In this study, we show significant improvements over conventional pressure-transient analysis on the basis of nonlinear regression by using total least squares (TLS), which minimizes errors in both pressure and time simultaneously. To our knowledge, TLS has not been applied to pressure-transient analysis before in this sense.TLS regression is not an easy problem to solve mathematically, especially for nonlinear pressure-transient-model functions. In this work, we compare four different versions of TLS. We formulate a robust approximation of the TLS solution, which can handle a variety of tests and reservoir models yet does not compromise the performance of the analysis. We show that our technique reduces ambiguity in the estimation of parameters to a large extent, especially in the presence of noise in time. Using our TLS algorithm, we obtain much narrower confidence intervals on the parameter estimates of a variety of real data sets, compared to the conventional least squares (LS) approach. For synthetic data sets, we observe that the TLS estimates are often closer to the true values than estimates made with LS, especially for poorly determined problems. When the deviation is in pressure only, TLS and LS results are comparable. However, in the presence of deviations in time in addition to pressure, the performance of TLS algorithms is substantially better. We, therefore, expect that our technique will provide more accurate estimation of reservoir parameters, allowing for better forecasting of reservoir performance.
Time/Pressure TLS Techniques for Pressure-Transient AnalysisTLS is a general term used to refer to techniques that take into account observational errors in both dependent (e.g., pressure) and independent (e.g., time) variables. Implementation of TLS is interpreted in terms of the orthogonal distances from the data points to the curve. The TLS algorithms minimize this distance. Since TLS combines in a single objective function errors from two different variables, which may have different units, it is necessary to multiply one of the variables by a weight factor. The relative weight between the two variables can be chosen to give more weight to minimize the errors in a particular variable. Furthermore, in traditional pressure-transient analysis, semilog and log-log graphs frequently are used to emphasize certain aspects of the data. TLSbased pressure-transient analysis can be formulated to correspond to the particular graph of choice. The orthogonal distance will be different when the log scaling of either of the axes is changed. For example, TLS regression on the semilog graph is expected to give more emphasis to early-time data. Consequently, there are three options for orthogonal distance regression (ODR) based on the graphs used in well testing-namely, linear p vs. linear t, linear p vs. log t, and log p vs. log t. Other than ODR, it is also possible to interpret TLS as a summation of vertical (pressure) and horizontal (time) errors in a single objective function, referred to here as summed distanc...