Field-scale rate optimization problems often involve highly complex reservoir models, production-and-facilities related constraints, and a large number of unknowns. These factors make optimal reservoir management through rate- and flood-front control difficult without efficient optimization tools. Some aspects of the optimization problem have been studied before mainly using an optimal control theory. However, the applications to date have been rather limited to 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 remained largely unexplored.

This paper proposes a practical and efficient approach for computing optimal injection and production rates, thereby managing the waterflood front to maximize sweep efficiency and delaying the arrival time to minimize water cycling. Our work relies on equalizing the arrival times of the waterflood front at all producers within selected subregions of a waterflood project. The arrival-time optimization has favorable quasilinear 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.

Several 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—particularly permeability contrast—can have an adverse impact on the performance of waterflooding. The presence of high-permeability streaks can severely reduce the sweep efficiency, leading to an early water arrival at the producers and bypassed oil. Also, an increased cost is associated with water recycling and handling. One approach to counteract the impact of heterogeneity and improve waterflood sweep efficiency is optimal rate allocation to the injectors and producers (Asheim 1988; Sudaryanto and Yortsos 2001; Brouwer et al. 2001; Brouwer and Jansen 2004; Grinestaff 1999; Grinestaff and Caffrey 2000). 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 to optimize waterflooding relied on optimal control theorem to allocate injection/production rates for fixed well configurations. Asheim (1988) 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 Yortsos (2001) 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. (2001) 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 can be obtained by reducing the distribution of water-arrival times at various segments along the producer. Subsequently, Brouwer and Jansen (2004) extended their work to dynamic optimization of waterflooding with smart wells using the 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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