A Generalized Pressure Derivative Analysis For Composite Reservoirs
Authors: Issaka M.B. , Ambastha A.K. 
Published: March 2010

Abstract
Pressure derivatives have been shown to be more sensitive to disturbances in the reservoir than pressure signals; resulting in more detail on derivative graphs than is apparent on pressure graphs. The semilog pressure derivative is widely used in well test analysis. One reason for its popularity is that, for radial systems, the response appears as a horizontal line during the infinite- acting radial flow period, resulting in easier identification. However, when the semilog pressure derivative is applied to flow geometries other than radial, the responses are not horizontal; making identification of flow regimes more difficult. Thus, a generalized pressure derivative is necessary to simplify the identification of flow regimes in any flow geometry.
In this study, a generalized pressure derivative is defined and used to identify the various flow regimes for composite systems in radial, elliptical, linear and spherical geometries. This generalized pressure derivative is of the power law type, and is characterized by a different exponent for each of the flow geometries. Using well test data from analytical solutions for radial, elliptical, linear and spherical composite reservoirs, a graph of the generalized pressure derivative versus time, for any of the flow geometries appears as a horizontal line during the primary flow regime characteristic of that geometry.
Design and analysis equations, based on the generalized pressure derivative, are presented for well testing of composite reservoirs in various flow geometries. Reservoir parameters estimated using these equations will add to the degree of confidence in the estimated parameters based on pressure analysis. The generalized pressure derivative is also used to investigate differences and similarities among the four flow systems. Results from this study confirm that for radial and elliptical systems, the long term pressure derivative behaviour is influenced only by the mobility ratio between the inner and outer regions of the composite system. For linear and spherical systems, however, long term derivative behaviour is governed by both the mobility ratio and the storativity ratio. This finding has a significant impact on the development of type curves for either manual or automated type curve matching for the various flow geometries.

Introduction
Pressure derivatives have been shown to be more sensitive to disturbances in the reservoir than pressure signals. This results in greater detail on a derivative graph than is apparent on a pressure graph. Pressure derivatives were first introduced by Tiab and Kumar(1), who presented the derivative of pressure with respect to time. Later, Bourdet et al.(2) introduced the semi-log pressure derivative, defined as the derivative of the well pressure with respect to the natural logarithm of time. The semilog pressure derivative response appears as a horizontal line during the infinite- acting radial flow period, resulting in an easy identification of the radial flow regime. As a result, the semilog pressure derivative is widely used in well test analysis of not only homogeneous, but also composite reservoirs.
To analyse well tests for thermal recovery projects, reservoirs have been idealized as composite reservoirs.