effective pressure interface law for transport phenomena ... · effective pressure interface law...

Effective pressure interface law fortransport phenomena between an

unconfined fluid and a porousmedium using homogenization

Andro Mikeli c

Departement de Mathematiques, Universite Lyon 1, FRANCE

Joint work with Anna Marciniak-Czochra (IWR and BIOQUANT, Universitat Heidelberg)

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. 1/60

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This research was partially supported by the GNR MOMASCNRS (Modélisation Mathématique et Simulationsnumériques liées aux problèmes de gestion des déchetsnucléaires) (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 surfacewhich separates a channel flow and a porous mediumis a classical problem.

Supposing a laminar incompressible and viscous flow,we find out immediately that the effective flow in aporous solid is described by Darcy’s law. In the freefluid we obviously keep the Navier-Stokes system.Hence we have two completely different systems ofpartial differential equations :

−µ∆u+∇p = f (1)

div u = 0 (2)

in the free fluid domain ΩF andTalk at the Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration”, RICAM, Linz , October 5, 2011 – p. 3/60

P1a

−µvF = K(f −∇p) (3)

div vF = 0 (4)

in the porous medium Ωp.

The orders of the corresponding differential operatorsare different and it is not clear what kind of conditionsone should impose at the interface between the freefluid and the porous part.

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


P2

Pression and the filtration velocity in a porous medium arethe averages over REVs. Consequently one shouldn’t applydirectly the first principles to obtain the interface laws.

CLASSICAL CONDITIONS :

an inviscid fluid : the pressure continuity + continuity of the normalvelocities at the interface Σ

a viscous flow : above conditions + vanishing of the tangentialvelocity 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


P4

i.e. a laminar incompressible flow through a 2D parallelchannel formed by an impermeable upper wall x2 = h and apermeable lower wall x2 = 0. The plane x2 = 0 defines aninterface between the porous medium and the free flow in ahorizontal channel.

A uniform pressure gradient (p0 − pb)/b is maintained inthe longitudinal direction x1 in both the channelΩ1 =]0, b[×]0, h[ and the permeable materialΩ2 =]0, b[×]−H, 0[.

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


P5

Σ

Ω1

x =−L

x =h

b

2

2

0x1

x2

p =p0

p =pb

ε

εu =0

εu =0

ε ε


P6

Beavers et Joseph proposed (and confirmedexperimentally) the following law

∂u1∂x2

(x1, 0) =α√K

(

u1(x1, 0)− vF1 (x1, 0))

(5)

where K is the permeability and vF = (K/µ)∇p is thefiltration velocity.

Analogously to the Poiseuille flow, we solve the problem

vF = −K

µ

p0 − pbb

~e1 = cte dans Ω2 (6)

ρ∂u∂t

+ (u∇)u − µ∆u+∇p = 0 in Ω1 (7)


P7

div u = 0 in Ω1 (8)

∂u1∂x2

=α√K

(

u1 − vF1)

on 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 − pb2µb

1

1 + αh/√K

·(

(1 + αh/√K)x22−

α√Kx2(h

2/K − 2)− h√K(h

√K + 2α)

)

(12)


P8

The mass flow rate M per unit width through channel isthen

M = −(1 + Φ)h3ρ

12µ

p0 − pbb

; σ =h√K

;

Φ =3(2α + σ)

σ(1 + ασ)(13)

The agreement between the measured values in theexperiment by Beavers and Joseph and the predictedvalues for Mexp/(µb) was good, with over 90% of theexperimental values having errors of less than 2%

A ” theoretical ” justification of the Beavers and Josephlaw , at a physical level of rigor, is in the article of P.G.Saffman (Studies in Applied Maths 1971) : He found


P9

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

u1 =

√K

α

∂u1∂x2

+O(K) (14)

He used a statistical approach to extend Darcy’s law tonon-homogeneous porous media and in order todeduce(14), made an ad hoc hypothesis about the representation ofthe averaged interfacial forces as a linear integral functional of thevelocity, with an unknown kernel.

In the article of G. Dagan (Water Resources Research 1979) wehave the same conclusion. He supposed Slattery’s relation betweenthe pressure gradient and the 1st and 2nd ordre derivatives of thefiltration velocity in order to get the law (14).


P9A

A numerical study of the hydrodynamic boundarycondition at the interface between a porous and a plainmedium is in Sahraoui and Kaviany (Int. J. Heat MassTransfer 1992).They calculated the slip coefficient andthey found that the Brinkman extension do notsatisfactory 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 thevolume averaging they obtained at the interface (a)continuity of the velocity and (b) the continuity of the ”modified ” normal stress. In order to perform theaveraging they had to suppose the Brinkman’s flow inthe porous part and a transition layer between twodomains.


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Laws proposed by H. Ene, T. Levy and E.Sanchez-Palencia in the articles Ene andSanchez-Palencia (J. de Mécanique 1975) and Levyand 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’spressure i.e.

[p] = O(√K) on Σ (15)


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Case B: The free fluid velocity and the filtration velocity are of thesame order. Then the pressure gradient is much larger inside theporous body than in the free fluid. The matching conditions to beimposed 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 whenthe characteristic pore size ε → 0 ? Let us note that the asymptoticexpansions in Ω1 and Ω2 are well-known.

If yes, are we able to prove convergence, i.e. to find a relationbetween 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 presentedin the article by Beavers and Joseph, for a periodic porous medium.It was published in the articleW.Jager, A.Mikeli c : On the interface boundary conditions byBeavers, Joseph and Saffman, SIAM J. Appl. Math. , 60 (2000), pp.1111 - 1127.

Construction is based on the results inW.Jager, A.Mikeli c : On the Boundary Conditions at the ContactInterface 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.Jager, A.Mikeli c, N.Neuß : Asymptotic analysis of the laminar viscousflow over a porous bed, SIAM J. on Scientific and Statistical Computing ,Vol. 22 (2001), p. 2006-2028.

For more details seeA.Mikeli c : Homogenization theory and applications to filtrationthrough porous media, chapter in ”Filtration in Porous Media andIndustrial Applications ” , Lecture Notes Centro InternazionaleMatematico Estivo (C.I.M.E.) Series, Lecture Notes in MathematicsVol. 1734, Springer, 2000, p. 127-214.

Formal derivation of the law using a 2-scales asymptotic expansionis explained in the articleW. Jager, A. Mikeli c : Modeling effective interface laws for transportphenomena between an unconfined fluid and a porous mediumusing homogenization, Transport in Porous Media, Volume 78,Number 3, 2009, p. 489-508.


J1

We suppose a periodic porous medium, obtained bytranslations of the cell Y ε = εY , where the squareY = (0, 1)2 contains an open Lipshitz set Z∗, strictlyincluded in Y .

Let YF = Y \ Z∗and let χ be the characteristic function

of YF , extended by periodicity to IR2. We setχε(x) = χ(xε ), x ∈ IR2, and define Ωε

2 byΩε2 = x | x ∈ Ω2, χ

ε(x) = 1. In addition,Ωε = Ω1 ∪ Σ ∪ Ωε

2 is the fluid part of Ω = Ω1 ∪ Σ ∪ Ω2. Wesuppose that (b/ε, L/ε) ∈ IN2. Consequently, our porousmedium contains a large number of channelsperiodically distributed and of the characteristic size ε,being small compared with the characteristic length ofthe macroscopic domain.


J2

Σ

Ω1

Ω2ε

−L

H

x=0y=0 x=b

ε

Y

Y*Z*


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A uniform pressure gradient is maintained in thelongitudinal direction in Ωε, as in the experiment byBeavers and Joseph. More precisely, for a fixed ε > 0,vε, pε are defined by

−µ4vε + 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)

vε2 = 0 on (0 ∪ b)× (−L, h), (22)

pε = p0 on 0 × (−L, h)

and pε = pb on b × (−L, h), (23)


J4

where µ > 0 is the viscosity and p0 and pb are givenconstants.Is there a solution for the problem (18)-(23)? Is it unique ?Is it possible to get uniform a priori estimates with respectto ε?

Let us note that the classical Poiseuille flow in Ω1,satisfying the no-slip condition on Σ, is given by

v0 =

(

pb − p02bµ

x2(x2 − h), 0

)

for 0 ≤ x2 ≤ h,

p0 =pb − p0

bx1 + 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|

µ2h2

8≤

1

16

(

1 +h

b√2

)−1/2max

1

2

√

3

2,

√

b√10

h

.

Then the Poiseuille flow is a unique solution for the ball

B =

z ∈ H1(Ω1)2| ‖z‖L4(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).


J5a

Proposition 1. Suppose that the Reynolds number satisfies

Re =ρ|pb − p0|

µ2h2

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| ‖z‖L4(Ωε)2 ≤

µ

154

√

1

2bh(1 +

h

b√2)−1/2

are equal to vε, pε.


J7

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

‖vε‖L2(Ωε2)2 ≤ Cε

√ε (27)

‖vε‖L2(Σ)2 ≤ Cε (28)

‖vε − v0‖L2(Ω1)2 ≤ Cε (29)

‖pε − p0‖L2(Ω1) ≤ C√ε (30)

Consequently, in the 1st approximation the free flow doesn’t see theporous medium.

In the estimate (26) the principal contribution was coming from thesurface term

∫

Σ ϕ1 and the law by Beavers and Joseph will come withorder ε.

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


Example 1D 1

Let Ω1 = (−∞, 0) and Ω2 = (0,∞). Interface between Ω1 andΩ2 is the point Σ = 0. Let Y = (0, 1) and Z∗ = (0, a),0 < a < 1. Then the "fluid" part of Ω1 is given byΩε1F = ∪∞

k=1ε(a− k, 1− k). The 1D "pore space" is nowΩε = Ωε

1F ∪ Σ ∪ Ω2.Let f ∈ C∞

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

−d2uε

dx2= f(x), in Ωε

uε = 0 on ∂Ωε, lim|x|→+∞

duε

dx= 0.

(31)


Example 1D 2

As in the derivation of Darcy’s law, using the 2-scalesexpansions, we have the following expansion for uε:

uε = −ε2f(x)2 (xε + k)(xε + k + 1− a) + O(ε3),

for − k + a− 1 ≤ x

ε≤ −k, k = 0, 1, . . .

uε =∫ x0 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 linkedthrough the interface Σ = 0. Without an interfacecondition, the approximation in Ω2 is not determined.


Example 1D 3

We search for effective interface conditions at Σ, leading toa good approximation of uε by some ueff .Classical way of finding interface conditions is by usingmatched asymptotic expansions (MMAE). A recentreference in asymptotic methods and boundary layers influid mechanics is the recent book by Zeytounian and forthe detailed explications, we invite reader to consult it andreferences therein.In the language of the MMAE, expansions in Ω1 and Ω2 giveus the outer expansions. We should supplement it by an (local)inner expansion in which the independent variable is stretchedout in order to capture the behavior in the neighborhood ofthe interface.The MMAE approach uses the limit matching rule, by whichasymptotic behavior of the outer expansion


Example 1D 4

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

εα , α > 0. The geometry ofΩε1F obliges us to take α = 1. Then the zero order term in

the expansion is linear in ξ and the limit matching ruleimplies 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 whenx → +∞. (34)


Example 1D 5

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

u0(x) =

∫ x

0

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)


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 = 0approximation for

duε

dxis not good and it differs at order

O(1).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. 31/60

Example 1D 7

Why the approximation deteriorates around the interface? Itis due to the fact that the MMAE method, as it is used inclassical textbooks, does not suit interface problems. Itmatches only the function values at the interface, but notthe values of the normal derivative. This difficulty is noteasy to circumvent because imposing matching of thevalues of the function and its normal derivative leads to anill posed problem for our 2nd order equation.In order to circumvent the difficulty, we propose thefollowing strategy, introduced in the papers Jäger et Mikelic(1996, 1998, 1999a,b, 2000) and Jäger et al (2001).:

1. STEP: We match the function values, as when usingthe MMAE method. In our particular example this means


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 todu0

dx=

∫ +∞

0

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)


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 obtainu1,eff (x) = u0(x) + εw(

x

ε). It is easy to see that


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) anddu0

dx (+0) + dwdy (+0) = du0

dx (+0). Consequently, we impose thefollowing effective interface condition :

ueff (+0) = ε(1− a)dueff

dx(+0) =

√

12k

1− a

dueff

dx(+0). (44)


Example 1D 11

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

−d2ueff

dx2= f, dansΩ2;

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,+∞) = supx≥0

|uε(x)− ueff (x)| ≤ Cε2. (46)

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


CL 1

The correction order ε and the law by Beavers and Joseph

In the estimate (26) the principal contribution was coming from thesurface 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


CL 2

−4yβbl +∇yω

bl = 0 in Z+ ∪ Z− (48)

divyβbl = 0 in Z+ ∪ Z− (49)

[

βbl]

S(·, 0) = 0 on S (50)

[

∇yβbl − ωblIe2

]

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−.


CL 3

S

Z+

Z−

Z1

Z0

Z−1

Z−2

Zbl

y1

y2


CL 4

The theory developed by Jäger et Mikelic in Ann. Sc.Norm. Sup. Pisa 1996 guarantees the existence ofγ0 ∈ (0, 1), Cbl

1 et Cblω such that

eγ0|y2|∇yβbl ∈ L2(ZBL)

4, eγ0|y2|βbl ∈ L2(Z−)2,

eγ0|y2|ωbl ∈ L2(Z−) and

| βbl(y1, y2)− (Cbl1 , 0) |≤ Ce−γ0y2 , y2 > y∗

| ωbl(y1, y2)− Cblω |≤ Ce−γ0y2 , y2 > y∗. .

(54)

In addition, constants Cbl1 and Cbl

ω are given by

Cblω =

∫ 10 ωbl(y1, a) dy1, ∀a ≥ 0

∫ 10 βbl

1 (y1, 0)dy1 =∫ 10 βbl

1 (y1, a)dy1 = Cbl1 < 0

(55)


CL 5

Now we introduce the " 2-scale " velocity by

v(ε) = v0 − βbl,ε ∂v01

∂x2(0) + εCbl

1∂v01∂x2

(0)H(x2)x2he1 (56)

A formal calculation gives

∂v(ε)1∂x2

=∂v01∂x2

(

1− ∂βbl1

∂y2(x

ε)

)

et1

εv(ε)1 = −βbl

1 (x

ε)∂v01∂x2

(57)

Averaging gives the law by Beavers and Joseph

ueff1 = −εCbl1

∂ueff1

∂x2on Σ (58)


CL 6

where ueff is the average of v(ε) and Cbl1 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)Cblω

)

µ∂v01∂x2

(0) (59)

Difficulty : v(ε) doesn’t satisfy the boundary conditionsand we need an exterior boundary layer around(0 ∪ b)×]−H,h[. It was constructed by Jäger andMikelic in the article SIAM J. Appl Math 2000. We skip ithere.


CL 11

Now we introduce the upscaled problem

−µ4ueff + (ueff∇)ueff +∇peff = 0 in Ω1, (67)

div ueff = 0 in Ω1, (68)

ueff = 0 on (0, b)× h, (69)

ueff2 = 0 on (0 ∪ b)× (0, h), (70)

pε = p0 on 0 × (0, h)

and pε = pb on b × (0, h), (71)

ueff2 = 0 and ueff1 + εCbl1

∂ueff1

∂x2= 0 on Σ. (72)


CL 12

Under the hypotheses of Proposition 1, the upscaledproblem has a unique solution

ueff =

(

pb−p02bµ

(

x2 − εCbl1 h

h−εCbl1

)(x2 − h), 0

)

0 ≤ x2 ≤ h,

peff = p0 = pb−p0b x1 + p0 0 ≤ x1 ≤ b.

(73)

The mass flow rate is then

M eff = b

∫ h

0

ueff1 (x2) dx2 = −pb − p012µ

h3h− 4εCbl

1

h− εCbl1

(74)


CL 13

Beavers−Joseph prof ilex =H2

0

− (H − )11

H−ε C H1bl

ε C Hbl

H2

4−11

H−ε C H1bl

ε C Hbl

14

x 1

p −pb 02bµ

p −pb 02bµ

Poiseuilleprofile

01

p −pb 02bµ 2 2v = x (x −H )

x 2

u (x ) =eff,01 2

(x −H) (x − )2 211

H−ε C H1

ε C Hbl

bl

p −pb 02bµ


CL 14

Proposition 8. We have

‖∇(vε − ueff )‖L1(Ω1)4 ≤ Cε| log ε|, (75)

‖vε − ueff‖H1/2−γ(Ω1)2 ≤ Cε3/2| log ε|, γ > 0, (76)

|M ε −M eff | ≤ Cε3/2| log ε|. (77)

Our interface is a mathematical one and it doesn’t exists as a physicalboundary. It is clear that we can take any straight line at the distanceO(ε) from the rigid parts as an interface. It is important to prove that thelaw by Beavers and Joseph doesn’t depend on the position of theinterface. We have


Invariance 1

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

a = (0, 1)× (a,+∞) and

Z−a = ZBL \ (Sa ∪ Z+

a ). Then we have

Ca,bl1 = Cbl

1 − a.

This simple result will imply the invariance of the obtained law on theposition of the interface:

Let Ωaε = (0, b)× (aε, h) for a < 0 and let ua,eff , pa,eff be asolution for (67)-(72) in Ωaε, with (72) replaced by

ua,eff2 = 0 and ua,eff1 +εCa,bl1

∂ua,eff1

∂x2= 0 on Σa = (0, b)×aε

(78)


Invariance 2

The unique solution ua,eff , pa,eff for (67)-(71), (78) isgiven by

ua,eff =(pb − p0

2bµ

(

(x2−aε)2−(x2−aε−εCa,bl1 )

(h− aε)2

h− aε− εCa,bl1

)

, 0)

for aε ≤ x2 ≤ h and

pa,eff = p0 =pb − p0

bx1 + p0 for 0 ≤ x1 ≤ b.

By Lemma 9., Ca,bl1 = Cbl

1 − a and

ua,eff (x) = ueff (x) +O(ε2).


Invariance 3

Therefore, a perturbation of the interface position for anO(ε) implies a perturbation in the solution of O(ε2).Consequently, there is a freedom in fixing position of Σ. Itinfluences the result only at the next order of the asymptoticexpansion.CAVEAT: Nevertheless, with interface position perturbationof order O(ε) , the proportionality constant in the law ofBeavers and Joseph changes also for an O(ε).

In the remaining part of the talk, we suppose that thelogarithmic pollution by the outer boundary layer iseliminated.


Pressure interface conditions 1

Detailed mathematical argument of the discussion whichfollows is in the preprintAnna Marciniak-Czochra, Andro Mikeli c : Effective pressureinterface law for transport phenomena between anunconfined fluid and a porous medium usinghomogenization, preprint, 2011.

and in this talk I will try to explain why the effective pressurehas a jump at the interface.Let us go back and recall the derivation of the law byBeavers and Joseph:

1. STEP Our first approximation was the impermeable

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



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

jump equal to −∂v01∂x2

|Σ. Contrary to the pressure difference,

which could be easily set to zero by the appropriate choiceof the effective porous medium pressure p0, the shearstress jump requires construction of the correspondingboundary layer. The natural stretching variable is given bythe geometry and reads y = x

ε . The correction w, pw isgiven by

−4yw +∇ypw = 0 in Ω1/ε ∪ Ωε2/ε, (79)

divyw = 0 in Ω1/ε ∪ Ωε2/ε, (80)

[

w]

(·, 0) = 0, [pw](·, 0) = 0, and[∂w1

∂y2

]

(·, 0) = ∂v01∂x2

|Σ on 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


Using periodicity of the geometry and independence of∂v01∂x2

|Σ of the fast variable y, we obtain

w(y) =∂v01∂x2

|Σβbl(y) and pw(y) =∂v01∂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 − (Cbl1 , 0) and ωbl − Cbl

ω areexponentially small for y2 > 0.



Stabilization of βbl,ε towards a nonzero constant velocityε(

Cbl1 , 0)

, at the upper boundary, generates a counterflow. Itis given by the following Stokes system in Ω1:

−4zσ +∇pσ = 0 in Ω1, (84)

div zσ = 0 in Ω1, (85)

zσ = 0 on x2 = h and z

σ =∂v01∂x2

|Σe1 on x2 = 0, (86)

In the setting of the experiment by Beavers and Joseph, zσ

was proportional to the two dimensional Couette flowd = (1− x2

h )e1 and pσ = 0.



Now we expected that the approximation for the velocityreads

vε = v0 − (βbl,ε − ε(Cbl

1 , 0))∂v01∂x2

|Σ − εCbl1 z

σ + O(ε2), (87)

Concerning the pressure, there are additional complicationsdue to the stabilization of the boundary layer pressure toCblω , when y2 → +∞. Consequently, the correction in Ω1 is

ωbl,ε −H(x2)Cblω∂v01∂x2

|Σ.

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



[pε]Σ = p0(x1,+0)−p0(x1,−0) = −Cblω∂v01∂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)Cblω

)∂v01∂x2

|Σ−

εCbl1 p

σH(x2) + O(ε). (89)

where the effective porous media pressure p0 is the functionsatisfying

div(

K(f(x)−∇p0)

)

= 0 in Ω2 (90)



p0 = peff + Cblω

∂ueff1

∂x2(x1, 0) on Σ;

K(f(x)−∇p0)|x2=−H · e2 = 0. (91)

We have

Proposition 10

1

ε2(

vε + βbl,ε ∂v01

∂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)



4. STEP Is it possible that the physical pressure pε isdiscontinuous? In the case of the experiment by Beavers

and Joseph it is easy to see that p0 = H(−x2)Cblω

∂v01∂x2

(x1, 0)

and we have

pε = (Cblω − ωbl,ε(x))

∂v01∂x2

(x1, 0) + O(ε).

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

ω to 0.Averaging leads to a discontinuous function.


Open problems

OPEN PROBLEMS :a) 3D non-tangential flows.b) Comparison with the results of Ochoa-Tapia andWhitaker ? They proposed the interface law involvingtangential derivatives of the normal stress and continuity ofvelocities. Then they argued that in most cases it reducesto i) pressure continuity and ii) law by Beavers and Joseph.c) Comparison with the models used by Discacciati, Miglioand Quarteroni (Applied Numerical Mathematics (2002)):

σeffnτ = 0 and σeffnn = −p0 + gravity.

d) Comparison with Payne and Straughan (J. Math. PuresAppl. (1998)):


Open problems

α√3

trKueff1 =

∂ueff1

∂x2+

∂ueff2

∂x1and σeffnn = −p0.

e) Interface laws for Biot’s consolidation theory? Recentpreprint by A. Mikelic and M. F. Wheeler"On the interface law between a deformable porous medium containing aviscous fluid and an elastic body".f) Recent simulations of the coupled flow +reaction-diffusion equations in the presence of the interfaceporous medium/unconfined fluid by T. Carraro, S. Goll, A.Marciniak-Czochra . . . (poster at this conference). Interfacelaws for the heat conducting fluid and the heat conductingsolid structure (P. Bastian, W. Jäger, A. Mikelic, . . . ).


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