A Semi-Analytical Model for Productivity Testing of Complex Well Configurations

This paper presents a semi-analytical method for the modeling of productivity testing of vertical, horizontal or multilateral wells. The method, which is applicable to both oil and gas reservoirs, automatically accounts for well interference. The use of analytical expressions ensures proper handling of transient short-time behavior and semi-steady-state long-time behavior, both close to the well and further into the reservoir. Calculation times are still very limited, in the order of a few minutes down to a few seconds when there are vertical wells only. This makes the tool suitable for well testing evaluation.<br><br>The approach is based on an earlier derived productivity prediction tool, in which the steady-state equations were solved. It has now been extended to solve the time-dependent diffusion equation and it is thus more rigorous than the extension to time-dependent behavior using solutions to the Laplace equation and moving pressure boundaries, which was presented recently. In our current method, the equations have first been transformed using the Laplace transformation. The expressions for the producing wells are combined with auxiliary sources outside the reservoir. The core of the semi-analytic method involves an adjustment of the positions and strengths of these sources in order to approximate the boundary conditions at the reservoir boundaries. The solution that is obtained is transformed back into the time domain using a Stehfest algorithm.<br><br>The new approach has been validated with numerical tools, including both reservoir simulators and welltest interpretation software. Validations were performed with artificial cases using both single-well and multiple-well production tests. The results of these tests were excellent. <br>