A New Look at Nonlinear Regression in Well Test Interpretation

Abstract

Abstract Nonlinear regression is a standard procedure in the oil industry; however, its practical applications can be constrained by issues observed commonly in real data sets. These issues include convergence performance that depends on starting guess, deviations in the time and pressure data, nonideality in the data, and ambiguity in parameter estimation – these issues limit the ability of well testing results to forecast future reservoir performance. We investigated a variety of strategies to improve nonlinear regression and reduce the susceptibility to these issues. The first strategy we investigated is Cartesian transformation of the parameter space. We developed transform pairs for common reservoir parameters and achieved performance improvement in terms of higher probability of convergence and smaller number of iterations for a random starting guess. In the second strategy, we applied the wavelet transform to the pressure data. We conducted nonlinear regression on a reduced wavelet basis to achieve improved likelihood of convergence and narrower confidence intervals. Wavelet analysis provided robust interpretation of dual-porosity reservoirs and reservoirs with boundary effects. In the third strategy we used total least squares for accurate estimation of reservoir parameters in the presence of deviations in the time data as well as deviations in the pressure data. We developed a robust approximation for total least squares (orthogonal distance regression) for the nonlinear pressure transient function. We tested the techniques rigorously by using a large matrix of test cases made up of real and generated well test data sets. The data sets include a selection of reservoir models and test scenarios, including dual-porosity and fractured reservoirs, reservoirs with rectangular boundaries, cyclic buildup-drawdown tests, and general multirate data. We determined the methods or combinations of methods that work best with a particular reservoir model or type of test. We have shown that these novel nonlinear regression algorithms greatly reduce the ambiguity (narrower confidence intervals) and provide robust parameter estimation even for large noise levels or data uncertainty. We demonstrated that our techniques provide faster convergence and are more robust to poor initial guesses. The techniques provide compression and automatic denoising of massive data sets from permanent downhole gauges. We expect that our techniques and analyses will allow for better forecasting of reservoir performance.