poroelastic wave equation including the biot/squirt mechanism...

Wave Motion 35 (2002) 223–245

Poroelastic wave equation including the Biot/squirt mechanismand the solid/fluid coupling anisotropy

Dinghui Yanga,∗, Zhongjie Zhangba Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China

b Institute of Geophysics, Chinese Academy of Sciences, Beijing 100101, PR China

Received 5 October 2000; received in revised form 2 July 2001; accepted 20 July 2001

Abstract

There is relative motion and inertial coupling between solids and fluids during seismic and acoustic propagation in rockswith fluids. This inertial coupling is anisotropic because of the microvelocity anisotropy of the fluid relative to solid in ananisotropic medium. The Biot mechanism and the squirt-flow mechanism are the two most important mechanisms of solid/fluidinteraction in rocks. We extend the Biot/squirt (BISQ) theory to include the solid/fluid coupling anisotropy and develop ageneral poroelastic wave equation including both mechanisms simultaneously. The new model estimates velocity/frequencydispersion and attenuation of waves propagating in the 2D PTL(periodic thin layers)+ EDA (extensive dilatancy anisotropy)medium with fluids. The attenuation and dispersion of the two quasi-P-waves and the quasi-SV-wave, which are related tothe solid/fluid coupling density and the permeability tensors, are anisotropic. The anisotropy are simultaneously affectedby the anisotropies of the solid skeleton, the permeability, and the solid/fluid coupling effect of the formation. Numericalmodeling suggests that variations of attenuation of both the fast quasi-P-wave and the quasi-SV-wave strongly depend on thepermeability anisotropy. In the low-frequency range, the maximum attenuation is in the direction of the maximum permeabilityfor the fast quasi-P-wave and in the direction of the minimum permeability for the quasi-SV-wave. The attenuation behaviorsof the two waves in the high-frequency range, however, are opposite to those in the low-frequency range. This paper alsopresents numerically how the attenuation and velocity dispersion of both the fast quasi-P-wave and the quasi-SV-waveare influenced by the anisotropic solid/fluid coupling density. The model results demonstrate when the wave propagation isperpendicular to the direction of maximum solid/fluid coupling density, the wave motion exhibits maximum attenuation(Q−1)

and maximum velocity dispersion for the fast quasi-P-wave, and minimum attenuation(Q−1)and minimum velocity dispersionfor the quasi-SV-wave. These phenomena may be applied in extracting anisotropic permeability and further determining thepreferential directions of fluid flow in a reservoir containing fluid-filled cracks from attenuation and dispersion data derivedfrom sonic logs and crosswell seismics. © 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

The Biot mechanism and the squirt-flow mechanism are the two most important mechanisms of solid/fluidinteraction in fluid-filled porous media [1,2]. The Biot mechanism is described through macroscopic rock properties,whereas the traditional description of the squirt mechanism is based on microscopic rock properties (e.g., anindividual pore geometry [3]) resulting in limited application of the squirt-flow theory. Biot [4] had established a

∗ Corresponding author.E-mail address: [email protected] (D. Yang).

0165-2125/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0165-2125(01)00106-8

224 D. Yang, Z. Zhang / Wave Motion 35 (2002) 223–245

theory of propagation of elastic waves in a fluid-filled porous solid. The theory has been widely applied to interpretvarious wave phenomena. Yet, it is well known that the Biot theory cannot adequately explain the large velocitydispersion and the strong attenuation in many rocks. Many investigators have shown that the squirt-flow mechanismcan be responsible for the observed large attenuation and velocity dispersion [3,5–7]. Dvorkin and Nur [1], Dvorkinet al. [2,8] have also shown that the squirt-flow mechanism results in much higher and realistic attenuation insaturated rocks than that predicted by the Biot mechanism. Traditionally, the Biot mechanism and the squirt-flowmechanism of solid/fluid interaction are treated separately. In fact, the two mechanisms occur simultaneously duringseismic and acoustic wave propagation in rocks with fluids, they effect and couple each other, as well as influencethe process of seismic energy propagation and attenuation.

Dvorkin and Nur [1] offered a consistent model dealing simultaneously with these two mechanisms of solid/fluidinteraction as coupled processes. This model combines the two mechanisms by considering the fluid’s motionboth parallel (the Biot mechanism) and transverse (the squirt-flow mechanism) to the direction of a planar P-wavepropagation. The BISQ model relates the dynamic poroelastic behavior of a saturated rock to traditional poroelasticconstants such as porosity, permeability, fluid compressibility and viscosity, and the characteristic squirt-flow length.Parra [9,10] extends this BISQ theory to include the transversely isotropic poroelastic medium. Parra’s derivationsare based on both the constitutive relations in the frequency domain [assuming exp(−jωt) variation] given byKazi-Aoual et al. [11] and the isotropic description of the solid/fluid coupling effect which is measured by masscoupling densityρa originally introduced by Biot [4]. For anisotropic porous media, however, the solid/fluid couplingeffect is anisotropic. Actually, the microvelocity anisotropy of the fluid relative to solid in pores results in theanisotropy of the inertial coupling effect of solid/fluid interaction in the anisotropic rock. Therefore, we describe theanisotropic solid/fluid coupling effect by the additional coupling densityρai (‘ i’ denotes theith direction) to developa new poroelasticity model relate wave propagation to two mechanisms and the solid/fluid coupling anisotropy inour present work.

An extension of the elastic wave propagation theory of Biot [4] to include anisotropy, viscoelasticity, and soliddissipation was given by Biot [12]. More generalized Biot theory of acoustic propagation in porous media waspresented in the quoted paper [13] by introducing the concept of viscodynamic operational tensor. Based on theequations of linear elasticity, the linearized equations of fluid dynamics, and the homogenization theory, morecommon anisotropic poroelasticity equations have been suggested by Burridge and Keller [14]. In this paper, we

Fig. 1. Schematic illustration of the PTL+ EDA model, which is a porous PTL medium with vertical aligned microcracks.

D. Yang, Z. Zhang / Wave Motion 35 (2002) 223–245 225

develop a generalized poroelastic model to relate wave propagation with the solid/fluid coupling anisotropy and bothmechanisms. The generalized poroelastic wave equation is derived from the kinetic energy of the two-phase systemper unit volume, the equation of the fluid’s mass conservation in the porous rock, the Lagrange’s equation, and the1D BISQ theory [1]. These equations include the solid/fluid coupling anisotropy and both the Biot and the squirtmechanisms and govern the propagation of waves in general case of porous anisotropy. From this system of equations,we obtain a dispersion equation of fourth degree that takes into account the coupling anisotropy and the Biot/squirtmechanisms by applying the BISQ theory to the PTL+ EDA model (Fig. 1). Its solution under the 2D coordinatesystem is discussed. Numerical models are given to evaluate the sensitivity of attenuation and phase velocity to thesolid/fluid coupling anisotropy and the permeability anisotropy from the low-frequency to the high-frequency ranges.

2. Dynamic relations based on the solid/fluid coupling anisotropy

Considering a unit cube of the fluid-filled material, Biot [4] showed that the kinetic energyT of the systemper unit volume could be expressed as

2T = ρ11ui ui + 2ρ12ui Ui + ρ22UiUi , (1)

in which

ρ11 = (1 − φ)ρs − ρ12, ρ22 = φρf − ρ12, ρ12 = −ρa,

andui = ∂ui/∂t , Ui = ∂Ui/∂t ; ui andUi(i = 1,2,3) denote the displacements of the solid and the fluid in thei-direction,ρs represents the density of the solid (grain) material,ρf is the density of the fluid,ρa is the additionalcoupling density introduced initially by Biot [4],φ is the porosity of the rock.

Expression (1) is based on the assumption that the porous medium is statistically isotropic hence the directionsx,y, z are equivalent. For the porous anisotropic medium with fluids, we assume that the additional coupling densityρapresented in formula (1) is not equivalent in the directionsx, y, andz. This condition is strictly valid in a fluid-filledanisotropic material because of the microvelocity anisotropy of the fluid relative to the solid. In this case underassumption, we have that these additional mass coupling densities areρa1, ρa2, ρa3 in thex-, y-, andz-directions,respectively. Therefore, the effective density parameters in the expression (1), had been discussed in the quotedpaper [4], can be corrected as

ρ11i = (1 − φ)ρs − ρ12i , ρ22i = φρf − ρ12i , ρ12i = −ρai , i = 1,2,3.

Based on the assumption of the anisotropy of the solid/fluid coupling effect, the kinetic energyT of per unittwo-phase material may be expressed as

2T = ρ11i ui ui + 2ρ12i ui Ui + ρ22i Ui Ui . (2)

The dissipation functionD, representing the power dissipated per unit volume of the two-phase material due to theviscosity of the fluid, assumed to be Newtonian, is

D = 1

2ηφ2

3∑i,j=1

rij(Ui − ui )(Uj − uj ), (3)

whereη is the viscosity of the fluid andrij the components of a flow resistive tensorr introduced by Biot [12].If we denote byfi the total force acting on the solid per unit volume in thei-direction and byFi the total force

on the fluid per unit volume, we derive from Lagrange’s equations

fi = ∂

∂t

(∂T

∂ui

)+ ∂D

∂ui= ∂2

∂t2(ρ11iui + ρ12iUi) − ηφ2

3∑j=1

rij(Uj − uj ), (4)


Fi = ∂

∂t

(∂T

∂Ui

)+ ∂D

∂Ui

= ∂2

∂t2(ρ12iui + ρ22iUi) + ηφ2

3∑j=1

rij(Uj − uj ). (5)

Therefore, the total force acting on the fluid-filled material per unit volume in thei-direction is

fi + Fi = ∂2

∂t2(ρ1ui + ρ2Ui), (6)

where the mass of solid per unit volume of the fluid-filled material isρ1 = (1 − φ)ρs, the mass of fluid per unitvolume of the fluid-filled material isρ2 = φρf .

In terms of stresses, the force components are expressed as stress gradients without considering extraneous forcessuch as the gravity forces, i.e.,

fi + Fi =3∑

j=1

∂τij

∂xj, (7)

Fi = −∂(φP )

∂xi, (8)

whereτij are the elements of the total stress tensor of the bulk material,P the fluid pressure.By substituting Eqs. (7) and (8) into Eqs. (6) and (5), we obtain the dynamic equations based on the solid/fluid

coupling anisotropy as follows:

3∑j=1

∂τij

∂xj= ∂2

∂t2(ρ1ui + ρ2Ui), (9)

− ∂

∂xi(φP ) = ∂2


3∑j=1

rij∂

∂t(Uj − uj ). (10)

3. Expression of fluid’s pressure from the Biot theory

For the isotropic case, the relation between the porosity differential dφ, the differentials of the skeleton’s defor-mation and of the fluid’s pressure has been presented in Ref. [15,16]. For anisotropic porous media, the relationbecomes

dφ = α : de + dP

Q, (11)

where dP is the differential of the fluid’s pressure, 1/Q the coefficient introduced initially by Biot [15] (given innext section),α the poroelastic coefficient tensor of the effective stress that can be expressed as (Appendix A)

α =

α11 α12 α13

α12 α22 α23

α13 α23 α33

, (12)

and de is a strain differential tensor defined as

de =

dexx dexy dexz

dexy deyy deyz

dexz deyz dezz

with eij = 1

2

(∂ui

∂xj+ ∂uj

∂xi

), i, j = 1,2,3.


In fact, this extension is based on the reason that both the differential of the skeleton’s deformation and the poroelasticcoefficientα of the effective stress are second-rank tensors.

For the 1D case, the relation (11) becomes as

dφ = α11 d

(duxdx

)+ 1

2α12 d

(duydx

)+ 1

2α13 d

(duzdx

)+ dP

Q, (13)

and the equation of the fluid’s mass conservation in the fluid-filled rock moving with deformation velocityux andthe fluid’s flow velocityUx

∂(ρfφ)

∂t+ ∂[ρfφ(Ux − ux)]

∂x= 0,

gives, after linearization and using the relation (13):

φ

ρf

∂ρf

∂t+ (α11 − φ)

∂2ux

∂x∂t+ 1

2α12

∂2uy

∂x∂t+ 1

2α13

∂2uz

∂x∂t+ ∂P

∂t

1

Q+ φ

∂2ux

∂x∂t= 0. (14)

We describe the fluid’s compressibility by a linear relation

dρf = dP

c20

= ρf dP

Kf, (15)

wherec0 is fluid acoustic velocity, andKf the fluid bulk modulus.Substituting Eq. (15) into Eq. (14), we relate pressureP to displacement componentsux, uy, uz andUx as:

∂P

∂t= −F1

(∂2Ux

∂x∂t+ α11 − φ

φ

∂2ux

∂x∂t+ α12

2φ

∂2uy

∂x∂t+ α13

2φ

∂2uz

∂x∂t

), (16)

whereF1 is stated in next section.

4. Expression of fluid’s pressure from the BISQ model

Considering 1D case and following the BISQ theory [1], we assume here that the solid skeleton of a rock deformsonly in the direction of P-wave propagation (x-direction). The fluid, however, can flow not only parallel (Biot’sflow), but also perpendicular (squirt-flow) to this direction (Fig. 2). The axis of the representative cylindrical modelis in the direction of wave propagation. Meanwhile, pore pressure on the external surface of this cylinder atr = R

(Fig. 2) does not change in time (constant pressure), so that pore fluid is squeezed from pore throats or thin cracksinto surrounding large pores or adjacent cracks. Dvorkin and Nur [1] and Dvorkin et al. [2] have presented that theradius of this cylinder is a characteristic squirt-flow length of the order of the average pore size. In addition, thephysical meaning of the characteristic squirt-flow length is the average length that produces the squirt-flow effectidentical to the cumulative effect of squirt-flow in pores of various shapes and sizes, so an individual pore geometryis not required. The microscale parameter that is assumed to be a fundamental rock property that does not dependon frequency and fluid characteristics, and thus can be determined experimentally [1,2]. To relate the characteristicsquirt-flow parameterR to pore-pressure, we have to examine the 2D axisymmetrical fluid flow.

For 1D case, using the 2D axisymmetrical equation of the fluid’s mass conservation and dynamic equation (10),we derive the average fluid pressure, based on the BISQ model as (Appendix B),

Psq = −F1

[1 − 2J1(λ1R1)

λ1R1J0(λ1R1)

](∂Ux

∂x+ α11 − φ

φ

∂ux

∂x+ α12

2φ

∂uy

∂x+ α13

2φ

∂uz

∂x

), (17)

whereJ0 andJ1 are the Bessel functions of zero-order and first-order, respectively; the parameterR1 represents theaverage characteristic squirt-flow length of fluids squirting in theyz-plane.


Fig. 2. The BISQ model [1]. A cylindrical representative volume of a rock,R is the squirt-flow length.

The only difference between the derivatives∂Psq/∂t based on the BISQ, derived easily from Eq. (17), and∂P/∂t

based on the Biot model is in the proportionality coefficient that relates the derivative of pressure to the derivativesof the fluid’s and the solid’s displacements. Let the proportionality coefficient

S1 = 1 − 2J1(λ1R1)

λ1R1J0(λ1R1),

be named a characteristic squirt-flow coefficient in theyz-plane. The coefficient relates to the anisotropic couplingdensity, the anisotropic permeability, and the poroelastic coefficient of the effective stress. Obviously, the squirt-flowcoefficients in different planes are different. In other words, the squirt-flow coefficients have to be a tensor. TheBiot-flow coefficientF1 is also not identical in thex-, y-, z-directions because of the directional effective stress.Thus, we define the Biot-flow coefficient tensor and the characteristic squirt-flow coefficient tensor as

F =

F1 0 0

0 F2 0

0 0 F3

, S =

S1 0 0

0 S2 0

0 0 S3

, (18)

where

Fj =(

1

Kf+ 1

φQj

)−1

, (18a)

1

Qj

=∑3

i=1αij − φ

Ks

, (18b)


Sj = 1 − 2J1(λjRj )

λjRjJ0(λjRj ), (18c)

λ2j = ρfω

2

Fj

(φ + (ρaj /ρf )

φ+ i

ωj

ω

), (18d)

ρaj = 1

2

3∑k=1k �=j

ρak, (18e)

ωj

ω= ηφrj

ρfω, (18f)

rk = 1

2

3∑i,j=1i �=k

βjkrij, (18g)

βjk ={βk, k = j,

1, k �= j.(18h)

The tensorS represents the effect of squirt-flow on the global Biot’s flow. Its derivation is assumed a representativecylindrical volume of rock geometries inx-, y-, andz-directions. For example, a sideways flow inyz-plane can berepresented by a cylinder with its axis parallel to the wave motion in thex-direction. The radius of this cylinder isequal to the average squirt-flow lengthR1, which is associated with the average permeability in theyz-plane. In asimilar manner, a flow in thexz-plane can be represented by a cylinder with its axis parallel to the wave motion inthey-direction. In this case, the radius of the cylinder is equal to the average squirt-flow lengthR2, associated withthe average permeability in thexz-plane. Similarly, the flow inxy-plane can be described by a cylinder, which itsaxis is parallel to the wave propagation inz-direction and its radius is equal to the average squirt-flow lengthR3. Theparameterβk in (18h) is the ratio of the fluid relative displacement in thekth direction to the average displacementof the fluid in the plane perpendicular to thekth direction. In particular, the ratio parameterβk(k = 1,2,3) isequal to zero for some special cases such as porous PTL, EDA, and PTL+ EDA media. The inverse Biot-flowcoefficient,F−1

j , represents the effect of the Biot-flow on the total compressibility of the solid/fluid coupling systemin thej -direction. The dynamic effect of the Biot/squirt-flow to the solid/fluid coupling material in thej -directionis represented by the expressionFjSj .

We extend Eq. (17) to including 3D wave motion, the total fluid pressure of the BISQ model in an anisotropicporous medium can be written as

P = −∇ · (F · S · U) −(F · S · α − φI

φ

): e, (19)

wheree = (eij) is the strain tensor of the porous medium,U = (Ux, Uy, Uz)T the particle displacement of the fluid,

I the second-rank unit tensor.

5. Poroelastic wave equations

We can express the stress–strain relation in terms of the effective stress or total stress as

τ = Ae − αP, (20)

whereτ = (τij) is the total stress of the bulk material,A the solid-frame stiffness tensor containing 21 independentdrained elastic coefficients for general anisotropic media.


Substituting the values ofτij into Eq. (9), we obtain

3∑j=1

∂

∂xj

3∑p,q=1

Aijpq∂uq

∂xp

−

3∑j=1

∂

∂xj(αijP) = ∂2

∂t2(ρ1ui + ρ2Ui), (21)

wherei = 1,2,3.Combination of Eqs. (10), (19) and (21) forms the poroelastic wave equations in fluid-filled anisotropic porous

media, which are based on the inertial coupling anisotropy and deal simultaneously with two mechanisms. Thesystem of wave equations can be simplified for some special cases such as the porous isotropic, PTL, EDA, andPTL+EDA media. For instance, we can write the PTL+EDA poroelastic wave equations by simplifying Eqs. (10),(19) and (21)(

A11∂2

∂x2+ A66

∂2

∂y2+ A55

∂2

∂z2

)ux + (A12 + A66)

∂2uy

∂x∂y+ (A13 + A55)

∂2uz

∂x∂z− α11

∂P

∂x

= ∂2

∂t2(ρ1ux + ρ2Ux), (22a)

(A12 + A66)∂2ux

∂x∂y+(A66

∂2

∂x2+ A22

∂2

∂y2+ A44

∂2

∂z2

)uy + (A23 + A44)

∂2uz

∂y∂z− α22

∂P

∂y

= ∂2

∂t2(ρ1uy + ρ2Uy), (22b)

(A55 + A13)∂2ux

∂x∂z+ (A23 + A44)

∂2uy

∂y∂z+(A55

∂2

∂x2+ A44

∂2

∂y2+ A33

∂2

∂z2

)uz − α33

∂P

∂z

= ∂2

∂t2(ρ1uz + ρ2Uz), (22c)

− ∂

∂xi(φP ) = ∂2


kii

∂

∂t(Ui − ui), i = 1,2,3, (22d)

P = −3∑

i=1

FiSi

(∂Ui

∂xi+ αii − φ

φ

∂ui

∂xi

). (22e)

6. Wave dispersion and attenuation

In this section, we investigate the properties of wave dispersion and attenuation in the PTL+ EDA poroelasticmedium with fluids.

6.1. Wave velocity and inverse quality factor

Applying the Fourier transformation with respect to space and time, we derive from Eqs. (22a)–(22e)D11k

2 − a11 D12k2 D13k

2 iD14k

D12k2 D22k

2 − a22 D23k2 iD24k

D13k2 D23k

2 D33k2 − a33 iD34k

iD41k iD42k iD43k −D44k2 + 1

Ax

Ay

Az

P0

= 0, (23)


where

aii = ω2(ρ − ρ2θi), i = 1,2,3, D11 = A11k2x + A66k

2y + A55k

2z , D12 = (A12 + A66)kxky,

D13 = (A13 + A55)kxkz, D22 = A66k2x + A22k

2y + A44k

2z , D23 = (A23 + A44)kykz,

D33 = A55k2x + A44k

2y + A33k

2z , Dj4 = (αjj − φθj )kj , D4j = FjSjφ

−1(αjj − φθj )kj , j = 1,2,3,

D44 = F1S1θ1k2x + F2S2θ2k

2y + F3S3θ3k

2z

ω2ρf, θj =

((ρaj /ρf ) + φ

φ+ i

ηφ

ωρf kjj

)−1

, j = 1,2,3,

wherek is the wave number,Ai andki(i = 1,2,3) the wave amplitude components and the wave number componentsof unit wave number in thei-direction,P0 the static pressure of the fluid,kjj denotes the permeability in thej -direction.

In terms of the algebraic theory, we derive the following wave number equation from Eq. (23)

b0k8 + b1k

6 + b2k4 + b3k

2 + b4 = 0, (24)

where

b0 =D44(−D11D22D33 + D11D223 + D2

12D33 − 2D12D23D13 + D22D213),

b1 = (D22D33 − D223)(a11D44 + D11) + D11D44(D22a33 + D33a22)

−D212D44a33 − D12(D12D33 − D23D13) − D2

13D44a22 + D13(D12D23 − D22D13)

+ (D14D12 − D24D11)(D43D23 − D42D33) + D42D34(D12D13 − D11D23)

+D43D22(D24D11 − D14D13) + D42D13(D14D23 − D24D13) + D43D12(D24D13 − D34D12)

+D41D12(D34D23 − D24D33) + D41D14(D22D33 − D223) + D41D13(D24D23 − D34D22),

b2 = −(D11 + a11D44)(D22a33 + a22D33) + D212a33 + D2

13a22 − D43D24(D11a22 + a11D22)

+D42D34D23a11 − D11D44a22a33 + a11(D223 − D22D33) + D43D14D13a22 + D42D14D12a33

+D41D34D13a22 − D41D14(D22a33 + a22D33) + D41D24D12a33,

b3 = a11a22(D33 + D43D24) + a11a33(D22 + D42D24) + a22a33(D11 + a11D44 + D41D14),

b4 = −a11a22a33.

The solutions of Eq. (24) gives four independent complex wave numbers associated with the quasi-compressionalwaves (including fast and slow quasi-P-waves), the quasi-SH-wave, and the quasi-SV-wave. Generally speaking,it is difficult to analysis the propagation characteristics of these waves in a porous anisotropic rock through theanalytical solutions of Eqs. (23) or Eq. (24). Therefore, we consider the plane wave propagation in thexz-plane.In this case, we can assume the unit wave number componentskx = sin(θ), kz = cos(θ), andky = 0 (θ is theplane-wave incident angle withz-axis). Substituting these values of the wave number components into Eq. (24), weobserve that one of the solutions of Eq. (24) is

k2SH = a22

D22= ω2(ρ − ρ2θ2)

A66 sin2θ + A44 cos2θ, (25)

therefore, the phase velocity and attenuation of the SH-wave are

vSH =√A66 sin2θ + A44 cos2θ

Real(√ρ − ρ2θ2)

, (26a)

Q−1SH = 2 Imag(

√ρ − ρ2θ2)

Real(√ρ − ρ2θ2)

. (26b)


These expressions (26a) and (26b) indicate that the SH-wave is anisotropic because its phase velocity depends onwave propagating direction, its attenuation is independent of the propagation direction but depends on the anisotropicsolid/fluid coupling effect and the anisotropic permeability only through the Biot mechanism. The formulae alsoshow that the quasi-SH-wave is not affected by the squirt-flow mechanism because it does not depend of thesquirt-flow coefficientsSj .

By substituting the value ofkSH into Eq. (23), we have the plane-wave amplitude�A = (0, Ay,0). It is aquasi-shear-wave because of the wave polarizing direction perpendicular to the direction of wave propagation in thexz-plane. For the three other waves (fast and slow quasi-P-waves and the quasi-SV-wave), wave number solutionsare determined from the following equation:

c0k6 + c1k

4 + c2k2 + c3 = 0, (27)

where

c0 = −D11D33D44 + D213D44,

c1 =D41D14D33 + D43D14D11 − D13D43D14 − D13D41D34 + D11(D33 + a33D44) + a11D33D44 − D213,

c2 = −D14D41a33 − a11D43D14 − a33D11 − a11(D33 + a33D44), c3 = a11a33.

Fig. 3. Comparison between the original BISQ model [1] and our present model for permeability 1.25, 2.5, and 5 md. (a) Compressional velocityvs. log angular frequency and (b) inverse attenuation quality factorQ−1 vs. log angular frequency as predicted by the original BISQ theory(dashed lines) and our BISQ model (solid lines).


Three coupled waves correspond to three wave number solutionski (i = 1,2,3) of Eq. (27), respectively. Thephase velocities and attenuation of these waves can be obtained directly from three complex wave number roots(ki , i = 1,2,3) as

vi = ω

Real(ki), (28a)

Q−1i = 2 Imag(ki)

Real(ki). (28b)

The complex solutions of the dispersion equation (27) give the wave numbers to determine the phase velocities (28a)and inverse attenuation quality factors (28b) of three waves associated with the Biot-flow tensor, the squirt-flowtensor, porous matrix tensor, and the anisotropic solid/fluid coupling density. The elements of the Biot-flow and thesquirt-flow tensors and the anisotropic coupling density are coupled through the coefficients of the cubic dispersionequation, namely,c0, c1, c2, andc3. This suggests that the phase velocity and attenuation of the fast and slowquasi-P-waves and the quasi-SV-wave will be influenced by both the Biot/squirt mechanisms and the solid/fluidcoupling anisotropy in the PTL+EDA porous rock with fluids. At the same time, The phase velocity and attenuationof three waves are anisotropic because they depend on the wave propagation direction. The anisotropy is thecombination of the solid skeleton anisotropy, the permeability anisotropy, and the solid/fluid coupling anisotropy.

Fig. 4. Comparison between the Parra’s BISQ model (1997) and our present model for frequency 500, 1500, 2500, and 4500 Hz with permeabilityk11 = k22 = 1000 md andk33 = 100 md. (a) Quasi-P-wave attenuation and (b) quasi-SV-wave attenuation as predicted by Parra’s BISQ model(dashed lines) and our present BISQ model (solid lines).


6.2. Numerical examples

To check our present model, the first step in the numerical calculations is to check the solution for the attenuationand dispersion equations given by Eq. (28a) and (28b), when the solid/fluid coupling anisotropy and two mechanismsare included simultaneously. In this case, we select the same material property and fluid property parameters usedin Dvorkin and Nur [1]. In other words, the porosityφ is 15%. The elastic characteristics of the skeleton are:K = 16 GPa, and Poisson’s ratioν = 0.15. The density of the solid isρs = 2650 kg/m3, its bulk modulusKs = 38 GPa. The fluid density isρf = 1000 kg/m3, and its viscosityη = 1 cps. The additional coupling densityis ρax = ρay = ρaz = 420 kg/m3. The characteristic squirt-flow length is chosenR1 = R2 = R3 = 1 mm. Thevelocity–frequency and attenuation curves of the P-wave for these three cases with permeability equal to 1.25, 2.5,5 md, are given in Fig. 3 for comparison, in where the solid lines and the dashed lines show the our BISQ-predictedand the original BISQ-predicted results, respectively. Fig. 3 displays that the result obtained using our present modelis the same as that obtained using the original BISQ-model [1], but slightly greater than that given by the BISQmodel of Dvorkin and Nur [1] at higher frequencies. In Fig. 4, the different curves, which the solid lines and thedashed lines show the our BISQ-predicted and the Parra’s BISQ-predicted results, respectively, correspond to valuesof the frequency that equal to 500, 1500, 2500, and 4500 Hz for the permeability parametersk11 = k22 = 1000 mdandk33 = 100 md given similarly in Parra’s study [9]. Comparing the two results, we observe from Fig. 4 thatboth results for the quasi-SV-wave are identical; the attenuation values of the quasi-P-wave predicted by the present

Fig. 5. The effect of frequency. (a) Phase velocity of the fast quasi-P-wave, (b) attenuation(Q−1) of the fast quasi-P-wave, (c) phase velocity ofthe slow quasi-P-wave, (d) attenuation(Q−1) of the slow quasi-P-wave for permeabilityk11 = 1,2,4, and 8 md; permeabilityk22 = k33 = 1 md.Squirt-flow lengthR1 = 2 mm,R3 = 2.5 mm.


model are slightly lower than those predicted by the Parra’s model in the directions near or parallel to the maximumpermeability (x-direction). The difference observed can be explained by using the average value of the anisotropicpermeabilities (k22 andk33) in the plane perpendicular to thex-direction when we calculate the ratioω1/omega viathe formula (18f). In addition, Fig. 4(a) displays that the quasi-P-wave attenuation curves obtained by our BISQmodel is consistent with those obtained using the Parra’s BISQ model in the direction of the minimum permeability.

Fig. 6. The effect of anisotropic coupling density and frequency on the fast quasi-P-wave. (a) Attenuation(Q−1); (b) phase velocity for couplingdensityρay = 400 kg/m3 andρaz = 200 kg/m3. Characteristic squirt-flow lengthR1 = 1 mm,R3 = 1.5 mm.


This consistency is clear because the quasi-P-wave (fast) attenuation in the vertical direction is mainly controlledby the transverse average permeability associated with the transverse squirt-flow mechanism, and in this case theaverage permeability in thexy-plane is just equal to thex-directional permeabilityk11 resulting in the same resultof ω3/ω presented in both our present model and the Parra’s BISQ model [9].

Next, we choose again four models to investigate the effects of frequency, solid/fluid coupling density tensor, andpermeability tensor on wave velocities and attenuation.

Model 1. We examine the effects of frequency on the attenuation and phase velocity of the wave propagation in thex-direction. In this case, we select the additional coupling densities such thatρax = ρay = ρaz = 420 kg/m3. Theporosity of the porous rock is 35%. The characteristic squirt-flow lengths are chosenR1 = 2 mm andR3 = 2.5 mm.The fluid characteristic parameters areρf = 1000 kg/m3, viscosity 4 cps, and fluid acoustic velocityVf = 1500 m/s.The solid-phase property parameters for the poroelastic PTL+ EDA medium are given in Table 1. Fig. 5 showsthe attenuation and dispersion curves for permeabilityk11 = 1,2,4, and 8 md withk22 = k33 = 1 md for thedifferent curves. Fig. 5(a) shows that the fast quasi-compressional wave velocity changes from its low-frequencylimit to the high-frequency limit as frequency increases. The transition zone shifts towards high frequencies aspermeability increases. This trend can also be observed onQ−1 versus frequency curves given in Fig. 5(b), wherethe attenuation peak shifts towards high frequencies with increasing the permeability. The physical significance of

Fig. 7. The effect of angle of propagation and frequency on attenuation(Q−1) of: (a) fast quasi-P-wave; (b) quasi-SV-wave. Anisotropiccoupling densityρax = 800 kg/m3, ρay = 100 kg/m3, andρaz = 100 kg/m3. Permeabilityk11 = k22 = k33 = 7000 md. Squirt-flow lengthR1 = R2 = R3 = 1 mm.


Table 1Solid-phase propertiesa

Ks (GPa) ρs (kg/m3) A11 A12 A13 A22 A23 A33 A44 A55 A66

38 2650 22.75 6.5 4.55 19.5 8.45 21.45 9.75 8.45 6.5

a The nine constantsAij (GPa) are independent drained elastic coefficients in the PTL+ EDA porous medium.

the result is that the squirting motion of the fluid becomes unrelaxed at lower frequencies for lower permeabilities.In the low-frequency range, Fig. 5(d) shows the large attenuation property of the slow quasi-P-wave (the values ofQ−1 are very large), and its phase velocity showed in Fig. 5(c) is not sensitive to frequency. But Fig. 5(c) shows thatthe slow quasi-P-wave velocity rapidly increases with increasing frequency and permeability in the high-frequencyrange.

Model 2. We investigate the effects of the solid/fluid coupling density tensor and frequency on attenuation anddispersion of the compressional wave. In this example, the fluid viscosity is 1 cps, permeability equal to 4000,

Fig. 8. The effect of angle of propagation and frequency on phase velocities of: (a) fast quasi-P-wave; (b) quasi-SV-wave. Anisotropiccoupling densityρax = 800 kg/m3, ρay = 100 kg/m3, andρaz = 100 kg/m3. Permeabilityk11 = k22 = k33 = 7000 md. Squirt-flowlengthR1 = R2 = R3 = 1 mm.


2000, and 400 md in thex-, y-, z-directions, respectively. The characteristic squirt-flow lengths areR1 = 1 mmandR3 = 1.5 mm. The sold/fluid mass coupling densities areρay = 400 kg/m3 andρax = 200 kg/m3. Theother parameters are the same as those of model 1 except the solid/fluid coupling densityρax . The phase velocityand attenuation surfaces of the quasi-P-wave are given in Fig. 6. Fig. 6(a) shows that as the coupling densityρaxin the x-direction increases, the attenuation values decrease and the attenuation peak slightly shifts towards lowfrequencies. The trend can also be observed on the phase velocity surface (Fig. 6(b)). The physical interpretationof the result is that increasing the solid/fluid coupling densityρax decreases the relative displacement between thefluid and solid phases implying that the formation is weak biphase. As a result, the high solid/fluid coupling limitsof the fast quasi-P-wave phase velocities and attenuation decrease.

Model 3. The goal of this example is to investigate further the effects of the solid/fluid coupling anisotropy onattenuation and dispersion of the quasi-P-wave and the quasi-SV-wave. Therefore, we choose the solid/fluid couplingdensities 800, 100, and 100 kg/m3 in x-, y-, andz-directions; the permeabilities arek11 = k22 = k33 = 7000 md;the porosity of the porous rock is 25%. The angle of propagation with respect to thez-direction varies from 0◦to 90◦. The rest parameters are chosen as those of the first example stated above which checks our present BISQ

Fig. 9. The effect of angle of propagation and frequency on attenuation(Q−1) of: (a) fast quasi-P-wave; (b) quasi-SV-wave. Anisotropic couplingdensityρax = 450 kg/m3, ρay = 400 kg/m3, andρaz = 200 kg/m3. Squirt-flow lengthR1 = 0.5 mm,R3 = 0.8 mm.


theory, and are the same as those parameters used in Dvorkin and Nur [1]. The phase velocity and attenuation of thequasi-P-wave and the quasi-SV-wave are shown in Figs. 7 and 8. In Fig. 7(a), we observe that attenuation(Q−1)

of the quasi-P-wave is maximum at all frequencies when the wave propagation is perpendicular to the direction(x-direction) of maximum solid/fluid coupling density. This suggests that the attenuation depends on the directionof the propagation. Alternatively, the quasi-SV-wave attenuation is maximum at all frequencies in the direction(x-direction) of the maximum solid/fluid coupling density (see Fig. 7(b)). This example suggests that variations ofattenuation of both the quasi-P-wave and the quasi-SV-wave depend on the anisotropy of the solid/fluid couplingeffect. In a similar manner, when the wave propagates perpendicular to the direction of maximum solid/fluidcoupling density, the wave motion exhibits maximum velocity dispersion for the quasi-P-wave (Fig. 8(a)) and

Fig. 10. The effect of angle of propagation and frequency on phase velocities of: (a) fast quasi-P-wave; (b) quasi-SV-wave. Anisotropic couplingdensityρax = 450 kg/m3, ρay = 400 kg/m3, andρaz = 200 kg/m3. Squirt-flow lengthR1 = 0.5 mm,R3 = 0.8 mm.


minimum velocity dispersion for the quasi-SV-wave (Fig. 8(b)). When the wave propagates parallel to the directionof maximum solid/fluid coupling density, the velocity dispersion is minimum for the quasi-P-wave and maximumfor the quasi-SV-wave (see Fig. 8).

Model 4. In the final example, we analyze the effects of the permeability anisotropy and wave propagationdirections on attenuation and dispersion of the quasi-P-wave and the quasi-SV-wave. The permeabilities are chosenk11 = 2000 md,k22 = 1000 md, andk33 = 200 md. The pore-fluid viscosity is 1 cps. We select the solid/fluidcoupling densities 450, 400, and 200 kg/m3 in x-, y-, andz-directions; the characteristic squirt-flow lengthsR1 =0.5 mm andR3 = 0.8 mm. The angle of propagation with respect to the vertical varies from 0◦ to 90◦. Therest parameters are chosen as those of model 1. Figs. 9 and 10 show the anisotropic characteristics of attenu-ation and phase velocities of the fast quasi-P- and the quasi-SV-waves propagating in the porous PTL+ EDAmedium with fluids. In Fig. 9, the surfaces show that the wave attenuation is maximum for the fast quasi-P-waveand is minimum for the quasi-SV-wave in the low-frequency range for the direction of propagation perpendicularto the minimum permeability. The attenuation characteristics in the high-frequency range, however, are oppo-site to those in the low-frequency range. This suggests that the attenuation depends on the direction of the wavepropagation and frequency. In other words, in the low-frequency range, the attenuation of the fast quasi-P-waveis maximum in the direction of maximum permeability and the quasi-SV-wave attenuation is maximum in thedirection of the minimum permeability. But in the high-frequency range, the maximum attenuation of the fastquasi-P-wave is in the direction of minimum permeability and the quasi-SV-wave maximum attenuation is inthe direction of the maximum permeability. In the intermediate frequency range, the attenuation behaviors ofthe fast quasi-P-wave and the quasi-SV-wave may be more complex. The example indicates that the anisotropicattenuation of the fast quasi-P-wave and the quasi-SV-wave depend on the permeability anisotropy of the for-mation. As the interpretation given by the authors [1], the results mean physically that the squirting motion ofthe fluid becomes more easily relaxed at lower frequencies and unrelaxed at higher frequencies for the higherpermeability.

7. Discussion and conclusions

The analysis of the solid/fluid interaction and coupling effect in anisotropic poroelastic media with fluids hasprovided a system of wave propagation equations that include the solid/fluid coupling anisotropy and both the Biotand the squirt mechanisms. The analysis of the plane wave solutions is the first step toward using the new poroelas-ticity equations to simulate synthetic seismograms and fluid’s pressure fields. The solutions of the poroelastic waveequations in the PTL+ EDA medium with a viscous fluid were used to demonstrate how the attenuation anisotropyand dispersion of the quasi-SV-wave and both quasi-P-waves (fast and slow) could be related to the permeabilityanisotropy and the anisotropy of the solid/fluid coupling effect. The anisotropic attenuation and dispersion depend onthe anisotropic mass coupling density and the anisotropic permeability. Two quasi-P-waves and the quasi-SV-waveare simultaneously affected by both the Biot and the squirt-flow mechanisms. The quasi-SH-wave is influenced onlyby the Biot mechanism in the 2D PTL+EDA medium with fluids. Model results show that in the low-frequency rangethe quasi-P-wave (fast) attenuation is maximum and the quasi-SV-wave attenuation is minimum in the direction ofthe maximum permeability. Yet in the high-frequency range, the attenuation characteristics are opposite to those inthe low-frequency range. Specially, in order to investigate further the effects of the solid/fluid coupling anisotropyon attenuation and dispersion of the shear SV-wave and the fast compressional wave, we select the isotropic rockmodel given in the original BISQ theory [1]. Modeling indicates that the anisotropy of the solid/fluid couplingeffect can cause the anisotropy of the attenuation and phase velocity of both the fast P-wave and the SV-wave. Inother words, the maximum attenuation and strong velocity dispersion depends on the minimum solid/fluid couplingeffect for the fast quasi-P-wave and on the maximum solid/fluid coupling effect for the quasi-SV-wave. The resultsmay be applied in extracting the anisotropic permeability, determining the fracture orientation, and estimating theproduction potential of the oil reservoir. Meanwhile, these examples show that the anisotropy of attenuation and


phase velocity of the poroelastic waves are caused by the solid skeleton anisotropy, permeability anisotropy, andthe solid/fluid coupling anisotropy.

In the previous studies, Dvorkin and coworkers [1,2] and Parra [9,10] have confirmed the BISQ theory bycomparing the BISQ-predicted phase velocity and the attenuation values with the experimental data from Klimentosand McCann [19] and crosswell seismic data. Comparing the present poroelastic wave equation with that of Parra[9] in the space-frequency domain, we can find that both results are essentially identical when assumingρax =ρay = ρaz in a 2D PTL porous medium (see Fig. 4). It shows that our result is an extension of Parra’s workand the extension is necessary because the solid/fluid coupling effect is practically anisotropic in an anisotropicporous medium. For the 1D isotropic case, our results are identical with those of Dvorkin and Nur [1] exceptingthe slightly difference at higher frequencies (see Fig. 3). In other words, the results in Refs. [1] and [9] are somespecial cases of our present results. Therefore, our present BISQ model is a generalized BISQ model, which thesolid/fluid coupling anisotropy and both mechanisms are simultaneously included in a unified poroelastic waveequation. The new poroelastic wave equation is an improved attempt to quantitatively and consistently relate wavepropagation in anisotropic poroelastic rocks with a viscous fluid to the anisotropic solid/fluid coupling effect andthe two important mechanisms of solid/fluid interaction. We believe that an important application of the presentporoelastic wave equation is in simulating wavefields and evaluating the preferential directions of fluid flow informations from attenuation and dispersion data. These further works are presently under development and will bereported in the near future.

We have noticed from Fig. 5(d) that the values of inverse attenuation quality factorQ−1 of the slow quasi-P-waveare very large in the low-frequency range. It represents the large attenuation characteristic of the slow compressionalwave. Thus, the slow P-wave will rapidly die out with increasing distance from the wave source. This is the mainreason why the slow P-wave is hardly detected in practical seismic observations. Actually, under this case, thewave equation becomes a diffusion equation. Therefore, the slow P-wave propagation shows the diffusivity or heatconductivity and its attenuation is very high (see Ref. [4]). As the interpretation given by the authors [17], it meansthat the second terms on the right-hand side of Eq. (10) are dominating and the solution of the poroelastic waveequation degenerates to that of a diffusion equation.

Acknowledgements

This work was financially supported by the National Natural Sciences Foundation of China (Grant Nos. 49704049and 4985108) and the Foundation of Tsinghua University (Grant No. JC1999044). We appreciate the suggestionsand revisions of the reviewers from Wave Motion. Special thanks to Dr. Wenyue Xu for helpful improvements ofthe paper.

Appendix A. Poroelastic coefficient tensor of the effective stress

For the most general case of porous anisotropy, the stress-strain relations have been expressed in the followingmatrix form [18]

σxx

σyy

σzz

σyz

σzx

σxy

−φP

=

C11 C12 C13 C14 C15 C16 C17

C22 C23 C24 C25 C26 C27

C33 C34 C35 C36 C37

C44 C45 C46 C47

symmetry C55 C56 C57

C66 C67

C77

exx

eyy

ezz

eyz

eyx

exy

ε

,


whereσij are the stress components acting on the solid part of the rock and−φP is the stress acting on the fluidpart of the rock.Cij are elastic coefficients.

Eliminating theε from the stress–strain relations stated above, we obtain:

σ = Ae − αP , (A.1)

whereσ is the stress acting on the solid part of the rock ande is the solid-phase strain;A = (Aij)6×6, Aij =Cij −Ci7Cj7/C77(i, j = 1,2, . . . ,6) are drained elastic coefficients; and the coefficient vectorα = (α1, α2, α3, α4,

α5, α6)T, in which αi = φCi7/C77, i = 1,2, . . . ,6.

On the other hand, the total stressesτ acting on the solid–fluid system can be separated into two parts: stressesσ acting on the solid part of the rock and stresses−φP acting on the fluid part (see Ref. [12,18]), i.e.,

τ = σ − φPδij, (A.2)

where

δij ={

0, i �= j,

1, i = j.

Substituting Eq. (A.1) into Eq. (A.2), we obtain

τ = Ae − (α + φδij)P . (A.3)

Rewritingα in double indices and defining the second-rank tensor

α = α + φI. (A.4)

This is the expression (12).For the isotropic case, we can easily obtain

αij = 0 (i �= j), α11 = α22 = α33 = Q + R

Rφ.

Obviously, the results are the same as those of Biot and Willis [18].An alternative method of determining the porous elastic coefficient tensorα is stated below. In fact, the total

stressesτ can directly be expressed by the solid-frame stiffness constants, the solid strain, porous elastic coefficients,and pressure, i.e.,

τ = Ae − αP. (A.5)

In an unjacketed compressibility test, let

τxx = τyy = τzz = −P ′, τxy = τxz = τyz = 0,

andP = −P ′ in Eq. (A.5), we can derive from Eq. (A.5)

αi = 1 − (Ai1δ′xx + Ai2δ

′yy + Ai3δ

′zz + Ai4δ

′yz + Ai5δ

′zx + Ai6δ

′xy), i = 1,2,3, (A.6a)

αj = −A1j δ′xx − A2j δ

′yy − A3j δ

′zz − A4j δ

′yz − A5j δ

′zx − A6j δ

′xy, j = 4,5,6, (A.6b)

whereP ′ is the external pressure on the jacket, andδ′ij = −eij/P

′ the strain coefficients. For the special case inwhich the principal axes of the ellipsoid are parallel to the coordinate axes, the coefficientsδ′

ii are given directly bythe principal strains and coefficientsδ′

yz, δ′zx, andδ′

xy are zero (see Ref. [18]). For some special anisotropic media(such as porous PTL, EDA, and PTL+ EDA), the formulae (A.6a) may be furthermore simplified if the solid isassumed to be isotropic. For example, we can obtain the poroelastic coefficients of the porous PTL+ EDA mediumfrom the relations (A.6a) and (A.6b):

αii = 1 − Ai1 + Ai2 + Ai3

3Ks, i = 1,2,3; αij = 0, i �= j, i, j = 1,2,3.


Appendix B. Fluid pressure of 1D BISQ model

The constitutive dynamic equation under the pillar coordinate(r, θ, x) can be written as

∂ρ

∂t+ ∂(ρVr)

∂r+ 1

r

∂(ρVθ )

∂θ+ ∂(ρVx)

∂x+ ρVr

r= 0, (B.1)

where(Vr , Vθ , Vx)T = (∂Φ/∂r, ∂Φ/∂θ, ∂Φ/∂x)T, Φ the velocity potential,Vr , Vθ , andVx the fluid velocitycomponents.

Consider axisymmetrical flows, the partial differential operator is given by∂/∂θ = 0 and thus Eq. (B.1) may besimplified as

∂ρ

∂t+ ∂(ρVr)

∂r+ ∂(ρVx)

∂x+ ρVr

r= 0. (B.2)

For the case under consideration, the fluid densityρ in the solid–fluid system equals toφρf ; the relative velocity of

the fluid to solid inx-direction yieldsVx = Ux − ux . Assuming ˙U = ∂U/∂t is an average velocity of the fluids

with the average displacementU in r-direction (Fig. 2), thereforeVr = ˙U under the assumptions in Section 4, wecan rewrite Eq. (B.2):

∂(ρfφ)

∂t+ ∂[ρfφ(Ux − ux)]

∂x+ ∂(ρfφ

˙U)

∂r+ ρfφ

˙Ur

= 0. (B.3)

In Eq. (B.3), the additive inverse of the first term,−(∂(ρfφ)/∂t), represents the fluid reduction of unit two-phasematerials and the second term indicates quantitatively the fluid flow out of unit volume in thex-direction. Boththe third and the fourth terms in Eq. (B.3) express the fluid mass flowing out of per unit volume inr-direction.Therefore, Eq. (B.3) shows the fluid’s mass conservation for the case under consideration. Retaining only linearterms in Eq. (B.3) yields:

φ

ρf

∂ρf

∂t+ ∂φ

∂t+ φ

∂2(Ux − ux)

∂x∂t+ φ

(∂2

∂r∂t+ 1

r

∂

∂t

)U = 0. (B.4)

By substituting Eqs. (13) and (15) into Eq. (B.4), we relate pressureP to displacementsUx , ux , andU :

∂P

∂t= −F1

[∂2Ux

∂x∂t+ α11 − φ

φ

∂2ux

∂x∂t+ α12

2φ

∂2uy

∂x∂t+ α13

2φ

∂2uz

∂x∂t+ ∂2U

∂r∂t+ 1

r

∂U

∂t

]. (B.5)

This equation is analogous to Eq. (16), which was derived for the case of 1D Biot-flow.Comparing Eqs. (B.5) and (16), it is clear that the fluid pressure relates not only to both the solid displacement and

the fluid displacement inx-direction but also the average fluid displacement inr-direction. To represent the pressurevariation induced by the transverse fluid squirt-flow through the solid and fluid displacements inx-direction, i.e.,in order to eliminate the average fluid displacementU , we have to combine Eqs. (B.5) and (10).

Assuming the solid–fluid relative displacement inx-direction and the fluid average displacement inr-directionyieldUx −ux = β1U , we can derive from the dynamic equation (10) via the average measurements of the solid/fluidcoupling effect and the power dissipation in theyz-plane:

− ∂

∂r(φP ) = (ρa1 + φρf )

∂2U

∂t2+ ηφ2r1

∂U

∂t, (B.6)

whereρa1 = 12(ρa2 + ρa3), r1 = 1

2[β1(r21 + r31) + r22 + 2r23 + r33].To find attenuation and phase velocity, we look for the solution of our problem in the following form:

U (x, r, t) = U0(r)exp[i(lx − ωt)], P (x, r, t) = P0(r)exp[i(lx − ωt)]. (B.7)


Substituting Eq. (B.7) into Eq. (B.6), we relate radial fluid displacement to pressure gradient as

∂P0(r)

∂r= U0ρfω

2(φ + (ρa1/ρf )

φ+ i

ω1

ω

), (B.8)

whereω1/ω = ηφr1/ωρf .Assuming that only average local flow parameters affect the global Biot’s flow, we can present the solid’s

displacement componentsuj (j = 1,2,3) and the fluid’s displacementUx in thex-direction as:

uj (x, t) = Cj exp[i(lx − ωt)], j = 1,2,3, Ux(x, t) = C4 exp[i(lx − ωt)]. (B.9)

Using Eqs. (B.7)–(B.9), we transform Eq. (B.5) into an ordinary differential equation which describes fluid pressuredependence on ther coordinate:

d2P0

dr2+ 1

r

dP0

dr+ P0

ρfω2

F1

(φ + (ρa1/ρf )

φ+ i

ω1

ω

)

= −il

(C4 + α11 − φ

φC1 + 1

2

α12

φC2 + 1

2

α13

φC3

)ρfω

2(φ + (ρa1/ρf )

φ+ i

ω1

ω

).

We solve this Bessel equation with a constant pressure boundary condition (e.g.,P0 = 0) atr = R1, whereR1is the radius of the representative cylinder with its axis parallel to the wave motion in thex-direction and is equalto the characteristic squirt-flow length. The solution is

P0(r) = −il

(C4 + α1 − φ

φC1 + 1

2

α6

φC2 + 1

2

α5

φC3

)F1

[1 − J0(λ1r)

J0(λ1R1)

],

whereJ0 is the Bessel function of zero-order and

λ21 = ρfω

2

F1

(φ + (ρa1/ρf )

φ+ i

ω1

ω

). (B.10)

We have assumed only average local flow parameters affecting the global Biot’s flow. In other words, the solidand the fluid displacements inx-direction are affected by values ofP andU averaged with respect tor. Therefore,we have to find an averaged fluid pressure. In fact, the averaged fluid pressure can be found from the previousequation and relations (B.7) and (B.9):

Psq= 1

πR21

∫ R1

02πrP(x, r, t)dr= − F1

[1 − 2J1(λ1R1)

λ1R1J0(λ1R1)

](∂Ux

∂x+α11 − φ

φ

∂ux

∂x+ α12

2φ

∂uy

∂x+α13

2φ

∂uz

∂x

).

This is the average fluid pressure (17) based on the BISQ theory for the 1D porous anisotropic case.

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poroelastic wave equation including the biot/squirt mechanism...

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