## 1. Introduction

[2] Groundwater flow entering a tunnel is a geotechnical problem commonly encountered during a progressive drilling in saturated geological formations. Some of engineering hazards occurring in a tunnel under excavation might be attributed to the problem of large flows entering the tunnel from highly fractured, water-saturated rocks. Since this heavy flow makes the operation of drilling a tunnel difficult and dangerous, the design and construction must be carefully exercised to stem the groundwater flow. The amount and rate of groundwater flow entering a tunnel needed to be accurately estimated in advance and a suitable drainage system can then be designed to draw off the water and avoid engineering disasters.

[3] To monitor drilling status, the types of the instant and progressive drilling must be considered in predicting the groundwater flow rate entering a tunnel. An install drilling type assumes that the tunnel is installed instantaneously through a whole length. However, this assumption results in the initial inflow rate being unrealistically high. In reality, the construction of a tunnel is drilled progressively and the flow rate into a tunnel increases from zero to maximum and is then followed by a period of decay. *Perrochet* [2005a] developed a simple analytical formula to evaluate the transient flow rate into a tunnel or well under constant drawdown. By using a straight-forward function, an alterative solution is suggested to replace the well function G(*τ*) of *Jacob and Lohman* [1952] by ln ^{−1}. In engineering applications, that solution can be used with great computational benefit and can avoid directly evaluating the integration in Jacob and Lohman's solution. *Perrochet* [2005b] further developed an analytical solution via a convolution integral to evaluate the transient, drilling speed-dependent discharge rate into a tunnel gradually excavated in a homogeneous, infinite, and confined aquifer. Based on these assumptions, he provided the type curves to estimate total discharge sensitivity during the drilling time and predict the maximum flow rates.

[4] There are several models seen in the literature for calculating the flows into a tunnel in a fully saturated, homogeneous, isotropic, and infinite aquifer system. Yet, those existing models all assume that the aquifer is a single permeable layer, whereas the excavation of a tunnel may in fact go through the formation with several different geological materials. Under such circumstances, the single-layer models cannot correctly calculate the groundwater flow entering a tunnel during the drilling. This paper presents a multi-layer model for predicting the transient inflow rate into a horizontal tunnel during progressive drilling. In addition, a unified numerical approach is provided to evaluate the newly derived closed-form solution of a multi-layer model. This approach includes a singularity removal scheme, the Gaussian quadrature, and the Shanks method.