Optimal Control Approach For Polymer Flooding In Enhanced Oil Recovery

Polymer flooding is a kind of technique in enhanced oil recovery, which has been widely applied in the eastern oilfields of China. Polymer is a very expensive material and the injection process is a long complex process. For making the injection schemes scientifically and economically, an optimal control based approach was investigated in this dissertation.The performance index of the optimal control problem (OCP) is chosen as the net present value (NPV) gained from oil recovery over a given time. The governed equations are fluid flow equations through porous medium which describe the dynamic procedure of polymer flooding. The maximum usage of polymer and bound constraints of control variables are considered as inequality constraints in the OCP. An optimal control model of a distributed parameter system with inequality constraints is established for polymer flooding.The control vector parameterization (CVP) method is illustrated in detail to deal with the unconstrained, control constrained and state constrained OCP. A time node parameterization (TNP) method is proposed to optimize injection switch time of polymer injection, which use a sigmoid function with time node parameter to approximate the step function in piecewise constant parameterization method. The derivatives of the performance to time grid-nodes can be obtained by differential chain rule and then the gradient based optimization methods can be used to deal with the nonlinear programming problem (NLP) transformed from the OCP. An improved subspace truncated-Newton algorithm is given for the bound constrained optimization problem during the iterations of penalty function method. Several numerical examples show the effectiveness of TNP and subspace truncated-Newton algorithm for solving the OCP. The necessary conditions of the OCP for polymer flooding were deduced by using calculus of variations. Both continuous and discrete cases are studied. For continuous case, the continuous adjoint equations and boundary conditions are derived for a 2-D model. For discrete case, the discrete adjoint equations and boundary conditions are obtained for a 3-D model.The OCP of polymer flooding is numerically solved based on piecewise constant parameterization method and the proposed TNP method. A full implicit finite difference method is presented for the polymer flooding model. The adjoint problem is solved based on discrete optimality conditions and the discrete adjoint equations are constructed by the coefficient matrixes calculated and stored during the solution of the forward model. A computation platform is built up for the OCP, which includes a 3-D numerical simulator, an optimal control solver and a few general algorithm packages.Two cases are studied to verify the proposed optimal control approach for polymer flooding. One is a reservoir consisting of a production well and 4 injection wells. To show the accuracy of the full implicit finite difference method, the water cut curves from simulation of water flooding and polymer flooding are compared with those by using commercial reservoir simulation software. The OCP is established for this case and solved under three conditions which are NPV without polymer usage constraints and different oil price, NPV under usage constraints and profit performance under usage constraints. The other case is a reservoir coming from a part of reality region of the Shengli oilfield, which consists of 5 production wells and 7 injection wells. The injection strategies are obtained based on OCP under usage constraints. The NPV, recovery efficiency and oil production are calculated respectively for each solution of the OCP in the two cases. Compared these results with those from the given injection strategies of the Shengli oilfield, it is illustrated that the injection schemes obtained by the proposed approach can obviously increase the NPV and oil production.