A Direct Deconvolution or Convolution Algorithm for Well Test Analysis
- A. Roumboutsos (Heriot-Watt U.) | G. Stewart (Heriot-Watt U.)
- Document ID
- Society of Petroleum Engineers
- SPE Annual Technical Conference and Exhibition, 2-5 October, Houston, Texas
- Publication Date
- Document Type
- Conference Paper
- 1988. Society of Petroleum Engineers
- 2.2.2 Perforating, 3.3.1 Production Logging, 4.3.4 Scale, 5.6.4 Drillstem/Well Testing, 3.3.3 Downhole and Wellsite Flow Metering, 4.1.2 Separation and Treating, 2.4.3 Sand/Solids Control, 5.5 Reservoir Simulation
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A numerical Laplace Transformation based on a piecewise linear approximation to a function has piecewise linear approximation to a function has been developed. The convolution of fundamental pressure and flow transforms yields variable pressure and flow transforms yields variable rate pressure responses. The deconvolution process involving the quotient of transforms process involving the quotient of transforms allows the equivalent constant rate response to be determined from measured pressure and rate data. This provides a general basis for the treatment of variable rate well tests in heterogeneous systems. The inversion of products or quotients of Laplace transforms is products or quotients of Laplace transforms is carried out by numerical inversion algorithms, usually the fast method due to Stehfest. The application of deconvolution to buildup tests in considered. Synthetic and field examples are used to illustrate the application of the techniques.
Knowledge of both the pressure response and the flow rate of a well is essential in well test interpretation. The inverse problem of parameter estimation is better posed when the parameter estimation is better posed when the sandface rate can be maintained constant and for this reason test design seeks to achieve constant rate during the drawdown period. This may be difficult to achieve in practice hence the preponderance of buildup testing which is not sensitive to rate variation during the preceding drawdown period, providing the rate is preceding drawdown period, providing the rate is stabilised before shutin. However it should be noted that a buildup is intrinsically inferior to a drawdown especially in heterogeneous systems since it is only an equilibration process with very small pressure effects at late process with very small pressure effects at late time.
There are three main reasons for sandface rate variation: these are:
(a) Wellbore storage (referred to an afterflow in buildup)
(b) Lack of control of the surface rate
(c) Crossflow between layers in multilayer testing.
The problem of wellbore storage has traditionally been tackled by the type curve matching method first proposed by Ramay et al and further developed by many authors e.g. McKinley, Earlougher and Gringarten. These are based on the assumption that the wellbore storage coefficient is constant. Recent evidence based on production logging measurements has shown that this approximation is not always valid. Problems also arise when the effect of external boundaries is felt before the end of wellbore storage influenced data. Recent advances in the technology of downhole flow measurement allow the sandface rate to be measured directly and therefore it is now worthwhile to develop methods for parameter estimation in the variable rate case.
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