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EXPONENTIAL EULER TIME INTEGRATOR FOR ISOTHERMAL INCOMPRESSIBLE TWO-PHASE FLOW IN HETEROGENEOUS POROUS MEDIA

DOI: 10.1615/ICHMT.2012.CHT-12.620
pages 1011-1024

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

Abstract

Accurate reservoir modelling requires stable and efficient time integrators. For quite a long time, standard time integrators (full implicit Euler, Crank–Nicolson, explicit Euler and implicit- explicit schemes) were mostly used for time integration. These standard time integrators only provide first or second order accuracy in time for simple linear problems, beside explicit Euler scheme suffers for time step constraints. In this paper, we present the exponential Euler time integrator combined with the finite volume (two-point or mult-point flux approximations) space discretization for simulating isothermal incompressible two-phase flow in heterogeneous porous media. This method 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 ODEs 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. Beside this computation can be done using the free-Jacobian technique. All our numerical examples demonstrate that our method can compete in terms of efficiency and accuracy with the standard time integrators for reservoir simulation in highly anisotropic and heterogeneous porous media.

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