Fast-Fourier-Transform-Based Deconvolution for Interpretation of Pressure Transient Test Data Dominated by Wellbore Storage
- Yueming Cheng (Texas A&M U.) | W. John Lee (Texas A&M U.) | Duane A. McVay (Texas A&M U.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- June 2005
- Document Type
- Journal Paper
- 224 - 239
- 2005. Society of Petroleum Engineers
- 4.1.5 Processing Equipment, 5.1.5 Geologic Modeling, 5.1.2 Faults and Fracture Characterisation, 4.1.2 Separation and Treating, 3.3.1 Production Logging, 5.6.11 Reservoir monitoring with permanent sensors, 5.6.3 Pressure Transient Testing, 5.6.4 Drillstem/Well Testing
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We present a deconvolution technique based on a fast-Fourier-transform (FFT)algorithm. With the new technique, we can deconvolve "noisy" pressure and ratedata from drawdown and buildup tests dominated by wellbore storage. Thewellbore-storage coefficient can be variable in the general case. In cases withno rate measurements, we use a "blind" deconvolution method to restore thepressure response free of wellbore-storage effects. Our technique detects theafterflow/unloading rate function with Fourier analysis of the observedpressure data.
The technique can unveil the early-time behavior of a reservoir systemmasked by wellbore-storage effects, and it thus provides a powerful tool toimprove pressure-transient-test interpretation. It has the advantages ofsuppressing the noise in the measured data, handling the problem of variablewellbore storage, and deconvolving the pressure data without ratemeasurement.
We demonstrate the applicability of the method with a variety of syntheticand actual field cases for both oil and gas wells. Some of the actual casesinclude measured sandface rates (which we use only for reference purposes), andothers do not.
Although this paper is focused on deconvolution of pressure-transient-testdata during a specific drawdown/buildup period corresponding to an abruptchange of surface flow rate, the deconvolution method itself is very generaland can be extended readily to interpret multirate test data.
In conventional well-test analysis, the pressure response to constant-rateproduction is essential information that presents the distinct characteristicsfor a specific type of reservoir system. However, in many cases, it isdifficult to acquire sufficient constant-rate pressure-response data. Therecorded early-time pressure data are often hidden by wellbore storage(variable sandface rates). In some cases, outer-boundary effects may appearbefore wellbore-storage effects disappear. Therefore, it is often imperative torestore the early-time pressure response in the absence of wellbore-storageeffects to provide a confident well-test interpretation.
Deconvolution is a technique used to convert measured pressure and sandfacerate data into the constant-rate pressure response of the reservoir. In otherwords, deconvolution provides the pressure response of a well/reservoir systemfree of wellbore-storage effects, as if the well were producing at a constantrate. Once the deconvolved pressure is obtained, conventional interpretationmethods can be used for reservoir system identification and parameterestimation.
However, mathematically, deconvolution is a highly unstable inverse problembecause small errors in the data can result in large uncertainties in thedeconvolution solution. In the past 40 years, a variety of deconvolutiontechniques have been proposed in petroleum engineering, such as directalgorithms, constrained deconvolution techniques, and Laplace-transform-basedmethods, but their application was limited largely because of instabilityproblems. Direct deconvolution is known as a highly unstable procedure. Toreduce solution oscillation, various forms of smoothness constraints have beenimposed on the solution.Coats et al. presented a linear programmingmethod with sign constraints on the pressure response and its derivatives.Kuchuk et al. used similar constraints and developed a constrained linearleast-squares method. Baygun et al. proposed different smoothness constraintsto combine with least-squares estimation. The constraints were anautocorrelation constraint on the logarithmic derivative of pressure solutionand 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 usingexponential and polynomial approximations to measured sandface rate andpressure data, respectively. Methods presented by Roumboutsos and Stewart andFair and Simmons used piecewise linear approximations to rate and pressuredata. All the Laplace-transform-based methods used the Stehfest algorithm toinvert the results in the Laplace domain back to the time domain.
Although the above methods may give a reasonable pressure solution at a lowlevel of measurement noise, the deconvolution results can become unstable anduninterpretable when the level of noise increases. Furthermore, existingdeconvolution techniques require simultaneous measurement of both wellborepressure and sandface rate. However, it is not always possible to measure ratein actual well testing. Existing techniques are, in general, not suitable forapplications without sandface rate measurement.
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