A Hybrid Numerical Method for High Contrast. Conductivity Problems. Liliana Borcea, and George C. Papanicolaou y. June 10, 1997.

Transcription

1 A Hbrid Numerical Method for High Contrast Conductivit Problems Liliana Borcea, and George C. Papanicolaou June, 997 Abstract We introduce and test etensivel a numerical method for computing ecientl current ow in media with high contrast conductivit. This method combines our analtical understanding of the form of the ow eld near narrow channels with standard numerical methods elsewhere in the ow regime and so it is a hbrid numerical method. Kewords: high contrast conductivit, resistor networs, hbrid numerical methods Introduction Natural porous media ehibit strong spatial variabilit in most of their properties, such as hdraulic conductivit, electrical conductivit, dielectric permeabilit, etc, even on mesoscopic length scales where some averaging has been incorporated into the modeling. Mathematicall this means that we have to solve partial dierential equations with coecients that tae ver large and ver small values in the domain. Forward problems for such partial dierential equations pose dicult analtical and computational questions. In numerical computations one needs ver ne meshes to capture the correct behavior of ow properties of the high contrast medium and this leads to large and ill conditioned matrices that must be inverted. Consequentl, iterative methods used in matri inversion have ver poor convergence rates. In this paper we develop a numerical method for high contrast conductivit problems which avoids the diculties encountered b other methods b combining standard numerical schemes with our analtical understanding of ow channeling [, 5] based on asmptotic methods. Flow channeling in media with high contrast conductivit was studied before. Keller [] and Batchelor and O'Brien [3] show that ow in media with high contrast conductive or resistive inclusions concentrates in narrow channels or gaps in between the inclusions. However, their analsis is based on assumptions about the shape of the inclusions. Our hbrid numerical method has been constructed with imaging applications in mind where the shape of the inclusions is not nown beforehand. In these applications it is convenient to assume that the high contrast arises in a simple, generic manner as a continuum [6]. Thus, we model the high contrast hdraulic conductivit K() b K() =K e, S() 2 ; (.) Applied Mathematics, Caltech 27-5, Pasadena, CA 925, borcea@ama.caltech.edu Dept of Mathematics, Stanford Universit, Stanford, CA 9435, papanico@math.stanford.edu

2 where K is a reference constant conductivit, S() is a smooth function and is a small parameter. Variations of the function S(), the scaled logarithm of the conductivit, are amplied b the parameter in (.), producing the high contrast of the hdraulic conductivit. Conductivities of form (.) were used before in the analsis of ow properties of media with high contrast [, 5, 8] and these studies show that the ow concentrates around regions where the conductivit has saddle points. Channels of strong ow near saddle points of the conductivit (.) are generalizations of the channels or gaps in high contrast media with conductive or resistive inclusions. In man applications we do not now precisel the local conductivit K(). We now it onl approimatel. In[5] we show that when the medium has high contrast, onl a limited amount of information about the local conductivit is needed for obtaining a good estimate of overall quantities lie the eective conductivit. The regions in the heterogeneous medium that reall matter are those where channels of strong ow develop. Thus, it is important that the ow in the channels is computed correctl. In general, in the vicinit of a channel, the conductivit K() is not a smooth function as (.) implies. However, as in [6], we can model the local conductivit K() in the vicinit of a channel b a smooth function of form (.) and obtain a good estimate of the overall ow behavior in the domain of interest. These conclusions are based on the results obtained for the inverse problem of inferring the conductivit in a domain from boundar measurements (see [6]). The ow channeling that develops at saddle points of the conductivit (.) leads to a resistor networ approimation of the high contrast medium (see [5]). However, the networ approimation is just the leading order term in the asmptotic approimation of the ow in the high contrast medium. In phsical problems where the contrast is nite and there are, in addition, regions of wea ow concentration, direct use of the asmptotics produces rather crude results. Our hbrid numerical method uses the asmptotic theor in regions where strong channeling develops, while standard numerical schemes are used elsewhere in the ow regime. The paper is organized as follows. In section 2 we formulate the problem, in section 3 we describe the hbrid method and in section 4 we test its performance in several situations and compare it to computations with standard numerical schemes as well as with the adaptive schemes of [2]. 2 Formulation of the problem We consider the isotropic stead state ow problem,r [K(; )r(; )] = f(; ) (2.) dened in a unit square domain (see g. 2.), where K is the hdraulic conductivit, (; ) is the hdraulic head and f(; ) is the recharge. We choose a simple domain in order to simplif the eposition of the hbrid method but the methodolog presented in this paper applies to regions of an shape. The u is obtained from Darc's law j =,Kr and the ow can be driven b either the forcing term f(; ) or b some inhomogeneous boundar conditions. In order to ideas we assume that the ow is conned in and impose the no-u =; (2.2) where n is the unit normal at the boundar, pointing outwards. The recharge f(; ) is prescribed so that the u injected equals the one taen out. For eample, for a point source at and sin at 2 we tae f(; ) =(, ), (, 2 ): (2.3)

3 = Figure 2.: Computational domain In the numerical computations we use an approimation of the recharge (2.3) where the delta functions are approimated b narrow Gaussians centered at and 2. Problem (2.)-(2.3) with K() > has a unique solution (; ) up to an additive constant. 3 The numerical method 3. Review of the asmptotic theor In this section we review some of the results obtained in [, 5] which appl to problem (2.)-(2.3) when the high contrast hdraulic conductivit is modeled b (.). The problem is analzed b using asmptotic and variational techniques and the results show that the ow concentrates in small neighborhoods of saddle points of the hdraulic conductivit K(). Moreover, b using the variational formulation of the eective conductivit and resistance K = min < K()r r > (3.) <r>=e K, = min < <j>=e; rj= K() j j >; it is shown that, in the asmptotic limit of innitel high contrast, the leading order term in the eective conductivit K is obtained b considering the ow onl in small neighborhoods of the saddle points of K(). In (3.) < > stands for the normalized average over, j() is the u and e is the unit vector in the direction of the average ow. To give a brief review of the asmptotic theor we consider the local ow problem in the vicinit of a saddle point S 2 of the conductivit. In a small neighborhood of the saddle point S the conductivit can be written as K() =K e, S() 2 K( S )e (, S )2 2 2, p(, S )2 2 2 ; (3.2) where and p are the curvatures of S() at the saddle point. In (3.2) we assume that the saddle point is oriented in the direction e. In general, the saddle point can be oriented in an direction and equation (3.2) still holds if at S weintroduce a local sstem of coordinates (; ) such that 2

4 direction coincides with the direction of the saddle. We tae as the small vicinit of S the square j, S j ; j, S j where the parameter! insuchawa that!as!. The local problem near the saddle point is rfk( S )ep[ (, S) 2, p(, S) 2 ]rg = for j, S j< and j, S j< r n = for j, S j= (3.3) =,C for = S + and = C for = S, ; which we solve b separation of variables. The constant C in equation (3.3) determines how much ow goes through the saddle point and it depends on the driving condition f(; ) given b (2.3), the contrast of K() and other factors. The solution of equation (3.3) is r (),Cerf[ 2 (, s)]: (3.4) 2 The analtical basis for (3.4) is a matched asmptotic epansion. The local analsis gives the leading term (3.4) of the inner epansion valid near the saddle point. The outer epansion deals with the diuse ow in the rest of the domain. The head () has, therefor, the form of an inner laer in the direction of the saddle, the direction in this case. The head gradient is r, q 2C e, (,s ) e (3.5) 2 and the u is given b j =,Kr K( S ) s p 2C q e, 2 p p(,s ) e : (3.6) We see from these epressions that when the contrast of K() is high ( is small) both the u and the head gradient are narrow Gaussians centered at S, which means that there is strong ow concentration around the saddle point of the conductivit. If there is onl one saddle point of K() in the domain, the results (3.5) and (3.6) impl that the overall, or eective, conductivit given b (3.) is (see [, 5]) K K( S ) s p : (3.7) Thus, the ow through the medium can be approimated b the ow through a networ consisting of one channel (resistor) with resistance r p R K( S ) : (3.8) In more general situations, where there are more saddle points of K in that are oriented in arbitrar directions, the ow still concentrates at the saddles but it follows a more complicated pattern, which is a networ of channels (resistors). Locall, around each saddle point of the conductivit, the ow is approimated b the current through a resistor given b (3.8). These resistors are connected in a networ that is identied as follows: the nodes of the networ are the maima of K() and the branches are paths connecting two adjacent maima over a saddle point. A detailed analsis is given in [5]. The asmptotic results summarized in this section 3

5 were derived for two-dimensional ows. However, the theor can easil be etended to three dimensions (see [, 5]) so the ideas presented in this paper appl to three-dimensional ow problems as well. The asmptotic networ theor reviewed in this section is onl for isotropic media. High contrast anisotropic media, where we ma have laers of ver dierent conductivit or fractures, have not been studied, et. 3.2 The hbrid method M m S m M Figure 3.: Region of high contrast of K(; ) in The method we introduce in this section uses the asmptotic result (3.4) in regions of strong ow concentration and matches it with the numericall computed solution elsewhere in the ow regime. The diuse ow awa from the channels can be computed with an standard numerical scheme but for simplicit wechoose a second order nite dierence method [4]. Thus, we introduce a uniform mesh with spacing h ==N and grid points We use the notation (i; j)! i =(i, )h; j =(j, )h; i; j =;:::N +: (3.9) i;j = ( i ; j ); K i;j = K( i ; j ); (3.) f i;j = f( i ; j ): and discretize equation (2.) with a central dierence scheme as K i;j ( i,;j + i;j,, 4 i;j + i;j+ + i+;j ) h 2 + (K i+;j, K i,;j ) ( i+;j, i,;j ) + (3.) (K i;j+, K i;j, ) ( i;j+, i;j, ) =,f i;j : 4

6 At the boundaries i =orn + and j =orn +wehave from (2.2) 2;j, ;j i;2, i; =; =; N+2;j, N;j i;n+2, i;n =; =: (3.2) D S Figure 3.2: Modied numerical domain We rst eplain the hbrid method in the simple case shown in g. 3.. We assume that the conductivit function has a high contrast region inside (shaded region in g. 3.) where K(; ) has onl one saddle point S,two minima and two maima denoted b m and M, respectivel. If the contrast in the shaded region is ver high, we can assume that K() =K ep(, S() )as 2 eplained in the introduction. E Figure 3.3: The u in domain D is matched with the u in n D E 5

7 We dene the modied domain n D as shown in g Inside the small bo D centered at the saddle point, the head (; ) has the epression (3.4). The constant C quanties the u through the saddle point and is treated as an unnown. To compute C beforehand, we would need the eective conductivit for the smooth regions of K, which is clearl unnown, and thus we must compute C numericall. The hbrid method solves (2.) in n D with Neumann boundar conditions (2.2) at the outer boundar and Dirichlet boundar conditions (3.4) at the inner boundar. The numerical method used in our computations is the nite dierence scheme described at the beginning of this section but clearl, an standard numerical method will do. In order to have aneven determined sstem of equations, an additional equation is needed. This additional equation determines the u constant C in terms of the nodal values of the head. We note that if we integrate (2.) over the shaded region shown in gure 3.3, we obtain, r(kr)d =, (Kr) e d = f(; )d; E whichwe tae as the additional equation needed. The discretization of this equation is done with the trapezoidal rule for integration. We thus obtain a linear sstem of equations with unnowns ( i ; j ); ( i ; j ) 2 n D and the u C which we solve with a direct solver for banded matrices (see [3]). 3.3 Choosing the size of the inner bo The accurac of the hbrid method depends on an appropriate choice of the size of the inner bo D (see g. 3.2). Inside the bo we assume that the asmptotic approimation (3.4) is accurate. Thus, it is important that D be small enough so that the asmptotic form (3.4) is valid. However, the bo cannot be too small because the standard numerical method used in n D cannot handle well the abrupt change in the head that occurs in the vicinit of the saddle. Thus, the size of the inner bo D must be chosen so as to balance these two conicting demands. The asmptotic approimations (3.5) and (3.6) for the head gradient and u suggest a good guess for the size of D. Both r and j are Gaussians centered at the saddle point. It is nown that at a distance equal to three standard deviations, the Gaussian is basicall zero. Since we must match the u entering D with the one inside, the bo should be smaller than this. In fact, numerical eperiments show that the appropriate size of the bo is close to two standard deviations. Thus, we determine initiall the inner bo D b r 2 2 j, s j= 2; r p 2 2 j, s j= 2: (3.4) With this choice for D, we compute the u constant C from equation (3.3). It is important to note that C is obtained b matching the u across a line that passes through the saddle point (see g. 3.2). At = S, epression (3.4) for the head () isver accurate and the computation of C is not aected b a small error in the size of the bo. However, the size of the bo aects the u close to the boundaries of D. We adjust iterativel the size of the inner bo from the initial guess (3.4) to a more accurate value b requiring that the u entering D equals the u at = S which isnown from the previous computation of C. Let us assume that the correct size of the bo in the ow direction is r 2 j, 2 s j= 2+; (3.5) where is the unnown adjustment parameter. When the inner bo size is chosen well, the q integrated horizontal u across the vertical line = L = s,(2+) 2 2 satises an equation 6

8 Figure 3.4: Eample of an integration domain needed for u matching in a problem with Dirichlet boundar conditions similar to (3.3). Thus, we have, p K( L ;)r( L ;) e d + p 2 2 j,sj>2 K( L ;) q 2C e,(2+) 2 d = 2 2 j,sj2 2 p p L f(; )dd; (3.6) where we used the asmptotic result (3.5) for the head gradient and equation (3.5). We now compute iterativel b using Newton's method for the nonlinear equation (3.6). The value of the adjustment parameter at iteration + is given b L f(; )dd + p K( L ;)r( L ;) e d + = p + [, ) 2 2 2(2 + e(2+ 2 j,sj>2 ] ) p K( L ;) p 2C 2 p d 2 2 j,sj2 where =; ; 2;::: and =: (3.7) This Newton iteration converges ver fast (two or three steps) because the initial guess (3.4) is ver good. Our numerical eperiments show that for contrasts of K() in the range 2 to 7, the appropriate size of the bo corresponds to (3.4) or =. Another interesting aspect that comes out of the numerical eperiments is that the vertical size of D given b (3.4) is appropriate for all contrasts higher than O( 2 ). However, in case an adjustment is needed, one can use a method similar to the above. 3.4 Generalization to arbitrar boundar conditions and multiple channels In general, when other than Neumann boundar conditions are prescribed the computation of the u constant C in the inner bo D cannot be done in a single step (as shown in 3:2). In such situations, C can be obtained iterativel. We can alwas obtain a good guess 7

9 Figure 3.5: Modied Domain (four channels) with regions needed for u matching for the initial C b assuming that all the ow passes through the saddle. Then, we improve it iterativel b requiring that the divergence theorem in a domain such as shown in g. 3.4 be satised. Thus, we must have f(; )dd =, Kr nds; where n is the outward normal For the portion included in D we use the asmptotic epression (3.5) of the head gradient, and after discretizing (3.8), we obtain a nonlinear equation in C that we solve with Newton's method. Our numerical eperiments show ver quic convergence (less than ve iterations) for contrasts of K() that are higher than O( 2 ). This is, of course, because we have a good initial guess for the u constant C. The hbrid method can be generalized to an number of saddle points and arbitrar orientation of the channels. We illustrate the generalization b considering a conductivit function that has two saddle points and 2 in the horizontal direction and two saddle points 3 and 4 in the vertical direction. The modied domain used for the numerical computations is shown in gure 3.5 where the shaded regions are the domains of integration for the four additional equations (similar to (3.3)) needed to determine the ues through each channel. 4 Performance of the hbrid method In this section we present results of numerical computations that test the performance of the hbrid method. In our tests we consider media with high contrast hdraulic conductivit that have one or four channels of ow, as described in 3:2. For comparison we also solved the ow problem (2.)-(2.3) with two additional numerical methods. One is a standard second order nite dierence scheme which uses a uniform discretization of the whole domain. We epect that the performance of this method will be rather poor because the regions of ow channeling need a ver ne discretization in order to capture the correct behavior of the solution. Thus, the scheme must use a ver ne mesh which leads to an etremel large and ill conditioned matri that must be inverted. We include in this section the eperiments performed with this numerical scheme in order to point out its shortcomings. The other method used for comparison is PLTMG (see [2]), a carefull designed general purpose solver for elliptic partial dierential equations. This is a multigrid solver with a nite 8

10 Hbrid M. Finite Di. PLTMG h = 32 h = 64 h ma = 32 ; h min = 256 Table 4.: Numerical grids element discretization based on continuous piecewise linear triangular elements and it provides for adaptive mesh renement. Even though this solver performs much better than the second order nite dierence scheme, we will show that in some cases it does not solve correctl the problem in the high contrast regions of the conductivit. In fact our hbrid solver can be used to estimate the limitations of PLTMG in high contrast situations. 4. One region of high contrast K Figure 4.: Conductivit function In this section we assume that the conductivit has a single channel (saddle point) of ow concentration (see g. 4.). The saddle point is located at (:5; :5) and oriented in the direction, and the high contrast region of K() isembedded in a fairl at bacground. We tested the performance of the hbrid method for constant curvatures at the channel ( = 86:76, p = 59:95), ed positions of the source (at (:2; :2)) and sin (at (:8; :8)) and dierent contrasts ranging from 2 to 8. The grids used in the computations are described in table 4.. The ow computed with the hbrid method for a contrast ma(k)=min(k) = 4 is shown in g. 4.2 and the strong ow concentration in the channel is evident. In order to test our method quantitativel, welooed at the net horizontal and vertical ues that can be obtained analticall from (2.)-(2.3). Thus, we must have and () = () = j e d, j e 2 d, 9 d d df(;) =: (4.) df(; ) =; (4.2)

11 Figure 4.2: Source / sin at (:2; :2) and (:8; :8), Contrast 4 where if <:2 or >:8 j e d = otherwise (4.3) and if <:2 or >:8 j e 2 d = otherwise: Since in our numerical computations the sources were not delta functions but rather ver narrow Gaussians, the net horizontal and vertical ues should be smoothed versions of (4.3) and (4.4). (4.4) :::Fin. Di.,PLTMG.2. ::: Fin. Di.,PLTMG.5.8 j () j.4.3 j () j Figure 4.3: Error in the net horizontal u for contrast 4 Figure 4.4: Error in the net vertical u for contrast 4 In g. 4.3 we plot the absolute error in the horizontal u j () j obtained with the three dierent numerical methods for a contrast 4. The solid line corresponds to the hbrid method, the dotted line to the nite dierence method and the dash-dot line to the PLTMG solver. The error given b all three methods is large (absolute error :, relative error 2%) around the \point" source and sin. Slightl awa from the sources the hbrid method gives a ver good result (the absolute and relative error is less than.3), whereas the nite dierence method has an error of 7% near the channel. PLTMG gives an error () that has an oscillator behavior in the high contrast region of K() (relative error 3%). The solution computed b PLTMG

12 required an adaptivel rened mesh with 5 vertices. Thus, even though the error given b PLTMG does not seem too bad, the computational cost was much higher than the cost of the hbrid method. Since in our eample ow channeling occurs onl in the horizontal direction, the net vertical u computed b all three methods is quite accurate (slightl awa from the sources), as can be seen from g The performance of the hbrid method remained unchanged for contrasts in the range 2 to 8 whereas the other two methods performed much better (worse) for contrasts lower (higher) than 4. We also tested the hbrid method for dierent positions of the source and sin and found that its performance is independent of the location of the source and sin provided that the are not in an neighborhood of the saddle point. In contrast, PLTMG gave quite inaccurate solutions (relative error 7%) when the source or sin were close to the channels and in some situations it failed to converge. Our eperiments show that PLTMG gives accurate solutions and uses reasonable meshes for contrasts no higher than O( 2 ) provided that the sources are far enough from the channel. The nite dierence method requires ver ne meshes in order to improve its results (h =2 for contrast 3 ) and is therefore quite inecient. We also tested the numerical convergence of the hbrid method for a variet ofcontrasts in the range 2 to 8. For eample, in an eperiment with a contrast of 6 and the same curvatures as before, we solved the problem with the hbrid method on two numerical grids. The coarse grid had spacing h c ==44 and the ne grid had a double number of grid points (i.e., h f ==88). The u constant C computed on the coarse grid is C =:937 and the ne grid solution is C = :325, which corresponds to a relative error of :86%. The solution f computed on the ne grid is also ver close to the coarse grid solution c. To be precise, the relative error is ma 2 conductivit K(). j c,f f 4.2 Four regions of high contrast j=2:85%. Similar results hold for dierent contrasts of the K Figure 4.5: The conductivit In this section we consider media with high contrast hdraulic conductivit that have four channels of ow (saddle points) of K( S ). The channels are located at =(:5; :225), 2 = (:5; :775), 3 =(:225; :5) and 4 =(:775; :5). Two channels (at ; 2 ) are horizontal and the other two are vertical. The curvatures of the saddles in the direction are =99:3 and p =75:4 and the curvatures of the saddles in the direction are =:9 and p =8:2. The conductivit used in the computations is similar to the one shown in g. 4.5, but we var the contrast (ma(k)=min(k)). We tested the performance of the hbrid method for dierent

13 Figure 4.6: Source / sin at (:; :) and (:9; :9), Contrast 6 contrasts ranging from 2 to 8 while the position of the source (at (:; :)) and sin (at (:9; :9)) were ept constant. The numerical grids used b the three methods have the same description as before (see table 4.) and the size of the four inner boes was chosen according t o (3.4) R J d.6.4 :::Fin. Di. R J d.5 :::Fin. Di..2,PLTMG.5,PLTMG Figure 4.7: Source / sin at (:; :) and (:9; :9), Contrast Figure 4.8: Source / sin at (:; :) and (:9; :9), Contrast 6 The ow computed with the hbrid method for an overall contrast ma(k)=min(k) = 6 is shown in g. 4.6 and it is strongl concentrated in the horizontal channels and onl weal concentrated in the vertical channels that are quite shallow. The error in the vertical and horizontal ues given b the hbrid method was as before 2% around the sources and ver small (relative and absolute error :3) elsewhere in the domain. In g. 4.7 and 4.8 we plot the horizontal and vertical ues given b all three methods and note that the hbrid method is more accurate than the other two schemes. The horizontal u computed b the nite dierence method is ver inaccurate in the high contrast region and PLTMG produces an undesired oscillator behavior similar to what we showed in 4:. The net vertical u computed b all three methods is more accurate because of the wea concentration near the shallow vertical saddles. 2

14 We also tested the hbrid method when the source and sin were displaced and the results reconrmed the conclusions of 4: that the performance of the method is independent of the position of the source and sin provided that the are not ver near the channels. The other two methods give quite inaccurate solutions when the source or sin are relativel close to an of the channels. For eample, an eperiment with contrast 6, the source located at (:4; :4) and the sin located at (:9; :9), shows that the error in the horizontal u given b the nite dierence method is 5%. The error given b PLTMG is 7%, where an adaptivel rened mesh consisting of 6 vertices was used. 5 Summar and conclusions We have introduced and tested etensivel a hbrid numerical method for solving conductivit problems in a heterogeneous medium with high contrast properties. The method combines analtical results of the asmptotic theor introduced in [, 5] with the use of a nite dierence numerical scheme, but other numerical schemes such as nite elements could also have been used. We carried out etensive numerical tests for dierent contrasts, dierent geometries and driving forces and showed that the hbrid method performs much better than standard ones, even the adaptive method of [2] where ver ne meshes are used locall. This is not surprising because a general purpose code cannot be epected to do as well as a speciall designed method that wors for a limited class of problems, lie the hbrid method for high contrast media. The use of the hbrid method is restricted to problems with high contrast of the conductivit that arises in a continuum manner, but the ideas presented in this paper can be generalized to an problem where an asmptoticall eact solution is nown. Our numerical computations show that the performance of the hbrid method is independent of the position of the source or sin in the ow region, ecept when the are situated ver near a saddle point of the conductivit. The other methods have serious diculties even when the source or sin are not so close to the region of high contrast. The hbrid method performs equall well for contrast of the conductivit ranging from 2 to 8. The size of the inner bo where the asmptotic solution is used is essential for good performance of the hbrid method and the initial choice (3.4) proved to be ver good for our computations. The computational cost of the hbrid method is low, hence its ecienc high, because we onl discretize the part of the domain where the ow is diuse and therefore well resolved b standard numerical schemes. At present, the hbrid method is implemented manuall, that is, we must now beforehand the position of the saddle points, their orientation and curvatures which are needed for specifing the saddle boes. Acnowledgements This wor was supported b grant AFF from AFOSR and b the NSF, DMS We wish to than Peter K. Kitanidis for his interest in our wor and man useful comments in the preparation of this paper. References [] Alumbaugh, D. E., Iterative Electromagnetic Born Inversion Applied to Earth Conductivit Imaging, PhD Thesis, Engineering - Material Science and Mineral Engineering, UC Berele,

15 [2] Ban, E. R. PLTMG: A Software Pacage for Solving Elliptic Partial Dierential Equations, SIAM, 99. [3] Batchelor, G. K., O'Brien, R. W., Thermal or electrical conduction through a granular material, Proc. R. Soc. London A, 977, 355, pp [4] Bensoussan, A., Lions, J. L., Papanicolaou, G. C., Asmptotic analsis for periodic structures, North-Holland, Amsterdam, 978. [5] Borcea, L., Papanicolaou G. C., Networ approimation for transport properties of high contrast materials, to appear in SIAM J. Appl. Math. [6] Borcea, L., Berrman, J. G., Papanicolaou G. C., High Contrast Impedance Tomograph, Inverse Problems, 2, -24, 996. [7] Daar, B. B., Kitanidis, P. K., Determination of the Eective Hdraulic Conductivit for Heterogeneous Porous Media Using a Numerical Spectral Approach. Method and 2. Results, Water Resources Research, Vol 28, No 4, pp and 67-78, April 992. [8] Golden, K. M., Kozlov, S. M., Critical path analsis of transport in highl disordered random media, preprint 995. [9] Jiov, V. V., Kozlov, S. M., Oleini, O. A., Homogenization of Dierential Operators and Integral Functional, Springer-Verlag, Berlin Heilderberg, 994. [] Keller, J. B., Conductivit of a Medium Containing a Dense Arra of Perfectl Conducting Spheres or Clinders or Nonconducting Clinders, J. Appl. Phs., 34 (4), 963, pp [] Kozlov, S. M., Geometric aspects of averaging, Russian Math. Surves 44:2, 989, pp [2] Kopli, J., Creeping ow in two-dimensional networs, J. Fluid Mech., 982, Vol. 9, pp [3] LAPACK User's Guide, SIAM, 992. [4] Mitchell, A. R., Computational methods in partial dierential equations, J. Wile, New Yor,