Non-local Multi-continua Upscaling for Flows in Heterogeneous Fractured Media

In this paper, we propose a rigorous and accurate non-local (in the oversampled region) upscaling framework based on some recently developed multiscale methods [10]. Our proposed method consists of identifying multi-continua parameters via local basis functions and constructing non-local (in the oversampled region) transfer and effective properties. To achieve this, we significantly modify our recent work proposed within Generalized Multiscale Finite Element Method (GMsFEM) in [10] and derive appropriate local problems in oversampled regions once we identify important modes representing each continua. We use piecewise constant functions in each fracture network and in the matrix to write an upscaled equation. Thus, the resulting upscaled equation is of minimal size and the unknowns are average pressures in the fractures and the matrix. We note that the use of non-local upscaled model for porous media flows is not new, e.g., in [14], the authors derive non-local approach. Our main contribution is identifying appropriate local problems together with local spectral modes to represent each continua. The model problem for fractures assumes that one can identify fracture networks. The resulting non-local equation (restricted to the oversampling region, which is several times larger compared to the target coarse block) has the same form as \cite{Hamdi_Nonlocal} with much smaller local regions. We present numerical results, which show that the proposed approach can provide good accuracy.