Generation of Low-Order Reservoir Models Using System-Theoretical Concepts

Abstract

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

Linearized reduced-order models for subsurface flow simulation

“…Specifically, as indicated in the Introduction, POD-based ROMs [9][10][11] achieved at most about a factor of 10 speedup. The much larger speedups observed here are due to the fact that the TPWL model avoids many of the computations required by the POD-based ROM procedures, as discussed in Section 2.3.…”

Reduced-order optimal control of water flooding using proper orthogonal decomposition

“…As described in [5], it may appear at first sight as if the reduced-order model does not lead to a reduction in simulation time. If we compute z(k + 1) explicitly, we may even obtain a slight increase in simulation time, because every time step, we have to perform two additional transformations (from z to x and back) because we need the original state vector x to compute the functions f. However, this increase will, in general, be offset by an increase in the minimum time step required for stability.…”

“…The time needed to calculate optimized controls increases with the number of grid blocks and the complexity of the reservoir model. Reduced-order modelling and reduced-order control may provide an alternative to this [5,9,10,19]. Moreover, parameters and variables of the model can be updated with history matching.…”

Complexity Reduction of Multi-Phase Flows in Heterogeneous Porous Media

“…This is usually the case in history matching and the optimization process. A variety of complexity-reduction techniques were proposed to ease this problem and to reduce the computational cost in the optimization under the uncertainty paradigm (Heijn et al 2004;Gildin 2010). In general, one can classify them in three broader areas if one is dealing with the forward simulations (production optimization) or the inverse problem (parameter estimation) (Oliver et al 2008):…”

“…In the case of nonintrusive methods, data-driven model reduction has been the choice in material-balance-type modeling such as the capacitance/resistance models (Yousef et al 2006) and flownetwork models (Lerlertpakdee et al 2014), and in the use of artificial intelligence, such as neural networks and fuzzy-logic techniques (Mohaghegh et al 2012). For the intrusive schemes, reduced-order modeling by projection has been used in the systems/controls-like framework, such as the balanced truncation (Heijn et al 2004), proper orthogonal decompositions (PODs) (Volkwein and Hinze 2005), and the trajectory-piecewise linear (TPWL) techniques (Cardoso and Durlofsky 2010), bilinear Krylov subspace methods , and quadratic bilinear model order reduction .…”