Ilk, Dilhan (2005-12). Deconvolution of variable rate reservoir performance data using B-splines. Master's Thesis.
This work presents the development, validation and application of a novel deconvolution method based on
B-splines for analyzing variable-rate reservoir performance data. Variable-rate deconvolution is a
mathematically unstable problem which has been under investigation by many researchers over the last 35
years. While many deconvolution methods have been developed, few of these methods perform well in
practice - and the importance of variable-rate deconvolution is increasing due to applications of
permanent downhole gauges and large-scale processing/analysis of production data. Under these
circumstances, our objective is to create a robust and practical tool which can tolerate reasonable
variability and relatively large errors in rate and pressure data without generating instability in the
We propose representing the derivative of unknown unit rate drawdown pressure as a weighted sum of Bsplines
(with logarithmically distributed knots). We then apply the convolution theorem in the Laplace
domain with the input rate and obtain the sensitivities of the pressure response with respect to individual
B-splines after numerical inversion of the Laplace transform. The sensitivity matrix is then used in a
regularized least-squares procedure to obtain the unknown coefficients of the B-spline representation of
the unit rate response or the well testing pressure derivative function. We have also implemented a
physically sound regularization scheme into our deconvolution procedure for handling higher levels of
noise and systematic errors.
We validate our method with synthetic examples generated with and without errors. The new method can
recover the unit rate drawdown pressure response and its derivative to a considerable extent, even when
high levels of noise are present in both the rate and pressure observations. We also demonstrate the use of
regularization and provide examples of under and over-regularization, and we discuss procedures for
ensuring proper regularization. Upon validation, we then demonstrate our deconvolution method using a variety of field cases.
Ultimately, the results of our new variable-rate deconvolution technique suggest that this technique has a
broad applicability in pressure transient/production data analysis. The goal of this thesis is to demonstrate
that the combined approach of B-splines, Laplace domain convolution, least-squares error reduction, and
regularization are innovative and robust; therefore, the proposed technique has potential utility in the
analysis and interpretation of reservoir performance data.