An Alternative to Superposition For Monotonically Changing Flow Rates
Authors: Moser Heinrich 
Published: March 2010

This paper presents a simple approximation method as an alternative to deconvolution (desuperposition) of pressure data obtained under variable rate conditions. The rates should vary monotonically; however, the method also works when the effects of perturbed rate changes are subsiding and rate variations are becoming monotonic (e.g., slug tests). In essence, this work is an improved method of the simplest form of deconvolution as proposed by Gladfelter et al. and by Weinstock and Colpitts. The proposed method, called "Time Correction Method," is also applicable in non-radial and heterogeneous flow problems.

On theoretical examples it has been demonstrated that this method delivers results accurate enough for practical purposes.

Field examples are presented for (a) pressure buildup for a wellbore storage problem where downhole rate and pressures were recorded simultaneously, (b) a slug test, and (c) a pressure buildup of a pumping well where pressure was recorded downhole.

Introduction

The method to convert a constant rate pressure solution into pressure behaviour of varying rates used to be known as "superposition"; the inverse procedure was, accordingly, "desuperposition." Modern literature use the terms "convolution" and "deconvolution," respectively, for the very same thing. For monotonically changing flow rates, a method suggesting that the drawdown or buildup pressure differential is to be divided by the instantaneous flow rate was proposed by Gladfelter et al.(1) and by Weinstock and Colpitts(2). After this "rate normalization" of pressure differentials, it is suggested that the data can be analysed by common semi-log procedures. In other words, this method represents the simplest form of deconvolution. Regardless of many recent published techniques(3–9) to solve the convolution integral, the Gladfelter and Weinstock/Colpitts method is still around and has even been recently evaluated(7). Although the method could be considered at best an approximation, it cannot be matched for its simplicity. Obviously, the new deconvolution techniques are not readily used by the well test analysts for the following reasons:The convolution integral (in order to deconvolve the variable rate data) must be solved numerically. This has inherent instability problems; albeit there is a way around it(6), noisy data disturb the apparent delicate solution technique again(3,8).The application is not user-friendly for spreadsheet analyses.Commercial "black box" software is not available.

The herein proposed method retains the simple features of the Gladfelter/Weinstock-Colpitt method but improves its accuracy dramatically with a few additional computations. Also, it is userfriendly for spreadsheet application. It is called an alternative to superposition (desuperposition) or deconvolution because it does not require a solution of the convolution integral with its tedious computation of a summation term. The proposed method is still an approximation but it is accurate enough for most practical applications.

Development and Applications

In order to convert a variable rate into a constant rate pressure solution, Horner(10) suggested dividing the cumulative production by the last rate before shut-in. In a similar way but in a continuous fashion, this method suggests that the drawdown time is corrected as

Equation 1–5 (available in full paper)