Numerical homogenization of well singularities in the flow transport through heterogeneous porous media

Multiscale Modeling and Simulation

Volumn 1, Issue 2, 2003, Pages 260-303

  Chen, Zhiming ^a   Yue, Xinye ^a,b  

^a INSTITUTE OF GEOLOGY AND GEOPHYSICS   (China)

^b SOOCHOW UNIVERSITY   (China)

Author keywords

Heterogeneous porous media; Numerical homogenization; Well singularity

Indexed keywords

FINITE ELEMENT METHOD; NUMERICAL METHODS; POROUS MATERIALS;

CONVERGENT ALGORITHMS; HETEROGENEOUS POROUS MEDIA; HOMOGENIZATION THEORY; MACROSCOPIC PARAMETERS; MULTISCALE FINITE ELEMENT METHOD; NUMERICAL EXPERIMENTS; NUMERICAL HOMOGENIZATION; SIMPLIFYING ASSUMPTIONS;

HOMOGENIZATION METHOD;

EID: 6344242235     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/S1540345902413322     Document Type: Article

References (30)

1. M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math., 40 (1987), pp. 803-847.
2. M. Avellaneda and F.-H. Lin, Lp bounds on singular integrals in homogenization, Comm. Pure Appl. Math., 44 (1991), pp. 897-910.
3. A. Bensoussan, J.L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.
4. Z. Chen and T.Y. Hou, A mixed multiscale finite element method for elliptic problem with oscillating coefficients, Math. Comp., 72 (2003), pp. 541-576.
5. G. Christakos, Random Field Models in Earth Sciences, Academic Press, San Diego, CA, 1992.
6. C. Conca, S. Natesan, and M. Vanninathan, Numerical solution of elliptic partial differential equations by Bloch waves method, in XVII CEDYA: Congress on Differential Equations and Applications/VII: Congress on Applied Mathematics (Spanish) (Salamanca, 2001), Departamento de Matematica Aplicada, Universidad de Salamanca, Salamanca, Spain, 2001, pp. 63-83.
7. L.J. Durlofsky, Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water Resources Research, 27 (1991), pp. 699-708.
8. L.J. Durlofsky, W.J. Milliken, and A. Bernath, Scale up in the near-well region, in Proceedings of the 15th SPE Symposium on Reservoir Simulation, Houston, TX, 1999.
9. B.B. Dykaar and P.K. Kitanidis, Transmissivity of a heterogeneous formation, Water Resource Research, 29 (1993), pp. 985-1001.
10. Y.R. Efendiev, T.Y. Hou, and X.-H. Wu, Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal., 37 (2000), pp. 888-910.
11. H.H. Gerke, U. Hornung, Y. Kelanemer, M. Slodiča, and S. Schumacher, Optimal Control of Soil Venting: Mathematical Modelling and Applications, Birkhäuser Verlag, Basel, 1999.
12. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
13. P. Grisvard, Elliptic Problems on Non-smooth Domains, Pitman, Boston, 1985.
14. M. Grüter and K.-O. Widman, The Green function for uniformly elliptic equations, Manuscripta Math., 37 (1982), pp. 303-342.
15. T.Y. Hou and X.H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), pp. 169-189.
16. T.Y. Hou, X.H. Wu, and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 68 (1999), pp. 913-943.
17. V.V. Jikov, S.M. Kozlov, and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.
18. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
19. D.Q. Li and S.X. Chen, On asymptotic behaviors of the equi-valued surface boundary value problem for the second order self-adjoint elliptic equations, Fudan Journal (Natural Sciences), 17 (1978), pp. 6-14 (in Chinese).
20. W. Littman, G. Stampacchia, and H.F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), pp. 43-77.
21. O. Mascarenhas and L.J. Durlofsky, Scale up in the vicinity of horizontal wells, in Proceedings of the 20th Annual International Energy Agency Workshop and Symposium, Paris, France, 1999.
22. S. Moskow and M. Vogelius, First order corrections to the homogenized eigenvalues of a periodic composite medium. A convergence proof, Proc. Royal Soc. Edinburgh Sect. A, 127 (1997), pp. 1263-1299.
23. A. Matache, I. Babuška, and C. Schwab, Generalized p-fem in homogenization, Numer. Math., 86 (2000), pp. 319-375.
24. D.W. Peaceman, Interpretation of well-block pressures in numerical reservoir simulations, Soc. Pet. Eng. J., 18 (1978), pp. 183-194.
25. D.W. Peaceman, Interpretation of well-block pressures in numerical reservoir simulations with non-square grid blocks and anisotropic permeability, Soc. Pet. Eng. J., 23 (1983), pp. 531-543.
26. Ph. Renard and G. de Marsily, Calculating equivalent permeability: A review, Adv. Water Res., 20 (1997), pp. 253-278.
27. A.H. Schatz and L.B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp., 31 (1977), pp. 414-442.
28. G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York, 1987.
29. L.B. Wahlbin, Local behavior in finite element methods, in Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet and J.L. Lions, eds., Elsevier, Paris, 1991, pp. 355-521.
30. P.M. De Zeeuw, Matrix-dependent prolongation and restrictions in a blackbox multigrid solver, J. Comput. Appl. Math., 33 (1990), pp. 1-27.