A simple, yet effective and remarkable, procedure is devised to handle discontinuities in flow rate when Stehfest's algorithm is used to solve for pressure responses. Example applications are presented to demonstrate the viability of our algorithm. We also demonstrate application of this method to completion schemes that have come into vogue recently, and for which no simple procedures are currently available.
Virtually every publication concerning the pressure behavior of wells producing a slightly-compressible liquid over the past decade has relied on the Stehfest algorithm. Like television in every-day life, in well-test analysis this algorithm is ubiquitous. Despite its popularity, as noted by Stehfest, this algorithm may not be used unless the unknown function which we seek to approximate does not have discontinuities. Thus, in considering variable rate problems, which are the norm in well-test analysis, it is not possible to use this algorithm directly. For example, rather than computing buildup responses by the relation
we use the relation
The purpose of this paper is to describe a procedure to use Eq. 1 directly.
To preview the problems we discuss, consider the computations displayed in Fig. 1 where we consider derivative responses for these are much more sensitive to the procedure used. The unbroken line is the drawdown solution. The line given as dashes and dots reflects computations obtained by applying Stehfest's algorithm to Eq. 1. The difficulties in using Eq. 1 are immediately apparent. It is for this reason we use Eq. 2 (circles). The squares represent computations by the method outlined in this paper.
Figure 1 clearly demonstrates the advantages of using our method.