Interference And Pulse Test Analysis By Nonlinear Regression
Authors: El-Khatib Noaman 
Published: March 2010

Abstract
A nonlinear regression technique based on the method of least squares is developed for pulse lest analysis. The values of the transmissivity, kh// µ and storativity h Φc, that minimize the sum of squares of differences between observed and calculated pressures are determined. The equations that represent the conditions for minimum are solved by an iterative procedure based Oil a combination of the Newton-Raphson and the Regula-falsi methods. The pressure at any point is estimated using the line-source solution and the principle of superposition. The method can be used for the analysis of pulse testing of equal and unequal periods as well as cases that could not be analyzed by the tangent construction method of the conventional pulse testing methods. Such cases include situations of unequal pulse ratio and variable flow rate in the different cycles. The method is applicable for the general case of multiple-rate interference testing with pulse testing as a special case.
The developed method was tested using simulated data. The method was found to converge rapidly. Excellent agreement was found between the assumed and estimated parameters for the different cases tested.

Introduction
Pulse testing is being widely considered as a method for obtaining interwell reservoir properties such as porosity and permeability(1–3). In pulse testing, the rate at an active welt is periodically changed between flowing and shut-in conditions while the pressure response at an observation well is recorded. The data is then analyzed graphically by drawing tangents to the pressure-time curve and determining the time lag and the pressure response amplitude of the different odd and even cycles. A set of correlations published by Kamal and Brigham(3) is then used to estimate the transmissivity, kh/ µ and storativity h Φc. Recently, Ogbe and Brigham(4) presented a method to correct for the skin and wellbore storage effects. They also found it necessary to correct the previously published Kamal-Brigham correlation curves. Moreover, the use of these correlations is limited to cases of constant pulse ratio, constant flow rate and constant cycle length of the different cycles. Cases where the lengths of the pulse and/or shut-in periods or flow rates are different in the different cycles cannot be analyzed by the conventional tangent construction method. Furthermore, the conventional graphical method cannot be used in a computer-based systematic analysis which is most desirable.
Some attempts were made to apply the methods of regression analysis for the determination of reservoir characteristics from well test data (5–7). These methods, however, are coupled with numerical reservoir simulators that solve the diffusivity equation. For a presumably homogeneous reservoir for which an analytical solution exists, such methods are unnecessarily complicated.
In a previous work, the method of least squares was applied for interference test analysis in the cases of constant rate and two rate tests(8). In this work, the method of least squares is used to develop a procedure for the analysis of the general case of multiplerate interference testing which includes pulse testing as a special case.