Carsten Völcker

Oil reservoir production optimization using optimal control

Practical oil reservoir management involves solution of large-scale constrained optimal control problems. In this paper we present a numerical method for solution of large-scale constrained optimal control problems. The method is a single-shooting method that computes the gradients using the adjoint method. We use an Explicit Singly Diagonally Implicit Runge-Kutta (ESDIRK) method for the integration and a quasi-Newton Sequential Quadratic Programming (SQP) algorithm for the constrained optimization. We use this algorithm in a numerical case study to optimize the production of oil from an oil reservoir using water flooding and smart well technology. Compared to the uncontrolled case, the optimal operation increases the Net Present Value of the oil field by 10%.

Adaptive Stepsize Control in Implicit Runge-Kutta Methods for Reservoir Simulation

Abstract This paper concerns predictive stepsize control applied to high order methods for temporal discretization in reservoir simulation. The family of Runge-Kutta methods is presented and in particular the explicit singly diagonally implicit Runge-Kutta (ESDIRK) methods are described. A predictive stepsize adjustment rule based on error estimates and convergence control of the integrated iterative solver is presented. We try to improve the predictive stepsize control by smoothing the stepsize sequence through combining the control of error with the control of convergence.

Oil Reservoir Production Optimization using Single Shooting and ESDIRK Methods

Abstract Conventional recovery techniques enable recovery of 10-50% of the oil in an oil field. Advances in smart well technology and enhanced oil recovery techniques enable significant larger recovery. To realize this potential, feedback model-based optimal control technologies are needed to manipulate the injections and oil production such that flow is uniform in a given geological structure. Even in the case of conventional water flooding, feedback based optimal control technologies may enable higher oil recovery than with conventional operational strategies. The optimal control problems that must be solved are large-scale problems and require specialized numerical algorithms. In this paper, we combine a single shooting optimization algorithm based on sequential quadratic programming (SQP) with explicit singly diagonally implicit Runge-Kutta (ESDIRK) integration methods and a continuous adjoint method for sensitivity computation. We demonstrate the procedure on a water flooding example with conventional injectors and producers.

Explicit Singly Diagonally Implicit Runge-Kutta Methods and Adaptive Stepsize Control for Reservoir Simulation

Simulation of fluid flow in petroleum reservoirs is an essential tool in understanding, predicting and controlling advanced oil recovery methods. The major computational effort in reservoir simulation comes from solving a very large system of differential equations describing the fluid flow and the complex behaviour of advanced oil recovery methods. Choosing an appropriate method in the numerical solution of a large system of differential equations involves deciding on factors such as the order of the integration scheme, stability properties and concern for computational efficiency. Current simulators normally uses first order integration schemes applied with heuristically guided strategies for controlling time-step sizes. In the solution process of complex recovery methods this can lead to unnecessary computations and inappropriately small time-steps. We establish a fully implicit integrator of high order applied with an adaptive time-step selection supported by error estimates. We describe the explicit singly diagonally implicit Runge-Kutta (ESDIRK) methods with an embedded scheme for error estimation. This class of methods is computationally efficient, A- and L-stable as well as stiffly accurate. The embedded method providing the error estimate is of different order than the method used for advancing the solution. Based on this error estimate, the time-step is computed by a predictive control law. The predicitive control law is designed based on a model of the numerical method (ESDIRK) itself. Implicit integration involves the solution of a system of coupled nonlinear residual equations which need to be solved iteratively. Fast convergence of the iterative solver is crucial and may be controlled by the time-step size. We present a strategy for adaptive stepsize selection that mitigates the trade off between the convergence rate and time-step size. Consequently, the stepsize selection rule keeps the error estimate bounded (i.e., close to a user-specified tolerance) and at the same time maintains a good convergence rate of the equation solver. In addition, the controller has the ability to combine the above stepsize selection rule with classical time-step control that limits maximum change in key variables.

NMPC for Oil Reservoir Production Optimization

Abstract In this paper, we use nonlinear model predictive control (NMPC) to maximize secondary oil recovery from an oil reservoir by controlling two-phase subsurface porous flow using adjustable down-hole control valves. The resulting optimal control problem is nonlinear and large-scale. We solve this problem numerically using a single shooting sequential quadratic programming (SQP) based optimization method. Explicit singly diagonally implicit Runge-Kutta (ESDIRK) methods are used for integration of the stiff system of differential equations describing the two-phase flow, and the adjoint method is used for sensitivity computations. We report computational experiences and oil recovery improvements for a standard test case.