A mixed finite element-characteristic mixed volume element is presented to solve three-dimensional incompressible Darcy-Forchheimer miscible displacement, and convergence analysis is shown in this paper. A mixed finite element approximation is applied to obtain the pressure and Darcy-Forchheimer velocity, and the accuracy of velocity is improved one order. The concentration is computed by a coupled scheme of characteristics and mixed volume element, where the diffusion is treated by the mixed volume element and the convection is treated by the method of characteristics. The method of characteristics has strong computation stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. Larger time-steps along the characteristics are shown to result in smaller time-truncation errors than those resulting from standard methods. More important in numerical simulation of seepage mechanics, mixed volume element has the property of conservation on each element and it can obtain numerical solution of the concentration and its adjoint vector function simultaneously. Using some techniques of priori estimates of differential equations, we show an optimal second order estimate in discrete L² norm. Numerical data are consistent with theoretical analysis, and the composite combination method could possibly become a powerful tool for solving the actual problems in porous media.

Darcy-Forchheimer model, 3-D incompressible miscible displacement, mixed finite element, characteristic mixed volume element, error estimates in L²-norm.

Received: August 21, 2023; Accepted: October 10, 2023; Published: April 3, 2024

How to cite this article: Yirang Yuan, Changfeng Li, Tongjun Sun and Qing Yang, A mixed finite element-characteristic mixed volume element and convergence analysis of Darcy-Forchheimer displacement problem, Far East Journal of Applied Mathematics 117(1) (2024), 67-100. http://dx.doi.org/10.17654/0972096024004

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

References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[2] 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.
[3] K. Aziz and A. Settari, Petroleum reservoir simulation, Applied Science Publishers, London, 1979.
[4] Z. Cai, J. E. Jones, S. F. Mccormilk and T. F. Russell, Control-volume mixed finite element methods, Comput. Geosci. 1 (1997), 289-315.
[5] 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), 187-206.
[6] 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.
[7] 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.
[8] 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.
[9] J. Douglas Jr., Finite difference method for two-phase incompressible flow in porous media, SIAM J. Numer. Anal. 4 (1983), 681-696.
[10] 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, Springer-Verlag, New York, 1986, pp. 119-132.
[11] J. Douglas Jr., P. J. Paes-Leme and T. Giorgi, Generalized Forchheimer flow in porous media, Boundary Value Problems for Partial Differential Equations and Applications, RMA Res. Notes Appl. Math., Masson, Paris 29 (1993), 99-111.
[12] R. E. Ewing, R. D. Lazarov, S. L. Lyons, D. V. Papavassiliou, J. Pasciak and G. Qin, Numerical well model for non-Darcy flow through isotropic porous media, Comput. Geosci. 3(3-4) (1999), 185-204.
[13] 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.
[14] P. Fabrie, Regularity of the solution of Darcy-Forchheimer’s equation, Nonlinear Anal. Theory. Methods Appl. 13(9) (1989), 1025-1049.
[15] V. Girault and M. F. Wheeler, Numerical discretization of a Darcy-Forchheimer model, Numer. Math. 110(2) (2008), 161-198.
[16] 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.
[17] H. Lopez, B. Molina and J. J. Salas, Comparison between different numerical discretization for a Darcy-Forchheimer model, Electron. Trans. Numer. Anal. 34 (2009), 187-203.
[18] E. J. Park, Mixed finite element method for generalized Forchheimer flow in porous media, Numer. Methods Partial Differential Equations 21(2) (2005), 213-228.
[19] H. Pan and H. X. Rui, Mixed element method for two-dimensional Darcy-Forchheimer model, J. Sci. Comput. 52 (2012), 563-587.
[20] H. Pan and H. X. Rui, A mixed element method for Darcy-Forchheimer incompressible miscible displacement problem, Comput. Methods Appl. Mech. Engrg. 264 (2013), 1-11.
[21] 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.
[22] H. X. Rui and W. Liu, A two-grid block-centered finite difference methods for Darcy-Forchheimer flow in porous media, SIAM J. Numer. Anal. 53(4) (2015), 1941-1962.
[23] T. F. Russell, Rigorous block-centered discretization on irregular grids: Improved simulation of complex reservoir systems, Project Report, Research Corporation, Tulsa, 1995.
[24] D. Ruth and H. Ma, On the derivation of the Forchheimer equation by means of the averaging theorem, Transport in Porous Media 7(3) (1992), 255-264.
[25] P. P. Shen, M. X. Liu and L. Tang, Mathematical model of petroleum exploration and development, Science Press, Beijing, 2002.
[26] 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.
[27] 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. Simulat. 107 (2015), 24-45.
[28] 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.
[29] Y. R. Yuan, Theory and Application of Reservoir Numerical Simulation, Science Press, Beijing, 2013.
[30] Y. R. Yuan, A. J. Cheng, D. P. Yang and C. F. Li, Applications, theoretical analysis, numerical method and mechanical model of the polymer flooding in porous media, Special Topics and Reviews in Porous Media - An International Journal 6(4) (2015), 383-401.
[31] Y. R. Yuan, A. J. Cheng and D. P. Yang, Theory and Actual Applications of Numerical Simulation of Oil Reservoir, Science Press, Beijing, 2016.
[32] Y. R. Yuan, A. J. Cheng, D. P. Yang and C. F. Li, Numerical simulation method, theory and applications of three-phase (oil, gas, water) chemical-agent oil recovery in porous media, Special Topics and Reviews in Porous Media - An International Journal 7(3) (2016), 1-28.
[33] 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 device detector, Appl. Math. Comput. 279 (2016), 1-15.