Discontinuous Galerkin methods for flow and transport problems in porous media

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1 T COMMUNICATIONS IN NUMERICA METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2; :1 6 [Verson: 2/3/22 v1.] Dscontnuous Galerkn methods for flow and transport problems n porous meda Béatrve Rvère and Mary F. Wheeler The Center for Subsurface Modelng, Texas Insttute for Computatonal and Appled Mathematcs, The Unversty of Texas at Austn, ACE 5.332, 21 East 24th Street, Austn TX SUMMARY Ths work presents a new scheme based on dscontnuous approxmaton spaces for solvng the mscble dsplacement problem n porous meda. Numercal comparsons are made between ths scheme and the well known mxed fnte element and hgher order Godunov methods. The smulatons clearly show the advantages of the dscontnuous Galerkn methods for stable or unstable flow. Copyrght c 2 John Wley & Sons, td. KEY WORDS: unstable flow mscble dsplacement, hgher order schemes, dscontnuous polynomals, unstructured meshes, 1. Introducton We consder the dsplacement of one ncompressble flud by another n a porous medum n. The nvadng and the dsplaced fluds are referred to as the solvent and the resdent flud; respectvely. et denote the tme nterval. The classcal equatons governng the mscble dsplacement n over are:!"$# %'& )(* +-,!,/. " 21 " 3 # 4 5 %'& 6(7 where the dependent varables are, the pressure n the flud mxture, and, the fracton volume of the solvent n the flud mxture. The permeablty of the medum measures the resstance of the medum to flud flow; s the vscosty of the flud mxture; " represents the Darcy velocty, + s the porosty of the medum and 1 " s the coeffcent of molecular dffuson and mechancal dsperson that depends on " n a nonlnear fashon. A survey on smulatng mscble dsplacement can be found n [3]. The boundary of the doman s decomposed nto a Drchlet part 8:9 and a Neumann part 8<; such that 8<9=?8; #A@ and 8<9CBD8; #,. We also defne the nflow part 8FECHGIKJ, M "NOQP SR Correspondence to: The Center for Subsurface Modelng, Texas Insttute for Computatonal and Appled Mathematcs, The Unversty of Texas at Austn, ACE 5.332, 21 East 24th Street, Austn TX Copyrght c 2 John Wley & Sons, td.

2 G + ^ < < # # O O # O 2 B. RIVIÈRE AND M.F. WHEEER and the outflow part 8 GI J, " /O R, where O denotes the unt outward normal vector to,. We assume Drchlet and Neumann boundary condtons for the pressure and Neumann and mxed boundary condtons for the concentraton. The vscosty of the flud mxture s assumed to follow the quarter-power mxng law, commonly applcable to hydrocarbon mxtures [5]: # E 3 E E where (resp. ) s the vscosty of the solvent (resp. resdent flud). The stablty of the flow s characterzed by the moblty rato,.e. the rato of the vscosty of the resdent flud to the vscosty of the solvent. Instabltes n the flow wll grow f the moblty rato s larger than unty. In that case, protusons referred to as vscous fngerng develop through the resdent flud. Another mportant physcal parameter s the Peclet number that quantfes the convectve effects wth respect to the dspersve effects. 2. Notaton and scheme and #!" ;$# % be two non-degenerate subdvsons (resp. ) are denoted by & ' ). On each edge et # ; of, that consst of trangles or quadrlaterals. The edges of (resp. & ' ). et ( (resp. ( ) denote the number of nteror edges n (resp. & ' (resp. & ' ), a unt normal vector O ' ( resp. O ' ) s arbtrarly fxed, except on the boundary, where t concdes wth + the outward unt normal vector. The drecton of O + ' unquely defnes the jump of the functon : ) # +*,.- +/*," f O ' s from to. The average of + s defned by R # +*,1- +*, 243. For 5 and 5 postve ntegers, one defnes the followng dscrete spaces # G : ;: *, < J=( 7 5@?$A R 6B7 # C # G : ;: *, < # JD( 7 # 5E?ARF We now formally defne two blnear forms. G ;H JI K:S # X ;H " I KY # ; M 1N, ; # Q R1TVU;W N Q R : PO G ' M N@Q R 1 " Y O ' : 3 3O ' RF): G ' M NSQ R Q R.TVU;W N Q R 3 : 3O ' : 3O ' RF) M 1N, # N, "C < # Y ' M N Q #R Z "C3O ' )Y "CO Q #R TVU;[ N\Q #R ' Y O G 1 " O ' M N@Q #R ' RF)Y G 1 " ' M N@Q #R Y 3O ' RF) ] where Z s the upwnd value of the concentraton on a gven edge. We also defne the functonals C:S # : 3O ' ^ 9Y # Q R TVU W N\Q R NE_ 4 Y Q #R TVU@` N@Q #R bac "CO ' Y where s the Drchlet datum for pressure and ac s the nflow datum for concentraton. The contnuous n tme Dscontnuous Galerkn (DG) [6] method s gven by the map d(fe.g he.g Q Copyrght c2 John Wley & Sons, td. Commun. Numer. Meth. Engng 2; :1 6

3 DG METHODS FOR MISCIBE DISPACEMENT 3 ) 687 C -( 687 # 9 determned by the relatons for any. n G ;$H<dh e.g. bi( e.g. K:S # ^ C:S5?: J 6 7 C 5 (1) +, he.g N _,. Y X ;H e.g. bi h e.g Y # ^ 9Y 5?Y J 6 7 # 9 b (2) The ntal concentratonh e.g s the^ projecton of the ntal concentraton and the Darcy velocty s defned by e.g # 2 ]he.g ( e.g We propose a tme-steppng procedure that reflects the fact that the velocty feld vares more slowly n tme than the concentraton for reasonable physcal data. Thus, the pressure tme step. wll be chosen to be tmes larger than the concentraton tme step.. The procedure has been shown to be effcent for other fnte element methods appled to the mscble dsplacement problem [4]. We denote. #. and. # A.. We now descrbe the algorthm for advancng of one pressure tme step. We assume that. #. for some fxed ndces and A and that the approxmaton h8e.g. s known. Then, the pressure at tme. can be calculated as the soluton of (1) for. #.. The dscretzaton n tme of the concentraton equaton for., where, s accomplshed by dervng a modfed verson of the standard backward-dfference scheme procedure N _ + he.g. he.g. E. Y X ;$H< e.g I h e.g. 5KY # ^ 9Y 5 e.g s a lnear extrapolaton of e.g. and e.g. 5?Y J 6 7 # 9 E. For the frst pressure step, ths where extrapolaton s not vald and we use a predctor-corrector technque. The use of slope lmters [2] s also needed to prevent from numercal overshoots and undershoots to occur n the neghborhood of the concentraton front. 3. Numercal Experments In all experments, the permeablty feld s randomly generated on the coarse mesh. We solve for quadratc approxmatons of the pressure and the concentraton. We frst consder the case where both flow and transport equatons are solved on structured meshes. In that case, we compare our smulatons wth those obtaned from the Parallel Subsurface Smulator (Parssm) developed at the Unversty of Texas at Austn [1]. In Parssm, flow s smulated usng the mxed fnte element method and transport s smulated usng a hgher order Godunov method. Fgure 1 shows the permeablty feld and the concentraton front obtaned wth the DG method n the case of moblty rato and a low Peclet number. The scontours # \ 3 obtaned wth DG and Parssm for low and hgh Peclet number are shown n Fg. 2. The legend DG h-h1 means that flow s solved on the coarse mesh and transport s solved on the mesh refned once. The DG concentraton fronts are comparable to those obtaned by Parssm even though coarser meshes have been used for DG. We also note that DG veloctes are accurate enough to be computed on a coarser mesh than DG concentratons. As we ncrease the moblty rato (see Fg. 3), nstabltes n the flow yeld more fngers n the concentraton front. We also observe that DG veloctes computed on the coarse mesh produce a much more detaled concentraton front than Parssm veloctes computed on the coarse mesh. We then consder the case where flow s solved on an unstructured mesh and transport s solved on a structured mesh. The motvaton for solvng each equaton on a dfferent mesh les n the fact that n realstc meda, the permeablty feld vares greatly n space n a very unstructured manner. However, Copyrght c2 John Wley & Sons, td. Commun. Numer. Meth. Engng 2; :1 6

4 4 B. RIVIÈRE AND M.F. WHEEER Z Y X PERM 9.233E E E E E E E E E E E E E E E-12 Fgure 1: Permeablty feld on coarse mesh. DG Concentraton front on mesh refned twce for moblty rato and Peclet number. DG h-h2 PARSSIM h3-h3 DG h-h2 PARSSIM h3-h3 Fgure 2: Comparsons between DG and Parssm concentraton socontours for moblty rato and Peclet number 4 and 16. the porosty of the medum s assumed to be unform, and structured meshes are prefered for solvng the concentraton equaton. Comparsons wth Parssm are not possble snce Parssm does not handle unstructured meshes. Fgure 4 shows the permeablty feld and the pressure feld obtaned n the case of moblty rato. We compare our numercal results to the DG soluton computed on the mesh refned twce for both pressure and concentraton, whch we refer to as the fne soluton. As the meshes for concentraton and pressure are successvely refned, we observe that the socontours converge to the fne soluton (see Fg. 5 ). In the case of hgh Peclet number, the front has more protusons (see Fg. 5 ). We repeat the experments for unstable flow (see Fg. 6) and the same phenomena are observed. However, the vscous fngers are more pronounced. Copyrght c2 John Wley & Sons, td. Commun. Numer. Meth. Engng 2; :1 6

5 DG METHODS FOR MISCIBE DISPACEMENT 5 DG h-h DG h-h1 PARSSIM h-h PARSSIM h1-h1 DG h-h DG h-h1 PARSSIM h-h PARSSIM h1-h1 Fgure 3: Comparsons between DG and Parssm concentraton socontours for moblty rato 3 and Peclet number 4 and 16. PERM 9.233E E E E E E E E E E E E E E E-12 Fgure 4: Permeablty feld on unstructured coarse mesh. DG Pressure feld on coarse unstructured mesh. 4. Conclusons In ths work, we have shown numercally that the DG method s well suted for flud flow problems n porous meda and n partcular that t s compettve wth other locally conservatve methods for solvng the mscble dsplacement problem. Frst, DG veloctes are accurate enough so that coarser meshes can be used for solvng the flow equaton. Second, the DG method can capture the nstabltes of the flow. Fnally, the DG method can handle unstructured meshes n an easy and natural way. REFERENCES Copyrght c2 John Wley & Sons, td. Commun. Numer. Meth. Engng 2; :1 6

6 6 B. RIVIÈRE AND M.F. WHEEER Fgure 5: Isocontours of the DG concentraton for moblty rato and for Peclet number. and Fgure 6: Isocontours of the DG concentraton for moblty rato and for Peclet number. and 1. Arbogast T. User s gude to Parssm1: the Parallel Subsurface Smulator, Sngle Phase. Texas Insttute for Computatonal and Appled Mathematcs 1998: Cockburn B, Shu CW. The Runge-Kutta dscontnuous Galerkn method for conservaton laws V. J. Comp. Physcs 1998; 141: Douglas J. The numercal smulaton of mscble dsplacement n porous meda. Comp. Meth. n Nonlnear Mech. 198: Douglas J, Ewng RE, Wheeler MF. A tme-dscretzaton procedure for a mxed fnte element approxmaton of mscble dsplacement n porous meda. R.A.I.R.O. Numercal Analyss 1983; 17(3): Koval EJ. A method for predctng the performance of unstable mscble dsplacement n heterogeneous meda Soc. Pet. Eng. J. 1963; 3: Rvère B, Wheeler MF, Grault V. A pror error estmates for fnte element methods based on dscontnuous approxmaton spaces for ellptc problems. SIAM J. Numer. Anal. 21; 39(3): Copyrght c2 John Wley & Sons, td. Commun. Numer. Meth. Engng 2; :1 6

VISCOELASTIC SIMULATION OF BI-LAYER COEXTRUSION IN A SQUARE DIE: AN ANALYSIS OF VISCOUS ENCAPSULATION

VISCOELASTIC SIMULATION OF BI-LAYER COEXTRUSION IN A SQUARE DIE: AN ANALYSIS OF VISCOUS ENCAPSULATION Mahesh Gupta Mchgan Technologcal Unversty Plastc Flow, LLC Houghton, MI 49931 Hancock, MI 49930 Abstract