Characteristic fractional step finite difference method for nonlinear section coupled system

doi:10.1007/s10483-014-1869-9

Abstract

Abstract: For the section coupled system of multilayer dynamics of fluids in porous media, a parallel scheme modified by the characteristic finite difference fractional steps is proposed for a complete point set consisting of coarse and fine partitions. Some tech- niques, such as calculus of variations, energy method, twofold-quadratic interpolation of product type, multiplicative commutation law of difference operators, decomposition of high order difference operators, and prior estimates, are used in theoretical analysis. Optimal order estimates in l² norm are derived to show accuracy of the second order approximation solutions. These methods have been used to simulate the problems of migration-accumulation of oil resources.

Key words: characteristic fractional step, numerical simulation, convergence, three-dimensional section coupled system, complete nonlinear equation

2010 MSC Number:

O241.82

Yi-rang YUAN;Chang-feng LI;Tong-jun SUN;Yun-xin LIU. Characteristic fractional step finite difference method for nonlinear section coupled system. Applied Mathematics and Mechanics (English Edition), 2014, 35(10): 1311-1330.

TrendMD

References

[1] Ewing, R. E. The Mathematics of Reservoir Simulation, SIAM, Philadelphia (1983)
[2] Bredehoeft, J. D. and Pinder, G. F. Digital analysis of areal flow in multiaquifer groundwater systems: a quasi-three-dimensional model. Water Resources Research, 6(3), 883-888 (1970)
[3] Don, W. and Emil, O. F. An iterative quasi-three-dimensional finite element model for heteroge- neous multiaquifer systems. Water Resources Research, 14(5), 943-952 (1978)
[4] Ungerer, P., Dolyiez, B., Chenet, P. Y., Burrus, J., Bessis, F., Lafargue, E., Giroir, G., Heum, O., and Eggen, S. A 2-D model of basin petroleum by two-phase fluid flow, application to some case studies. Migration of Hydrocarbon in Sedimentary Basins (ed. Dolyiez, B.), Editions Techniq, Paris, 414-455 (1987)
[5] Ungerer, P. Fluid flow, hydrocarbon generation and migration. AAPG Bull, 74(3), 309-335 (1990)
[6] Douglas, J. Jr. and Russell, T. F. Numerical method for convection-dominated diffusion prob- lems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal., 19(5), 871-885 (1982)
[7] Douglas, J. Jr. Finite difference methods for two-phase incompressible flow in porous media. SIAM J. Numer. Anal., 20(4), 681-696 (1983)
[8] Bermudez, A., Nogueriras, M. R., and Vazquez, C. Numerical analysis of convection-diffusion- reaction problems with higher order characteristics/finite elements, part I: time diseretization. SIAM J. Numer. Anal., 44(5), 1829-1853 (2006)
[9] Bermudez, A., Nogueriras, M. R., and Vazquez, C. Numerical analysis of convection-diffusion- reaction problems with higher order characteristics/finite elements, part II: fuly diseretized scheme and quadratare formulas. SIAM J. Numer. Anal., 44(5), 1854-1876 (2006)
[10] Peaceman, D. W. Fundamental of Numerical Reservoir Simulation, Elsevier, Amsterdam (1980)
[11] Yuan, Y. R. C-F-D method for moving boundary value problem. Science in China, Ser. A, 24(10), 1029-1036 (1994)
[12] Yuan, Y. R. C-F-D method for enhanced oil recovery simulation. Science in China, Ser. A, 36(11), 1296-1307 (1993)
[13] Marchuk, G. I. Splitting and alternating direction method. Handbook of Numerical Analysis (eds. Ciarlet, P. G. and Lions, J. L.), Elesevior Science Publishers, Paris, 197-460 (1990)
[14] Douglas, J. Jr. and Gunn, J. E. Two order correct difference analogous for the equation of multi- dimensional heat flow. Math. Comp., 17(81), 71-80 (1963)
[15] Douglas, J. Jr. and Gunn, J. E. A general formulation of alternating direction methods, part 1. parabolic and hyperbolic problems. Numer. Math., 6(5), 428-453 (1964)
[16] Yuan, Y. R. Fractional steps method. Science in China, Ser. A, 28(10), 893-902 (1998)
[17] Yuan, Y. R. Upwind finite difference fractional steps method. Science in China, Ser. A, 45(5), 578-593 (2002)
[18] Yuan, Y. R. Characteristic fractional step finite difference method. Science in China, Ser. A, 35(11), 1-27 (2005)
[19] Lazarov, R. D., Mischev, I. D., and Vassilevski, P. S. Finite volume methods for convection- diffusion problems. SIAM J. Numer. Anal., 33(1), 31-55 (1996)
[20] Ewing, R. E. Mathematical modeling and simulation for multiphase flow in porous media. Nu- merical Treatment of Multiphase Flows in Porous Media, Springer-Verlag, New York (2000)
[21] Yuan, Y. R. and Han, Y. J. Numerical simulation of migration-accumulation of oil resources. Comput. Geosi., 12, 153-162 (2008)
[22] Yuan, Y. R., Wang, W. Q., and Han, Y. J. Theory, method and application of a numerical simulation in an oil resources basin method of numerical solution of aerodynamic problems. Special Topics & Reviews in Porous Media-An International Journal, 1, 49-66 (2012)
[23] Samarskii, A. A. and Andreev, B. B. Finite Difference Methods for Elliptic Equation, Science Press, Beijing (1994)
[24] Ewing, R. E., Lazarov, R. D., and Vassilev, A. T. Finite difference scheme for parabolic problems on composite grids with refinement in time and space. SIAM J. Numer. Anal., 31(6), 1605-1622 (1994)