AIP/Mikeli´c, Wheeler

Theory of the dynamic Biot-Allard equations and their link to the quasi-static Biot systema) Andro Mikeli´c1, b) and Mary F. Wheeler2, c) 1)

Universit´e de Lyon, CNRS UMR 5208,

Universit´e Lyon 1, Institut Camille Jordan, 43, blvd. du 11 novembre 1918, 69622 Villeurbanne, France 2)

The Center for Subsurface Modeling,

Institute for Computational Engineering and Sciences The University of Texas at Austin, 201 East 24th Street Austin, TX 78712, U. S. A. (Dated: 16 August 2012)

We undertake establishing well-posedness of the dynamic Biot-Allard equations. It is obtained using the precise properties of the dynamic permeability matrix following the homogenization derivation of the model. By taking the singular limit of the contrast coefficient, the quasi-static Biot system can be obtained from the dynamic Biot equations. These results can be used to formulate an efficient computational algorithm for solving dynamic Biot-Allard equations for subsurface flows with the characteristic reservoir time scales larger than the intrinsical characteristic time. This result appears to be completely new in the literature on Biot’s theory. We conclude by showing that in the case of periodic deformable porous media the dynamic permeability has the required properties. PACS numbers: 02.30.Jr ; 02.30.Vv ; 43.20.Bi ; 43.20.Tb ; 47.56.+r ; 92.40.ke ; Keywords: poroelasticity, Biot-Allard’s equations, vector-valued Laplace transform, quasi-static equations

a)

The research of A.M. was partially supported by the GNR MOMAS (Mod´elisation Math´ematique et

Simulations num´eriques li´ees aux probl`emes de gestion des d´echets nucl´eaires) (PACEN/CNRS, ANDRA, BRGM, CEA, EDF, IRSN). He would like to thank The Center for Subsurface Modeling, ICES, UT Austin for hospitality in April 2009, 2010 and 2011. The research by M. F. Wheeler was partially supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences through DOE Energy Frontier Research Center: The Center for Frontiers of Subsurface Energy Security (CFSES) under Contract No. DE-SC0001114. b) c)

Electronic mail: [email protected]

Electronic mail: [email protected]

1

I.

INTRODUCTION Effective deformation and filtration in a deformable porous medium is described by the

Biot-Allard equations

∫

t

ρ∂tt u − Div {A e(u) − αp} + ∂t

A(

H

0

( ∂t

(t − ζ)η )(ρf F(x, ζ) ρf ℓ2

−∇p(x, ζ) − ρf ∂ζζ u(x, ζ)) dζ = ρF, (1) ∫ t ( ) (t − ζ)η 1 )(F(x, ζ) − ∇p(x, ζ) − ∂ζζ u(x, ζ)) dζ} = 0 M p + div αu + div { A( ρf ℓ2 ρf 0 )

(2) where e(∗) stands for the symmetrized gradient (strain tensor), u is the effective solid phase displacement, p is the effective pressure and the parameters are defined in Table I SYMBOL

QUANTITY

UNITY

ρs

solid grain density

kg/m3

ρf

pore fluid density

kg/m3

φ

porosity

0 λ0 > 0}. We apply it to the system (10), (11) to have 7

Theorem 1. Let us suppose F ∈ C0∞ (R+ ; L2 (Ω)3 ) and : (H1) A0 is a symmetric positive definite 4th order tensor. : (H2) M 0 is a positive constant. : (H3) α is a symmetric matrix. : (H4) The Laplace transform of Aˆ of the dynamic permeability matrix A is a complex symmetric (but not Hermitian) matrix and satisfies ˆ )ξξ} ≥ C λ1 + ℜτ |ξ|2 , ∀ξ ∈ C3 , ℜ{A(τ |λ1 + τ |2

(12)

ˆ )ξξ} ≥ C|ξ|2 , for some c1 ≥ 0 and ∀ξ ∈ C3 , ℜ{(τ + c1 )A(τ

(13)

ˆ κf ))ξξ + A(κ ˆ f τ )ββ + (α − τ κf A(τ ˆ κf ))βξ − (α − τ κf A(τ ˆ κf ))ξβ} ℜ{τ (κI − τ κ2f A(τ κf ℜτ |β|2 , ∀ξ, β ∈ C3 , |λ1 + κf τ |2 1 ˆ )||∞ ≤ C ||A(τ , |λ1 + τ |

≥ κs (1 − φ)ℜτ |ξ|2 + C

(14) (15)

for some λ1 > 0. Then we have the following a priori estimate: ∫

ℜτ κf ℜτ {κs |τ u ˆ | + |e(ˆ u )| + |ˆ p | } dx + |λ1 + τ κf |2 Ω 0 2

0

2

∫ ˆ 2 2 . (16) |∇ˆ p0 |2 dx ≤ C||(1 + |τ |)F|| L (Ω)

0 2

Ω

Remark 1. As discussed in the Appendix, the hypothesis (H1)-(H4) are natural. In particular, the matrix Aˆ is not uniformly positive definite with respect to τ and (14)-(15) give its precise behavior. It is known from the literature, that in the case ℜτ = 0, the impedance ˆ Im τ ) goes to zero for large values of Im τ . For details we refer to9 and10 . A(i Proof.

Application of the Laplace transform to (10) and (11) with τ ∈ C+ yields ˆ κf ))ˆ p0 }+ − Div {A0 e(ˆ u0 )} + Div {(α − τ κf A(τ ˆ κf ))F(x, ˆ κf ))τ 2 u ˆ τ ), ˆ 0 = (ψI − τ κf ψf A(τ (κI − τ κ2f A(τ

(17)

ˆ κf ))τ u ˆ κf )∇ˆ ˆ κf )ψf F(x, ˆ τ )}. (18) M 0 τ pˆ0 + div {(α − τ κf A(τ ˆ 0 } − div {A(τ p0 } = − div {A(τ 8

Now we test equation (17) with τ u ˆ 0 , take the complex conjugate of (18), test it with pˆ0 and sum up the obtained variational equalities, to get ∫ ∫ ∫ 2 ˆ 0 0 0 0 0 0 τ ((κI − τ κf A(τ κf ))τ u ˆ τu ˆ dx + τ u ) dx + τ M A e(ˆ u ) : e(ˆ |ˆ p0 |2 dx+ Ω Ω Ω ) ∫ ( ∫ 0 0 0 0 0 0 ˆ ˆ ˆ (α − τ κf A(τ κf ))∇ˆ p τu A(τ κf )∇ˆ p ∇ˆ p dx + ˆ − (α − τ κf A(τ κf ))τ u ˆ ∇ˆ p dx Ω Ω ∫ ∫ 0 ˆ ˆ κf ))F(x, ˆ ˆ τ )τ u = ψf A(τ κf )F(x, τ )∇ˆ p dx + (ψI − τ κf ψf A(τ ˆ 0 dx. (19) Ω

Ω

Using (14)-(15) yields ∫ ∫ ℜτ κf 0 2 0 2 0 2 |∇ˆ p0 |2 dx ≤ ˆ | + |e(ˆ u )| + |ˆ p | } dx + ℜτ {κs |τ u 2 |λ + τ κ | 1 f Ω Ω ( √ ) 0 ℜτ κ ∇ˆ p f 0 ˆ L2 (Ω) || C||F|| ||L2 (Ω) + ||τ u ˆ ||L2 (Ω) . |λ1 + τ κf |

(20)

(20) implies estimate (16).  In order to return to the original time variable we need to invert the Laplace transform. In the vector valued setting we use the following result: Let X be a Hilbert space, and let H 2 (C+ , X) be the subset of the space of holomorphic functions defined by H 2 (C+ , X) = { h : C+ → X such that ∫ 2 ||h||H 2 (C+ ,X) = sup ||h(x + is)||2X ds < +∞}. x>0

R

Then we have Theorem 2. (vector valued Paley-Wiener theorem from11 , page 48) Let X be a Hilbert space. Then the map f → fˆ|C is an isometric isomorphism of L2 (R+ , X) onto H 2 (C+ , X). In our situation X = L2 (Ω) and by Theorem 2 estimate (16) yields Corollary 1. We have the following a priori estimate ∫ T∫ {κs |∂t u0 |2 + |e(u0 )|2 + |p0 |2 } dxdt+ 0 Ω ∫ +∞ ∫ ∫ T∫ ∫ t 1 2 −λ1 (t−ζ)/κf 0 p (x, ζ) dζ| dxdt ≤ C (|F|2 + |∂t F|2 ) dxdt. |∇ √ e κf 0 0 Ω 0 Ω

(21)

In order to avoid negative Sobolev norm in time for ∇p0 in (21) we do further calculations. Namely we multiply the unknowns {ˆ u0 , pˆ0 } in (17) and (18) by (λ1 + κf τ ), respectively, and apply the same argument as in Theorem 1 to them. Using estimate (16) we obtain 9

Proposition 1. Under the assumptions of Theorem 1, we have ∫ ∫ C(κf , κs ) 0 2 2 0 2 0 2 ˆ 2 2 . (22) {κs |τ u ˆ | + |e(τ u ˆ )| + |τ pˆ | } dx + |∇ˆ p0 |2 dx ≤ ||(1 + |τ |2 )F|| L (Ω) ℜτ Ω Ω and ∫

T

∫ (

) κs |∂tt u | + |e(∂t u )| + |∂t p | 0 2

0

Ω

0 2

0

2

∫

+∞

C(κf , κs ) 0

∫

T

|∇p0 |2 dxdt ≤

dxdt + 0

2 ∫ ∑

∫ Ω

|∂tm F|2 dxdt.

(23)

Ω

m=0

It is important to control derivatives independently of κf . It is achieved by proving the estimate (16) for {τ u ˆ 0 , τ pˆ0 }. Proposition 2. Let

∫ Ω

F dx = 0, for every t > 0. Under the assumptions of Theorem 1, we

have ∫ {κs |τ 2 u ˆ 0 |2 + |e(τ u ˆ 0 )|2 + |e(ˆ u0 )|2 + |ˆ u0 |2 + Ω

ˆ 22 . τ pˆ | + |ˆ p0 |2 + |∇ˆ p0 |2 } dx ≤ C||(1 + |τ |2 )F|| L (Ω) 0 2

(24)

and ∫

T

∫ {κs |∂tt u0 |2 + |e(∂t u0 )|2 + |e(u0 )|2 + |u0 |2 + |∂t p0 |2 + |∇p0 |2 } dxdt

0

Ω

∫

+∞

≤C 0

2 ∫ ∑ m=0

|∂tm F|2 dxdt.

(25)

Ω

Proof. Arguing as in the proof of Theorem 1, we obtain ∫ ˆ 22 . ℜτ {κs |τ 2 u ˆ 0 |2 + |e(τ u ˆ 0 )|2 + |e(ˆ u0 )|2 + |τ pˆ0 |2 + |ˆ p0 |2 } dx ≤ C||(1 + |τ |2 )F|| L (Ω)

(26)

Ω

∫

ˆ dx = 0, after integrating equation (17) over Ω and using (77), we obtain Since F Ω ∫ u ˆ 0 dx = 0. Korn’s inequality implies Ω

∫

∫ |ˆ u | dx ≤ C

|e(ˆ u0 )|2 dx.

0 2

Ω

Ω 0

Next, we test equation (18) with pˆ and use (26) and (12) to conclude that ∫ λ1 + ℜτ κf ˆ 22 . |∇ˆ p0 |2 dx ≤ C||(1 + |τ |2 )F|| L (Ω) |λ1 + τ κf |2 Ω 10

(27)

0

Finally, testing equation (18) with τ pˆ and using (26) and (13) yields ∫ ˆ 22 . |∇ˆ p0 |2 dx ≤ C||(1 + |τ |2 )F|| L (Ω)

(28)

Ω

 With the estimates (21), (23) and (25) proving existence and uniqueness for the dynamic Biot system (10), (11) with homogeneous initial conditions and periodic boundary conditions, is straightforward. We conclude the Section by stating our main result Theorem 3. Under the assumptions of Theorem 1, the problem (10), (11), supplemented by homogeneous initial conditions and periodic boundary conditions, has a unique variational solution {u0 , p0 } ∈ H 1 ((0, T ) × Ω)3 ∩ H 2 (0, T ; L2 (Ω))3 × H 1 (0, T ; L2 (Ω)) ∩ L2 (0, T ; H 1 (Ω)).

III.

FROM THE DYNAMIC BIOT SYSTEM TO THE QUASI-STATIC

BIOT SYSTEM In this Section we show that the quasi-static Biot equations can be used iteratively to solve the dynamic Biot equations. We study the behavior of the system (10), (11) in the limit κf , κs → 0. Note that κf (and consequently also κs ) stands for the ratio between the intrinsical characteristic time and the characteristic reservoir time scale. In reservoir engineering they are small. An exemple is given at Table II.

For the data from Table II

SYMBOL QUANTITY

CHARACTERISTIC VALUE

Λ

Young’s modulus

7e9 Pa

ρf

fluid density

1e3 kg/m3

ρs

solid grain density

2.65e3 kg/m3

η

fluid viscosity

1e−3 kg/m sec

ℓ

typical pore size

1e−5 m

L

observation length

5000 m

ε

small parameter

ε = ℓ/L = 0.2e−8

Ef

pore fluid bulk modulus 1e6 Pa

TABLE II. Typical reservoir data description

11

one finds Tc = 0.41 days , κf = 0.28e − 8, κs = 0.742e − 8,

κco = O(1) = ψf = O(1) = ψs .

For more details we refer to12 . We note that in simulation of the noise protection tiny poroelastic layers κf = O(1) = κs and one has to consider the full hyperbolic-parabolic Biot-Allard system with memory. Formally in the singular limit κf , κs → 0 one obtains the quasi-static Biot system: − Div {A0 e(uQS )} + Div {αpQS } = ψF(x, t),

(29)

M 0 ∂t pQS + div {K(ψf F − ∇pQS ) + α∂t uQS } = 0

(30)

with periodic boundary conditions and the homogeneous initial condition for the pressure ∫ +∞ pQS |t=0 = 0. Appearance of the Darcy permeability tensor K = A(z) dz is linked to 0

the fact that for every t positive and a bounded continuous function g defined for t ≥ 0, and with values in R3 one has ∫ t ∫ +∞ t − z g(z) lim A( ) dz = ( A(z) dz)g(t) = Kg(t). κf →0 0 κf κf 0

(31)

Proving that the quasi-static Biot system has a unique solution with high regularity in time is straightforward. We have Lemma 1. Let F ∈ C0∞ (R+ ; L20 (Ω)3 ), A0 be a symmetric positive definite 4th order tensor and K be a positive definite symmetric matrix. Assume M 0 is a positive constant and α be a symmetric matrix. Then the problem (29)-(30) has a unique solution {uQS , pQS } ∈ ∫ 1 H k (0, T ; Hper (Ω))4 , Ω uQS dx = 0, for all k ∈ N. In order to obtain the iterative procedure we introduce the unknowns uc =

u0 − uQS κf

and

pc =

p0 − pQS . κf

Using (10), (11) and (29), (30) we see that {uc , pc } satisfy the system ∫ t t−z c 0 c c A( κ∂tt u − Div {A D(u ) − αp } − ∂t )(∇pc (x, z) + κf ∂zz uc (x, z)) dz = κf 0 ( ) ∫ t t−z κ 1 QS QS QS A( − ∂tt u + ∂t ) ∇p + κf ∂zz u − ψf F(x, z) dz, (32) κf κf κf 0 ∫ t t−z 1 0 c M ∂t p − div { A( )( ∇pc + ∂zz uc ) dz} + div {α∂t uc } = κ κf f 0 ∫ t 1 t−z 1 ψf 1 div {K(ψf F − ∇pQS )} + div { A( )( ∇pQS + ∂zz uQS − F) dz}. (33) κf κf κf κf κf 0 with homogeneous initial conditions and periodic boundary conditions. 12

Theorem 4. Let us suppose that the assumptions of Theorem 1 and Lemma 1 are satisfied and in addition assume Aˆ is an analytic function such that ∫ +∞ d ˆ A(0) = − tA(t) dt. (34) dτ 0 Then in the limit κf , κs → 0 we have u0 − uQS → ucor in H 1 ((0, T ) × Ω)3 κf p0 − pQS pc = → pcor in H 1 ((0, T ) × Ω), κf

uc = and

(35) (36)

where {ucor , pcor } is the solution for the problem − Div {A0 e(ucor )} + Div {αpcor } = −(φ + (1 − φ)

ρs )∂tt uQS − ρf

∂t K(ψf F(x, t) − ∇pQS (x, t)),

(37)

M 0 ∂t pcor + div {−K∇pcor + α∂t ucor } = div {K∂tt uQS } ∫ +∞ + div {( zA(z) dz)∂t (ψf F(x, t) − ∇pQS (x, t))}.

(38)

0

ˆ Remark 2. Using K = A(0), we see that (34) implies ∫ t ∫ +∞ 1 t − z g(z) lim { A( ) dz − Kg(t)} = −( τ A(z) dz)∂t g, κf →0 κf κf κf 0 0 for every g ∈ H01 (R+ ). Corollary 2. {uc , pc } = {uQS , pQS }+κf {ucor , pcor }+o(κf ) in the topology of H 1 ((0, T )×Ω)4 . ∫ Proof: First we remark that integrating the equation (17) over Ω gives Ω u0 dx = 0. Application of the Laplace transform to (32) and (33) with τ ∈ C+ yields ˆ κf ))τ 2 u ˆ κf ))ˆ ˆc = − Div {A0 e(ˆ uc )} + Div {(α − τ κf A(τ pc } + (κI − τ κ2f A(τ κ ˆ κf ))τ 2 u ˆ κf )F(x, ˆ κf )ˆ ˆ τ ) + τ div {A(τ −( I − τ κf A(τ ˆ QS − τ ψf A(τ pQS }, κf

(39)

ˆ κf ))τ u ˆ κf )∇ˆ ˆ c } − div {A(τ pc } = M 0 τ pˆc + div {(α − τ κf A(τ −

1 ˆ κf ) − K)(ψf F(x, ˆ κf )τ 2 u ˆ τ ) − ∇ˆ div {(A(τ pQS )} + div {A(τ ˆ QS }. κf

Using the arguments of Proposition 2 we conclude that ∫ {κs |τ 2 u ˆ c |2 + |e(τ u ˆ c )|2 + |e(ˆ uc )|2 + |ˆ uc |2 + |τ pˆc |2 + |ˆ pc |2 + Ω ) ( 2 QS 2 QS 2 c 2 2 2 2 ˆ 2 + ||τ ∇ˆ ˆ ||L2 (Ω) , p ||L2 (Ω) + ||τ u |∇ˆ p | } dx ≤ C(1 + |τ | ) ||F|| L (Ω) 13

(40)

(41)

where C is independent of κf . Therefore the sequence {ˆ uc , pˆc } contains a H 2 (C+ , L2 (Ω))3 weakly convergent subsequence which converges to the quadruple {ˆ uc,0 , pˆc,0 } in the following sense

  τu ˆc ⇀ τ u ˆ c,0 , u ˆc ⇀ u ˆ c,0 and e(τ u ˆ c ) ⇀ e(τ u ˆ c,0 ),  ∇ˆ pc ⇀ ∇ˆ pc,0 , pˆc ⇀ pˆc,0 and τ pˆc ⇀ τ pˆc,0 ,

(42)

as κf → 0. In addition we have the following convergences in H 2 (C+ , L2 (Ω))3 as κf → 0: ρs κ ˆ κf ))τ 2 u I − τ κf A(τ ˆ QS → (φ + (1 − φ) )τ 2 uQS , κf ρf

(43)

ˆ κf )F(x, ˆ ˆ τ ) → ψf A(0)τ ˆ τ ), τ ψf A(τ F(x,

(44)

ˆ κf )ˆ ˆ τ div {A(τ pQS } → div {A(0)τ pˆQS },

(45)

d ˆ 1 ˆ κf ) − K)(ψf F(x, ˆ τ ) − ∇ˆ ˆ τ ) − ∇ˆ pQS )} → div { A(0)τ div {(A(τ pQS )}, (46) (ψf F(x, κf dτ 2 QS ˆ κf )τ 2 u ˆ div {A(τ ˆ QS } → div {A(0)τ u ˆ }.

(47)

Therefore {ˆ uc,0 , pˆc,0 } satisfies the system (37)-(38), with homogeneous initial conditions and periodic boundary conditions. Because of uniqueness for the quasi-static Biot system, we conclude that {ˆ uc,0 , pˆc,0 } = {ˆ ucor , pˆcor } and the convergence of the whole sequence. Strong convergences follows as usual, by using the solutions as test functions.  Remark 3. We see that the quasi-static Biot equations solver could be used to solve the dynamic Biot equations. The quasi-static Biot equations are solved efficiently using iterative coupling procedures, where either the flow or the mechanics is solved first followed by solving the other problem using the latest solution information. We refer to13 for presentation of the four widely used methods and their von Neumann stability analysis. The convergence and convergence rates for two widely used schemes, the undrained split method and the fixed stress split method was recently proved in15 . Remark 4. It is easy to see that {uW , pW } = {uQS + κf ucor , pQS + κf pcor } satisfies at order O(κf ) the system κ∂tt uW − Div {A0 e(uW )} + Div {αpW − κf K∂t pW } = ψF − κf ψf K∂t F, (48) ∫ +∞ M 0 ∂t pW + ∂t div {αuW − κf K∂t uW } − div {K∇pW − κf ( zA(z) dz)∂t ∇pW } = 0 ∫ +∞ ψf div {κf ( zA(z) dz)∂t F − KF}. (49) 0

14

IV.

APPENDIX: ELLIPTICITY ESTIMATES FOR THE DYNAMIC

PERMEABILITY As stated above, the Biot-Allard equations can be obtained using homogenization if one supposes the statistical homogeneity of the pore structure.

In particular we suppose a

periodic porous medium defined by (A1)-(A2). Proposition 3. (see7 ) Let us suppose (A1)-(A2). Then the hypotheses (H1)-(H3) from Theorem 1 are valid. We now demonstrate the properties of the dynamic permeability assumed as the hypothesis (H4) in Theorem 1 are valid. In order to calculate the dynamic permeability, we consider the permeability auxiliary problem: ∂t qi − ∆qi + ∇πqi = 0 in Yf × (0, T ) div qi = 0 in Yf × (0, T ),

qi = 0 on (∂Yf \ ∂Y) × (0, T )

{qi , πqi } are 1-periodic in y,

qi |{t=0} = ei on Yf .

(50) (51) (52)

We note that qi depends only on the geometry. We calculate qi using the corresponding spectral problem: −△w + ∇ρ = λw div w = 0

Yf ,

in

Yf

(53)

w = 0 on ∂Yf \∂Y

(54)

in

w is H 1 (Y) − periodic and ρ is L2 (Y) − periodic.

(55)

By the elementary spectral theory the eigenfunctions for (53)-(55) form an orthonormal 1 basis {f k }k≥1 for L2 (Yf )3 and an orthogonal basis for Vper = {z ∈ Hper : z = 0 on ∂Yf \∂Y 1 = {z ∈ H 1 (Yf )3 : z is H 1 (Y)-periodic }. and div z = 0 in Yf }. Note that Hper

Furthermore λ1 , the minimum eigenvalue of the Stokes operator (53), is positive and λk → +∞, as k → +∞. By the elementary variational parabolic theory we have the following separation of variables expansion for qj : j

q (y, t) =

+∞ ∑

e

−λk t k

∫

f (y) Yf

k=1

fjk (z) dz.

The series (56) converges in C([0, T ]; L2 (Yf )) and in L2 (0, T ; Vper ). 15

(56)

Consequently we have ||qi (t)||L2 (Yf ) ≤ φ1/2 e−λ1 t ,

1 ≤ i ≤ 3,

φ = |Yf |.

(57)

The dynamic permeability matrix is defined by ∫ Aij (t) = ∫

Yf

qji (y, t) dy

(58)

+∞

and Kij = Aˆij (0) =

Aij (t) dt is Darcy’s permeability.

(59)

0

The Laplace transform of the matrix A, given by (58) is ∫ Aˆij (τ ) = ∫ and

Yf

qˆji (y, τ ) dy, ∫

+∞

+∞

tAij (t) dt = 0

0

with Kij = Aˆij (0)

(60)

∫ Yf

qji (y, t)wik (y) dydt.

(61)

We have ∫ ∫ ∫ i j i ˆ q ˆq ˆ dy + ∇y q ˆ i ∇y q ˆj dy = Aij (τ ) = qˆj (y, τ ) dy = τ Yf Yf Y ∫ f ∫ τ q ˆi q ˆj dy + ∇y q ˆ i ∇y q ˆj dy, 1 ≤ i, j ≤ 3, Yf

(62)

Yf

which implies that Aˆ is a complex symmetric matrix (but not a Hermitian matrix). Furthermore, we have ∫ τ Aˆij (τ ) = |τ |2 Let ξ ∈ C3 , let eξ =

∑3

j j=1 ξj e

∫ q ˆi q ˆj

dy + τ

Yf

Yf

and let q ˆξ (τ ) =

∇y q ˆ i ∇y q ˆj dy.

∑3

ˆj (τ ). j=1 ξj q

(63)

Then q ˆξ is the solution to

the problem τq ˆξ − ∆ˆ qξ + ∇ˆ πqξ = eξ in Yf , div q ˆξ = 0 in Yf ,

q ˆξ = 0 on (∂Yf \ ∂Y),

{ˆ qξ , π ˆqξ } are 1-periodic in y.

(64) (65) (66)

We use the spectral basis {f k }k≥1 to write expansions for q ˆξ , wj = q ˆj (0) and for the matrices 16

Aˆ and K:

∫ +∞ ∑ f k (y) eξ · f k (z) dz, q ˆ (y, τ ) = λ + τ k Yf k=1 ∫ +∞ ∑ f k (y) wj (y) = fjk (z) dz, λ k Yf k=1 ∫ +∞ ∑ 1 ˆ )ξξ = eξ · f k (z) dz|2 , A(τ | λ + τ k Yf k=1 ∫ +∞ ∑ 1 and Kξξ = | eξ · f k (z) dz|2 . λ k Yf k=1 ξ

(67) (68) (69) (70)

Proposition 4. Let A be given by (58) and its Laplace transform Aˆ by (60). Then the estimate (15) from Theorem 1 holds true. Furthermore, the matrix Aˆ is positive definite for every τ ∈ C+ = {τ ∈ C : ℜτ > 0} in the sense that the following estimates hold:

∫ +∞ ∑ λk ℜτ 2 |ξ| + | eξ · f k (z) dz|2 , ∀ξ ∈ C3 , 2 2 |λ1 + τ | |λk + τ | Yf k=1

(71)

ˆ )ξξ} ≥ C λ1 + ℜτ |ξ|2 , ∀ξ ∈ C3 , ℜ{A(τ |λ1 + τ |2

(72)

ˆ )ξξ} + ℜ{A(τ ˆ )ξξ} ≥ Kξξ, ∀ξ ∈ C3 . ℜ{τ A(τ

(73)

ˆ )ξξ} ≥ C ℜ{A(τ

Proof:

Estimate (15) is obvious. Next we prove (71): ∫ +∞ ∑ λ + ℜτ k ˆ )ξξ} = | ℜ{A(τ eξ · f k (z) dz|2 ≥ 2 |λ + τ | k Yf k=1 ∫ ∫ +∞ +∞ ∑ λk λ21 ℜτ ∑ 1 ξ k 2 | e · f (z) dz| + | eξ · f k (z) dz|2 , 2 |λ1 + τ |2 k=1 λ2k Yf |λ + τ | k Yf k=1

∀ξ ∈ C3 .

(74)

Estimate (74) implies estimate (71). Next we have ˆ )ξξ} = ℜ{A(τ

∫ +∞ ∑ λ1 + ℜτ λk + ℜτ eξ · f k (z) dz|2 ≥ C | Kξξ. 2 2 |λ |λ k + τ| 1 + τ| Y f k=1

(75)

Since the permeability tensor K is positive definite, (75) implies (72). Finally, the remaining lower bound is

∫ +∞ 2 2 ∑ |τ | + λ ℜτ λ + λ ℜτ 1 k k k ˆ )ξξ} = ℜ{(1 + τ )A(τ ( + )| eξ · f k (z) dz|2 ≥ 2 2 λ |λ + τ | |λ + τ | k k k Yf k=1 ∫ +∞ ∑ 1 | eξ · f k (z) dz|2 ≥ Kξξ. λ k Yf k=1 17

(76)

 Lemma 2. Let the matrix A be given by (58) and let its Laplace’s transform Aˆ be given by ˆ f τ )) is positive definite for every τ ∈ C+ and we have (60). Then the matrix τ κI − τ κ2 A(κ f

ˆ κf ))ξξ} ≥ κs (1 − φ)ℜτ |ξ|2 + ℜ{τ (κI − τ κ2f A(τ ∫ +∞ ∑ λk (Im τ )2 2 κf | eξ · f k (z) dz|2 , ∀ξ ∈ C3 . 2 |λ k + κf τ | Yf k=1 Proof:

(77)

ˆ f τ )ξξ, with τ ∈ C+ : We estimate the sesquilinear form τ 2 κ2f A(κ

∫ +∞ ∑ λk ℜτ 2 + κf ℜτ |τ |2 = | eξ · f k (z) dz|2 = 2 |λ + κ τ | k f Yf k=1 ∫ ∫ +∞ +∞ ∑ ∑ λk κf ℜτ + κ2f |τ |2 λk (Im τ )2 ξ k 2 2 | e · f (z) dz| −κf | eξ · f k (z) dz|2 ≤ (ℜτ κf ) 2 2 |λ + κ τ | |λ + κ τ | k f k f Yf Yf k=1 k=1 ∫ ∫ +∞ +∞ ∑ ∑ λk (Im τ )2 | ℜτ κf eξ · f k (z) dz|2 −κ2f | eξ · f k (z) dz|2 2 |λ + κ τ | k f Yf Yf k=1 k=1 ∫ +∞ ∑ λk (Im τ )2 | = ℜτ φκf |ξ|2 −κ2f eξ · f k (z) dz|2 , ∀ξ ∈ C3 , (78) 2 |λ + κ τ | k f Y f k=1 ˆ f τ )ξξ} ℜ{τ 2 κ2f A(κ

κ2f

which implies (77).  Corollary 3. Estimates (12)-(15) from the hypothesis (H4) of Theorem 1 hold true. Proof:

It remains only to prove (14). Let ξ, β ∈ C3 . Then we have ∫ ˆ ˆ ∇y q ˆ β ∇y q ˆξ dy and τ A(τ )βξ − τ A(τ )ξβ = 2i Im τ Yf

ˆ κf ))ξξ + A(κ ˆ f τ )ββ + (α − τ κf A(τ ˆ κf ))βξ − (α − τ κf A(τ ˆ κf ))ξβ} ≥ ℜ{τ (κI − τ κ2f A(τ ∫ +∞ ∑ ( ℜτ λk 2 2 2 eξ · f k (z) dz|2 + |β| + (κf Im τ ) | κs (1 − φ)ℜτ |ξ| + Cκf 2 |λ1 + κf τ |2 |λ k + κf τ | Yf k=1 ∫ ∫ ∫ | eβ · f k (z) dz|2 − 2 Im τ κf | eξ · f k (z) dz|| eβ · f k (z) dz|), Yf

Yf

Yf

which yields (14). Remark 5. For the definition of the dynamic permeability in a general porous medium we refer to16 . Heuristic estimates for a random porous medium are in17 and18 . For computation of the dynamic permeability for periodic, random and fractal porous media we refer to9 and10 . 18

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20