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

Abstract

Abstract 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.