In this paper, we extend the weighted discontinuous Galerkin finite element method (WDG) on polygonal grids for solving the dual-porosity-Navier-Stokes model. The Navier-Stokes model describes the free flow in conduits, while the dual-porosity model describes the fluid flow in a medium composed of matrix and microfractures. These two models are coupled through four physically meaningful interface conditions. We obtain the existence and local uniqueness of the solution, as well as the optimal error estimate, under appropriate small data conditions that maintain physical properties. Through numerical experiments, the advantages of the numerical method are verified, such as the optimal convergence rate of the numerical solution to different mesh types and numerical schemes, the performance of the classical upwind scheme combined with the Picard iteration method in handling small viscosity problems, the flow around a horizontal production wellbore with open-hole completion, the different application simulation of multistage hydraulic fractured horizontal wellbore with cased hole completion, as well as the simulation of fluid flow characteristics around macro-fractures.