Spatial-temporal tensor decompositions for characterizing control-relevant flow profiles in reservoir models

This paper considers the use of spatial-temporal (tensor) decompositions for the compact representations of saturation patterns in reservoir models. Reservoir flow patterns, in the sense of evolution of saturation patterns over time, can be considered to drive the economic performance of reservoirs as they drive the ultimate recovery. This makes the reservoir flow pattern a natural dissimilarity measure between models in the context of production optimization. We show that the application of multilinear algebra techniques allows the construction of low-complexity representations of the essential saturation patterns. The reservoir flow patterns are stored in large-scale multidimensional arrays, and tensor decompositions can be effectively used to describe the spatial-temporal behavior of the reservoir flow patterns. The dimensionality of the reservoir flow patterns can be substantially reduced, showing that a small number of spatial-temporal basis functions are required to characterize the dominant features. When applying the tensor decompositions to flow profiles of an ensemble of realizations they can be used for clustering models with similar dynamical properties, by allowing a fast calculation of a flow-relevant dissimilarity measure between realizations. For large ensembles of realizations they can lead to considerable computational advantages in (robust) optimization. This is illustrated by using a gradient-based technique to maximize Net Present Value (NPV) in water flooding for an ensemble of realizations. Besides in reducing ensemble sizes the resulting tools have potential use in constructing reduced order models.