MIXED VOLUME ELEMENT-CHARACTERISTIC MIXED FINITE ELEMENT METHOD OF CHEMICAL OIL RECOVERY SEEPAGE DISPLACEMENT WITH CAPILLARY FORCE
Physical model of numerical simulation of three-dimensional chemical oil recovery with capillary force is described by a system of seepage displacement, whose mathematical model is described by several partial differential equations. The flow equation is an elliptic equation for the pressure, the concentration is defined by a convection-diffusion equation and the saturations of different components are defined by a nonlinear system of convection-diffusion equations. Darcy velocity, determined by the pressure, appears in the concentration and the saturations of different components and controls their motions. The flow equation is solved by the conservative mixed volume element and one order of accuracy is improved for the computation of Darcy velocity. The mixed volume element with the characteristics is applied to approximate the concentration, that is to say, that the diffusion is discretized by mixed volume element and the convection is computed by the method of characteristics. The method of characteristics can confirm the high computation stability at the sharp fronts and can avoid numerical dispersion and nonphysical oscillation. Though it adopts a large step, it has small time truncation error and has more high order of accuracy. The mixed volume element is applied to approximate the diffusion. It can compute the concentration and the adjoint vector function simultaneously, and it confirms the locally conservative law of mass. The saturations of different components are solved by the method of characteristic fractional step difference and the computational work is shortened greatly by decomposing a three-dimensional problem into three successive one-dimensional problems and using the algorithm of speedup. Using the theory and technique of priori estimates of differential equations, we obtain the convergence of optimal second order in L2-norm. This method gives a powerful tool to solve international famous problems.
chemical oil recovery, capillary force, mixed volume element-characteristic mixed volume element, local conservation of mass, second-order error estimate in L2-norm.
Received: February 2, 2021; Accepted: March 2, 2021; Published: March 19, 2021
How to cite this article: Yuan Yirang, Cheng Aijie, Li Changfeng and Yang Qing, Mixed volume element-characteristic mixed finite element method of chemical oil recovery seepage displacement with capillary force, Far East Journal of Applied Mathematics 109(2) (2021), 167-211. DOI: 10.17654/AM109020167
This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License
References:
[1] R. E. Ewing, Y. R. Yuan and G. Li, Finite element for chemical-flooding simulation, Proceeding of the 7th International Conference Finite Element Method in Flow Problems, The University of Alabama in Huntsville, Huntsville, Alabama: UAHDRESS, 1989, pp. 1264-1271.[2] Y. R. Yuan, Theory and Application of Numerical Simulation of Energy Sources, Science Press, Beijing, 2013, pp. 257-304.[3] Y. R. Yuan, D. P. Yang and L. Q. Qi et al., Research on algorithms of applied software of the polymer, Qinlin Gang (Editor-in-Chief), Proceedings on Chemical Flooding, Petroleum Industry Press, Beijing, 1998, pp. 246-253.[4] J. Douglas, Jr., Finite difference method for two-phase in compressible flow in porous media, SIAM J. Numer. Anal. 4 (1983), 681-696.[5] J. Douglas, Jr. and T. F. Russell, Numerical method for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal. 19(5) (1982), 871-885.[6] J. Douglas, Jr., Simulation of miscible displacement in porous media by a modified method of characteristic procedure, Numerical Analysis, Dundee, 1981, Lecture Notes in Mathematics, 912, Springer-Verlag, Berlin, 1982.[7] 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.[8] J. Douglas, Jr. and J. E. Roberts, Numerical methods for a model for compressible miscible displacement in porous media, Math. Comp. 41(164) (1983), 441-459.[9] Y. R. Yuan, Time stepping along characteristics for the finite element approximation of compressible miscible displacement in porous media, Math. Numer. Sin. 14(4) (1992), 385-400.[10] Y. R. Yuan, Finite difference methods for a compressible miscible displacement problem in porous media, Math. Numer. Sin. 15(1) (1993), 16-28.[11] R. E. Ewing, The Mathematics of Reservoir Simulation, SIAM, Philadelphia, 1983.[12] Y. R. Yuan, Characteristic finite difference methods for moving boundary value problem of numerical simulation of oil deposit, Sci. Sin. Math. 37(12) (1994), 1442-1453.[13] Y. R. Yuan, The characteristic mixed finite element method and analysis for three-dimensional moving boundary value problem, Sci. Sin. Math. 39(3) (1996), 276-288.[14] Y. R. Yuan, Finite difference method and analysis for three-dimensional semiconductor device of heat conduction, Sci. Sin. Math. 39(11) (1996), 1140-1151.[15] O. Axelsson and I. Gustafasson, A modified upwind scheme for convective transport equations and the use of a conjugate gradient method for the solution of non-symmetric systems of equations, J. Inst. Math. Appl. 23 (1979), 321-337.[16] R. E. Ewing, R. D. Lazarov and A. T. Vassilevski, Finite difference scheme for parabolic problems on composite grids with refinement in time and space, SIAM J. Numer. Anal. 31(6) (1994), 1605-1622.[17] R. D. Lazarov, I. D. Mishev and P. S. Vassilevski, Finite volume method for convection-diffusion problems, SIAM J. Numer. Anal. 33(1) (1996), 31-55.[18] D. W. Peaceman, Fundamental of Numerical Reservoir Simulation, Elsevier, Amsterdam, 1980.[19] J. Douglas, Jr. and J. E. Gunn, Two order correct difference analogues for the equation of multidimensional heat flow, Math. Comp. 17(81) (1963), 71-80.[20] J. Douglas, Jr. and J. E. Gunn, A general formulation of alternating direction methods, Part 1. Parabolic and hyperbolic problems, Numer. Math. 6(5) (1964), 428-453.[21] G. I. Marchuk, Splitting and alternating direction method, P. G. Ciarlet, J. L. Lions, eds., Handbook of Numerical Analysis, Elsevier Science Publishers BV, Paris, 1990, pp. 197-460.[22] M. A. Cella, T. F. Russell, I. Herrera and R. E. Ewing, A Eulerian-Lagrangian localized adjoint method for the advection-diffusion equations, Adv. Water Res. 13(4) (1990), 187-206.[23] T. Arbogast and M. F. Wheeler, A characteristics-mixed finite element methods for advection-dominated transport problems, SIAM J. Numer. Anal. 32(2) (1995), 404-424.[24] T. J. Sun and Y. R. Yuan, An approximation of incompressible miscible displacement in porous media by mixed finite element method and characteristics-mixed finite element method, J. Comput. Appl. Math. 228(1) (2009), 391-411.[25] Z. Cai, On the finite volume element method, Numer. Math. 58 (1991), 713-735.[26] R. H. Li and Z. Y. Chen, Generalized Difference of Differential Equations, Jilin University Press, Changchun, 1994.[27] 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-Verlag, Berlin, 1977.[28] 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.[29] 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.[30] T. F. Russell, Rigorous block-centered discretization on irregular grids: improved simulation of complex reservoir systems, Project Report, Research Corporation, Tulsa, 1995.[31] 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.[32] J. E. Jones, A mixed finite volume element method for accurate computation of fluid velocities in porous media, Ph.D. Thesis, University of Colorado, Denver, Co., 1995.[33] Z. Cai, J. E. Jones, S. F. Mccormilk and T. F. Russell, Control-volume mixed finite element methods, Comput. Geosci. 1 (1997), 289-315.[34] S.-H. Chou and D. Y. Kwak, Mixed covolume methods on rectangular grids for elliptic problem, SIAM J. Numer. Anal. 37 (2000), 758-771.[35] S.-H. Chou, D. Y. Kwak and P. S. Vassilevski, Mixed covolume methods for elliptic problems on triangular grids, SIAM J. Numer. Anal. 35 (1998), 1850-1861.[36] S.-H. Chou and P. S. Vassilevski, A general mixed covolume framework for constructing conservative schemes for elliptic problems, Math. Comp. 68 (1999), 991-1011.[37] 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.[38] H. Pan and H. X. Rui, Mixed element method for two-dimensional Darcy-Forchheimer model, J. Sci. Comput. 52(3) (2012), 563-587.[39] Y. R. Yuan, A. J. Cheng, D. P. Yang and C. F. Li, Convergence analysis of an implicit upwind difference fractional step method of three-dimensional enhanced oil recovery percolation coupled system, Sci. Sin. Math. 44(10) (2014), 1035-1058.[40] Y. R. Yuan, A. J. Cheng, D. P. Yang and C. F. Li, Theory and application of fractional steps characteristic-finite difference method in numerical simulation of second order enhanced oil production, Acta Math. Sci. 35(4) (2015), 1547-1565.[41] P. P. Shen, M. X. Liu and L. Tang, Mathematical Model of Petroleum Exploration and Development, Science Press, Beijing, 2002.[42] J. Nitsche, Linear splint-funktionen and die methoden von Ritz for elliptishce randwert probleme, Arch. Rational Mech. Anal. 36 (1968), 348-355.[43] L. S. Jiang and Z. Y. Pang, Finite Element Method and its Theory, People’s Education Press, Beijing, 1979.[44] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
