Unit Response Function From Varying-Rate Data
Authors: Jargon J.R. , Van Poollen H.K. 
Published: December 2007

Abstract
Flow of slightly compressible fluids through porous media can be described by linear equations. Many mathematical solutions are available for the boundary conditions of constant terminal pressures or constant flow rates, or a combination of both. Various techniques of formation evaluation use drastic rate changes, such as sudden shutin. Other methods require that the well be produced at a constant rate. In practice, one is not always able to fulfill these rigid requirements. This report shows a method by which varying-rate data can be converted to a constant-rate pressure response, called the unit response function. After obtaining the unit response function, one may analyze it, using techniques described in the literature. The method uses the principle of superposition. Necessary data is a time history of pressure and production. If pressure changes and/or rate changes are very drastic, the unit response function becomes erratic and oscillates. The erratic and oscillating results are obvious, and under those circumstances, no answer can be obtained. It appears, however, that a response function without oscillation will not contain any appreciable errors - one gets either good results or no results. This report includes a number of examples of converting constant-rate data to constant terminal-pressure data and vice versa. In addition, a number of field-test applications are given. This method should be useful in converting mathematical solutions from one boundary condition to another. It already has proven useful in converting varying-rate data to constant-rate conditions.

Introduction
Pressure drawdown tests are performed on production wells to determine various reservoir characteristics, including permeability, distances to discontinuities, reservoir continuity and wellbore conditions. An ideal drawdown test is performed at a constant production rate with pressure observation during the drawdown period. A typical drawdown curve for a constant rate of production is shown in Fig. 1.In practice, a constant production rate cannot always be maintained. Fig. 2 shows a pressure drawdown curve for a varying rate. Usual methods of analysis are not applicable to tests such as this. It is, therefore, necessary to convert the varying-rate data somehow, before usual analytical techniques can be applied. This report describes a method of obtaining a response function. A response function can be defined as the relationship between pressure and time for a well producing at a constant unit rate. Such a response function can be analyzed by the usual analytical techniques. Hutchinson and Sikora first approximated a resistance function (response function), and by applying the superposition principle, pressures were calculated using the varying rates. If no match was obtained, adjustments were made to the response function and the closer approximation was used again in an attempt to match calculated and observed pressure behavior. If a good match was obtained, the response function was then used to extrapolate the future aquifer behavior.
JPT
P. 965ˆ