Continuous-time finite element analysis of multiphase flow in groundwater hydrology. (English) Zbl 0847.76030

A nonlinear differential equation describing an air-water system in groundwater hydrology is studied. The system is presented in a fractional flow formulation, while a continuous-time version of the finite element method is developed and analyzed for the approximation of the saturation and pressure. More specifically, the saturation equation is treated numerically by a Galerkin finite element method, and the pressure equation is analyzed by a mixed finite element method.


76M10 Finite element methods applied to problems in fluid mechanics
76T99 Multiphase and multicomponent flows
76S05 Flows in porous media; filtration; seepage
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
86A05 Hydrology, hydrography, oceanography
Full Text: EuDML


[1] J. Bear: Dynamics of Fluids in Porous Media. Dover, New York, 1972. · Zbl 1191.76001
[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. · Zbl 0631.65107
[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. · Zbl 0689.65065
[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. · Zbl 0599.65072
[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. · Zbl 0603.76101
[7] Z. Chen: Analysis of mixed methods using conforming and nonconforming finite element methods. RAIRO Modèl. Math. Anal. Numér. 27 (1993), 9-34. · 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. · Zbl 0760.65093
[10] Z. Chen and J. Douglas, Jr.: Prismatic mixed finite elements for second order elliptic problems. Calcolo 26 (1989), 135-148. · Zbl 0711.65089
[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. · Zbl 0732.76081
[13] P. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. · 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. · 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. · Zbl 0537.76062
[16] J. Douglas, Jr. and J. Roberts: Global estimates for mixed methods for second order elliptic problems. Math. Comp. 45 (1985), 39-52. · Zbl 0624.65109
[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. · Zbl 0607.76103
[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, Society for Industrial and Applied Mathematics, Philadelphia, 1984, pp. 40-58. · Zbl 0551.76079
[19] K. Fadimba and R. Sharpley: A priori estimates and regularization for a class of porous medium equations. Preprint, submitted to Nonlinear World. · Zbl 0809.35037
[20] K. Fadimba and R. Sharpley: Galerkin finite element method for a class of porous medium equations. Preprint. · Zbl 1050.76030
[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.
[23] H. J. Morel-Seytoux: Two-phase flows in porous media. Advances in Hydroscience 9 (1973), 119-202.
[24] J. C. Nedelec: Mixed finite elements in \(\operatorname{Re} ^3\). Numer. Math. 35 (1980), 315-341. · Zbl 0419.65069
[25] J. Nitsche: \(L_\infty \)-Convergence of Finite Element Approximation. Proc. Second Conference on Finite Elements, Rennes, France, 1975.
[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. · Zbl 0505.76100
[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.
[29] M. Rose: Numerical Methods for flow through porous media I. Math. Comp. 40 (1983), 437-467. · Zbl 0518.76078
[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. · Zbl 0414.65066
[31] R. Scott: Optimal \(L^\infty \) estimates for the finite element method on irregular meshes. Math. Comp. 30 (1976), 681-697. · Zbl 0349.65060
[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.
[33] 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. · Zbl 0232.35060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.