Non-oscillatory central differencing for hyperbolic conservation laws
Journal of Computational Physics
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
Journal of Computational Physics
A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
Comparison of various formulations of three-phase flow in porous media
Journal of Computational Physics
Stochastic methods for the prediction of complex multiscale phenomena
Quarterly of Applied Mathematics - Special issue on current and future challenges in the applications of mathematics
Weighted essentially non-oscillatory schemes on triangular meshes
Journal of Computational Physics
Prediction and the quantification of uncertainty
Physica D - Special issue originating from the 18th Annual International Conference of the Center for Nonlinear Studies, Los Alamos, NM, May 11&mdash ;15, 1998
Corrected Operator Splitting for Nonlinear Parabolic Equations
SIAM Journal on Numerical Analysis
Journal of Computational Physics
A technique of treating negative weights in WENO schemes
Journal of Computational Physics
Capillary instability in models for three-phase flow
Zeitschrift für Angewandte Mathematik und Physik (ZAMP)
Finite-volume WENO schemes for three-dimensional conservation laws
Journal of Computational Physics
Three-phase immiscible displacement in heterogeneous petroleum reservoirs
Mathematics and Computers in Simulation - Special issue: Applied and computational mathematics - selected papers of the fifth PanAmerican workshop - June 21-25, 2004, Tegucigalpa, Honduras
High-Order Relaxation Schemes for Nonlinear Degenerate Diffusion Problems
SIAM Journal on Numerical Analysis
Adaptive Semidiscrete Central-Upwind Schemes for Nonconvex Hyperbolic Conservation Laws
SIAM Journal on Scientific Computing
Journal of Computational Physics
A Fast Explicit Operator Splitting Method for Passive Scalar Advection
Journal of Scientific Computing
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In this paper is introduced a new numerical formulation for solving degenerate nonlinear coupled convection dominated parabolic systems in problems of flow and transport in porous media by means of a mixed finite element and an operator splitting technique, which, in turn, is capable of simulating the flow of a distinct number of fluid phases in different porous media regions. This situation naturally occurs in practical applications, such as those in petroleum reservoir engineering and groundwater transport. To illustrate the modelling problem at hand, we consider a nonlinear three-phase porous media flow model in one- and two-space dimensions, which may lead to the existence of a simultaneous one-, two- and three-phase flow regions and therefore to a degenerate convection dominated parabolic system. Our numerical formulation can also be extended for the case of three space dimensions. As a consequence of the standard mixed finite element approach for this flow problem the resulting linear algebraic system is singular. By using an operator splitting combined with mixed finite element, and a decomposition of the domain into different flow regions, compatibility conditions are obtained to bypass the degeneracy in order to the degenerate convection dominated parabolic system of equations be numerically tractable without any mathematical trick to remove the singularity, i.e., no use of a parabolic regularization. Thus, by using this procedure, we were able to write the full nonlinear system in an appropriate way in order to obtain a nonsingular system for its numerical solution. The robustness of the proposed method is verified through a large set of high-resolution numerical experiments of nonlinear transport flow problems with degenerating diffusion conditions and by means of a numerical convergence study.