Research Article A Combination Method of Mixed Multiscale Finite-Element and Laplace Transform for Flow in a Dual-Permeability System

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1 Internatonal Scholarly Research Network ISRN Appled Mathematcs Volume 22, Artcle ID 22893, pages do:.542/22/22893 Research Artcle A Combnaton Method of Mxed Multscale Fnte-Element and Laplace ransform for Flow n a Dual-Permeablty System ang-we Lu,, 2, 3 He-Hua Xu, and Xue-Ln Qu CAS Key Laboratory of Margnal Sea Geology, South Chna Sea Insttute of Oceanology, Guangzhou 53, Chna 2 Graduate Unversty, CAS, Bejng 49, Chna 3 Faculty of Scence, East Chna Insttute of echnology, Fuzhou 344, Chna Correspondence should be addressed to Xue-Ln Qu, xlqu@scso.ac.cn Receved 28 Aprl 22; Accepted 28 May 22 Academc Edtors: H. Du and G. Kyracou Copyrght q 22 ang-we Lu et al. hs s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. An effcent combnaton method of Laplace transform and mxed multscale fnte-element method for couplng partal dfferental equatons of flow n a dual-permeablty system s present. Frst, the tme terms of parabolc equaton wth unknown pressure term are removed by the Laplace transform. hen the transformed equatons are solved by mxed FEMs whch can provde the numercal approxmaton formulas for pressure and velocty at the same tme. Wth some assumptons, the multscale bass functons are constructed by utlzng the effects of fnescale heterogenetes through bass functons formulaton computed from local flow problems. Wthout tme step n dscrete process, the present method s effcent when solvng spatal dscrete problems. At last, the assocated pressure transform s nverted by the method of numercal nverson of the Laplace transform.. Introducton In recent years, there have been some mportant developments n numercal methods of flow n fractured porous meda. And an mportant numercal method s mxed fnte-element method. Due to the heterogeneous propertes and the multscale nature of meda, a complete analyss of these problems s extremely dffcult, and effcent numercal solvers usually requre an extremely large amount of computer storage and CPU tme. Developng multscale methods allow us to overcome above dffculty whle retanng a satsfactory accuracy. he popular multscale methods nclude multscale fnte-element method, numercal upscalng method, and multscale fnte volume method, 2. In general, multscale fnte-element methods are regarded as numercal methods and strateges n whch bass functons are

2 2 ISRN Appled Mathematcs computed by solvng local homogeneous PDEs subject to specal boundary condtons, 3. Multscale FEMs are effcency and convenence for ellptc equaton of steady flow model. Despte ther advantage, they stll have some practcal lmtatons n solvng parabolc equatons for nonsteady flow model 2. Frst, ther major drawback s that t s necessary to take small tme steps. A severe lmtaton on the tme step may requre an excessve amount of computer tme. Second, all the nteror velocty or pressure must be computed at each tme step. We can see that numercal methods for parabolc equaton wth general FEMs requre the soluton of some smultaneous algebrac equatons at each tme step. On the other hand, these approaches tend to ncrease cost when the solutons must be carred out over long tme perods 9. he present method, the combned use of the Laplace transform and the multscale fnte-element method, s used to solve some problems of flow n fractured porous meda. It s effcent to overcome the above dffculty. 2. Mathematcal Model and Laplace ransform Incompressble flow n fractured porous meda can be descrbed by some coupled partal dfferental equatons. We consder the followng mathematcal model consstng of four couplng equatons for pressure p, p 2 and Darcy velocty u, u 2 n a spatal doman 3, 4: Φ C p x, t t Φ 2 C 2 p 2x, t t dvu x, t λα ( p 2 x, t p x, t ), x, t>, 2. dvu 2 x, t λα ( p 2 x, t p x, t ), x, t>, 2.2 u x, t λk x p x, t, u 2 x, t λk 2 x p 2 x, t, p x, p, p 2 x, p 2, where s a doman n R d d 2, C, 2 denote the compressblty, φ, 2 are the porosty, and k x, 2 denote the heterogeneous feld permeablty tensor whch have scale separaton and perodcty. We assume that only u,p, 2 are unknown expressons. And here we assume that the equatons are equpped wth Neumann boundary condtons u n on, 2.7 u 2 n on, 2.8 where n s the outward normal of.

3 ISRN Appled Mathematcs 3 In order to remove tme dependence from the governng equaton and boundary condtons, the scheme of the Laplace transform wll be utlzed. he Laplace transform of the functon p and ts nverson formulas are defned as Px, s L [ px, t px, t L [ px, s 2π e st px, tdt, c c e st px, sds, 2.9 where s c ω, c, ω R. he Laplace transform of s: Φ C ( spx, s p x, ) dvu x, s λα ( p 2 p ), Φ 2 C 2 ( sp2 x, s p 2 x, ) dvu 2 x, s λα ( p 2 p ) he expresson can be wrtten as Φ C s λαp λαp 2 Φ C p dvu, Φ 2 C 2 s λαp 2 λαp Φ 2 C 2 p 2 dvu Now the unknown functon u x, s and p x, s,, 2, wll be solved. By the Laplace transform of , we can get the followng: u x, t λk x p x, t, u 2 x, t λk 2 x p 2 x, t By the Laplace transform of Neumann boundary condton , we can know u n, on,, Next we consder mxed multscale fnte-element method to Mxed Fnte-Element Method he mxed fnte-element method s based on a mxed formulaton 6 8, 5. For our model problem , the mxed formulaton s as follows. Fnd u, p, 2 H,dv L 2, such that v k λ u dx v p dx, 3. dvu wdx ( Φ C s λαp λαp 3 Φ C p ) wdx, 3.2

4 4 ISRN Appled Mathematcs whch hold for all v H,dv and w L 2, { ( d } H,dv v L ) 2 :dvv L 2, v n on, 3.3 where L 2 s the space of square ntegral functon n. he above formulas can provdes the numercal results of p x, s and u x, s, 2 at the same tme. In mxed fnte-element method, L 2 and H,dv are approxmated by fntedmensonal subspaces W and V, respectvely, 6. For nstance, L 2 s replaced by W: W { } w L 2 : w k const k. 3.4 And H,dv s replaced by V : V { v H,dv :dvv L 2,, ( v n j ) γj const, γ j, and v n j s contnuous aross γ j }. 3.5 Here n j s the unt normal to γ j pontng from to j. Denotng the approxmaton of p, u, 2 ndvdually by p h, u h, 2,the dscrete formulaton reads as follows. Fnd p h, u h, 2 W V, such that hold for all w W and v V. By applyng Green formulaton for 3., we can conclude the followng: v k λ u h dx v k 2 λ u h 2dx p h vdx, 3.6 p h 2 vdx. 3.7 Now lettng V span{ψ j } and w span{ϕ j },weletv be equal to velocty bass functon ψ k and w equal pressure bass functon ϕ k k, 2,...,N.. And the approxmatons of u h, u 2 h and p h, p 2 h can be wrtten as follows: u h Ũ j s ψ jx, p h P j s ϕ jx, u h 2 Ũ j 2 s ψ jx, p h 2 P j 2 s ϕ jx. 3.8

5 ISRN Appled Mathematcs 5 hen formulas can show that ψ k k λ N ψ k k 2 λ N dv N Ũ j s ψ jx ϕ k dx Ũ j s ψ jx dx N P j s ϕ jx ψ k dx, Ũ j 2 s ψ jx dx N P j 2 s ϕ jx ψ k dx, Φ C s λα P j s ϕ jx λα P j 2 s ϕ jx Φ C p ϕ k dx, dv N Ũ j 2 s ψ jx ϕ k dx Φ2 C 2 s λα P j 2 s ϕ jx λα P j s ϕ jx Φ 2 C 2 p 2 ϕ k dx, k, 2,...,N. 3.9 he above four formulas can be rearranged as follows: ψ k k λ N Ũ j s ψ jx dx N P j s ϕ jx ψ k dx, dv N Ũ j s ψ jx ϕ k dx Φ C s λα P j s ϕ jx ϕ k dx λα P j 2 s ϕ jx dx Φ C p ϕ kdx, λα P j s ϕ jx dx Φ2 C 2 s λα P j 2 s ϕ jx ϕ k dx dv N Ũ j 2 s ψ jx ϕ k dx

6 6 ISRN Appled Mathematcs Φ 2 C 2 p 2 ϕ kdx N P j 2 s ϕ jx ψ k dx ψ k k 2 λ N Ũ j 2 s ψ jx dx, k, 2,...,N. 3. hen, we obtan the approxmatons: u h Ũ j s ψ jx, p h P j s ϕ jx, u h 2 Ũ j 2 s ψ jx, p h 2 P j 2 s ϕ jx, 3. where the coeffcents Ũ j, P j, P j 2 and Ũj 2 are obtaned by solvng the followng lnear system: A B B 2 D D 2 D 3 D 4 B 3 B 4 A 2 Ũ j P j P j 2 Ũ j 2 F, 3.2 F 2 where [ A [ ψ k λk ψ j dx, A 2 ψ k λk 2 ψ j dx, [ B B 4 ϕ j dv ( [ ) ψ k dx, B 2 B 3 ϕ k dv ( ) ψ j dx, [ [ D Φ C s λαϕ j ϕ k dx, D 2 D 3 λαϕ j dx, D 4 F [ [ ( k, j, 2,...,N ). Φ 2 C 2 s λαϕ j ϕ k dx, Φ C p ϕ kdx, F 2 [ Φ 2 C 2 p 2 ϕ kdx 3.3

7 ISRN Appled Mathematcs 7 hen the lnear system 3.2 can be wrtten as A B B 2 D D 2 D 2 D 4 B 2 B A 2 Ũ j P j P j 2 Ũ j 2 F. 3.4 F 2 4. Multscale Bass Functons he man dea of the MMsFEM Mxed Multscale Fnte-Element Method s to construct specal local bass functons that are adaptve to the local propertes of the ellptc dfferental operator 6. In ths paper, two sets of grds are consdered: a fne grd and a coarsened grd n whch each coarse block H conssts of connected cells from the underlyng fne grd. Local bass functons ϕ H for pressure and ψ H for velocty n each coarse block H generally are constructed by solvng { ψ H H x, for x H,, else, ψ H λk ϕ H, 4. ψ H n, on H, where subscrpt denotes the order number of the bass functon. he weghtng functon H x plays an mportant role to dstrbute dvu, 2 onto the fne grd approprately. It can be chosen on the form H x lx/ H lxdx wth varous choces of lx6. Now denote V ms span{ψ H } the approxmaton space of V and W span{ϕ H j }, we let v H equal velocty bass functon ψ H and w H equal pressure bass functon ϕ H, j, 2,...N.. And the approxmatons of u H, u H 2 and p H, ph 2 can be wrtten as follows: u H N s ψ j x, u H N 2 p H N P H,j s ϕ j x, p H 2 N 2 s ψ j x, P H,j 2 s ϕ j x. 4.2

8 8 ISRN Appled Mathematcs hen the MMsFEM seeks p H, u H, 2 W V ms,such that hold for all v H V ms and w H W. We can get the followng formulas on a smlar concluson from to expresson 3.4: A B B 2 D D 2 D 4 D 4 B 2 B A 2 P H,j P H,j 2 2 F, 4.3 F 2 A [ B B 4 [ ϕ H j ψ kh λk ψ H j [ dx, A 2 ψ H k λk 2 ψ H j dx, ( ) [ dv ψ H k dx, B 2 B 3 ϕ H k (ψ dv H j ) dx, [ D [ Φ C s λαϕ H j ϕ H k dx, D 2 D 3 λαϕ H j dx, D 4 [ Φ 2 C 2 s λαϕ H j ϕ H k dx F 2 [ Φ 2 C 2 p 2 ϕh k, dx F [ Φ C p ϕh k, dx 4.4 ( k, j, 2,...,N ). 5. Numercal Inverson of the Pressure and Velocty erm In ths secton we compute the numercal value of U H,j t, 2; j, 2,...,N and P H,j t, 2; j, 2...,N by s, 2; j, 2...N and P H,j s, 2; j, 2...N usng the numercal nverson formula of Laplace transform. In accordance wth the method of Durbn 6, the Fourer seres expanson of U H,j t can be derved as follows 9, 7: U H,j t evt { 2 Ũ v, t,, Re [ŨH,j v { Re k [ŨH,j S k cos kπt [ŨH,j Im S k sn kπt } } S k v kπ, k,,...,n. 5.

9 ISRN Appled Mathematcs 9 And here the dscrete error s gven by Ũ v, t, e 2vk 2k t. 5.2 k So the numercal approxmate value of U H,j t s,n t evt { 2 Re [ŨH,j v } { evt Re k [ŨH,j S k cos kπt [ŨH,j Im S k sn kπt }, 5.3 where 2 s the perod of Fourer seres approxmate nverson functon on the nterval, 2 and N and v are free parameters. All s are computed by formulaton 4.3. Obvously, there occurs a truncaton error gven by,a t evt { { [ŨH,j Re S k cos kπt [ŨH,j Im S k sn kπt } }. 5.4 kn Because numercal nverson formula requres the value of transformed functon at each S k k,, 2, 3,...,N, t s needed to solve the equatons for any S S k. he approxmate value of P H,j t, 2; j, 2,...,N can be computed n the same way. P H,j,N t { evt [ } 2 Re P H,j v { [ evt Re P H,j k S k cos kπt [ Im P H,j S k sn kπt }. 5.5 he soluton at a specfc node n the gven doman can be obtaned from 4.3 and hs strategy s dfferent from that of classcal FEM, whch must compute all nodal values for each tme step untl the specfc tme s reached. he present method can compute the specfc nodal value at a specfc tme. In addton, t s obvous that the present method takes less computer tme f lengthy solutons are requred. 6. Concluson hs paper focuses on effcent numercal method whch combnes the numercal nverson of Laplace transform wth mxed FEM for couplng partal dfferental equatons of flow n a dual-permeablty system. he present combned method can save computng cost by multscale method n spatal doman, and overcome some dsadvantages comng from numercal processng of dfferent tme step sze by the numercal nverson of Laplace transform. However, the present method can be extended to other two- or three-dmensonal lnear tme-dependent problems.

10 ISRN Appled Mathematcs Acknowledgments hs research was supported by Natural Scence-echnology Major Specal ask of Chna 2ZX , Natural Scence Foundaton of Chna 62, and Natural Scence Foundaton of Jangx provnce 24BAB26. References V. Kppe, J. E. Aarnes, and K.-A. Le, A comparson of multscale methods for ellptc problems n porous meda flow, Computatonal Geoscences, vol. 2, no. 3, pp , L. Dng, B. Han, and J.-Q. Lu, A wavelet multscale method for nverson of Maxwell equatons, Appled Mathematcs and Mechancs, vol. 3, no. 8, pp , J. E. Aarnes and Y. Efendev, Mxed multscale fnte element methods for stochastc porous meda flows, SIAM Journal on Scentfc Computng, vol. 3, no. 5, pp , C.-C. Chu, I. G. Graham, and.-y. Hou, A new multscale fnte element method for hgh-contrast ellptc nterface problems, Mathematcs of Computaton, vol. 79, no. 272, pp , 2. 5 S. Krogstad, K.-A. Le, H. M. M. Nlsen, J. R. Natvg, B. Skaflestad, and J. E. Aarnes, SINEF. A multscale mxed dnte-element solver for three-phase black-ol flow, n Proceedngs of the Socety of Petroleum Engneers (SPE 8993), pp. 3, Y. Efendev and. Y. Hou, Multscale Fnte Element Methods: heory and Applcatons, Sprnger, New York, NY, USA, X. Ma and N. Zabaras, A stochastc mxed fnte element heterogeneous multscale method for flow n porous meda, Computatonal Physcs, vol. 23, no. 2, pp , Arbogast, Mxed Multscale Methods for Heterogeneous Ellptc Problems, vol. 83 of Lecture Notes n Computatonal Scence and Engneerng, L. Ren, A hybrd Laplace transform fnte element method for solute radal dsperson problem n subsurface flow, Hydrodynamcs A, vol. 9, no., pp , 994 Chnese. X. G. He and L. Ren, Adaptve multscale fnte element method for unsaturated flow n heterogeneous porous meda I. Numercal scheme, Shul Xuebao, vol. 4, no., pp , 29 Chnese. Y. Q. Xue, S. J. Ye, C. H. Xe, and Y. Zhang, Applcaton of mult-scale fnte element method to smulaton of groundwater flow, Shul Xuebao, vol. 7, pp. 7 3, 24 Chnese. 2 H.. Chen and C. K. Chen, Hybrd laplace transform/fnte element method for two dmensonal heat conducton problem, Internatonal hermophyscs, vol. 2, no., pp. 3 36, J. L. Ge, he Modern Mechancs of fluds Flow n Ol Reservor (I), Petroleum Industry Press, Bejng, Chna, Vogel, H. H. Gerke, R. Zhang, and M.. Van Genuchten, Modelng flow and transport n a two-dmensonal dual-permeablty system wth spatally varable hydraulc propertes, Hydrology, vol. 238, no. -2, pp , 2. 5 A. Huang and. X. Zhou, heores and Methods of Fnte Element (I), Scence Press, Bejng, Chna, F. Durbn, Numercal nverson of Laplace transforms: an effcent mprovement to Dubner and Abate s method, he Computer Journal, vol. 7, pp , G. Hong and U. Hrdes, A method for the numercal nverson of Laplace transforms, Computatonal and Appled Mathematcs, vol., no., pp. 3 32, 984.

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