Asymptotic expansion for a flow in a periodic porous medium

C. R. Acad. Sci. Paris, t. 325, Sdrie II b, p. 369-374, 1997 Milieux granulaires, sols, milieux poreux/Granularmedia, soils, porousmedia

Asymptotic expansion for a flow in a periodic porous medium Eduard MARUSI(~-PALOKA Department of Mathematics, University of Zagreb, Bijenifika 30, 10000 Zagreb, Croatia.

Abstract.

We consider a fluid flow in a porous medium f2~ with a characteristic pore size g. Starting from the Navier-Slokes equations on the microscopic level and using the homogenization method we lind the correctors for the ,classical Darcy's law governing the macroscopic flow. Keywords: fluid flow / porous medium / homogenization / periodic medium

D$veloppement asymptotique pour un ~coulement clans un milieu poreux p(~riodique Rdsumd.

On consid~re l'&'oulement d'un fluide clans un milieu poreux 0~, de faille caract~ristique des pores ~. A partir de l'dcoulement microscopique ddcrit par le systkme de Navier-Stokes et en utilisant la mdthode d'homogdn~isation, on obtient les correcteurs pour la loi de Darcy classique caractgrisant l'dcoulement global.

Mots elds : ~coulement / milieu poreux / homog~n~isation / milieu p~riodique

Version fran~aise abrdg~e On 6tudie par homogdnrisation l'rcoulement d'un fluide visqueux incompressible dans un milieu poreux pdriodique. L'dcoulement microscopique est rdgi par les 6quations de Navier-Stokes (2). Divers auteurs (Ene et Sanchez-Palencia, 1975 ; Sanchez-Palencia, 1980 ; Marugid-Paloka et Mikelid, 1996), en utilisant des mdthodes diffdrentes, ont montr6 que, pour e << 1 trbs petit, en omettant les termes en O( e ), la loi macroscopique est celle de Darcy donnre par une relation linraire entre le gradient de la pression et la vitesse (7). Le but de ce travail est de tenir compte des effets d'ordre, O ( e ) , O ( e 2) .... et de calculer les con'ecteurs de la loi de Darcy de la forme (1). Deux 6chelles spatiales apparaissent naturellement el on cherchera par consrquent ( u ~ , f f ) sous la forme d'un drveloppement asymptotique (3). On obtient alors une suite cl'6quations (4) drfinissant les termes

Note prdsentde par Evariste SANCHEZ-PALENCIA. 1251-8069/97/03250369 © Acadrmie des Sciences,rElsevier,Paris

369

E. MarugiGPaloka

( u ~,p~) dans le d6veloppement (3). En utilisant la s6paration des variables, on trouve les solutions des 6quations (4) sous la forme •

ue(x,y) :

ff~(Y) (D ~2 ....~ f ~ , ( x ) _ D ~ , p O ( x ) ) Ioq=¢+1

+

~

ff~"(y) D~'p°'2(x) + ..... + ~

I'~1= ~- 1

pe(x, y)

i= 1

ff/(y) OP°'e ( x ) ; Oxi

(El)

ft'~(y) (1) ~ ....'~'f,~,(x) - O°'p°(x) )

=

I,xl

=t+l

+ ~

I~1= e -

~(y)D~p°'~(x) + .... + ~ ~'(y) ~ ( x ) 1

i= 1

..... ~k), I ~ l =OLI-t- - • • probl~me auxiliaire pos6 sur la p6riode OU O ~ = ( ~ 1

/.tA~ '~ + VR ~

-t-Odk, D ' ~ -

2 p 0ff~'~' " ~e ~ + Oy~e

div ~° = - G ; - ~ , _ 1 +

I~1

Oy,~ l

~e

... Oy,~

Oxi

et ( w ~, r(~) est une solution du

..... ~te- 1wOdl

[

dans

= 0 sur S, ( ~", ~'~ ) est Y-p6riodique, [ ~" = 0 ,/

avec H} = La pression dans un syst~me de Stokes est d6finie ~ une constante pr~s. Pour fixer cette constante dans (6) on a pos6 une condition /

#

condition [

d

~i = 0. Cette condition peut ~tre 6videmment remplac6e par la

~i= ci, o6 ci est un nombre r6el quelconque. Mais, selon la d6finition (9), les

fonctions ( w 0, x 'j) d6pendent de cette constante. N6anmoins, on d6montre (proposition 1) que les correcteurs ul(x, y), p2(x, y) donn6s par (13) ne ddpendent pas du choix de la constante c i. De plus, on d6montre l'antisym6trie H~ = - H ~ des coefficients homog6n6is6s d6finissant le correcteur d'ordre O ( e ) [6q. (1)]. Finalement on obtient l'estimation d'erreur (16).

1.

Introduction

The engineering model for a flow of an incompressible viscous fluid in a porous medium is the well-known Darcy's law giving a linear relation between the filtration velocity and the pressure drop. In the case of a periodic porous medium Darcy's law can be rigorously derived using the method of homogenization (Ene and Sanchez-Palencia, 1975; Sanchez-Palencia, 1980; Maru~iGPaloka and

370

Fluid flow in a porous medium

Mikelid, 1996). The justification of that model in all cited references was based on the assumption that the characteristic pore size e is a very small parameter. However in a real-life situation e is sometimes not so small and lower-order effects (O( e ), O( e 2 ), ...) should be taken into account. In the present paper we find the corrections for Darcy's law in the form

v=K(f-Vp)+e

~ ~,j,i=O

Hi~c- -OOxj(f._Ox3~.i) ee+e2

~_2 ~,j,i,k=O

ue "-qk ~ 0 2

( f i _ O Oxi)e~ P + ....

(1)

Moreover, we carry out a qualitative study of the homogenized coefficients H~. Finally, we estimate the error between the exact solution ( u ~, p~) and our correctors.

2. Microscopic equations Let eYk = e(k + Y ) , k e Z ", Y= ]0, 1[ ". The cube Y is divided into two disjoint parts, A, the solid part, ~ , the fluid part, by a smooth surface S. We suppose that the fluid part of the lattice E ~= uk~ z . e(k + ~ ) is connected and has a smooth boundary SL The microscopic equations now read -/.tAu ~ + ( u~ V ) u ~ + Vff =--f, div u ~ = 0 in E ~, u ~ = 0 on S ~

(2)

We have considered the case of an infinite domain in order to avoid the boundary layer effects, which are not the subject of this work (for a treatment of boundary layers consult Marugid-Paloka and Mikelid (1996)). Supposing that f is L-periodic with L/8 = m ~ N the problem (2) admits at least one solution 1 (d',pe)~Hper(~e)nXL2(g-2~.) where f 2 ~ = E ~ n I 2 , I 2 = ] 0 , L[ ". Moreover, for f and[ /J independent of e, and e << 1, that solution is unique.

3. Derivation of the macroscopic model via homogenization Using the technique of a two-scale asymptotic expansion [see Sanchez-Palencia (1980) and Flahvalov and Panasenko (1989)], we seek the solution in the form

Substituting (3) into (2) and collecting equal powers of e we obtain a recursive sequence of equations

I

- I_tAy u¢ + Vy p eÈ l = 2 ~A~, u ~ - l + p A ~ u e - 2 - V x p ¢ -

[-,-j-~

4 (ukVx)ud'

divyue+divxue-l=0inl2x~,

~2 (ukVy)d k+j~-3

(4)

ue=0onsQxS

where we have defined ue= 0 for f < 0. For ~ = 0 we can solve (4) by separating the variables [see for instance Sanchez-Palencia (1980)], i.e. by posing

i= 1

(z--

Oxi / "

i= 1

Oxi J

(5) 371

E. Maru~,iE-Paloka

where ( w ~, rt ~) is the solution of the auxiliary cell problem

- pAw i + Vrt i = ei,

div

0 in ~ ,

Wi =

( w i,

w i = 0 on S,

")T,i )

is Y-periodic

(6)

Taking the average over °3/, equation (5) leads to the classical Darcy's law Vo

? = Jee u°(., y ) dy = K ( f - Vp ° ),

div v° = 0 in f2

(7)

with the symetric and positive permeability tensor K 0 = f wj.i Our next goal is to solve equation (4) for f = 1. We plug in equation (4) an ansatz in the fol~a e

I u l ( x ' Y ) = Z°

0

wij(y)~O-~7--( f ~ ( x ) - O ~ - ( x ) ' ] +

i, j = 1

L p2(x' y )

E

"

OXj ~,,

rtiJ(Y)

OXi

fi(x)-

wi(y) O-P~-(x)

"/

(x)

i= 1

+

i,j = 1

OXi

(8)

xi(y)OpO'l(x) i= 1

OXi

with ( w ij, rt 0 ) being a solution of the second auxiliary problem

I

- ,uAw ij + Vrt 'j = 2 p

'3~j•

Kij - rt' ej, div w';" _ - -~vj,i + T ~ in q/ (9)

w 0 = 0 on S,

( w 0, x0 ) is Y-periodic

Taking again the average over ~ in equation (8) gives the corrector for the macroscopic velocity of the form

Vl=~ ul(x'y)dy=

k

B~Y 3 (

k,i,j=l

~Xj

where the tensor B/5 is defined by B} = ]

fi(x)--

-3P°(x))ek+k OX i

i,j=l

Kij OpO'l aXi ej

(10)

ij ) dy. After integrating over 02/the incompressibility wk(y

d

condition now reads

div(KVp°'l)=

~

B'Sff~-xk-~xj

3xi/

i,j,k=l

which with the L-periodicity condition gives a well-posed elliptic boundary value problem for p0,1. We note now that the pressure rt i in the Stokes system (6) is determined up to a constant. Since x i is appearing on the right-hand side of equation (9) the solution ( w i;, xij ) depends on that constant and so does the tensor B~. LEMMA 1. - Let B k be the rank-three tensor defined above. Then B~=2p

372

f

' ~ 7ow w - K k ' ] - -1~

f

~

B} +

=

2K'f x'

(12)

Fluid flow in a porous medium

Proof - Multiplying the first equation in (9) by wk, the second equation in (9) by xJ, equation (6) by w~j and integrating over @, we obtain

B/5 = 2/-t L - ~ y j w -

e,

=: 2 p ~ ;9.-~jw - ~ - - [

giving equation (12). Second assertion :follows by partial integration.

[]

However, the dependence of B} on f~xJ does not imply the dependence of u ~ and p2 on that constant. PROPOSITION 1. - The definition (8) of the veloci~" corrector u 1 does not depend on the choice of

constant in the auxiliary pressure hi. Moreover

f,(x)- o ; ( x ) , p2(x,y)= i,j = I

(x)-OP°(x) i,j = 1

where ~° = wiJ- wi f

03)

OXi

rg, fta = nij- ni I ~J.

Using the recursive sequence of equations (4) and assuming f E C=( R" ) we can proceed with our calculation and compute u t, pe for any f. If we neglect the inertial term ( u ~ V )u ~ in (2) we can solve (4) using the ansatz (El). Functions p0, e are defined by

- div (KVp °'t) = div { I~1=e+l~-~It=(D ~ 2 ~'ef~, - D~p °) (14) + I~l=dE 1 g~'D~p°'2+ .... + ~!=2 ~ H~D~'P°'e 1)

In the case of the full Navier-Stokes system our ansatz (El) fi?r f > 3 has to be corrected by adding the nonlinear terms containing the products of the derivatives of f - Vp ° and Vp °°. For example in the case t = 3 the term ( u°(x, y ) V s ) u°( x, y ) appears on the right-hand side and (u 3, p4) defined by (El) has to be corrected by adding

U3Ns(X,Y) = ~

OfJ(fe(x)-'opO(x))(fj(x)--O--~(X))

f, j = 1

pN (,y)

2, j= l

OXt

A J(y)

(x)

+2

OXj

0_p2'

OX~, ( x )

w~(y) O---q~(x)

k=l

op°(x ) +

( X ) -- OXj

OXk

k= l

OXk

)

where ( cJ j, A ej ) is the solution of the auxiliary problem

- t.u~o.)eJ+ VAeJ= - ~1 [(weV)v~d + ( w / V ) w t ] ,

divcoej = O i n ~ t

(15) [co ej = 0 on S,

(co ~j, /I ej ) is Y-periodic, f ~ A tj = 0 373

E. Maru~i~-Paloka

Function q0, 1 is defined from the problem

f - d i v ( K V q ° ' l ) = d i v ~ t_k,e,j:l ~

l kM fO=t :

~/c°~(y)dy=M/t'k

M~j(fe-OO-~x°e)(f) -Op°'] Oxj/ ek } ",

k j M,k= 0 Mej+Mkt+

For a detailed study of the inertial effects we refer to Bourgeat et al. (1996).

4. Error estimate

In the last section we prove the error estimate for the first correction of Darcy's law. Using the same method the result can be easily generalized to the higher-order corrections. THEOREM 1. - Denoting opt(x) = (p( x, x ) , we get

u~-{u°+eu~} L2(O~)~+ Ip~- {p° + ep,~}IL2<~,) +

~ - e{u ° + eu~} H~(O~)~<_Ce2

(16)

Proof is based on the same ideas as in Bourgeat et aI. (]L996). Note remise le 14 f6vrier 1997, acceptde le 5 mai 1997.

References Bahvalov N. S., Panasenko G., 1989. Homogenization: Averaging Processes in Periodic Media, Kluwer, Dordrecht. Bourgeat A., Maru~i~-Paloka E., Mikelifi A., 1996. Weak non-linear corrections for Darcy's Law, Math. Models Methods Appl. Sci., 6, 1143-1155. Ene H., Sanchez-Palencia E., 1975. l~quations et phfnom~nes de surface pour l'rcoulement dans un modrle de milieu poreux, J. Mgc., 14, 73-108. Maru~ifi-Paloka E., Mikelifi A., 1996. An error estimate for homogenization of Stokes and Navier-Stokes Equations, Boll.U.M.L, 10-A, 661-671. Sanehez-Palencia E., 1980. Non-Homogeneous Media and Vibration TheoD; Springer LNP 127, Springer-Verlag, Berlin.

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