We used optimal control theory as an optimization algorithm for the valve settings in smart wells. We focused on their use in injectors and producers for the waterflooding of heterogeneous reservoirs. As a followup to an earlier intuitive optimization approach, a systematic dynamic optimization approach based on optimal control theory was developed. The objective was to maximize recovery or net present value of the waterflooding process over a given time period. We investigated the scope for optimization under purely pressure- and purely rate-constrained operating conditions, and concluded that:

  1. for wells operating on bottomhole-pressure constraints, the benefit of using smart wells is mainly reduced water production rather than increased oil production, and

  2. for wells operating on rate constraints, there is generally a large scope for accelerating production and increasing recovery, in combination with a drastic reduction in water production.


The development of wells containing permanent downhole measurement and control equipment (in the remainder of this paper referred to as smart wells) potentially enables significant improvement of the oil production process. In an earlier study, we investigated the static optimization of waterflooding with smart wells, using heuristic algorithms.1 Static implies that the injection and production rates of the inflow control valves (ICVs) in the wells were kept constant during the displacement process, until water breakthrough at the producers occurred. Significant improvements were realized for simple reservoir models. Results suggested that more improvement could be expected by dynamically optimizing injection and production rates. Later, we therefore addressed the same problem using an optimization technique known as optimal control theory.2 In addition to being more systematic, enabling optimization also for more complex heterogeneities, this technique allowed for dynamic waterflooding control. Optimal control theory has been used before in reservoir engineering. Fathi and Ramirez used it to optimize surfactant flooding processes,3,4 Mehos to optimize CO2 flooding, 5 Liu and Ramirez to optimize steamflooding,6 and Zakirov et al. to optimize production from a thin oil rim.7 Furthermore, the same technique has been used in history matching reservoir models with production data. Optimization of waterflooding using optimal control theory has been studied before by Asheim,8 Virnovski,9 Sudaryanto and Yortsos,10-12 and Dolle et al.2 The optimization objective was either to maximize water breakthrough time at given field rates, or to maximize cumulative oil production or net present value (NPV) within a given time. In all these waterflood-optimization cases, the flow in the reservoir was controlled with wells that operated at constant field injection and production rates (i.e., the total injection and production rates were kept constant but the distribution over the wells was changed over time). In real life, however, a production strategy with constant field rates will often not be feasible, because it may require unrealistic bottomhole pressures, and associated sandface pressures. These could be too low pressures at the producers, resulting in lift die-out, or too high at the injectors, exceeding maximum allowable formation or equipment pressures. In the present study, we therefore investigated the scope for dynamic optimization for two extreme well-operating conditions. The first is completely rate-constrained injection and production, in which field injection and production rates are always at the maximum. The second is completely pressure-constrained injection and production, in which the injectors always inject at the maximum allowable injection pressure and the producers always produce at the minimum allowable well-flowing pressure.

Problem Formulation
Dynamic System Model.

We considered a heterogeneous, horizontal, 2D, two-phase (oil/water) reservoir with two horizontal smart wells, an injector and a producer, at opposite sides (see Fig. 1). The reservoir has no-flow boundaries at all sides. Each well is divided into segments with ICVs, allowing for individual inflow control of the segments. Alternatively, the two horizontal wells can be interpreted as rows of vertical injectors and producers. We used a conventional finite-difference approximation to describe the reservoir, with details as given in Appendix A. The resulting numerical model can be represented as a discrete-time dynamic system model:

Equation 1

where g is a nonlinear vector function, x is the vector of state variables with elements corresponding to the oil pressures and water saturations in each gridblock, k=0, . . ., K is the timestep, and u is the vector of the input variables or control variables, with elements that correspond to the water injection or liquid production rates in those gridblocks that are penetrated by a well.

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