{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:42:23Z","timestamp":1787330543752,"version":"build-2736575974"},"reference-count":39,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>Standard reservoir simulation schemes employ first order upwind schemes for approximation of the convective fluxes when multiple phases or components are present. These schemes introduce both coordinate-line numerical diffusion and cross-wind diffusion into the solution that is grid and geometry dependent. New locally conservative higher-order multidimensional upwind schemes that significantly reduce both directional and cross-wind diffusion are presented for convective flow approximation. The new schemes are composed of two steps; (a) higher order approximation that corrects the directional diffusion of the approximation and (b) truly multidimensional upwind approximation, which involves flux approximation using upwind information obtained by upstream tracing along wave-vector paths where wave information travels in multiple dimensions. This approximation reduces cross-wind diffusion. Conditions on the tracing direction and Courant\u2013Friedrichs\u2013Levy number lead to a local maximum principle that ensures stable solutions free of spurious oscillations. The schemes are coupled with full-tensor Darcy flux approximations. Benefits of the resulting schemes are demonstrated for classical convective test cases in reservoir simulation, including cases with full tensor permeability fields, where the methods prove to be particularly effective. The test cases involve structured and unstructured grids with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher dimensional formulations are shown to effectively reduce numerical cross-wind diffusion effects, leading to improved resolution of concentration and saturation fronts.<\/jats:p>","DOI":"10.1137\/080727750","type":"journal-article","created":{"date-parts":[[2010,4,14]],"date-time":"2010-04-14T18:13:19Z","timestamp":1271268799000},"page":"1119-1139","source":"Crossref","is-referenced-by-count":20,"title":["Higher Order Multidimensional Upwind Convection Schemes for Flow in Porous Media on Structured and Unstructured Quadrilateral Grids"],"prefix":"10.1137","volume":"32","author":[{"given":"Sadok","family":"Lamine","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael G.","family":"Edwards","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,4,14]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2000.6672"},{"key":"R2","unstructured":"K. Aziz and A. Settari,\n                      Petroleum Reservoir Simulation\n                      , Applied Science Publishers, London, 1979."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(89)90054-5"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142902405394"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9991(05)80015-4"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.2118\/72469-PA"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(90)90233-Q"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1993.1072"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"M.G. Edwards,\n                      A higher order Godunov scheme coupled with dynamic local grid refinement for flow in a porous medium\n                      , Comput. Methods. Appl. Mech. Engrg. (1996), pp. 287\u2013308.","DOI":"10.1016\/0045-7825(95)00935-3"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2000.6418"},{"key":"R11","doi-asserted-by":"crossref","unstructured":"M.G. Edwards,\n                      Unstructured, control-volume distributed, full tensor finite volume schemes with flow based grids\n                      , Comput. Geosci. (2002), pp. 433\u2013452.","DOI":"10.1023\/A:1021243231313"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"M.G. Edwards,\n                      Higher dimensional wave oriented upwind schemes with minimal cross-wind diffusion\n                      , in Proceedings of the SPE Reservoir Simulation Symposium, Houston, TX, Society of Petroleum Engineers, 2003.","DOI":"10.2118\/79689-MS"},{"key":"R13","unstructured":"M.G. Edwards,\n                      Higher-order hyperbolic-coupled-elliptic flux-continuous CVD finite volume schemes in two and three dimensions\n                      , in Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, Egmond aan Zee, The Netherlands, 2006, pp. 273\u2013277."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1002\/fld.1289"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"M.G. Edwards,\n                      Higher-resolution hyperbolic-coupled-elliptic flux continuous CVD schemes on structured and unstructured grids in 3-d\n                      , Internat. J. Numer. Methods Fluids, (2006), pp. 1079\u20131095.","DOI":"10.1002\/fld.1289"},{"key":"R16","unstructured":"M.G. Edwards,\n                      Global and local central non-upwind finite volume schemes for hyperbolic conservation laws in porous media\n                      , Internat. J. Numer. Methods Fluids, http:\/\/www3.interscience.wiley.com\/journal\/122655605\/abstract."},{"key":"R17","doi-asserted-by":"crossref","unstructured":"M.G. Edwards and M.A. Christie,\n                      Dynamically adaptive Godunov schemes with renormalization for reservoir simulation\n                      , in Proceedings of the 12th SPE Reservoir Simulation Symposium, New Orleans, LA, Society of Petroleum Engineers, 1993.","DOI":"10.2118\/25268-MS"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/S106482750343028X"},{"key":"R19","unstructured":"C. Hirsch and P. van Ransbeek,\n                      Cell centered multidimensional upwind algorithm and structured meshes\n                      , in Proceedings of the 1st European CFD Conference, Vol. 1, Amsterdam, The Netherlands, Elsevier, 1992, pp. 53\u201360."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1997.5823"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"A. Jameson,\n                      Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence in transonic and hypersonic flows\n                      , in Proceedings of the 11th Computational Fluid Dynamics Conference, Orlando, FL, American Institute of Aeronautics and Astronautics 1993, 93-3359.","DOI":"10.2514\/6.1993-3359"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1007\/s11242-008-9281-1"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1002\/nme.2335"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"S. Lamine and M.G. Edwards,\n                      Higher-dimensional wave oriented upwind convection schemes for flow in porous media on structured and unstructured grids\n                      , in Advanced Computational Methods in Engineering, Liege, Belgium, 2008.","DOI":"10.3997\/2214-4609.20146379"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"S. Lamine and M.G. Edwards,\n                      Higher dimensional upwind schemes for flow in porous media on unstructured grids\n                      , in Proceedings of the 11th European Conference on the Mathematics of Oil Recovery, Bergen, Norway, 2008.","DOI":"10.3997\/2214-4609.20146379"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"S. Lamine and M.G. Edwards,\n                      Higher order multidimensional wave oriented upwind schemes for flow in porous media on unstructured grids\n                      , in Proceedings of the SPE Reservoir Simulation Symposium, The Woodlands, TX, Society of Petroleum Engineers, 2009.","DOI":"10.2118\/119187-MS"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021287013566"},{"key":"R28","doi-asserted-by":"crossref","unstructured":"R.J. Leveque,\n                      Finite Volume Methods for Hyperbolic Problems\n                      , Cambridge University Press, London, 2002.","DOI":"10.1017\/CBO9780511791253"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1007\/BF02818932"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1002\/fld.1211"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(76)90058-X"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1137\/0729089"},{"key":"R33","unstructured":"P.L. Roe, H. Deconinck, and R.J. Struijs von Karman,\n                      Fluctualtion Splitting for Multidimensional Convection Problems: An Alternative to Finite Volume and Finite Element Methods\n                      , VKI Computational Fluid Dynamics Lecture Series 2, von Karman Institute, 1990."},{"key":"R34","doi-asserted-by":"crossref","unstructured":"P.L. Roe, H. Deconinck, and R.J. Struijs von Karman,\n                      Recent progress in multidimensional upwinding\n                      , in Numerical Methods in Fluid Dynamics, K.W. Morton, ed., Lecture Notes in Phys. 371, Springer-Verlag, Berlin, 1990, pp. 273\u2013277.","DOI":"10.1007\/3-540-53619-1_182"},{"key":"R35","unstructured":"D. Sidilkover,\n                      A Genuinly Multidimensional Upwind Scheme and an Efficient Multigrid for Compressible Euler Equations\n                      , Technical report, ICASE, NASA Langley Research Center, Hampton, VA, 1994."},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2004.12.021"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1137\/0721062"},{"key":"R38","doi-asserted-by":"crossref","unstructured":"M. Thiele and M.G. Edwards,\n                      Physically based higher order Godunov schemes for reservoir simulation for compositional simulation\n                      , in Proceedings of the SPE Reservoir Simulation Symposium, Houston, TX, Society of Petroleum Engineers, 2001.","DOI":"10.2523\/66403-MS"},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(79)90145-1"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080727750","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:11:35Z","timestamp":1787328695000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080727750"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":39,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/080727750"],"URL":"https:\/\/doi.org\/10.1137\/080727750","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}