The three-dimensional displacement of two-phase flow in porous media is a basic problem of numerical simulation of energy science and mathematics. The mathematical model is formulated by a system of nonlinear partial differential equations to describe incompressible miscible case. The pressure equation is elliptic and the concentration is defined by a convection-dominated diffusion equation. The pressure generates Darcy velocity and controls the dynamic change of concentration. We adopt a conservative block-centered scheme to approximate the pressure and Darcy velocity, and the accuracy of Darcy velocity is improved by one order. We use a block-centered upwind method to solve the concentration, where the diffusion term and convection term are approximated by a block-centered scheme and an upwind scheme, respectively. The composite algorithm is effective to solve convection-dominated problem, since numerical oscillation and dispersion are avoided and computational accuracy is improved. Block-centered method is conservative, and the concentration and the adjoint function are computed simultaneously. This physical nature is important in numerical simulation of seepage fluid. Using the convergence theory and techniques of priori estimates, we derive optimal estimate error. Numerical experiments and data show the support and consistency of theoretical result. The argument in the present paper shows a powerful tool to solve the well-known model problem.

three-dimensional incompressible miscible displacement, block-centered upwind method, local conservation, convergence analysis, numerical computation.

Received: February 18, 2022; Accepted: April 4, 2022; Published: November 7, 2022

How to cite this article: Li Changfeng, Yuan Yirang, Song Huailing and Sun Tongjun, Block-centered upwind method for two-phase displacement and convergence analysis, Far East Journal of Applied Mathematics 115 (2022), 25-59. http://dx.doi.org/10.17654/0972096022017

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] R. E. Ewing, The Mathematics of Reservoir Simulation, SIAM, Philadelphia, 1983.
[2] J. Douglas, Jr., Finite difference method for two-phase in compressible flow in porous media, SIAM J. Numer. Anal. 4 (1983), 681-696.
[3] T. F. Russell, Time stepping along characteristics with incomplete interaction for a Galerkin approximation of miscible displacement in porous media, SIAM J. Numer. Anal. 22(5) (1985), 970-1013.
[4] R. E. Ewing, T. F. Russell and M. F. Wheeler, Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics, Comput. Methods Appl. Mech. Engrg. 47(1-2) (1984), 73-92.
[5] J. Douglas, Jr. and Y. R. Yuan, Numerical simulation of immiscible flow in porous media based on combining the method of characteristics with mixed finite element procedure, Numerical Simulation in Oil Rewvery, New York, Springer-Berlag, 1986, 119-132.
[6] Y. R. Yuan, Theory and application of numerical simulation of energy sources, Beijing, Science Press, 2013, 257-304.
[7] C. N. Dawson, T. F. Russell and M. F. Wheeler, Some improved error estimates for the modified method of characteristics, SIAM J. Numer. Anal. 26(6) (1989), 1487-1512.
[8] Z. Cai, On the finite volume element method, Numer. Math. 58 (1991), 713-735.
[9] R. H. Li and Z. Y. Chen, Generalized difference of differential equations, Changchun: Jilin University Press, 1994.
[10] P. A. Raviart and J. M. Thomas, A mixed finite element method for second order elliptic problems, in: Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics, 606, Springer, 1977.
[11] J. Douglas, Jr., R. E. Ewing and M. F. Wheeler, Approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Anal. Numer. 17(1) (1983), 17-33.
[12] J. Douglas, Jr., R. E. Ewing and M. F. Wheeler, A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media, RAIRO Anal. Numer. 17(3) (1983), 249-265.
[13] T. F. Russell, Rigorous block-centered discretization on inregular grids: Improved simulation of complex reservoir systems, Project Report, Research Corporation, Tulsa, 1995.
[14] A. Weiser and M. F. Wheeler, On convergence of block-centered finite difference for elliptic problems, SIAM J. Numer. Anal. 25(2) (1988), 351-375.
[15] J. E. Jones, A mixed volume method for accurate computation of fluid velocities in porous media, Ph. D. Thesis, University of Colorado, Denver, Co. 1995.
[16] Z. Cai, J. E. Jones, S. F. Mccormilk and T. F. Russell, Control-volume mixed finite element methods, Comput. Geosci. 1 (1997), 289-315.
[17] S. H. Chou, D. Y. Kawk and P. Vassileviki, Mixed volume methods on rectangular grids for elliptic problem, SIAM J. Numer. Anal. 37 (2000), 758-771.
[18] S. H. Chou, D. Y. Kawk and P. Vassileviki, Mixed volume methods for elliptic problems on triangular grids, SIAM J. Numer. Anal. 35 (1998), 1850-1861.
[19] S. H. Chou and P. Vassileviki, A general mixed covolume frame work for constructing conservative schemes for elliptic problems, Math. Comp. 12 (2003), 150-161.
[20] H. X. Rui and H. Pan, A block-centered finite difference method for the Darcy-Forchheimer Model, SIAM J. Numer. Anal. 50(5) (2012), 2612-2631.
[21] H. Pan and H. X. Rui, Mixed element method for two-dimensional Darcy-Forchheimer model, J. of Scientific Computing 52(3) (2012), 563-587.
[22] Y. R. Yuan, Y. X. Liu, C. F. Li, T. J. Sun and L. Q. Ma, Analysis on block-centered finite differences of numerical simulation of semiconductor detector, Appl. Math. Comput. 279 (2016), 1-15.
[23] Y. R. Yuan, Q. Yang, C. F. Li and T. J. Sun, Numerical method of mixed finite volume-modified upwind fractional step difference for three-dimensional semiconductor device transient behavior problems, Acta. Mathematica Scientia 37B(1) (2017), 259-279.
[24] P. P. Shen, M. X. Liu and L. Tang, Mathematical model of petroleum exploration and development, Beijing: Science Press, 2002.
[25] T. Arbogast, M. F. Wheeler and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal. 34(2) (1997), 828-852.
[26] J. Nitsche, Linear splint-funktionen and die methoden von Ritz for elliptishce randwert probleme, Arch. for Rational Mech. and Anal. 36 (1968), 348-355.
[27] L. S. Jiang and Z. Y. Pang, Finite element method and its theory, Beijing: People’s Education Press, 1979.
[28] Y. R. Yuan, A. J. Cheng and D. P. Yang, Theory and actual applications of numerical simulation of oil reservoir, Beijing: Science Press, 2016.