Automated Well Test Analysis Using Robust (LAV) Nonlinear Parameter Estimation

Abstract

ABSTRACT Most of the nonlinear regression algorithms used in automated well test interpretation perform a minimization of an objective function formed by the sum of the squares of the differences between the data and the model values. This constitutes a typical nonlinear least-squares problem. One of the drawbacks of the least squares approach is that it is significantly affected by outliers, which are data points that can be considered bad observations. This study proposes a robust algorithm that uses the LAV (Least Absolute Value) as the criterion for the minimization. One of the greatest advantages of the LAV approach is that it provides a smooth transition between full acceptance and total rejection of a given observation, providing a systematic way of rejecting outliers by automatically assigning them less weight in the objective function. The technique described in this work is based on the expansion of the model function in a Taylor series up to the first order terms. This poses a new problem which is solved iteratively after being transformed into a multiple linear regression parameter estimation. The procedure becomes therefore a sequence of multiple linear regression problems, which are solved using the least absolute value criterion. Because the equations of condition are solved, instead of the so-called normal equations (as in several implementations of nonlinear parameter estimation algorithms discussed in the literature), the method avoids unnecessary exacerbation of the ill-conditioning of the design matrix. Examples where no large errors in the data are present show that this method produces results similar to least-squares minimization. When outliers are introduced into the observations, however, the present technique yields much better estimates. Also, the least absolute value approach works better than the least-squares criterion when estimating parameters whose sensitivity is small, as in the case of obtaining the storativity ratio and the interporosity flow coefficient in a naturally fractured reservoir with double-porosity behavior.