Deconvolution of Variable-Rate Reservoir Performance Data Using B-Splines

Abstract

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. iv 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.

Numerical Inversion of Laplace Transforms in the Solution of Transient Flow Problems With Discontinuities

“…for a given rate function, ( ) t q (Ilk et al, 2005). In deconvolution, this causes small errors in data, especially in ( ) t q , to yield large variations (errors) in the deconvolved constant-rate response.…”

“…In other words, it is possible to improve the success of a given deconvolution algorithms by improving the numerical inversion of Laplace transforms of discontinuous or piecewise differentiable functions. Along these lines, Ilk et al (2005) have noted the importance of the numerical inversion algorithm used to invert the Laplace transform of the sampled rate function in their deconvolution approach using B-splines. Our renewed interest in the problem was driven by a new algorithm presented by Iseger (2006) that claims to handle singularities and discontinuities.…”

Diagnosis of nonlinear reservoir behaviour for correctly applying the superposition principle and deconvolution

“…Based on this method, Levitan (2005) and Levitan et al (2006) made some improvements with practical considerations. Ilk et al (2006) presented B-spline deconvolution and applied this algorithm on the gas well test. With the installation of permanent down-hole pressure and flow-rate measure systems, deconvolution becomes more important because it can process pressure and rate data simultaneously and can obtain the underlying reservoir properties, which change with time (Onur et al, 2008).…”

“…Levitan (2005) and Houze et al (2010) demonstrated that deconvolution fails for the changes of the wellbore storage and skin case. Ilk et al (2010) discussed several scenarios where deconvolution is invalid, such as gas flow, multiphase flow, and non-Darcy flow. These nonlinearities usually are caused by the changes in reservoir properties and well conditions.…”

“…All pressure build-ups are plotted in the same plot and changes in reservoir properties and flow conditions can be detected, but there are a limited number of pressure build-ups and therefore nonlinearity cannot be diagnosed in real-time. Ilk et al (2010) and Li et al (2011) used deconvolution as a diagnosis tool, but there are theory uncertainties when applying the deconvolution algorithm in nonlinear systems. Another effective diagnosis method should be developed.…”