Characteristic mixed volume element for compressible two-phase displacement in porous media

References

• R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
   Google Scholar
• T. Arbogast, and M.F. Wheeler, A characteristics-mixed finite element methods for advection-dominated transport problems. SIAMJ. Numer. Anal 32(2) (1995), pp. 404–424.
   Web of Science ®Google Scholar
• R.B. Bird, W.E. Lightfoot, and E.N. Stewart, Transport Phenomenon, John Wiley and Sons, New York, 1960.
   Google Scholar
• G. Bretti, and B. Piccoli, A tracking algorithm for car paths on road networks. SIAM J. Appl. Dyn. Syst 7(2) (2008), pp. 510–531.
   Web of Science ®Google Scholar
• J. Cai, and H. Yang, A finite volume–alternating direction implicit method for the valuation of American options under the Heston model. Int. J Comput. Math 97(3) (2020), pp. 700–724.
   Web of Science ®Google Scholar
• Z. Cai, On the finite volume element method. Numer. Math 58(1) (1990), pp. 713–735.
   Web of Science ®Google Scholar
• Z. Cai, J.E. Jones, S.F. Mccormilk, and T.F. Russell, Control-volume mixed finite element methods. Comput. Geosci 1 (1997), pp. 289–315.
   Google Scholar
• M.A. Cella, T.F. Russell, I. Herrera, and R.E. Ewing, An Eulerian–Lagrangian localized adjoint method for the advection–diffusion equations. Adv. Water Resour 13(4) (1990), pp. 187–206.
   Web of Science ®Google Scholar
• C.J. Chen, W. Liu, and C.J. Bi, A two-grid characteristic finite volume element method for semilinear advection-dominated diffusion equations. Numer. Meth. Part. D. E 29 (2013), pp. 1543–1562.
   Web of Science ®Google Scholar
• 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), pp. 1850–1861.
   Web of Science ®Google Scholar
• 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), pp. 758–771.
   Web of Science ®Google Scholar
• S.H. Chou, and P. Vassileviki, A general mixed covolume framework for constructing conservative schemes for elliptic problems. Math. Comp 68(227) (1999), pp. 991–1011.
   Web of Science ®Google Scholar
• A. Cutolo, B. Piccoli, and L. Rarita, An upwind-Euler scheme for an ODE–PDE model of supply chains. SIAM J. Scient. Computing 33(4) (2011), pp. 1669–1688.
   Web of Science ®Google Scholar
• 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), pp. 1487–1512.
   Web of Science ®Google Scholar
• J.J. Douglas, and J.E. Roberts, Numerical methods for a model for compressible miscible displacement in porous media. Math. Comp 41(164) (1983), pp. 441–459.
   Web of Science ®Google Scholar
• J.J. Douglas, 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. 119-132, Springer-Berlag, New York, 1986.
   Google Scholar
• 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), pp. 17–33.
   Google Scholar
• Douglas, Jr. Jim, Finite difference methods for two-phase incompressible flow in porous media. SIAM J. Numer. Anal 20(4) (1983), pp. 681–696.
   Web of Science ®Google Scholar
• Ewing R. E. The Mathematics of Reservoir Simulation. SIAM, Philadelphia, 1983.
   Google Scholar
• 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), pp. 73–92.
   Web of Science ®Google Scholar
• R.E. Ewing, M.F. Wheeler, Galerkin methods for miscible displacement problems with point sources and sinks-unit mobility ratio case. In: Mathematical methods in energy research, SIAM, Philadelphia. (1984), pp. 40–58.
   Google Scholar
• 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, 1264-1271. The University of Alabama in Huntsville, Huntsville, AL: UAHDRESS, 1989.
   Google Scholar
• L.S. Jiang, and Z.Y. Pang, Finite Element Method and its Theory, People's Education Press, Beijing, 1979.
   Google Scholar
• Jones J.E., A mixed volume method for accurate computation of fluid velocities in porous media. Ph. D. Thesis. University of Colorado, Denver, Co. 1995.
   Google Scholar
• R.H. Li, and Z.Y. Chen, Generalized Difference of Differential Equations, Jilin University Press, Changchun, 1994.
   Google Scholar
• J. Nitsche, Lineare spline-funktionen und die methoden von Ritz fur elliptische randwert probleme. Arch. Rational Mech. Anal 36(5) (1970), pp. 348–355.
   Google Scholar
• H. Pan, and H.X. Rui, Mixed element method for two-dimensional Darcy-Forchheimer model. J Sci. Comput 52(3) (2012), pp. 563–587.
   Web of Science ®Google Scholar
• 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-Verlag, Berlin, 1977.
   Google Scholar
• H.X. Rui, and H. Pan, A block-centered finite difference method for the Darcy–Forchheimer model. SIAM J. Numer. Anal 50(5) (2012), pp. 2612–2631.
   Web of Science ®Google Scholar
• T.F. Russell, Time stepping along characteristics with incomplete interaction for a Galerkin approximation of miscible displacement in porous media. SLAM J. Numer. Anal 22(5) (1985), pp. 970–1013.
   Web of Science ®Google Scholar
• T.F. Russell, Rigorous block-centered discretization on irregular grids: Improved simulation of complex reservoir systems. Project Report, Research Corporation, Tulsa, 1995.
   Google Scholar
• P.P. Shen, M.X. Liu, and L. Tang, Mathematical Model of Petroleum Exploration and Development, Science Press, Beijing, 2002.
   Google Scholar
• 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), pp. 391–411.
   Web of Science ®Google Scholar
• T.J. Sun, and Y.R. Yuan, Mixed finite method and characteristics-mixed finite element method for a slightly compressible miscible displacement problem in porous media. Math. Comput. Simul. 107 (2015), pp. 24–45.
   Web of Science ®Google Scholar
• Y.R. Yuan, Time stepping along characteristics for the finite element approximation of compressible miscible displacement in porous media. Math. Numer. Sinica 14(4) (1992), pp. 385–400.
   Google Scholar
• Y.R. Yuan, Finite difference methods for a compressible miscible displacement problem in porous media. Math. Numer. Sinica 15(1) (1993), pp. 16–28.
   Google Scholar
• Y.R. Yuan, Characteristic finite element scheme and analysis the three-dimensional two-phase displacement semi-definite problem. Chin. Sci. Bull 42(1) (1997), pp. 17–32.
   Google Scholar
• R. Yuan Y, The characteristic finite difference fractional steps method for compressible two-phase displacement problem. Sci. Sin. Math 42(1) (1999), pp. 48–57.
   Google Scholar
• Y.R. Yuan, Characteristic finite difference methods for positive semidefinite problem of two phase miscible flow in porous media. J. Systems Sci. Math. Sci 12(4) (1999), pp. 299–306.
   Google Scholar
• Y.R. Yuan, The modified method of characteristics with finite element operator-splitting procedures for compressible multi-component displacement problem. J Syst. Sci. Complex 16(1) (2003), pp. 30–45.
   Google Scholar
• R. Yuan, The upwind finite difference fractional steps methods for two-phase compressible flow in porous media. Numer. Meth. Part. D. E 19(1) (2003), pp. 67–88.
   Web of Science ®Google Scholar
• Y.R. Yuan, The characteristic finite element alternating direction method with moving meshes for nonlinear convection-dominated diffusion problems. Numer. Meth. Part. D. E 22(3) (2006), pp. 661–679.
   Web of Science ®Google Scholar
• Y.R. Yuan, Theory and Application of Reservoir Numerical Simulation, Science Press, Beijing, 2013.6.
   Google Scholar
• Y.R. Yuan, A.J. Cheng, P. Yang D, C.F. Li, and Q. Yang, Convergence analysis of mixed volume element-characteristic mixed volume element for three-dimensional chemical oil-recovery seepage coupled problem. Acta Math. Sci (2018), 38B. (2): 519–545.
   Web of Science ®Google Scholar
• Y.R. Yuan, M. Cui, C.F. Li, T.J. Sun, The method of mixed volume element-characteristic mixed volume element and its numerical analysis for groundwater pollution in binary medium. Appl. Math. Comput. 362 (2019), pp. 124536.
   Google Scholar
• Y.R. Yuan, C.F. Li, T.J. Sun, and Q. Yang, Mixed volume element-characteristic mixed volume element method and its analysis for compressible flow of contamination from nuclear waste in porous media. Int. J Comput. Math. (2020). Published online, https://doi.org/https://doi.org/10.1080/00207160.2020.1734795.
   Google Scholar
• Y.R. Yuan, T.J. Sun, C.F. Li, Y.X. Liu, and Q. Yang, Mixed volume element combined with characteristic mixed finite volume element method for oil–water two phase displacement problem. J Comput. Appl. Math 340 (2018), pp. 404–419.
   Web of Science ®Google Scholar
• Y.R. Yuan, T.J. Sun, C.F. Li, and Q. Yang, The method of mixed volume element-characteristic mixed volume element and its numerical analysis for three-dimensional slightly compressible two-phase displacement. Numer Methods Partial Differ Eq 34(2) (2018), pp. 661–685.
   Web of Science ®Google Scholar
• Y.R. Yuan, D.P. Yang, 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.
   Google Scholar