This paper describes the application of regression analysis for obtaining a two - dimensional areal description of heterogeneous reservoirs from short-term pressure-time data such as that obtained in interference tests. The method replaces the time-consuming trial-and-error procedure commonly used to match field data on an electric analyzer or digital computer with a systematic search which is programmed for a computer. The computer program adjusts the properties of a reservoir model automatically until a least-squares fit is obtained between observed and calculated pressure data. The reservoir is simulated by a single-phase, compressible, two-dimensional model. It is divided into a number of homogeneous blocks whose transmissibility (kh/f) and storage (fch) values are varied to obtain the least-squares fit. The reliability of these values is determined from their standard deviations and correlation coefficients. Although the method is rigorously applicable to single-phase flow only, multiphase flow can be handled provided saturation changes are small during the test. Possibly the method can also be used to obtain a reservoir description from pressure-production history, but this application is outside the scope of this work. The paper includes, in addition to a description of the numerical procedure, a discussion of some of the problems associated with the method. Rules are given to help in selecting the number of homogeneous blocks and deciding upon their arrangement. The uniqueness of a reservoir description is considered. Finally, the use of the method is illustrated by the interpretation of field data from two interference tests. INTRODUCTION Pressure data from short-term transient tests, such as single-well and interference tests, are widely used to obtain reservoir properties. These tests are usually analyzed by assuming a simple reservoir model; very often, a homogeneous one is used. As a result, analysis of the transient data from each well frequently gives different values for reservoir properties. The problem then arises to combine all these differing results into a more detailed picture of the reservoir. One technique is to simulate the reservoir with a digital computer or with an electrical analyzer and to adjust the reservoir parameters by trial and error until the simulated pressure data are in reasonable agreement with the observed pressure data for all wells. Although this method has been used for both transient tests and pressure-history data, it is time-consuming and subjective. A second technique uses regression analysis to replace the trial-and-error procedure with a systematic search that can be programmed for a digital computer. Use of regression analysis in reservoir description was proposed recently by Jacquard and Jain.1 They divided the reservoir into a number of homogeneous blocks whose properties are varied until a least-squares fit is obtained between observed and calculated pressures. However, they did not consider their technique to be operational, mainly because of "...the lack of experience in using the method. . . notably for the improvement of convergence; andlimitations imposed by the insufficiency of available computers".1 While the analysis presented in this paper applies the same general principle used by Jacquard and Jain, the specific method is significantly different. Some differences arethe regression problem is solved in a different way which requires less computer time in most cases;a stepwise solution, in which the detail in the reservoir description is increased from step to step, is used to improve convergence; andthe reliability of the estimated reservoir properties, as measured by their standard deviation and correlation coefficient, is estimated.

This paper describes a preliminary study of the linear flow of a non-Newtonian fluid, a water solution of Dextran (a polysaccharide), in porous media. A modification of Darcy's law, which uses capillary rheology data, is developed to describe non-Newtonian flow in underground reservoirs. The generalization, in effect, replaces the porous media with a capillary of equivalent radius proportional to the square root of the ratio of permeability to porosity. The constant of proportionality a . should be independent of permeability and porosity for a given type of rock. This has been partially confirmed experimentally. In principle, a capillary rheogram and a single core test permit evaluation of ao. Then non-Newtonian flow can be predicted in this type of rock regardless of porosity, permeability, or flow rate.The flow of a Newtonian fluid (a fluid with a constant viscosity) through porous media is in good agreement with Darcy's law as expressed by Equation (1) with gravitational forces neglected.(1)where V, is the fluid superficial velocity vector, V p is the pressure gradient, f i is the fluid viscosity, and k, a property of the structure of the porous medium, is the permeability with consistent units of squared length. In the petroleum industry, it is customary to measure V, in cm./sec., f i in centipoise, and V p in atm./cm. with the resulting units of k called the darcy. The conversion is 1 darcy = 9.87 X 10-e sq. cm. This paper describes an extension of Darcy's equation (1) to the case in which the fluid does not follow Newton's law of a constant viscosity (ratio of shear stress to deformation rate); rather, the fluid belongs to a class called Stokesian fluids in the literature (1 ) . Such fluids have the property that the nonhydrostatic components of the fluid stress are unique nonlinear functions of the deformation rate components with the result that the measured "viscosity" varies with the shear stress prevailing in the viscometer. The bulk of the paper develops a one-dimensional theory and compares it with preliminary experimental results for an aqueous solution of a sugar polymer available commercially under the trade name Dextran. The model is developed by direct analogy with the results obtained for flow of the fluid through a uniform capillary and utilizes the rheogram obtained from the latter. A twodimensional analysis, not given here, showed that the dispersive or nonaxial component of the velocity enters the flow equation only as a product with an unknown normal coefficient of viscosity. It should also be pointed out that our choice of the model was guided by two restrictions:( 1 ) The model must give good predictions over a wide shear stress range, and (2) the model must be in a form suitable for routine calculations. The first restriction eliminates some of the stress-deformation rate relations proposed in the literature for Stokesian fluids such as the power law model; the second eliminates others such as the Eyring model.Finally, the flow of non-Newtonian fluids in porous media is important to th...

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