A New Numerical Method of Simulating Two-dimensional Two-phase Flow Through Media With Single And Double Porosity
Authors: Liu M.X. , Chen Z.X. 
Published: March 2010

Abstract
The method proposed in this paper overcomes the effect of numerical dispersion, which is inherent in conventional numericalimulation method. In simulation, the procedure of finding the field of stream junction by the finite element method with varying mesh, and that of tracing the advance of saturation along the streamlines are carried out alternately. Two-dimensional interpolation is avoided because the streamlines are used as mesh lines, and there/ore, the procedure of drawing streamlines is simplified.

Introduction
Several papers have dealt with the numerical simulation of two-dimensional two-phase flow through a medium with double porosity(1–4). These papers are based on conventional numerical methods such as finite difference(5) and finite element(6) which provides the values of saturation at fixed grid points at each time step, and the geometry of the advancing oil-water front cannot be given precisely, especially when there is a strong discontinuity in the saturation distribution. The fact that the position and configuration of the discontinuity surface cannot be given accurately is caused by numerical dispersion at the discontinuity surface. The purpose of this paper is to give a new numerical method which can overcome numerical dispersion when simulating two-phase plane flow through a medium with double porosity (including a medium with single porosity as a special case). The theoretical model of fluid displacement in a medium with double porosity proposed by Chen and Liu(7) is adopted in this paper. This model yields a differential equation imilar to that of Buckley and Leverett, but an imbibition term depending on the history of waterflooding is added. The resulting algorithm for simulation of flow through media with single porosity, and that for media with double porosity are unified.
In 1962, Higgins and Leighton(8) presented a stream tube method for single-porosity media that eliminated numerical dispersion. The drainage domain is divided into a number of tubes which are assumed not to change with time, and then the existing solution of frontal displacement for one-dimensional stream tube is used to determine the advance of saturation within each independent stream tube. With this approach the saturation in the whole drainage domain is obtained approximately. In fact, the method of Higgins and Leighton is just the realization of the method proposed in 1958 by Efros(9) in computer programming ater, this method was improved by Chen(10) and Martin et al.(11) by introducing varying stream tubes; namely, the distribution and form of stream tubes are redetermined at each time step in the process of saturation advance. In 1964, several theoretical relationships and the electrical analogy using the treamline approach were proposed by Chen and Yuan(12). Streamlines are used instead of stream tubes, thus, allowing the advance of saturation to be determined by a more rigorous method. In 1983, based on Ref. 12, a new method was presented by Chen et al. (13) to establish a procedure for numerical simulation which overcomes numerical dispersion.
In the case of a medium with double porosity, certain difficulties are encountered in both the stream tube method and the streamline method due to the presence of imbibition.