Use of Tikhonov Regularization in Pressure and Rate Transient Derivative Analysis from Noisy Data
Abstract
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.
Citation
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 https : / /hdl .handle .net /1969 .1 /174313.