Field scale rate optimization problems often involve highly complex reservoir models, production and facilities related constraints and a large number of unknowns. All these make optimal reservoir management via rate and flood front control difficult without efficient optimization tools. Some aspects of the optimization problem have been studied before using mainly optimal control theory. However, the applications to-date have been limited to rather small problems because of the computation time and the complexities associated with the formulation and solution of adjoint equations. Field-scale rate optimization for maximizing waterflood sweep efficiency under realistic field conditions has still remained largely unexplored.

We propose a practical and efficient approach for computing optimal injection and production rates and thereby manage the waterflood front to maximize sweep efficiency and delay the arrival time to minimize water cycling. Our work relies on equalizing the arrival times of the waterfront at all producers within selected sub-regions of a water flood project. The arrival time optimization has favorable quasi-linear properties and the optimization proceeds smoothly even if our initial conditions are far from the solution. Furthermore, the sensitivity of the arrival time with respect to injection and production rates can be calculated analytically using a single flow simulation. This makes our approach computationally efficient and suitable for large-scale field applications. The arrival time optimization ensures appropriate rate allocation and flood front management by delaying the water breakthrough at the producing wells.

Multiple examples are presented to support the robustness and efficiency of the proposed optimization scheme. These include several 2D synthetic examples for validation purposes and a 3D field application. In addition, we demonstrate the potential of the approach to optimize the flow profile along injection/production segments of horizontal-smart wells


Waterflooding is by far the most commonly used method to improve oil recovery after primary depletion. In spite of its many favorable characteristics, reservoir heterogeneity, permeability contrast in particular, can adversely impact the performance of waterflooding. It is well known that the presence of high permeability streaks can severely reduce the sweep efficiency leading to an early water arrival at the producers and by-passed oil. Also, there is an increased cost associated with water re-cycling and handling. One approach to counteract the impact of heterogeneity and to improve waterflood sweep efficiency is through optimal rate allocation to the injectors and producers. Through optimal rate control, we can manage the propagation of the flood front, delay water breakthrough at the producers and also increase the recovery efficiency.

Previous efforts on optimization of waterflooding relied on optimal control theorem to allocate injection/production rates for fixed well configurations. Asheim1 investigated the optimization of waterflood based on maximizing net present value (NPV) for multiple vertical injectors and one producer where the rate profiles change throughout the optimization time. Sudaryanto and Yortsos2 used maximizing the displacement efficiency at water breakthrough as the objective for the optimization with two injectors and one producer. The optimal injection policy was found to be ‘bang bang’ type. That is, the injectors were operated only at their extreme values, either at the maximum allowable injection rate or fully shut. The optimization then involved finding the switch time between the two injectors to ensure simultaneous water arrival at the producing well. Brouwer et al.3 studied the static optimization of waterflooding with two horizontal smart wells containing permanent downhole well control valves and measurement equipment. The static optimization implies that the flow rates of the inflow control valves (ICVs) along the well segments were kept constant during the waterflooding process until the water arrived at the producer. Various heuristic algorithms were utilized to minimize the impact of high permeability streaks on the waterflood performance through rate control. The results indicated that the optimal rate allocation amounts to reducing the distribution of water arrival times at various segments along the producer. Subsequently, Brouwer and Jansen4 extended their work to dynamic optimization of water flooding with smart wells using optimal control theory. The optimization was performed on one horizontal producer and one horizontal injector. Each well is equipped with 45 ICVs. The objective was to maximize the NPV, and it was achieved through changing the rate profile along the well segments throughout the optimization period. Both rate constrained and bottomhole pressure constrained well conditions were studied.

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