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Effective pressure interface law for transport phenomena between an unconfined fluid and a porous medium using homogenization

Andro Mikeli´ c D´ epartement de Math´ ematiques, Universit´ e Lyon 1, FRANCE Joint work with Anna Marciniak-Czochra (IWR and BIOQUANT, Universit¨ at Heidelberg)

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This research was partially supported by the GNR MOMAS CNRS (Modélisation Mathématique et Simulations numériques liées aux problèmes de gestion des déchets nucléaires) (PACEN/CNRS, ANDRA, BRGM, CEA, EDF , IRSN) and by the Romberg professorship at IWR, Universität Heidelberg, 2011-1013.

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• 1. INTRODUCTION

Finding effective boundary conditions at the surface which separates a channel flow and a porous medium is a classical problem. Supposing a laminar incompressible and viscous flow, we find out immediately that the effective flow in a porous solid is described by Darcy’s law. In the free fluid we obviously keep the Navier-Stokes system. Hence we have two completely different systems of partial differential equations :

−µ∆u + ∇p = f

(1)

div u = 0

(2)

in the free fluid domain ΩF and

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−µvF = K(f − ∇p)

(3)

div vF = 0

(4)

in the porous medium Ωp. The orders of the corresponding differential operators are different and it is not clear what kind of conditions

• ne should impose at the interface between the free

fluid and the porous part. We search for the correct interface conditions between a porous medium Ωp and a free fluid ΩF. (Navier-Stokes ⇌ Darcy)

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Pression and the filtration velocity in a porous medium are the averages over REVs. Consequently one shouldn’t apply directly the first principles to obtain the interface laws. CLASSICAL CONDITIONS :

an inviscid fluid : the pressure continuity + continuity of the normal velocities at the interface Σ a viscous flow : above conditions + vanishing of the tangential velocity at the interface Σ.

NON-CLASSICAL CONDITIONS :

Interface condition of Beavers et Joseph(J. Fluid Mech. 1967 ) : We consider a 2D Poiseuille’s flow over a naturally permeable block,

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"""""""""""""""" """""""""""""""" """""""""""""""" """""""""""""""" """""""""""""""" """""""""""""""" """"""""""""""""

porous medium channel

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i.e. a laminar incompressible flow through a 2D parallel channel formed by an impermeable upper wall x2 = h and a permeable lower wall x2 = 0. The plane x2 = 0 defines an interface between the porous medium and the free flow in a horizontal channel. A uniform pressure gradient (p0 − pb)/b is maintained in the longitudinal direction x1 in both the channel

Ω1 =]0, b[×]0, h[ and the permeable material Ω2 =]0, b[×] − H, 0[.

Problem : Find the effective flow in Ω1 ∪ Σ ∪ Ω2.

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Σ Ω1

x =−L

x =h b 2 2 x 1 x 2 p =p0 p =pb

ε

ε u =0 ε u =0 ε ε

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Beavers et Joseph proposed (and confirmed experimentally) the following law

∂u1 ∂x2 (x1, 0) = α √ K

• u1(x1, 0) − vF

1 (x1, 0)

• (5)

where K is the permeability and vF = (K/µ)∇p is the filtration velocity. Analogously to the Poiseuille flow, we solve the problem

vF = −K µ p0 − pb b

• e1 = cte dans Ω2

(6)

ρ{∂u ∂t + (u∇)u} − µ∆u + ∇p = 0 in Ω1

(7)

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div u = 0 in Ω1

(8)

∂u1 ∂x2 = α √ K

• u1 − vF

1

• n Σ

(9)

u2 = 0 on Σ ∪

• {0} ∪ {b}
• ×]0, h[

(10)

p = p0 on {0}×]0, h[; p = pb on {b}×]0, h[

(11)

We find u2 = 0 and

u1 = p0 − pb 2µb 1 1 + αh/ √ K ·

• (1 + αh/

√ K)x2

2−

α √ Kx2(h2/K − 2) − h √ K(h √ K + 2α)

• (12)

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The mass flow rate M per unit width through channel is then

M = −(1 + Φ)h3ρ 12µ p0 − pb b ; σ = h √ K ; Φ = 3(2α + σ) σ(1 + ασ)

(13)

The agreement between the measured values in the experiment by Beavers and Joseph and the predicted values for Mexp/(µb) was good, with over 90% of the experimental values having errors of less than 2% A ” theoretical ” justification of the Beavers and Joseph law , at a physical level of rigor, is in the article of P .G. Saffman (Studies in Applied Maths 1971) : He found

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that the tangential velocity on Σ is proportional to the shear stress i.e.

u1 = √ K α ∂u1 ∂x2 + O(K)

(14)

He used a statistical approach to extend Darcy’s law to non-homogeneous porous media and in order to deduce(14), made an ad hoc hypothesis about the representation of

the averaged interfacial forces as a linear integral functional of the velocity, with an unknown kernel. In the article of G. Dagan (Water Resources Research 1979) we have the same conclusion. He supposed Slattery’s relation between the pressure gradient and the 1st and 2nd ordre derivatives of the filtration velocity in order to get the law (14).

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A numerical study of the hydrodynamic boundary condition at the interface between a porous and a plain medium is in Sahraoui and Kaviany (Int. J. Heat Mass Transfer 1992).They calculated the slip coefficient and they found that the Brinkman extension do not satisfactory model the flow field in the porous medium. Next we have the articles by J.A. Ochoa-Tapia and S. Whitaker (Int. J. Heat Mass Transfer, Vol. 14 (1995), 2635 - 2655 and J. Porous Media 1998). Using the volume averaging they obtained at the interface (a) continuity of the velocity and (b) the continuity of the ” modified ” normal stress. In order to perform the averaging they had to suppose the Brinkman’s flow in the porous part and a transition layer between two domains.

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Laws proposed by H. Ene, T. Levy and E. Sanchez-Palencia in the articles Ene and Sanchez-Palencia (J. de Mécanique 1975) and Levy and Sanchez-Palencia (Int. J. Eng. Sci. 1975):

Case A: The velocity of the free fluid u is much larger than the filtration velocity vF in the porous medium. They concluded that

vF = O( √ K) on Σ. Another condition they found was that the

pressure of the free fluid on Σ could be equal to the Darcy’s pressure i.e.

[p] = O( √ K)

• n Σ

(15)

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Case B: The free fluid velocity and the filtration velocity are of the same order. Then the pressure gradient is much larger inside the porous body than in the free fluid. The matching conditions to be imposed are the following :

u · ν = vF · ν

continuity of the normal velocities on Σ (16)

p = cte on Σ

(17) WHAT ARE THE TRUE INTERFACE CONDITIONS ? Is it possible to find the interface conditions on Σ in the limit when the characteristic pore size ε → 0 ? Let us note that the asymptotic expansions in Ω1 and Ω2 are well-known. If yes, are we able to prove convergence, i.e. to find a relation between the ε-problems and the effective problem when ε → 0?

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JUSTIFICATION OF THE LAW BY BEAVERS AND JOSEPH We will present the justification for the physical situation presented in the article by Beavers and Joseph, for a periodic porous medium. It was published in the article W.J¨ ager, A.Mikeli´ c : On the interface boundary conditions by Beavers, Joseph and Saffman, SIAM J. Appl. Math. , 60 (2000), pp. 1111 - 1127. Construction is based on the results in W.J¨ ager, A.Mikeli´ c : On the Boundary Conditions at the Contact Interface between a Porous Medium and a Free Fluid, Ann. Sc.

• Norm. Super. Pisa, Cl. Sci. - Ser. IV, Vol. XXIII (1996), Fasc. 3, p.

403 - 465. Constants are calculated in

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W.J¨ ager, A.Mikeli´ c, N.Neuß: Asymptotic analysis of the laminar viscous flow over a porous bed, SIAM J. on Scientific and Statistical Computing ,

• Vol. 22 (2001), p. 2006-2028.

For more details see A.Mikeli´ c : Homogenization theory and applications to filtration through porous media, chapter in ”Filtration in Porous Media and Industrial Applications ” , Lecture Notes Centro Internazionale Matematico Estivo (C.I.M.E.) Series, Lecture Notes in Mathematics

• Vol. 1734, Springer, 2000, p. 127-214.

Formal derivation of the law using a 2-scales asymptotic expansion is explained in the article

• W. J¨

ager, A. Mikeli´ c : Modeling effective interface laws for transport phenomena between an unconfined fluid and a porous medium using homogenization, Transport in Porous Media, Volume 78, Number 3, 2009, p. 489-508.

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We suppose a periodic porous medium, obtained by translations of the cell Y ε = εY , where the square

Y = (0, 1)2 contains an open Lipshitz set Z∗, strictly

included in Y . Let YF = Y \ Z

∗ and let χ be the characteristic function

• f YF, extended by periodicity to I
• R2. We set

χε(x) = χ(x

ε), x ∈ I

R2, and define Ωε

2 by

Ωε

2 = {x | x ∈ Ω2, χε(x) = 1}. In addition,

Ωε = Ω1 ∪ Σ ∪ Ωε

2 is the fluid part of Ω = Ω1 ∪ Σ ∪ Ω2. We

suppose that (b/ε, L/ε) ∈ I

• N2. Consequently, our porous

medium contains a large number of channels periodically distributed and of the characteristic size ε, being small compared with the characteristic length of the macroscopic domain.

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Σ Ω1

Ω2

ε

−L

H

x=0 y=0

x=b

ε

Y Y* Z*

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A uniform pressure gradient is maintained in the longitudinal direction in Ωε, as in the experiment by Beavers and Joseph. More precisely, for a fixed ε > 0,

{vε, pε} are defined by −µ△vε + ρ(vε∇)vε + ∇pε = 0 in Ωε,

(18)

div vε = 0 in Ωε,

(19)

vε = 0 on ∂Ωε \ ∂Ω,

(20)

vε = 0 on (0, b) × ({−L} ∪ {h}),

(21)

vε

2 = 0 on ({0} ∪ {b}) × (−L, h),

(22)

pε = p0 on {0} × (−L, h)

and pε = pb on {b} × (−L, h),

(23)

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where µ > 0 is the viscosity and p0 and pb are given constants. Is there a solution for the problem (18)-(23)? Is it unique ? Is it possible to get uniform a priori estimates with respect to ε? Let us note that the classical Poiseuille flow in Ω1, satisfying the no-slip condition on Σ, is given by     

v0 =

pb − p0

2bµ x2(x2 − h), 0

• for 0 ≤ x2 ≤ h,

p0 = pb − p0 b x1 + p0

for 0 ≤ x1 ≤ b.

(24)

We extend this solution to Ω2 by v0 = 0. Suppose that the Reynolds number Re satisfies

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Re = |pb − p0|

µ2 h2 8 ≤ 1 16

• 1 +

h b √ 2

−1/2 max

• 1

2

• 3

2,

• b

√ 10 h

• .

Then the Poiseuille flow is a unique solution for the ball

B =

• z ∈ H1(Ω1)2| zL4(Ω1)2 ≤

µ 4ρ

4

√ 2bh (1 + h b √ 2 )−1/2

Our idea is to construct a solution for the system (18)-(23) as a non-linear perturbation of the Poiseuille’s flow (24).

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Proposition 1.Suppose that the Reynolds number satisfies Re = ρ|pb − p0|

µ2 h2 8 ≤ 3 50

• b

h √ 2

• 1 +

h b √ 2

−1/2

(25) Then for

ε ≤ ε0 = max

b

π 25 12 √ 3 (1 − |Y ∗|), (1 − |Y ∗|) √π · h4L √ 2 (

4

√ 8h2 + 2bL)2, 1 3840 1 − |Y ∗| √ 2π b2 h(Re)2(1 + h b √ 2 )−2

problem (18)-(23) has a solution {vε, pε} ∈ H2(Ωε)2 × H1(Ωε) satisfying

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∇(vε − v0)L2(Ωε)4 ≤ 8

4

√ 2πh2 µb √ b

• 1 − |Y ∗|

|pb − p0|√ε.

(26)

In addition, all solutions contained in the ball

B0 =

• z ∈ H1(Ωε)2| zL4(Ωε)2 ≤

µ 15

4

• 1

2bh(1 + h b √ 2 )−1/2

are equal to {vε, pε}.

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Proposition 2. For the solution to problem (18)-(23), satisfying (26), we have the following a priori estimates :

vεL2(Ωε

2)2 ≤ Cε√ε

(27)

vεL2(Σ)2 ≤ Cε

(28)

vε − v0L2(Ω1)2 ≤ Cε

(29)

pε − p0L2(Ω1) ≤ C√ε

(30) Consequently, in the 1st approximation the free flow doesn’t see the porous medium. In the estimate (26) the principal contribution was coming from the surface term

• Σ ϕ1 and the law by Beavers and Joseph will come with
• rder ε.

We will explain the main ideas on an 1D example.

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Example 1D 1

Let Ω1 = (−∞, 0) and Ω2 = (0, ∞). Interface between Ω1 and

Ω2 is the point Σ = {0}. Let Y = (0, 1) and Z∗ = (0, a), 0 < a < 1. Then the "fluid" part of Ω1 is given by Ωε

1F = ∪∞ k=1ε(a − k, 1 − k). The 1D "pore space" is now

Ωε = Ωε

1F ∪ Σ ∪ Ω2.

Let f ∈ C∞

0 (R) be a given function. We consider the

problem           

−d2uε dx2 = f(x),

in Ωε

uε = 0

• n ∂Ωε,

lim

|x|→+∞

duε dx = 0.

(31)

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Example 1D 2

As in the derivation of Darcy’s law, using the 2-scales expansions, we have the following expansion for uε:               

uε = −ε2f(x)

2

(x

ε + k)(x ε + k + 1 − a) + O(ε3),

for − k + a − 1 ≤ x

ε ≤ −k, k = 0, 1, . . . uε =

x

0 tf(t) dt + x ∞ x f(t) dt + Cε, in Ω2,

(32)

where Cε is an unknown constant. The corresponding "permeability" is k = ε2(1 − a)3/12. Two domains are linked through the interface Σ = {0}. Without an interface condition, the approximation in Ω2 is not determined.

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Example 1D 3

We search for effective interface conditions at Σ, leading to a good approximation of uε by some ueff. Classical way of finding interface conditions is by using matched asymptotic expansions (MMAE). A recent reference in asymptotic methods and boundary layers in fluid mechanics is the recent book by Zeytounian and for the detailed explications, we invite reader to consult it and references therein. In the language of the MMAE, expansions in Ω1 and Ω2 give us the outer expansions. We should supplement it by an (local)

inner expansion in which the independent variable is stretched

• ut in order to capture the behavior in the neighborhood of

the interface. The MMAE approach uses the limit matching rule, by which asymptotic behavior of the outer expansion

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Example 1D 4

in the neighborhood of the interface has to be equal to asymptotic behavior of the inner expansion outside interface. The stretched variable is ξ = x

εα, α > 0. The geometry of

Ωε

1F obliges us to take α = 1. Then the zero order term in

the expansion is linear in ξ and the limit matching rule implies that, at the leading order,

u0 = 0

at the interface

Σ = {0}.

(33)

In Ω1 we have u0 = 0. In Ω2

−d2u0 dx2 = f

and

du0 dx → 0 when x → +∞.

(34)

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Example 1D 5

The system (33)-(34) determines u0. It is easy to find out that

u0(x) =

       x

tf(t) dt + x

∞

x

f(t) dt, x ≥ 0; u0 = 0,

in

Ω1;

(35)

and

uε(x) =

x

ε(a−1)

(t + ε(1 − a))f(t) dt + (x + ε(1 − a))

∞

x

f(t) dt,

for x ≥ −ε(1 − a));

(36)

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Example 1D 6

uε(x) =

x

ε(a−1)−kε

(t + ε(1 − a + k))f(t) dt + (x + ε(k + 1 − a))·

−εk

x

f(t) dt − 1 ε(1 − a)

−εk

ε(a−1)

(t + ε(1 − a))f(t) dt

• ,

for − kε ≥ x ≥ ε(a − 1) − kε, k = 1, 2, . . .

(37)

Now we see that

uε(x) = u0(x) + O(ε)

in

Ω1.

(38)

Nevertheless in the neighborhood of the interface Σ = {0} approximation for duε

dx is not good and it differs at order O(1).

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Example 1D 7

Why the approximation deteriorates around the interface? It is due to the fact that the MMAE method, as it is used in classical textbooks, does not suit interface problems. It matches only the function values at the interface, but not the values of the normal derivative. This difficulty is not easy to circumvent because imposing matching of the values of the function and its normal derivative leads to an ill posed problem for our 2nd order equation. In order to circumvent the difficulty, we propose the following strategy, introduced in the papers Jäger et Mikeli´ c (1996, 1998, 1999a,b, 2000) and Jäger et al (2001).:

• 1. STEP:

We match the function values, as when using the MMAE method. In our particular example this means

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Example 1D 8

that the first approximation u0,eff is given by the problem (33)-(34).

• 2. STEP:

At Σ = {0} we have the derivative jump equal to

du0 dx =

+∞

f(t) dt. Natural stretching variable is given by

the geometry and reads y = x

ε. Therefore, the correction w

is given by

−d2w dy2 = 0 in (0, +∞); [w]Σ = w(+0) − w(−0) = 0;

(39)

[dw dy ]Σ = dw dy (+0) − dw dy (−0) = −du0 dx (+0) on Σ

(40)

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Example 1D 9

−d2w dy2 = 0 in (a−1, 0); w(a−1) = 0; dw dy → 0, y → +∞. (41)

Hence

w(y) =

        

du0 dx (+0)(1 − a),

for

y > 0; du0 dx (+0)(1 − a + y),

for

a − 1 < y ≤ 0. 0,

pour

y ≤ −1.

(42)

We add this correction to u0 and obtain

u1,eff(x) = u0(x) + εw(x ε). It is easy to see that

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Example 1D 10

    

uε(x) = u0(x) + εw(x ε) + O(ε2); duε dx (x) = du0 dx (x) + dw dy (x ε) + O(ε).

(43)

Next we find out that u0(+0) + εw(+0) = ε(1 − a)du0

dx (+0) and du0 dx (+0) + dw dy (+0) = du0 dx (+0). Consequently, we impose the

following effective interface condition :

ueff(+0) = ε(1 − a)dueff dx (+0) =

• 12k

1 − a dueff dx (+0).

(44)

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Example 1D 11

In (0, +∞), ueff satisfies the original PDE:

−d2ueff dx2 = f, dans Ω2; dueff dx → 0, quand x → +∞.

(45)

By easy direct calculation, we calculate the solution ueff for (44)-(45) and find out that

||uε − ueff||L∞(0,+∞) = sup

x≥0

|uε(x) − ueff(x)| ≤ Cε2.

(46)

Clearly, in the case of a porous medium things are more complicated and w should be calculated using the corresponding boundary layer problem.

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CL 1

The correction order ε and the law by Beavers and Joseph In the estimate (26) the principal contribution was coming from the surface term

• Σ ϕ1. In order to eliminate this term, we use the functions

βbl,ε(x) = εβbl(x ε) and ωbl,ε(x) = ωbl(x ε), x ∈ Ωε,

(47) where {βbl, ωbl} are given by

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CL 2

−△yβbl + ∇yωbl = 0 in Z+ ∪ Z−

(48)

divyβbl = 0 in Z+ ∪ Z−

(49)

• βbl

S(·, 0) = 0 on S

(50)

• {∇yβbl − ωblI}e2
• S(·, 0) = e1 sur S

(51)

βbl = 0 on ∪∞

k=1 (∂Z∗ − {0, k}),

(52)

{βbl, ωbl} is y1 − periodic,

(53)

where S = (0, 1) × {0}, Z+ = (0, 1) × (0, +∞),

Z− = (0, 1) × (−∞, 0) \ ∪∞

k=1(Y ∗ − {0, k}) and

ZBL = Z+ ∪ S ∪ Z−.

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CL 3 S Z + Z − Z1 Z0 Z −1 Z −2

Zbl

y1 y2

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 39/60

CL 4

The theory developed by Jäger et Mikeli´ c in Ann. Sc.

• Norm. Sup. Pisa 1996 guarantees the existence of

γ0 ∈ (0, 1), Cbl

1 et Cbl ω such that

eγ0|y2|∇yβbl ∈ L2(ZBL)4, eγ0|y2|βbl ∈ L2(Z−)2, eγ0|y2|ωbl ∈ L2(Z−)

and

• | βbl(y1, y2) − (Cbl

1 , 0) |≤ Ce−γ0y2, y2 > y∗

| ωbl(y1, y2) − Cbl

ω |≤ Ce−γ0y2, y2 > y∗.

.

(54)

In addition, constants Cbl

1 and Cbl ω are given by

• Cbl

ω = 1 0 ωbl(y1, a) dy1, ∀a ≥ 0

1

0 βbl 1 (y1, 0)dy1 = 1 0 βbl 1 (y1, a)dy1 = Cbl 1 < 0

(55)

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 40/60

CL 5

Now we introduce the " 2-scale " velocity by

v(ε) = v0 − βbl,ε ∂v0

1

∂x2 (0) + εCbl

1

∂v0

1

∂x2 (0)H(x2)x2 h e1

(56)

A formal calculation gives

∂v(ε)1 ∂x2 = ∂v0

1

∂x2

• 1 − ∂βbl

1

∂y2 (x ε)

• et 1

εv(ε)1 = −βbl

1 (x

ε)∂v0

1

∂x2

(57)

Averaging gives the law by Beavers and Joseph

ueff

1

= −εCbl

1

∂ueff

1

∂x2

• n Σ

(58)

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 41/60

CL 6

where ueff is the average of v(ε) and Cbl

1 is given by (55).

We’ll rigorously justify (58). Now we define the " 2-scale " pressure " p(ε) by

p(ε) = p0H(x2) + p1,εH(−x2)−

• ωbl,ε − H(x2)Cbl

ω

• µ∂v0

1

∂x2 (0)

(59)

Difficulty : v(ε) doesn’t satisfy the boundary conditions and we need an exterior boundary layer around

({0} ∪ {b})×] − H, h[. It was constructed by Jäger and

Mikeli´ c in the article SIAM J. Appl Math 2000. We skip it here.

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 42/60

CL 10

Theorem 1. Let

Uε(x) = uε − v(ε) − sε ∂v0

1

∂x2 (0)

(60)

Pε = pε − p(ε) − ϑεµ∂v0

1

∂x2 (0),

(61) Then we have the following estimates

∇UεL2(Ωε)4 ≤ Cε| log ε|

(62)

UεL2(Ωε

2)2 ≤ Cε2| log ε|

(63)

UεL2(Σ)2 ≤ Cε3/2| log ε|

(64)

UεL2(Ω1)2 ≤ Cε3/2| log ε|

(65)

PεL2(Ω1) ≤ Cε| log ε|

(66)

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 43/60

CL 11

Now we introduce the upscaled problem

−µ△ueff + (ueff∇)ueff + ∇peff = 0 in Ω1,

(67)

div ueff = 0 in Ω1,

(68)

ueff = 0 on (0, b) × {h},

(69)

ueff

2

= 0 on ({0} ∪ {b}) × (0, h),

(70)

pε = p0 on {0} × (0, h)

and

pε = pb on {b} × (0, h),

(71)

ueff

2

= 0 and ueff

1

+ εCbl

1

∂ueff

1

∂x2 = 0 on Σ.

(72)

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 44/60

CL 12

Under the hypotheses of Proposition 1, the upscaled problem has a unique solution   

ueff =

• pb−p0

2bµ

• x2 −

εCbl

1 h

h−εCbl

1 )(x2 − h), 0

• 0 ≤ x2 ≤ h,

peff = p0 = pb−p0

b

x1 + p0 0 ≤ x1 ≤ b.

(73)

The mass flow rate is then

Meff = b

h

ueff

1

(x2) dx2 = −pb − p0 12µ h3h − 4εCbl

1

h − εCbl

1

(74)

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 45/60

CL 13

Beavers−Joseph profile x =H 2 − (H− ) 1 H−ε C H 1

bl

ε C H

bl

H2 4 − 1 1 H−ε C H 1

bl

ε C H

bl

1 4 x 1 p −p b 2bµ p −p b 2bµ Poiseuille profile 1 p −p b 2bµ 2 2 v = x (x −H) x 2 u (x ) = eff,0 1 2 (x −H) (x − ) 2 2 1 H−ε C H 1 ε C H

bl bl

p −p b 2bµ

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 46/60

CL 14

Proposition 8.We have

∇(vε − ueff)L1(Ω1)4 ≤ Cε| log ε|,

(75)

vε − ueffH1/2−γ(Ω1)2 ≤ Cε3/2| log ε|, γ > 0,

(76)

|Mε − Meff| ≤ Cε3/2| log ε|.

(77) Our interface is a mathematical one and it doesn’t exists as a physical

• boundary. It is clear that we can take any straight line at the distance

O(ε) from the rigid parts as an interface. It is important to prove that the

law by Beavers and Joseph doesn’t depend on the position of the

• interface. We have

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 47/60

Invariance 1

Lemma 9. Let a < 0 and let βa,bl be the solution for (48)-(53) with S replaced by Sa = (0, 1) × {a}, Z+ by Z+

a = (0, 1) × (a, +∞) and

Z−

a = ZBL \ (Sa ∪ Z+ a ). Then we have

Ca,bl

1

= Cbl

1 − a.

This simple result will imply the invariance of the obtained law on the position of the interface: Let Ωaε = (0, b) × (aε, h) for a < 0 and let {ua,eff, pa,eff} be a solution for (67)-(72) in Ωaε, with (72) replaced by

ua,eff

2

= 0

and

ua,eff

1

+εCa,bl

1

∂ua,eff

1

∂x2 = 0

• n

Σa = (0, b)×aε

(78)

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 48/60

Invariance 2

The unique solution {ua,eff, pa,eff} for (67)-(71), (78) is given by

ua,eff =

pb − p0

2bµ

• (x2−aε)2−(x2−aε−εCa,bl

1

) (h − aε)2 h − aε − εCa,bl

1

• , 0
• for aε ≤ x2 ≤ h and

pa,eff = p0 = pb − p0 b x1 + p0

for 0 ≤ x1 ≤ b. By Lemma 9., Ca,bl

1

= Cbl

1 − a and

ua,eff(x) = ueff(x) + O(ε2).

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 49/60

Invariance 3

Therefore, a perturbation of the interface position for an

O(ε) implies a perturbation in the solution of O(ε2).

Consequently, there is a freedom in fixing position of Σ. It influences the result only at the next order of the asymptotic expansion.

CAVEAT: Nevertheless, with interface position perturbation

• f order O(ε) , the proportionality constant in the law of

Beavers and Joseph changes also for an O(ε). In the remaining part of the talk, we suppose that the logarithmic pollution by the outer boundary layer is eliminated.

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 50/60

Pressure interface conditions 1

Detailed mathematical argument of the discussion which follows is in the preprint

Anna Marciniak-Czochra, Andro Mikeli´ c : Effective pressure

interface law for transport phenomena between an unconfined fluid and a porous medium using homogenization, preprint, 2011. and in this talk I will try to explain why the effective pressure has a jump at the interface. Let us go back and recall the derivation of the law by Beavers and Joseph:

• 1. STEP Our first approximation was the impermeable

interface approximation, where we had v0 = 0 on Σ.

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 51/60

Pressure interface conditions 2

• 2. STEP At the interface Σ we obtain the shear stress

jump equal to −∂v0

1

∂x2 |Σ. Contrary to the pressure difference,

which could be easily set to zero by the appropriate choice

• f the effective porous medium pressure ˜

p0, the shear

stress jump requires construction of the corresponding boundary layer. The natural stretching variable is given by the geometry and reads y = x

ε. The correction {w, pw} is

given by

−△yw + ∇ypw = 0

in

Ω1/ε ∪ Ωε

2/ε,

(79)

divyw = 0

in

Ω1/ε ∪ Ωε

2/ε,

(80)

• w
• (·, 0) = 0, [pw](·, 0) = 0, and

∂w1

∂y2

• (·, 0) = ∂v0

1

∂x2 |Σ

• n

Σ/ε,

(81)

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 52/60

Pressure interface conditions 3

Using periodicity of the geometry and independence of

∂v0

1

∂x2|Σ of the fast variable y, we obtain

w(y) = ∂v0

1

∂x2 |Σβbl(y)

and

pw(y) = ∂v0

1

∂x2 |Σωbl(y),

(82)

where {βbl, ωbl} is a boundary layer given by (48)-(53). Now we set

βbl,ε(x) = εβbl(x ε)

and

ωbl,ε(x) = ωbl(x ε), x ∈ Ωε,

(83)

• 3. STEP Here we recall that βbl − (Cbl

1 , 0) and ωbl − Cbl ω are

exponentially small for y2 > 0.

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 53/60

Pressure interface conditions 4

Stabilization of βbl,ε towards a nonzero constant velocity

ε

• Cbl

1 , 0

• , at the upper boundary, generates a counterflow. It

is given by the following Stokes system in Ω1:

−△zσ + ∇pσ = 0

in Ω1,

(84)

div zσ = 0

in Ω1,

(85)

zσ = 0 on {x2 = h} and zσ = ∂v0

1

∂x2 |Σe1 on {x2 = 0},

(86)

In the setting of the experiment by Beavers and Joseph, zσ was proportional to the two dimensional Couette flow

d = (1 − x2

h )e1 and pσ = 0.

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 54/60

Pressure interface conditions 5

Now we expected that the approximation for the velocity reads

vε = v0 − (βbl,ε − ε(Cbl

1 , 0))∂v0 1

∂x2 |Σ − εCbl

1 zσ + O(ε2),

(87)

Concerning the pressure, there are additional complications due to the stabilization of the boundary layer pressure to

Cbl

ω , when y2 → +∞. Consequently, the correction in Ω1 is

ωbl,ε − H(x2)Cbl

ω

∂v0

1

∂x2 |Σ.

At the flat interface Σ, the normal component of the normal stress reduces to the pressure field. Subtraction of the stabilization pressure constant at infinity leads to the pressure jump on Σ:

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 55/60

Pressure interface conditions 6

[pε]Σ = p0(x1, +0)−˜ p0(x1, −0) = −Cbl

ω

∂v0

1

∂x2 |Σ+O(ε) for x1 ∈ (0, L).

(88)

Therefore the pressure approximation is

pε(x) = p0H(x2) + ˜ p0H(−x2) −

• ωbl,ε(x) − H(x2)Cbl

ω

∂v0

1

∂x2 |Σ− εCbl

1 pσH(x2) + O(ε).

(89)

where the effective porous media pressure ˜

p0 is the function

satisfying div

• K(f(x) − ∇˜

p0)

• = 0 in Ω2

(90)

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 56/60

Pressure interface conditions 7

˜ p0 = peff + Cbl

ω

∂ueff

1

∂x2 (x1, 0) on Σ; K(f(x) − ∇˜ p0)|{x2=−H} · e2 = 0.

(91)

We have

Proposition 10

1 ε2

• vε + βbl,ε ∂v0

1

∂x2 (x1, 0)

• − K(f(x) − ∇˜

p0) ⇀ 0

weakly in L2(Ω2)2, as ε → 0;

(92)

pε − ˜ p0 → 0 strongly in L2(Ω2),

as ε → 0;

(93)

||pε − peff||H−1/2(Σ) ≤ C√ε.

(94)

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 57/60

Pressure interface conditions 8

• 4. STEP

Is it possible that the physical pressure pε is discontinuous? In the case of the experiment by Beavers and Joseph it is easy to see that ˜

p0 = H(−x2)Cbl

ω ∂v0

1

∂x2(x1, 0)

and we have

pε = (Cbl

ω − ωbl,ε(x))∂v0 1

∂x2 (x1, 0) + O(ε).

The leading part of pε is a continuous function but changing very rapidly in the neighborhood of Σ from Cbl

ω to 0.

Averaging leads to a discontinuous function.

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 58/60

Open problems

OPEN PROBLEMS : a) 3D non-tangential flows. b) Comparison with the results of Ochoa-Tapia and Whitaker ? They proposed the interface law involving tangential derivatives of the normal stress and continuity of

• velocities. Then they argued that in most cases it reduces

to i) pressure continuity and ii) law by Beavers and Joseph. c) Comparison with the models used by Discacciati, Miglio and Quarteroni (Applied Numerical Mathematics (2002)):

σeffnτ = 0

and

σeffnn = −˜ p0 + gravity.

d) Comparison with Payne and Straughan (J. Math. Pures

• Appl. (1998)):

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 59/60

Open problems

α √ 3 trK ueff

1

= ∂ueff

1

∂x2 + ∂ueff

2

∂x1

and

σeffnn = −˜ p0.

e) Interface laws for Biot’s consolidation theory? Recent preprint by A. Mikeli´ c and M. F . Wheeler "On the interface law between a deformable porous medium containing a

viscous fluid and an elastic body".

f) Recent simulations of the coupled flow + reaction-diffusion equations in the presence of the interface porous medium/unconfined fluid by T. Carraro, S. Goll, A. Marciniak-Czochra . . . (poster at this conference). Interface laws for the heat conducting fluid and the heat conducting solid structure (P . Bastian, W. Jäger, A. Mikeli´ c, . . . ).

Talk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 60/60