Abstract

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 works also when the effects of perturbed rate changes are subsiding and rate variations are becoming monotonical (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 behavior of varying rates is used to be known as "Superposition"; the inverse procedure is accordingly "Desuperposition". Modem literature call it "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 all and by Weinstock and Colpitts2. After this "rate normalisation" 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 techniques3,9 to solve the convolution integral, the Gladfelter and Weinstock/Colpitts method is still around and even has been recently evaluated7. 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:

  1. 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 it6 but noisy data disturb the apparent delicate solution technique again3,8.

  2. The application is not user-friendly for spreadsheet analyses.

  3. Commercial "black box" software is not available.

The herein proposed method retains the simple features of the Gladfelter/Wcinstock-Colpitt method but improves its accuracy dramatically with few additional computations. Also it is user-friendly 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, Homer10 suggested to divide 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) Available In Full Paper

or

Equation (2) Available In Full Paper

and the rate normalised pressure differential for the drawdown

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