Proceedings of Czech–Japanese Seminar in Applied Mathematics 2004 August 4-7, 2004, Czech Technical University in Prague http://geraldine.fjfi.cvut.cz pp. 23–35 EVALUATION OF SATURATION-DEPENDENT FLUX ON TWO-PHASE FLOW USING GENERALIZED SEMI-ANALYTIC SOLUTION RADEK FU ˇ C ´ IK 1 , TISSA H. ILLANGASEKARE 2 , AND JI ˇ R ´ I MIKY ˇ SKA 1 Abstract. An important step in model development protocol is the verification of the model code. This is performed by comparing the code performance with existing analytical solutions under simplified conditions (e.g. geometry, flow dimensionality, parameter distribution and boundary condi- tions). Two closed form solutions are commonly used as benchmarks for the verification of two-phase flow in porous media codes: (1) Buckley-Leverett solution and (2) McWhorter and Sunada solution. The first assumes the capillary effects are negligible, and the second formulation includes capillary effects but assumes a particular functional form for the boundary flux that starts from a physically unrealistic infinity value. This paper presents a derivation that generalizes the boundary flux term by allowing it to start from a finite value. To expand the class of admissible boundary and initial conditions, we offer a numerical algorithm that solves the transport equation for phase saturations using the Finite-Difference Method in space and time. The use of the algorithm is demonstrated by conducting a series of computations in one-dimensional spatial domain. (see also our article in [2]) Key words. Multiphase flow code verification, McWhorter-Sunada’s closed form solution, transport equation, Finite-Difference Method, heterogeneous media. AMS subject classifications. 65M06, 65M20, 35K55, 76T99 1. Model of multiphase flow. McWhorter and Sunada [6] formulated the solution for a semi-infinite problem domain with specific boundary conditions to arrive at a close form analytical solution of the multiphase flow. For general setting see [4], for an application see [5]. The non-wetting phase (indexed n) is displaced by the wetting liquid (water, indexed w) horizontally with no gravity effect and assumed incompressibility of fluids. This section contains the derivation of the multiphase flow equation. The Darcy law applied to the two fluid phases yields q w = − k w μ w ∂p w ∂x , q n = − k n μ n ∂p n ∂x , (1.1) where p α is the pressure, q α is the volumetric flux, k α is the hydraulic conductivity and μ α is the dynamic viscosity for the phase denoted by α. The total flux is denoted by q t = q w + q n and the capillary pressure p c (S w )= p n − p w . (1.2) The aim is to eliminate variables respective to the non-wetting phase i.e. the flux q n and the pressure p n . Subtraction of formulas in (1.1) yields ∂p n ∂x − ∂p w ∂x = −q n μ n k n + q w μ w k w . 1 Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Tech- nical University in Prague, Trojanova 13, 120 00 Prague, Czech Republic. 2 Department of Environmental Science and Engineering, Colorado School of Mines, 1400 Illinois Street, Golden CO 80401, Colorado, USA. 23