FINITE ELEMENT APPROXIMATION OF OPTIMAL CONTROL FOR SYSTEM GOVERNED BY IMMISCIBLE DISPLACEMENT IN POROUS MEDIA

In this work, we study the finite element approximation of a model optimal control problem governed by the system describing the two-phase incompressible flow in porous media, with the aim to maximize production of oil from petroleum reservoirs. We first give the proof for the existence of the solutions of the control problem. The optimality conditions are then obtained and the existence of the solution of the adjoint equations is shown. After that we consider its finite element approximation. We have obtained the a priori error estimates with the optimal orders and minimum regularity requirements. Finally, we carry out some numerical tests. 1. Motivation The field of petroleum engineering is concerned with the search for ways to extract more oil and gas from the earths subsurface. In a world in which an increase in production of tenths of a percentage may result into a growth in profit of millions of dollars, no stone is left unturned. A common technique in oil recovery, known as makes use of two types of wells: injection and production wells. The production wells are used to transport liquid and gas from the reservoir to the subsurface. The injection wells inject water into the oil reservoir with the aim to push the oil towards the production wells and keep up the pressure difference. The oil-water front progresses toward the production wells until water breaks through into the production stream. An increasing amount of water is used, while the oil production rate diminishes, until at some time the recovery is no longer profitable and production is brought to an end. Using water flooding, up to about 35 percent of the oil can be recovered economically. Due to the strongly heterogeneous nature of oil reservoirs, the oil- water front does not travel uniformly towards the production wells, but is usually irregularly shaped. As a result, large amounts of oil may be still trapped within the reservoir as water breakthrough occurs and production is brought to an end. Recent advances in petroleum engineering allow for advanced well downhole measurement and control devices, which expand the possibilities to manipulate and control fluid flow paths through the oil reservoir. The ability to manipulate the progression of the oil-water front provides the possibility to search for a con- trol strategy that will result in maximization of oil recovery. A straightforward approach to find such a control strategy is to use the optimal control technique to increase recovery by delaying water breakthrough and increasing sweep, based on a predictive reservoir model. Obviously, this problem can be described as an optimal control problem of PDEs where the goal is to find a control q over a time interval