An Analytical Model for Water Inflow into a Karst Tunnel in Vuggy and Fractured Porous Rock Aquifers

Vugs and fractures are the main channels within porous rock aquifers and play a dominant role in the water inrush of the karst tunnel. In this paper, a mathematical model of the coupled free-fluid seepage-fluid with second-order accuracy was established to analyze the flow field characteristics of vuggy and fractured porous rock aquifers. The Navier–Stokes equation was used to govern the free fluid flow in vugs and fractures, while the Darcy–Brinkman equation was employed to describe the seepage fluid motion in the porous rock matrix. The analytical solutions for the flow velocity and water discharge were derived under the requirement of continuous velocity and shear stress at the interface between free and seepage fluids. These solutions can be successfully reduced to those of the classical Poiseuille’s law, assuming that the rock matrix is impervious to water. The numerical, experimental, and analytical results were compared to validate the present analytical model, and the results of the analytical model showed greater accuracy compared with the classical Poiseuille’s law and cubic law. Furthermore, the water inrush of the Xiangpingshan karst tunnel was selected as a real engineering case. Perforated bricks were used to stack a vuggy and fractured porous physical tunnel model (1:100) with the same hydraulic conductivity as the Xiangpingshan tunnel. The relative errors between current theoretical and physical tests obtained for the water inflow discharges are within 18%. The current model can provide a reference for predicting the water inflow of karst tunnels in vuggy and fractured porous rock aquifers.