Estimating the Specific Productivity Index in Horizontal Wells From Distributed-Pressure Measurements Using an Adjoint-Based Minimization Algorithm

Abstract

Summary Recent developments in the deployment of distributed-pressure-measurement devices in horizontal wells promise to lead to a new, low-cost, and reliable method of monitoring production and reservoir performance. Practical applicability of distributed-pressure sensing for quantitative-inflow detection will strongly depend on the specifications of the sensors, details of which were not publicly available at the time of publication. Therefore, we theoretically examined the possibility of identifying reservoir inflow from distributed-pressure measurements in the well. The wellbore and nearwellbore region were described by semianalytical steady-state models, and a gradient-based inversion method was applied to estimate the specific productivity index (SPI) as a function of along-well position. We employed the adjoint method to obtain the gradients, which resulted in a computationally efficient inversion scheme. With the aid of two numerical experiments (one of which was based on a real well and reservoir), we investigated the effects of well and reservoir parameters, sensor spacing, sensor resolution, and measurement noise on the quality of the inversion results. In both experiments, we generated synthetic measurements with the aid of a high-resolution reservoir-simulation model and used these to test the semianalytical inversion algorithm. In the first experiment, we considered a 2000-m horizontal well passing through two 300-m high-permeability streaks in a background with a permeability that was 10 times lower. The location of the streaks and the SPIs along the well were detected with fair accuracy using 20 unknown parameters (SPI values) and 20 pressure measurements. Decreasing the number of measurements resulted in a poorer detection of the streaks and their SPIs. The detection performance also decreased for increasing noise levels and deteriorated sensor resolution, though the negative effect of random measurement noise was cancelled out primarily by stacking multiple measurements. The detrimental effects of measurement noise and low sensor resolution were strongest in areas where the inflow was lowest (usually close to the toe). The second experiment concerned a high-rate near-horizontal well with slightly varying inclination that intersected a dipping package of formations with strongly variable permeabilities. Additionally, a satisfactory detection of SPIs was obtained even though the heterogeneities were no longer perpendicular to the well as in the first experiment. As a result of using the simple semianalytical forward model and the adjoint method, the inversions typically required less than 90 seconds on a standard laptop. This offered the opportunity to extend the algorithm to multiphase flow and dynamic applications (pressure-transient testing), while still maintaining sufficient computational speed to perform the inversion in real time.