The three-dimensional displacement problem of oil and water incompressible miscible flow in porous media is preliminary and important in numerical simulation of energy science and mathematics, and its mathematical model is formulated by a nonlinear system of partial differential equations. The pressure and the concentration are defined by an elliptic equation and a convection-dominated diffusion equation, respectively. The pressure determines Darcy velocity and affects the whole physical process. We adopt a conservative block-centered scheme to approximate the pressure and Darcy velocity, and the accuracy of Darcy velocity is improved one order. We use a block-centered upwind method on changing meshes to solve the concentration, where the changing partition is used on the whole computational region. 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 problems, 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 a priori estimates, we derive optimal order estimate errors. Numerical experiments show the feasibility and efficiency, and numerical data are consistent with theoretical results. Thus, this method can be applied to solve some actual problems.

three-dimensional incompressible miscible displacement, block-centered upwind differences on changing meshes, local conservation, convergence analysis, numerical computation.

Received: July 1, 2022; Accepted: January 24, 2023; Published: April 19, 2023

How to cite this article: Yuan Yirang, Song Huailing and Changfeng Li, Block-centered upwind difference on changing meshes for numerical simulation of seepage flow, Far East Journal of Applied Mathematics 116(2) (2023), 173-213. http://dx.doi.org/10.17654/0972096023010

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 incompressible flow in porous media, SIAM J. Numer. Anal. 20(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 Recovery, New York, Springer-Berlag, 1986, pp. 119-132.
[6] Y. R. Yuan, Theory and Application of Numerical Simulation of Energy Sources, Beijing, Science Press, 2013, pp. 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(1) (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, 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 irregular 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(3) (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(3) (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(5) (1998), 1850-1861.
[19] S. H. Chou and P. Vassileviki, A general mixed covolume framework for constructing conservative schemes for elliptic problems, Math. Comp. 68(227) (1999), 991-1011.
[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. Sci. Comput. 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] M. Ohlberger, Convergence of a mixed finite elements-finite volume method for the two phase flow in porous media, East-West J. Numer. Math. 5(3) (1997), 183 210.
[25] C. Chainais-Hillairet and J. Droniou, Convergence analysis of a mixed finite volume scheme for an elliptic-parabolic system modeling miscible fluid flows in porous media, SIAM J. Numer. Anal. 45(5) (2007), 2228-2258.
[26] B. Amaziane and M. El Ossmani, Convergence analysis of an approximation to miscible fluid flows in porous media by combining mixed finite element and finite volume methods, Numer. Methods Partial Differential Equations 24(3) (2008), 799-832.
[27] C. N. Dawson and R. Kirby, Solution of parabolic equations by backward Euler-mixed finite element method on a dynamically changing mesh, SIAM J. Numer. Anal. 37(2) (2000), 423-442.
[28] P. P. Shen, M. X. Liu and L. Tang, Mathematical Model of Petroleum Exploration and Development, Beijing, Science Press, 2002.
[29] 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.
[30] J. Nitsche, Lineare spline-funktionen und die methoden von Ritz für elliptische randwertprobleme, Arch. for Rational Mech. Anal. 36(5) (1970), 348-355.
[31] L. S. Jiang and Z. Y. Pang, Finite Element Method and its Theory, Beijing, People’s Education Press, 1979.
[32] Y. R. Yuan, A. J. Cheng and D. P. Yang, Theory and Actual Applications of Numerical Simulation of Oil Reservoir, Beijing, Science Press, 2016.