Summary
We present a deconvolution technique based on a fast-Fourier-transform (FFT)
algorithm. With the new technique, we can deconvolve “noisy” pressure and rate
data from drawdown and buildup tests dominated by wellbore storage. The
wellbore-storage coefficient can be variable in the general case. In cases with
no rate measurements, we use a “blind” deconvolution method to restore the
pressure response free of wellbore-storage effects. Our technique detects the
afterflow/unloading rate function with Fourier analysis of the observed
pressure data.
The technique can unveil the early-time behavior of a reservoir system
masked by wellbore-storage effects, and it thus provides a powerful tool to
improve pressure-transient-test interpretation. It has the advantages of
suppressing the noise in the measured data, handling the problem of variable
wellbore storage, and deconvolving the pressure data without rate
measurement.
We demonstrate the applicability of the method with a variety of synthetic
and actual field cases for both oil and gas wells. Some of the actual cases
include measured sandface rates (which we use only for reference purposes), and
others do not.
Although this paper is focused on deconvolution of pressure-transient-test
data during a specific drawdown/buildup period corresponding to an abrupt
change of surface flow rate, the deconvolution method itself is very general
and can be extended readily to interpret multirate test data.
Introduction
In conventional well-test analysis, the pressure response to constant-rate
production is essential information that presents the distinct characteristics
for a specific type of reservoir system. However, in many cases, it is
difficult to acquire sufficient constant-rate pressure-response data. The
recorded early-time pressure data are often hidden by wellbore storage
(variable sandface rates). In some cases, outer-boundary effects may appear
before wellbore-storage effects disappear. Therefore, it is often imperative to
restore the early-time pressure response in the absence of wellbore-storage
effects to provide a confident well-test interpretation.
Deconvolution is a technique used to convert measured pressure and sandface
rate data into the constant-rate pressure response of the reservoir. In other
words, deconvolution provides the pressure response of a well/reservoir system
free of wellbore-storage effects, as if the well were producing at a constant
rate. Once the deconvolved pressure is obtained, conventional interpretation
methods can be used for reservoir system identification and parameter
estimation.
However, mathematically, deconvolution is a highly unstable inverse problem
because small errors in the data can result in large uncertainties in the
deconvolution solution. In the past 40 years, a variety of deconvolution
techniques have been proposed in petroleum engineering, such as direct
algorithms, constrained deconvolution techniques, and Laplace-transform-based
methods, but their application was limited largely because of instability
problems. Direct deconvolution is known as a highly unstable procedure. To
reduce solution oscillation, various forms of smoothness constraints have been
imposed on the solution. Coats et al. presented a linear programming
method with sign constraints on the pressure response and its derivatives.
Kuchuk et al. used similar constraints and developed a constrained linear
least-squares method. Baygun et al. proposed different smoothness constraints
to combine with least-squares estimation. The constraints were an
autocorrelation constraint on the logarithmic derivative of pressure solution
and an energy constraint on the change of logarithmic derivative.
Efforts also were made to perform deconvolution in the Laplace domain.
Kuchuk and Ayestaran developed a Laplace-transform-based method using
exponential and polynomial approximations to measured sandface rate and
pressure data, respectively. Methods presented by Roumboutsos and Stewart and
Fair and Simmons used piecewise linear approximations to rate and pressure
data. All the Laplace-transform-based methods used the Stehfest algorithm to
invert the results in the Laplace domain back to the time domain.
Although the above methods may give a reasonable pressure solution at a low
level of measurement noise, the deconvolution results can become unstable and
uninterpretable when the level of noise increases. Furthermore, existing
deconvolution techniques require simultaneous measurement of both wellbore
pressure and sandface rate. However, it is not always possible to measure rate
in actual well testing. Existing techniques are, in general, not suitable for
applications without sandface rate measurement.
© 2005. Society of Petroleum Engineers
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History
- Original manuscript received:
31 March 2004
- Revised manuscript received:
21 January 2005
- Manuscript approved:
31 March 2005
- Version of record:
15 June 2005
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