Determination of Reservoir Properties from Sinusoidal and Multirate Flow Tests in One or More Wells
Authors: Kuo C.H. 
Published: December 2007

Abstract
This theoretical research deals with the use of periodic variable-rate flow tests for obtaining reservoir parameters. During the test period, the flow rate at a test well is varied periodically, either sinusoidally or in a repeated sequence of constant flow rates, while the accompanying resultant pressures are measured at the test well itself (single-well test) and/or at nearby responding wells (multiwell test). Theoretical pressure equations due to these periodic rates of fluid flow at a test well are derived. Methods of interpreting the test results are proposed.
For the general periodic variable-rate flow tests, reservoir properties can be calculated from the slope and the intercept in a linear plot of the measured pressure responses vs a known time function. In cases where a sinusoidal rate is imposed at a test well, it is only necessary to measure the maximum pressure response amplitude and the phase lag in order to calculate the formation parameters.
A single-well test yields the kh value and the skin factor. A multiwell test yields the average kh value and the storage capacity between wells. When the same wells are used, differences in the kh values from these tests would be indicative of reservoir heterogeneities. Since the tests can be carried out simultaneously, this combination provides more information about the reservoir than would be available from either test alone. A frequency analysis of the pressure responses also offers the possibility of determining the heterogeneity distribution within the reservoir.

Introduction
Pressure buildup analysis is one of the most common means of determining reservoir properties from well tests. The method is operationally simple, and the theory is well developed. Unfortunately, interpretation of pressure buildup test results is often difficult for reservoirs of very high or very low permeability. Another method involving use of two or more wells to evaluate reservoir properties is the interference test. This method, however, has not been used often because of the long interruption of normal field operations usually required to obtain useful data. To eliminate this drawback, a new method of interference testing, called pulse testing, recently was proposed by Johnson et al. The test utilizes a sensitive pressure gauge at a responding well to measure the response generated by a series of flow-rate changes at a test well. A key feature of these pulse test is that, because of the cyclic nature of the pressure response, the arrival of the response can be distinguished from the background pressure. Thus, the time required to obtain a diagnostic pressure response is very short (usually a few hours or less) compared with conventional interference testing. The field applications of this pulse-testing technique, made to determine the distribution of reservoir properties, also have been reported.
This theoretical research investigates the use of periodically varying flow rates to obtain reservoir parameters. Two types of flow rates, sinusoidal and periodic multiple rates, are considered in this study. The pressure response can be measured simultaneously at both the test well and a nearby responding well. In this report, when the pressure response is measured at the test well, the test will be called a "single-well test"; if the pressure is measured at a responding well some distance away from the test well, the test will be termed a "multiwell test". Theoretical pressure response equations for these tests are derived in this report. Interpretation methods for evaluating the formation parameters are also developed. The advantages and the drawbacks of these tests will be discussed.

THEORETICAL DEVELOPMENT
In a manner analogous to that used in developing the theoretical pressure response for many other types of well tests, the reservoir is considered to be a homogeneous and isotropic porous medium. This reservoir has a finite thickness h. The porosity phi and the permeability k are assumed to be constants. A test well is located in this porous medium of infinite radial extent.
SPEJ
P. 499ˆ