System-Theoretical Model Reduction for Reservoir Simulation and optimization

Abstract

This thesis is concerned with low-order modelling of heterogeneous reservoir systems for the purpose of efficient simulation and optimization of flooding processes with multiple injection and production (smart) wells. Typically, one is initially equipped with a physics-based ('white-box') model consisting of O(103-106) equations and parameters representing a (coupled) system of discretized PDEs defined on a geometric grid. The model-order reduction (MOR) methodology undertaken in this research is fundamentally different from the traditional, 'grid-coarsing' approximation methods, in that no coarse-grid approximation of the fine-grid problem is employed at all. Instead, the reduced-order models are here based on 'system-theoretic' and dynamically intrinsic properties of the fine-scale system. In single-phase flow problems that can be modelled as linear time-invariant state-space systems these properties are, e.g., the system's transfer function in the Laplace domain, the eigenstructure of the system matrix, or controllability and observability of the (particular state-space realization of the) system. For multi-phase flow problems resulting in nonlinear state-space models, intrinsic information needs to be sought in data obtained by simulating the fine-scale model. The contribution of this thesis can be divided into three themes: 1) Standard 'projection-based' MOR: assessment of the performance of modal truncation, singular perturbation, balanced truncation, transfer function moments maching (inc. Krylov-subspaces), and proper orthogonal decomposition (POD), 2) Acceleration of solving the fine-scale problem: use of MOR as a 'shadow simulation' to determine an improved fine-scale initial guess, and 3) Acceleration of waterflooding optimization: use of POD in the inner-loop of an adjoint-based optimization scheme.