Rate optimization of fractional flow reservoir model based on the continuous adjoint method

Abstract

In this paper, continuous adjoint method is applied to research the rate optimization of fractional flow reservoir model. The continuous adjoint equations are derived by the variational theory and the stable numerical schemes of adjoint equations are obtained through numerical methods. Firstly, the continuous adjoint equations are linear equations and evolve from the end simulation time. The convection velocity direction of the continuous adjoint saturation equation is opposite to the convection direction of the saturation equation. Secondly, the total simulation time has significant impact on the adjoint variables and the adjoint gradient. After water breakthrough in production well, the adjoint variables exhibit spatial variety that gradually changes from production well to injection well. With this observation, we find that this variety is related to the fractional flow function and its derivative value. The sharp change of fractional flow function and its derivative at the end of the simulation time can lead to large fluctuation of the adjoint gradient along the time step. Thirdly, the influence of time interval on optimization results is studied. Increasing the number of time steps will increase the optimal value, but the value of the difference between optimal and initial value will not increase significantly. When the variable number of time steps is adopted, this difference decreases as the number of time steps increases, which indicates that the coarse time step optimization can provide a better initial control value for the fine time step optimization. An adaptive time step from coarse step to fine step is thus recommended for optimization.