Summary
In this work, we present an investigation of recent deconvolution methods proposed by von Schroeter et al. (2002, 2004), Levitan (2005) and Levitan et al. (2006), and Ilk et al. (2006a, b). These works offer new solution methods to the long-standing deconvolution problem and make deconvolution a viable tool for well-test and production-data analysis. However, there exists no study presenting an independent assessment of all these methods, revealing and discussing specific features associated with the use of each method in a unified manner. The algorithms used in this study for evaluating the von Schroeter et al. and Levitan methods represent our independent implementations of their methods based on the material presented in their papers, not the original algorithms implemented by von Schroeter et al. and Levitan. Three synthetic cases and one field case are considered for the investigation.
Our results identify the key issues regarding the successful and practical application of each method. In addition, we show that with proper care and attention in applying these methods, deconvolution can be 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 important because it provides the equivalent constant rate/pressure response of the well/reservoir system affected by variable rates/pressures (von Schroeter et al. 2002, 2004; Levitan 2005; Levitan et al. 2006; Ilk et al. 2006a, b; Kuchuk et al. 2005). 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 a constant-pressure rate system response have emerged with development of robust pressure/rate (von Schroeter et al. 2002, 2004; Levitan 2005; Levitan et al. 2006; Ilk et al. 2006a, b) and rate/pressure (Kuchuk et al. 2005) deconvolution algorithms. In this work, we focus on the pressure/rate deconvolution for analyzing well-test data.
For over a half century, 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 (Hutchinson and Sikora 1959; Coats et al. 1964; Jargon and van Poollen 1965; Kuchuk et al. 1990; Thompson and Reynolds 1986; Baygun et al. 1997). A thorough review and list of the previous deconvolution algorithms can be found in von Schroeter et al. (2004). 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, as stated by Levitan, the deconvolved response would remove the constraints of conventional analysis techniques (Earlougher 1977; Bourdet 2002) that have been built around the idea of applying a special time transformation [e.g., the logarithmic multirate superposition time (Agarwal 1980)] to the test pressure data so that the pressure behavior observed during individual flow periods would be similar in some way to the constant-rate system response. As also stated by Levitan, the superposition-time transform does not completely remove all effects of previous rate variations and often complicates test analysis because of residual superposition effects.
Unfortunately, deconvolution is an ill-posed inverse problem and will usually not have a unique solution even in the absence of noise in the data. Even if the solution is unique, it is quite sensitive to noise in the data, meaning that small changes in input (measured pressure and rate data) can lead to large changes in the output (deconvolved) result. Therefore, this ill-posed 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. Although there exists a variety of different deconvolution algorithms proposed in the past, only those developed by von Schroeter et al., Levitan, and Ilk et al. appear to offer the necessary robustness to make deconvolution a viable tool for well-test and production data analysis. In this paper, our objectives are to conduct an investigation of these three deconvolution methods and to establish the advantages and limitations of each method.
As stated in the abstract, the algorithms used in this study for evaluating the von Schroeter et al. and Levitan methods represent our independent implementations based on the material presented in their papers; therefore, our implementations may not be identical to their versions. However, as is shown later, validation conducted on the simulated (test) data sets (von Schroeter et al. 2004; Levitan 2005) sent to us directly by von Schroeter and Levitan shows that our implementations reproduce almost exactly the same results generated by their original algorithms for these simulated data sets.
The paper is organized as follows: First, we describe the pressure/rate deconvolution model and error model considered in this work. Then, we provide the mathematical background of the von Schroeter et al., Levitan, and Ilk et al. methods together with their specific features. We compare the performance of each method by considering three synthetic and one field well-test data sets. Finally, we provide a discussion of our results obtained from this investigation.
An improved deconvolution algorithm using B-splines for well-test data analysis in petroleum engineering