Model-based optimal control of water flooding generally involves multiple reservoir simulations, which makes it into a time-consuming process. Furthermore, if the optimization is combined with inversion, i.e., with updating of the reservoir model using production data, some form of regularization is required to cope with the ill-posedness of the inversion problem. A potential way to address these issues is through the use of proper orthogonal decomposition (POD), also known as principal component analysis, KarhunenYLoève decomposition or the method of empirical orthogonal functions. POD is a model reduction technique to generate low-order models using Fsnapshots_ from a forward simulation with the original high-order model. In this work, we addressed the scope to speed up optimization of water-flooding a heterogeneous reservoir with multiple injectors and producers. We used an adjoint-based optimal control methodology that requires multiple passes of forward simulation of the reservoir model and backward simulation of an adjoint system of equations. We developed a nested approach in which POD was first used to reduce the state space dimensions of both the forward model and the adjoint system. After obtaining an optimized injection and production strategy using the reduced-order system, we verified the results using the original, high-order model. If necessary, we repeated the optimization cycle using new reduced-order systems based on snapshots from the verification run. We tested the methodology on a reservoir model with 4050 states (2025 pressures, 2025 saturations) and an adjoint model of 4050 states (Lagrange multipliers). We obtained reduced-order models with 20Y100 states only, which produced almost identical optimized flooding strategies as compared to those obtained using the high-order models. The maximum achieved reduction in computing time was 35%.

Over the past few years, more and more systems and control concepts have been applied in reservoir engineering, such as optimal control, Kalman filtering, and model reduction. The success of these applications is determined by the controllability, observability, and identifiability properties of the reservoir at hand. The first contribution of this paper is to analyze and interpret the controllability and observability of single-phase flow reservoir models and to investigate how these are affected by well locations, heterogeneity, and fluid properties. The second contribution of this paper is to show how to compute an upper bound on the number of identifiable parameters when history matching production data and to present a new method to regularize the history matching problem using a reservoir's controllability and observability properties. Keywords

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In spite of large uncertainties in the actual reservoir structure, structural parameters of a reservoir model are usually fixed during history matching and only the flow properties of the model are allowed to vary. This often leads to unlikely or even unfeasible property updates and possibly to a poor predictive capability of the model. In those cases it may be expected that updating of the structural parameters will improve the quality of the history match. Preferably such structural updates should be implemented in the static (geological) model, and not just in the dynamic (flow) model. In this paper we use a gradientbased history matching method to update structural properties of the static model. We use an adjoint method to efficiently compute the derivatives of the data mismatch with respect to grid block porosities in the dynamic model and convert the corresponding volume changes to structural updates (layer thicknesses) in the static model. This method is suitable for structural updating of large scale reservoir models using production data and/or time-lapse seismics or other spatially distributed data. The method is tested on a 3D synthetic model, where time-lapse as well as production data have been used to update depth of the reservoir's bottom horizon. We obtained significant improvements in the history match quality and the predictive capability of the model.

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