TWO-SCALE CONVERGENCE OF A MODEL FOR FLOW IN A PARTIALLY FISSURED MEDIUM

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1 Electronic Journal of Differential Equations, Vol ), No., pp. 1. ISSN: URL: or ftp ejde.math.swt.edu ejde.math.unt.edu login: ftp) TWO-SCALE CONVERGENCE OF A MODEL FOR FLOW IN A PARTIALLY FISSURED MEDIUM G. W. CLARK & R.E. SHOWALTER Abstract. The distributed-microstructure model for the flow of single phase fluid in a partially fissured composite medium due to Douglas-Peszyńska- Showalter [1] is extended to a quasi-linear version. This model contains the geometry of the local cells distributed throughout the medium, the flux exchange across their intricate interface with the imbedded fissure system, and the secondary flux resulting from diffusion paths within the matrix. Both the exact but highly singular micro-model and the macro-model are shown to be well-posed, and it is proved that the solution of the micro-model is two-scale convergent to that of the macro-model as the spatial parameter goes to zero. In the linear case, the effective coefficients are obtained by a partial decoupling of the homogenized system. 1. Introduction A fissured medium is a structure consisting of a porous and permeable matrix which is interlaced on a fine scale by a system of highly permeable fissures. The majority of fluid transport will occur along flow paths through the fissure system, and the relative volume and storage capacity of the porous matrix is much larger than that of the fissure system. When the system of fissures is so well developed that the matrix is broken into individual blocks or cells that are isolated from each other, there is consequently no flow directly from cell to cell, but only an exchange of fluid between each cell and the surrounding fissure system. This is the totally fissured case that arises in the modeling of granular materials. In the more general partially fissured case of composite media, not only the fissure system but also the matrix of cells may be connected, so there is some flow directly within the cell matrix. The developments below concern this more general model with the additional component of a global flow through the matrix. An exact microscopic model of flow in a fissured medium treats the regions occupied by the fissure system and by the porous matrix as two Darcy media with different physical parameters. The resulting discontinuities in the parameter values 1991 Mathematics Subject Classification. 35A15, 35B7, 76S5. Key words and phrases. fissured medium, homogenization, two-scale convergence, dual permeability, modeling, microstructure. c 1999 Southwest Texas State University and University of North Texas. Submitted October 8, Published January 14,

2 G. W. CLARK & R. E. SHOWALTER EJDE 1999/ across the matrix-fissure interface are severe, and the characteristic width of the fissures is very small in comparison with the size of the matrix blocks. Consequently, any such exact microscopic model, written as a classical interface problem, is numerically and analytically intractable. For the case of a totally fissured medium, these difficulties were overcome by constructing models which describe the flow on two scales, macroscopic and microscopic; see [, 4, 5, 13, 3]. A macro-model for flow in a totally fissured medium was obtained as the limit of an exact micro-model with properly chosen scaling of permeability in the porous matrix. It is an example of a distributed microstructure model. Derivations of these two scale models have been based on averaging over the exact geometry of the region see [, 3]) or by the construction of a continuous distribution of blocks over the region as in [3] or by assuming some periodic structure for the domain that permits the use of homogenization methods [8, 9]. See [15] or [16] for a review, and for more information on homogenization see [7, 1].) This model was extended in [1] to the partially fissured case. The novelty in this construction was to represent the flow in the matrix by a parallel construction in the style of [6, 4]. Thus, two flows are introduced in the exact micro-model for the matrix, one is the slow scale flow of [5] which leads to local storage, and the additional one is the global flow within the matrix. A formal asymptotic expansion was used in [1] to derive the corresponding distributed microstructure model. See [1, 11] for another approach to modeling flow in a partially fissured medium and [15] for further discussion and related works. Here we extend the considerations to a quasi-linear version, and we use two-scale convergence to prove the convergence of the micro-model to the corresponding macro-model. Our plan for this project is as follows. In the remainder of this section, we briefly recall the partial differential equations that describe the flow through a homogeneous medium in order to introduce some notation. Then we describe in turn various function spaces of L p or of Sobolev type, the two-scale convergence procedure, and basic results for weak and strong formulations of the Cauchy problem in Banach space. In Section we describe a nonlinear version of the micro-model from [1] for flow through a partially fissured medium and show that this system leads to a well-posed initial-boundary-value problem. In Section 3 we show that this micro-model has a two-scale limit as the parameter, and this limit satisfies a variational identity. The point of Section 4 is to establish that this limit satisfies additional properties which collectively comprise the homogenized macromodel. These results on the well-posedness of the macro-model are sumarized and completed in Section 5. There we relate the weak and strong formulations of the macro-model problem to the corresponding realizations as a Cauchy problem for a nonlinear evolution equation in Banach space. We also develop a simpler and useful reduced system to describe this limit, and we show that it agrees with the usual homogenized model from [1] in the linear case. The authors would like to acknowledge the considerable benefit obtained from discussions with M. Peszyńska [1, 16, 17, 18, 19, ] on the homogenization method for modeling of flow through porous media. These led to many substantial improvements in the manuscript. We begin with a review of notation in the context of the flow of a single phase slightly compressible liquid through a homogeneous medium. Thus the density ρx, t) and pressure px, t) are related by the state equation ρ = ρ e κp,andthe

3 EJDE 1999/ FLOW IN A PARTIALLY FISSURED MEDIUM 3 equation for conservation of mass is given by cx) ρ N t ρk j ρ p ) p )=fx, t). x j x j = κρ p x j, so the conservation equa- The state equation yields the relationship ρ x j tion can be written cx) ρ N t k j 1 ρ ) 1 ρ )=fx, t). κ x j κ x j Finally, by introducing the flow potential uw) = w ρdp,wehave cx) u µ u)=fx, t), t where the flux is given componentwise by the negative of the function µ u) 1 N κ k j u x j ) u x j. We shall assume below that this is a monotone function of the gradient. The classical Forchheimer-type corrections to the Darcy law for fluids lead to such functions with growth of order p = 3. Various spaces of functions on a bounded for simplicity) domain in R N with smooth boundary Γ will be used. For each 1 < p <, L p ) is the usual Lebesgue space of equivalence classes of) p-th power summable functions, and W 1,p ) is the Sobolev space of functions which belong to L p ) together with their first order derivatives. The trace map γ : W 1,p ) L p Γ) is the restriction to boundary values. Let Y = [,1] N denote the unit cube. Corresponding spaces of Y -periodic functions will be denoted by a subscript #. For example, C # Y ) is the Banach space of functions which are defined on all of R N and which are continuous and Y -periodic. Similarly, L p # Y ) is the Banach space of functions in Lp loc RN ) which are Y -periodic. For this space we take the norm of L p Y ) and note that L p #Y )is equivalent to the space of Y -periodic extensions to R N of the functions in L p Y ). Similarly, we define W 1,p # Y ) to be the Banach space of Y -periodic extensions to R N of those functions in W 1,p Y ) for which the trace or boundary values) agree on opposite sides of the boundary, Y, and its norm is the usual norm of W 1,p Y ). The linear space C# Y ) C #Y ) C R N )isdenseinbothofl p # Y)andW1,p # Y ). Various spaces of vector-valued functions will arise in the developments below. If B is a Banach space and X is a topological space, then CX; B) denotes the space of continuous B-valued functions on X with the corresponding supremum norm, and for any measure space we let L p ; B) denote the space of p-th power norm-summable equivalence classes of) functions on with values in B. When X =[,T]or=,T) is the indicated time interval, we denote the corresponding evolution spaces by C,T;B)andL p,t;b), respectively. Next we quote some definitions and results on two-scale convergence from [1] slightly modified to allow for homogenization with a parameter which we denote by t ). These changes do not affect the proofs from [1] in any essential way.

4 4 G. W. CLARK & R. E. SHOWALTER EJDE 1999/ Definition 1.1. A function, ψx, t, y) L p,t),c # Y)), which is Y -periodic in y and which satisfies lim ψ x, t, x ) p dx dt = ψx, t, y) p dy dx dt,,t ),T ) Y is called an admissible test function. Here p is the conjugate of p, thatis, 1 p 1 p =1. Definition 1.. A sequence u in L p,t) ) two-scale converges to u x, t, y) L p,t) Y) if for any admissible test function ψx, t, y), 1.1) lim u x, t)ψ x, t, x ) dt dx =,T ) u x, t, y)ψx, t, y) dy dt dx.,t ) Theorem 1.1. If u is a bounded sequence in L p,t) ), then there exists a function u x, t, y) inl p,t) Y) and a subsequence of u which two-scale converges to u. Moreover, the subsequence u converges weakly in L p,t) ) to ux, t) = Y u x, t, y) dy. When the sequence, u,isw 1,p -bounded, we get more information. Theorem 1.. Let u be a bounded sequence in L p,t;w 1,p )) that converges weakly to u in L p,t); W 1,p )). Then u two-scale converges to u, andthereis a function Ux, t, y) inl p,t) ; W 1,p # Y )/R) such that, up to a subsequence, x u two-scale converges to x ux, t) y Ux, t, y). Theorem 1.3. Let u and x u be two bounded sequences in L p,t) )). Then there exists a function U x, t, y) inl p,t) ; W 1,p # Y )/R) such that, up to a subsequence, u and x u two-scale converge to U x, t, y) and y Ux, t, y), respectively. Finally, we formulate the Cauchy problem or initial-value problem for an evolution equation in Banach space in a form that will be convenient for our applications below. Let V be a reflexive Banach space with dual V ; we shall set V = L p,t;v) for 1 <p<, and its dual is V = L p,t;v ). Let V be dense and continuously embedded in a Hilbert space H, sothatv H and we can identify H V by restriction. Proposition 1.4. The Banach space W p,t) {u V:u V }is contained in C[,T],H). Moreover, if u W p,t)then u ) H is absolutely continuous on [,T], d dt ut) H =u t)ut) ) a.e. t [,T], and there is a constant C for which u C[,T ],H) C u Wp,T ), u W p. Y Corollary 1.5. If u, v W p,t)thenu ),v )) H is absolutely continuous on [,T] and d ) ut),vt) dt H =u t)vt) ) v t)ut) ), a.e. t [,T].

5 EJDE 1999/ FLOW IN A PARTIALLY FISSURED MEDIUM 5 Suppose we are given a not necessarily linear) function A : V V and u H, f V. Then consider the Cauchy Problem to find 1.) u V:u t)a ut) ) =ft) in V, u) = u in H. It is understood that u V in 1.), so it follows from Proposition 1.4 that u is continuous into H and the condition on u) is meaningful. If A is known to map V into V, i.e., the realization of A : V V as an operator on V has values in V, then 1.) is equivalent to the variational formulation 1.3) u V: for every v V with v V and vt )= T ut),v t) ) T H dt A ut) ) T vt) dt = ft)vt) dt u,v) ) H. The equivalence of the strong and variational formulations of the Cauchy problem will be used freely in all of our applications below. See Chapter III of [] for the above and related results on the Cauchy problem.. The Micro-Model We consider a structure consisting of fissures and matrix periodically distributed in a domain in R N with period Y, where >. Let the unit cube Y =[,1] N be given in complementary parts, Y 1 and Y, which represent the local structure of the fissure and matrix, respectively. Denote by χ j y) the characteristic function of Y j for j =1,, extended Y -periodically to all of R N. Thus, χ 1 y)χ y)=1. We shall assume that both of the sets {y R N : χ j y) =1},, are smooth. With the assumptions that we make on the coefficients below to obtain coercivity estimates, it is not necessary to assume further that these sets are also connected. The domain is thus divided into the two subdomains, 1 and, representing the fissures and matrix respectively, and given by x j {x :χ j =1}, j =1,. ) Let Γ 1, 1 be that part of the interface of 1 with that is interior to, and let Γ 1, Y 1 Y Y be the corresponding interface in the local cell Y. Likewise, let Γ, Ȳ Y and denote by Γ, its periodic extension which forms the interface between those parts of the matrix which lie within neighboring Y -cells. The flow potential of the fluid in the fissures 1 is denoted by u 1 x, t) and the corresponding flux there is given by µ x 1, 1) u. The flow potential in the matrix is represented as the sum of two parts, one component u x, t) with flux µ x, ) u which accounts for the global diffusion through the pore system of the matrix, and the second component u 3 x, t) with flux µ x 3 3), u and corresponding very high frequency spatial variations which lead to local storage in the matrix. The total flow potential in the matrix is then αu βu 3. Here α β =1withα andβ>.) In the following, we shall set Y 3 = Y and likewise set χ 3 = χ in order to simplify notation. For j =1,,3,let µ j : R N R N R N and assume that for every ξ R N,µ j, ξ ) is measurable and Y -periodic and for a.e. y Y, µ j y, ) is continuous. In addition, assume that we have positive constants k, C, c and

6 6 G. W. CLARK & R. E. SHOWALTER EJDE 1999/ 1 <p< such that for every ξ, η R N and a.e. y Y µ j y, ξ) C ξ p 1.1) k µ j y, ξ) ) ).) µ j y, η) ξ η µ j y, ξ ) ξ c ξ p.3) k. Let c j C # Y )begivensuchthat.4) <c c j y) C, 1 j 3. Since these are given on R N, we can define for j =1,,3 the corresponding scaled coefficients at x j, ξ R N by x ) c j x) c j, µ j x, ξ ) x µ j, ξ ). The exact micro-model introduced in [1] for diffusion in a partially fissured medium is given by the system.5).6).7).8).9).1) t c 1 x)u 1 x, t)) µ 1 x, u ) 1 x, t) = in 1 t c x)u x, t)) µ x, u ) x, t) = in t c 3 x)u 3 x, t)) µ 3 x, u ) 3 x, t) = in u 1 = αu βu 3 on Γ 1, αµ 1 x, u ) 1 x, t) ν 1 = µ x, u ) x, t) ν 1 on Γ 1, βµ 1 x, u ) 1 x, t) ν 1 = µ 3 x, u ) 3 x, t) ν 1 on Γ 1, where ν 1 is the unit outward normal on 1. We shall similarly let ν denote the unit outward normal on, so ν 1 = ν on Γ 1,. The first equation is the conservation of mass in the fissure system. In the matrix,,wehavetwo components of the flow potential. The first is the usual flow through the matrix, and the second component is scaled by p to represent the very high frequency variations in flow that result from the relatively very low permeability of the matrix. Each of these is assumed to satisfy a corresponding conservation equation. The total flow potential in the matrix is given by the convex combination αu βu 3 where α, β > denote the corresponding fractions of each, so α β =1. Thus, the first interface condition is the continuity of flow potential, and the remaining conditions determine the corresponding partition of flux across the interface. Since the boundary conditions will play no essential role in the development, we shall assume homogeneous Neumann boundary conditions µ 1 x, u ).11) 1 x, t) ν 1 = on 1, µ x, u ).1) x, t) ν = and x, u ).13) 3 x, t) ν = on. µ 3

7 EJDE 1999/ FLOW IN A PARTIALLY FISSURED MEDIUM 7 The system is completed by the initial conditions.14) u 1, ) = u 1 ), u,) = u ), u 3,) = u 3 ) in H. Next we develop the variational formulation for the initial-boundary-value problem.5)-.14) and show that the resulting Cauchy problem is well posed in the appropriate function space. Define the state space H L 1 ) L ) L ), a Hilbert space with the inner product [u 1,u,u 3 ],[ϕ 1,ϕ,ϕ 3 ]) H c 1 x) u 1 x) ϕ 1 x) dx 1 [c x) u x) ϕ x)c 3x)u 3 x)ϕ 3 x)] dx. Let γ j : W 1,p j ) Lp j) be the usual trace maps on the respective spaces for j =1,,>, and define the energy space V H {[u 1,u,u 3 ] W 1,p 1) W 1,p ) W 1,p ): γ1 u 1=αγ u βγ u 3 on Γ 1, }. Note that V is a Banach space when equipped with the norm [u 1,u,u 3 ] V χ 1u 1 L ) χ u L ) χ u 3 L ) χ L χ 1 u 1 u L χ L u3. p ) p ) p ) If we multiply each of.5),.6),.7) by the corresponding ϕ 1 x), ϕ x),ϕ 3 x) for which [ϕ 1,ϕ,ϕ 3 ] V, integrate over the corresponding domains, and make use of.9)-.13), we find that the triple of functions u ) [u 1 ),u ),u 3 )] in L p,t;v ) satisfies ) t [u 1t),u t),u 3t)], [ϕ 1,ϕ,ϕ 3 ] A [u 1t),u t),u 3t)]) [ϕ 1,ϕ,ϕ 3 ]) = H for all [ϕ 1,ϕ,ϕ 3 ] V, where we define the operator A : V V ) by A [u 1,u,u 3 ]) [ϕ 1,ϕ,ϕ 3 ]) x, u ) 1 x) ϕ 1 x) dx 1 { µ x, ) u x) µ 1 ϕ x)µ 3 x, u ) 3 x) ϕ } 3 x) dx for [u 1,u,u 3 ], [ϕ 1,ϕ,ϕ 3 ] V. Thus, the variational form of this problem is to find, for each >and[u 1,u,u 3] H a triple of functions u ) [u 1 ),u ), u 3 )] in Lp,T;V ) such that d dt u )A u )=inl p.15),t;v ) ) and.16) u ) = u in H. Conversely, a solution of.15) will satisfy.5)-.8), and if that solution is sufficiently smooth, then it will also satisfy.5)-.13).

8 8 G. W. CLARK & R. E. SHOWALTER EJDE 1999/ The assumptions.1),.) and.3) guarantee that A satisfies the hypotheses of [, Proposition III.4.1], so there is a unique solution u [u 1,u,u 3]in L p,t;v )of.15) and.16). Note that since d dt u L p,t;v ) ) that u C [,T]; H )andso.16) is meaningful by Proposition 1.4 above. 3. Two-scale limits We introduce the scaled characteristic functions x ) χ j x) χ j, j =1,. These will be used to denote the zero-extension of various functions. In particular, for any function w defined on j the product χ j w is understood to be defined on all of as the zero extension of w. Similarly, if w is given on Y j,thenχ j wis the corresponding zero extension to all of Y. Our starting point is a preliminary convergence result for the solutions described above. Lemma 3.1. There exist a pair of functions u j in L p,t;w 1,p ) ),,,and triples of functions U j in L p,t) ; W 1,p # Y )/R), g j in L p,t) Y N )), u j L Y )for,,3, and a subsequence taken from the sequence of solutions of.15)-.16) above, hereafter denoted by u =[u 1,u,u 3], which twoscale converges as follows: 3.1) 3.) 3.3) 3.4) 3.5) 3.6) 3.7) 3.8) 3.9) 3.1) χ 1u 1 χ 1 y) u 1 x, t) χ [ 1 u 1 χ 1 y) u1 x, t) ] y U 1 x, y, t) χ u χ y) u x, t) χ [ u χ y) u x, t) ] y U x, y, t) χ u 3 χ y) U 3 x, y, t) χ u 3 χ y) y U 3 x, y, t) ) χ 1µ 1 u 1 χ1 y) g 1 x, y, t) ) χ µ u χ y) g x, y, t) u 3) χ y) g 3 x, y, t) χ µ 3 χ j u j,t) χ j y)u j x), j =1,,3. Proof. Using Proposition 1.4 and.15) we can write 1 d dt [u 1,u,u 3],[u 1,u,u 3]) H A [u 1,u,u 3]) [u 1,u,u 3]) =. Integrating in t gives 3.11) 1 u t) H 1 u ) H t A [u 1,u,u 3]) [u 1,u,u 3]) dt =

9 EJDE 1999/ FLOW IN A PARTIALLY FISSURED MEDIUM 9 which, with the assumption.3) yields 3.1) 1 t χ u t) H c 1 u 1 p χ u p ) χ u p 3 dt L p ) L p ) L p ) 1 [χ1 u 1,χ u,χ u 3] t k, t T. H Thus, u ) is bounded in L,T;H ),and so χ 1u 1, χ u,andχ u 3are bounded in L,T;L )). Also, χ 1 u 1, χ u and χ u 3 are bounded in Lp,T; L p ) N ). We obtain 3.1) through 3.4) exactly as in [1, Theorem.9] by Theorem 1.. Statements 3.5) and 3.6) follow from Theorem 1.3. Finally, from.1) and the bounds already established, we have that χ j µ j x, u ) j x, t) for j =1,) and χ µ 3 x, u ) ) 3 x, t) are bounded in L p [,T],L p ) due to.3), 3.1) and T χ x j µ j, ξ p x)) T dxdt χ j ξ x) p 1)p dxdt T = χ j ξ x) p dxdt. Thus χ j µ j x, u ) j x, t) and χ µ 3 x, u ) 3 x, t) converge as stated. Define the flow potential u χ 1 u 1 χ αu βu 3 ) Lp,T;W 1,p ) ) for each >, and note that on Γ 1, Thus γ 1u = γ 1u 1 = αγ u βγ u 3 = γ u. u = χ 1 u 1 χ and from Lemma 3.1) we see that and Now let ϕ C u α u β u ) 3 L p [,T] ) χ1 y) u 1 x)χ y)αu x, t)βu 3 x, y, t)) u χ y) β y U 3 x, y, t).,c )) # Y N and note that u x, t) ϕ x, x ) dx = [ u x, t) ϕ x, x ) y ϕ x, x )] dx. Taking two-scale limits on both sides yields 3.13) βχ y) y U 3 x, y, t) ϕ x, y) dxdy = Y χ 1 y) u 1 x, t)χ y)αu x, t)βu 3 x, y, t))) y ϕ x, y) dxdy. Y

10 1 G. W. CLARK & R. E. SHOWALTER EJDE 1999/ The divergence theorem shows that the left hand side of 3.13) is simply β y U 3 x, y, t) ϕ x, y) dxdy = βu 3 x, y, t) y ϕ x, y) dxdy Y Y βu 3 x, s, t) ϕ x, s) ν dxds Y while the right hand side of 3.13) can be written u 1 x, t) y ϕ x, y) dxdy Y 1 αu x, t)βu 3 x, y, t)) y ϕ x, y) dxdy. Y We see that 3.13) yields βu 3 x, s, t) ϕ x, s) ν dxds = Y u 1 x, t) y ϕ x, y) dxdy αu x, t) y ϕ x, y) dxdy Y 1 Y = u 1 x, t) ϕ x, s) ν 1 dxds αu x, t) ϕ x, s) ν dxds. Y 1 Y Since U 3 and ϕ are periodic on Γ,, this shows that 3.14) βu 3 αu = u 1 on Y 1 Y Γ 1,. Next we seek a variational statement which is satisfied by the limits obtained in Lemma 3.1. Choose smooth functions ϕ j L p,t;w 1,p ) ),,, Φ j L p,t) ; W 1,p # Y ) ), j =1,,3, such that ϕ j t L p,t;w 1,p ) ), j =1,, Φ 3 t L p,t) ; W 1,p # Y ) ), and βφ 3 x, y, t) =ϕ 1 x, t) αϕ x, t) for y Γ 1,. In the following we shall use the notation ),t to represent the time derivative t ). Apply.15) to the triple [ϕ 1 x, t)φ 1 x, x,t),ϕ x, t)φ x, x,t), Φ 3 x, x,t) ]inl p,t;v ), where we define Φ 3 x, y, t) Φ 3 x, y, t) β Φ 1x, y, t) α β Φ x, y, t). Then integrate by

11 EJDE 1999/ FLOW IN A PARTIALLY FISSURED MEDIUM 11 parts in t to obtain T 3.15) T j j T c j u j ϕ j,t Φ j,t ) dxdt c 3 u 3 Φ 3,t dxdt c j u j x, T ) ϕ j x, T )Φ j x, x,t )) dx c ju j j T µ 3 c 3 u 3 x, T )Φ 3 x, x,t ) dx ϕ j x, ) Φ j x, x )), dx c 3u 3Φ 3 x, x ), dx µ j j x, u ) j x, t) ϕ j x, t)φ j x, x )),t dxdt x, u ) 3 x, t) Letting in3.15) now yields [ Φ 3 x, x ),t 1 y Φ 3 x, x ) ],t dxdt =. T 3.16) c j y) u j x, t) ϕ j,t x, t) dydxdt Y j T c 3 y) U 3 x, y, t)φ 3,t x, y, t) dydxdt Y c j y) u j x) ϕ j x, T ) dydx c 3 y) u 3 x)φ 3x, y, T ) dydx Y j Y c j y) u j x) ϕ j x, ) dydx c 3 y) u 3 x)φ 3x, y, ) dydx Y j Y T [ g j x, y, t) ϕj x, t) y Φ j x, y, t)] dydxdt Y j T g 3 x, y, t) y Φ 3 x, y, t) dydxdt =. Y We can summarize the preceding as follows. Define the energy space ) 3 W {[u 1,u,U 1,U,U 3 ] W 1,p ) L p ; W 1,p # Y ) : βu 3 x, y) =u 1 x) αu x) fory Γ 1, }. We have shown that the limit obtained in Lemma 3.1 satisfies [u 1,u,U 1,U,U 3 ] L p,t;w) and by density, 3.16) holds for all [ϕ 1,ϕ,Φ 1,Φ,Φ 3 ] L p,t;w) such that d dt [ϕ 1,ϕ,,,Φ 3 ] L p,t;w ). It remains to find the strong form of the problem and to identify the flux terms g j.

12 1 G. W. CLARK & R. E. SHOWALTER EJDE 1999/ 4. The Homogenized Problem We shall decouple the variational identity 3.16) in order to obtain the strong form of our homogenized system. This will be accomplished by making special choices of the test functions [ϕ 1,ϕ,Φ 1,Φ,Φ 3 ] as above, and the strong form will be displayed below in Corollary 5.. First we choose ϕ 1,ϕ,Φ 1,Φ all equal to zero, and choose Φ 3 as above and to vanish at t =andt=t and on Γ 1,. Together with the identity 3.14) from above, this gives at a.e. x thecell system 4.1) c 3 y) U 3 x, y, t) t y g 3 x, y, t) =, y Y, 4.) U 3 and g 3 ν are Y -periodic on Γ,, 4.3) βu 3 = u 1 αu on Γ 1,. Next let ϕ 1 be as above and vanish at t =andt=t,andchooseφ 3 by the requirement that βφ 3 x, y, t) =ϕ 1 x, t) fory Y. With the remaining test functions all zero, this yields the macro-fissure equation 4.4) ) u1 x, t) c 1 y) dy 1 c 3 y)u 3 x, y, t) dy Y 1 t β t Y = g 1 x, y, t) dy. Y 1 Similarly we choose ϕ as above and vanishing at t =andt=t and let Φ 3 be determined by βφ 3 x, y, t) = αϕ x, t) fory Y 1 to obtain the macro-matrix equation 4.5) ) u x, t) c y) dy α c 3 y)u 3 x, y, t) dy Y t β t Y = g x, y, t) dy. Y Finally, by setting the test functions ϕ 1,ϕ,Φ 3 all equal to zero and by choosing Φ 1,Φ as above, we obtain the pair of systems 4.6) y g j x, y, t) = y Y j, 4.7) g j ν =onγ 1, and g j ν is Y -periodic on Y j Y for j =1,. Note that 4.1) and 4.4) and 4.5) hold in L p,t) ; W 1,p # Y ) )andl p,t; W 1,p ) ), respectively. Substituting 4.1)-4.7) in 3.16) gives the boundary conditions 4.8) g 1 x, y, t) dy ν 1 = and Y 1 4.9) g x, y, t) dy ν = Y on and the initial and final conditions 4.1) U 3 x, y, ) = u 3 x),u 3x, y, T )=u 3 x, y) and 4.11) u j x, ) = u j x), u j x, T )=u jx) for j =1,

13 EJDE 1999/ FLOW IN A PARTIALLY FISSURED MEDIUM 13 in L Y )andl ) respectively. The final conditions appearing above will be used only to identify the functions g i x, y, t) below; they are not part of the problem. Note also that using 4.1) and 4.11), integrating by parts in t in 3.16), and replacing the test functions ϕ j for j =1,) and Φ j for j =1,,3) with sequences converging to u j and U j gives the following homogenized version of 3.11), 4.1) 1 c j y) u j x, T ) dydx 1 c 3 y) U 3 x, y, T ) dydx Y j Y 1 c j y) u j x) dydx 1 c 3 y) u Y j 3 x) dydx Y T [ g j x, y, t) uj x, t) y U j x, y, t)] dydxdt Y j T g 3 x, y, t) y U 3 x, y, t) dydxdt =. Y It remains to find g 1, g and g 3 in terms u 1,u,U 1, U and U 3. To this end, let φ and ξ ) N ) be in C [,T] ; C# Y ) and Φ1, Φ, Φ 3 C [,T] ; C# Y ) and for >,define the triple of functions η j x, t) =χ j x ) uj x ) x, t)χ Φj j x, x ),t λφ x, x ),t, j =1,, and η 3 x, t) =χ x 3 x, ) Φ x ),t λξ x, x )),t. Note that each ηj x, t) and because of the continuity assumption) µ x j,η j x, t)) j = 1,,3) arises from an admissible test function, and we have the two-scale convergence η j η j x, y, t) χ j y) u j x, t)χ j y) y Φ j x, y, t)λ φx, y, t), j =1,, η 3 By.) we have ) η 3 x, y, t) χ y) y Φ 3 x, y, t) λξ x, y, t). 4.13) T j µ j x, u ) j µ ) ) j x, η j u j ηj) dxdt T µ 3 x, u ) ) 3 µ 3 x, η3) u ) 3 η3 dxdt.

14 14 G. W. CLARK & R. E. SHOWALTER EJDE 1999/ Expanding 4.13) and employing 3.11) at t = T gives 1 { j c j u j } u j x, T ) ) dx 1 T T j c 3 u 3 u 3 x, T ) ) dx { µ ) ) j x, η j u j ηj µ j x, u ) } j ηj dxdt { µ 3 x, η3) u ) 3 η3 µ 3 x, u ) } 3 η3 dxdt. Let and apply the two-scale convergence results above to obtain ) c j y) u j x) dydx 1 c 3 y) u 3 Y j x) dydx Y 1 lim c j u j x, T ) dx 1 c 3 u 3 x, T ) dx j T µ j y, η j x, y, t)) y U j x, y) y Φ j x, y, t) λφ x, y, t)) dydxdt Y j T g j x, y, t) uj x, t) y Φ j x, y, t)λφx, y, t)) dydxdt Y j T µ 3 y, η 3 x, y, t)) y U 3 x, y, t) y Φ 3 x, y, t) λξ x, y, t)) dydxdt Y T g 3 x, y, t) y Φ 3 x, y, t)λξx, y, t)) dydxdt. Y Set φ = χ 1θ1 χ θ where θ j C [,T],C Y j )) and χ j θj is Y -periodic. Following [1] we note that since each µ j is continuous in the second variable, we may replace Φ j by sequences converging strongly in L p,t) ; W 1,p # Y )/R) to χ 1 y)u 1 x, y, t), χ y) U x, y, t) andχ y)u 3 x, y, t) for, and 3, respectively. Thus 4.14) becomes 4.15) 1 c j y) u j x) dydx 1 c 3 y) u 3 Y j x) dydx Y 1 lim c j u j x, T ) dx 1 c 3 u 3 x, T ) dx j

15 EJDE 1999/ FLOW IN A PARTIALLY FISSURED MEDIUM 15 T Y j µ j y, u j x, t) y U j x, y, t)λ ) θ j x, y, t) λ θ j x, y, t) dydxdt T g j x, y, t) Y j T µ 3 Y T g 3 x, y, t) Y We now employ 4.1), 4.11) and 4.1) in 4.15) to obtain 4.16) T T 1 µ j Y j T uj x, t) y U j x, y, t)λ θ j x, y, t)) dydxdt y, y U 3 x, y, t)λξx, ) y, t) λξ ) x, y, t) dydxdt y U 3 x, y, t)λ ξx, y, t)) dydxdt. y, u j x, t) y U j x, y, t)λ ) θ j x, y, t) λ θ j x, y, t) dydxdt Y µ 3 y, y U 3 x, y, t)λξx, ) y, t) λξ x, y, t) dydxdt T g 3 x, y, t) λξ x, y, t) dydxdt Y g j x, y, t) λ θ j x, y, t) dydxdt Y j 1 lim j c j u j x, T ) dx 1 Y j c j y) u j x, T ) dydx 1 c 3 u 3 x, T ) dx Y c 3 y) U 3 x, y, T ) dydx. The right hand side of 4.16) is non-negative by [1, Proposition 1.6], so dividing by λ and letting λ gives 4.17) T T Y j [µ j y, u j x, t) ) ] y U j x, y, t) g j x, y, t) θ j x, y, t) dydxdt Y [µ 3 y, ) ] y U 3 x, y, t) g 3 x, y, t) ξ x, y, t) dydxdt. This holds for all θ 1, θ, and ξ,so µ j y, u j x, t) ) y U j x, y, t) = g j x, y, t) in Y j, j =1,, and µ 3 y, ) y U 3 x, y, t) = g 3 x, y, t) in Y. These identities complete the strong form of the homogenized problem. We shall summarize and complement these results in the following section.

16 16 G. W. CLARK & R. E. SHOWALTER EJDE 1999/ 5. The Main Result Theorem 5.1. Assume that.1)-.4) hold, that β>, and that u 1,u, and u 3 L ) are given. Then the limits [u 1,u,U 1,U,U 3 ] established above in Lemma 3.1 are the unique solution u j L p,t;w 1,p )), j =1,, U j L p,t) ; W 1,p # Y j)/r),,,3, with βu 3 x, y, t) =u 1 x, t) αu x, t) fory Γ 1, of the homogenized system T 5.1) c j y) u j x, t) ϕ j,t x, t) dydxdt Y j T c 3 y) U 3 x, y, t)φ 3,t x, y, t) dydxdt Y c j y) u j x) ϕ j x, ) dydx c 3 y) u 3 x)φ 3x, y, ) dydx Y j Y T y, u j x, t) ) y U j x, y, t) [ ϕj x, t) ] y Φ j x, y, t) dydxdt for all Y j µ j T µ 3 y, ) y U 3 x, y, t) y Φ 3 x, y, t) dydxdt =, Y ϕ j L p,t;w 1,p ) ), j =1,, Φ j L p ),T) ; W 1,p # Y j), j =1,,3, for which and ϕ j t L p,t;w 1,p )) ) Φ 3,,, L p,t) ; W 1,p # t Y j)) ), βφ 3 x, y, t) =ϕ 1 x, t) αϕ x, t) for y Γ 1,, ϕ 1 x, T )=ϕ x, T )=Φ 3 x, y, T )=. Only the uniqueness needs yet to be verified, and this will follow below. In particular, U 1 and U are determined within a constant for each t,t), so each of these is unique up to a corresponding function of t. We shall check that 5.1) is just the variational form of the Cauchy problem for an appropiate evolution equation in Banach space, and that the corresponding strong problem is described as follows. The state space is given by H {[ϕ 1,ϕ,Φ 3 ] L ) L ) L ; L Y ) ) with the scalar product u,ϕ) H c j y) dy u j x) ϕ j x) dx Y j Y c 3 y) U 3 x, y)φ 3 x, y) dy dx.

17 EJDE 1999/ FLOW IN A PARTIALLY FISSURED MEDIUM 17 Define the energy space )) V {[ϕ 1,ϕ,Φ 3 ] H W 1,p ) W 1,p ) L p ; W 1,p # Y ) : βφ 3 x, y) =ϕ 1 x) αϕ x) fory Γ 1, } and the corresponding evolution space by V = L p,t;v). Corollary 5.. The triple u ) [u 1 ), u ),U 3 )] is the unique solution u ) V with u ) V of the strong homogenized system ) u1 x, t) c 1 y) dy 1 c 3 y)u 3 x, y, t) dy Y 1 t β t Y = y, u 1 x, t) ) y U 1 x, y, t) dy, ) u x, t) c y) dy α Y t β t c 3 y) U 3 x, y, t) t Y 1 µ 1 c 3 y)u 3 x, y, t) dy Y = Y µ U 3 x, y, t) andµ 3 y, y U 3 x, y, t) y, u x, t) y U x, y, t) y µ 3 y, ) y U 3 x, y, t) =, y Y, ) ν are Y -periodic on Γ,, βu 3 = u 1 αu on Γ 1,, with the boundary conditions µ 1 y, u 1 x, t) ) y U 1 x, y, t) dy ν 1 =and Y 1 y, u x, t) ) y U x, y, t) dy ν =on, Y µ and the initial conditions u j x, ) = u j x) for j =1,, U 3 x, y, ) = u 3 x), ) dy, where U 1 L p,t) ; W 1,p # Y 1)/R), U L p,t) ; W 1,p # Y )/R) are solutions of the local problems y µ j y, y U j x, y, t) u ) j x, t) =, y Y j, µ j y, y U j x, y, t) u ) j x, t) ν =onγ 1,, Y-periodic on Γ, j =1,. Proof. Define an operator A : V V by 5.) Au,ϕ Y j {µ j y, u j x) y U j x, y)} ϕ j x)) dydx {µ 3 y, y U 3 x, y)} y Φ 3 x, y)) dydx, Y 3 u =[u 1,u,U 3 ], ϕ =[ϕ 1,ϕ,Φ] V,

18 18 G. W. CLARK & R. E. SHOWALTER EJDE 1999/ where U 1 x, y) andu x, y) are determined by ) 5.3) U j L p ; W 1,p # Y j) : {µ j y, y U j x, y) u j x)} y Φx, y)) dydx =, Y j Φ L p ) ; W 1,p # Y j), for j =1,. It has been already shown in Section 4 that [u 1,u,U 1,U,U 3 ] satisfies the homogenized system of Theorem 5.1, and from this it follows that u ) satisfies the variational form of the Cauchy problem 1.3). It is easy to check that A is monotone and bounded V V,sou ) is the unique solution as well of the strong problem 1.), and this is realized as the strong homogenized system of Corollary 5.. Remark 5.1. Theorem 5.1 describes the limiting form of the original micro-model from Section as a system for the five unknowns u 1, u, U 1, U,andU 3. This system can also be realized from an evolution equation based on the variational identity 3.16) on the space W, but this would be of degenerate type: the time derivatives of U 1 and U do not occur in the system. However, by following the suggestion implicit in Corollary 5. we were able to incorporate the local functions U 1 and U in the definition of the operator A and thereby to write our limiting system as a non-degenerate evolution equation on the space V with three components. In the linear case, one can carry this decoupling even further and represent each of the functions U 1 and U in terms of the corresponding u 1 or u in order to obtain a closed system for the remaining three unknowns. Suppose that we have symmetric Y -periodic coefficient functions a 1 ij y) C Y 1)anda ij y) C Y ) which are zero off their respective domains). We assume that there is a c >, independent of y, such that N a k ijy)ξ i ξ j c ξ, y Y k for k =1,. i, Extending each a k ij to all of R by periodicity, we define for k =1,and ξ R N µ k N x, ξ )i = a k ij x ) ξ j. Then with p =,the results developed above apply. For each of k =1,weisolate from 5.1) the following problem for U k )x, y, t): Find U k L,T) ; W 1, # Y k) such that T 5.4) µ k y, u k x, t) ) y U k x, y, t) y Φ k x, y, t) dydxdt = Y k ) for all Φ k L,T) ; W 1, # Y k). The input to this problem is u k x, t), independent of y, so this permits us to separate variables with the following construction:

19 EJDE 1999/ FLOW IN A PARTIALLY FISSURED MEDIUM 19 For 1 i N, define Wi k y) to be the solution of [ y µ k y, ] y Wi k y) e i ) = in Y k, µ k y, y Wi k y) e i) n= on Y k Y Wi k ) isy-periodic. Then by linearity we can write U k x, y, t) = N u k x j x, t)w j y). If we substitute this into 5.1) with Φ k x, y, t) = N ϕ k x,t) x j Wj k y), we obtain the decoupled homogenized system T c k u k x, t) ϕ k,t x, t) dxdt c k u k x) ϕ k x, ) dx k=1 k=1 T N Y k i, k=1 A k u k ij x, t) ϕ k x, t) dydxdt = x i x j T c 3 y)u 3 x, y, t)φ 3,t x, y, t) dydxdt Y T fork=1, c 3 y) u 3 x)φ 3 x, y, ) dydx Y y, ) y U 3 x, y, t) y Φ 3 x, y, t) dydxdt =. Y µ 3 where the coefficients are given by c k = c k dy, A k ij = µ k y, y Wi k y) e i ) y Wj k y) e j )dy. Y k Y k These are the usual effective coefficients which are constants that result from the averaging due to the homogenization procedure. References [1] Grégoire Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal ), [] T. Arbogast, Analysis of the simulation of single phase flow through a naturally fractured reservoir, SIAM Jour. Numer. Anal , 1 9. [3] T. Arbogast, The double porosity model for single phase flow in naturally fractured reservoirs, in Numerical Simulation in Oil Recovery, The IMA Volumes in Mathematics and its Applications, vol. 11, M. F. Wheeler, ed.), Springer-Verlag, Berlin, New York, 1988, pp [4] T. Arbogast, J. Douglas, Jr., and U. Hornung, Modeling of naturally fractured reservoirs by formal homogenization techniques, Frontiers in Pure and Applied Mathematics, R. Dautray, ed.), Elsevier, Amsterdam, 1991, [5] T. Arbogast, J. Douglas, Jr., and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 1 199), [6] G. I. Barenblatt, I. P. Zheltov, and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata], Prikl. Mat. Mekh., 4 196), ; J. Appl. Math. Mech., 4 196),

20 G. W. CLARK & R. E. SHOWALTER EJDE 1999/ [7] A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, [8] A. Bourgeat, S. Luckhaus and A. Mikelić, Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow, SIAM J. Math. Anal., 7, 1996), [9] G.W. Clark, Derivation of Microstructure Models of Fluid Flow by Homogenization, J. Math. Anal. Appl., ), [1] G.W. Clark and R.E. Showalter, Fluid flow in a layered medium, Quarterly of Applied Mathematics, ), [11] J. D. Cook and R. E. Showalter, Microstructure diffusion models with secondary flux, J. Math. Anal. Appl ), [1] J. Douglas, Jr., M. Peszyńska and R. E. Showalter, Single phase flow in partially fissured media, Transport in Porous Media ), [13] U. Hornung and R. E. Showalter, Diffusion models for fractured media, J. Math. Anal. Appl ), [14] U. Hornung, Miscible displacement in porous media influenced by mobile and immobile water, Rocky Mtn. Jour. Math ), ; Corr [15] U. Hornung, Homogenization and Porous Media, Springer-Verlag, Berlin, New York, 1997 [16] M. Peszyńska, Fluid flow through fissured media. Mathematical analysis and numerical approach, Ph.D. Thesis, University of Augsburg, 199. [17] M. Peszyńska, Finite element approximation of a model of nonisothermal flow through fissured media, Finite Element Methods: fifty years of the Courant element, Marcel Dekker, New York, NY, pp ; Lecture Notes in Pure and Applied Mathematics, vol. 164 M. Křížek, P. Neittaanmäki, and R. Stenberg, eds.), [18] M. Peszyńska, Analysis of an integro differential equation arising from modelling of flows with fading memory through fissured media, J. Partial Diff. Eqs ), [19] M. Peszyńska, On a model of nonisothermal flow through fissured media. Differential Integral Equations ), no. 6, [] M. Peszyńska, Finite element approximation of diffusion equations with convolution terms, Math. Comp ), no. 15, [1] E. Sanchez Palencia, Non-Homogeneous Media and Vibration Theory, inlecture Notes in Physics, vol 17, Springer Verlag, Berlin, New York, 198. [] R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, in Mathematical Surveys and Monographs, vol 49, American Mathematical Society, Providence, [3] R. E. Showalter and N. J. Walkington, Micro-structure models of diffusion in fissured media, Jour. Math. Anal. Appl ), 1. [4] J. E. Warren and P. J. Root, The behavior of naturally fractured reservoirs, Soc. Petroleum Engr. J ), G. W. Clark Department of Mathematical Sciences, Virginia Commonwealth University Richmond, VA 384 USA address: gwclark@saturn.vcu.edu R. E. Showalter Department of Mathematics, University of Texas at Austin Austin, TX 7871 USA address: show@math.utexas.edu