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A hydraulic fracturing system with super-hydrophobic proppants ischaracterized by a transient triple-porosity Navier-Stokes model. For thiscomplex multiphysics system, particularly in the context of three-dimensionalspace, a local parallel and non-iterative finite element method based ontwo-grid discretizations is proposed. The underlying idea behind utilizing thelocal parallel approach is to combine a decoupled method, a two-grid method anda domain decomposition method. The strategy allows us to initially capturelow-frequency data across the decoupled domain using a coarse grid. Then ittackles high-frequency components by solving residual equations withinoverlapping subdomains by employing finer grids and local parallel proceduresat each time step. By utilizing this approach, a significant improvement incomputational efficiency can be achieved. Furthermore, the convergence resultsof the approximate solutions from the algorithm are obtained. Finally, weperform 2D/3D numerical experiments to demonstrate the effectiveness andefficiency of the algorithm as well as to illustrate its advantages inapplication.

A Local Parallel Finite Element Method for Super-Hydrophobic Proppants  in a Hydraulic Fracturing System Based on a 2D/3D Transient Triple-Porosity  Navier-Stokes Model