A semi-empirical model for water inflow into a tunnel in fractured-rock aquifers considering non-Darcian flow

Abstract

Water inflow into a tunnel is usually estimated by assuming that the flow follows the linear Darcy’s law. However, fluid flow in fractured-rock aquifers is prone to deviation from Darcy’s law and gives rise to nonlinear flow phenomena due to significant inertial losses. This non-Darcian effect is seldom considered when evaluating water inflow into the tunnel in a fractured-rock aquifer. In this study, we investigated an analytical method for predicting water inflow into a tunnel considering the non-Darcian effect. A series of numerical simulations for water inflow into a tunnel were conducted with different combinations of parameters, where non-Darcian flow was considered and characterized by the Forchheimer’s law. Based on the numerical results and an ingenious parameter integration method, a semi-empirical model was developed to analytically predict water inflow into a tunnel where the flow is under non-Darcian flow regime. The semi-empirical model includes the classic theoretical formula for predicting water inflow and the Forchheimer equation for describing non-Darcian flow. In this context, the non-Darcian effect on water inflow into a tunnel can be quantitatively estimated. This developed model has a concise expression satisfying dimensional consistency with the parameters having sound physical meanings, and it can be successfully reduced to the classic formula under the Darcy flow regime. The validity of this semi-empirical model was confirmed by numerical results, showing a high fidelity in predicting water inflow into a tunnel. By means of this proposed model, the error induced by neglecting non-Darcian effect was estimated for a large number of real engineering cases, which rapidly increased as a power function with the magnitude of water inflow into the tunnel. The necessity of considering the non-Darcian effect when calculating water inflow was thus proven, especially for massive tunnel gushing. The proposed model and the method used in this study can provide guidance for accurately predicting water inflow into a tunnel under a complex non-Darcian flow regime.