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analytical solution for steady-state groundwater inow into a drained circular tunnel in a semi-innite aquifer: a revisitkyung-hopark a,*,adisornowatsiriwong a,joo-gonglee baschool of engineering and technology, asian institute of technology, p.o.box4, klongluang, pathumthani 12120, thailandbdodam 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 available online 6 april 2007abstractthis study deals with the comparison of existing analytical solutions for the steady-state groundwater inow into a drained circular tunnel in a semi-innite aquifer. two dierent 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 dierent boundary conditions by using the conformal mapping technique. the water inow predictions are compared to investigate the dierence among the solutions. the correct use of the boundary condition along the tunnel circumference in a shallow drained circular tunnel is emphasized. 2007 elsevier ltd. all rights reserved.keywords:analytical solution; tunnels; groundwater ow; semi-innite aquifer1. introductionprediction of the groundwater inow 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 inow 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 dierence in the prediction of groundwater inow by the solutions. kolymbas and wagner (2007)also presented the analytical solution for the groundwater inow, 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 inow into a circular tunnel can be found in the literature,they cannot be easily compared with each other because of the use of dierent notations, assumptions of boundary conditions, elevation reference datum,and solution methods. in this study, we shall revisit the closed-form analytical solution for the steady-state groundwater inow into a drained circular tunnel in a semi-innite aquifer with focus on two dierent 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 dierent boundary conditions are re-derived within a common theoretical framework by using the conformal mapping technique. the dierence in the water inow predictions among the approximate and exact solutions is re-compared to show the range of appli-cability of approximate solutions.2. denition of the problemconsider a circular tunnel of radius r in a fully saturated,homogeneous,isotropic,and semi-innite porous aquifer with a horizontal water table (fig.1).the surrounding ground has the isotropic permeability k and a steady-state groundwater ow condition is assumed.fig.1.circular tunnel in a semi-infinite aquifer.according to darcys law and mass conservation, the two-dimensional steady-state groundwater ow around the tunnel is described by the following laplace equation: (1)where=total head (or hydraulic head), being given by the sum of the pressure and elevation heads, or (2)p =pressure, =unit weight of water, z =elevation 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 e1 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. (1),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 (3)in the case of a drained tunnel, however, two dierent boundary conditions along the tunnel circumference can be found in the literature:(fig.1)(1)case 1:zero water pressure, and so total head=elevation head (el tani,2003) (4) (2)case 2:constant total head, ha(lei, 1999; kolymbas and wagner,2007) (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 these two dierent boundary conditions along the tunnel circumference, two dierent solutions for the steady-state groundwater rinow into a drained circular tunnel are re-derived in the next.3.analytical solutions3.1.conformal mappingthe ground surface and the tunnel circumference in the z-plane can be mapped conformally onto two circles of radius 1 and ,in the transformed -plane by the analytic function (fig.2) (verruijtandbooker,2000) （6）where a = h(1-2)/(1+2 ), h is the tunnel depth and is a parameter dened as or (7)then, eq. (1) can be rewritten in terms of coordinate - (8)by considering the boundary conditions, the solution for the total head on a circle with radius in the -plane can be expressed as (9)where c1, c2, c3 and c4 are constants to be determined from the boundary conditions at the ground surface and along the tunnel circumference.3.2.the surface boundary conditionthe constant c1 can be obtained by considering the boundary condition at the ground surface with =1 in the -plane, (10)fig.2. plane of conformal mapping.3.3.the tunnel boundary conditionthe other constants can be obtained by considering two dierent tunnel boundary conditions.(1)case 1:zero water pressure.by considering = aexp()in the -plane, the elevation head around the tunnel circumference can be expressed as (11a)or in the series form (verruijt,1996) (11b)and then applying the boundary condition of eq. (4) gives (12)so, (13)note that eq. (13) is the same form as eq. (4.1) in el tani(2003) for the case of h =0.(2) case 2:constant total head, ha.applying the boundary condition of eq. (5) gives (14)so, (15)3.4.the solution for groundwater inow the solution for the groundwater inow, which is the volume of water per unit tunnel length,into a drained circular tunnel can be obtained for two dierent cases as (16) (17)note that eq. (16) is the same solution as el tani (2003) with h =0,whereas eq. (17) is the same solution as kolymbas and wagner (2007). there is a clear dierence between eqs. (16) and (17): a(=h (1-2)/(1+ 2) in eq. (16) and ha in eq.(17) due to the dierent boundary conditions along the tunnel circumference. it is also noted that the solutions (16) and (17) are used for the case in which the water table is above the ground surface. if the groundwater table is below the ground surface,the groundwater level is used as the elevation reference datum.the solutions (16) and (17) should be used with h =0 and h = the groundwater depth (not tunnel depth).4.comparison with approximate solutionsfrom the exact solution eq. (17),the previous approximate solutions can be obtained with the assumption that the total head everywhere at the tunnel circumference is equal to the total head at (x = r, y = -h), i.e. ha =-h (lei, 1999; el tani, 2003).(1) approximate solution by assuming ha = -h.by simply assuming ha = -h and h =0,eq. (17) can be simplied as (18)where subscript a means approximate solution. eq. (18) was indicated as the solution by rat, schleiss, leiin table 1 of el tani (2003).(2) approximate solution in the case of hr (deep tunnel)for hr, we have h+, and hence eq. (18) can be further simplied as (19)eq. (19) was indicated as the solution by muskat, goodman et al.in table 1 of el tani (2003).4.1.dierence in water inow predictionsin order to investigate the dierence in water inow predictions among the exact and approximate solutions and the range of applicability of approximate solutions, the relative error, previously shown in fig. 3 of el tani (2003), are obtained again from or 1 and 2 show the dierences between q1 (case 1) andqa1, qa2 (approximate solutions of case 2) respectively. here, h = 0 is used, and so this case is that the groundwater level is at/below the ground surface.fig.3. diffierence among solutions (e1 t ani,2003)from fig. 3, 1 and 2 indicate that the approximate solutions, qa1 and qa2, overestimate the inow rate by about 1015% when r/h =0.5. interestingly the overestimation by the approximate solution qa1 increases drastically as r/h 1. this may because the term and as r/h 1. thus, the approximate solution qa2 seems to give better prediction of groundwater inow than qa1.since, the term as r/h1, q1 gives stable results. if h 0, however the term could cause instability of q1 as r/h1.this eect is investigated in the next.4.2.eect of h in the underwater tunnelthe eect of h on the water inow prediction in the underwater tunnel is investigated by using the approximate and exact solutions. fig. 4 shows the results of water inow with respect to r/h with dierent b (=h/h). the inow is obtained from eq. (16) for q1 or eq. (19) for qa2 considering ha = -h and h r. or the solid line represents the results for q1,whereas dotted line indicates the result for qa2.it can be seen from fig. 4 that the water inow increases with increasing b. for b =0.5 and 1, the inow rate by q1 increases greatly as r/h1 ,as expected. the approximate solution qa2 slightly overestimates the inow rate for r/h 0.6, but the results are stable.generally,for the tunnel with r/h 0.4 (h 2.5 r), the existing exact and approximate solutions can be used without considering the boundary condition along the tunnel circumference. for the tunnel with r/h 0.4, the exact solutions should be used with the correct consideration of the boundary condition along the tunnel circumference. the approximate solution qa2 seems to give stable results in the case that the water table is above the ground surface.fig.4. water inflow predictions with different b.5.conclusionssimple closed-form analytical solutions for the steady-state groundwater inow into a drained circular tunnel in a semi-innite aquifer have been revisited by re-deriving the solutions within a common theoretical framework for two dierent boundary conditions (one for zero water pressure and the other for a constant total head) along the tunnel circumference.the approximate solutions can be used for the tunnel with r/h 0.4 (h 2.5r). correctly estimating boundary condition along the tunnel circumference is shown to be important in a shallow drained circular tunnel. if the water table is above the ground surface, the approximate solution qa2 seems to be better for practical use.acknowledgementsthe rst author thanks two scholars for their invaluable comment on the correctness of eq. (9).references1 el tani, m., 2003. circular tunnel in a semi-innite aquifer.tunn.undergr.space technol.18(1),4955.2 kolymbas,d.,wagner,p.,2007.groundwater ingress to tunnelsthe exact analytical solution.tunn.undergr.space technol.22(1),2327.3 lei,s.,1999.ananalytical solution for steady ow into a tunnel. ground water37,2326.4 verruijt,a.,1996.complex variable solutions of elastic tunneling problems.geotechnical laboratory,delft university of technology.5 verruijt,a.,booker,j.r.,2000.complex variable analysis of mindlins tunnel problem. development of theoretical geomechanics. balkema,sydney,pp.322.半无限含水层中圆形排水隧道稳定地下水流涌水量计算的解析法摘要本文研究半无限含水层中圆形排水隧道稳定地下水流涌水量计算现有的解析法对比。提到了适用于隧道掌子面两个不同边界条件下(一个无水头压力，另外一个为恒定水头)现有的解决方案 。利用保角映射技术，在两种不同边界条件共同的理论框架下,重新推导出简单封闭条件下的解析解。涌水量的预测就变成研究边界条件的差异。强调浅排水圆形隧道隧道掌子面的边界条件的合理运用。2007 elsevier股份有限公司版权所有。关键词:解析解、隧道、地下水流量、半无限含水层1.论文简介隧道的排水系统设计和排水系统的环境影响评价都需要隧道的涌水量预测值。最近, el tani (2003) 在莫比乌斯（德国数学家）变换公式和傅立叶级数的基础上提出了地下水涌水的解析解。许多研究人员(muscat, goodmanet al., karlsrud, rat, schleiss, lei, 以及 lombardi),收集精确的和近似的解决方法, el tani (2003)列明了这些地下水涌水量预测方法的巨大差异。kolymbas 和 wagner (2007) 也提出了对于深隧道和浅隧道以及隧道掌子面和地表总水头不恒定同样有效地解析解。尽管在文献中可以找到数种圆形隧道中涌水量的解析方法，但是由于他们给使用的符号,假设的边界条件,参考资料标准和解决问题的研究方法使得他们彼此之间不能轻易进行对比。在本研究中, 我们应当通过集中研究隧道掌子面两个不同边界条件 (一个无水头压力，另外一个为恒定水头)重新梳理半无限含水层中圆形排水隧道稳定地下水流涌水量计算的现有解决方法。通过在两种不同边界条件共同的理论框架内利用保角映射技术，重新推导两种不同边界条件的解决方法。重新对比近似的涌水量预测方法与精确的方法的差异并给出近似的预测方法的适用范围。2.问题定义假定一个半径为r的圆形隧道处于一完全饱和、均质、各向同性、水位水平的多孔介质含水层中（图一）。假定周围土层渗透系数k各向同性，地下水流为稳定流。fig.1.半无限含水层中的圆形隧道根据达西定律和质量守恒定律，隧道周围的二维稳态地下水流由下面的拉普拉斯方程描述。 (1)其中，为总水头（或水头压力），等于压力水头与位置水头的和。 (2)p为压力，为水的重度，z为位置水头，它是某一高于或低于基准面的给定点到基准面的垂直距离。此处，地表被用作是参考基准面以考虑地下水位在地表上的情况。注意e1 tani (2003)将水平面作为参考基准面，而kolymbas和wagner (2007)使用了地面。为了解式 (1), 需要两种边界条件：一个在地表而另一个在隧道掌子面。在地表的边界条件 (y = 0)可确定的表达为 (3)在这个圆形隧道的例子中，隧道掌子面的两种不同边界条件同样可以在文献中发现(图1)。(1)例1：零水头压力,因此总水头=位置水头(el tani,2003) (4)(2)例2：总水头恒定，ha(lei, 1999; kolymbas 和 wagner,2007) (5)应该注意到公式(5)中边界条件假设为总水头恒定，而公式（4）中则给出的是沿隧道掌子面变化的总水头。通过考虑这两个隧道掌子面的不同边界条件，重新推导出两种圆形排水隧道中稳态地下水涌水量计算不同的解决方案如下。3.解析解3.1保角映射在z平面的地面和隧道掌子面可以被保角映射到两个半径分别为1和的圆上, 由解析函数在这个轴上转变 (图2)(verruijtandbooker,2000年)。表2 保角映射平面 （6）其中a = h(1-2)/(1+2 ),h为隧道的深度, 为一个参数，被定义如下： 或 (7)则，公式（1）可以依据-坐标系改写为 (8)考虑到边界条件,轴上半径为的圈上总水头的解可表达为： (9)其中，c1,c2,c3与c4为常数,取值取决于地表和隧道掌子面的边界条件。3.2地面的边界条件常数c1可以通过考虑地表的边界条件,即轴上= 1，确定。 (10)3.3隧道的边界条件另一个常量可以通过考虑两种不同的隧道的边界条件确定。（1）例 1：无水头压力考虑到在轴上 = aexp(),隧道掌子面的位置水头可以表示为 (11a)或者以下面这系列的形式(verruijt,1996) (11b)然后应用公式（4）给的边界条件 (12)则， (13)注意公式（13）与el tani(2003)公式(4.1) h = 0时的表达式相同。（2）例2:总水头恒定，ha。应用公式（5）给的边界条件 (14)则， (15)3.4涌水量计算方法圆形排水隧道单位长度涌水量的体积可有两种不同情况获得： (16) (17)注意公式(16)是el tani (2003)h = 0的表达式,而公式(17) 与kolymbas和wagner (2007)是相同的解决方法。公式(16)和公式(17)之间一个明显的区别就是：公式(16)中是a(=h (1-2)/(1+ 2)而公式(17)中的是ha，这是由于隧道掌子面的边界条件不同。还应指出, 地下水位在地面之上的这种情况也用到公式(16)与(17)。如果对地下水位低于地面，则将潜水面作为参考基准面。公式(16)与(17)在h = 0和h为地下水埋深(不是隋道深度)时适用。4.与近似解的比较由公式(17),由隧道掌子面的总水头处处等于(x = r, y = -h)的总水头也即ha =-h的假定，先前的近似解可确定(lei, 1999; el tani, 2003).(1) 通过假定ha = -h的近似解。通过简单假设ha = -h 且 h =0。公式(17)可以简化为 (18)其中下标a表示近似解。