Application of the Unit Step Function to Unusual Well Test Problems
Authors: Correa A.C. 
Published: December 2007

SPE Members

Abstract
Determination of reservoir parameters by the analysis of well-bore pressure data is based upon solutions to the diffusivity equation. Complex solutions may be written as a combination of basic solutions by means of the superposition theorem. However, because the implementation of superposition requires a flow model to be known a priori, the results obtained from the analysis of test data may be affected by the model choice. This may be a serious drawback when information on the reservoir geometry is sought from well tests. Furthermore, in many cases the use of superposition may not be practical.
This work describes a general procedure to solve transient flow problem with boundary conditions which depend upon time. The method uses combination of unit step functions to write a boundary condition which is valid for all times. Solution is then obtained by Laplace transformation. In general, the time-dependent boundary conditions which are combined with the step functions can be of different kinds, like production at constant pressure following production at constant rate.
The method is applied to solve two well test problems: a changing wellbore storage case and the problem of constant-rate production following the blowdown of a gas cushion. production following the blowdown of a gas cushion. In both cases the wellbore pressure behavior may be written as combinations of the skin effect and wellbore storage constant-rate solution and its time derivative. Solution was developed for a generic flow model. but application of the resulting equations for radial flow is promptly obtained.
The procedure described in this paper is useful to generate customized solutions to complex well test problems in order to allow application of non-linear regression methods for automatic analysis of field data.
The method prevented here is not restricted to solution of the diffusivity equation and may be applied to a variety of other interesting and useful problems.

Introduction
Solutions to the diffusivity equation are used to develop methods of interpretation to estimate reservoir parameters from the pressure behavior of wells. Solutions to complex problems are pressure behavior of wells. Solutions to complex problems are often composed from basic solutions by means of the superposition theorem. However, practical application of the superposition integral requires the boundary conditions in the decomposed problem and in the original problem to be of the same kind (e.g. - solution to a variable rate problem may be decomposed as series of constant rate solutions).
The unit step function is a powerful method that may be used to generate solutions to the diffusivity equation for problems with mixed or time-dependent boundary conditions. For instance, an analytical solutions to problems of constant-rate production followed by constant-pressure production, or vice-versa, may be generated by direct application of the unit step function. The use of the unit step function yields boundary conditions that are valid for all times, allowing solution by Laplace transformation.
This paper describes the application of the unit step function to unusual well test problems. A procedure to generate customized solutions to complex problems is presented. Two cases are discussed in detail:Constant-rate production with changing wellbore storage,Blowdown gas cushion on liquid to start constant-rate production.
The first case describes the pressure solution to a changing wellbore storage process in wells either producing or injecting at a constant wellhead flow rate. Step changes in the wellbore storage coefficient may occur clue to change from a compressibility dominated wellbore storage mechanism to a changing liquid level process, or vice-versa. Although it an approximate solution to process, or vice-versa. Although it an approximate solution to this problem is available in the literature, it is not accurate enough to be used with non-linear regression methods for automatic interpretation of well tests.
The second case discusses the problem of constant-rate liquid production following the blowdown of a gas cushion. production following the blowdown of a gas cushion.