Multiscale finite element methods for stochastic porous media flow equations and application to uncertainty quantification

Computer Methods in Applied Mechanics and Engineering

Volumn 197, Issue 43-44, 2008, Pages 3445-3455

  Dostert, P ^a   Efendiev, Y ^a   Hou, T Y ^b  

^a TEXAS A AND M UNIVERSITY   (United States)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Multiscale; Porous media; Two phase flow; Uncertainty quantification; Upscaling

Indexed keywords

CHAOTIC SYSTEMS; MANAGEMENT SCIENCE; MARKOV PROCESSES; MATHEMATICAL MODELS; MONTE CARLO METHODS; POROUS MATERIALS; STOCHASTIC PROGRAMMING;

ANALYTICAL COMPUTATIONS; BASIS FUNCTIONS; DIFFUSION CO-EFFICIENT; FINITE DIMENSIONAL; FINITE-ELEMENT METHODS; INTEGRATED RESPONSE; LANGEVIN; MARKOV CHAIN MONTE CARLO METHODS; MULTI SCALING; MULTI-SCALE METHODS; MULTISCALE; NUMERICA L RESULTS; POROUS MEDIA; POROUS MEDIUMS; POROUS-MEDIA FLOW; PRODUCTION DATA; RESPONSE SURFACES; SCALE MODELING; TWO-PHASE FLOW; UNCERTAINTY QUANTIFICATION; UPSCALING;

FINITE ELEMENT METHOD;

EID: 49049087504     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2008.02.030     Document Type: Article

References (31)

1. Aarnes J. On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation. SIAM MMS 2 (2004) 421-439
2. J. Aarnes, Y. Efendiev, Mixed multiscale finite element methods for stochastic porous media flows, in press.
3. Arbogast T. Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow. Comput. Geosci. 6 (2002) 453-481 Locally conservative numerical methods for flow in porous media
4. Arbogast T., Pencheva G., Wheeler M.F., and Yotov I. A multiscale mortar mixed finite element method. SIAM J. Multiscale Model. Simul. 6 1 (2007) 319-346
5. Babuška I., Caloz G., and Osborn E. Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31 (1994) 945-981
6. Babuška I., and Osborn E. Generalized finite element methods: Their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20 (1983) 510-536
7. Barthelmann V., Novak E., and Ritter K. High dimensional polynomial interpolation on sparse grids. Adv. Comput. Math. 12 (2000) 273-288
8. Brezzi F. Interacting with the subgrid world. Numerical Analysis 1999 (Dundee) (2000), Chapman & Hall/CRC, Boca Raton, FL 69-82
9. Y. Chen, L. Durlofsky, An ensemble level upscaling approach for efficient estimation of fine-scale production statistics using coarse-scale simulations, SPE paper 106086, Presented at the SPE Reservoir Simulation Symposium, Houston, February 26-28, 2007.
10. A. Christen, C. Fox, MCMC using an approximation. Technical report, Department of Mathematics, The University of Auckland, New Zealand.
11. P. Dostert, Uncertainty quantification using multiscale methods for porous media flows, Ph.D. Thesis, Texas A&M University, 2007.
12. Dostert P., Efendiev Y., Hou T., and Luo W. Coarse-gradient Langevin algorithms for dynamic data integration and uncertainty quantification. J. Comput. Phys. 217 (2006) 123-142
13. Durlofsky L.J. Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resour. Res. 27 (1991) 699-708
14. Efendiev Y., Ginting V., Hou T., and Ewing R. Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys. 220 1 (2006) 155-174
15. Efendiev Y., Hou T., and Ginting V. Multiscale finite element methods for nonlinear problems and their applications. Comm. Math. Sci. 2 (2004) 553-589
16. Efendiev Y., Hou T., and Luo W. Preconditioning Markov chain Monte Carlo simulations using coarse-scale models. SIAM Sci. 28 2 (2006) 776-803
17. Efendiev Y.R., Hou T.Y., and Wu X.H. Convergence of a nonconforming multiscale finite element method. SIAM J. Numer. Anal. 37 (2000) 888-910
18. Y. Efendiev, A. Datta-Gupta, V. Ginting, X. Ma, B. Mallick, An efficient two-stage Markov chain Monte Carlo method for dynamic data integration. Water Resour. Res., 41, W12423. doi:10.1029/2004WR003764.
19. Efendiev Y., and Pankov A. Numerical homogenization of nonlinear random parabolic operators. SIAM Multiscale Model. Simul. 2 2 (2004) 237-268
20. Ganapathysubramanian B., and Zabaras N. Modeling diffusion in random heterogeneous media: Data-driven models, stochastic collocation and the variational multiscale method. J. Comput. Phys. 226 (2007) 326-353
21. Grenander U., and Miller M.I. Representations of knowledge in complex systems (with discussion). J. R. Statist. Soc. B 56 (1994) 549-603
22. Hou T.Y., and Wu X.H. A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189
23. Hughes T., Feijoo G., Mazzei L., and Quincy J. The variational multiscale method - a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3-24
24. Jenny P., Lee S.H., and Tchelepi H. Multi-scale finite volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187 (2003) 47-67
25. Liu J. Monte Carlo Strategies in Scientific Computing (2001), Springer, New York
26. Loève M. Probability Theory. fourth ed. (1977), Springer, Berlin
27. Oliver D., Cunha L., and Reynolds A. Markov chain Monte Carlo methods for conditioning a permeability field to pressure data. Math. Geol. 29 (1997)
28. Robert C., and Casella G. Monte Carlo Statistical Methods (1999), Springer-Verlag, New York
29. Smolyak S.A. Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 4 (1963) 240-243
30. Wong E. Stochastic Processes in Information and Dynamical Systems (1971), McGraw-Hill
31. Xiu D., and Hesthaven J. High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. Wol. 27 3 (2005) 1118-1139