A stochastic mixed finite element heterogeneous multiscale method for flow in porous media

Journal of Computational Physics

Volumn 230, Issue 12, 2011, Pages 4696-4722

  Ma, Xiang ^a   Zabaras, Nicholas ^a  

^a School of Operations Research and Industrial Engineering   (United States)

Author keywords

Adaptivity; Flow in porous media; High dimensional model representation; Mixed finite element method; Sparse grids; Stochastic collocation method; Stochastic multiscale method; Stochastic partial differential equations

Indexed keywords

FINITE ELEMENT METHOD; NUMERICAL METHODS; POROUS MATERIALS; STOCHASTIC SYSTEMS;

ADAPTIVITY; FLOWS IN POROUS MEDIA; HIGH DIMENSIONAL MODEL REPRESENTATION; HIGH-DIMENSIONAL MODEL REPRESENTATIONS; MIXED FINITE ELEMENT METHODS; MULTISCALE METHOD; SPARSE GRID; STOCHASTIC COLLOCATION METHOD; STOCHASTIC MULTISCALE METHOD; STOCHASTIC PARTIAL DIFFERENTIAL EQUATION; STOCHASTICS;

STOCHASTIC MODELS;

EID: 79955652744     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2011.03.001     Document Type: Article

References (63)

1. Hou T.Y., Wu X.-H. A multiscale finite element method for elliptic problems in composite materials and porous media. Journal of Computational Physics 1997, 134(1):169-189.
2. Hou T.Y., Wu X.-H., Cai Z. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Mathematics of Computation 1999, 68:913-943.
3. Efendiev Y., Hou T.Y. Multiscale Finite Element: Theory and Applications 2009, Springer, New York.
4. Hughes T.J.R. Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Computer Methods in Applied Mechanics and Engineering 1995, 127:387-401.
5. Hughes T.J.R., Feijo G.R., Mazzei L., Quincy J.-B. The variational multiscale method - a paradigm for computational mechanics. Computer Methods in Applied Mechanics and Engineering 1998, 166:3-24.
6. Weinan E., Engquist B. The heterogeneous multi-scale methods. Communications in Mathematical Physics 2002, 1:87-132.
7. Weinan E., Engquist B., Li X., Ren W., Vanden-Eijnden E. Heterogeneous multiscale methods: a review. Communications in Computational Physics 2007, 2:367-450.
8. Chen Z., Hou T.Y. A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Mathematics of Computation 2003, 72:541-576.
9. Raviart P., Thomas J. A mixed finite element method for 2-nd order elliptic problems. Lecture Notes in Mathematics 1977, vol. 606:292-315. Springer-Verlag. Galligani, E. Magenes (Eds.).
10. Aarnes J.E. On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation. Multiscale Modeling and Simulation 2004, 2:421-439.
11. Aarnes J.E., Kippe V., Lie K.-A. Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels. Advances in Water Resources 2005, 28:257-271.
12. Jenny P., Lee S.H., Tchelepi H.A. Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. Journal of Computational Physics 2003, 187:47-67.
13. Jenny P., Lee S.H., Tchelepi H.A. Adaptive multiscale finite-volume method for multiphase flow and transport in porous media. Multiscale Modeling and Simulation 2005, 3:50-64.
14. Jenny P., Lee S., Tchelepi H. Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media. Journal of Computational Physics 2006, 217:627-641.
15. Arbogast T. Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow. Computational Geosciences 2002, 6:453-481.
16. Arbogast T. Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM Journal on Numerical Analysis 2004, 42:576-598.
17. Juanes R., Dub F.-X. A locally conservative variational multiscale method for the simulation of porous media flow with multiscale source terms. Computational Geosciences 2008, 12:273-295.
18. Kippe V., Aarnes J.E., Lie K.-A. A comparison of multiscale methods for elliptic problems in porous media flow. Computational Geosciences 2008, 12:377-398.
19. Ren W. Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. Journal of Computational Physics 2005, 204:1-26.
20. Li X. Multiscale modeling of the dynamics of solids at finite temperature. Journal of the Mechanics and Physics of Solids 2005, 53:1650-1685.
21. Abdulle A., Schwab C. Heterogeneous multiscale FEM for diffusion problems on rough surfaces. Multiscale Modeling and Simulation 2005, 3:195-220.
22. Abdulle A., Nonnenmacher A. A short and versatile finite element multiscale code for homogenization problems. Computer Methods in Applied Mechanics and Engineering 2009, 198:2839-2859.
23. Abdulle A. The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs. GAKUTO International Series Mathematical Sciences and Applications 2009, 31:135-184.
24. E W., Ming P., Zhang P. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. Journal of the American Mathematical Society 2004, 18:121-156.
25. Abdulle A. On a priori error analysis of fully discrete heterogeneous multiscale FEM. Multiscale Modeling and Simulation 2005, 4:447-459.
26. Abdulle A., Nonnenmacher A. A posteriori error analysis of the heterogeneous multiscale method for homogenization problems. Comptes Rendus Mathematique 2009, 347:1081-1086.
27. Chen Z. Multiscale methods for elliptic homogenization problems. Numerical Methods for Partial Differential Equations 2006, 22:317-360.
28. Ghanem R. Probabilistic characterization of transport in heterogeneous media. Computer Methods in Applied Mechanics and Engineering 1998, 158:199-220.
29. Ghanem R. Scales of fluctuation and the propagation of uncertainty in random porous media. Water Resources Research 1998, 34:2123-2136.
30. Ghanem R., Dham S. Stochastic finite element analysis for multiphase flow in heterogeneous porous media. Transport in Porous Media 1998, 32:239-262.
31. Zhang D. Stochastic Method for Flow in Porous Media: Coping with Uncertainties 2002, Academic Press, San Diego.
32. Zhang D., Tchelepi H. Stochastic analysis of immiscible two-phase flow in heterogeneous media. SPE Journal 1999, 4:380-388.
33. Zhang D., Li L., Tchelepi H. Stochastic formulation for uncertainty analysis of two-phase flow in heterogeneous reservoirs. SPE Journal 2000, 5:60-70.
34. Jarman K.D., Russell T.F. Eulerian moment equations for 2-d stochastic immiscible flow. Multiscale Modeling and Simulation 2003, 1:598-608.
35. Ganis B., Klie H., Wheeler M.F., Wildey T., Yotov I., Zhang D. Stochastic collocation and mixed finite elements for flow in porous media. Computer Methods in Applied Mechanics and Engineering 2008, 197:3547-3559.
36. Li H., Zhang D. Probabilistic collocation method for flow in porous media: comparisons with other stochastic methods. Water Resources Research 2007, 43:W09409.
37. Lin G., Tartakovsky A. An efficient, high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media. Advances in Water Resources 2009, 32:712-722.
38. Lin G., Tartakovsky A. Numerical studies of three-dimensional stochastic Darcys equation and stochastic advection-diffusion-dispersion equation. Journal of Scientific Computing 2010, 43:92-117.
39. Babuška I., Nobile F., Tempone R. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Journal on Numerical Analysis 2007, 45:1005-1034.
40. Xiu D., Hesthaven J.S. High-order collocation methods for differential equations with random inputs. SIAM Journal on Scientific Computing 2005, 27:1118-1139.
41. Ganapathysubramanian B., Zabaras N. Sparse grid collocation schemes for stochastic natural convection problems. Journal of Computational Physics 2007, 225:652-685.
42. Nobile F., Tempone R., Webster C.G. A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM Journal on Numerical Analysis 2008, 46:2309-2345.
43. Ma X., Zabaras N. An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. Journal of Computational Physics 2009, 228:3084-3113.
44. Ma X., Zabaras N. An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations. Journal of Computational Physics 2010, 229:3884-3915.
45. Li G., Rosenthal C., Rabitz H. High dimensional model representations. The Journal of Physical Chemistry A 2001, 105:7765-7777.
46. Asokan B.V., Zabaras N. A stochastic variational multiscale method for diffusion in heterogeneous random media. Journal of Computational Physics 2006, 218:654-676.
47. Ganapathysubramanian B., Zabaras N. Modeling diffusion in random heterogeneous media: data-driven models, stochastic collocation and the variational multiscale method. Journal of Computational Physics 2007, 226:326-353.
48. Xu X.F. A multiscale stochastic finite element method on elliptic problems involving uncertainties. Computer Methods in Applied Mechanics and Engineering 2007, 196:2723-2736.
49. Ganapathysubramanian B., Zabaras N. A stochastic multiscale framework for modeling flow through random heterogeneous porous media. Journal of Computational Physics 2009, 228:591-618.
50. Aarnes J.E., Efendiev Y. Mixed multiscale finite element methods for stochastic porous media flows. SIAM Journal on Scientific Computing 2008, 30:2319-2339.
51. Dostert P., Efendiev Y., Hou T. Multiscale finite element methods for stochastic porous media flow equations and application to uncertainty quantification. Computer Methods in Applied Mechanics and Engineering 2008, 197:3445-3455.
52. Chen Z., Du Q. An upwinding mixed finite element method for a mean field model of superconducting vortices. Mathematical Modelling and Numerical Analysis 2000, 34:687-706.
53. L.J. Durlofsky, Upscaling of geocellular models for reservoir flow simulation: a review of recent progress, in: 7 th International Forum on Reservoir Simulation, Baden-Baden, Germany, 2003.
54. Ghanem R., Spanos P.D. Stochastic Finite Elements: A Spectral Approach 1991, Springer-Verlag, New York.
55. Brezzi F., Fortin M. Mixed and Hybrid Finite Element Method 1991, Springer-Verlag, New York.
56. Yue X., E W. The local microscale problem in the multiscale modeling of strongly heterogeneous media: effects of boundary conditions and cell size. Journal of Computational Physics 2007, 222:556-572.
57. Wen X.H., Durlofsky L.J., Edwards M.G. Use of border regions for improved permeability upscaling. Mathematical Geology 2003, 35:521-547.
58. Gautier Y., Blunt M., Christie M. Nested gridding and streamline-based simulation for fast reservoir performance prediction. Computational Geosciences 1999, 3:295-320.
59. S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc Web page, 2009. http://www.mcs.anl.gov/petsc.
60. Chen Y., Durlofsky L.J., Gerritsen M., Wen X.H. A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations. Advances in Water Resources 2003, 26:1041-1060.
61. Zabaras N., Samanta D. A stabilized volume-averaging finite element method for flow in porous media and binary alloy solidification processes. Advances in Water Resources 2004, 60:1103-1138.
62. Christie M.A., Blunt M. Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reservoir Evaluation and Engineering 2001, 4:308-317.
63. X. Ma, N. Zabaras, Kernel principal component analysis for stochastic input mode generation, Journal of Computational Physics, under review.