Optimal Control of Petroleum Reservoirs

Abstract

The flow of fluids in a petroleum reservoir can be modeled using a set of nonlinear second order partial differential equations. To produce the fluids, especially oil, a number of production wells are drilled through the reservoir formations. The well model, which describes the interaction between the well and the reservoir, can be considered as the boundary conditions for the flow equations. The amount of produced fluid is usually controlled by changing the bottom hole pressures via control valves. Because the reservoir could contain undesirable fluids, e.g., water or gas, setting the bottom hole pressures to the maximum production limit could cause high water or gas production, which would decrease the oil revenue. The present paper describes an optimal control method for maximizing the oil revenue in petroleum reservoir systems. The flow equation is discretized in a space variable to yield a state space representation model. The valve openings are considered as the control variables and are written as piecewise constant functions. Furthermore, the gradient of the oil revenue with respect to the control variable is computed using the adjoint method and the optimal control setting is obtained using a line search method. A numerical example of a water-flooding case is presented to illustrate the application of the method.