Sensitivity Calculation by Adjoint Method: An Application to Reservoir Simulation

Scope of this paper is the sensitivity calculation by using Adjoint method. We have a 9 x 9 two phase, two dimension, five spot model with one injector and one producer. The high uncertainty of the permeability field gives us the data mismatch in the model, and we set the permeability as model parameter (uncertainty). The data distribution we get is bottom-hole pressure at injector and water cut at producer for 10 time steps. To get the optimal permeability field, we first calculate the sensitivity coefficients for permeability by using Adjoint method. We already have a forward simulator for this problem that is the fully implicit black-oil simulator. Hence, we need to extract necessary information from the forward simulator, i.e. Jacobian matrix, transmissibility, and accumulation etc. Advantage of the Adjoint method is that it enables us to reduce the considerable amount of computation time for calculating the sensitivity matrix compared with gradient simulator method. The forward simulation we need at each minimization step is only one time for calculating the sensitivity. Then we minimize the objective function by Levenberg-Marquardt algorithm