎦⎥⎥L L I I S1 1 .

2d

i

i i0

11

ϕ ϕ(=∑ )ii is the total porosity. ϕi is the volume fraction of the ith pore. L0 is the solid grain elastic modulus tensor. Si is the

Eshelby tensor of ith pore. I is the fourth-order identity tensor. σf is the average fluid stress over all pore space. ϕe is theaverage pore strain over all pore space. σi is the fluid stress in ith pore. The superscript �1 denotes inverse operator. Forsimplicity, throughout this paper, double-dot operation symbols are neglected, double-dot product of two tensors X and Y issimply written as XY .

This paper is restricted to isotropic poro-elastic rocks in which the frame and solid grain are also isotropic. Solid grain elasticmodulus tensor, drained modulus tensor and effective elastic modulus tensor can be written as

= + ( )K GL I I3 2 , 3h d0 0 0

= + ( )K GL I I3 2 , 4d dh

dd

and

= + ( )K GL I I3 2 , 5h d

where K0 and G0 are solid grain bulk and shear moduli. Kd and Gd are drained bulk and shear moduli. K and G are effective bulkand shear moduli to be determined. Ih and Id are two fourth-order tensors with components δ δ=Iijkl

hij kl

13

and

δ δ δ δ δ δ= + −Iijkld

ik jl il jk ij kl12

12

13

,

respectively. For simplicity, we denote = ⊗I I Ih 13 2 2, where I2 is second-order identity tensor with components δ( ) =I ij ij2 .

The feature of the modified Mori–Tanaka scheme is that the dependence of effective elastic properties on fluid propertiesis converted to the dependence on pore pressure. The advantage of the pore-pressure dependent scheme is that the problemof prediction of dynamic modulus tensor due to SF is converted to find the induced pore pressure and pore deformation.This is the main idea of our SF model. Song et al. (submitted to Geophysics) showed that the pore-stress-dependent schemeis rigorously consistent with Gassmann's equation when pore pressure is equilibrated within the whole pore system, and itis consistent with original Mori–Tanaka scheme in the absence of pore mass communication. These results provide low- andhigh-frequency limits for our model.

3. Analysis of local flow

Now the problem of local flow is changed to determine the induced pore pressures. In other words, the problem comesdown to the transform from applied loadings to induced pore responses. To obtain the transform relation, we need to first


describe the pore microstructure and then solve the corresponding governing equations.Following David and Zimmerman (2012), we assume that the pore space of the rock consists of a distribution of cracks

(compliant/soft pores) with different aspect ratios and one family of stiff pores having uniform aspect ratio. The stiff poresand cracks form the whole interconnected pore space. The cracks are assumed to be spheroidal pores of small aspect ratio(penny-shaped cracks). The cracks can vary in aspect ratio, orientation and size. Each crack is characterized by individualporosity ϕc

i, aspect ratio γ i and volume vci . Stiff pores are assumed to be spheroidal pores but with aspect ratios greater than

0.01 (Walsh, 1965; David and Zimmerman, 2012). For simplicity, we assume that stiff pores have the uniform aspect ratio γp

and the uniform volume vpi , but can vary in orientation. Total porosity consists of two parts:

∑ ∑ϕ ϕ ϕ= +

( )ϕ ϕ

,

6

ici

ipi

c p

where ϕc and ϕp are total crack porosity and total stiff pore porosity, respectively. ϕci is the volume fraction of the ith crack. ϕp

i

is the volume fraction of the ith stiff pore. For a representative volume element (RVE) whose scale is smaller than wave-length but larger than size of pores, let the volume of the RVE be v. ϕc

i and ϕpi are therefore expressed as

ϕ = ( )vv

, 7ci c

i

ϕ = ( )v

v. 8p

i pi

The total pore volume ϕv is

∑ ∑ ϕ= + =( )

ϕv v v v.9i

ci

ipi

The average fluid stress σf tensor is

∑ ∑ϕ

ϕ ϕσ σ σ= +( )

⎛⎝⎜⎜

⎞⎠⎟⎟1,

10f

ici

ci

ipi

pi

where σci is the fluid stress tensor in the ith crack, σp

i is the fluid stress tensor in the ith stiff pore.The average pore strain ϕe tensor is

∑ ∑ϕ

ϕ ϕ= +( )

ϕ

⎛⎝⎜⎜

⎞⎠⎟⎟e e e

1,

11ici

ci

ipi

pi

where eci is the strain tensor of the ith crack, ep

i is the strain tensor of the ith stiff pore.

3.1. Description of pores

Now let us consider the stress and strain of the pores and their infilled fluid.

3.1.1. CrackFor a given crack, the infilled fluid satisfies the following mass conservation equation,

( )σ= = − ( )K pL e I eor , 12f cfi

ci

f cfi

ci

2

where ecfi is the fluid strain in the ith crack. The fluid stress σc

i is related to fluid pressure via σ = − pIci

ci

2 , where pci is the fluid

pressure in the ith crack. Lf is the fluid elastic modulus tensor that can be written as = KL I3f fh, where Kf is the fluid bulk

modulus.Upon Eshelby (1957)'s solution, Song et al. (submitted to Geophysics) expressed the fluid-saturated pore strain as a

superposition of dry pore strain and a perturbation term due to actual fluid stress. For non-interaction approximationscheme (including the MT scheme), the resulting relation is

( ) ( ) σ= − − − ( )−

∞− −e I S e I S S L , 13c

ici

ci

ci

ci1 1

01

where eci is the strain of the ith crack, ∞e is the externally applied strain, and Sc

i is the Eshelby tensor of the crack. Sci depends

on Poisson's ratio, crack of the solid grain, the crack shape as well as coordinate system rotation.Within the crack, the fluid volume content decrement (fluid volume squeezed out of the crack due to passing wave) qc

i

can be related to ecfi and ec

i via


− =( )

q

vI e I e ,

14cfi

ci c

i

ci2 2

where = ( )I e etrcfi

cfi

2 and = ( )I e etrci

ci

2 are the fluid volume strain and crack volume strain, respectively. In the high-frequency

limit, there is no time for the fluid pressure to equilibrate such that the magnitude of qci should be zero, while in the low-

frequency limit, there is enough time for fluid pressure to be in equilibrium such that the magnitude of qci should be greater

than that at any other frequencies.

3.1.2. Stiff poreSimilarly, we can write the equations for stiff pores as follows:

( )σ= = − ( )K pL e I eor . 15f pfi

pi

f pfi

pi

2

( ) ( ) σ= − − − ( )−

∞− −e I S e I S S L , 16p

ipi

pi

pi

pi1 1

01

and

− =( )

q

vI e I e ,

17pfi

pi p

i

pi2 2

where epfi is the fluid strain in the ith stiff pore, pp

i is the fluid pressure. qpi is the fluid volume content decrement in the ith

stiff pore. Spi is the corresponding Esheby tensor.

Here, we need to clarify that Eshelby's solution shows that the elastic fields in an arbitrary ellipsoidal pore (inclusion)embedded in an elastic matrix material subjected to displacement with uniform strain field at infinity are homogeneous. Forarbitrary materials (rocks), the inclusions (pores) will not, in general, be dilute. Throughout the paper, σc

i , σpi , ec

i , epi , ecf

i andepf

i are essentially volume-averaged variables over individual subpores. Using average stress and strain to replace Eshelby'shomogeneous stress and strain is a common treatment for the effective medium method. Although the actual pores areseldom spheroidal, these spheroidal pores represent a wide variety of pore shapes, and are amenable to analytic treatmentthrough the formalism of Eshelby (1957).

3.2. Fluid pressure relaxation

We next study pore-fluid pressure relaxation due to fluid transport or mass communication.

3.2.1. Crack relaxationTo analyze local fluid transport caused by crack squirt, we need to consider a particular geometrical configuration for

cracks. For a given crack characterized by aspect ratio γ =i aband volume π=v abc

i 43

2, where a and b are two semiaxes, weassume the its infilled fluid is a Newtonian fluid with a viscous fluid of viscosity η, density ρf , and bulk modulus Kf . Ac-cording to Eshelby's solution, significant fluid pressure can be produced when an externally compressive loading is appliedto the direction normal to the crack faces. SF will occur if there exists a pressure difference between the crack and adjacentpore space.

To model the pressure relaxation and fluid flow, we assume that: (1) the crack with small aspect ratio is equivalent to aparallel gap so that cylindrical coordinate systems θ( )r z, , can be employed to solve for the fluid dynamics Eq. (2) the parallelgap has the same aspect ratio and volume as the crack so that its effective radius R and thickness h are characterized by

= ( )R b2/313 and = ( )h a2 2/3

13 , respectively; and (3) fluid mass exchange occurs at =r R.

Adopting an ω(− )i texp dependence and considering an axisymmetric radial flow, i.e., = =θ θ∂∂ v 0, for a crack, we write the

linear governing equations controlling fluid flow as (e.g., Tang, 2011)

Fig. 1. A crack and its adjacent pores. The crack may open to stiff pores and/or other cracks.

Fig. 2. Radial flow in a crack.


ω− ˜ + ∂∂

( ) + ∂∂

=( )

⎡⎣⎢

⎤⎦⎥i p K

r rrv

vz

10,

18ci

f rz

and

ωρη η

∂∂

+ ∂∂

− + ∂∂

+ =∂ ˜∂ ( )

vr r

vr

vr

vz

i vp

r1 1

,19

r r r r fr

ci2

2 2

2

2

where p̃ci is the deformation-induced fluid pressure. vr and vz are the radial and axial fluid particle velocity components,

respectively. Fig. 2 shows a sketch of the radial flow in a crack. Eq. (18) is the compressibility law, and Eq. (19) is the Navier–Stokes equation for radial flow in which nonlinear convective acceleration is neglected in the small-strain limit.

The boundary conditions are

ω= ∓ Δ( )=±

vi

h2

,20

zz h/2

= ( )=±v 0, 21r z h/2

and

˜ = ( )=p p , 22c

ir R f

where Δh is the opening distance of crack upper and lower surfaces. For real rock configuration, the crack may open to stiffpores or other cracks (see Fig. 1). Thus, the expected value of fluid pressure at =r R should be represented by the volume-average pore pressure throughout all pore space. As is shown in Eq. (22) (according to Hill (1963)'s average principle),

=∑ + ∑ϕϕ

ϕ

ϕ

⎛⎝⎜

⎞⎠⎟p p pf i c

ii p

ici

pi

is selected as the squirt-flow fluid pressure boundary. In Gurevich et al. (2010)'s squirt-flow model,

soft pores were assumed to be fully filled with the fluid, whereas the stiff pore is drained. Consequently, they adopted zeropore pressure boundary condition. But this is not a case for real fully saturated rocks. In Tang et al. (2012)'s model, theyadopted stiff-pore fluid pressure as the pressure boundary condition, this indeed corresponds to a real case but may not beproper for a large volume ratio of soft and stiff pores. In Chapman et al. (2002)'s model, each pore was assumed to be locatedwithin an element/lattice connecting to six other elements. They suggested that a so-called element-number-average porepressure could denote the expected pore pressure of all pore space without being aware of that their expected value wasdifferent from the actual average value. In contrast, we select volume average value of fluid pressure as the boundarycondition. In Pride et al. (2004)'s model, all cracks respond uniformly to externally compressive loadings and thus aretreated as a whole. Their analysis of pore pressure relaxation changes to relaxation between stiff pores and one re-presentative isotropic crack. In contrast to Pride et al.'s model, the analysis of this paper treats one single local crack.

Because of the dominant flow occurs in −r direction, we let the fluid pressure p̃ci be independent of z , i.e., ˜ = ˜ ( )p p rc

ici .

Using of variable separation method, we obtain that the general solution of Eq. (19) is

( ) ( )η

= + +˜

( )⎡⎣ ⎤⎦v

kC k z C k z

dp

dr1

1 cos sin ,23

rz

z zci

2 1 2

where

( )˜= ( )

dp

drC J k r , 24

ci

r3 1

and


ωρη

+ =( )

k ki

.25r z

f2 2

J1 is Bessel function of the first order. C1, C2 and C3 are constants to be determined. Use of the boundary condition (21) andthe symmetry condition ( ) = ( − )v r z v r z, ,r r leads to

( )( )η

= −˜

( )

⎡⎣⎢⎢

⎤⎦⎥⎥v

k

k z

k h

dp

dr1

1cos

cos 0.5.

26r

z

z

z

ci

2

Substituting Eq. (26) into Eq. (18) and then applying an integral ∫ (⋅)−

dzh h

h10.5

0.5operation to the result yield

~+

~+

~ ~ +~ Δ =

( )d p

dr r

dp

drk p K k

hh

10,

27ci

ci

ci

f

2

2

2 2

where ∫ ω= −−

∂∂

Δdz ih h

h vz

hh

10.5

0.5 z is employed and

( )ωη˜ =

− ( )⎡⎣⎢

⎤⎦⎥

ki k

K 1.

28

z

fk h

k h

2 2

tan 0.5

0.5z

z

Eq. (27) is an inhomogeneous Bessel equation of zero order. Using the pressure boundary condition (22), we arrive at

( )( )

( )( )

δ˜ = − −˜

˜ +˜

˜( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥p K

hh

J kr

J kRp

J kr

J kR1 ,

29ci

f f0

0

0

0

where J0 is Bessel function of the zero order. Comparing Eq. (29) with Eq. (24), we obtain

= ˜ ( )k k. 30r

For frequencies ω η ρ< ( )h/ f2 , applying Taylor expansion with respect to zero point to the term ( )k htan 0.5 z of Eq. (28)

yields that the wave number kr can be approximated as

ωη≈( )

kiK h

3 2.

31r

f

For water, the ratio η ρ ≈ − −/ 10 m sf6 2 1. For a water-saturated crack with thickness < μh 1 m, this approximation is indeed

appropriate even at 1 MHz frequencies.Alternatively, the mass conservation equation can be written in an integral form,

∫− ~ = Δ +( )v K

p dvv

v

q

v

1,

32ci

f v ci c

i

ci

ci

ci

c

where Δvci is the crack volume dilatation. Eq. (32), which is equivalent to Eq. (14), relates fluid flow to crack deformation. The

crack volume dilatation Δvci can also be well approximated by the motion of the upper and lower surfaces,

∫ πΔ ≈ ⋅ = Δ( )

v h Rn u ,33c

i

Sf

2

c

where the normal vector n is directed outward from crack fluid. The area Sc consists upper and lower crack surfaces. uf is the

fluid particle displacement vector. The average pressure ∫= ˜p p dvci

v v ci1

ci c

i can be approximated by

∫≈ ˜ ( )( )

pR

p r rdr2

.34c

iR

ci

2 0

In Eq. (32), the SF qci can be expressed by the lateral volumetric flow at =r R,

∫ω= − |

( )−

+=q

iv dz

1.

35ci

h

h

r r R/2

/2

Substituting Eq. (29) into Eq. (34) and using (31) leads to

( ) ( )ς ς= − Δ − + ( )⎡⎣ ⎤⎦p K

hh

f p f1 , 36ci

f ci i

f ci i

where


( ) ( )( )ςς

ς ς=

( )f

J

J

2,

37ci i

i

i i

1

0

and

ςγ

ωη=( )

i K1

3 / .38

ii f

Multiplying I2 on both sides of Eq. (36) leads to a second-order tensor form,

( )σ σ= − + ( )f fL e1 , 39ci

ci

f ci

ci

f

where σ = − pIci

ci

2 and σ = − pIf f2 . Remember that pci denotes the average fluid pressure in an individual crack.

We call Eq. (39) the crack relaxation equation. In the low-frequency limit (ω = 0), =f 1ci , we obtain a desired result:

σ σ=ci

f , i.e., the crack pressure is in equilibrium with average pore pressure. In the high-frequency limit (ω → ∞), =f 0ci , we

obtain another desired result: σ = L eci

f ci (i.e., crack strain is identical to crack fluid strain), which implies =q 0c

i .To illustrate the SF relaxation, we use Eq. (14) to eliminate ec

i and then use (12) to eliminate ecfi . The result is

σ σ−= −

− −=

−

( )

⎛⎝⎜⎜

⎞⎠⎟⎟K

f

f

q

v

p p

K

f

f

q

vI

1or

1.

40

ci

f

f

ci

ci

ci

ci

ci

f

f

ci

ci

ci

ci2

Eq. (40) shows that the squirt flow qci is controlled by the pressure difference between ith crack and the averaged pore

pressure.

3.2.2. Stiff-pore relaxationThe stiff pores often occupy the main pore space, storing the fluid. We assume that the fluid transport in stiff pores is

associated with fluid mobility and can be described by Darcy's law,

( ) ( )κη

−= ∂

−

( )

p p

L

q q

S,

41pi

pt

pi

p

where L is the effective stiff-pore distance, S is the effective stiff-pore cross section area. ∂t is the partial differential withrespective to time. κ is the permeability of stiff-pore space (i.e., the permeability of a rock without cracks). pp is the averagefluid pressure over all stiff pores. qp is the averaged fluid volume flowing out of stiff pores.

∑ ∑ϕ

ϕ= =

( )=

⎛⎝⎜⎜

⎞⎠⎟⎟p p

Np

1,

42p

i

pi

ppi

p i

N

pi

1

p

∑=( )

qN

q1

,43

pp i

pi

where Np is the total number of stiff pores.Since the pores are interconnected, for a given stiff pore the fluid transport is allowed to communicate with more than

one other stiff pore. We therefore use pressure difference −p ppi

p to measure the fluid transport between stiff pores. Inessence, Eq. (41) describes a statistical feature for stiff pores. The term qp in right-hand side of Eq. (41) guarantees∑ ( − ) == p p 0i

Npi

p1p . We call Eq. (41) the stiff-pore relaxation equation.

3.3. Macroscopic fluid mass conservation

In fact, we need one more equation to form the complete governing equations. This paper is restricted to squirt dis-persion and neglects Biot global flow. The “closure relation” can be obtained from the fluid mass conservation at macro-scopic scale. In the absence of global flow, the fluid mass within a RVE is constant. We therefore obtain

∑ ∑+ =( )

q q 0.44i

ci

ipi

Eq. (44) is the desired closure relation. An alternative interpretation of the macroscopic mass conservation is that fluidstrain ≡ ∑ + ∑ϕ

ϕ

ϕ

ϕe e ef i cf

ii pf

ici

pi

is identical to the pore strain ϕe .

3.4. Governing equations and transform relation

We now summarize the final form of governing equations of our model. They are:


� Stress-strain relations: (12)–(17)� Fluid relaxation (Eqs. (39) and 41)� Mass conservation law: (44)� Average principles: (10), (42) and (43)

Solving these equations, we obtain the following transform relations (see detailed derivation in Appendix A):

=( )

∞q

v

fI Q e

3,

45ci

ci

ci

ci

2

σ = ( )∞K D e , 46ci

f ci

σ = ( )∞K D e , 47pi

f pi

σ = ( )∞K Ce , 48f f

= ˜ ( )∞e T e , 49ci

ci

= ˜ ( )∞e T e , 50pi

pi

= ˜ ( )ϕ ϕ ∞e T e , 51

where a series of dimensionless transform tensors are

= ˜ ( )−AC B, 521

( ) ( ) ( )( ) ( ) ( )

= + − + − −

− + − − − ( )

− − − −−

− −− −

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

f

f

Q I L I S S L I L I S S L C

I L I S S L I I S

1

3 1 , 53

ci

f ci

ci

ci

f ci

ci

ci

f ci

ci h

ci

10

1 10

11

10

11 1

( ) ( ) ( ) ( )= + − − − − + ( )− −

− −⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥f f fD I L I S S L I I S C1 3 1 , 54c

ici

f ci

ci

ci h

ci

ci1

01

1 1

( ) ( )∑ ∑ωω φ

φ φ ωω

φφ

= − + − − − − −( )

− −−

−

⎪ ⎪

⎪ ⎪⎧⎨⎩⎡⎣⎢

⎤⎦⎥

⎫⎬⎭⎧⎨⎩

⎛⎝⎜⎜

⎞⎠⎟⎟

⎡⎣⎢⎢

⎤⎦⎥⎥

⎫⎬⎭

i ifD I I L I S S L I C D I S Q

13 ,

55pi

sf p

ipi h

p ici

ci

spi

i

ci

pci

ci1

01

11

( ) ( )˜ = − − ( )− −KT I S I S L D , 56c

ici

f ci

ci1

01

( ) ( )˜ = − − ( )− −KT I S I S L D , 57p

ipi

f pi

pi1

01

∑ ∑ϕ

ϕ ϕ˜ = ˜ + ˜( )

ϕ

⎛⎝⎜⎜

⎞⎠⎟⎟T T T

1,

58ici

ci

ipi

pi

and where

( ) ( ) ( )

( ) ( )

∑

∑

ϕ

ϕ ϕ

= + − + − −

+ − + − −( )

− − − −−

− −−

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

A M f f

f f

I I L I S S L I L I S S L I

I I L I S S L I

1

3 1 ,59

ip c

ici

f ci

ci

ci

f ci

ci

ici

ci

ci

f ci

ci

21

01 1

01

1

2

21

01

1

2

( ) ( ) ( ) ( )

( ) ( )

∑

∑

ϕ

ϕ

˜ = + − + − − −

+ + − −( )

− −− −

− −− −

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

M f f fB I I L I S S L I I S

I I L I S S L I I S

9 1 1

9 ,60

h

ici

p ci

ci

ci

f ci

ci h

ci

h

ipi

f pi

pi h

pi

10

11 1

10

11 1


( )= + − ( )− −

−⎡⎣⎢

⎤⎦⎥M I I L I S S L I

13

, 61p f pi

pi

21

01

1

2

ω δη

=( )

K,

62s

f

and where

δ κ=( )v

SL

.63p

i

Under any orthogonal transformation, Mp is one of two linear invariants of fourth-order tensor [ + ( − ) ]− − −I L I S S Lf pi

pi1

01 1;

thus the superscript i in Eq. (61) can be dropped. δ is a dimensionless parameter which measures the stiff-pore spaceconnectivity. We call ωs the characteristic frequency of stiff-pore pressure relaxation.

4. Dynamic bulk and shear moduli

4.1. Dynamic effective elastic moduli

To obtain the effective elastic moduli, we substitute (Eqs. (46–48) and 51) into Eq. (1), and then eliminate ∞e . Rearrangingthe result, we obtain that the effective elastic modulus tensor can be expressed as

( )( ) ( )ω( ) = − ˜ − + ˜ − ( )ϕ ϕ−

KL L L T C L E L T L L , 64f d d0 0 01

0

or

( )( ) ( )ω( ) = + + + ˜ − ( )ϕ−

KL L L E C L E L T L L , 65d d f d d01

0

where

∑ ∑ϕϕ

ϕ

ϕ= +

( )E E E ,

66i

ci

ci

i

pi

pi

and where

( )= − ( )− −KE I S S L D , 67c

if c

ici

ci1

01

and

( )= − ( )− −KE I S S L D . 68p

if p

ipi

pi1

01

4.2. Crack aspect ratio distribution

For isotropic elastic materials, one may hope to know bulk and shear moduli directly. Moreover, in real rocks a widerange of different crack aspect ratios may be present. Therefore, an important issue is how to express the effective elasticmoduli in terms of distributions of crack aspect ratio. Now, we present details in calculating the effective bulk and shearmoduli for the case of distribution of crack aspect ratio.

In (Eqs. (55), 58–60) and (66), there exist sum operations ∑i for tensors over cracks, for example, ϕ∑ Di ci

ci and ϕ∑ T̃i c

ici . For

a rock sample containing ′N families of randomly orientated cracks, these sums can be uniformly determined as follows:

∑ ∑ϕ ϕ= ¯( )α

α α

=

′( ) ( )X X ,

69ici

ci

N

c1

where ϕ α( )c is the porosity of αth family of cracks whose aspect ratio is denoted by γ α( ). ¯ α( )X is calculated via orientation

average rule (Dvorak, 2012),

¯ = + ( )α α α( ) ( ) ( )X XX I I , 70Bh

Sd

where

= ( )α α( ) ( )X X

13

, 71B iijj

1


= −( )

α α α( ) ( ) ( )⎛⎝⎜

⎞⎠⎟X X X

15

13

.72S ijij iijj

There also exist sums with respect to stiff pores in (Eqs. (58), (60) and 66). They can be determined in the same way.For the case of a continuous distribution of crack aspect ratio, the effective elastic modulus tensor can be determined by

introducing the concept of crack porosity distribution function γ( )c , which can be related to the total crack porosity via

∫ϕ γ γ= ( ) ( )c d , 73c

where γ is the crack aspect ratio. The sum operation over all cracks can be reformulated in the form:

∫∑ ϕ γ γ γ= ( ) ¯ ( )( )

c dX X ,74i

ci

ci

where γ¯ ( )X , which depends on aspect ratio, is an isotropic tensor that can be written as follows:

γ¯ ( ) = + ( )X XX I I , 75Bh

Sd

where

γ γ( ) = ( ) ( )X X13

, 76B iijj

γ γ γ( ) = ( ) − ( )( )

⎛⎝⎜

⎞⎠⎟X X X

15

13

.77S ijij iijj

If a rock sample contains only one kind of randomly orientated cracks, γ i is simply written as γc and ϕγ α( )c is identical to ϕc .

4.3. Low- and high-frequency limits

To better understand the frequency-dependent behavior of effective elastic moduli, we consider two limiting cases.

) Low-frequency limit (i.e., ω = 0)In this limit, we have ς( ) =f 1c

i , → ∞ωω

s . From (Eqs. (52–61), 67) and (68), we obtain

( ) ( )( ) ( )

ϕ ϕ

ϕ ϕ=

∑ − + ∑ −

∑ + − + ∑ + − ( )

− −

− − − −⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

CI I S I I S

I I L I S S L I I I L I S S L I

9 9,

78

i ci h

ci h

i pi

pi

i ci

f ci

ci

p i f pi

pi

1 1

21

01

2 21

01

2

( ) ( )= + − − − ( )− − −⎡

⎣⎢⎤⎦⎥Q I L I S S L C I I S3 , , 79c

if c

ici h

ci1

01 1

= = ( )D D C, 80ci

pi

and

( ) ( )( ) ( )

( )( )

˜ = − −

˜ = − −

= −

= − ( )

− −

− −

− −

− −

K

K

K

K

T I S I S L C

T I S I S L C

E I S S L C

E I S S L C

.

81

ci

ci

f ci

pi

pi

f pi

ci

f ci

ci

pi

f pi

pi

10

1

10

1

10

1

10

1

According to Eq. (2), we rewrite drained elastic modulus tensor as

( ) ( )∑ ∑ϕ ϕ ϕ ϕ= ( − ) ( − ) + − + −( )

− −−⎡

⎣⎢⎢

⎤⎦⎥⎥L L I I S I S1 1 .

82d

ici

ci

ipi

pi

01 1

1

Rearranging Eq. (82), we obtain that

( ) ( ) ( )∑ ∑ϕ ϕ ϕ− + − = ( − ) −( )

− − −I S I S L L I1 .83i

ci

ci

ipi

pi

d1 1 1

0

Inserting Eq. (83) into Eq. (78), we obtain

( )ϕ

ϕ α ϕ=

( − ) −

+ ( − ) ( )

−

KC I

L L I3

1

84

h d

f K

10

1

d

where

2


α = −( )

KK

185

d

0

α is the so-called Biot-Willis coefficient (Biot and Willis, 1957).Substituting Eq. (81) into Eq. (65) and then using Eq. (83) to eliminate all terms associated with Eshelby's tensors, weobtain that

( ) ϕ ϕ( ) = + − ( − ) − ( − ) −( )

− − − −−⎧⎨⎩

⎡⎣ ⎤⎦⎫⎬⎭K

L L I L L C L L L L I L C01

186

d df

d d01 1

01

0 01

1

Substituting Eq. (84) into Eq. (86) and using Eq. (5), we obtain

α( ) = + ( )K K M0 , 87d2

( ) = ( )G G0 , 88d

where

α ϕ ϕ= − +( )

−⎛⎝⎜

⎞⎠⎟M

K K.

89s f

1

M is the so-called Biot storage modulus. It is clear that (Eqs. (87) and 88) are exactly Gassmann's formula. It is noteworthythat Eq. (80) shows that in the low-frequency limit the fluid pressure is in equilibrium within the whole pore system.

) Extremely high-frequency limit (i.e., ω → ∞)

In this limit, ς( ) =f 0ci and =ω

ω0s . Moreover, we obtain that

=

=

˜ =˜ = ( )

K

K

D L T

D L T

T T

T T 90

f ci

f ci

f pi

f pi

ci

ci

pi

pi

and

( )( )

∑ ∑

∑ ∑

ϕϕ

ϕ

ϕ

ϕϕ

ϕ

ϕ

= −

= −

= +

˜ = +( )

ϕ

− −

− −

K

E T S L L I S

E T S L L I S

C L T L T

T T T91

ci

ci

ci

f ci

pi

pi

pi

f pi

fi

ci

f ci

i

pi

f pi

i

ci

ci

i

pi

pi

01 1

01 1

where

( )= − + ( )− −

T I S S L L , 92ci

ci

ci

f01 1

and

( )= − + ( )− −

T I S S L L . 93pi

pi

pi

f01 1

Substituting (Eqs. (91) and 82) into Eq. (64), we obtain

( ) ∑ ∑ ∑ ∑ϕ ϕ ϕ ϕ ϕ(∞) = − − + ( − ) + +( )

−

⎪ ⎪

⎪ ⎪⎛⎝⎜⎜

⎞⎠⎟⎟

⎧⎨⎩

⎫⎬⎭

L L L L T T I T T194

fi

ci

ci

ipi

pi

ici

ci

ipi

ci

0 0

1

Eq. (94) is the original Mori–Tanaka scheme which describes the effective elastic properties for isolated pores. In thiscase, fluid mass exchange has no enough time to take place, resulting in locking the fluid in each pore. The induced fluidpressure may vary from pore to pore, depending on the shape and orientation of each pore.

4.4. Simplified model for squirt flow dispersion

Grechka (2009) showed numerically that Gassmann's formula are excellent approximation for isolated pores of aspectratio larger than 0.2. This implies that the relaxation within the stiff pore is negligible for stiff pore larger than 0.2. Also, as


suggested by Mavko and Jizba (1991), the assumption of uniform stiff-pore pressures are applicable at ultrasonic fre-quencies. Moreover, the well-established observation (Gurevich et al., 2010) shows that the SF dispersion between seismicand ultrasonic frequencies is caused mainly by cracks and is neglibible at high effective stress. Thus, neglecting the stiff-porerelaxation leads to a simplified model for the SF dispersion. The simplified model can be obtained by equivalently lettingω ω< < s. Upon Eqs. (55) and (A33), we obtain that

∑ϕ

ϕ ϕ= − −( )

⎛⎝⎜⎜

⎞⎠⎟⎟D I D C

1,

95pi

p

h

ici

ci

( )∑ϕ

ϕ ϕ= − − − −( )

−−⎛

⎝⎜⎜

⎞⎠⎟⎟

MQ D C I S

3.

96pi p

p ici

ci

pi

11

Eq. (96) shows that fluid transport values are generally different from pore to pore, since the second term in the right-hand side of Eq. (96) depends on pore orientation. The only pore geometry for which stiff pores respond uniformly toapplied loadings is the case of a sphere. It is easy to demonstrate that the fluid pressure in stiff pores is in equilibrium byusing (Eqs. (95) and 47).

For the simplified model, when ω γ> > ( )η

K i3 max

2f , we obtain by using Eq. (52), (54) and (55) that

( )

∑ ∑

∑

∑

ϕϕ

ϕ

ϕ

ϕ

ϕ

ϕ

ϕ

= +

=

=

˜ =

˜ = − −( )

− −⎛⎝⎜⎜

⎞⎠⎟⎟

K

K

K

C L T L T

D L T

D L T

T T

T I S I S L L T97

fi

ci

f ci

i

pi

f pi

f ci

f ci

f pi

fi

pi

ppi

ci

ci

pi

pi

pi

fi

pi

ppi1

01

In this relatively high-frequency case, let the effective elastic bulk and shear moduli be *K and *G . The fourth subequationin Eq. (97) illustrates that the fluid is trapped in each crack. The third subequation in Eq. (97) shows that the average stiff-pore pressure is equal to these in the extreme high-frequency limit. When a sample is subjected to bulk compression, theaverage pore strain and pore pressure are equal to these in the extreme high-frequency limit, resulting in * = (∞)K K . Incontrast, when the sample is subjected to shear loadings, applying orientation rule to (97) shows that the induced stiff-porepressure is zero. This is different from extreme high-frequency limit.

5. Numerical examples

In order to illustrate the dispersion caused by pore microstructures, some numerical examples are presented. For thesolid grain elastic properties, we take =K 38GPa0 (quartz) and ν = 0.20 as fixed constants. For the pore fluid properties, wetake =K 2.2GPaf and η = 0.001Pas (water) as fixed constants. The pore microstructure parameters used in the calculationsare listed in Table 1, in which all the parameters are dimensionless. In Table 1, γp is the stiff pore aspect ratio. The Eshelbytensors used for calculation are shown in Appendix B.

The calculated inverse quality factors of the bulk and shear moduli are respectively defined by

= − ( )( ) ( )

−QKK

ImRe

,98K

1

and

Table 1Pore microstructure parameters for calculating the dynamic bulk and shear moduli. All the parameters are dimensionless.

Parameter Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7 Fig. 8

ϕp 0.15 0.18 0.18 0.04 0.11 0

γp 0.15 0.07 0.07 0.05 0.05 —

ε 0.15 0.03∼0.15 0.12 0.24 0.15 0.2γc 5E�4 5E�5∼5E�3, 1e�3 2E�4 3E�4δ — — — 1E�9�1E�3 1E�9∼1E�4 —

Fig. 3. Real part of K and real part of G versus ω.

Fig. 4. (a) Real part of the bulk modulus, (b) inverse quality factor of the bulk modulus, (c) real part of the shear modulus and (d) inverse quality factor ofthe shear modulus versus frequency for water-saturated rocks with different crack densities using the simplified model.


Fig. 5. (a) Real part of the bulk modulus, (b) inverse quality factor of the bulk modulus, (c) real part of the shear modulus and (d) inverse quality factor ofthe shear modulus versus frequency for water-saturated rocks with different crack aspect ratios using the simplified model.


= − ( )( ) ( )

−QGG

ImRe

,99G

1

where Re and Im denote the real and imaginary parts, respectively.

5.1. Results of the simplified SF model

In this subsection, the dispersion curves are calculated using the simplified model mentioned in subsection 4.4.First of all, we would like to compare our model with Gurevich et al. (2010)’s SF model. Their model agrees well with

experimental data for different effective stress. In their double-porosity-structure model, the effective bulk and shearmoduli in frequency domain are given by

= ++

( )ϕ − −⎛

⎝⎜⎜

⎞⎠⎟⎟

K K1 1 1

,

100

s1 1

p Kf KsKmf Ks

1 11 1

− = −( )

⎛⎝⎜

⎞⎠⎟G G K K

1 1 415

1 1,

101d mf d

where Kd and Gd are dry bulk and shear moduli, respectively. Ks is the bulk modulus of the solid grain. Kf is the bulk modulusof the pore fluid. ϕp is the porosity of the stiff pores. Kh is the dry bulk modulus at the highest confining pressure availablewithin the elastic regime such that all cracks are closed or filled with the frame mineral. Here, we use the dry bulk modulusof a rock without cracks to estimate its value. Kmf is the modified frame bulk modulus:

Fig. 6. (a) Real part of the shear modulus and (b) real part of the bulk modulus versus frequency for water-saturated rocks having different stiff porerelaxation parameters using the intact model. The bulk dispersion is independent of the stiff pore relaxation. (For interpretation of the references to colorin this figure legend, the reader is referred to the web version of this article.)


= ++

( )ϕ

−−

*

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K K1 1 1

,

102

mf h1 1

Kd Kh c K f Ks

1 11 1

where *K f is the effective bulk modulus of the fluid saturating cracks:

ςς ς

* = −( )( ) ( )

⎡⎣⎢

⎤⎦⎥K K

JJ

12

103f f

1

0

with

ςγ

ωη=( )

i K1

3 /104

f

γ is the effective aspect ratio of the cracks.Fig. 3 compares the dispersion curves (black lines) predicted by the present model with those (red circles) predicted by

Gurevich et al. (2010)'s SF model. The comparison illustrates that the effective bulk and shear moduli predicted by the

Fig. 7. Real part of the shear modulus versus pore relaxation parameter for a water-saturated rock at frequencies of: ω = 10 rad/s5 (black line) andω = 10 rad/s6 (red line) using the intact model.


simplified model developed by us agrees with those predicted by Gurevich et al. (2010)'s model very well in the frequencydomain.

Next, Fig. 4 shows (a) the real part of the bulk modulus, (b) inverse quality factor of the bulk modulus, (c) real part of theshear modulus and (d) inverse quality factor of the shear modulus in frequency domain for various crack densities. The crackaspect ratio and stiff porosity are taken as fixed constants. The crack density ε is related to crack porosity ϕc via

ϕ πεγ= ( )43

, 105c c

where γc is the crack aspect ratio. The different crack densities could be thought of as being due to the application of effective stress.Fig. 4(a) and (c) shows that the moduli increase with increasing frequency ω. As the crack density decreases, there is a

drop in the real parts of moduli at each frequency and an increase in the dispersion magnitude between low- and high-frequency moduli. Fig. 4(b) and (d) clearly shows that the bulk and shear attenuation magnitudes are controlled by the crackaspect ratio. It demonstrates that the dispersion magnitudes and attenuation values are sensitive to the crack aspect ratio.

Furthermore, Fig. 4(a) shows that the high-frequency bulk moduli are very close to the bulk modulus of a samplecontaining no crack. In contrast, Fig. 4(c) shows that the high-frequency shear moduli are obviously lower than the shearmodulus of the sample containing no crack.

Fig. 5 shows (a) the real part of the bulk modulus, (b) inverse quality factor of the bulk modulus, (c) real part of the shearmodulus and (d) inverse quality factor of the shear modulus in frequency domain for various crack aspect ratios. The crackdensity and stiff porosity are taken as fixed constants. Fig. 5(a) and (c) also shows that the moduli increase with increasingfrequency ω. As the crack aspect ratio decreases, there is an increase in the real parts of moduli at each frequency. Fig. 5(b) and (d) shows that the SF characteristic frequency at which the maximum values of attenuation occurs depends on thecube of the crack aspect ratio. This is also consistent with previous interpretation (O’Connell and Budiansky, 1977).

5.2. Results based on the full model

In this subsection, we use the full context of the model to quantify the effect of stiff-pore pressure relaxation on thedispersive behavior of cracked rocks.

In order to display the stiff-pore relaxation, we take a relatively small value γ = 0.05p for stiff-pore aspect ratio. Fig. 6shows (a) the real part of the shear modulus and (b) real part of the bulk modulus in frequency domain for various stiff-porerelaxation parameters δ . Fig. 6(a) shows that the real part of the shear modulus increases with increasing frequency. The γcvalue is chosen so that there would be a significant peak in squirt-flow attenuation at ultrasonic frequencies and is taken tobe fixed. Fig. 6(a) also shows that the stiff-pore relaxation affects the shear modulus and the stiff-pore relaxation frequencylinearly depends on δ . For δ = −1E 3 and δ = −1E 5, the stiff-pore relaxation frequencies are higher than SF relaxationfrequency, the corresponding curves (red and black lines) might describe the dispersive behavior for high-permeabilityrocks. The real part of the shear modulus increase to *G at first, and then increase to (∞)G as frequency further increases. Forδ = −1E 7 (blue line), the SF relaxation frequency and stiff-pore relaxation frequency lie in the same frequency range. Forδ = −1E 9, the corresponding curve (magenta line) describes the dispersive behavior for a low-permeability rock. When thefrequency is greater than squirt-flow relaxation frequency, the shear modulus is identical to (∞)G .

Fig. 8. Normalized real part of the bulk modulus (blue line) and normalized real part of the shear modulus (red line) versus frequency for a cracked water-saturated rocks with no stiff pore. The simplified model and the intact model predicts the same results. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)


In contrast, Fig. 6(b) shows that the stiff-pore relaxation parameter has no effect on bulk dispersion. This indicates thatthe stiff-pore relaxation affects shear waves more than compressional waves. The relaxation mechanism may not be neg-ligible for low-permeability rocks.

In Fig. 7 we show the dependence of shear moduli on the stiff-pore parameter δ , for two values of frequency. This figureanalyzes the effects of stiff-porosity connectivity on shear modulus within ultrasonic frequency band. In this case the realparts of shear moduli decrease with increasing δ .

5.3. The effects of stiff-pore volume fraction on dispersive behavior

In this subsection, we investigate the effects of stiff-pore volume fraction on attenuation.First, let's consider a special pore space consisting entirely of water-saturated cracks. This is the case first considered by

O’Connell and Budiansky (1977). Fig. 8 plots ( ) ( )K KRe / 0 and ( ) ( )G GRe / 0 versus frequency for a pore space consisting of onlyone family of randomly orientated water-cracks. The bulk modulus is a constant, whereas the shear modulus is obviouslydispersive. When such a sample is subjected to bulk compression, no pore-fluid pressure gradient is created. Thus the bulkmodulus is frequency-independent. Unlike the bulk compression case, the induced pore-fluid pressure depends on or-ientation, resulting in local fluid flow and dispersive shear behaviors.

Second, let us predict the dispersive behavior of samples containing two families of randomly orientated cracks for variousstiff-pore volume fractions. Fig. 9(a) and (b) plots −Q K

1 and −Q G1 versus ω for different ϕp, respectively. We take

ε γ= = −0.1, 1E 3c1 1 for the first kind of crack, ε γ= = −0.1, 1E 4c

2 2 for the second kind of crack, and δ γ= − =1E 2, 0.05p forthe stiff-pore. We take δ so large that the stiff-pore relaxation frequency is much higher than ultrasonic frequencies, as we aimat analyzing the crack SF mechanism. Solid grain properties and fluid properties are the same as these used in Figs. 4–8.

For the frequency-dependent bulk attenuation, Fig. 9(a) shows that there are two peaks in each curve. The peaks cor-respond to SF relaxation frequencies of the two kinds of crack, respectively. Moreover, the attenuation values in the blue line(ϕ = 0.1p ) are higher than these in other lines at most frequencies. This indicates that there exists an optimal ratio of thecrack volume fraction and the stiff pore volume fraction for the bulk attenuation.

Fig. 9(b) shows there are also two peaks in each shear attenuation curve. Moreover, it shows the more stiff pores, the lessattenuation. The result leads to that the existence of stiff pores can weaken the shear attenuation. The feature is differentfrom the bulk attenuation.

5.4. Squirt flow dispersion upon inverted pore aspect ratio distribution

To better estimate the dispersive behaviors of a realistic rock which contains a distribution of cracks with different aspectratios. It is of great interests to calculate the dispersion curves upon the inverted pore distribution. The input parameterscome from the inverted data by David and Zimmerman (2012). The mechanical properties and pore structures of two water-saturated sandstones (Vosges and Fontainebleau) are shown in Table 2. In their study, the crack aspect ratio distributions areinverted from the laboratory measurements at ultrasonic frequencies (�MHz) under the assumption that the fluid istrapped in the individual pores. To carry out comparison, we use simplified SF model to calculate the dispersion curves,neglecting stiff-pore relaxation. This is valid for stiff-pore aspect ratio larger than 0.2 (Grechka, 2009).

Fig. 9. (a) inverse quality factor of bulk moduli and (b) inverse quality factor of the shear moduli versus frequency for water-saturated rocks with differentstiff-pore volume fractions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 2Water-saturated rock parameters (i)).

Sandstone K0 G0 ϕ ϕc γp Dry density

Vosgesa 39 GPa 24 GPa 0.25 0.14% 0.13 1950kg/m3

Fontainebleaub 37 GPa 44 GPa 0.04 0.016% 0.6 2544kg/m3

a With fluid properties: =K 2.24GPaf , η = −10 Pas3 , and density ρ = 1000kg/mf3.

b With fluid properties: =K 2.20GPaf , η = −10 Pas3 , and density ρ = 1000kg/mf3.


Fig. 10(a) and (b) plots −Q K1 and −Q G

1 against ω for the Vosges standstone, respectively. Red lines are calculated upon theinverted pore aspect ratio distribution (see Fig. 7 of David and Zimmerman, 2012), while the black lines are calculated upondouble-porosity-structure model in which cracks have the uniform aspect ratio characterized by one effective crack aspectratio. In the black lines, we take ϕ = 0.14%c , which is the same as that in distribution model, and take × −5 10 4 as the effectivecrack aspect ratio, at which crack porosity distribution function reaches maximum.

Fig. 10(a) shows that while the attenuation peak of the continuous distribution model is lower than that of the discretedouble-porosity model in frequency domain, the continuous model predicts that the attenuation occurs in a broader fre-quency band than the case of single crack aspect ratio. Both of the continuous model and the discrete model predict the

Fig. 10. Comparison of (a) bulk and (b) shear modulus attenuations between the continuous distribution model and the discrete double-porosity model.The crack porosity distribution in red lines is given by (David and Zimmerman, 2012), while the effective crack aspect ratio used in black lines is × −5 10 4 atwhich the crack porosity distribution function reaches maximum value. ϕ = 0.14%c in both of the red and black lines. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. The comparison of P-wave attenuation between the prediction by the present squirt-flow model and the measured data of Sams et al. (1997)(rectangular boxes). These data correspond to various measurements coming from different depth ranges at their test site. Within these rectangular boxes,there are 40 VSP (30–280 Hz), 69 corsshole (0.2–2.3 kHz), 854 sonic logging (8–24 kHz), and 46 laboratory core (300–900 kHz) measurements. The at-tenuation curves are calculated upon parameters in Table 2 and inverted pore aspect ratio distribution (David and Zimmerman, 2012). The frequency

ω π=f /2 . The inverse quality factor = −− ( )( )

Q PHH

1 ImRe

, where = +H K G43

.


attenuation peaks lie in sonic logging band of frequencies (10–100 kHz). It is noteworthy that the continuous model showsthat the SF mechanism contribute a certain amount of attenuation to seismic frequencies (1–1000 Hz). Previous researchers(Pride et al., 2004; Gurevich et al., 2010; Tang et al., 2012), however, were always working with just the effective crack aspectratio instead of the crack aspect ratio distribution. Consequently, they showed that the SF mechanismwould not be involvedin explaining the observed levels of intrinsic attenuation at seismic frequencies. In contrast, this study calculates the SFattenuation from the inverted data and shows evidence that the SF creates non-negligible attenuation in exploration work.Fig. 10(b) shows that the shear attenuation has similar behaviors to the bulk attenuation.

Finally, in Fig. 11, we compare the P-wave attenuations (Vosges and Fontainebleau) predicted by the present SF model tothe data of Sams et al. (1997), who used different seismic measurements (VSP, corsswell, sonic log, and ultrasonic lab) todetermine attenuation over a wide band of frequencies. The data come from the measurements for various rocks at differentdepths. The geological structures at their test is a sequence of limestones, sandstones, siltstones and mudstones. Besides theSF process, heterogeneous solids are likely to attenuate waves as well. Nevertheless, the comparison may tell us that how


much attenuation the SF mechanism contributes to the total intrinsic attenuation. The comparison shows that the SFmechanism causes significant attenuation at the sonic and seismic frequencies.

Recently, laboratory measurements by Pimienta et al. (2015a, b) show that pore microstructural effect (specifically, thesquirt-flow mechanism) are likely to play a role in producing seismic attenuation in fluid-saturated rock samples (e.g., seedata of Fo9 in Pimienta et al., 2015a; see also Pimienta et al., 2015b).

6. Conclusions

We have developed a dynamic elastic modulus model for the grain-scale local fluid flow in cracked fluid-saturatedporous rocks. Mori–Tanaka scheme is extended for the consideration of fluid flow due to pore microstructure effects. Theresults are exactly consistent with Gassmann's theory in the low-frequency limit and with original (isolated-pore) Mori–Tanaka scheme in the extreme high-frequency limit. In the model, there are two relaxation frequencies: crack squirt-flowrelaxation frequency and stiff-pore relaxation frequency. The squirt-flow relaxation frequency depends on the cube of crackaspect ratio, while the stiff-pore relaxation frequency depends on the stiff-pore connectivity. In particular, a simplifiedmodel is also proposed to consider SF dispersion, neglecting the stiff-pore relaxation.

A number of different SF models have been proposed previously, but most of them restrict to the case in which all crackshave the same crack aspect ratio. In contrast, the present model can predict the velocity dispersion and attenuation for thedistribution of crack aspect ratio.

A practical application of the SF model is the interpretation of elastic wave attenuation and velocity dispersion in se-dimentary rocks. Using rock parameters inverted by (David and Zimmerman, 2012), the SF model are used to quantifymodulus dispersion and attenuation in frequency domain. It is shown that the SF mechanism contributes to observedattenuation at seismic frequencies as it does at acoustic logging frequencies.

More measurements of SF dispersion over a much wider frequency range are needed to validate the squirt flow model.

Acknowledgment

Yongjia Song thanks China Scholarship Council (CSC) for supporting his two-year visit at Northwestern University. Thiswork is partially supported by National Natural Science Foundation of China under the contracts (No. 11372091).

Appendix A : Derivation of transform relations

In this appendix, we derive the transformation relations between induced fluid pressures, pore strains, squirt flow andexternally applied loadings. We hope to express pore-scale variables σ σ( )q qe e, , , , ,c

ipi

ci

pi

ci

pi and macroscopic-scale variables

σ( )ϕe,f in terms of ∞e .Substituting Eq. (13) into Eq. (14) and then eliminating ecf

i by using (12) leads to

( ) ( )σ = + − + −( )

− −− −

∞⎡⎣⎢

⎤⎦⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

q

vI L I S S L L I L I S e

3 A1ci

f ci

ci c

i

ci f f c

i10

11

21

where =K I L If f213 2 is used.

Substituting Eq. (13) into Eq. (39) leads to

( ) ( ) ( ) ( )σ σ= + − − − − + ( )− −

− −∞

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥f f fI L I S S L L I S e1 1 A2c

ici

f ci

ci

ci

f ci

ci

f1

01

1 1

Eliminating σci by combining (Eqs. (A1) and A2) leads to

( ) ( ) ( )

( ) ( ) ( )

σ= + − + − −

− + − − − ( )

− − − −−

− −− −

∞

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

q

vf f

f f

L I I L I S S L I L I S S L

I L I S S L L I S e

31

1 A3

ci

ci f f c

ici

ci

f ci

ci

ci

f

ci

ci

f ci

ci

f ci

21

01 1

01

1

10

11 1

Multiplying I2 on both sides of Eq. (A3) leads to

( ) ( ) ( )

( ) ( ) ( )

σ= + − + − −

− + − − −( )

− − − −−

− −− −

∞

⎧⎨⎩⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎫⎬⎭

qv f

Kf

f

I I L I S S L I L I S S L

I L I S S L L I S e

31

1A4

ci c

ici

ff c

ici

ci

f ci

ci

f

ci

f ci

ci

f ci

21

01 1

01

1

10

11 1


where = KI L I 9f f2 2 is used. Substituting Eq. (A2) into Eq. (10) gives

( ) ( ) ( ) ( )

∑

∑

σ σ

σ

=

− + − − − − +( )

ϕ

− −− −

∞⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

v v

v f f fI L I S S L L I S e1 1A5

ipi

pi

f

ici

ci

f ci

ci

ci

f ci

ci

f1

01

1 1

Substituting Eq. (16) into Eq. (17) and then eliminating epfi by using Eq. (15) leads to

( ) ( )σ = + − + −( )

− −− −

∞⎡⎣⎢

⎤⎦⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

q

vI L I S S L L I L I S e

3 A6pi

f pi

pi p

i

pi f f p

i10

11

21

or

( ) ( )σ= + − − −( )

− − −∞

⎡⎣⎢

⎤⎦⎥

q

vL I I L I S S L L I S e

3 A7f

pi

pi f p

ipi

pi

f pi

21

01 1

Substituting Eq. (A6) into Eq. (A5) leads to

( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

∑

∑

∑

∑

σ σ

+ −

= − + − −

− − + − − −

− + − −( )

φ

− −−

− −−

− −− −

∞

− −− −

∞

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

q

v v f f

f v f

v

I L I S S L L I

I L I S S L

I L I S S L L I S e

I L I S S L L I S e

3

1

1 1

A8

if p

ipi

fpi

fi

ci

ci

ci

f ci

ci

f

ici

ci

ci

f ci

ci

f ci

ipi

f pi

pi

f pi

10

11

2

10

11

10

11 1

10

11 1

Let all stiff pores be equivalent by only one kind of randomly orientated spheroidal pore with the same size. Thenmultiplying I2 on both sides of Eq. (A8) leads to

( )∑ ∑+ − =( )

− −−⎡

⎣⎢⎤⎦⎥ q M K qI I L I S S L L I 9

A9if p

ipi

f pi

p fi

pi

21

01

1

2

and

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

∑ ∑

∑

∑

σ= − + − −

− − + − − −

− + − −( )

φ− −

−

− −− −

∞

− −− −

∞

⎪

⎪

⎪

⎪

⎧⎨⎩

⎡⎣⎢⎢

⎡⎣⎢

⎤⎦⎥

⎤⎦⎥⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎫⎬⎭

qM K

v v f f

f v f

v

I I I L I S S L

I L I S S L L I S e

I L I S S L L I S e

13

1

1 1

A10

ipi

p f ici

ci

ci

f ci

ci

f

ici

ci

ci

f ci

ci

f ci

ipi

f pi

pi

f pi

21

01

1

10

11 1

10

11 1

and where

( )= + − ( )− −

−⎡⎣⎢

⎤⎦⎥M I I L I S S L I

13 A11p f p

ipi

21

01

1

2

Under any orthogonal transformation, Mp is one of two linear invariants of fourth-order tensor [ + ( − ) ]− − −I L I S S Lf pi

pi1

01 1.

The inverse of Mp is

( )= + − ( )− − −⎡

⎣⎢⎤⎦⎥M I I L I S S L I

13 A12p f p

ipi1

21

01

2

Substituting Eq. (A10) and Eq. (A4) into fluid mass conservation Eq. (44) leads to

− = ( )∞Ap I Be A13f 2

where σ = − pIf f2 is applied and


( ) ( ) ( ) ( )

( ) ( )

∑

∑

ϕ

ϕ

= + − + − − −

+ + − −( )

− −− −

− −− −

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

K M f f f

K

B I L I S S L I I S

I L I S S L I I S

3 1 1

3A14

fi

ci

p ci

ci

ci

f ci

ci h

ci

fi

pi

f pi

pi h

pi

10

11 1

10

11 1

( ) ( ) ( )

( ) ( )

∑

∑

ϕ

ϕ ϕ

= + − + − −

+ − + − −( )

− − − −−

− −−

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

A M f f

f f

I I L I S S L I L I S S L I

I I L I S S L I

1

3 1A15

ip c

ici

f ci

ci

ci

f ci

ci

ici

ci

ci

f ci

ci

21

01 1

01

1

2

21

01

1

2

where δ δ=Iijklh

ij kl13

and = KL I3f fh is used.

pf denotes average pore pressure. Then we write σf as

σ = ( )∞K Ce A16f f

with

= ˜ ( )−AC B A171

where

( ) ( ) ( ) ( )

( ) ( )

∑

∑

ϕ

ϕ

˜ = + − + − − −

+ + − −( )

− −− −

− −− −

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

M f f fB I I L I S S L I I S

I I L I S S L I I S

9 1 1

9A18

h

ici

p ci

ci

ci

f ci

ci h

ci

h

ipi

f pi

pi h

pi

10

11 1

10

11 1

Substituting Eq. (A16) back into Eq. (A2), we have

σ = ( )∞K D e A19ci

f ci

where

( ) ( ) ( ) ( )= + − − − − + ( )− −

− −⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥f f fD I L I S S L I I S C1 3 1 A20c

ici

f ci

ci

ci h

ci

ci1

01

1 1

Substituting Eq. (A19) into Eq. (13) leads to

= ˜ ( )∞e T e A21ci

ci

where

( ) ( )˜ = − − ( )− −KT I S I S L D A22c

ici

f ci

ci1

01

Substituting Eq. (A16) into Eq. (A4) leads to

= ( )∞qv f

I Q e3 A23c

i ci

ci

ci

2

where

( ) ( ) ( )( ) ( ) ( )

= + − + − −

− + − − − ( )

− − − −−

− −− −

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

f

f

Q I L I S S L I L I S S L C

I L I S S L I I S

1

3 1 A24

ci

f ci

ci

ci

f ci

ci

ci

f ci

ci h

ci

10

1 10

11

10

11 1


∑ ∑= −( )

∞qv f

I Q e3 A25i

pi

i

ci

ci

ci

2

Substituting (Eqs. (A16) and A19) into Eq. (10) leads to

∑ ∑ϕ ϕ ϕσ = −( )

∞

⎛⎝⎜⎜

⎞⎠⎟⎟K KC D e

A26ipi

pi

fi

ci

f ci

Substituting (Eqs. (A26) and A25) to (Eqs. (42) and 43), we obtain that


∑ ∑ϕ

ϕ ϕϕ

ϕ ϕσ= − = −( )

∞ ∞

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟p

KK KI D C e C D e

3or

1

A27p

f

p ici

ci

pp

fi

ci

f ci

2

and

∑= −( )

∞qN

v f I Q e1

3 A28p

p ici

ci

ci

2

Substituting (Eqs. (A27) and A28) into Eq. (41) leads to

∑ ∑κη ϕ

ϕ ϕ ω− − = − +( )

∞ ∞

⎡⎣⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤⎦⎥⎥

⎛⎝⎜⎜

⎞⎠⎟⎟L p

Ki q

Nv fI D C e I Q e

31

3 A29pi f

p ici

ci

pi

p ici

ci

ci

2 2

where ω∂ = − it is used. Combining Eqs. (A29) with (A7), we obtain

σ = ( )∞K D e A30pi

f pi

and

= ( )∞q v I Q e A31pi

pi

pi

2

where

( ) ( )∑ ∑δωη

ϕϕ

δωηϕ

ϕ ϕ= − + − − − − −( )

− −−

−⎧⎨⎩⎡⎣⎢

⎤⎦⎥

⎫⎬⎭⎡⎣⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤⎦⎥⎥

Ki

fK

iD I I L I S S L I Q I I S I D C3

A32pi f

f pi

pi h

i

ci

pci

ci h

pi f

p

h

ici

ci1

01

11

( )∑ ∑ωηδ ϕ

ϕ ϕωη

ϕ δϕ= − − − + − −

( )

− − − −−⎛

⎝⎜⎜

⎞⎠⎟⎟

⎡⎣⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤⎦⎥⎥

i M

K

M i M

KfQ D C Q I S1

3 3 A33pi p

f

p

p ici

ci p

f p ici

ci

ci

pi

1 1 1 11

and where

δ κ=( )v

SL

.A34p

i

=N v vp pi

p, ϕ = v v/p p and = ⊗I I Ih 13 2 2 are employed to arrive at Eqs. (A30–A33).


= ˜ ( )∞e T e A35pi

pi

where

( ) ( )˜ = − − ( )− −KT I S I S L D A36p

ipi

f pi

pi1

01

Substituting (Eqs. (A21) and A36) into Eq. (11), we obtain the average pore/fluid strain,

= ˜ ( )ϕ ϕ ∞e T e A37

where

∑ ∑ϕ

ϕ ϕ˜ = ˜ + ˜( )

ϕ

⎛⎝⎜⎜

⎞⎠⎟⎟T T T

1

A38ici

ci

ipi

pi

The effective fluid elastic modulus tensor L̄f is determined by

σ = ¯ ( )ϕL e A39f f

Substituting (Eq. (A37) and A16) into Eq. (A39), we obtain

∑ ∑ϕ ϕ ϕ¯ = ˜ + ˜( )

−⎛⎝⎜⎜

⎞⎠⎟⎟KL C T T

A40f f

ici

ci

ipi

pi

1


Appendix B. : Eshelby tensor for spheroids

The derivation of Eshelby tensor in isotropic materials was initially performed by Eshelby (1957). Consider a spheroid,imbedded in an infinite isotropic solid having Possion's ratio ν and occupying region

{ }( ) ( ) ( )Ω = + + = ( )x y z x a y a z a, , ; / / / 1 . B112

12

32

The aspect ratio of the spheroid is defined by γ = aa

3

1, where a1, a3 are two semiaxes. If γ ≠ 1, then components of the

Eshelby tensor are (Kachanov et al., 2003)

( ) ( )( ) ( )γ

ν γ νν

γ= = −

− −+

−− +

− ( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥S S g

3

8 1 1

14 1

1 29

4 1 B21111 2222

2

02

02

( ) ( )ν

νγ ν

νγ

=−

− −−

+−

− − +− ( )

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥S g

11

21

11

2 12 2

31 B3

33330

0 20

0 2

( ) ( ) ( )ν γ ν

νγ

= =−

−−

+−

− − +− ( )

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥S S g

18 1

11

11

16 14 1 2

31 B4

1122 22110

20

0 2

( )( ) ( )γ

ν γ νν γ

γ= =

− −−

−− +

− ( )

⎡⎣⎢

⎤⎦⎥S S g

2 1 1

14 1

1 23

1 B51133 2233

2

02

00

2

2

( ) ( ) ( ) ( )ν

νγ ν

νγ

= =−

− − +−

+−

− −− ( )

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥S S g

12 1

1 21

11

4 12 1 2

31 B6

3311 33220

0 20

0 2

( )( ) ( ) ( )γν γ ν

νγ

= = −− −

+−

− +− ( )

⎡⎣⎢

⎤⎦⎥S S g

8 1 1

116 1

4 1 23

1 B71212 2121

2

02

00 2

( ) ( )νν γ

γ νν γ

γ= =

−− + +

−−

−− + +

− ( )

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟S S g

14 1

1 211

18 1

1 2 311 B8

1313 23230

0

2

20

0

2

2

where

( )( )

( )

( )

γ

γγ γ γ γ

γ

γγ γ γ γ

=−

− − <

−− − >

( )

⎧

⎨⎪⎪

⎩⎪⎪

g1

arccos 1 for 1

11 arccosh for 1

B9

2 1.52

2 1.52

The other non-null components are obtained using the symmetry relations for Eshelby tensor, = =S S Sijkl jikl ijlk.If γ = 1 (i.e., for a sphere), the Eshelby tensor has the simple form

( ) ( )νν

νν

= +−

+ −− ( )

S I I1

3 18 10

15 1.

B10h d0

0

0

0

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