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.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
5 July 2006
- Meeting paper published:
24 September 2006
- Revised manuscript received:
8 December 2007
- Manuscript approved:
12 December 2007
- Version of record:
25 June 2008
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