Reduced-Order Modeling of Coupled Flow-Geomechanics Problems

Abstract

A reduced-order modeling framework is developed and applied to simulate coupled flow-geomechanics problems. The reduced-order model is constructed using POD-TPWL, in which proper orthogonal decomposition (POD), which enables representation of the solution unknowns in a low-dimensional subspace, is combined with tra jectory piecewise linearization (TPWL), where solutions with new sets of well controls are represented via linearization around previously simulated (training) solutions. The over-determined system of equations is pro jected into the lowdimensional subspace using a least-squares Petrov-Galerkin procedure, which has been shown to maintain numerical stability in POD-TPWL models. The states and derivative matrices required by POD-TPWL, generated by an extended version of Stanford's Automatic-Differentiation-based General Purpose Research Simulator, are provided in an offline (pre-processing or training) step. Offline computational requirements correspond to the equivalent of 5-8 full-order simulations, depending on the number of training runs used. Runtime (online) speedups of O(100) or more are typically achieved for new POD-TPWL test-case simulations. The POD-TPWL model is tested extensively for a 2D coupled problem involving oil-water flow and geomechanics. It is shown that POD-TPWL provides predictions of reasonable accuracy, relative to full-order simulations, for well-rate quantities, global pressure and saturation fields, global maximum and minimum principal stress fields, and the Mohr-Coulomb rock failure criterion, for the cases considered. A systematic study of POD-TPWL error is conducted using various training procedures for different levels of perturbation between test and training cases. The use of randomness in the well bottom-hole pressure profiles used in training is shown to be beneficial in terms of POD-TPWL solution accuracy. The procedure is also successfully applied to a prototype 3D example case.