Fast and efficient sensitivity calculation using adjoint method for three-phase field-scale history matching

Abstract

Adjoint method-based sensitivity for field-scale history matching with large numbers of parameters suffers from several limitations. First, the CPU time depends on the data points which are large for any brown fields of long history; second, it requires large memory to save the gridblock pressure and saturation per each time step used in the forward model. Third, it is computationally expensive as it requires solving the Adjoint system of equations backward in time per each forward time step which is usually of high magnitude in case of field scale applications of long history. Lastly, the solver used for solving the Adjoint system of equations needs to be efficient for large-scale applications. We propose an efficient and fast approach for sensitivity calculation based on the Adjoint method to overcome much of the current limitations. First, we use a commercial finite difference simulator, ECLIPSE, as a forward model, which is general and can account for complex physical behavior that dominates most field applications. Second, the production data misfit is represented by a single generalized travel time misfit per well, thus effectively reducing the number of data points into one per well. Third, we solve the Adjoint system of equations backward in time in a larger time step that is equivalent to the time of severe changes in pressure and saturation due to changing well conditions or introducing new infill wells rather than using the forward model time steps. This approach reduces the computational effort and memory allocated for the sensitivity calculation. Fourth, we use an iterative sparse matrix solver, LSQR, for solving the Adjoint system of equations which shows high stability for field-scale applications. We demonstrate the power and utility of our approach using synthetic and pseudo field examples. The synthetic examples show the robustness and efficiency of our sensitivity calculation approach compared to the perturbation. The pseudo-field example had 10 years of production history with an original gas cap and a strong aquifer support. Using well log data, core data, water cut and gas–oil ratio history from producing wells; we characterize the permeability at each cell, thus demonstrating the feasibility of our approach for field applications.