Abstract:

In this work we made significant improvements to nonlinear regression used in well test interpretation. Nonlinear regression was introduced to well testing more than three decades ago and quickly became a standard practice in the industry. However, limited improvement has been achieved for some time. This widely-used technique is vulnerable to issues commonly observed in real data sets, namely sensitivity to noise, parameter uncertainty (ambiguity), and dependence on starting guess. We developed several different methods that improved nonlinear regression significantly. We investigated the performance of these methods on a variety of field data to determine which method (or combination of methods) works best in particular well test situations.

The techniques we developed can be considered in three groups:

In the first group we considered parameter transformations. We developed techniques to find robust Cartesian transform pairs that worked very well with a variety of reservoir models. The Cartesian parameter transformations we proposed provided faster convergence, doubled the probability of convergence for a random starting guess, and revealed the ambiguities inherent in the data.

In the second group, data space transformations, we analyzed the wavelet transform and the pressure derivative. We developed four different strategies to form a reduced wavelet basis and conducted nonlinear regression in the reduced basis rather than the original pressure data points. Using these strategies we achieved improved performance in terms of likelihood of convergence and narrower confidence intervals (reduced uncertainty). We also developed a novel interpretation technique for cyclic data analysis. The technique is based on the two-dimensional wavelet transform and takes into account the correlation between subsequent cycles for error correction.
We also considered derivative curve analysis as another form of data space transformation. Derivative fitting was found to improve confidence intervals significantly and provide faster convergence for dual-porosity reservoirs. We also showed the necessity of using the Monte Carlo simulation technique for accurate computation of confidence intervals for dual-porosity reservoirs.

In the third group of nonlinear regression techniques we considered alternative objective functions to regular least squares. We developed a robust total least squares (TLS) algorithm that considers and minimizes deviations in both time and pressure simultaneously, hence making interpretation results more accurate and more stable. When there are deviations in the time data TLS performs substantially better than least squares, giving much narrower confidence intervals. In addition, the total least squares approach was found to be less prone to time-shift errors and errors in the early time data.

We also considered the least absolute value (LAV) technique as an alternative to the least squares objective function. Using orthogonal distance regression together with the least absolute value criterion, we achieved a robust estimator for data with time deviations and outliers. We developed an analysis technique based on the sum of square roots. The least square root technique was found to be robust against nonideality in data.

We tested the techniques rigorously by using a large matrix of test cases made up of real and generated well test data sets. In the test matrix all possible combinations of different methods were applied to 20 real well test data sets from 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. We expect that our techniques will provide more accurate estimation of reservoir parameters, allowing for better forecasting of reservoir performance.

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