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

Abstract

Previous deconvolution algorithms based on B-splines are much easier to be understood and programmed for academic researchers and engineers. However, due to the use of a linear regularization, their stability is weaker than that of the commonly used von Schroeter et al.'s deconvolution algorithm in which a nonlinear regularization is used; the linear regularization can make the deconvolution algorithms less tolerant to data errors. Good stability for the deconvolution algorithms is very important in order to make deconvolution as a viable tool for well-test analysis. In the paper, in order to improve the stability of the deconvolution algorithms based on Bsplines, a nonlinear regularization by minimizing the curvature of pressure derivative response, as used in von Schroeter et al.'s algorithm, is appended instead of the linear regularization. And the corresponding nonlinear regularization equations are appropriately deduced. In particular, the improved algorithm is based on the Duhamel principle directly, and the complex transformation by the nonlinear z function, as used in von Schroeter et al.'s algorithm, is avoided; it does simplify the whole deconvolution process; moreover, the sensitivity matrix of an involved basic linear system from the measured pressure and rate data can also be solved directly by the piecewise analytical integration method, which can largely improve the deconvolution computation speed. Ultimately, in combination with the nonlinear regularization equations, a nonlinear least-squares problem is formulated for the stability-improved deconvolution algorithm based on B-splines. Besides, a constraint condition for tuning the parameter values of the B-spline base and an involved smooth factor is presented for restricting the nonlinear regularization process. Through a simulated case study, it is found that the nonlinear least-squares problem can be solved stably by the advanced Powell's Dog Leg method due to its great convergence ability and numerical stability; and the solution accuracy is also verified. Then the effects of the two parameters on the type curves of the deconvolution results are analyzed. And the effect of the error in the initial formation pressure on the type curves of the deconvolution results is also analyzed. Then a statement on how to perform the nonlinear regularization is presented specifically. Furthermore, through the study on two simulated cases with added data errors and an actual case, it is demonstrated that when the nonlinear regularization is appended, the stability of the deconvolution algorithm based on B-splines can be largely improved for mitigating the effect of data errors; besides, the stability-improved algorithm based on B-splines even exhibits higher stability than von Schroeter et al.'s algorithm that takes the same nonlinear regularization method, and the reason can be attributed to the superior properties of the representation of the wellbore pressure derivative (to be deconvolved) by B-spline functions in the numerical stability of computations an...

Fracture Spacing Optimization Method for Multi-Stage Fractured Horizontal Wells in Shale Oil Reservoir Based on Dynamic Production Data Analysis

“…where p is the bottom hole pressure under variable flow rate, atm; t is the production time, s; p u is the flow response per unit flow rate, atm. According to Duhamel's principle, the flow rate function under variable bottom hole pressure is obtained as follows [40]:…”

“…In the case of known variable flow rate q and bottom hole pressure p under the variable flow q rate, Equation ( 11) can be used to obtain the transient pressure response p u in the oil reservoir for the whole production time [24]). It is worth noting that when using deconvolution calculations in practical applications, it is necessary to exclude the influence of stimulation measures during production, and the changes in reservoir physical property, fluid property and variable wellbore storage effect [25,26]. A series of studies on the deconvolution algorithm for inversion of reservoir production data [25][26][27][28][29][30][31] were carried out by many scholars.…”

“…It is worth noting that when using deconvolution calculations in practical applications, it is necessary to exclude the influence of stimulation measures during production, and the changes in reservoir physical property, fluid property and variable wellbore storage effect [25,26]. A series of studies on the deconvolution algorithm for inversion of reservoir production data [25][26][27][28][29][30][31] were carried out by many scholars. In 2004, based on Tikhonov regularization objective function [27], a new deconvolution algorithm was proposed by Schroeter et al [28].…”

“…Therefore, based on the dynamic production data inversion, a new production-oriented optimization method for the fracture spacing of multi-stage fractured horizontal wells in shale oil is proposed. In particular, deconvolution algorithm [24][25][26][27][28][29][30][31] is introduced to normalize the pressure data. Not only can the data of variable pressure and variable flow rate be directly converted into pressure data under unit flow, but also the regularization of deconvolution calculation can be performed, which can eliminate the influence of data error and expand the investigation distance of production data analysis.…”

A Study on Optimization of CBM Water Drainage by Well-Test Deconvolution in the Early Development Stage

“…Good stability for the deconvolution algorithms is very important in order to make deconvolution a viable tool for well-test analysis. To improve the stability of the deconvolution algorithms based on B-splines, a nonlinear regularization by minimizing the curvature of pressure derivative response, as used in von Schroeter et al's algorithm [11,12], is appended instead of the linear regularization [24]. A spline function can be represented by the linear combination of B-spline functions.…”