Analytical solution for steady-state groundwater inflow into a drainedcircular tunnel in a semi-infinite aquifer: A revisitKyung-HoPark 比.AdisornOwatsiriwong a,Joo-GongLeeaSchool of Engineering and Technology, Asian Institute of Technology, P.O.Box% KlongLuang, Pathumthani12120, ThailandhDODAM E&C Co.,Ltd., 3F. 799, Anyang-Megavalley, Gwanyang-Dong, Dongan-Gu, Anyang, Gyeonggi-Do,Republic of KoreaReceived 19 November 2006;received in revised form 13 February 2007;accepted 18 February 2007 Availableonline 6 April 2007AbstractThis study deals with the comparison of existing analytical solutions for the steady-state groundwater inflow into a drained circular tunnel in a semi-infinite aquifer. Two different boundary conditions (one for zero water pressure and the other for a constant total head) along the tunnel circumference, used in the existing solutions, are mentioned. Simple closed-form analytical solutions are re-derived within a common theoretical framework for two different boundary conditions by using the conformal mapping technique The water inflow predictions are compared to investigate the difference among the solutions. The correct use of the boundary condition along the tunnel circumference in a shallow drained circular tunnel is emphasized 6 2007 Elsevier Ltd. All rights reserved.Keywords: Analytical solution; Tunnels; Groundwater flow; Semi-infinite aquiferL IntroductionPrediction of the groundwater inflow into a tunnel is needed for the design of the tunnel drainage system and the estimation of the environmental impact of drainage Recently, El Tani (2003) presented the analytical solution of the groundwater inflow based on Mobius transformation and Fourier series. By compiling the exact and approximate solutions by many researchers (Muscat, Goodmanet al.? Karlsrud, Rat, Schleiss, Lei, and Lombardi), El Tani(2003) showed the big difference in the prediction of groundwater inflow by the solutions. Kolymbas and Wagner (2007)also presented the analytical solution for the groundwater inflow, which is equally valid for deep and shallow tunnels and allows variable total head at the tunnel circumference and at the ground surface.While several analytical solutions for the groundwater inflow into a circular tunnel can be found in the literature,they cannot be easily compared with each other because of the use of different notations, assumptions of boundary conditions, elevation reference datum.and solution methods.In this study, we shall revisit the closed-fonn analytical solution for the steady-state groundwater inflow into a drained circular tunnel in a semi-infinite aquifer with focus on two different boundary conditions (one for zero water pressure and the other for a constant total head) along the tunnel circumference, used in the existing solutions. The solutions for two different boundary conditions are re-derived within a common theoretical framework by using the conformal mapping technique. The difference in the water inflow predictions among the approximate and exact solutions is re-compared to show the range of appli-cability of approximate solutions.2. Definition of the problemConsider a circular tunnel of radius r in a fully saturated,homogeneousjsotropic.and semi-infinite porous aquifer with a horizontal water table (Fig. 1).The surrounding ground has the isotropic permeability k and a steady-state groundwater flow condition is assumed.According to Darcys law and mass conservation, the two-dimensional steady-stategroundwater flow around the tunnel is described by the following Laplace equation:a?=0where (f) =total head (or hydraulic head), being given by the sum of the pressure and elevation heads, or p =prcssurc, yw =unit weight of water, Z =clcvation head,which is the vertical distance of a given point above or below a datum plane Here,the ground surface is used as the elevation reference datum to consider the case in which the water table is above the ground surface. Note that El Tani (2003) used the water level as the elevation reference datum,whereas Kolymbas and Wagner (2007) used the ground surface.In order to solve Eq. (l),two boundary conditions are needed:one at the ground surface and the other along the tunnel circumference.The boundary condition at the ground surface (y =0) is clearly expressed as血十H(3)In the case of a drained tunnel, however, two different boundary conditions along the tunnel circumference can be found in the literature:(Fig.l )(l )Case 1:zero water pressure, and so total head=elevation head (El Tani,2003)%)= y(4)(2)Case 2:constant total head, /ia(Lei9 1999; Kolymbas and Wagner,2007) 如=心(5)It should be noted that the boundary condition of Eq.(5) assumes a constant total head, whereas Eq.(4) gives varying total head along the tunnel circumference. By considering thesetwo different boundary conditions along the tunnel circumference, two different solutions for the steady-state groundwater rinflo