Deconvolution of Well-Test Data as a Nonlinear Total Least-Squares Problem

Abstract

Summary We present a new time-domain method for the deconvolution of well test data which is characterized by three novel features:Instead of the rate-normalized pressure derivative itself, we estimate its logarithm, which makes explicit sign constraints necessary;the formulation accounts for errors in both rate and pressure data, and thus amounts to a Total Least Squares (TLS) problem; andregularization is based on a measure of the overall curvature of its graph. The resulting separable nonlinear TLS problem is solved using the Variable Projection algorithm. A comprehensive error analysis is given. The paper also includes tests with a simulated example and an application to a large field example. Introduction With current trends towards permanent downhole instrumentation, continuous bottomhole well pressure monitoring is becoming the norm in new field developments. The resulting well-test data sets, recorded mainly during production, can consist of hundreds of flow periods and millions of pressure data points stretched over thousands of hours of recording time. Such data sets contain information about the reservoir at distances from the well which can be several orders of magnitude larger than the radius of investigation of a single flow period. Conventional derivative analysis is thus ill equipped to access the full potential information content. What is required is an analysis method which can extract the response which the reservoir would exhibit when subjected to a single drawdown at constant rate over any period of time up to the entire production period. In mathematical terms, this is a deconvolution problem. Since its first formulation by Hutchinson and Sikora in 1959,1 it has received sporadic, but recurring attention.2-17 This paper presents a new approach which is based on a regularized, nonlinear TLS formulation. It is an update on earlier versions which were presented at the SPE Annual Meetings in 200118 and 200219 (henceforth referred to as "Paper I" and "Paper II"). More recently, our approach was taken up by Levitan,20 who subjected it to a critical evaluation and suggested some modifications. In terms of the usual classification into time-domain and spectral approaches, ours is a time-domain approach. It differs from earlier approaches in this category in three important ways:The solution is encoded in terms of the logarithm of the rate-normalized pressure derivative, which automatically ensures strict positivity of the derivative itself at the expense of rendering the problem nonlinear. However, we are thus able to avoid explicit constraints on the solution space which made previous constrained approaches so difficult, yet still cannot prevent zeros in the deconvolved derivative.A new error measure accounts for uncertainties not only in the pressure, but also in the rate data, which are usually much less accurately known. Thus, provided sufficient data are available, our method can provide a joint estimate of initial pressure, rates, and response parameters; the time-consuming manual correction of rate errors is rendered obsolete. The mathematical formulation is an instance of what is known as a TLS problem in the numerical analysis literature and as an "Errors-In-Variables" problem in statistics. TLS has become a standard approach in parameter estimation problems, but its application to well-test analysis seems to be new.Regularization is based on a measure of the total curvature of the deconvolved pressure derivative, instead of its average slope, as in an earlier approach 15 and Paper I. Here, the motivation is that slopes provide important information about the flow regime and should therefore be preserved as much as possible. The paper is organized as follows: The first two sections are introductory and give a summary of the deconvolution problem in well-test analysis and a concise survey of its treatment in the petroleum engineering and hydrology literature. Based on the mathematical framework developed in these sections, we then give a comprehensive account of our own approach. We also derive analytic expressions for bias and variance of the estimated parameter set based on simple Gaussian models for the measurement errors in pressure and rate signals. We illustrate our method with a small simulated data set, demonstrating the effect of varying levels of regularization on the confidence intervals. The final section presents an application to a large field example which allows a direct comparison of our method with conventional derivative analysis.

Understanding diagnostic plots for well-test interpretation

“…If the flow rates vary continuously in time, other more sophisticated algorithms are required (e.g. Levitan 2005;von Schroeter et al 2001).…”

Numerical Modeling of Periodic Pumping Tests in Wells Penetrating a Heterogeneous Aquifer

“…Two issues remain which need further attention in well test analysis: the inclusion of reservoir heterogeneity and the treatment of rate variations. Reservoir heterogeneity which cannot be handled with traditional analytical methods is usually approached using numerical techniques Copty and Findikakis, 2004;Ye et al, 2004;Houze et al, 2010); rate variations are approached with deconvolution techniques (Schroeter et al, 2004;Gringarten, 2010).…”

Flow Modeling of Well Test Analysis for Porous–Vuggy Carbonate Reservoirs

“…Savitzky and Golay (1964) researched the least-squares procedures usefulness for smoothing data. This technique has since been validated for use as a smoothing algorithm (Gorry 1991;Green et al 1985) and is often used to smooth pressure derivatives derived from well-testing (Lane et al 1991;Pirard and Bocock 1986;Von Schroeter et al 2004). Figure 18 shows the smoothed derivative curve, which is concave.…”