Abstract

For the case of an infinite radial system operating at constant terminal rate, the reservoir engineer often uses the "point source" solution of the diffusivity equation to study pressure interference effects. At early times and at short distances from the inner boundary these solutions are invalid. The amount of error is often not precisely defined because of mathematical difficulties. Work presented here shows that, if dimensionless time is defined appropriately, previous solutions of the pressure equation can be displayed as a family of curves on one chart. These solutions include the point source solution (referred to in the field of hydrology as the Theis solution) and other solutions obtained with digital computer methods. With these curves, an exact evaluation of the pressure drop within a reservoir or an aquifer can be made by the engineer. Examples of field problem solutions are presented. In most reservoirs the error involved when the Theis solution is employed is often negligible; whereas, in the calculation of interference effects in an aquifer, a substantial error can occur through such an approach.

Introduction

Flow equations are used in petroleum engineering to study the behavior of individual wells and reservoirs. In the case of wells, the pressure response at the wellbore face is the major point of interest; whereas, in the case of reservoirs, the pressure response at the interface of the aquifer boundary is sought. To aid in such studies, the flow equations have been solved in terms of the behavior at these two inner boundaries. Only limited work has been published in regard to the pressure conditions away from these points, i.e., within the reservoir or aquifer. Theis and Mortada are among the few who have reported on this problem. The Theis approach employs the exponential integral and is valid for pressure conditions that occur some distance away from the flow disturbance. It is derived from the concept of a point source, as opposed to a flow across some finite area. The Mortada results, on the other hand, are valid at all points within the reservoir or aquifer. They are presented in terms of dimensionless ratios of the radius where the pressure is desired to the radius where the flow rate is measured. Their main use, in the past, has been in aquifer studies. The published results are presented in the form of graphs that are limited to a maximum radius ratio of 64. These graphical results are cumbersome to interpolate at non-integral radius ratios, so that one may be forced to utilize a rather involved analytical expression presented by Mortada.

BASIC EQUATIONS

The solutions of Mortada and Theis are both based on the diffusivity equation as applied to the case of an infinite radial system subject to a constant terminal rate. The equation is obtained by combining the material balance equation with Darcy's flow equation. The assumptions implicit in the use of this equation are as follows:

1. a single fluid is present that occupies the entire pore volume;

2. the reservoir is horizontal, homogeneous, uniform in thickness, and of infinite radial extent;

3. compressibility and viscosity of the fluid remain constant at all pressures; and

4. fluid density obeys the equation

(1)

Using the diffusivity equation in situations where the above conditions do not hold will result in errors. These errors (not discussed here) but only the errors which arise in the solution of the equation itself. The diffusivity equation for the homogeneous reservoir conditions cited above can be written in cylindrical coordinates, as

(2)

To obtain a dimensionless equation, so a single solution can be used for applications of different porosity, permeability and fluid properties, the following transformations are usually made:

(3)

(4)

(5)

After these transformations are made, Eq. 3 can be written in dimensionless form, as

(6)

JPT

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