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%.

SUMMARYWe propose the use of reduced-order models to accelerate the solution of systems of equations using iterative solvers in time stepping schemes for large-scale numerical simulation. The acceleration is achieved by determining an improved initial guess for the iterative process based on information in the solution vectors from previous time steps. The algorithm basically consists of two projection steps: (1) projecting the governing equations onto a subspace spanned by a low number of global empirical basis functions extracted from previous time step solutions, and (2) solving the governing equations in this reduced space and projecting the solution back on the original, high dimensional one. We applied the algorithm to numerical models for simulation of two-phase flow through heterogeneous porous media. In particular we considered implicit-pressure explicit-saturation (IMPES) schemes and investigated the scope to accelerate the iterative solution of the pressure equation, which is by far the most time-consuming part of any IMPES scheme. We achieved a substantial reduction in the number of iterations and an associated acceleration of the solution. Our largest test problem involved 93 500 variables, in which case we obtained a maximum reduction in computing time of 67%. The method is particularly attractive for problems with time-varying parameters or source terms.

Summary We present five methods to derive low-order numerical models of two-phase (oil/water) reservoir flow, and illustrate their features with numerical examples. Starting from a known high-order model, these methods apply system-theoretical concepts to reduce the model size. Using a simple but heterogeneous reservoir model, we illustrate that the essential information of the model can be captured by a limited number of state variables (pressures and saturations). Ultimately, we aim at developing computationally efficient algorithms for history matching, optimization, and the design of control strategies for smart wells. In this study we applied (1) modal decomposition, (2) balanced realization, (3) a combination of these two methods, (4) subspace identification, and (5) proper orthogonal decomposition (POD), also known as principal component analysis, Karhunen-Loève decomposition, or the method of empirical orthogonal functions. Methods 1 through 4 result in linear low-order models, which are only valid during a limited time span. However, the POD results in a nonlinear model that remains valid over a much longer period. Methods that result in linear low-order models are not very promising for speeding up reservoir simulation. POD, however, has the potential to improve computational efficiency in the case of multiple simulations of the same reservoir for different well operating strategies, but further research is required to quantify this scope. The potential benefit of low-order models is therefore mainly in the development of low-order control algorithms, and in history matching, where the use of reduced models may form an alternative to classical regularization methods. Introduction Smart wells have the potential to increase oil recovery through controlling the pressures or flow rates in the smart well segments. Optimization techniques for reservoir models containing smart wells have been developed to investigate this potential1,2 and Fig. 1 represents a waterflooding example of a simple 2D heterogeneous reservoir taken from Ref. 1. At one side of the reservoir, a horizontal smart injection well is installed, and at the opposite side, a horizontal smart production well; the optimization problem involves maximizing oil recovery or net present value over a given time interval by adjusting the flow rates in the smart well segments. A reservoir model is called a high-order model if it consists of a large number (typically 103 to 106) of variables (pressures and saturations). Optimization of high-order reservoir models is computationally very intensive and thus time-consuming and expensive. Therefore, we are looking for methods to reduce high-order models to low-order models (typically 101 to 103 variables) before optimization. The dynamics of high-order reservoir models are usually captured in a smaller degree space than the models initially may imply. Therefore low-order models, found by, for instance, projecting the original state dynamics onto lower-order subspaces, are often sufficiently accurate to describe reservoir dynamics. Based on these low-order models, which contain the most relevant features, low-order controllers can be constructed. Note that controllers also need to be of relatively low order to be of practical value. In addition, low-order models are of relevance for updating of reservoir model parameters based on measured data from, for example, production tests or time-lapse seismic. This inverse problem, also known as history matching or data assimilation, is well known to be ill-posed because high-order models typically contain many more parameters than can be uniquely determined from the measurements. The inverse problem usually involves minimizing an objective function that represents the difference between modeled and measured data, and a classic way to overcome the ill-posedness is to impose constraints on the solution space for the model parameters through the addition of regularization terms to the objective function. The use of low-order models provides an alternative to classic regularization. There are two main approaches for deriving low-order models: mathematical reduction of high-order white-box models, and the identification of low-order black-box models directly; respectively, they are illustrated in the upper and the lower branch of Fig. 2. White-box models explicitly take the physics of the system into account, whereas black-box models are based on measured input/output behavior only. We will discuss mathematical reduction of a white-box model using modal decomposition, balanced realization, a combination of the two, and POD. While the first three methods result in linear low-order models, the model obtained by the latter method remains nonlinear. Afterward, we will discuss identification of a black-box model. Early attempts to use black-box models in reservoir engineering have been reported by Rowan and Clegg3 and Chierici.4 We will use a more recently developed identification method, which is one of the many methods that are available in the measurement and control community at present. Although identification is typically a black-box modeling method, it can also be applied to input/output data of a white-box high-order model. We are not always able to access and to derive low-order models from the mathematical high-order models used in (commercial) reservoir simulators. Therefore, identification can be seen as a useful fifth method of mathematical reduction. Reduction and identification have already successfully been applied to single-phase linear 2D reservoir models.5 The use of POD to derive low-order proxies of reservoir models was described by Gharbi et al.,6,7 while the use in groundwater flow modeling was described by Vermeulen et al.8 We briefly reported a comparison of the various methods in an earlier publication.9

We present five methods to derive low-order numerical models of two-phase (oil-water) reservoir flow, and illustrate their features with numerical examples. Starting from a known high order model these methods apply system-theoretical concepts, originally developed in measurement and control theory, to reduce the model size. Using a simple but heterogeneous reservoir model, we illustrate that the essential information of the model can be captured by a very limited amount of state variables (pressures and saturations). Ultimately we aim at developing computationally efficient algorithms for history matching, optimization and in particular the design of control strategies for smart wells. In this study we applied 1) modal decomposition, 2) balanced realization, 3) a combination of these two methods, 4) subspace identification, and 5) proper orthogonal decomposition (POD), also known as principal component analysis or the method of empirical orthogonal functions. Methods 1 to 4 result in linear low-order models, which are only valid during a limited timespan. However, the 5th method (POD) results in a non-linear model that remains valid over a much longer period. Subspace identification requires only input-output data and no knowledge of the system itself, and could therefore, in theory, also be applied to measured data. In particular POD and identification are promising methods to generate low-order models. Introduction Smart wells have the potential to increase oil recovery through water flooding in heterogeneous reservoirs by controlling the pressures or flow rates in the smart well segments. Optimization techniques for a two-dimensional (2D) reservoir model containing two smart wells have been developed to investigate this potential1. At one side of the reservoir a horizontal smart injection well is installed, at the opposite side a horizontal smart production well; see Fig. 1. The optimization techniques aim at maximizing oil recovery or net present value over a given time interval by adjusting the flow rates in the smart well segments. A reservoir model is called a high-order model if it consists of a large number (typically 103-106) of variables (pressures and saturations). Optimization of high-order reservoir models is computationally very intensive and thus time-consuming and expensive. Therefore we are looking for methods to reduce high-order models to low-order models (typically 101-103 variables) before optimization. The behavior of high-order reservoir models is usually determined by only a small part of the information it contains. Therefore low-order models are often sufficiently accurate to describe reservoir dynamics. Based on these low-order models containing the most relevant features controllers can be constructed. The controllers also need to be of relatively low-order to be of practical value. In this article we will present the application of system theoretical concepts to select the relevant (‘dominant’) information and to derive efficient low-order models.

In this paper we present a number of data-drieen approaches to obtain non-linear low-order models of heterogeneous reservoirs. Relying on the `proper orthogonal decomposition' (POD) method of snapshots, the proposed approaches scale-up an existing high-order dynamic reservoir model to a model of lower order. Combined with so-called `balanced' model order reduction techniques for linear systems the POD methods yield an intuitively motivated and systematic procedure for construction of low-order models for complex, high-dimensional nonlinear systems. The reduced-order nonlinear models are obtained in way that resembles standard model reduction methods for linear systems, such as balanced truncation, by preserving only those features of the dynamics that are most relevant to the control design, i .e. the most controllable and observable states. In one of the methods matrices are obtained that can be interpreted as generalized gramians for the nonlinear system. In addition we propose the use of subspace identification techniques to generate a linearized minimal balanced realization based on reservoir input-output data (flow rates and pressures), rather than Taylor-like linearization of the original full-order nonlinear model. The main advantage of this approach is the possibility to create linearizations of commercial reservoir simulation models without having acces to the code. Although the reduced-order models resulting from our approach are nonlinear, the methods used in their construction are inherently linear ; they produce a linear coordinate transformation by constructing and decomposing matrices obtained from `numerical experiments' (simulation) .