A Semi-Implicit Method for Simulating Reservoir Behavior

Abstract

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Abstract A method to simulate multiphase flow behavior in reservoirs is presented that employes spatially varying degree of implicitness with respect to fluid saturations. By suitably defining the implicit region, the technique preserves the stability associated with simultaneous solution and computationally superior to it. Furthermore, the time truncation error is lower than a fully implicit formulation. Comparisons are presented for water-oil and 3-phase (with dissolved gas) coning systems and a gas percolation problem. INTRODUCTION: Multiphase flow behavior in reservoirs is described by a set of non-linear differential equations that are obtained by combining the continuity equations with the Darcy's law. These equations are generally solved numerically by variants of one of the following formulations: 1. Implicit Pressure Explicit-Saturation Method (IMPES) In this approach the governing differential equations are reduced to a set of finite-difference equations with single dependent variable as the oil-phase pressure. Such a reduction to a pressure equation is carried out by combining the flow equation of each phase in a manner that eliminates the time derivatives or differences of saturations. The procedure treats the saturation dependent terms explicitly that include the capillary-pressures and the transmissibilities to each phase. The set of equations so obtained is solved implicitly for pressure with iterative updating of the pressure pressure with iterative updating of the pressure dependent terms. Having obtained the pressures, the saturations are computed directly from the water and/or oil flow equations. Iterative updating of the saturation dependent coefficients impairs the computing efficiency and the stability of the system. The implicit pressure entails simultaneous solution of N-equations for an N-grid-block system per iteration. The IMPES procedure by virtue of its simplicity is amenable to a number of efficient matrix inversion techniques, and thus involves minimum computing time per time step. The explicit dating of the saturation dependent terms, however, imposes a conditional stability that limits the time steps to an impractically small size in many applications of interest. In particular, wellbore coning systems, the gas percolation and the water drainage problems, percolation and the water drainage problems, reservoirs with steep capillary-pressure slopes, etc. are difficult to simulate in the framework of IMPES, without resorting to artifices. 2. Simultaneous Solution Method (SS) SS approach readily permits implicit treatment of the saturation dependent terms and offer a high computational stability.