Introduction: The safe and cost-effective design and construction of tunnels need comprehensive geological, hydrological, environmental, geotechnical, and seismological information. Among the most critical challenges in tunneling is groundwater inflow into the underground excavation, particularly when the tunnel alignment lies below the water table. Several studies have focused on predicting the water inflow into tunnels using analytical methods (Freeze and Cherry 1979; Heuer 1995; El Tani 2003; Moon and Fernandez 2010), numerical simulations (Jing 2003; Aryafar et al. 2009; Butscher 2012; Aalianvari et al. 2013; Farhadian and Bahmani Shahraki 2024), and empirical approaches (McFeat-Smith et al. 1998; Katibeh and Aalianvari 2009; Farhadian and Katibeh 2017). Although extensive research has been conducted on tunnel inflow under the simplifying assumption of isotropic soil, only a limited number of studies have quantified tunnel inflow in the presence of anisotropic permeability. Accordingly, this study aims to determine the groundwater inflow rate into a circular tunnel embedded in an anisotropic medium across a wide range of influencing parameters using the finite element method. In addition, an equation is developed, using the nonlinear least square method, to estimate the inflow rate to a circular tunnel, without the need for numerical modeling of the domain.
Methodology: This study aims at modeling flow into a circular tunnel and estimating the overall flowrate to the tunnel, using finite element method. The initial position of the groundwater table was defined by imposing fixed-head flow boundary conditions along the vertical boundaries. In contrast, a no-flow condition was applied at the bottom boundary. Regarding mechanical constraints, the vertical boundaries were restricted to vertical displacement only, thereby preventing any horizontal movement. The base boundary was fully fixed, restricting displacements in both the horizontal and vertical directions.
Model dimensions are another critical factor that can significantly influence the computed flow rate. As the model extent increases, the water inflow rate initially decreases and then gradually converges toward a constant value (Moon and Fernandez 2010; Butscher 2012; Nikvar Hassani et al. 2016). Su et al. (2017), indicated that boundary effects can be neglected when the model extent, defined as the perpendicular distance from the tunnel center to the vertical and bottom boundaries, exceeds 50 times the tunnel diameter. In the present study, the maximum tunnel diameter considered was 12 m; therefore, a model extent of 600 m was adopted consistently for all cases. In the present study, the initial and final number of elements were set to 5000 and 50000, respectively, over five adaptive iterations. This means the analysis began with a coarse mesh of 5000 elements, which was automatically refined in successive iterations, ultimately reaching a mesh density of 50000 elements in the final step.
Results and Discussion: The results indicated that increasing the horizontal to vertical permeability ratio (Kh/Kv) markedly enhances water flow rate (q); for instance, when Kv = 1 × 10⁻⁶ m/s, the inflow rate at Kh/Kv = 20 is nearly six times greater than Kh/Kv = 1. Thus, ignoring the permeability anisotropy could lead to a significant underestimation of the flow rate to the tunnel.
It was seen that for a constant Kv, an increase in the anisotropy ratio results in a smoother and more gradual phreatic surface around the tunnel. In contrast, lower values of Kh/Kv lead to a steeper drawdown, indicating a more pronounced reduction in pore water pressure close to the tunnel boundary. I was also observed that for the highest anisotropy ratio, the phreatic line may even extend above the tunnel crown.
Using a dataset produced by this verified model, a predictive model was developed to estimate the inflow rate (q) as a function of the anisotropy ratio using nonlinear least-squares regression. Statistical evaluation of the proposed model yielded the coefficient of determination R2 = 0.986, the root mean square error RMSE = 0.014, and the mean absolute percentage error MAPE = 5.56%, demonstrating the high predictive accuracy of the model. However, to quantitatively assess the goodness-of-fit of the proposed model, a hypothesis test was performed on the normalized residuals. In the present study, the z-test was employed, assuming the null hypothesis that the normalized residuals follow a normal distribution with a mean of zero and a standard deviation of one. The resulting p-value was 0.93, which is considerably higher than the significance level of α = 0.05, indicating that there is insufficient evidence to reject the null hypothesis.
Conclusion: A total number of 500 finite element analyses were conducted to evaluate the effect of permeability anisotropy, considering anisotropy ratios (Kh/Kv) ranging from 1 to 20. The results indicated that as Kh/Kv increases, the inflow rate also increases. The anisotropy ratio was also found to affect the configuration of the phreatic line. Higher anisotropy ratios promote more pronounced horizontal flow, resulting in a gentler groundwater drawdown near the tunnel.
Subsequently, a nonlinear least-squares regression was employed to develop a model for the water inflow rate in anisotropic media. The obtained results demonstrated a high level of accuracy of the model.