Use of Tikhonov Regularization in Pressure and Rate Transient Derivative Analysis from Noisy Data
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Use of the pressure derivative for pressure transient test analysis has been a crucial tool in the analysis of reservoir and well performance data since it was formally proposed by Bourdet, Ayoub, and Pirard in 1989. Despite the diagnostic advantage of using the pressure derivative for well test interpretation, the key drawback of the pressure derivative is its calculation. Bourdet et al. proposed a simple and very consistent numerical method to calculate the pressure derivative. This method is based on a weighted central-difference scheme. The so-called "Bourdet" algorithm is the most common pressure derivative calculation used in the petroleum literature. Even with its wide acceptance, the Bourdet pressure derivative calculation method has limitations, particularly for noisy data. The goal of this work is to provide an alternate derivative calculation to the Bourdet method. The method we propose is Tikhonov Regularization. Tikhonov Regularization can be regarded as regression with the addition of a penalty term. The goal is to balance between goodness-of-fit and the roughness of fitted data to calculate a smooth derivative function from the result of regularization. In contrast to the Bourdet method, the regularization parameter can be calculated mathematically by generalized cross-validation and does not require manual manipulation from the analyst to determine the optimum regularization value. This study includes the development and implementation of the Tikhonov Regularization method in order to calculate the pressure derivative using the MATLAB software program. We studied the effectiveness of the Tikhonov Regularization method for calculating the pressure derivative as compared to the Bourdet algorithm from this developed module. The effectiveness of derivative calculation is validated using both synthetic pressure data and field pressure data. We employ the Root-Mean-Square (RMS) and Mean-Absolute-Error (MAE) as statistical measures of effectiveness. Our results show that the Tikhonov Regularization method yields a significantly better derivative calculation than Bourdet method in all the cases in this study, particularly those cases with elevated levels of noise. Based on the results obtained in this study, we propose that the Tikhonov Regularization method should be used to calculate the pressure derivative for data cases exhibiting high levels of noise.
Lortong, Nattapon (2018). Use of Tikhonov Regularization in Pressure and Rate Transient Derivative Analysis from Noisy Data. Master's thesis, Texas A & M University. Available electronically from