A Numerical Method for Computing the Dynamical Behavior of Fluid-Fluid Interfaces in Permeable Media
- J.W. Sheldon (Computer Usage Co., Inc.) | E.W. Dougherty (Standard Oil Co. Of California)
- Document ID
- Society of Petroleum Engineers
- Society of Petroleum Engineers Journal
- Publication Date
- June 1964
- Document Type
- Journal Paper
- 158 - 170
- 1964. Original copyright American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Copyright has expired.
- 4.3.4 Scale, 4.1.2 Separation and Treating, 5.4.2 Gas Injection Methods
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A numerical method for computing the dynamical behavior of fluid-fluid interface is described. Results of studies made to assess the accuracy and economy of the method on a computer are reported. If is concluded that it is practical to obtain digital computer solutions for a fluid-fluid interface problems with two space dimensions.
We consider two-dimensional flow in a rectangle as shown in Fig. 1, where a fluid-fluid interface divides the rectangle into two regions. The pressure is specified along the lower boundary (the injection boundary) and upper boundary (the withdrawal boundary). There is to be no flow of fluid across the side boundaries, which are walls of symmetry for the flow, and px = 0 along these boundaries. The fluids are incompressible, the absolute permeability and porosity is constant, and each fluid has its own constant mobility. Thus the pressure satisfies Laplace's equation in each region. Across the interface, pressure is continuous and the normal component of fluid flux is 1. These conditions are shown in Fig. 1. The velocity of a point on the interface in the direction of the normal from Fluid 1 into Fluid 2 is , where is the normal component of volumetric flux at the point on the interface, i.e.,
For brevity, we call the problem defined above "Problem A". Problem A is a simple example of a dynamical fluid-fluid interface problem. However, the same numerical methods which we use to solve Problem A can be extended to the solution of a large number of fluid-fluid interface problems which are mathematical models for various oil recovery processes, hence our interest in Problem A.
OUTLINE OF METHOD OF SOLUTION
We define the interface by prescribing the x, y coordinates of a finite set of points. These points are called "interface meshpoints" and are illustrated by triangles or asterisks in the figures. Fig. 2 is a procedural (flow) chart displaying the main steps in the solution. We comment briefly on some of these steps in this section and then give further details in subsequent sections.
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