Transport in Porous Media 12:73 106, 1993. 73 9 1993 Kluwer Academic Publishers. Printed in the Netherlands. Immiscible Displacement in Vertically Fractured Reservoirs JIM DOUGLAS, JR. a, TODD ARBOGAST 2, PAULO J. PAES-LEME 3, JEFFREY L. HENSLEY 4, and NECI P. NUNES 5 Department of Mathematics, Purdue University, West Lafayette, IN 47907, U.S.A. 2Department of Computational and Applied Mathematics, Rice University, Houston, TX77251-1892, U.S.A. 3Departamento de Matemdtica, Pont~'cia Universidade Cat6lica do R J, 22453 Rio de Janeiro, R J, Brazil 4Center for Parallel and Scientific Computing, The University of Tulsa, Tulsa, OK 74104, U.S.A. Slnstituto Polit~cnico do Rio de Janeiro, 28600 Nova Fribur9o, R J, Brazil (Received: 17 September 1991; in final form: 21 December 1992) Abstract. A dual-porosity model is defined for saturated, two-phase, compressible, immiscible flow in a vertically fractured reservoir or aquifer. This model allows detailed simulation of the matrix-fracture interaction as well as the matrix flow itself. This is accomplished by directly coupling the matrix and fracture systems along the vertical faces of the matrix blocks, incorporating gravitational effects directly, and simulating flow inside the block. Thus fluid segregation due to gravitational effects and heterogenei- ties can be simulated. We show that our model can be derived via homogenization techniques. The model (in incompressible form for simplicity of exposition) is then approximated by a computationally efficient finite difference scheme. Calculations are presented to show the convergence of the scheme and to indicate the behavior of the model as a function of several parameters. Key-words. Naturally fractured, dual-porosity, immiscible displacement, homogenization. Nomenclature B a block q source of sink D areal extent (horizontal cross-sec- Q cross-section of a block tion) of the reservoir domain qm matrix-to-fracture source (i.e., the e Cartesian unit vector transfer function) E scaling tensor for the homogeniz- s, S saturation (with no phase subscript, ation s = Sw) 9, g gravitational constant sr residual saturation h, H grid spacing t time g{' reservoir thickness w auxiliary function used to solve the I = (0,•) vertical extent of the reservoir do- closure problem main x, y position k, K, K absolute permeability x', y' horizontal position kr relative permeability L, l grid nodal numbers Greek n, N number of grid cells ~, gravity-density term ~lr grid nodal positions F boundary of a matrix block p, P pressure ~ij i if i = j, 0 otherwise Pc, Pc capillary pressure function dB, OQ, Of~ boundary of the given domain p;-1 inverse of capillary pressure func- Vt time step tion ~ homogenization parameter