Robust Well-Test Interpretation by Using Nonlinear Regression With Parameter and Data Transformations

Abstract

Summary Nonlinear regression is a well-established technique in well-test interpretation. However, this widely used technique is vulnerable to issues commonly observed in real data sets—specifically, sensitivity to noise, parameter uncertainty, and dependence on starting guess. In this paper, we show significant improvements in nonlinear regression by using transformations on the parameter space and the data space. Our techniques improve the accuracy of parameter estimation substantially. The techniques also provide faster convergence, reduced sensitivity to starting guesses, automatic noise reduction, and data compression. In the first part of the paper, we show, for the first time, that Cartesian parameter transformations are necessary for correct statistical representation of physical systems (e.g., the reservoir). Using true Cartesian parameters enables nonlinear regression to search for the optimal solution homogeneously on the entire parameter space, which results in faster convergence and increases the probability of convergence for a random starting guess. Nonlinear regression using Cartesian parameters also reveals inherent ambiguities in a data set, which may be left concealed when using existing techniques, leading to incorrect conclusions. We proposed suitable Cartesian transform pairs for common reservoir parameters and used a Monte Carlo technique to verify that the transform pairs generate Cartesian parameters. The second part of the paper discusses nonlinear regression using the wavelet transformation of the data set. The wavelet transformation is a process that can compress and denoise data automatically. We showed that only a few wavelet coefficients are sufficient for an improved performance and direct control of nonlinear regression. By using regression on a reduced wavelet basis rather than the original pressure data points, we achieved improved performance in terms of likelihood of convergence and narrower confidence intervals. The wavelet components in the reduced basis isolate the key contributors to the response and, hence, use only the relevant elements in the pressure-transient signal. We investigated four different wavelet strategies, which differ in the method of choosing a reduced wavelet basis. Combinations of the techniques discussed in this paper were used to analyze 20 data sets to find the technique or combination of techniques that works best with a particular data set. Using the appropriate combination of our techniques provides very robust and novel interpretation techniques, which will allow for reliable estimation of reservoir parameters using nonlinear regression.