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

Abstract

A B S T R A C TIlk 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 i...

A stability-improved efficient deconvolution algorithm based on B-splines by appending a nonlinear regularization

“…The inverse problem can provide the equivalent constant unit production rate pressure response of the well in a reservoir system that is affected by the variable production rates for the entire duration of the production history. The relevant deconvolution algorithms have attracted big attentions over the past forty years (Liu et al, 2017). Due to the commonly existent errors of wellbore pressure and production rate data in the fields, the deconvolution computation is always ill-conditioned inherently (Çınar et al, 2006).…”

“…Due to the commonly existent errors of wellbore pressure and production rate data in the fields, the deconvolution computation is always ill-conditioned inherently (Çınar et al, 2006). As far as we know, although many deconvolution algorithms have been proposed, just several ones appear to exhibit the stability of data error tolerance; they are proposed by von Schroeter et al (von Schroeter et al, 2002;von Schroeter et al, 2004), Levitan et al (Levitan, 2005;Levitan et al, 2006) and Ilk et al (Ilk, 2005;Ilk et al, 2005;Liu et al, 2017), respectively. Here, these aforementioned different deconvolution algorithms will be introduced in details.…”

“…As for Levitan et al's deconvolution algorithm (Levitan, 2005;Levitan et al, 2006), their ideas are also from von Schroeter et al's deconvolution algorithm. They are both based on the same concept of minimizing a nonlinear weighted least-square objective function, involving the sum of three mismatch terms including the pressure, the rate and the curvature, for reconstructing the deconvolved pressure drop and its logarithmic derivative (Liu et al, 2017). The difference of the two algorithms mainly lies in the aspects of model assumption and the specific definition of objective functions.…”

“…In combination with Laplace transform and numerical Laplace inversion, the formulated linear least-squares problem can be solved. What's more, Ilk et al's algorithm is further improved by Liu et al (2017) through a technique of piecewise analytical integration for calculating the involved sensitivity matrix (Liu et al, 2017) in the real time space instead of the Laplace space; then the success of the deconvolution computation based on B-splines can be guaranteed, and the improved deconvolution algorithm exhibits big advantage in the fast computational speed due to the use of the analytical solution method.…”

“…Good stability of deconvolution algorithms is very necessary in order to make deconvolution as a viable tool for well-test analysis; and stability improvement is also the main difficulty in the development of deconvolution algorithms. Çınar et al (2006) have ever conducted a comparative study on these deconvolution algorithms mentioned above: Significantly, it is found that the weaker linear regularization method applied in the deconvolution algorithms based on B-splines (including Ilk et al's deconvolution algorithm (Ilk, 2005;Ilk et al, 2005) and its improved version by Liu et al (2017)) can make the algorithms less tolerant to data errors; in contrast, von Schroeter et al's deconvolution algorithm shows relatively higher stability by using the nonlinear regularization method, and the deconvolution algorithm has been implemented into Saphir as the pressure transient analysis module of KAPPA software due to its good stability. However, in von Schroeter et al's deconvolution algorithm, the transformed deconvolution equation of Duhamel principle i. e. Eq.…”