Acoustic Wave Propagation in a Composite of Two DifferentPoroelastic Materials with a Very Rough Periodic Interface: a

Homogenization Approach

Robert P. GilbertDepartment of Mathematical Sciences

University of Delaware, Newark, DE 19716, U.S.A.Miao-jung Ou

Institute for Mathematics and its Applications,University of Minnesota, Minneapolis, MN 55455, USA

October 18, 2002

Abstract

Homogenization is used to analyze the system of Biot-type partial differential equations in a domain of two differentporoelastic materials with a very rough periodic interface. It is shown that by using homogenization, such a rough interfacecan be replaced by an equivalent flat layer within which a system of modified differential equations holds. The coefficientsof this new system of equations are certain “effective” parameters. These coefficients are determined by solutions of theauxiliary problems which involve the detailed structure of the interface. In this paper, the auxiliary problems are derivedand the homogenized system of equations is given.

1 Introduction

Poroelasticity, the mechanics of porous elastic solids with fluid-filled pores, has received attention in the last fewdecades for its important role in oil recovery, the study of the triggering of earthquakes, liquid waste disposed byunderground seepage into pores, and underwater acoustics involving propagation in the water-saturated, porousbottom of the ocean, etc.

In underwater acoustics, a poroelastic seabed model is more realistic than a rigid seabed or an elastic seabed.In each of these three models, the sea bottom and the interface between different layers are usually assumed tobe flat. In [2], acoustic wave propagation in a shallow ocean with a flat poroelastic seabed was studied. Here incontrast, we take into account the geometry of the interface and study its effect on acoustic wave propagation.We will assume the interface between the two poroelastic layers to be very rough. For simplicity, we assume thatthe interface has a periodic geometry, i.e. the ratio of the amplitude to the “wavelength” is large.

Problems involving rough boundaries or interfaces can usually be analyzed using perturbation methods whenthe amplitude/wavelength ratio is small. For problems with a large ratio, other methods of analysis are required.For example, the homogenization method used by Kohler, Papanicolaou and Varadhan [3], or Nevard and Keller[4]. In their works, it is shown that for certain partial differential equations, the problem of the PDE in the regionwhich contains the rough interface can be replaced by another problem of a homogenized PDE in an equivalentflat layer. The basic assumption underlying the method is that the scale length, or “wavelength”ε of the roughnessis small compared with all other relevant lengths, especially the roughness amplitude.

1

In this paper, we use a homogenization method, similar to the one mentioned above, to study the time-harmonic acoustic wave propagation in a composite of two different poroelastic materials. The auxiliary problemsand the homogenized equations in the equivalent layer will be derived.

2 The Constitutive Equations

Figure 1: Schematic representation of the rough interface

The poroelastic equations derived by Biot [5, 6, 7] have long been regarded as standard in solving problemsin poroelasticity. However, the validity of these equations has recently been questioned. For this reason, variousauthors have used homogenization methods to derive anew the governing equations of linear poroelasticity bystarting with the detailed micro-structure of the pores. For example, Burridge and Keller [8], Gilbert and Mikelic[9], Auriault [10], and Auriault et al. [11]. In these papers, the pores are assumed to have periodic structure, andthe linearized equations of elasticity and the linearized Navier-Stokes equations are used to describe the behaviorof the solid part and the fluid part, respectively. These works have shown that the newly derived equationscoincide with Biot’s equations when the dimensionless viscosity of the fluid is small. Furthermore, their worksalso enable us to perform a complete calculation of the effective parameters in Biot’s equations.

Since the pore fluid in the seabed can be regarded as Newtonian and incompressible , we will adapt theequations given by Auriault et al. in [11]. For time harmonic motionu(x, t) = u(x)eiωt, p(x, t) = p(x)eiωt inan isotropic poroelastic medium, they are

Σ = Ce(u)− αpI, (1)

div (Σ) = −ω2 {ρs(1− f)u + fρlU l} , (2)

Ifω (U l − u) = K(ω2ρlu− gradp

), (3)

f div (U l − u) = −α div u− βp. (4)

In these equations,I is the unit tensor andI =√−1. Σ represents the total stress tensor;C is the elasticity tensor

of the skeleton,p the fluid pressure ( positive for compression),ω the acoustic frequency,u the displacement ofthe solid part,U l the displacement of the fluid part,e (u) the strain tensor,f the porosity,ρl the density of thepore fluid,ρs the density of the skeleton andK is the generalized Darcy permeability tensor, which is introducedby the homogenization theory, symmetric andω-dependent. The effective parametersα andβ can be computed.

2

In [12], using Helmholtz’s decomposition theorem, it is shown that the degree of freedom of (1)-(4) is onlyfour. Therefore, we may reformulate this system in terms ofu andp. This gives (5) and (6). The interfacebetween the two different poroelastic materials is described byx3 = h(x1, x2), whereh is ε-periodic in boththex1 andx2 direction ( See Fig. 1). The physical parameters are assumed to be constants above and below theinterface and have a jump discontinuity across the interface. In what follows, we adopt the summation convention,Latin subscripts taking the values 1, 2, 3 and Greek subscripts taking the values 1, 2.

∂xj (Cijkl∂xluk)− αδijp + ω2(ρijuj) = −αij∂xjp, x3 6= h(x1, x2), (5)

−Iω

Kij

(∂xi∂xjp− ρlω

2∂xiuj

)− α∂xiui = βp, x3 6= h(x1, x2), (6)

[u] = 0, (7)

[nj (Cijkl∂xluk − αδijp)] = 0, (8)

[p] = 0, (9)[niKij

(∂xjp− ρlω

2∂xiuj

)]= 0, (10)

where

αij := IωρlKij ,

ρij := ρs (1− f) δij + ρl {−αij + fδij}

and[·] denotes the jump across the interfacex3 = h(x1, x2), i.e.

[F (x∗)] := limY −3x→x∗∈Γ

F (x) − limY +3x→x∗∈Γ

F (x).

In (5)-(10), δ is the Kronecker delta tensor andn is the unit normal vector of the interfacex3 = h(x1, x2)

Y+

Y−

Γ

X

X

1

2

ε

0 ε

Figure 2: Schematic representation of thex1x2 profile of the periodicity cellY .

pointing in the positivex3 direction. The interface conditions (7), (8), (9) and (10) represent continuity ofdisplacement, normal stress, pressure and the total velocity across the interface, respectively.

3 Formal Expansions and the Homogenized Equations

We introduce a new independent variabley, whereyα :=xα

ε, α = 1, 2. Using the fact thatn is proportional to

(ε−1∂y1h, ε−1∂y2h,−1), (8) and (10) become[ε−1

(∂yβ

h)(Ciβkl∂xl

uk − αpδiβ)− (Ci3kl∂xluk − αpδi3)

]= 0, (11)[

ε−1 Kαj

(∂xjp− ρlω

2uj

)(∂yαh)−K3j

(∂xjp− ρlω

2uj

)]= 0. (12)

3

Define new dependent variableswi andq such that

wi(x,y, t, ε) : = ui(x, t, ε),q(x,y, t, ε) : = p(x, t, ε).

Note that∂xαui = (∂xα + ε−1∂yα)wi and∂xαp = (∂xα + ε−1∂yα)q.

In terms of the new variables, (5) and (6) can be written as

ε−2∂yβ (Ciβkδ∂yδwk)

+ ε−1{∂xj (Cijkδ∂yδ

wk) + ∂yβ(Ciβkl∂xl

wk) + αiγ∂yγq − αδiγ∂yγq}

+ ε0{∂xj (Cijkl∂xl

wk) + ω2ρijwj + αij∂xjq − αδij∂xjq}

= 0, (13)

ε−2

(−I

ωKαβ∂yα∂yβ

q

)+ ε−1

{IωρlKαj∂yαwj − α∂yαwα −

2 Iω

Kαj∂xj∂yαq

}+ ε0

{IωρlKij∂xiwj −

Iω

Kij∂xi∂xjq − α∂xiwi − βq

}= 0. (14)

Now we assume thatw andq have the following asymptotic expansions forε small:

w(x,y, t, ε) = u(0)(x,y, t) +2∑

k=1

u(k)(x,y, t)εk +O(ε3), (15)

q(x,y, t, ε) = p(0)(x,y, t) +2∑

k=1

p(k)(x,y, t)εk +O(ε3). (16)

We also assume that each term in the expansions isY -periodic and that the asymptotic forms of the derivativesof w andq are given by term by term differentiation of (15) and (16), respectively. By equating the terms of thesame order inε, we get the following equations:

O(ε−2) :

∂yβ

(Ciβkδ∂yδ

u(0)k

)= 0, (17)

Kαβ∂yα∂yβp(0) = 0, (18)[

Ciβkδ∂yδu

(0)k ∂yβ

h]

= 0, (19)[Kαβ∂yβ

p(0)∂yαh]

= 0. (20)

4

O(ε−1) :

∂yβ

(Ciβkδ∂yδ

u(1)k

)+ ∂xj

(Cijkδ∂yδ

u(0)k

)+ ∂yβ

(Ciβkl∂xl

u(0)k

)+αiγ∂yγp(0) − αδiγ∂yγp(0) = 0, (21)

− Iω

Kαβ∂yα∂yβp(1) + IωρlKαj∂yαu

(0)j − α∂yαu(0)

α

− 2 Iω

Kαj∂yα∂xjp(0) = 0, (22)

[(Ciβkδ∂yδ

u(1)k + Ciβkl∂xl

u(0)k − αp(0)δiβ

)(∂yβ

h)− Ci3kδ∂yδu

(0)k

]= 0, (23)[(

Kαβ∂yβp(1) + Kαj

(∂xjp

(0) − ρlω2u

(0)j

))(∂yαh)−K3β∂yβ

p(0)]

= 0. (24)

O(ε0) :

∂yβ

(Ciβkδ∂yδ

u(2)k

)+ ∂xj

(Cijkδ∂yδ

u(1)k

)+ ∂yβ

(Ciβkl∂xl

u(1)k

)+αiγ∂yγp(1) − αδiγ∂yγp(1) + ∂xj

(Cijkl∂xl

u(0)k

)+ω2ρiju

(0)j + αij∂xjp

(0) − αδij∂xjp(0) = 0, (25)

−Iω

Kαβ∂yα∂yβp(2) + IωρlKαj∂yαu

(1)j − α∂yβ

u(1)β − 2 I

ωKαj∂xj∂yαp(1)

+IωρlKij∂xiu(0)j − α∂xiu

(0)i − I

ωKij∂xi∂xjp

(0) − βp(0) = 0, (26)

[(Ciβkδ∂yδ

u(2)k + Ciβkl∂xl

u(1)k − αp(1)δiβ

)(∂yβ

h)

−Ci3kδ∂yδu

(1)k − Ci3kl∂xl

u(0)k + αp(0)δi3

]= 0, (27)[(

Kαβ∂yβp(2) + Kαj

(∂xjp

(1) − ρlω2u

(1)j

))(∂yαh)

− K3β∂yβp(1) −K3j

(∂xjp

(0) − ρlω2u

(0)j

)]= 0, (28)[

u(0)]

= 0, (29)[p(0)

]= 0. (30)

We now consider equations (17), (19) and (29), which correspond tou(0):

∂yβ

(Ciβkδ∂yδ

u(0)k

)= 0, (31)[

Ciβkδ∂yδuk

(0)∂yβh]

= 0, (32)[u(0)

]= 0. (33)

By scalar multiplying (31) byu(0), then integrating the product over the periodicity cellY ( see Figure 2 )

5

and applying theY -periodic condition tou(0), we have∫Y

Ciβkδ∂yδu

(0)k ∂yβ

u(0)i dy

= −∫

Γ

[u

(0)i Ciβkδ∂yδ

uk(0)∂yβ

h

]dσ

|∇yh|, (34)

where

∇yh := (∂y1h, ∂y2h)T .

We apply the jump conditions (32) and (33) to (34) to obtain∫Y

Ciβkδ∂yδu

(0)k ∂yβ

u(0)i dy = 0.

By the symmetry ofCijkl and∂yβu

(0)i , we have

∂yβu

(0)i = 0, (35)

i.e. u(0) does not depend ony.

Equations (18), (20) and (30) constitute a system of equations forp(0). Multiplying (18) by p(0), integratingthe result overY and applying (20), (30) together with theY -periodic condition onp(0), we get∫

YKαβ∂yαp(0) ∂yβ

p(0)dy = 0.

This implies

∂yβp(0) = 0, (36)

i.e. p(0) does not depend ony.

Next, we consider theO(ε−1) equations. By using (35) and (36), (22) and (24) reduce to

(P1)

{Kαβ∂yα∂yβ

p(1) = 0, x3 6= h(y1, y2),[(Kαβ∂yβ

p(1) + Kαj

(∂xjp

(0) − ρlω2u

(0)j

))(∂yαh)

]= 0.

(37)

To solve (P1), we introduce a new variableφj(x3,y) by writing p(1) in the form

p(1)(x,y, t) = φj(x3,y)(∂xjp

(0)(x, t)− ρlω2u

(0)j (x, t)

). (38)

Substituting (38) into (P1), we can see thatp(1)(x,y, t) in (38) will solve (P1) ifφj(x3,y) satisfies

(AP1){

Kαβ∂yα∂yβφj = 0, x3 6= h(y1, y2),[(

Kαβ∂yβφj + Kαj

)(∂yαh)

]= 0.

(39)

We also requireφj(x3,y) to beY -periodic in they variable, continuous inY and to have zero average overY .These conditions uniquely determineφj .

6

Similarly, by applying (35) and (36) to equations (21) and (23), we get the following system of equations foru(1):

(P2) :

∂yβ

(Ciβkδ∂yδ

u(1)k

)= 0, x3 6= h(y1, y2),[(

Ciβkδ∂yδu

(1)k + Ciβkl∂xl

u(0)k − αp(0)δiβ

)(∂yβ

h)]

= 0.

To solve (P2), we introduce a new variableχkmn by writing u(1)k in the form

u(1)k (x,y, t) = χkmn(x3,y)

(∂xnu(0)

m (x, t) + Amnp(0)(x, t))

, (40)

whereAmn is an absolute constant matrix such that{C+

iβmnAmn = −α+δiβ,

C−iβmnAmn = −α−δiβ.

By direct calculation, it can be seen that

A11 = A22 =α− (λ+ + µ+)− α+ (λ− + µ−)

(λ− + µ−) (2λ+ + 3µ+)− (λ+ + µ+) (2λ− + 3µ−),

A33 =α− (2λ+ + 3µ+)− α+ (2λ− + 3µ−)

(λ− + µ−) (2λ+ + 3µ+)− (λ+ + µ+) (2λ− + 3µ−),

Amn = 0 if m 6= n,

whereλ+, µ+, λ− andµ− are the Lame coefficients of the material inY + andY −, respectively.

Substituting (40) into (P2) gives{Ciβkδ

(∂yβ

∂yδχkmn

) (∂xnu

(0)m + Amnp(0)

)= 0, x3 6= h(y1, y2),[

(Ciβkδ∂yδχkmn + Ciβmn) ( ∂yβ

h)](∂xnu

(0)m + Amnp(0)) = 0.

Therefore, (40) solves (P2) ifχkmn satisfies

(AP2){

Ciβkδ

(∂yβ

∂yδχkmn

)= 0, x3 6= h(y1, y2),[

(Ciβkδ∂yδχkmn + Ciβmn) ( ∂yβ

h)]

= 0.(41)

We also requireχkmn to be continuous inY , Y -periodic in they variable and to have zero average overY . Theseconditions uniquely determineχkmn.

Next, we consider theO(ε0) equations. We integrate (25) with respect toy overY and divide it by the areaof Y , which is denoted byA. The first and the third term can be converted to integrals aroundΓ and∂Y by thedivergence theorem. The integrals along∂Y vanish because of the assumedY -periodicity ofu(1) andu(2). Wethus obtain

−A−1

∫Γ

[(Ciβkδ∂yδ

u(2)k + Ciβkl∂xl

u(1)k − αδiβp(1)

)(∂yβ

h)] dσ

|∇yh|

+ A−1

∫Y

∂xj

(Cijkδ∂yδ

u(1)k

)dy + A−1

∫Y

αiγ∂yγp(1)dy

+ A−1

∫Y

∂xj

(Cijkl∂xl

u(0)k

)dy + ω2 < ρij > u

(0)j

+ < αij > ∂xjp(0)− < α > ∂xip

(0) = 0, (42)

7

where< · > is the “averaging operator” defined as

< · >:= A−1

∫Y· dy.

By using (27), the first integral in (42) can be written as

−A−1

∫Γ

[Ci3kδ∂yδ

u(1)k + Ci3kl∂xl

u(0)k − αδiβp(0)

] dσ

|∇yh|

= −A−1

∫Γ

[Ci3kδ∂yδ

u(1)k

] dσ

|∇yh|−A−1

([Ci3kl ] ∂xl

u(0)k − [α]δi3p

(0)) ∫

Γ

dσ

|∇yh|

= −A−1

∫Γ

[Ci3kδ∂yδ

u(1)k

] dσ

|∇yh|−A−1

([Ci3kl ] ∂xl

u(0)k − [α]δi3p

(0))

∂x3A1, (43)

whereA1 is the area ofY +. The last equality in (43) follows from the coarea formula [1], which states that∫Γ

dσ

|∇yh|= ∂x3A1.

By direct calculation and using the facts thatY + andY − are functions ofx3 only, andu(0), Amn andp(0) do notdepend ony, the second integral in (42) can be converted to

< Cijkδ∂yδχkmn > ∂xj

(∂xnu(0)

m + Amnp(0))

+ < Ci3kδ∂x3∂yδχkmn >

(∂xnu(0)

m + Amnp(0))

. (44)

The fourth integral in (42) is equal to∂xj∂xlu

(0)k < Cijkl > becauseu(0) is not a function ofy andCijkl are

constants inY + andY −. Realizing that< Cijkl > depends only onx3 and

< Cijkl > = A−1(A1C

+ijkl + (A−A1)C−

ijkl

),

we may further write(∂xj∂xl

u(0)k

)< Cijkl > as(

∂xj∂xlu

(0)k

)< Cijkl >

= ∂xj

(∂xl

u(0)k < Cijkl >

)+ A−1 (∂x3A1) [Ci3kl ] ∂xl

u(0)k . (45)

Similarly, the last term in (42) can be written as

< α > ∂xip(0) = ∂xi

(< α > p(0)

)+ A−1 (∂xiA1)δi3[α]. (46)

Finally, we substitute (43), (44), (45), (46), (38) and (40) into (42). Note that the last term in (43) cancels withthe last term in (45) and (46). This yields a new equation

−(

A−1

∫Γ

[Ci3kδ∂yδχkmn]

dσ

|∇yh|

) (∂xnu(0)

m + Amnp(0))

+ < Cijkδ∂yδχkmn > ∂xj

(∂xnu(0)

m + Amnp(0))

+ < Ci3kδ∂x3∂yδχkmn >

(∂xnu(0)

m + Amnp(0))

+(∂xjp

(0) − ρlω2u

(0)j

)< αiγ∂yγφj >

+ ∂xj

(∂xl

u(0)k < Cijkl >

)+ ∂xi

(< α > p(0)

)+ ω2 < ρij > u

(0)j + < αij > ∂xjp

(0) = 0. (47)

8

Introduce the “effective” parameters

Ceffijmn : = < Cijkδ∂yδ

χkmn > + < Cijnm >, (48)

M effimn : = −A−1

∫Γ

[Ci3kδ∂yδχkmn]

dσ

|∇yh|− ∂x3 < Cijkδ∂yδ

χkmn > + < Ci3kδ∂x3∂yδχkmn >, (49)

αeffij : = < αiγ∂yγφj + αij > +Amn < Cijkδ∂yδ

χkmn >, (50)

ρeffij : = < ρij > −ρl < αiγ∂yγφj > . (51)

Then (47) can be rewritten as

∂xj

(Ceff

ijmn∂xnu(0)m

)+ ω2ρeff

ij u(0)j −M eff

imn∂xnu(0)m = −αeff

ij ∂xjp(0), (52)

which is the homogenized equation of (5).

Similarly, we first apply< · > to (26). Using the divergence theorem on the first and fifth term, we convert theintegrals to be boundary integrals of jumps aroundΓ. Secondly, we apply equation (28) to rewrite the boundaryintegral of the jump. Thirdly, we replaceu(1)

k andp(1) by using (38) and (40). Finally, (26) becomes(Iω

) (A−1

∫Γ

[K3β∂yβ

φj + K3j

] dσ

|∇yh|

) (∂xjp

(0) − ρlω2u

(0)j

)−

(Iω

)< Kij > ∂xi∂xjp

(0)− < α∂yβχβmn >

(∂xnu(0)

m + Amnp(0))

−(Iω

) (∂xjp

(0) − ρlω2u

(0)j

)< K3β∂x3(∂yβ

φj) >

−(Iω

) (∂xi∂xjp

(0) − ρlω2∂xiu

(0)j

)< Kiβ∂yβ

φj >

− < α > ∂xiu(0)i − < β > p(0) + Iωρl < Kij > ∂xiu

(0)j = 0. (53)

Introducing another set of “effective” parametersKeffij , Leff

nm , Qeffj andβeff ,

Keffij :=< Kij > + < Kiβ∂yβ

φj >, (54)

Leffnm :=< α∂yβ

χβmn > + < α > δmn, (55)

Qeffj := −A−1

∫Γ

[K3β∂yβ

φj + K3j

] dσ

|∇yh|+ < K3β∂x3(∂yβ

φj) >, (56)

βeff :=< β > +Amn < α∂yβχβmn >, (57)

(53) can be written as (−I

ω

)Keff

ij

(∂xi∂xjp

(0) − ρlω2∂xiu

(0)j

)− Leff

nm ∂xnu(0)m

= βeffp(0) +Iω

Qeffj

(∂xjp

(0) − ρlω2u

(0)j

).

This is the homogenized equation of (6).

We summarize our discussion as follows.

9

Theorem 3.1. Let u(x, t, ε) and p(x, t, ε) satisfy the dynamic equation (5) and the continuity equation (6)on both sides of the periodic surfacex3 = h(x1, x2) with constant solid densityρs, fluid densityρl, DarcypermeabilityK, porosityf , effective parametersα and β on each side. Supposeu(x, t, ε) andp(x, t, ε) alsosatisfy the continuity conditions (7)-(10) across the interfacex3 = h(x1, x2).

If u(x, t, ε) andp(x, t, ε) have the asymptotic forms (15) and (16), respectively. Thenu(0)(x, t) andp(0)(x, t)satisfy the following system of homogenized equations in0 < x3 < a ,

∂xj

(Ceff

ijmn∂xnu(0)m

)+ ω2ρeff

ij u(0)j −M eff

imn∂xnu(0)m = −αeff

ij ∂xjp(0), (58)(

−Iω

)Keff

ij

(∂xi∂xjp

(0) − ρlω2∂xiu

(0)j

)− Leff

nm ∂xnu(0)m = βeffp(0)

+Iω

Qeffj

(∂xjp

(0) − ρlω2u

(0)j

). (59)

Herea is as defined in Figure 1 ,< · > the average operator, and the effective parameters are defined in (48) to(51) and (54) to (57).

4 A Numerical Example

Silty ClayFine Sand

0.25 ε

ε

ε

ε

Figure 3: Single cell of the periodic interface

In this section, we will construct the effective parameters of a special case of the previous discussion. Forsimplicity, the interface is assumed to consist of very narrow truncated uniform cylinders with radius equal0.25 ε( See Figure 3). In this case,h = x2

1 + x22 = (0.25ε)2, ε = 0.1, and the unit normal vectors on the interface are

proportional to(ε−1∂y1h, ε−1∂y2h, 0) rather than(ε−1∂y1h, ε−1∂y2h,−1). Accordingly, the effective parameterM eff

inm andQeffj are modified to be

M effimn : = −∂x3 < Cijkδ∂yδ

χkmn > + < Ci3kδ∂x3∂yδχkmn >,

Qeffj : = < K3β∂x3(∂yβ

φj) >,

whereas the other effective parameters remain unchanged.

Suppose the upper layer is of fine sand and the lower layer is of silty clay. Using the experimental data in[13] and the formulas (58)-(71) in [14], we list the Lame coefficients and Darcy tensor of each layer in Table 1.The frequencyω is chosen to be 200 Hz.

The effective parameters are computed using the solutions of the two auxiliary problems(AP1) and(AP2).Part of the solution of these problems is given in Figure 4 - Figure 6. The effective parameters are listed below.

10

0.052567 −0.051842

(a) Real part

2.7185e−06−2.9571e−06

(b) Imaginary part

Figure 4: Solution to(AP1): φ1

Ceff1111 Ceff

1112 Ceff1113

Ceff1121 Ceff

1122 Ceff1123

Ceff1131 Ceff

1132 Ceff1133

= 109 ×

1.19242− 0.00047 I 0.00000− 0.00000 I 0.00000 + 0.00000 I0.00000− 0.00000 I 1.18068− 0.00009 I 0.00000 + 0.00000 I0.00000 + 0.00000 I 0.00000 + 0.00000 I 1.41625− 0.00010 I

;

Ceff1211 Ceff

1212 Ceff1213

Ceff1221 Ceff

1222 Ceff1223

Ceff1231 Ceff

1232 Ceff1233

=

Ceff2111 Ceff

2112 Ceff2113

Ceff2121 Ceff

2122 Ceff2123

Ceff2131 Ceff

2132 Ceff2133

= 106 ×

0.00000− 0.00000 I 5.87099− 0.18940 I 0.00000 + 0.00000 I5.87099− 0.18940 I 0.00000− 0.00000 I 0.00000 + 0.00000 I0.00000 + 0.00000 I 0.00000 + 0.00000 I 0.00000 + 0.00000 I

;

Table 1: Physical parameters of the poroelastic media

fine sand silty clayµ 7.12× 106 − 2.3× 105 I 7.86× 106 − 2.5× 104 Iλ 1.68× 109 − 1.04263× 105 I 3.5× 108 − 9.6× 104 I

Kij (3.09× 10−11 − 5.55× 10−16 I) δij (5.15× 10−11 − 2.34× 10−15 I) δij

11

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) Real part

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Imaginary part

Figure 5: Solution to(AP1): Vector plot of(φ1, φ2)

Ceff1311 Ceff

1312 Ceff1313

Ceff1321 Ceff

1322 Ceff1323

Ceff1331 Ceff

1332 Ceff1333

=

Ceff3111 Ceff

3112 Ceff3113

Ceff3121 Ceff

3122 Ceff3123

Ceff3131 Ceff

3132 Ceff3133

= 106 ×

0.00000 + 0.00000 I 0.00000 + 0.00000 I 5.18393− 0.16727 I0.00000 + 0.00000 I 0.00000 + 0.00000 I −0.00000 + 0.00000 I5.18393− 0.16727 I −0.00000 + 0.00000 I 0.00000 + 0.00000 I

;

Ceff2211 Ceff

2212 Ceff2213

Ceff2221 Ceff

2222 Ceff2223

Ceff2231 Ceff

2232 Ceff2233

= 109 ×

1.18068− 0.00009 I 0.00000− 0.00000 I 0.00000 + 0.00000 I0.00000− 0.00000 I 1.19242− 0.00047 I 0.00000 + 0.00000 I0.00000 + 0.00000 I 0.00000 + 0.00000 I 1.41625− 0.00010 I

;

12

0.11171−0.11171

(a) Real part

4.0495e−06

−4.0494e−06

(b) Imaginary part

Figure 6: Solution to(AP2): χ111

Ceff2311 Ceff

2312 Ceff2313

Ceff2321 Ceff

2322 Ceff2323

Ceff2331 Ceff

2332 Ceff2333

=

Ceff3211 Ceff

3212 Ceff3213

Ceff3221 Ceff

3222 Ceff3223

Ceff3231 Ceff

3232 Ceff3233

= 106 ×

0.00000 + 0.00000 I 0.00000 + 0.00000 I −0.00000 + 0.00000 I0.00000 + 0.00000 I 0.00000 + 0.00000 I 5.18393− 0.16727 I−0.00000 + 0.00000 I 5.18393− 0.16727 I 0.00000 + 0.00000 I

;

Ceff3311 Ceff

3312 Ceff3313

Ceff3321 Ceff

3322 Ceff3323

Ceff3331 Ceff

3332 Ceff3333

= 109 ×

1.18180− 0.00013 I 0.00000− 0.00000 I 0.00000 + 0.00000 I0.00000− 0.00000 I 1.18180− 0.00013 I 0.00000 + 0.00000 I0.00000 + 0.00000 I 0.00000 + 0.00000 I 1.42800− 0.00048 I

;

13

M effimn = 0, ∀i, m, n;

αeffij = 103 ·

−1.77942− 0.05891 I 0.00000 + 0.00000 I 0.00000 + 0.00000 I0.00000 + 0.00000 I −1.77942− 0.05891 I 0.00000 + 0.00000 I0.00000 + 0.00000 I 0.00000 + 0.00000 I −1.77170− 0.05891 I

;

ρeffij = 103 ·

1.87055− 0.00001 I 0.00000 + 0.00000 I 0.00000 + 0.00000 I0.00000 + 0.00000 I 1.87055− 0.00001 I 0.00000 + 0.00000 I0.00000 + 0.00000 I 0.00000 + 0.00000 I 1.87055− 0.00001 I

;

Keffij = 10−11 ·

3.40805− 0.00009 I −0.00000− 0.00000 I 0.00000 + 0.00000 I−0.00000− 0.00000 I 3.40805− 0.00009 I 0.00000 + 0.00000 I0.00000 + 0.00000 I 0.00000 + 0.00000 I 3.49347− 0.00009 I

;

Leffnm =

1.90292 + 0.00003 I −0.00000 + 0.00000 I 0.00000 + 0.00000 I−0.00000 + 0.00000 I 1.90292 + 0.00003 I 0.00000 + 0.00000 I0.00000 + 0.00000 I 0.00000 + 0.00000 I 1.59361 + 0.00000 I

;

Qeffj =

000

;

βeff = −5.7549× 10−7 − 1.9171× 10−8I.

AcknowledgementThis work was supported in part by grant INT-9726213.

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