| Coupling problems of multi-region and multi-physical models are widely present in the fields of hemodynamics,engineering filtration,and shale gas exploitation.This paper considers the dual-porosity-Stokes coupling model formed by the combination of the seepage flow in the underground shale matrix and the seepage process in the fracture with the pipeline flow.More specifically,the governing equation in the free fluid subregion of the pipeline is the Stokes equation.In the dual porosity subregion where both the shale matrix and the fracture are considered,the governing equations are two Darcy's laws with material exchange.The two sub-regions are coupled through the mass balance condition on the interface,the normal force balance condition and the classic Beavers-Joseph-Saffman condition.In addition,we also assume that there is no direct material exchange between the substrate and the pipe.The research on the numerical solution of this model is of great significance in the fields of fluid mechanics,reservoir numerical simulation and energy calculation.Aiming at the unsteady dual-porosity-Stokes coupling model with the Beavers-Joseph-Saffman condition,this paper first derives the weak form of the model,then analyzes the well-posedness of the model,and then constructs two second-order numerical methods.The Crank-Nicolson algorithm which solves the model directly and the CNLF algorithm which decouples the coupled model to solve the problem.Theoretically,the unconditional stability of Crank-Nicolson algorithm and the conditional stability of CNLF algorithm are analyzed.Numerically,the feasibility and effectiveness of the two algorithms are verified through numerical examples.The comparison shows that both can achieve second-order convergence,but the CNLF format has more advantages in terms of calculation time.The goals of reducing the scale of calculation and improving the efficiency of calculations to be solved respectively coincide with each other. |
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