Multistage optimization of a petroleum production system with material balance model

In this paper, we propose a mathematical formulation for the optimal management over time of an oil production network as a multistage optimization problem. The proposed model differs from common practice where the reservoir of the oil production network is approximated by decline curves or by black-box simulators. We model the reservoir as a controlled (non-linear) dynamical system by using material balance equations, under the standard assumptions that the fluids follow a black-oil model and that the reservoir has a tank-like behavior. The state of the dynamical system has five dimensions: the total volume of respectively oil, gas, and water in the reservoir; the total pore volume; and the reservoir pressure. We use a dynamic programming algorithm to numerically solve the multistage optimization problem on two specific instances of the general optimization problem where the state dimension can be reduced from dimension five to dimension one or two. More precisely, the first numerical application consists in optimizing the production of a dry gas reservoir which is subdivided in two tanks and which leads to a two-dimensional state (one dimension per tank), whereas the second numerical application tackles an oil reservoir with water injection which leads to a two-dimensional state. The two applications illustrate that our approach handles interconnected tanks (in the dry gas case) and that our approach allows optimization beyond first recovery of oil (in the oil with water injection case). We also provide numerical and theoretical comparisons with decline curves in the dry gas application.