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.
Explicit Deconvolution of Well Test Data Dominated by Wellbore Storage