Novel Well Testing Applications of Laplace Transform Deconvolution

Abstract

Abstract Well-test solutions are formulated typically as a dimensionless pressure change that occurs in response to a single change in flow rate. Ideally, the rate change is a step change in sandface flow rate; however, in practice wellbore storage and poor control of surface flow rate often lead to variable-rate tests. Deconvolution methods have been shown to improve analysis of such data by removing the effects of undesired rate variations from the measured pressure response. In 1988, Roumboutsos and Stewart1 proposed deconvolving well-test data by numerically transforming the discrete pressure and rate data into the Laplace transform domain. In Laplace space, the often difficult deconvolution process is reduced to simple division. Roumboutsos and Stewart also introduced a method of synthetically deconvolving constant wellbore-storage pressure data without explicitly measuring the flow rate. This paper extends the concepts introduced by Roumboutsos and Stewart and demonstrates that when the wellbore storage is constant, a trial and error method may be used to estimate storage and progressively strip away the afterflow effects. Moreover, when storage is not constant, a series of synthetic deconvolutions may be used to strip away the constant components of storage to reveal the variable components of the total wellbore storage. A new concept of afterflow extraction is introduced. In this technique, an analytic reservoir solution is substituted in the Laplace domain and used to explicitly calculate afterflow. This technique can be used to evaluate flow meter performance. Afterflow extraction provides a unique opportunity to investigate the sensitivity of deconvolution to errors in flow rate measurement. Examples are presented in which afterflow is extracted from a single set of pressure data using multiple reservoir solutions (i.e., radial, dualporosity, etc.). Each of the extracted afterflow profiles are then used to deconvolve the recorded pressure data. The examples illustrate that extremely slight differences in the afterflow can significantly alter the deconvolved response. These examples show how small errors in afterflow measurement can result in an incorrect diagnosis of the reservoir model.