Generation of low-order reservoir models using system-theoretical concepts

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-Loeve 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.