Flow in Heterogeneous Hele-Shaw Models
Authors: Greenkorn R.A. , Haring R.E. , Jahns Hans O. , Shallenberger L.K. 
Published: December 2007

Abstract
This paper is a study of the effects of heterogeneity on flow in an analog of porous media, the Hele-Shaw model. A set of experiments in heterogeneous Hele-Shaw models showed streamlines through and around heterogeneities of various sizes, shapes and levels. (A level, we define as the ratio between the transmissibility of the heterogeneity and that of the rest of the model.) The heterogeneities were restrictions or expansions of the flow stream analogous to variations in the transmissibility of porous media. The experimental data agreed well with numerical results and with an analytical solution, which we derived for a circular heterogeneity in an infinite field. This study considers the flow-stream distortion due to the shape, size and level of heterogeneities. Size and level are much more important than shape provided the heterogeneity is not long and narrow. Our analytical solution shows that a circular heterogeneity in a large field can be replaced by an equivalent circle of either zero or infinite permeability. The radius of the. equivalent circle is a simple function of size and level of the actual circle.
Introduction
With the availability of high-speed computers and numerical procedures to predict reservoir behavior, we are faced with an important question. How much information about the reservoir do we need to justify the cost of the computer in any given case? To answer this question, we have to know how various reservoir parameters affect flow behavior. Reservoir heterogeneity is one of these parameters. In this study, we used a simple, two-dimensional model of porous media, the Hele-Shaw model, to investigate the effect of heterogeneity on flow behavior. We restricted ourselves to linear, single-phase, steady-state flow in a rectangular field with a single heterogeneity at its center.
ANALOGY BETWEEN FLOW IN HELE-SHAW MODELS AND IN POROUS MEDIA
HOMOGENEOUS HELE-SHAW MODELS
The analogy between flow in Hele-Shaw models and in porous media is easily verified. Let us first consider a homogeneous Hele-Shaw model with constant plate separation h. (The Hele-Shaw model is constructed by placing two plates, usually glass, very close together and allowing liquid to flow between them. Streamlines are made visible by introducing colored fluid into the space between the plates at a number of points across the model.)We assume a cartesian coordinate system with its origin in the middle between the plates and the z axis directed perpendicular to the glass plates (Fig. 1). The fluid flow is always in a direction parallel to the glass plates and varies from a maximum value to zero in the very small distance from the middle (z = 0) to either plate (z = h/2).For slow motion of an incompressible fluid, neglecting inertia and body forces, we have the viscous flow equation:
......................(1)
where p is the fluid pressure, mu the viscosity, andthe velocity vector with components u, v and w in the x, y and z directions, respectively. In our case
and the derivatives of with respect to x and y are small as compared with the derivative in the z direction. Therefore, approximately,
.............(2)
with phi p and v being two-dimensional vectors in the x-y plane.
SPEJ
P. 307ˆ