Homogenization of a pseudoparabolic system

Applicable Analysis Vol. 88, No. 9, September 2009, 1265–1282 Homogenization of a pseudoparabolic system Malgorzata Peszyńska, Ralph Showalter* and Son-Young Yi Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA Communicated by R. Gilbert (Received 3 December 2008; final version received 4 August 2009) Downloaded At: 01:53 6 January 2010 Pseudoparabolic equations in periodic media are homogenized to obtain upscaled limits by asymptotic expansions and two-scale convergence. The limit is characterized and convergence is established in various linear cases for both the classical binary medium model and the highly heterogeneous case. The limit of vanishing time-delay parameter in either medium is included. The double-porosity limit of Richards’ equation with dynamic capillary pressure is obtained. Keywords: homogenization; pseudoparabolic equations; fractured porous media; dynamic capillary pressure AMS Subject Classifications: Primary 35B27; 35K70; Secondary 74Q10; 76S05 1. Introduction Pseudoparabolic equations arise in a range of applications from radiation with timedelay [1], degenerate double-diffusion and heat-conduction models [2,3] and resolution of ill-posed problems [4] through recently developed applications in level set methods [5] and models of lightning propagation [6]. They were first analysed in [7–9]; see [10] for an extensive review and bibliography. Here we are interested in a degenerate pseudoparabolic equation arising from modelling dynamic capillary pressure in unsaturated flow; specifically, we study the case of flow in heterogeneous media in which the coefficients are periodic on a fine scale. The classical Richards equation for flow through a partially saturated porous medium with porosity (x) and permeability K(x) takes the form ðxÞ @uðt, xÞ kw ðuðt, xÞÞ þ r  KðxÞ rðPc ðuðt, xÞÞ  GDðxÞÞ ¼ 0, @t w ð1Þ where u denotes saturation, and gravitational effects depend on depth D(x) and (constant) density . Here kw(u), Pc(u) denote relative permeability and capillary pressure relationships, respectively. This standard model follows from Darcy’s law extended to multiphase flow and conservation of mass [11,12] with the assumption *Corresponding author. Email: show@math.oregonstate.edu ISSN 0003–6811 print/ISSN 1563–504X online ß 2009 Taylor & Francis DOI: 10.1080/00036810903277077 http://www.informaworld.com 1266 M. Peszyńska et al. that atmospheric pressure of air is constant. The model has been analysed in [13–15] and elsewhere. The experimental determination of the pressure–saturation relationship p ¼ Pc(u) is based on the assumption that this is an instantaneous process, although in reality it requires substantial time to approach an equilibrium before measurements can be taken. This led to the introduction of dynamic capillary pressure [16] in which Pc(u) is replaced by Pc,dyn ðuÞ  Pc ðuÞ   @u @t with 40. Other dynamic models had been introduced earlier [17,18]; see [19–22] for supporting experimental evidence. A similar model was derived in [23] by homogenization from standard two-phase models with special interface conditions. The dynamic capillary pressure model of [16] leads to the nonlinear pseudoparabolic equation @uðt, xÞ kw ðuðt, xÞÞ þ r  KðxÞ rðPc ðuðt, xÞÞ  GDðxÞÞ @t w kw ðuðt, xÞÞ @uðt, xÞ ¼ 0:  r  KðxÞ rðxÞ w @t Downloaded At: 01:53 6 January 2010 ðxÞ ð2Þ When written in terms of pressure u ° Pc(u) (see Section 4) and linearized about a  known solution u0, with ðxÞ  KðxÞ kwðuw0 Þ,  replaced by  @u @p ju0 and  by , Equation (2) takes the form   @uðt, xÞ @uðt, xÞ  r  ðxÞr uðt, xÞ þ ðxÞðxÞ ¼ r  ðxÞGDðxÞ: ð3Þ ðxÞ @t @t If the convective term is dropped, i.e. set D(x) ¼ 0, we obtain   @uðt, xÞ @uðt, xÞ  r  ðxÞr uðt, xÞ þ ðxÞðxÞ ðxÞ ¼ 0: @t @t ð4Þ In realistic porous media there is substantial variation of (x) and K(x), as well as the nonlinear relationships kw(), Pc(), () in (2). Consequently the coefficients in linearized models (3) and (4) vary similarly. In this article we derive homogenized models for (2) and (4), and in particular for the special case of binary media in which (x), K(x), (x) and consequently (x) oscillate between two respective constant values. See [24,25] for further discussion of heterogeneous dynamic capillary pressure models, references and numerical results. The multiscale analysis is aided by the structure of the pseudoparabolic system
@uðt, xÞ 1 þ uðt, xÞ  vðt, xÞ ¼ 0, @t ðxÞ

1 vðt, xÞ  uðt, xÞ ¼ 0,  r  ðxÞrvðt, xÞ þ ðxÞ ð5aÞ ðxÞ x2 : ð5bÞ This system is equivalent to a single equation: if we eliminate v we obtain the pseudoparabolic equation (4) for the variable u(t, x); v satisfies a similar equation. It is supplemented with corresponding boundary and initial conditions. Here we take homogeneous Dirichlet boundary conditions vðt, sÞ ¼ 0, a.e. s 2 @ , ð5cÞ Applicable Analysis 1267 and the initial condition ðxÞuð0, xÞ ¼ ðxÞu ðxÞ, a.e. x 2 : ð5dÞ The well-posedness of the system (5) follows from very general assumptions on the coefficients and initial function. The following suffices for our purposes here. THEOREM 1.1 Assume that functions (), (), () 2 L1( ) are given, each with a strictly positive lower bound, and let u() 2 L2( ). Then there is a unique pair u() 2 H1((0, T ); L2( )) and vðÞ 2 L2 ðð0, T Þ; H10 ð ÞÞ such that u(0, ) ¼ u() and Z 

@uðt, xÞ 1 ’ðxÞ þ uðt, xÞ  vðt, xÞ ’ðxÞ  ðxÞ ðxÞ @t ðxÞ  þ ðxÞrvðt, xÞ  r ðxÞ dx ¼ 0 Downloaded At: 01:53 6 January 2010 for all ’() 2 L2( ) and ð6Þ ðÞ 2 H10 ð Þ. Corresponding results hold under much more general conditions of non-negativity of the coefficients. See [10,26–29]. The initial value u need be chosen only with ()1/2u() 2 L2( ). Also, the a priori estimates show explicitly that u  v ! 0 as  ! 0. Our objective is to homogenize the system (5) and thereby the corresponding pseudoparabolic equation (4) when the coefficients depend (periodically) on a small parameter ". The precise description of coefficients will follow below. Bensoussan et al. [30] briefly investigated the homogenization of pseudoparabolic equations as an example for which the limiting problem is of a different type, and perhaps non-local, not even a partial differential equation. (See [30] Chapter II, Section 3.9, pp. 318, 338.) We shall see below that this occurs when certain variables are eliminated or hidden. The limited regularity and estimates for solutions of the corresponding pseudoparabolic equation (4) makes the homogenization more delicate. Only in special cases there is a purely upscaled limit. In Section 2, we obtain the formal asymptotic expansion of the solution for the linear equation (4) in the classical case and find the dependence of the limit on  and . The analysis and homogenization of the linear system (5) by two-scale convergence is developed in Section 3 for "-periodic binary coefficients and includes cases of  ! 0 with parabolic or first-order kinetic systems as limits. Finally, Section 4 contains the asymptotic expansion for a nonlinear highly heterogeneous case arising from Richards’ equation with dynamic capillary pressure. 2. Asymptotic expansion First we introduce periodic coefficients into the pseudoparabolic system (5) and use formal asymptotic expansions to obtain the limiting problem as the period scale "40 tends to zero. Let Y denote the unit cube in RN, let there be given the Y-periodic functions (y), (y), (y) and then define " ðxÞ ¼ ðx" Þ,  " ðxÞ ¼ ðx" Þ, " ðxÞ ¼ ðx" Þ. The three functions ",  ", " are the respective "-periodic coefficients in (5), so the 1268 M. Peszyńska et al. corresponding solution u", v" to (5) depends on ". We write these as formal asymptotic expansions u" ðt, xÞ ¼ 1 X p¼0 " p up ðt, x, yÞ, v" ðt, xÞ ¼ 1 X " p vp ðt, x, yÞ, p¼0 x y¼ , " ð7Þ with each up(t, x, ), vp(t, x, ) being Y-periodic. Substitute (7) into (5) and collect terms by powers " p for p  2. Note that the gradient r ¼ rx þ 1" ry is used in calculations where y ¼ x/". The ordinary differential equation (5a) gives (at p ¼ 0) ð yÞ @u0 ðt, x, yÞ 1 þ ðu0 ðt, x, yÞ  v0 ðt, x, yÞÞ ¼ 0: @t ð yÞ The initial condition will always be assumed to be independent of the local variable, y 2 Y. The procedure for the elliptic equation (5b) is standard [30–32]. Equating to zero the coefficient of "2 in the expansion of (5b) gives Downloaded At: 01:53 6 January 2010 ry  ð yÞry v0 ðt, x, yÞ ¼ 0, y 2 Y: With the Y-periodic boundary conditions on v0, we conclude that ryv0(t, x, y) ¼ 0, and so v0 ¼ v0(t, x) is independent of y 2 Y. From the combined coefficients of "1 in the expansion of (5b) we obtain ry  ð yÞðry v1 ðt, x, yÞ þ rx v0 ðt, xÞÞ  rx  ð yÞry v0 ðt, xÞ ¼ 0: The last term is null, so the function v1(t, x, y) is the solution of an elliptic periodic boundary-value problem on Y, and we can represent it in terms of Y-periodic solutions !j (y) of the cell problem (see (17))
ry  ð yÞ ry !j þ ej ¼ 0, j ¼ 1 . . . N: P @ This representation v1 ðt, x, yÞ ¼ N j¼1 !j ð yÞ @xj v0 ðt, xÞ (up to a function of x) will be  used to compute the effective tensor  below. Finally, collecting terms with "0 in the expansion of (5b) gives  ry  ð yÞðry v2 þ rx v1 Þ  rx  ð yÞðrx v0 ðt, xÞ þ ry v1 ðt, x, yÞÞ 1 ðv0 ðt, xÞ  u0 ðt, x, yÞÞ ¼ 0: þ ð yÞ Integrate this equation over Y. The first term vanishes due to Y-periodicity of each vr, and the second becomes the effective elliptic contribution with the tensor . The third term gets averaged to yield the second equation of the system @u0 ðt, x, yÞ 1 þ ðu0 ðt, x, yÞ  v0 ðt, xÞÞ ¼ 0, @t ð yÞ Z 1 ðv0 ðt, xÞ  u0 ðt, x, yÞÞdy ¼ 0, r   rv0 ðt, xÞ þ Y ð yÞ ð yÞ ð8aÞ ð8bÞ the first being copied from above. The effective tensor  is obtained in this R  calculation as ij ¼ Y ð yÞðry !i ð yÞ þ ei Þ  ðry !j ð yÞ þ ej Þdy. 1269 Applicable Analysis Only if the product ()() is constant we get u0(t, x, y) ¼ u0(t, x) independent of y 2 Y, and in that case we can eliminate v0 from the system to obtain the upscaled pseudoparabolic equation @u0 ðt, xÞ @u0 ðt, xÞ ð9Þ  r   ru0 ðt, xÞ  r   r   ¼ 0: @t @t R  The homogenized porosity is the average R 1 ¼ Y1(y)dy and the homogenized  time-delay is the harmonic average  ¼ ð Y ð yÞ dyÞ . In the general situation, u0 depends also on the local variable y 2 Y, and then the limit system (8) is partially upscaled, a combination of the local equation (8a) and the upscaled (8b). We will make similar but much more interesting calculations below when () and () are piecewise constant.  3. The pseudoparabolic system Downloaded At: 01:53 6 January 2010 Next we extend the models to include binary media of classical or highly heterogeneous type, and then we obtain the homogenized limit problems by two-scale convergence. 3.1. The heterogeneous micro-models We use a binary medium to emphasize the dependence of singularities on geometry. Let the unit cube Y be given in open disjoint complementary parts, Y1 and Y2, so Y1 \ Y2 ¼ ; and Y is the interior of Y1 [ Y2 . We denote by j (y) the characteristic function of Yj for j ¼ 1, 2, extended Y-periodically to all of RN. Thus, 1(y) þ 2(y) ¼ 1 for a.e. y in RN. It is assumed that the sets {y 2 RN : j (y) ¼ 1} for j ¼ 1, 2, have smooth boundary, but we do not require these sets to be connected. The corresponding "-periodic characteristic functions are defined by x "j ðxÞ  j , x 2 RN , j ¼ 1, 2, " and these naturally partition the global domain into two sub-domains, "1 and "2 by "j  fx 2 : "j ðxÞ ¼ 1g, j ¼ 1, 2: We use the characteristic functions as multipliers to denote the zero-extension of various functions. Let  @Y1 \ @Y2 \ Y be the part of the interface between Y1 and Y2 that is interior to the local cell Y. Then "  @ "1 \ @ "2 \ represents the corresponding interface between "1 and "2 that is interior to . We denote by  j the boundary trace of functions on Yj to and by j" the boundary trace of functions on "j to ". (See [29,33].) 3.1.1. The classical case Let the strictly positive lower-bounded functions j (, ), j (, ), j ð, Þ 2 L1 ð ; CðYj ÞÞ be given, and define Y-periodic functions in L1 ð ; L2# ðY ÞÞ by ðx, yÞ  j ðx, yÞ, ðx, yÞ  j ðx, yÞ, ðx, yÞ  j ðx, yÞ, y 2 Yj , j ¼ 1, 2, x2 : The subscript # denotes the subspace of Y-periodic functions in any function space. Corresponding functions on "j are defined by  x  x  x "j ðxÞ  j x, , "j ðxÞ  j x, , j" ðxÞ  j x, , x 2 "j , j ¼ 1, 2, " " " 1270 M. Peszyńska et al. and the coefficients for the pseudoparabolic system (5) are given by " ðxÞ  "1 ðxÞ"1 ðxÞ þ "2 ðxÞ"2 ðxÞ, ð10aÞ " ðxÞ  "1 ðxÞ"1 ðxÞ þ "2 ðxÞ"2 ðxÞ, ð10bÞ  " ðxÞ  "1 ðxÞ1" ðxÞ þ "2 ðxÞ2" ðxÞ: ð10cÞ These are "-periodic on the fine scale. Theorem 1.1 gives a unique solution of the "-problem: u"() 2 H1((0, T ); L2( )) and v" ðÞ 2 L2 ðð0, T Þ; H10 ð ÞÞ satisfy Z 

@u" ðt, xÞ 1 ’ðxÞ þ " u" ðt, xÞ  v" ðt, xÞ ’ðxÞ  ðxÞ " ðxÞ @t  ðxÞ  þ " ðxÞrv" ðt, xÞ  r ðxÞ dx ¼ 0 ð11Þ Downloaded At: 01:53 6 January 2010 for all ’() 2 L2( ) and ðÞ 2 H10 ð Þ, together with the initial condition u"(0, ) ¼ u(). The initial value u is independent of ". If the coefficients "j are continuous on "j , the strong form of (11) is the transmission problem " ðxÞ
@u" ðt, xÞ 1 þ " u" ðt, xÞ  v" ðt, xÞ ¼ 0, @t  ðxÞ x2 ,

1 r  "1 ðxÞrv" ðt, xÞ þ " v" ðt, xÞ  u" ðt, xÞ ¼ 0, 1 ðxÞ

1 r  "2 ðxÞrv" ðt, xÞ þ " v" ðt, xÞ  u" ðt, xÞ ¼ 0, 2 ðxÞ 1" v" ðt, sÞ ¼ 2" v" ðt, sÞ, "1 ðsÞrv" ðt, sÞ  ¼ "2 ðsÞrv" ðt, sÞ  , where denotes the unit outward normal on @ boundary conditions v" ðt, sÞ ¼ 0 " 1. s2 " ð12aÞ x2 " 1, ð12bÞ x2 " 2, ð12cÞ ð12dÞ , ð12eÞ We have homogeneous Dirichlet a.e. s 2 @ , ð12fÞ and the initial condition u"(0, x) ¼ u(x), a.e. x 2 . This is the exact micro-model. If " is continuous on ", there are no interface conditions and (12) reduces to the single system (5) over . Even then, the fine-scale dependence on the coefficients and geometry make it numerically intractable for realistically small values of "40. 3.1.2. The highly heterogeneous case In the highly heterogeneous case, the permeability is scaled by "2 in the second region " 2 " " " 2 , so the flux is given by " 2 ðxÞrv in 2: " ðxÞ  "1 ðxÞ"1 ðxÞ þ "2 "2 ðxÞ"2 ðxÞ: ð13Þ 1271 Applicable Analysis Then the system (11) becomes " ðxÞ
@u" ðt, xÞ 1 þ " u" ðt, xÞ  v" ðt, xÞ ¼ 0, @t  ðxÞ x2 ,

1 r  "1 ðxÞrv" ðt, xÞ þ " v" ðt, xÞ  u" ðt, xÞ ¼ 0, 1 ðxÞ x2

1 r  "2 "2 ðxÞrv" ðt, xÞ þ " v" ðt, xÞ  u" ðt, xÞ ¼ 0, 2 ðxÞ x2 ð14aÞ " 1, " 2, 1" v" ðt, sÞ ¼ 2" v" ðt, sÞ, "1 ðsÞrv" ðt, sÞ  ¼ "2 "2 ðsÞrv" ðt, sÞ  , s2 " ð14bÞ ð14cÞ ð14dÞ : ð14eÞ The "-problem for the model developed by Arbogast et al. [34] is recovered by letting  " ! 0. Downloaded At: 01:53 6 January 2010 3.2. Homogenization of the classical case 3.2.1. The two-scale limit Let the coefficients in (5) be given by (10). Denote the gradient in the y-variable 2 by ry, and use the symbol ‘!’ to denote two-scale convergence [35]. LEMMA 3.1 For each "40, let u"(), v"() denote the unique solution to the pseudoparabolic "-problem (11). These satisfy the estimates ku" kL2 ðð0, T Þ Þ þ kv" kL2 ðð0, T Þ;H1 ð 0 ÞÞ  C, so there exist (i) a function U in L2 ðð0, T Þ  ; L2# ðY ÞÞ, (ii) a function v in L2 ðð0, T Þ; H10 ð ÞÞ, (iii) a function V in L2 ðð0, T Þ  ; H1# ðY Þ=RÞ, and a subsequence, hereafter denoted by u", v", which two-scale converges as follows: 2 u" ! Uðt, x, yÞ, 2 ð15aÞ v" ! vðt, xÞ, ð15bÞ rv" ! rvðt, xÞ þ ry Vðt, x, yÞ: ð15cÞ 2 This suggests use of the corresponding test functions ~ ðxÞ ¼ ðxÞ þ " ðx, x="Þ,
; C1 # ðY Þ . Setting these in (11), we obtain ~ ’ðxÞ ¼ ðx, x="Þ, where 2 H10 ð Þ, , 2 C01 Z

@u" ðt, xÞ 1 ðx, x="Þ þ " " ðxÞ u" ðt, xÞ  v" ðt, xÞ ðx, x="Þ  ð ðxÞ þ " ðx, x="ÞÞ @t  ðxÞ
" " þ  ðxÞrv ðt, xÞ  rð ðxÞ þ " ðx, x="ÞÞ dx ¼ 0: 1272 M. Peszyńska et al. Take the limit as " ! 0 to obtain the two-scale limit system Z Z 

@Uðt, x, yÞ 1 ðx, yÞ þ ðx, yÞ Uðt, x, yÞ  vðt, xÞ ðx, yÞ  ðxÞ @t ðx, yÞ Y 

þ ðx, yÞ rvðt, xÞ þ ry Vðt, x, yÞ  r ðxÞ þ ry ðx, yÞ dy dx ¼ 0, ð16Þ for all , , as above, and U(0, x, y) ¼ u(x). From the uniqueness of the solution of the initial-value problem for (16), it follows that the original sequence u", v" two-scale converges as above. In order to eliminate the function V(t, x, y) from this system, we use the periodic cell problem: for each k ¼ 1, 2, . . . , N, define !k by !k 2 L2 ð ; H1# ðY ÞÞ : Z
ðx, yÞ ry !k ðx, yÞ þ ek  ry ðx, yÞdy ¼ 0 for all 2 L2 ð ; H1# ðY ÞÞ: ð17Þ Downloaded At: 01:53 6 January 2010 Y R y)dy ¼ 0 to fix the constant.) Then we have the (Let us ask that Y !k(x,P @vðt, xÞ representation Vðt, x, yÞ ¼ N Specify similar test functions i¼1 @xi !i ðx, yÞ: PN @ ðxÞ ðx, yÞ ¼ j¼1 @xj !j ðx, yÞ above to obtain the following theorem. THEOREM 3.2 The limits U, v in Lemma 3.1 are the solution of the partially homogenized pseudoparabolic system


U 2 H1 ð0, T Þ; L2 ; L2# ðY Þ , v 2 L2 ð0, T Þ; H10 ð Þ :  Z Z 

@Uðt, x, yÞ 1 ðx, yÞ þ ðx, yÞ Uðt, x, yÞ  vðt, xÞ ðx, yÞ  ðxÞ dy dx @t ðx, yÞ Y ! Z X N
@vðxÞ @ ðxÞ dx ¼ 0, for all  2 L2 ; L2# ðY Þ , 2 H10 ð Þ, ð18Þ ij ðxÞ þ @x @x i j i,j¼1 and U(0, x, y) ¼ u(x), where the effective coefficients are given by Z  ij ðxÞ ¼ ðx, yÞðry !i ðx, yÞ þ ei Þ  ðry !j ðx, yÞ þ ej Þdy: Y 3.2.2. Summary The strong formulation of the system (18) is ðx, yÞ Z
@Uðt, x, yÞ 1 þ Uðt, x, yÞ  vðt, xÞ ¼ 0, @t ðx, yÞ y 2 Y,
1 vðt, xÞ  Uðt, x, yÞ dy  r   rvðt, xÞ ¼ 0: Y ðx, yÞ ð19aÞ ð19bÞ This extends (8) from "-periodic coefficients to those which depend also on the slow variable, x 2 . Consider the case of a binary medium in which each of j,  j 2 L1( ) is independent of y 2 Yj. Then the same is true of  U1 ðt, xÞ, y 2 Y1 , Uðt, x, yÞ  U2 ðt, xÞ, y 2 Y2 , 1273 Applicable Analysis and we have the homogenized binary system jY1 j1 ðxÞ jY2 j2 ðxÞ
@U1 ðt, xÞ jY1 j þ U1 ðt, xÞ  vðt, xÞ ¼ 0, @t 1 ðxÞ ð20aÞ
@U2 ðt, xÞ jY2 j þ U2 ðt, xÞ  vðt, xÞ ¼ 0, @t 2 ðxÞ ð20bÞ
jY2 j
jY1 j vðt, xÞ  U1 ðt, xÞ þ vðt, xÞ  U2 ðt, xÞ  r   rvðt, xÞ ¼ 0: 1 ðxÞ 2 ðxÞ ð20cÞ This is the binary medium analogue of (9). 3.3. Homogenization of the highly heterogeneous case 3.3.1. The two-scale limit Here the permeability is given by (13), so we obtain weaker a priori estimates and correspondingly weaker convergence results. Downloaded At: 01:53 6 January 2010 LEMMA 3.3 For each "40, let u"(), v"() denote the unique solution to the pseudoparabolic "-problem (11). These satisfy the estimates ku" kL2 ðð0, T Þ Þ þ kv" kL2 ðð0, T Þ Þ þ kv" kL2 ðð0, T Þ;H1 ð " ÞÞ 1 þ k"v" kL2 ðð0, T Þ;H1 ð " ÞÞ 2  C, so there exist (i) a function U in L2 ðð0, T Þ  ; L2# ðY ÞÞ, (ii) a function v1 in L2 ðð0, T Þ; H10 ð ÞÞ, (iii) a pair of functions Vj in L2 ðð0, T Þ  ; H1# ðYj Þ=RÞ, j ¼ 1, 2, and a subsequence, hereafter denoted by u", v", which two-scale converges as follows: 2 u" ðt, xÞ ! Uðt, x, yÞ, 2 ð21aÞ "1 v" ! 1 ð yÞv1 ðt, xÞ, ð21bÞ "1 rv" ! 1 ð yÞ½rv1 ðt, xÞ þ ry V1 ðt, x, yÞ , ð21cÞ 2 2 ð21dÞ 2 ð21eÞ "2 v" ! 2 ð yÞV2 ðt, x, yÞ, ""2 rv" ! 2 ð yÞry V2 ðt, x, yÞ: The function V2 satisfies  2(V2(t, x, y) ¼ v1(x), y 2 . (See [36].) These suggest use of the corresponding test functions  : x 2 "1 , 1 ðxÞ þ " 1 ðx, x="Þ ’ðxÞ ¼ ðx, x="Þ, ðxÞ ¼ : x 2 "2 , 2 ðx, x="Þ þ " 1 ðx, x="Þ 1 1 1 1 1 and with where 1 2 H0 ð Þ, , 1 2 C0 ð ; C# ðY ÞÞ 2 2 C0 ð ; C# ðY2 ÞÞ  2 2(x, ) ¼ 1(x) on . Setting these in (11) yields  Z 
@u" ðt, xÞ " ðxÞ " ðx, x="Þ þ "1 " u ðt, xÞ  v" ðt, xÞ ðx, x="Þ  ð 1 ðxÞ þ " 1 ðx, x="ÞÞ @t 1 ðxÞ

"2 ðxÞ " þ " u ðt; xÞ  v" ðt; xÞ ðx; x="Þ  ð 2 ðx; x="Þ þ " 1 ðx; x="ÞÞ 2 ðxÞ 1274 M. Peszyńska et al. þ "1 ðxÞ"1 ðxÞrv" ðt; xÞ  rð þ "2 ðxÞ"2 ðxÞ"rv" ðt; xÞ 1 ðxÞ  "rð þ" 1 ðx; x="ÞÞ 2 ðx; x="Þ þ"
1 ðx; x="ÞÞ dx ¼ 0: Downloaded At: 01:53 6 January 2010 Take the limit as " ! 0 to obtain the two-scale limit system Z Z 
@Uðt, x, yÞ 1 ð yÞ ðx, yÞ þ Uðt, x, yÞ  v1 ðt, xÞ ðx, yÞ  ðx, yÞ @t 1 ðx, yÞ Y

2 ð yÞ þ Uðt, x, yÞ  V2 ðt, xÞ ðx, yÞ  2 ðx, yÞ 2 ðx, yÞ

þ 1 ð yÞ1 ðx, yÞ rv1 ðt, xÞ þ ry V1 ðt, x, yÞ  r 1 ðxÞ þ ry 1 ðx, yÞ  þ 2 ð yÞ2 ðx, yÞry V2 ðt, x, yÞ  ry 2 ðx, yÞ dy dx ¼ 0,
1 ðxÞ ð22Þ for all , 1, 1, 2 as above, and U(0, x, y) ¼ u(x). The uniqueness of the solution to the corresponding initial-value problem shows that the original sequence converges to it. As before, we can represent each V1(t, x, ) by a cell problem: define !k(x, y) by Z
2 1 1 ðx, yÞ ry !k ðx, yÞ þ ek  ry 1 ðx, yÞdy ¼ 0 !k 2 L ð ; H# ðY1 ÞÞ : Y1 Z 2 for all 1 2 L ð ; H1# ðY1 ÞÞ, !k ðx, yÞdy ¼ 0: ð23Þ Y1 PN @v1 ðt, xÞ i¼1 @xi !i ðx, yÞ, Then we have 1 ðt, x, yÞ ¼ PN V @ 1 ðxÞ ðx, yÞ ¼ 1 j¼1 @xj !j ðx, yÞ above to obtain and we specify the test functions THEOREM 3.4 The limits U, v1, V2 in Lemma 3.3 are the solution of the partially homogenized pseudoparabolic system


U 2 H1 ð0, T Þ; L2 ; L2# ðY Þ , v1 2 L2 ð0, T Þ; H10 ð Þ ,
V2 2 L2 ð0, T Þ  ; H1# ðY2 Þ with V2 j ¼ v1 : Z Z

@Uðt, x, yÞ 1 ð yÞ ðx, yÞ þ ðx, yÞ Uðt, x, yÞ  v1 ðt, xÞ ðx, yÞ  1 ðxÞ @t 1 ðx, yÞ Y


2 ð yÞ þ Uðt, x, yÞ  V2 ðt, x, yÞ ðx, yÞ  2 ðx, yÞ dy dx 2 ðx, yÞ  Z X N @v1 ðt, xÞ @ 1 ðxÞ  dx þ ij ðxÞ @xi @xj i,j¼1 Z Z 2 ðx, yÞry V2 ðt, x, yÞ  ry 2 ðx, yÞdy dx ¼ 0, þ Y2

2 for all  2 L2 ; L2# ðY Þ , 1 2 H10 ð Þ, ; H1# ðY2 Þ with  2 j ¼ 1 , 2 2L ð24Þ and U(0, x, y) ¼ u(x), where the effective coefficients are given by Z ij ðxÞ ¼ 1 ðx, yÞðry !i ðx, yÞ þ ei Þ  ðry !j ðx, yÞ þ ej Þdy: Y1 1275 Applicable Analysis Next we separate the components of the system. First write the part over Y2
@Uðt, x, yÞ 1 þ Uðt, x, yÞ  V2 ðt, x, yÞ ¼ 0 and @t 2 ðx, yÞ
V2 ðt, x, yÞ  Uðt, x, yÞ  ry  2 ðx, yÞry V2 ðt, x, yÞ ¼ 0, y 2 Y2 , 2 ðx, yÞ 1 2 ðx, yÞ V2 ðt, x, yÞ ¼ v1 ðt, xÞ, y 2 , and then substitute these back into (24) and use Stokes’ theorem on Y2 to get  Z Z 

@Uðt, x, yÞ 1 ðx, yÞ þ 1 ðx, yÞ Uðt, x, yÞ  v1 ðt, xÞ ðx, yÞ  1 ðxÞ dy dx @t 1 ðx, yÞ Y1  Z Z Z X N @v1 ðt, xÞ @ 1 ðxÞ 2 ðx, yÞry V2 ðt, x, yÞ  dS 1 ðxÞdx ¼ 0: ij ðxÞ dx þ þ @xi @xj i,j¼1 Downloaded At: 01:53 6 January 2010 3.3.2. Summary The strong form of the partially homogenized system (24) is
@Uðt, x, yÞ 1 þ Uðt, x, yÞ  v1 ðt, xÞ ¼ 0, @t 1 ðx, yÞ
1 v1 ðt, xÞ  Uðt, x, yÞ dy  r   rv1 ðt, xÞ Y1 1 ðx, yÞ Z þ 2 ðx, yÞry V2 ðt, x, yÞ  dS ¼ 0, 1 ðx, yÞ Z y 2 Y1 , ð25aÞ and for each x 2 ,
@Uðt, x, yÞ 1 þ Uðt, x, yÞ  V2 ðt, x, yÞ ¼ 0, @t 2 ðx, yÞ
V2 ðt, x, yÞ  Uðt, x, yÞ  ry  2 ðx, yÞry V2 ðt, x, yÞ ¼ 0, y 2 Y2 , 2 ðx, yÞ 1 2 ðx, yÞ V2 ðt, x, yÞ ¼ v1 ðt, xÞ, y2 : ð25bÞ Note the coupling in the system: the function v1 from (25a) is input to (25b), and the total flux from (25b) is the distributed source in (25a). Suppose now that 1 and  1 are independent of y 2 Y1, and therefore so also is u(t, x)  U(t, x, y), y 2 Y1. Then (25a) is homogenized:
@uðt, xÞ 1 þ uðt, xÞ  v1 ðt, xÞ ¼ 0, @t 1 ðxÞ
1 1 r   rv1 ðt, xÞ v1 ðt, xÞ  uðt, xÞ  1 ðxÞ jY1 j Z 1 þ 2 ðx, yÞry V2 ðt, x, yÞ  dS ¼ 0, jY1 j 1 ðxÞ ð26aÞ 1276 M. Peszyńska et al. and for each x 2 ,
@Uðt, x, yÞ 1 þ Uðt, x, yÞ  V2 ðt, x, yÞ ¼ 0, @t 2 ðx, yÞ
V2 ðt, x, yÞ  Uðt, x, yÞ  ry  2 ðx, yÞry V2 ðt, x, yÞ ¼ 0, y 2 Y2 , 2 ðx, yÞ 1 2 ðx, yÞ V2 ðt, x, yÞ ¼ v1 ðt, xÞ, y 2 : ð26bÞ Note that (26a) is the upscaled fissured medium system, and (26b) is the local fissured medium system at each x 2 . Downloaded At: 01:53 6 January 2010 3.4. Vanishing time-delay Suppose that 1" ¼ oð"Þ in the classical system (12). Then ku"  v"kL2(Y1) ¼ o("1/2), so in the limit we obtain U(t, x, y)jY1 ¼ v(t, x). Choose test functions (x, y) ¼ (x) þ " (x, y) in the weak form, with the equations added, and take the limit to get the homogenized mixed parabolic–pseudoparabolic system (compare (20)) Z
@vðt, xÞ 1  r   rvðt, xÞ þ 1 ðxÞ vðt, xÞ  Uðt, x, yÞ dy ¼ 0, ð27aÞ @t Y2 2 ðx, yÞ
@Uðt, x, yÞ 1 þ ð27bÞ Uðt, x, yÞ  vðt, xÞ ¼ 0, y 2 Y2 , @t 2 ðx, yÞ R with effective porosity 1 ðxÞ ¼ Y1 1 ðx, yÞdy. Then (27a) is a parabolic equation with a memory term determined by (27b). See Peszyńska [37] for results and additional references to memory functionals in parabolic equations; also see [31] for first-order kinetic models. Suppose that 1" ¼ oð"Þ in the highly heterogeneous system (14). Then U(t, x, y)jY1 ¼ v1(t, x) and instead of the system (25a) we obtain the homogenized parabolic equation Z @v1 ðt, xÞ  r   rv1 ðt, xÞ þ 2 ðx, yÞry V2 ðx, yÞ  dS ¼ 0: 1 ðxÞ ð28aÞ @t 2 ðx, yÞ Suppose that 2" ¼ oð"Þ in (14). Then U(t, x, y)jY2 ¼ V2(t, x, y) and instead of the system (25b) we obtain the local parabolic equations 2 ðx, yÞ @V2 ðt, x, yÞ  ry  2 ðx, yÞry V2 ðt, x, yÞ ¼ 0, @t V2 ðx, yÞ ¼ v1 ðxÞ, y 2 : y 2 Y2 , ð28bÞ ð28cÞ If both vanish in the limit, then we recover the Arbogast–Douglas–Hornung [34] double-porosity model (28) of a fractured porous medium. 4. Partially saturated flow with dynamic capillary pressure 4.1. Microscopic equations Let us consider the unsaturated flow in a highly heterogeneous medium with the "-periodic structure of Section 3. Here Y2 is the matrix block and Y1 is the 1277 Applicable Analysis surrounding fracture domain. Each of the subdomains "i is characterized by a rock permeability tensor Ki, a porosity i, the relative permeability kiw ðui Þ and the capillary pressure function Pic ðui Þ. Here ui denotes the saturation in "i . The fluid has constant viscosity  and density . It has been observed that the dynamic effects in capillary pressure equilibrium are much more significant in media with low conductivity than those with high conductivity, so we assume that the unsaturated flow can be locally described by the original Richards equation (1) in the fracture domain "1 and by the pseudoparabolic Richards equation (2) in the porous matrix "2 :
@u1 1 þ r  K1 k1w ðu1 Þr P1c ðu1 Þ  GDðxÞ ¼ 0, x 2 "1 ,  @t   2 1 2 2 2 @u2 2 2 2 2 @u þ " r  K kw ðu Þr Pc ðu Þ    GDðxÞ ¼ 0, x 2   @t @t 1 ð29aÞ " 2: ð29bÞ Downloaded At: 01:53 6 January 2010 Hereafter for simplicity we set depth D(x) ¼ x3. Introduce pi ¼ Pic ðui Þ, ui ¼ i ð pi Þ, i ð pi Þ ¼ 1 Ki kiw ðui Þ, so i() is inverse to Pic ðÞ, and Equations (29a) and (29b) can be rewritten as 1 2
@ 1 ð p1 Þ  r  1 ð p1 Þ rp1 þ Ge3 ¼ 0, @t ð30aÞ
@ 2 ð p2 Þ @ 2 ð p2 Þ  "2 r  2 ð p2 Þ rp2 þ r þ Ge3 ¼ 0, @t @t ð30bÞ and are subject to the interface conditions p1 ¼ p2 þ  @ 2 ð p2 Þ , @t x2 " ð30cÞ ,  
@ 2 ð p2 Þ þ Ge3  , 1 ð p1 Þ rp1 þ Ge3  ¼ "2 2 ð p2 Þ rp2 þ r @t where " out of " 2, p ðx, 0Þ ¼ pi ðxÞ, x2 is the unit normal on i x2 " , ð30dÞ and the initial conditions are " i, ð30eÞ i ¼ 1, 2: 4.2. Asymptotic expansions We shall expand the solution in powers of " in the form pi ðt, xÞ ¼ pi0 ðt, x, yÞ þ "pi1 ðt, x, yÞ þ "2 pi2 ðt, x, yÞ þ    , i ¼ 1, 2, ð31Þ where pik are Y-periodic in y 2 Yi for k ¼ 0, 1, 2, . . . . Following methods of [38,39], we develop various nonlinear quantities (p) in powers of " by ð pi Þ ¼ ð pi0 Þ þ 0 ð pi0 Þð pi  pi0 Þ þ 00 ð pi0 Þð pi  pi0 Þ2 =2 þ    ¼ ð pi0 Þ þ 0 ð pi0 Þð"pi1 þ "2 pi2 þ   Þ þ 00 ð pi0 Þð"pi1 þ "2 pi2 þ   Þ2 =2 þ    ¼ ð pi0 Þ þ " 0 ð pi0 Þ pi1 þ "2 ð 0 ð pi0 Þ pi2 þ 00 ð pi0 Þð pi1 Þ2 =2Þ þ    ¼ ð pi0 Þ þ " ^1i þ "2 ^2i þ    , for appropriate ^1i , ^2i , . . . , i ¼ 1, 2: 1278 M. Peszyńska et al. Now, we substitute (31) into the microscopic model and expand the gradient according to the relation r ¼ rx þ 1" ry . Then, we collect terms by powers of ". From (30a) we obtain three equations for the combined "2, "1 and "0 terms when x 2 , y 2 Y1:
ry  1 ð p10 Þry p10 ¼ 0, ð32aÞ  
ry  1 ð p10 Þðrx p10 þ ry p11 þ Ge3 Þ þ ^ 11 ry p10 þ rx  1 ð p10 Þry p10 ¼ 0, 1   @ 1 ð p10 Þ  rx  1 ð p10 Þðrx p10 þ ry p11 þ Ge3 Þ þ ^ 11 ry p10  ry  1 ð p10 Þðrx p11 þ ry p12 Þ @t  þ ^ 11 ðrx p10 þ ry p11 þ Ge3 Þ þ ^ 12 ry p10 ¼ 0: ð32cÞ First, equations for "0 from (30b) and (30c) for x 2 are   @ 2 ð p20 Þ @ 2 ð p20 Þ 2 2 2  ry   ð p0 Þry p0 þ   ¼ 0, @t @t 2 Downloaded At: 01:53 6 January 2010 ð32bÞ p20 þ  @ 2 ð p20 Þ ¼ p10 , @t y 2 Y2 , ð33bÞ y2 : The "1, "0 and "1 equations of (30d) for x 2 , y 2 ð33aÞ are 1 ð p10 Þry p10  ¼ 0, ð34aÞ   1 ð p10 Þðrx p10 þ ry p11 þ Ge3 Þ þ ^ 11 ry p10  ¼ 0, ð34bÞ   1 ð p10 Þðrx p11 þ ry p12 Þ þ ^ 11 ðrx p10 þ ry p11 þ Ge3 Þ þ ^ 12 ry p10    @ 2 ð p20 Þ ¼ 2 ð p20 Þry p20 þ   : @t ð34cÞ Equations (32a) and (34a) form an elliptic system for p10 in terms of y. Since its solution is independent of y, it follows that p10 ¼ p10 ðt, xÞ, so all terms with ry p10 vanish. Equations (32b) and (34b) form a linear elliptic system in y whose solution p11 can be represented in terms of p10 . Define !j (y) for j ¼ 1, 2, 3 as the Y-periodic solution of the cell problem (compare (23)) ry2 !j ¼ 0 for y 2 Y1 , ry !j  ¼ ej  ¼  j ð35aÞ for y 2 : ð35bÞ Then from Equation (32b) we obtain the representation p11 ðx, y, tÞ ¼ 3 X j¼1 !j ð yÞ  1 @p0 ðx, tÞ þ G @xj 3j  : ð36Þ 1279 Applicable Analysis Now, we locally average (32c) by integrating it over Y1 to remove the y-variable and get Z @ 1 ð p10 Þ  jY1 j rx  1 ð p10 Þðrx p10 þ ry p11 þ Ge3 Þdy @t Y1 Z   1 1 ¼ ry   ð p0 Þðrx p11 þ ry p12 Þ þ ^ 11 ðrx p10 þ ry p11 þ Ge3 Þ dy: 1 ð37Þ Y1 Apply the divergence theorem to the second integral above, use (34c), make a second application of the divergence theorem, and use (33a) to obtain Z   ry  1 ð p10 Þðrx p11 þ ry p12 Þ þ ^ 11 ðrx p10 þ ry p11 þ Ge3 Þ dy Y1 Z   ¼ 1 ð p10 Þðrx p11 þ ry p12 Þ þ ^ 11 ðrx p10 þ ry p11 þ Ge3 Þ  dS @Y1  @ 2 ð p20 Þ ¼  dS  þ @t @Y2   Z @ 2 ð p20 Þ ¼ ry  2 ð p20 Þry p20 þ  dy @t Y2 Z @ 2 ð p20 Þ dy: ¼ 2 @t Y2 Downloaded At: 01:53 6 January 2010 Z 2 ð p20 Þry  p20 The first integral in (37) is evaluated using (36). Its integrand becomes (with implied summation) rx  1 ð p10 Þðrx p10 þ ry p11 þ Ge3 Þ     1 @ @p0 @!j @p10 1 1 ¼ þ þ G 3j þ G  ð p0 Þ @xk @xk @yk @xj   1   @ @!j @p0 1 1 þ jk  ð p0 Þ þ G 3j : ¼ @xk @yk @xj 3k  Define the effective fracture permeability tensor K ¼ fKjk g and the macroscopic fracture porosity  by Kjk 1 ¼K Z  Y1 @!j þ @yk jk  dy,  ¼ jY1 j1 : We also define  ð pÞ ¼ 1  1 1 K kw ð ð pÞÞ:  Then, the equation for p10 is @ 1 ð p10 Þ  rx   ð p10 Þðrx p10 þ Ge3 Þ ¼   @t  Z Y2 2 @ 2 ð p20 Þ dy: @t 1280 M. Peszyńska et al. 4.3. Summary The complete system of flow equations for p10 ðx, tÞ, p20 ðx, y, tÞ is given by Z @ 1 ð p10 Þ @ 2 ð p20 Þ  þ dy  rx   ð p10 Þðrx p10 þ Ge3 Þ ¼ 0, x 2 , 2 @t @t Y2   @ 2 ð p20 Þ @ 2 ð p20 Þ 2  ry  2 ð p20 Þry p20 þ  ¼ 0, y 2 Y2 , @t @t p20 þ  @ 2 ð p20 Þ ¼ p10 , @t p10 ðx, 0Þ ¼ p1init ðxÞ, ð38bÞ ð38cÞ y2 , p20 ðx, y, 0Þ ¼ p2init ðxÞ, ð38aÞ y 2 Y2 : ð38dÞ Downloaded At: 01:53 6 January 2010 This is the double-porosity model consisting of the upscaled equation (38a) together with the distributed family of local boundary-value problems (38b), (38c) for x 2 . It is a nonlinear analogue of the system (28a), (26b). References [1] E. Milne, The diffusion of imprisoned radiation through a gas, J. London Math. Soc. 1 (1926), pp. 40–51. [2] G.I. Barenblatt, I.P. Zheltov, and I.N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata), J. Appl. Math. Mech. 24 (1960), pp. 1286–1303. [3] L.I. 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