An improved deconvolution algorithm using B-splines for well-test data analysis in petroleum engineering

Abstract

Ilk et al.’s deconvolution algorithm using B-splines involves the Laplace transformation of the convolution equation with respect to production rate and wellbore pressure based on Duhamel principle. However, for common cases, the production rate function has “discontinuity” with respect to production time; it does not satisfy the precondition that the function to be transformed by Laplace transformation should be continuous. This inherent defect may directly cause enormous amount of computational time or even the failure of the numerical Laplace inversion in the deconvolution process. Based on these concerns, a fundamentally improved deconvolution algorithm using B-splines is presented here. In the convolution equation, the wellbore pressure derivative corresponding to constant unit production rate as the target of deconvolution is still represented by weighted summation of second-order B-splines; however, the computation process of the deconvolution is kept in the level of integral in the real time space instead of the Laplace space, for the reason that there will be no continuity requirement for the production rate function in the application of Duhamel principle for the deconvolution computation problem. According to the real production rate history, a technique of piecewise analytical integration is adopted for obtaining the elements of sensitivity matrix of a linear system with respect to weight coefficients; the linear system is generated by substituting the measured wellbore pressure data and corresponding variable production rate data into the convolution equation containing B-splines. The proposed direct analytical solution method of the integration for calculating the elements of the sensitivity matrix can not only guarantee the success of the deconvolution computation, but also can largely enhance the deconvolution computation speed. Moreover, in order to further improve the computation speed, a binary search method is also applied to find which production segments (with constant production rate) the measured wellbore pressure data points locate at in the deconvolution computation process. Another linear system with respect to weight coefficients for the regularization from Ilk et al.’s deconvolution algorithm is appended in order to overcome the effect of data errors. The two linear systems are combined together as an over-determined linear system, which can be solved by the least square method. Eventually, the reconstructed wellbore pressure and its derivative by B-splines corresponding to the constant unit production rate can be obtained.Numerical experimental tests demonstrate that the improved deconvolution algorithm exhibits good accuracy, computation speed and stability of data error tolerance. And the statement on how to perform the regularization when data error exists is also made in order to deconvolve the correct wellbore pressure derivative. The improved deconvolution algorithm is also applied into an actual field example. It is found that the deconvolution results by the improved deconvolution algorithm have good agreement with the ones by Von Schroeter et al.’s deconvolution algorithm and by Levitan et al.’s deconvolution algorithm as a whole; and the feature of typical log-log curves of the wellbore pressure drop and the wellbore pressure derivative corresponding to the improved algorithm is very close to the one of typical log-log curves calculated directly from the wellbore pressure data in the well shut-in period. In addition, through many numerical experimental tests, it is also concluded that as the quantity of data largely increases, the improved Ilk et al.'s deconvolution algorithm exhibits the big advantage in fast computational speed over von Schroeter et al.’s algorithm and Levitan et al.’s algorithm.