A Comparative Study of Recent Robust Deconvolution Algorithms for Well-Test and Production-Data Analysis

Abstract

Abstract In this work, we provide a comparative study of recently proposed deconvolution algorithms which were designed to function in the presence of reasonable levels of noise in both the rate and pressure input data. The algorithms considered for comparison are those presented by von Schroeter et al.,1,2 Levitan,3,4 and Ilk et al.5,6 These works offer robust solution algorithms to the long-standing deconvolution problem and make deconvolution a viable tool to well-test and production data analysis. However, there exists no comparative study revealing and discussing specific features associated with the use of each algorithm in a unified manner. We have independently reproduced the von Schroeter etal. and Levitan algorithms to assess the specific advantages and limitations of each method (as well as the Ilk et al. method), and we provide a comparative study of these algorithms using synthetic and field case examples. Our results identify the key issues regarding the successful and practical application of each algorithm. In addition, we show that with proper care and attention in applying these methods, deconvolution can be used as an important tool for the analysis and interpretation of variable rate/pressure reservoir performance data. Introduction Applying deconvolution for well test and production data analysis is an important challenge because deconvolution is an ill-conditioned inverse problem in the presence of noise in pressure/rate measurements.1–7 Deconvolution provides the equivalent constant rate/pressure response of the well/reservoir system affected by variable rates/pressures. With the implementation of permanent pressure and flow-rate measurement systems, the importance of deconvolution has increased because it is now possible to process the well test/production data simultaneously and obtain the underlying well/reservoir model (in the form of a constant rate pressure response). New methods of analyzing well test data in the form of a constant-rate drawdown system response and production data in the form of constant-pressure rate system response have emerged with development of robust pressure/ rate1–6 and rate/ressure7 deconvolution algorithms. In this work, we focus on the pressure/rate deconvolution,1–6 rather than rate/pressure deconvolution7 for analyzing well test and production data. Over the past 40 years, pressure/rate deconvolution techniques have been applied to well test pressure and rate data as a means to obtain the constant-rate behavior of the system.8–12 (A through review and list of the previous deconvolution algorithms can be found in the paper by von Schroeter et al.2) The primary objective of applying pressure/rate deconvolution is to convert the pressure data response from a variable-rate test or production sequence into an equivalent pressure profile that would have been obtained if the well were produced at a constant rate for the entire duration of the production history. If such an objective could be achieved with some success then the deconvolved response would remove the constraints of conventional analysis techniques13,14 that have been built around the idea of applying a special time transformation (based on the logarithmic multi-rate superposition time) to the test pressure data so that the pressure behavior observed during individual flow periods would be similar in some way to constant-rate system response. As is well known,14 the superposition-time transform does not completely remove all effects of previous rate variations and often complicates test analysis due to residual superposition effects. Due to these reasons, pressure/rate deconvolution problem has attracted considerable interest over the past 40 years. Unfortunately, deconvolution requires the solution of an ill-conditioned problem; meaning that small changes in input (measured pressure and rate data) can lead to large changes in the output (deconvolved) result. Therefore, this ill-conditioned nature of the deconvolution problem combined with errors that are inherent in pressure and rate data makes the application of deconvolution a challenge; particularly so in terms of developing robust deconvolution algorithms which are error-tolerant.