Deconvolution of variable rate reservoir performance data using B-splines
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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 deconvolution process. 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.
Ilk, Dilhan (2005). Deconvolution of variable rate reservoir performance data using B-splines. Master's thesis, Texas A&M University. Texas A&M University. Available electronically from