In the energy industry, reservoir simulators enable oil companies to optimize oil and gas production. I analyze the accuracy of the discontinuous Galerkin method when solving an optimal control problem for the miscible displacement equations, which model a tertiary oil recovery process. Miscible displacement is a process in a porous media when one fluid is injected into a well to displace another fluid. An optimal control problem is a way to find a solution to a given constraint as well as optimize some control function. For the miscible displacement problem, the constraint is a partial differential equation that models the miscible displacement, where the state variables are the fluid mixture pressure and velocity, as well as the concentration of the injected fluid. The control variables are the flow rates at the injection wells, which are the variables we want to optimize. To approximate the PDE, I use two numerical methods, a discontinuous Galerkin method and a finite element method and then compare the results. Discontinuous Galerkin methods and finite element methods are numerical methods to solve PDEs using weak derivatives. In the finite element method, we assume the approximation is continuous over the domain, while in the discontinuous Galerkin method, we assume the approximation is discontinuous over the domain. In our problem, the domain is time and space. Since this PDE is time dependent, I also use a time integrator method when solving the problem. I combine these methods with optimization theory to solve the PDE while simultaneously solving the optimal control problem.
