Implementation of Adjoint Solution for Optimal Control of Smart Wells

Abstract

Abstract Practical production optimization problems typically involve large, highly complex reservoir models, thousands of unknowns and many nonlinear constraints, which makes the numerical calculation of gradients for the optimization process impractical. This work explores a new algorithm for production optimization using optimal control theory. The approach is to use the underlying simulator as the forward model and its adjoint for the calculation of gradients. Direct coding of the adjoint model is, however, complex and time consuming, and the code is dependent on the forward model in the sense that it must be updated whenever the forward model is modified. We investigate an adjoint procedure that avoids these limitations. For a fully implicit forward model and specific forms of the cost function and nonlinear constraints, all information necessary for the adjoint run is calculated and stored during the forward run itself. The adjoint run then requires only the appropriate assembling of this information to calculate the gradients. This makes the adjoint code essentially independent of the forward model and also leads to enhanced efficiency, as no calculations are repeated. Further, we present an efficient approach for handling nonlinear constraints that also allows us to readily apply commercial constrained optimization packages. The forward model used in this work is the General Purpose Research Simulator (GPRS), a highly flexible compositional/black oil research simulator developed at Stanford University. Through two examples, we demonstrate that the linkage proposed here provides a practical strategy for optimal control within a general-purpose reservoir simulator. These examples illustrate production optimization with conventional wells as well as with smart wells, in which well segments can be controlled individually. Introduction Most of the existing major oilfields are already at a mature stage, and the number of new significant discoveries per year is decreasing [1]. In order to satisfy increasing worldwide demand for oil and gas, it is becoming increasingly important to produce existing fields as efficiently as possible, while simultaneously decreasing development and operating costs. Optimal control theory is one possible approach that can be deployed to address these difficult issues. The main benefit of the use of optimal control theory is its efficiency, which makes it suitable for application to real reservoirs simulated using large models, in contrast to many existing techniques. The above-mentioned problem is essentially an optimization problem, wherein the objective is to maximize or minimize some cost function J(u) such as net present value (NPV) of the reservoir, sweep efficiency, cumulative oil production, etc. Here, u is a set of controls including, for example, well rates and bottom hole pressures (BHP), which can be manipulated in order to achieve the optimum. In other words, u is anything that can be controlled. It should be understood that the optimization process results in control of future performance to maximize or minimize J(u), and thus the process of optimization cannot be performed on the real reservoir, but must be carried out on some approximate model. This approximate model is usually the simulation model of the reservoir. This simulation model is a dynamic system that relates the controls u to the cost function J(u). Consider for example the simple schematic of a reservoir shown in Fig. 1, where the cost function is cumulative oil production and the control is the injection rate. Changing the injection rate changes the dynamic states of the system (pressures, saturations), which changes the oil production rate of the producer, which in turn impacts the cost function. Thus, the controls u are related to J(u) through the dynamic system. The dynamic system can also be thought of as a set of constraints that determine the dynamic state of the system given a set of controls. Further, the controls u themselves may be subject to other constraints that dictate the feasible or admissible values of the controls, such as surface facility constraints or fracture pressure limits. It is these additional constraints that in many cases complicate the problem and the solution process.