Oil and Gas Well Evaluation Using the Reciprocal Productivity Index Method
Authors: Crafton J.W. 
Published: December 2007

Abstract
The ability to estimate descriptive engineering parameters, such as permeability, and to generate production forecasts and estimated ultimate recovery based on those parameters, without the cost of full numerical simulation or extended build-up tests, is provided by the Reciprocal Productivity Index (RPI) graphical production analysis method. The method's theoretical basis arises from the fact that the traditional constant rate or constant pressure boundary conditions are sufficient, but that the necessary boundary condition only requires that the outflow face transmissibility remain constant over time. With that difference, it is possible to accurately evaluate production histories, in which both the rate and the producing pressure are varying over time, using traditional well testing methods. Examples for both oil and gas wells demonstrate the interpretive capability and limitations. The parameters derived from the RPI method are testable for sensitivity and reasonableness. The forecasts can also be used to test the benefits of changes in operating pressure, pump inlet level or stimulation. When relatively noise-free data sets are available, it is possible to derive geologic and other production engineering information from them.

Introduction
The traditional log of rate versus time (Ref. 1) method of examining oil and gas well production data provides no direct method for determination of effective permeability-thickness, effective wellbore radius or drained area. Type curve comparison procedures (Refs. 2 through 5, among many others) are in some cases only partially theoretically rigorous or in all cases prone to wide ranges of apparently feasible values. The type curve procedures can even yield invalid solutions, because cross-checking procedures such as routinely used in pressure transient testing had not been shown to apply. Reitman, under what is now known to be an unnecessarily restrictive constant pressure inner boundary, has demonstrated that the use of well testing-type evaluation procedures, such as a modified Miller-Dyes-Hutchinson plot are appropriate evaluation tools. A predecessor to Reitman's work conducted by Neal and Mian demonstrated the qualitative effectiveness in a low permeability gas reservoir, but did not present the quantitative use. Upon demonstrating a more general theoretical basis, Crafton showed that production data containing time variant rates and pressures could be interpreted by similar methods in the proper setting.
A large body of literature, eat also exists on the theme of automatic history matching as a precursor to numerical simulation of wells and reservoirs. The sensitivity of those methods to noisy data is well established. Automatic history matching or even the manual matching methods must respond appropriately to the embedded character of the well/reservoir system. Usually, the history matching procedure implicitly predetermines system character, evidenced by the selection of a certain simulation tool or geometry. The process then is to vary the descriptive parameters such as permeability until the calculated and observed dependent variables agree within an acceptable tolerance. Unfortunately, this procedure does not present the information in a manner consistent with the system's theoretical form.

Theoretical Basis for RPI
In order to treat the broadest possible range of production data and to extract the most information from it, a procedure must be demonstrated, which incorporates both rate and pressure time-dependence, allows for direct determination of system descriptive parameters, does not require intensive computing and allows the investigator to directly examine the data in a theoretically consistent setting. A theoretical formulation which fulfills those requirements is shown in Appendix A. It is an abbreviated presentation of the formulation of the "inner boundary condition", usually the constant mass rate sandface constraint, for the radial geometry (the linear geometry has been previously presented). This is the step in which the second constant of integration for the solution is defined. P. 199^