Previous |  Up |  Next


mixed method; finite element; compressible flow; porous media; error estimate; air-water system
A nonlinear differential system for describing an air-water system in groundwater hydrology is given. The system is written in a fractional flow formulation, i.e., in terms of a saturation and a global pressure. A continuous-time version of the finite element method is developed and analyzed for the approximation of the saturation and pressure. The saturation equation is treated by a Galerkin finite element method, while the pressure equation is treated by a mixed finite element method. The analysis is carried out first for the case where the capillary diffusion coefficient is assumed to be uniformly positive, and is then extended to a degenerate case where the diffusion coefficient can be zero. It is shown that error estimates of optimal order in the $L^2$-norm and almost optimal order in the $L^\infty $-norm can be obtained in the nondegenerate case. In the degenerate case we consider a regularization of the saturation equation by perturbing the diffusion coefficient. The norm of error estimates depends on the severity of the degeneracy in diffusivity, with almost optimal order convergence for non-severe degeneracy. Existence and uniqueness of the approximate solution is also proven.
[1] J. Bear: Dynamics of Fluids in Porous Media. Dover, New York, 1972.
[2] F. Brezzi, J. Douglas, Jr., R. Durán, and M. Fortin: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987), 237–250. DOI 10.1007/BF01396752 | MR 0890035
[3] F. Brezzi, J. Douglas, Jr., M. Fortin, and L. Marini: Efficient rectangular mixed finite elements in two and three space variables. RAIRO Modèl. Math. Anal. Numér 21 (1987), 581–604. DOI 10.1051/m2an/1987210405811 | MR 0921828
[4] F. Brezzi, J. Douglas, Jr., and L. Marini: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985), 217–235. DOI 10.1007/BF01389710 | MR 0799685
[5] M. Celia and P. Binning: Two-phase unsaturated flow: one dimensional simulation and air phase velocities. Water Resources Research 28 (1992), 2819–2828.
[6] G. Chavent and J. Jaffré: Mathematical Models and Finite Elements for Reservoir Simulation. North-Holland, Amsterdam, 1978.
[7] Z. Chen: Analysis of mixed methods using conforming and nonconforming finite element methods. RAIRO Modèl. Math. Anal. Numér. 27 (1993), 9–34. DOI 10.1051/m2an/1993270100091 | MR 1204626 | Zbl 0784.65075
[8] Z. Chen: Finite element methods for the black oil model in petroleum reservoirs. IMA Preprint Series $\#$ 1238, submitted to Math. Comp.
[9] Z. Chen and J. Douglas, Jr.: Approximation of coefficients in hybrid and mixed methods for nonlinear parabolic problems. Mat. Aplic. Comp. 10 (1991), 137–160. MR 1172090
[10] Z. Chen and J. Douglas, Jr.: Prismatic mixed finite elements for second order elliptic problems. Calcolo 26 (1989), 135–148. DOI 10.1007/BF02575725 | MR 1083050
[11] Z. Chen, R. Ewing, and M. Espedal: Multiphase flow simulation with various boundary conditions. Numerical Methods in Water Resources, Vol. 2, A. Peters, et als. (eds.), Kluwer Academic Publishers, Netherlands, 1994, pp. 925–932.
[12] S. Chou and Q. Li: Mixed finite element methods for compressible miscible displacement in porous media. Math. Comp. 57 (1991), 507–527. DOI 10.1090/S0025-5718-1991-1094942-7 | MR 1094942
[13] P. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. MR 0520174 | Zbl 0383.65058
[14] J. Douglas, Jr.: Finite difference methods for two-phase incompressible flow in porous media. SIAM J. Numer. Anal. 20 (1983), 681–696. DOI 10.1137/0720046 | MR 0708451 | Zbl 0519.76107
[15] J. Douglas, Jr. and J. Roberts: Numerical methods for a model for compressible miscible displacement in porous media. Math. Comp. 41 (1983), 441–459. DOI 10.1090/S0025-5718-1983-0717695-3 | MR 0717695
[16] J. Douglas, Jr. and J. Roberts: Global estimates for mixed methods for second order elliptic problems. Math. Comp. 45 (1985), 39–52. MR 0771029
[17] N. S. Espedal and R. E. Ewing: Characteristic Petrov-Galerkin subdomain methods for two phase immiscible flow. Comput. Methods Appl. Mech. Eng. 64 (1987), 113–135. DOI 10.1016/0045-7825(87)90036-3 | MR 0912516
[18] R. Ewing and M. Wheeler: Galerkin methods for miscible displacement problems with point sources and sinks-unit mobility ratio case. Mathematical Methods in Energy Research, K. I. Gross, ed., Society for Industrial and Applied Mathematics, Philadelphia, 1984, pp. 40–58. MR 0790511
[19] K. Fadimba and R. Sharpley: A priori estimates and regularization for a class of porous medium equations. Preprint, submitted to Nonlinear World. MR 1376946
[20] K. Fadimba and R. Sharpley: Galerkin finite element method for a class of porous medium equations. Preprint. MR 2025071
[21] D. Hillel: Fundamentals of Soil Physics. Academic Press, San Diego, California, 1980.
[22] C. Johnson and V. Thomée: Error estimates for some mixed finite element methods for parabolic type problems. RAIRO Anal. Numér. 15 (1981), 41–78. DOI 10.1051/m2an/1981150100411 | MR 0610597
[23] H. J. Morel-Seytoux: Two-phase flows in porous media. Advances in Hydroscience 9 (1973), 119–202. DOI 10.1016/B978-0-12-021809-7.50009-2
[24] J. C. Nedelec: Mixed finite elements in $\Re ^3$. Numer. Math. 35 (1980), 315–341. DOI 10.1007/BF01396415 | MR 0592160
[25] J. Nitsche: $L_\infty $-Convergence of Finite Element Approximation. Proc. Second Conference on Finite Elements, Rennes, France, 1975. MR 0568857
[26] D. W. Peaceman: Fundamentals of Numerical Reservoir Simulation. Elsevier, New York, 1977.
[27] O. Pironneau: On the transport-diffusion algorithm and its application to the Navier-Stokes equations. Numer. Math. 38 (1982), 309–332. DOI 10.1007/BF01396435 | MR 0654100
[28] P.A. Raviart and J.M. Thomas: A mixed finite element method for second order elliptic problems. Lecture Notes in Math. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555
[29] M. Rose: Numerical Methods for flow through porous media I. Math. Comp. 40 (1983), 437–467. DOI 10.1090/S0025-5718-1983-0689465-6 | MR 0689465
[30] A. Schatz, V. Thomée, and L. Wahlbin: Maximum norm stability and error estimates in parabolic finite element equations. Comm. Pure Appl. Math. 33 (1980), 265–304. DOI 10.1002/cpa.3160330305 | MR 0562737
[31] R. Scott: Optimal $L^\infty $ estimates for the finite element method on irregular meshes. Math. Comp. 30 (1976), 681–697. MR 0436617
[32] D. Smylie: A near optimal order approximation to a class of two sided nonlinear degenerate parabolic partial differential equations. Ph. D. Thesis, University of Wyoming, 1989.
[32] M. F. Wheeler: A priori $L_2$ error estimates for Galerkin approximation to parabolic partial differential equations. SIAM J. Numer. Anal. 10 (1973), 723–759. DOI 10.1137/0710062 | MR 0351124
Partner of
EuDML logo