Spatial-Temporal Tensor Decompositions for Characterizing Control-Relevant Flow Profiles in Reservoir Models

Abstract

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.

Use of reduced-order models in well control optimization

“…In that case the state snapshots are not stored as vectors and combined in a matrix, but are instead stored as matrices (2D or 3D, depending on the simulation performed) and combined in 3D or 4D tensors. For a reservoir application, see Insuasty et al (2015).…”

Clustering Techniques for Value-of-information Assessment in Closed-loop Reservoir Management

“…E.g., we could construct a 3D tensor  by stacking up the two-dimensional permeability fields (matrices) of an ensemble of 2D model realizations. We then perform the following minimization (Insuasty et al, 2015)   1: 1:J 1:…”

“…This can be extended for tensors with more dimensions. For more information, we refer to Insuasty et al (2015), who also show that the solution for tensor decomposition in (6) can be approximated by performing high-order SVD (HOSVD). In this case, the tensor is flattened in a planar matrix structure where we can operate similarly to classical SVD.…”

“…We can use the HOSVD coefficients associated with this dimension to perform clustering. Insuasty et al (2015) show that this approach allows us to compare model realizations based on very rich datasets, such as the temporal evolution of the spatial ECMOR XV -15 th European Conference on the Mathematics of Oil Recovery 29 August -1 September 2016, Amsterdam, Netherlands distribution of pressures and saturations inside the reservoir. They are able to select a subset of realizations representative in terms of dynamic flow patterns and form reduced ensembles to perform robust production optimization more efficiently.…”

Multidimensional Feature Explorer for Unbalanced Spatiotemporal Data

“…The first is the multidimensional method that is based on the tensor structure, which is a high‐dimensional generation of a matrix (Acar & Yener, ). Tensor methods (e.g., tensor decomposition) can be used to analyze a multidimensional coupling relationship in multiple dimensions (Insuasty et al, ) and have been widely used in various fields of geoscience (Luo et al, ; Martin‐Herrero & Ferreiro‐Arman, ; Zhang et al, ). In tensor analysis, all dimensions of data are treated equally, and the signals are extracted as a whole (Beckmann & Smith, ).…”