Analyzing the Pressure Response during the Afterflow Period

Abstract

Several theoretical methods (Russell’s equation, McKinley method) have been presented for analyzing the pressure response during the afterflow period in order to determine formation permeability and skin factor. It is known, however, that the results obtained from these techniques are less accurate than those from the Horner analysis. In some cases (e.g. Middle East) the afterflow period can completely dominate the pressure buildup period and the afterflow analysis is the only method of determining the essential reservoir parameters. A buildup pressure test with afterflow can be considered as a variable flow rate test. In this case the Duhamel integral (the superposition theorem) can be used to derive solutions for time-dependent boundary conditions. Many investigators consider that the exponential function of time as a good approximation of the sandface rate during afterflow. We developed a direct method of calculating the above-mentioned integral. In our study we assumed that only afterflow bottomhole pressure was recorded during a build-up test. Using the solution of the Duhamel integral we developed a method of the reconstruction of the sandface flow rate during the afterflow period. This allows to determine the formation permeability and skin factor, evaluate the duration of the wellbore storage period. In cases when the Middle-Time Region (infinite-acting reservoir) is very short and representative data (to apply Horner’s technique) can not be obtained, the solution of the of the Duhamel integral can be utilized to estimate approximate values of formation permeability and skin factor. Simple computer programs are prepared to process field data. In the paper are presented three examples (field cases) of calculations of the formation permeability and skin factor.