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Journal of Porous Media
IF: 1.49 5-Year IF: 1.159 SJR: 0.43 SNIP: 0.671 CiteScore™: 1.58

ISSN Print: 1091-028X
ISSN Online: 1934-0508

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Journal of Porous Media

DOI: 10.1615/JPorMedia.v16.i5.10
pages 381-393

EFFICIENT NUMERICAL SIMULATION OF INCOMPRESSIBLE TWO-PHASE FLOW IN HETEROGENEOUS POROUS MEDIA BASED ON EXPONENTIAL ROSENBROCK−EULER METHOD AND LOWER-ORDER ROSENBROCK-TYPE METHOD

Antoine Tambue
Department of Mathematics, University of Bergen, P.O. Box 7800, N-5020 Bergen, Norway

ABSTRACT

In this paper, we present the exponential Rosenbrock−Euler method and the lowest-order Rosenbrock-type method combined with the finite volume (two-point or multi-point flux approximations) space discretization to simulate isothermal incompressible two-phase flow in heterogeneous porous media. The exponential Rosenbrock−Euler method (EREM) linearizes the saturation equation at each time step and makes use of a matrix exponential function of the Jacobian, then solves the corresponding stiff linear ordinary differential equations exactly in time up to the given tolerance in the computation of a matrix exponential function of the Jacobian from the space discretization. Using a Krylov subspace technique makes this computation efficient. Besides, this computation can be done using the free-Jacobian technique. The lowest-order Rosenbrock type method, also called linearly implicit method, is deduced from EREM by approximating the exponential function of the Jacobian by the appropriate rational function of the Jacobian. As a result, this scheme is L-stable and only one linear system is normally solved at each time step. All our numerical examples demonstrate that our methods can compete in terms of efficiency and accuracy with the standard time integrators for reservoir simulation in highly anisotropic and heterogeneous porous media. Simulations are performed up to 1.1 million of unknowns with and without capillary pressure.