Non-Intrusive Reduced Basis two-grid method for flow and transport problems in heterogeneous porous media

Due to its non-intrusive nature and ease of implementation, the Non-Intrusive Reduced Basis (NIRB) two-grid method has gained significant popularity in numerical computational fluid dynamics simulations. The efficiency of the NIRB method hinges on separating the procedure into offline and online stages. In the offline stage, a set of high-fidelity computations is performed to construct the reduced basis functions, which is time-consuming but is only executed once. In contrast, the online stage adapts a coarse-grid model to retrieve the expansion coefficients of the reduced basis functions. Thus it is much less costly than directly solving a high-fidelity model. However, coarse grids in heterogeneous porous media of flow models are often accompanied by upscaled hydraulic parameters (e.g. hydraulic conductivity), thus introducing upscaling errors. In this work, we introduce the two-scale idea to the existing NIRB two-grid method: when dealing with coarse-grid models, we also employ upscaled model parameters. Both the discretization and upscaling errors are compensated by the rectification post-processing. The numerical examples involve flow and heat transport problems in heterogeneous hydraulic conductivity fields, which are generated by self-affine random fields. Our research findings indicate that the modified NIRB method can effectively capture the large-scale features of numerical solutions, including pressure, velocity, and temperature. However, accurately retrieving velocity fields with small-scale features remains highly challenging.