An Analytical Model for the Coupling Between Seepage and Mud Inrush in the Karst Tunnel

The phenomenon of water inrush and mud outburst in the Karst tunnels usually occurs simultaneously. This study proposes a theoretical model that takes into consideration the coupling between the seepage in the porous rock mass and the mud in Karst conduits. The Darcy–Brinkman equation is used to model the flow of limpid water (a Newtonian fluid) through the porous rock mass. The Herschel–Bulkley constitutive equation is adopted to characterize the flow of turbid water (a non-Newtonian fluid) through a Karst conduit. Under the condition of continuous velocity and stress at the conduit–matrix interface, the expressions of the velocity distribution and discharge for the present model are derived. The results demonstrate that classic models (i.e., the Power law model, Poiseuille’s law, Bingham model, Herschel–Bulkley model, and Darcy’s law) are special cases of the present model. Furthermore, the present model is verified by Finite Element Method (FEM) simulations. Sensitivity analysis indicates that the velocity of the turbid water in the conduit is positively correlated with the permeability and negatively correlated with rheological parameters (i.e., yield stress, consistency coefficient, and rheological index). Conversely, the seepage velocity of the limpid water in the porous rock mass is only positively associated with permeability.