Development and application of reduced‐order modeling procedures for subsurface flow simulation

Abstract

SUMMARYThe optimization of subsurface flow processes is important for many applications, including oil field operations and the geological storage of carbon dioxide. These optimizations are very demanding computationally due to the large number of flow simulations that must be performed and the typically large dimension of the simulation models. In this work, reduced-order modeling (ROM) techniques are applied to reduce the simulation time of complex large-scale subsurface flow models. The procedures all entail proper orthogonal decomposition (POD), in which a high-fidelity training simulation is run, solution snapshots are stored, and an eigen-decomposition (SVD) is performed on the resulting data matrix. Additional recently developed ROM techniques are also implemented, including a snapshot clustering procedure and a missing point estimation technique to eliminate rows from the POD basis matrix. The implementation of the ROM procedures into a general-purpose research simulator is described. Extensive flow simulations involving water injection into a geologically complex 3D oil reservoir model containing 60 000 grid blocks are presented. The various ROM techniques are assessed in terms of their ability to reproduce high-fidelity simulation results for different well schedules and also in terms of the computational speedups they provide. The numerical solutions demonstrate that the ROM procedures can accurately reproduce the reference simulations and can provide speedups of up to an order of magnitude when compared with a high-fidelity model simulated using an optimized solver.

Fracture Characterization in Enhanced Geothermal Systems by Wellbore and Reservoir Analysis

“…Researchers at the Stanford Department of Energy Resources Engineering are actively working on the topic, specifically within the Smart Fields Consortium. A few publications of 45 interest are those by Sarma et al (2006), Cardoso et al (2009) andEcheverria Ciaurri et al (2011). Professor Jan-Dirk Jansen at the Delft University of Technology also runs an active research group with an interest in production optimization.…”

Model-reduced gradient-based history matching

“…It was noticed in [21,38] and [7] that the pressure behavior can be well represented by a few patterns but that the situation is worse for the saturation behavior, which is caused by the moving fluid interface. A significantly larger number of patterns is therefore required to satisfy the chosen accuracy level.…”

“…Next, the (tangent-) linear model equations are projected on the low-order basis formed by the basis functions. It results in a reduced-order model, in terms of coefficients multiplying the basis functions, which is still able to reproduce the dominant dynamic behavior of the original model, see, e.g., [7] or [38] for further details. If the original model is nonlinear, the matrix coefficients of the tangent-linear high-order model need to be recomputed every time step before applying the reduction, which significantly reduces the computational advantage of the POD method as compared to the linear case.…”

“…The application of the POD method reduces the CPU time of the model simulation but does not change the complexity of the problem and, consequently, does not solve the implementation problem of the adjoint model. The POD method has been successfully applied to increase the computational efficiency of reservoir simulation by Heijn et al [21], Markovinovic and Jansen [30], and Cardoso et al [7]. Use of the POD method in combination with adjoint-based optimization has been investigated by, e.g., Cao et al [6] and Fang et al [17] in oceanography and by Van Doren et al [38] in reservoir simulation.…”

Projection-Based Model Reduction for Finite Element Approximation of Shallow Water Flows

“…It has been applied to a host of general, large-scale parameterized dynamical systems. Fundamentally, it is based on approximating the high-dimensional solution to the dynamical system in a lowdimensional subspace that is chosen by considering modes that capture a targeted portion of the system's energy [4,10]. To obtain a dynamical system of low dimension, POD proceeds by projecting the original high-dimensional equations on the solution subspace.…”