A Semianalytic Model for the Productivity Testing of Multiple Wells

We present a semi-analytic method for modeling the productivity testing of vertical, horizontal, slanted, or multilateral wells. The method is applicable to both oil and gas reservoirs and automatically accounts for well interference. The use of analytic expressions ensures that short-time transient behavior and long-time semi-steady-state behavior are handled appropriately, whether close to the well or further into the reservoir. Calculation times are still very limited: in the order of a few minutes to a few seconds when all wells are vertical. This makes the tool suitable for evaluating well testing and determining well productivity. 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. 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 crux 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 obtained is transformed back into the time domain using a Stehfest algorithm. The new approach has been validated with numeric tools, including both reservoir simulators and well-test interpretation software. Validations were performed with artificial cases and with field production test data, using both single-well and multiple-well production tests. The results of these tests were excellent.