A Closed-Form Solution for Free- and Seepage-Flow Field in Axisymmetric Infilled Conduit

Numerous continuous conduits exist in rock masses. These conduits affect the specific discharge distribution of the bulk rock mass. In this study, a modified theoretical model is presented to explore the specific discharge distribution in bedrock with an axisymmetric infilled conduit. The movement of fluid in the free region abides by the Navier-Stokes equation, and the seepage flow in the filling and rock regions complies with the Brinkman-extended Darcy equation. The analytical solution for the specific discharge distribution is derived by requiring flow continuity at the domain interface. This solution can be reduced to Darcy's law and Poiseuille's law. Sensitivity analysis shows that both the relative aperture of the free region and the permeability of the filling or rock regions positively influence the specific discharge. The specific discharge distribution curves seem to have an intersection for different porosity values. Moreover, the width of the transition layer where the porous media flow differs from traditional Darcy's flow is positively correlated with the permeability. However, the width is not directly correlated with the relative aperture and the porosity.