In this paper, a hybrid model reduction method for solving flows in fractured media is proposed. The approach integrates the Generalized Multiscale Finite Element Method (GMsFEM) with a novel variable-separation (VS) technique. Compared with many widely used variable-separation methods, the proposed model reduction method shares their merits but has lower computation complexity and higher efficiency. Within this framework, we can get the low-rank variable-separation expansion of dual-continuum model solutions in a systematic enrichment manner. No iteration is performed at each enrichment step. The expansion is constructed using two sets of basis functions: stochastic basis functions and deterministic physical basis functions, both derived from offline, model-oriented computations. To efficiently construct the stochastic basis functions, the original model is used to learn stochastic information. Meanwhile, the deterministic physical basis functions are trained using solutions obtained by applying an uncoupled GMsFEM to the dual-continuum system at a select number of optimal samples. Once these bases are established, the online evaluation for each new random sample becomes highly efficient, allowing for the computation of a large number of stochastic realizations at minimal cost. To demonstrate the performance of the proposed method, two numerical examples for dual-continuum models with random inputs are presented. The results confirm that the hybrid model reduction method is both efficient and achieves high approximation accuracy.
Lingling Ma (Fri,) studied this question.