The theory of optimal control of distributed-parameter systems is presented for determining the best possible injection policies for EOR processes. The optimization criterion is to maximize the amount of oil recovered at minimum injection costs. Necessary conditions for optimality are obtained through application of the calculus of variations and Pontryagin's weak minimum principle. A gradient method is proposed for the computation of optimal injection policies.
In recent years, EOR techniques have received much attention. This has been motivated by the rapid escalation in the price of crude oil, uncertainties of supply from foreign sources, and low efficiencies of current recovery technologies. There are four main EOR methods considered viable for the production of light crudes. These are polymer floods, caustic floods, CO2 floods, and micellar-surfactant floods. All methods require the injection of rather expensive fluids into oil-bearing reservoir formations. Commercial application of any EOR process relies on economic projections that show a decent return on investment. The key factors in an economic projection for emerging tertiary oil recovery processes are investment requirements such as chemical and development costs, oil recovered, time required to obtain the oil, oil price, and tax load. Because of high chemical costs, it is important to optimize EOR processes to provide the greatest recovery at the lowest cost. The goal is to determine the theoretical basis for computing the best way of injecting an EOR fluid into a reservoir formation. It is important to note that this is only a part (although an important part) of the complete optimization of an EOR system for a given reservoir. The purpose of an optimization problem is to determine the control policy that will minimize (or maximize) a specific performance criterion, subject to the constraints imposed by the physical nature of the problem. Techniques available for the solution of dynamic-optimization problems, which contain differential or integral equality constraints, are the classical calculus of variations, the minimum principle of Pontryagin, and the dynamic programming of Bellman. Each of these techniques is equivalent to the others, but each manifests itself in a different way. The fundamental theorem of the calculus of variations is applied to problems with unconstrained states and controls. Consideration of the effect of control constraints leads to Pontryagin's minimum principle. Dynamic programming is suited to the optimization of serial structures. Although there are many books on the subject of optimal control theory, only a few cover the optimal control of distributed-parameter systems. Application of these techniques to chemical and petroleum engineering operations is beginning, and dynamic-process optimization is becoming a valuable tool in the design of modern systems. Refs. 5 and 6 present surveys of recent applications of distributed-parameter systems theory. Optimization objectives can be expressed as a cost functional or performance index to be minimized. The control on the system is the composition or physical state of the injected fluids. Thus, the optimization problem is concerned with determining the injection policy that leads to a minimum in the cost functional, subject to differential equality constraints that describe the system dynamics. Mathematical models have been formulated for each of the major EOR methods. Polymer flooding is described by Chauveteau, Hirasaki, Thakur, and Shah. Caustic flooding modeling is considered by Breit and deZabala. CO2 flooding fundamentals are presented by Henderson, Metcalfe, Pontious, Todd, and Leach. Micellar-surfactant modeling concepts are given by Ramirez et al., Pope, Fleming et al., and Hirasaki. Each of these models consists of conservation relations needed to describe the dynamic state of the process given by the chemical compositions and the fluid saturation. We denote the fluid compositions by the vector 1 and the fluid saturation in a two-phase system by either the water- or oil-phase saturation. Here, x2 denotes the water-phase saturation. The state of the system is therefore given by the vectorX 1X = ...................................(1)X 2