Implicit–explicit methods for models for vertical equilibrium multiphase flow

In this paper we provide a derivation of a 1D N-phase flow model in a porous medium under the condition of vertical equilibrium which generalizes the three-phase flow model developed in Guerrero et al. (2013). We identify, from a mathematical point of view, a set of conditions on the capillary pressures that ensure that the linearized, purely parabolic, initial value problem is well-posed.For the numerical simulation of the model, we advocate the use of Implicit–Explicit (IMEX)–Runge–Kutta (RK) schemes for time evolution with a Weighted-Essentially Non-Oscillatory (WENO) spatial discretization of the convective terms. In previous papers (Donat et al., 2013), (Guerrero et al., 2013), we showed that IMEX–RK methods can be useful in the numerical simulation of 1D two-phase and three-phase flows, since the stability restrictions for the time-step of these schemes are less severe than those of fully explicit schemes. On the other hand, their implementation requires, in general, a fairly intensive use of a nonlinear system solver, so that the efficiency and robustness of IMEX schemes for multi-phase flow is directly related to this nonlinear technique.In this paper we describe an efficient nonlinear system solver, based on an appropriate fixed-point iteration technique, in order to find the solution of the nonlinear systems that result from the implicit discretization of the nonlinear diffusive terms in the model. In addition, we implement zero-flux boundary conditions fully consistent with the vertical equilibrium assumptions.A set of numerical examples confirm the efficiency, robustness and reliability of the proposed numerical technique.