This article, written by Technology Editor Dennis Denney, contains highlights of paper SPE 95571, "Deconvolution of Variable- Rate Reservoir-Performance Data Using B-Splines," by D. Ilk, P.P. Valko, SPE, and T.A. Blasingame, SPE, Texas A&M U., prepared for the 2005 SPE Annual Technical Conference and Exhibition, Dallas, 9–12 October.
This work presents the development, validation, and application of a deconvolution method that uses B-splines to analyze variable-rate reservoir-performance data. Variable-rate deconvolution is a mathematically unstable problem. B-splines represent the derivative of unknown unit-rate drawdown pressure, and numerical inversion of the Laplace transform is used in this formulation. When significant errors and inconsistencies exist in the data functions, direct and indirect regularization methods are used (i.e., mathematical "uniformity" processes). The new deconvolution method has broad applicability in variable-rate/pressure problems and can be implemented in typical well-test and production data analysis.
The constant-rate drawdown-pressure behavior of a well/reservoir system is the primary signature used to classify/establish the characteristic reservoir model. Transient-well-test procedures typically are designed to create a pair of controlled flow periods (a pressure-drawdown/-buildup sequence) and to convert the last part of the response (the pressure buildup) into an equivalent constant-rate drawdown by use of special time transforms. However, the presence of wellbore storage, previous flow history, and rate variations may mask or distort characteristic features in the pressure and rate responses.
With accurate downhole data, the variable-rate deconvolution should be viable for traditional well-test methods because deconvolution can provide an equivalent constant-rate response for the entire time span of observation.
Variable-rate deconvolution is mathematically ill-conditioned. While many methods are applied to deconvolve ideal data, very few deconvolution methods perform well in practice. The ill-conditioned nature of the deconvolution problem means that small changes in the input data cause large variations in the deconvolved constant-rate pressures. Mathematically, this method attempts to solve a first-kind Volterra equation, which is ill-posed. However, in this case, the kernel of the Volterra-type equation is the flow-rate function (i.e., the generating function), which is not known analytically but approximated from the observed flow rates, adding to the complexity of the problem.