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Analytical solution for the limiting drainage of a mountain tunnel based on area-well theory
Tunnelling and Underground Space Technology 84 (2019) 22–30
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Analytical solution for the limiting drainage of a mountain tunnel based on area-well theory
T
Pan Chenga, Lianheng Zhaob, , Zhenbing Luoa, Liang Lib, Qiao Lia, Xiong Denga, Wenqiang Penga ⁎
a b
Department of Applied Mechanics, College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, China School of Civil Engineering, Central South University, Changsha, China
ARTICLE INFO
ABSTRACT
Keywords: Groundwater limiting drainage Groundwater table drawdown Area-well theory Ecological balance Tunnel
Water seepage from a tunnel may induce the drawdown of the groundwater table and destroy vegetation and the ecological environment in the tunnel area. One of the primary objectives in tunnel engineering is to develop effective methods to estimate the amount of the limiting drainage of a mountain tunnel. Based on the area-well theory, a groundwater seepage model of a tunnel is established. Due to the model and groundwater dynamics, the equation describing the relationship between the amount of water inflow and drawdown of the groundwater table for a tunnel is derived under unsteady flow conditions. After the ecological groundwater table for vegetation is introduced, a method of calculating the amount of the limiting drainage for a mountain tunnel is proposed to prevent the drawdown of groundwater table from exceeding the ecological groundwater table required to support vegetation. According to the proposed method, the three-dimensional form of the groundwater drawdown funnels and the influence scope of tunnel water inflow are determined. The rainfall recharge amount Wt within the influence scope is obtained based on the rainfall infiltration coefficient method, and the value is compared to the total groundwater discharge Q. These values Wt and Q are then equated through adjusting the unit groundwater inflow q0. The values less than or equal to q0 can be determined as the limiting drainage value of groundwater for the tunnel, and the groundwater table will not drop and balances the groundwater and the vegetation ecological environment. The results and case study show that the proposed method exhibits good agreement with actual conditions and is suitable for use in tunnel construction.
1. Introduction Highways and railways are often prioritized for economic development (Li et al., 2017), e.g., the “the Belt and Road” strategy proposed by the Chinese government aims to boost economic development along these routes and achieve common prosperity. However, during highway and railway construction, especially in mountain areas, tunnels are inevitably encountered. Generally, a tunnel is buried underground and often surrounded by groundwater. Nevertheless, this groundwater can be detrimental to underground engineering. If treated inappropriately, the groundwater seepage could cause groundwater table drawdown and rock collapse, even destroy the ecological environment of the tunnel site (Li et al., 2017). For example, after the Taoshan Tunnel in the JingTong (Beijing-Tongliao) railway was excavated, a reservoir with a volume of 3.0 × 104 m3 near the tunnel dried up, and the water source for a village on the mountain was cut off (Ministry of Railways of the People's Republic of China, 2001). Additionally, a large quantity of water burst into the Zhongliangshan Tunnel of the Xiang-Yu
⁎
(Xiangyang-Chongqing) railway and created 29 regions of collapse at the top of the mountain. Moreover, 48 springs dried up, which seriously endangered the drinking water and irrigation of hundreds of households near the tunnel site (Gan et al., 2008). If the discharge of groundwater near the tunnel is too large, serious consequences similar to those described above can occur. However, if the groundwater is not discharged, that is, measures are taken to contain the groundwater, the above consequences will not happen, but the tunnel lining will be subjected to an enormous water pressure. If the water pressure is too large, the lining can break. The hydrostatic pressure at the Yuanlianshan Tunnel of the Yu-Huai (ChongqingHuaihua) railway has reached 4.42–4.6 MPa (Gan et al., 2008), while the groundwater pressure bearing capacity for a normal lining is 0.3–0.7 MPa (Wang et al., 2009). Thus, controlled drainage principle is applied when mountain tunnels are located in areas with high groundwater head. The controlled drainage principle focuses on decreasing the water inflow via grouting methods or other measures, and the groundwater that seeps into the tunnel is discharged. This principle
Corresponding author at: School of Civil Engineering, Central South University, Changsha, Hunan 410075, China. E-mail address: [email protected] (L. Zhao).
https://doi.org/10.1016/j.tust.2018.10.014 Received 16 April 2018; Received in revised form 20 September 2018; Accepted 28 October 2018 0886-7798/ © 2018 Elsevier Ltd. All rights reserved.
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can control the drawdown of the groundwater table and prevent the ecological environment and the tunnel lining from being destroyed. Some experts have studied the problem of the controlled drainage principle. For instance, the standard of allowable water seepage for Norwegian subsea tunnel was 0.432 m3·m−1·d−1 (Grφv, 2002; Eivind and Bjφrn, 2007; Lu et al., 2005), and the allowable water seepage in the Seikan Tunnel in Japan was 0.2736 m3·m−1·d−1 (Kitamura, 1986). The Oslofjord Tunnel is a subsea tunnel with sections under residential and recreational areas, and the allowable water inflow was defined as 0.288 m3/(m·d) (Blindheim and Øvstedal, 2002). In engineering practice in China, such as for the Yuanliangshan Tunnel (Zhang et al, 2007), the Qiyueshan Tunnel (Zhu, 2011), the Zhongliangshan Tunnel (Jiang, 2012), etc., the allowable amount of water seepage is determined based on protecting the ecological environment. However, the limiting drainage value is based on practical experience and lacks a consistent calculation method; thus, it is not suitable for other tunnels. To prevent the lining structure from being destroyed by the hydraulic pressure of groundwater, He et al. (2010) proposed a method for determining the limiting drainage. However, the method cannot ensure that the ecological environment will not be influenced. Cheng et al. (2013, 2014) introduced the concept of ecological water requirements of vegetation and presented the method for determining the limiting drainage criterion for water discharge from a tunnel during steady flow situations based on the groundwater dynamics method. However, there is no one systematic and mature method available to calculate the amount of the groundwater drainage. Thus, a method for calculating the amount of the drainage water of a tunnel is proposed in this study based on the groundwater and ecological balance.
known, the groundwater table can be controlled above the minimum value by limiting the drainage of the groundwater from the tunnel and reaching a balance between groundwater and the ecological environment. The species and growth state of vegetation are influenced by many factors, such as climate, rainfall, soil and groundwater. For a mountain tunnel, the dynamic balance among the vegetation, the ecological environment and these factors forms after a long period of evolution, and the ecological environment of vegetation will not be destroyed unless geological disasters or extreme climate effects occur. However, the original groundwater discharge and runoff conditions are changed by tunnel construction. If the amount of groundwater discharge exceeds the infiltration replenishment of the groundwater, the groundwater table will draw down and form a drainage funnel. In general, an appropriate groundwater depth exists at which typical vegetation can thrive in a certain region, and this groundwater depth is named the ecological groundwater table (Wan et al., 2005). In the influence scope of the funnel, if the drawdown of the groundwater table exceeds the ecological groundwater table, the vegetation can be destroyed due to a lack of water. Maitre et al. (1999) studied the root system and root depth of shrubs and woody plants. The root depth can reach 60 m for some plants, and vegetation is dependent on groundwater. Plant growth and the species assemblage can be influenced by the availability of groundwater, which can be in the form of a water table declining at a rate faster than root growth or an alteration in the annual fluctuations of the water table. Groundwater extraction can also result in these changes. Horton et al. (2001) investigated the physiological response of three vegetation species to groundwater availability along different groundwater depth gradients. The result showed that to effectively conserve or restore forests, it is necessary to reduce human activities that lower the groundwater beyond the rooting depth of tree species, because deep groundwater has a greater negative physiological impact on vegetation. Newman et al. (2006) proposed that the relationship between groundwater and vegetation growth should be considered according to the viewpoint of ecohydrology. Jin et al. (2007), Yan et al. (2007), Zhao et al. (2008), Jin and Jin(2009), Jin (2010), Cheng et al. (2012) studied the relationship between the groundwater table and vegetation. They found that the groundwater controls the vegetation growth conditions and that vegetation has a suitable groundwater table depth. The above analysis shows that the vegetation is closely related to the groundwater table. If the groundwater table drawdown caused by tunnel seepage does not exceed the ecological groundwater table, the vegetation can grow normally. Thus, the ecological groundwater table in a certain region can be determined by in situ investigation. Based on the relationship between groundwater inflow and the drawdown of the groundwater table, the tunnel water inflow can be determined to ensures that the drawdown of the groundwater table does not exceed the depth of the ecological groundwater table to avoid the destruction of the ecological environment.
2. Analysis model and method From the theory and practice of groundwater inflow, in well flow theory (Xue, 1997; Chen and Lin, 1999), the groundwater table (GWT) will drop after water is pumped from a well, and the drawdown area will be funnel-shaped. The situation for a tunnel will be similar that for a well after groundwater drainage from the tunnel. The funnel-shaped area will form at the top of the tunnel area, as shown in Fig. 1. The initial groundwater table of the tunnel is horizontal. Then, the groundwater table drops after a large amount of water being discharged out of the tunnel, which forms a drainage funnel. Inside the funnel area, vegetation is abundant before tunnel excavation. The vegetation will decline and even die due to a lack of water when the groundwater table drops after the construction of the tunnel. Thus, if the minimum groundwater table that can ensure the normal growth of vegetation is
GWT after drainage
3. Method of calculating drawdown of groundwater table for a tunnel
Original GWT
3.1. Water inflow forecasting for a tunnel Many methods, such as groundwater dynamics methods, empirical formulas and hydrogeological analogue method, are available for estimating tunnel water inflow. The first two methods above have been widely applied in engineering practice because the physical meanings of each parameter are clear (Huang and Yang, 1999; Wang et al., 2004; Zhu and Li, 2000; Li et al., 2012). For partially penetrating tunnel in a phreatic aquifer, the Goodman formula and the Kuniaki Sato method (Zhu and Li, 2000) can be used to calculate the maximum water inflow q0 (m3·m−1·d−1) and the water inflow (less than the maximum flow) at a certain time t.
root system
aquifer
tunnel Fig. 1. Calculation model. 23
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q0
0
=
q0
1
=
2 KH ln(2H r0 )
(1)
2 mKH
{
ln tan
(2H r0) 4h c
r
cot 4h0
c
}
(2)
K 2t r0 Kr0 µB q0
qt = q0
(3)
where K is the hydraulic conductivity of the aquifer; H is the distance from the groundwater table to the centreline of the tunnel; r0 is the radius of the tunnel; m is the conversion coefficient, which is generally taken as 0.86; hc is the thickness of the aquifer; ε is an empirical factor, which is generally taken as 12.8; μ is the specific yield of the aquifer; and B is the width of the tunnel before being lined. For the tunnel buried deeply in a confined aquifer, the Hiroshi Oshima, Ochiai Toshiro and empirical formula recommended in the standard “Code for Hydrogeological Investigation of Railway Engineering” (National Railway Administration of the People's Republic of China, 2015) are used to predict water inflow.
q0
2
=
2 K (H r0 ) m ln[2(H r0 )/r0 ]
(4)
q0
3
= 0.0255 + 1.9224KH
(5)
Fig. 2. Sketch map of the vertical section for the tunnel.
well that is pumping from the well-group area. Water seepage in a tunnel is similar to pumping from a well. Thus, the calculation methods for water inflow and the drawdown of groundwater after drainage are generally based on well flow theory. However, the assumptions in well flow theory are that the diameter of the well is infinitely small and the well is vertical. In this case, a groundwater drawdown funnel appears near the mouth of the well. Because a tunnel is similar to a horizontal well, the drawdown funnel will appear at the top of the tunnel. However, the length of the tunnel is much larger than the width, when computing the drawdown of groundwater, it differs from that for a single well, and the tunnel wall can be assumed to be composed of an infinite number of pumping wells. Then, considering the superposition of drawdown for the infinite number of pumping wells, as shown in Fig. 2, a method of calculating groundwater drawdown for a tunnel can be obtained. Therefore, the area-well method is introduced to calculate the drawdown of groundwater in this study. After the drawdown of the groundwater table is calculated, based on the perspective of protecting the vegetation and ecology, the groundwater table is allowed to decrease to a certain depth that enables the vegetation to survive, and a reasonable drainage quantity is determined.
Because of the diversity and complexity of engineering geological and hydrogeological conditions for different tunnels, there is no general method of predicting the exact water inflow. Thus, it is better to use various methods to predict and obtain the average water inflow q0AVG based on the actual situation at a specific tunnel. 3.2. Relationship between water inflow and drawdown of groundwater table Calculating groundwater table drawdown is a classical problem in groundwater dynamics. For steady flow and unsteady flow, the equations for the drawdown funnel curve under confined water and phreatic water in well flow are established based on the Dupuit well-flow model (Xue, 1997; Chen and Lin, 1999). Generally, groundwater flow is unsteady after a tunnel is excavated. Thus, the unsteady flow model is applied in this paper. The following formulas have been widely applied to calculate the unsteady flow of groundwater (Xue, 1997; Chen and Lin, 1999):
s=
q W (u ) 4 T
(6)
r 2µ 4Tt
(7)
3.2.1. Assumptions (1) The surrounding tunnel rock is homogeneous and isotropic. (2) The tunnel cross section is circular, and the tunnel is cylindrical in shape. (3) The water inflow at each point on the tunnel wall is uniform, stable, and equal. (4) The fluid is incompressible and can be instantaneously released.
and
u=
W (u) =
x
e u
x
dx =
0.577216
( 1)n
ln u + u n= 2
3.2.2. Derivation process of groundwater table drawdown for the tunnel The tunnel wall is expanded to a plane, as shown in Fig. 3. The plane is assumed to be composed of an infinite number of tiny water seepage surfaces, and the amount of water inflow is equal everywhere on the surface. A single surface can be regarded as a well-point, and the influence of a single surface is equal to that of one well-point. Therefore,
un n ·n!
(8)
where s is the drawdown of the groundwater table (m); q is the tunnel water inflow per unit length (m3·m−1·d−1); T is the transmissibility coefficient of the aquifer (m3·m−1·d−1); a is the coefficient of pressure conductivity, where a = T/μ, and μ is the specific yield; t is time from the start of leakage (d); R is the radius of the tunnel (m); and W (u) is the well function. Theis formula can be directly applied to a single well (Chen and Lin, 1999). For multiple wells, a superposition method can be used to obtain reliable results. The superposition method applied in this paper is called the area-well method, and it can be explained as follows (Chen and Lin, 1999). There are many single wells constitute various geometric shapes, such as rectangular and circular. When the amount of the water inflow from a single well is similar to that of other wells, the well-group can be regarded as an entity, and the single well can be regarded as an area
Fig. 3. Sketch map of area well. 24
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by integrating all the water seepage surfaces in the plane, the drawdown at an arbitrary point P can be obtained. We assume that a tunnel exists with length L and radius r0 and transform the tunnel wall into a plane of length L and width 2πr0, assuming that 2 lx = L and 2ly = 2πr0. The amount of water inflow per linear meter in the tunnel is q, and total water inflow Qa for the plane can be expressed as follows.
2
4a
we can obtain the following equation.
d =d
1
x + lx 4a
a
x lx 4a
0 x lx 4a
a
2d
e
2d
e
+
0
x + lx 4a
2d
e
(17) x 2dx .
x + lx 4a
x
erf
lx (18)
4a
1
(y y )2 4a
e
ly
y + ly
dy = a erf
erf
4a
y
ly (19)
4a
Substituting Eqs. (18) and (19) into (15) and setting T = aμ yields the following equation.
d 2
=
x
d
0
dx =
q 4 T
s=
(11)
e
r2
4a
x
t
·x
d
=
t
e
4a
0
d
t
e
r2 4a
0
dx dy 4 T
t
=
e
0
d =
dx dy 4 T
t
e
(x x )2 4a
0
·
e
(y y )2 4a
s=
0
lx lx
e
(x x )2 4a
dx ·
1
lx lx
e
(y y )2 4a
ly
d
4a
(20)
4µ
1 0
{
t 4µ
erf
(
1 0
)
erf
(
x
lx 4at
)
ly + y
lx + x 4at
4at
1 0
ly + y
lx x 4at
( ) } td erf ( ) erf ( ) dt erf ( ) erf ( ) dt
× erf
( ) erf ( ) dt + erf ( ) erf ( ) dt +
1 erf 0
+
x + lx 4at
( ) y + ly 4at
lx + x 4at
1 0
4at
lx x 4at
y
erf
ly
4at
ly
y
4at ly
y
4at
(21) Then, a function is defined as follows.
S ( , )=
1 0
erf
erf
d
(22)
The drawdown equation for an arbitrary point in the rectangular well area can be expressed as follows.
d
S=
t Ar (l x , l y , x , y , at ) 4µ
(23)
where
The groundwater table drawdown at the point P (x, y) that is influenced by the total inflow through the water seepage surface can be obtained by taking the integral of the entire planar area.
1
y
erf
4a
lx 4a
t
=
(14)
t
y + ly
x
erf
and substitute variables into Eq. (20) to yield the following expression.
(13)
(x x )2 (y y )2 4a
x + lx 4a
erf
0
Next, we set
(12)
d
t
4µ
× erf
r2
For an arbitrary point P (x, y) in the tunnel area, when the amount of water inflow through the water seepage surface is Qa, the groundwater table drawdown s at point P that is influenced by the water seepage surface can be calculated as follows. Taking the arbitrary point (x', y') as the centre and selecting the differential surface dx'dy', the total amount of water inflow through the differential surface area is ωdx'dy'. The groundwater table drawdown ds at point P (x, y) that is influenced by the differential surface dx'dy' can be obtained based on Eq. (13), as follows.
4
2
· 4a d
In the same way, we can obtain Eq. (19).
For a well point with a specific water inflow q in the plane, the drawdown at the arbitrary point in the infinite plane that is influenced by the well-point can be expressed as follows.
s=
2
2
x + lx 4a
dx = a erf
ly
r2 4a
ds =
(x x )2 4a
e
lx lx
Therefore, the Eq. (8) can be transformed to following equation.
s=
x lx 4a e
1
e The Gaussian error function is expressed as erf ( ) = 0 Therefore, Eq. (17) can be transformed into the following formula.
where
x
dx =
2
r2 x= 4a
r2 4at
(x x )2 4a
lx e lx
=
Exchanging the variables in well functions Eq. (8) yields the following formula. Setting
x
(16)
(10)
r2 u= 4at
e
dx 4a
=
=
The coordinate system of the plane is shown in Fig. 3. The location of the differential surface in the seepage area is expressed by the coordinate system x'oy', and the location of the calculation point for drawdown of the groundwater table is expressed by the coordinate system xoy. Due to a = T/μ, Eq. (7) can be changed to
dx =
x) 4a
1
Therefore, the water inflow per unit area is given by the following equation.
qL Qa = = 2l x ·2l y 2l x ·2l y
(x
Therefore, the following formulas can be derived.
(9)
Qa = qL
x )2
(x
=
Ar (lx , l y , x , y, at ) = S +S
dy d
(
( lx
)+S ( )+S (
lx + x ly + y , 4at 4at
x ly + y , 4at 4at
lx
lx + x ly y , 4at 4at
x ly y , 4at 4at
)
) (24)
Here, Ar is the well function of the well in the rectangular area. The value of the function S*(α, β) can be obtained from reference (Chen and Lin, 1999) when the ranges of values are α = 0.02–3.0 and
(15)
By substituting variables and defining 25
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β = 0.02–3.0. However, the length of a tunnel is typically larger than the width; therefore, α and β often exceed the ranges given in the literature (Chen and Lin, 1999). Thus, a computational procedure in MATLAB is developed to determine S*(α, β) for arbitrary scopes of α, β. If we set (x = 0, y = 0) in Eq. (24), the drawdown equation of the centre of the tunnel is as follow.
sc =
t S µ
lx , 4at
(2) Geological surveys can be conducted to identify impervious boundaries, recharge boundaries, obvious catchment depressions, and other characteristics around the tunnel. If the scope of an impervious boundary or recharge boundary is smaller than that of the calculated influence scope of water inflow for the tunnel, the scope of the impervious boundary or recharge boundary is used to be the influence scope of the tunnel water inflow. If the scope of the impervious boundary or recharge boundary is larger than the calculated influence scope of water inflow for the tunnel, another method should be adopted. For example, if the tunnel passes through a catchment basin or depression, the area can be regarded as the catchment area of the tunnel, and the average width of the area can be considered the width of influence of tunnel water inflow.
ly (25)
4at
The drawdown is largest at the tunnel centre, and Eq. (25) is used to calculate the largest drawdown of the groundwater table. Based on Eq. (25), the t value can be achieved when sc and q are given. 4. The shape and expansion of the drawdown funnel
Therefore, if the tunnel corresponds to the above two situations, the influence scope of tunnel water inflow can be determined based on the above methods. Nevertheless, if the tunnel does not correspond these cases, e.g., if the boundary cannot be determined and there is no existing tunnel for reference, specific empirical formulas can be used to determine the influence scope. These formulas include the Kusakin formula, the Jihaerte formula and the formula recommended in the standard “Code for Hydrogeological Investigation of Railway Engineering” (National Railway Administration of the People's Republic of China, 2015). However, these formulas are typically associated with a single factor only and do not consider temporal changes or the expansion of the drawdown of the groundwater table. A drawdown funnel will form around the well after pumping for a long time. Based on the flow characteristic of the well, a drawdown funnel will form above the tunnel when the amount of drainage is too high. However, unlike the well, the tunnel length is much larger than the width; therefore, the drawdown funnel is shaped like an inverse elliptical cone. The elliptical scope of the drawdown funnel is the influence scope of the tunnel and the rainfall infiltration recharge area. Eq. (21) is the groundwater table drawdown formula, and the drawdown at the edge of the influence scope is 0, that is sc = 0, so Ar = 0. If x and y are known, the major and minor axes of the ellipse funnel can be determined, and the influence scope can then be obtained. The crosswise drawdown at the central of the tunnel can be expressed as follows when x = 0 and y = b.
Eq. (21) is the formula for groundwater drawdown, and the parameters for the tunnel are given as follows. The length of the water-bearing section of the tunnel is L = 1020 m; the equivalence circle radius of the tunnel is r0 = 6.85 m; the width of the tunnel is B = 12.8 m; the hydraulic conductivity of the surrounding rock is k = 0.0258 m/d; the specific yield is μ = 0.056; the groundwater hydraulic head is H = 450 m; and q0AVG = 9.12 m3·m−1·d−1 based on the Eqs. (1), (2), (4) and (5); T = 11.61 m2/d; a = T/μ = 207.3 m2/d. The drawdown of the groundwater table from 1 to 3 days is shown in Figs. 4–6. (1) Two-dimensional cross-sectional plot of the drawdown funnels for the tunnel (2) Three-dimensional plot of the drawdown funnel for the tunnel As shown in Figs. 4–6, the drawdown of the groundwater table increases, and the influence scope gradually expands as t increases. 5. Determination of the influence scope of tunnel water inflow There is no broadly used prediction method for the influence scope of tunnel water inflow. Hydrogeological analogue methods and geologic survey methods are recommended in the standard “Code for Hydrogeological Investigation of Railway Engineering” (National Railway Administration of the People's Republic of China, 2015). (1) If the engineering geological and hydrogeological conditions for a new tunnel are similar to those of an existing tunnel, the influence scope of the water inflow of the new tunnel can be based on that of the existing tunnel. This approach is called hydrogeological analogue method.
s =
t S 2µ
ly - b lx , 4at 4at
ly + b lx , +S 4at 4at
(26)
For s = 0, the minor axis b of the elliptical funnel is obtained. The lengthways drawdown at the central of the tunnel can be
0
Drawdown of GWT (m)
2
4
0 day 1st day 2nd day 3rd day
6
8
10
-60
-40
-20
0
Distance of cross direction (m)
20
40
Fig. 4. The plot of groundwater table (GWT) drawdown of the tunnel centre cross section. 26
60
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Fig. 5. Three-dimensional plot of the drawdown funnel for a tunnel on the 1st day.
expressed as follows when x = l and y = 0.
s =
t S 2µ
lx + l , 4at
ly 4at
+S
lx - l , 4at
portion of the total rainfall W. By dividing these values, we can obtain the rainfall infiltration coefficients p (Xiao et al., 2010), i.e., p = Wr/W. If p and W are known, the recharge amount can be obtained. In plain, hill and mountain areas, the rainfall infiltration coefficient p of an aquifer can be approximately determined from historical values, as shown in Table 1, when there is a lack of experimental data (Zhu and Li, 1995). Rainfall amount can be obtained according to local meteorological data. When the values of F, W and p are known, the recharge amount Wr of groundwater per day can be obtained as follows:
ly 4at
(27)
For s = 0, the major axis of the elliptical funnel l can be obtained. Strictly speaking, a steady flow is unable to form when tunnel seepage occurs without any recharge in an infinite aquifer. On the contrary, the groundwater hydraulic head decreases everywhere in the aquifer (Chen and Lin, 1999). However, if vertical recharge occurs in the aquifer, the amount of recharge increases with the expansion of the funnel area of the tunnel. When the amount of the vertical recharge equals the tunnel water inflow, the drawdown funnel will stop expanding. Then, the groundwater flow becomes a steady flow. Therefore, we take the distance from the central of the tunnel to the distance where groundwater table drawdown can be neglected as the influence scope of the tunnel water inflow. The influence scope can be obtained after the major and minor axes of the ellipse funnel are determined, and is expressed as follow:
Wr = p · W · F 365
Wr is the daily average amount of rainfall infiltration recharge (m3/d); p is the rainfall infiltration coefficient; W is the annual rainfall (m); and F is the rainfall infiltration area (m2). However, the amount of water inflow per unit time for the tunnel is q, and the total water inflow Q within t can be defined as follows.
Q = qLt
(28)
F = lb
(29)
F is the area of rainfall infiltration recharge.
(30)
When the total water inflow Q and the recharge amount Wt within t are known for specific groundwater table drawdown conditions, if Q = Wt, the discharge amount and recharge amount are balanced, and the groundwater table will no longer continue to fall. If Q > Wt, groundwater drainage can result in aquifer dewatering, which can negatively affect the ecological environment. Thus, the tunnel water inflow q should be properly reduced and to achieve a balance between recharge and discharge. If Q < Wt, the water inflow is sufficient and does not need to be addressed.
6. Analysis of the groundwater balance To keep the groundwater balance, the total rainfall infiltration recharge of groundwater must be equals to the total drainage groundwater. In general, the primary recharge source for mountain tunnel groundwater is rainfall. However, the recharge amount Wr is only a
Fig. 6. Three-dimensional plot of the drawdown funnel for a tunnel on the 3rd day. 27
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daily recharge is Wr = 339.1 m3/d according to Eq. (29), and the total recharge is Wt = 3.39 × 103 m3 for 10 d.
Table 1 Experimental data for the rainfall infiltration coefficient. Strata
p
Strata
p
Silty clay Silty soil Silty sand Fine sand Medium sand Coarse sand Round gravel (sand) Pebble (sand) Intact rock
0.01–0.02 0.02–0.05 0.05–0.08 0.08–0.12 0.12–0.18 0.18–0.24 0.24––0.30 0.30–0.35 0.01–0.10
Relatively intact rock Relatively broken rock Broken rock Extremely broken rock Very weak karst development Weak karst development Karst medium development Karst development
0.10–0.15 0.15–0.18 0.18–0.20 0.20–0.25 0.01–0.10 0.10–0.15 0.15–0.20 0.20–0.50
(6) Analysis of groundwater balance From above analysis, Q > Wt, which means that for a long time, the aquifer will be dewatered, and vegetation could be destroyed under these conditions. Therefore, the amount of drainage water should be reduced to achieve the balance between drainage and recharge. (7) Determination method of the amount of limiting drainage to maintaining a groundwater balance Due to the above analysis, the amount of water inflow drainage is too high to maintain a groundwater balance; therefore, the water inflow q0 should decrease until the amount of drainage water equals the amount of recharge water, i.e., Q = Wt, and the water inflow q0 is reasonable amount of drainage water. Because the scope of the q0 is unknown before the calculations, and the computational efficiency will seriously decrease if the searching scope is too large. Therefore, it is better to select several values of q0 and determine a preliminary scope, and then perform loop computing to obtain a reasonable q0. In this paper, values of q0 = 2.7 m3·m−1·d−1, 2.86 m3·m−1·d−1 and 3.0 m3·m−1·d−1 were separately used to assess the relationships between drainage and recharge, drainage and time, and drainage and influence scope of water inflow, as shown in Figs. 7–9. Fig. 7 illustrates that a q0 value between 2.86 and 2.88 m3·m−1·d−1 can make Q equal Wt. Thus, the search interval for loop computing is set from 2.86 m3·m−1·d−1 to 2.88 m3·m−1·d−1 to determine the exact q0 value. According to Fig. 8, when q0 = 2.86–2.88 m3·m−1·d−1, the t value is in the range of 120–125 d. Similarly, the major and minor axes of tunnel drawdown funnel are in the range of l = 1450–1480 m, and b = 1150–1200 m, respectively, according to Fig. 9. However, Fig. 7 cannot accurately reflect the q0 values, so the scopes of the q0, t, l and b values should be appropriately expanded to avoid the result exceeding the scope. As shown in Figs. 7–9, in the case of a given drawdown, the time required to achieve a given depth and influence scope of water inflow gradually decreases as the unit amount of water inflow increases. Moreover, total groundwater discharge and recharge also decrease. The amount of total discharge is related to the unit discharge and time, although the unit discharge increases, but the time required to reach a given depth is reduced, and the total drainage decreases. However, the recharge amount is related to the influence scope of water inflow, and the decreasing scope causes the recharge amount to decrease. Because the decreasing range of the total recharge amount is greater than the
7. Case study A case study was investigated for a tunnel with a total length of 850 m and an aquifer composed of mudstone and siltite. The groundwater hydraulic head is H = 180 m. The parameters of the aquifer are as follows: the aquifer thickness is hc = 380 m, the hydraulic conductivity K = 0.0386 m/d, the storage coefficient is μ = 0.048, and the transmissibility coefficient is T = 13.51 m2/d. The equivalent circle radius of the tunnel cross section r0 is 5.94 m, and the width of the tunnel is B = 11.6 m. Based on the investigation and analysis of the vegetation and ecological conditions at the tunnel site, the maximum allowable drawdown of the groundwater table is 20 m. (1) The amount of water inflow without plugging Based on Eqs. (1), (2), (4) and (5), the water inflow of the tunnel is q0-0 = 12.80 m3·m−1·d−1, q0-1 = 9.10 m3·m−1·d−1, q03 −1 −1 ·d , and q0-3 = 13.89 m3·m−1·d−1. Thus, the average 2 = 8.92 m ·m value q0AVG = 11.18 m3·m−1·d−1 is considered the water inflow. (2) The time t when the allowable drawdown is reached The drawdown of the groundwater table at the tunnel centre reaches 20 m after time t based on Eq. (3). Therefore, the following equation can be established: qt = 11.18–0.0014 t. Considering Eq. (25) and given that sc = 20 m, t can be determined. Taking the integral, a value of t = 10 d is obtained. Thus, the drawdown of groundwater table at the central tunnel will reach 20 m after 10 d. (3) Influence scope of tunnel water inflow
5
3
Drainage Q and discharge Wt (×10 m )
After 10 d, sc < 0.0001 m at the location 341 m away from the central of the crosswise tunnel. The drawdown of the groundwater table can be neglected at this distance; therefore, the minor axis of the influence scope of the tunnel is set to b = 341 m. Similarly, the minor axis is l = 772 m. Thus, the rainfall infiltration recharge area is F = 7.73 × 105 m2 based on Eq. (28). (4) The total amount of groundwater inflow The amount of water inflow after 10 d, q10d = 11.17 m3·m−1·d−1 is obtained using Eq. (3), and q = (q0 + q10d)/2 = 11.18 m3·m−1·d−1. Thus, the total groundwater inflow after 10 d is Q = 9.50 × 104 m3 according to Eq. (30). (5) The total amount of rainfall infiltration recharge The average annual rainfall at the tunnel site is 1000 mm according to local meteorological data, and the rainfall infiltration recharge area is 7.73 × 105 m2. The rock surrounding the tunnel is fractured, so the rainfall infiltration coefficient p is selected as 0.16 from Table 1. The
4.0 3.9 3.8 3.7 3.6 3.5 3.4 3.3 3.2 3.1 3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.65
Q
Wt 2.70
2.75
2.80
2.85
2.90
2.95 3
-1
3.00
-1
Unit water inflow of tunnel q0 (m .m .d ) Fig. 7. Relationship between total drainage and recharge. 28
3.05
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The method presented is used to calculate the limiting drainage of the Yuanliangshan Tunnel (Jiang, 2005), and the allowable drawdown is set to 20 m. Based on calculations, when q0 = 2.70 m3·m−1·d−1, QWt = 1.8 × 103 m3, which is able to maintain a sufficient groundwater balance. However, the limiting drainage standard of the Yuanliangshan Tunnel is 5.0 m3·m−1·d−1 (Wang et al., 2005) which is larger than the calculated value in practical engineering, and result in groundwater table drawdown, ground subsidence and soil erosion problems (Gan et al., 2008). If the limiting drainage is lower than 2.70 m3·m−1·d−1, the problems will be avoided, and the vegetation and ecological environment can be effectively protected. 8. Conclusions The ecological groundwater table of vegetation is introduced in this study, and the limiting drainage of a tunnel to maintain a balance between groundwater and the ecological environment is analysed based on the groundwater dynamic method that considers groundwater recharge by rainfall. The main conclusions of the study are as follows:
Fig. 8. Relationship between water inflow and t. 1550
(1) Because the length of a tunnel is much larger than the width, the superposition of the influence of water seepage should be considered. Based on the area-well theory, a groundwater seepage model of a tunnel is established. Due to the model and groundwater dynamics, the equation describing the relationship between the amount of water inflow and drawdown of the groundwater table for a tunnel is derived under unsteady flow conditions. Due to the formula, two-dimensional and three-dimensional plots of groundwater table drawdown are created, and the plots show the intuitional acquaintance of the characteristics of the drainage funnel. (2) The method for determining the limiting drainage value of groundwater is presented. This limit must be not being exceeded to maintain the balance between groundwater and the vegetation environment. When the allowable drawdown of the groundwater table sc, the time t when the maximum allowable drawdown is reached, and the rainfall infiltration recharge area F are determined by calculating, the total amount of rainfall infiltration recharge Wt is obtained. Based on the comparation of Wt and the total groundwater drainage Q, the analytical solution for the limiting drainage of groundwater q0 can be obtained. If Wt ≥ Q, the amount of recharge is larger than the discharge, and the groundwater table will not drop; therefore, the vegetation ecological environment will not be destroyed. In this case, the initial water inflow by forecasting is the limiting drainage mount of the tunnel. However, if Q > Wt, the drawdown of groundwater table will exceeds the ecological groundwater table of vegetation and causing vegetation destroy, and the water inflow q0 of the tunnel must be gradually decreased, and the calculation must be repeated until Wt ≥ Q, then the value of water inflow is the limited drainage of groundwater for the tunnel, and the local vegetation and ecological environment will be protected and preserved. (3) The results and case study show that the proposed method exhibits good agreement with actual conditions and is suitable for use in tunnel construction.
1500
Influence scope (m)
1450
l
1400 1350 1300 1250 1200 1150 1100 2.65
b 2.70
2.75
2.80
2.85
2.90
3
2.95 -1
-1
3.00
3.05
Unit water inflow of tunnel q0 (m .m .d )
Fig. 9. Relationship between water inflow and the influence scope.
decreasing range of the total discharge amount, a water inflow value exists that can equate the recharge amount and discharge amount. Based on calculations, when q0 = 2.872438 m3·m−1·d−1, t = 124 d, Q = 29361.0 m3, Wt = 29405.9 m3, and the recharge amount is slightly larger than the discharge amount, with a difference of only 442.6 m3. When the difference value is converted to daily discharge over the total length of the tunnel, there is only 3.6 m3/d, which can be considered negligible. Typically, when the resulting numbers of q0 are down to the 6th or even more decimal points, the difference between Q and Wt is approximately equal to zero. However, such exact q0 value have no engineering significance, and a resulting q0 value down to the 2nd decimal point is more reasonable. If the resulting numbers of q0 is down to the 2nd decimal point, the difference between Q and Wt will reach an order magnitude of 103, even though in this situation, the daily discharge amount is only 8.06 m3/d, which can be also ignored. Thus, related measures can be terminated under the condition that |Q Wt | < 1000 . In this case, when q0 = 2.87 m3·m−1·d−1, the total groundwater recharge is approximately equal to the total groundwater recharge, and a groundwater balance is achieved. Therefore, the drawdown of the groundwater table will not continually decrease when it reaches 20 m, and the local vegetation and ecological environment will be protected and preserved. Of course, if the limiting drainage is less than 2.87 m3·m−1·d−1, groundwater recharge will be larger than the groundwater discharge, and the groundwater table drawdown will be smaller than 20 m, which is more beneficial for vegetation and the ecological environment. Therefore, the limiting drainage for the present case can be determined as q0 < 2.87 m3·m−1·d−1.
Acknowledgements This research project was jointly supported by the National Natural Science Foundation of China (51809271, 61505259, 51478477, 11572349, 11602299) and the Scientific Research Program of National University of Defense Technology (ZK2017-03-40). Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.tust.2018.10.014. 29
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Blindheim, O.T., Øvstedal, E., 2002. Design principles and construction methods for water control for subsea road tunnels in rock. In: Publication No.12, [s. n.], 43-49, Oslo. Chen, C.X., Lin, M., 1999. Groundwater Dynamics. China University of Geosciences Press, Wuhan. Cheng, D.H., Wang, W.K., Hou, G.C., Yang, H.B., Li, Y., Zhang, E.Y., 2012. Relationship between vegetation and groundwater in Mu Us Desert. J. Jilin Univ. (Earth Sci. Ed.) 42 (1), 184–189. Cheng, P., Li, L., Zou, J.F., Zhao, L.H., Luo, W., 2013. Determination method for water discharge of tunnel based on the ecological water requirement of vegetation. J. China Railway Soc. 35 (07), 107–113. Cheng, P., Zhao, L.H., Li, L., Zou, J.F., Luo, W., 2014. Limiting drainage criterion for groundwater of mountain tunnel. J. Cent. South Univ. 21 (12), 4660–4668. Eivind, G., Bjφrn, N., 2007. Subsea tunnel projects in hard rock environment in Scandinavia. Chin. J. Rock Mech. Eng. 26 (11), 2176–2192. Gan, K.R., Jiang, S., Li, Y.H., 2008. Environmental farm and counter measures for water outburst in mountain tunnel construction. J. Railway Eng. Soc. 9, 71–76. Grφv. E., 2002. Introduction to water control in Norwegian tunnelling. In: Publication No. 12. [s. n.], 5-11, Oslo. He, B.G., Zhang, Z.Q., Zhu, Y.Q., 2010. Study on the reference of limited discharge in subject to mountain tunnel. In: The Second National Conference on Engineering Safety and Protection. Chinese Society for Rock Mechanics and Engineering of Engineering Safety and Protection Branch, pp. 111–117. Horton, J.L., Kolb, T.E., Hart, S.C., 2001. Physiological response to groundwater depth varies among species and with river flow regulation. Ecol. Appl. 11 (4), 1046–1059. Huang, T., Yang, L.Z., 1999. A prediction study of water-gush yield in fractured tunnels under coupling between seepage and stress. J. China Railway Soc. 21 (6), 75–80. Jiang, J., 2012. Mountain Tunnel Lining Stress Mechanism and Structure Design in High Pressure and Rich Water Stratum. Chongqing Jiaotong University, Chongqing. Jiang, Z.X., 2005. Interaction between tunnel engineering and water environment. Chin. J. Rock Mech. Eng. 24 (1), 121–127. Jin, X.M., 2010. Quantitative relationship between the desert vegetation and groundwater depth in Ejina Oasis, the Heihe River Basin. Earth Sci. Front. 17 (6), 181–186. Jin, X.M., Jin, T., 2009. Study on the relationship between groundwater and vegetation growth in lower Heihe River region. Adv. Sci. Technol. Water Res. 29 (1), 1–4. Jin, X.M., Wan, L., Zhang, Y.K., 2007. A study of the relationship between vegetation growth and groundwater in the Yinchuan plain. Earth Sci. Front. 14 (3), 197–203. Kitamura, A., 1986. Technical development for the Seikan tunnel. Tunn. Undergr. Space Technol. 1 (3), 341–349. Li, S., Shi, S., Bu, L., 2017. China: Rail network must protect giant pandas. Nature 545 (7654) 289 289.
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Analytical solution for the limiting drainage of a mountain tunnel based on area-well theory
T
Pan Chenga, Lianheng Zhaob, , Zhenbing Luoa, Liang Lib, Qiao Lia, Xiong Denga, Wenqiang Penga ⁎
a b
Department of Applied Mechanics, College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, China School of Civil Engineering, Central South University, Changsha, China
ARTICLE INFO
ABSTRACT
Keywords: Groundwater limiting drainage Groundwater table drawdown Area-well theory Ecological balance Tunnel
Water seepage from a tunnel may induce the drawdown of the groundwater table and destroy vegetation and the ecological environment in the tunnel area. One of the primary objectives in tunnel engineering is to develop effective methods to estimate the amount of the limiting drainage of a mountain tunnel. Based on the area-well theory, a groundwater seepage model of a tunnel is established. Due to the model and groundwater dynamics, the equation describing the relationship between the amount of water inflow and drawdown of the groundwater table for a tunnel is derived under unsteady flow conditions. After the ecological groundwater table for vegetation is introduced, a method of calculating the amount of the limiting drainage for a mountain tunnel is proposed to prevent the drawdown of groundwater table from exceeding the ecological groundwater table required to support vegetation. According to the proposed method, the three-dimensional form of the groundwater drawdown funnels and the influence scope of tunnel water inflow are determined. The rainfall recharge amount Wt within the influence scope is obtained based on the rainfall infiltration coefficient method, and the value is compared to the total groundwater discharge Q. These values Wt and Q are then equated through adjusting the unit groundwater inflow q0. The values less than or equal to q0 can be determined as the limiting drainage value of groundwater for the tunnel, and the groundwater table will not drop and balances the groundwater and the vegetation ecological environment. The results and case study show that the proposed method exhibits good agreement with actual conditions and is suitable for use in tunnel construction.
1. Introduction Highways and railways are often prioritized for economic development (Li et al., 2017), e.g., the “the Belt and Road” strategy proposed by the Chinese government aims to boost economic development along these routes and achieve common prosperity. However, during highway and railway construction, especially in mountain areas, tunnels are inevitably encountered. Generally, a tunnel is buried underground and often surrounded by groundwater. Nevertheless, this groundwater can be detrimental to underground engineering. If treated inappropriately, the groundwater seepage could cause groundwater table drawdown and rock collapse, even destroy the ecological environment of the tunnel site (Li et al., 2017). For example, after the Taoshan Tunnel in the JingTong (Beijing-Tongliao) railway was excavated, a reservoir with a volume of 3.0 × 104 m3 near the tunnel dried up, and the water source for a village on the mountain was cut off (Ministry of Railways of the People's Republic of China, 2001). Additionally, a large quantity of water burst into the Zhongliangshan Tunnel of the Xiang-Yu
⁎
(Xiangyang-Chongqing) railway and created 29 regions of collapse at the top of the mountain. Moreover, 48 springs dried up, which seriously endangered the drinking water and irrigation of hundreds of households near the tunnel site (Gan et al., 2008). If the discharge of groundwater near the tunnel is too large, serious consequences similar to those described above can occur. However, if the groundwater is not discharged, that is, measures are taken to contain the groundwater, the above consequences will not happen, but the tunnel lining will be subjected to an enormous water pressure. If the water pressure is too large, the lining can break. The hydrostatic pressure at the Yuanlianshan Tunnel of the Yu-Huai (ChongqingHuaihua) railway has reached 4.42–4.6 MPa (Gan et al., 2008), while the groundwater pressure bearing capacity for a normal lining is 0.3–0.7 MPa (Wang et al., 2009). Thus, controlled drainage principle is applied when mountain tunnels are located in areas with high groundwater head. The controlled drainage principle focuses on decreasing the water inflow via grouting methods or other measures, and the groundwater that seeps into the tunnel is discharged. This principle
Corresponding author at: School of Civil Engineering, Central South University, Changsha, Hunan 410075, China. E-mail address: [email protected] (L. Zhao).
https://doi.org/10.1016/j.tust.2018.10.014 Received 16 April 2018; Received in revised form 20 September 2018; Accepted 28 October 2018 0886-7798/ © 2018 Elsevier Ltd. All rights reserved.
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can control the drawdown of the groundwater table and prevent the ecological environment and the tunnel lining from being destroyed. Some experts have studied the problem of the controlled drainage principle. For instance, the standard of allowable water seepage for Norwegian subsea tunnel was 0.432 m3·m−1·d−1 (Grφv, 2002; Eivind and Bjφrn, 2007; Lu et al., 2005), and the allowable water seepage in the Seikan Tunnel in Japan was 0.2736 m3·m−1·d−1 (Kitamura, 1986). The Oslofjord Tunnel is a subsea tunnel with sections under residential and recreational areas, and the allowable water inflow was defined as 0.288 m3/(m·d) (Blindheim and Øvstedal, 2002). In engineering practice in China, such as for the Yuanliangshan Tunnel (Zhang et al, 2007), the Qiyueshan Tunnel (Zhu, 2011), the Zhongliangshan Tunnel (Jiang, 2012), etc., the allowable amount of water seepage is determined based on protecting the ecological environment. However, the limiting drainage value is based on practical experience and lacks a consistent calculation method; thus, it is not suitable for other tunnels. To prevent the lining structure from being destroyed by the hydraulic pressure of groundwater, He et al. (2010) proposed a method for determining the limiting drainage. However, the method cannot ensure that the ecological environment will not be influenced. Cheng et al. (2013, 2014) introduced the concept of ecological water requirements of vegetation and presented the method for determining the limiting drainage criterion for water discharge from a tunnel during steady flow situations based on the groundwater dynamics method. However, there is no one systematic and mature method available to calculate the amount of the groundwater drainage. Thus, a method for calculating the amount of the drainage water of a tunnel is proposed in this study based on the groundwater and ecological balance.
known, the groundwater table can be controlled above the minimum value by limiting the drainage of the groundwater from the tunnel and reaching a balance between groundwater and the ecological environment. The species and growth state of vegetation are influenced by many factors, such as climate, rainfall, soil and groundwater. For a mountain tunnel, the dynamic balance among the vegetation, the ecological environment and these factors forms after a long period of evolution, and the ecological environment of vegetation will not be destroyed unless geological disasters or extreme climate effects occur. However, the original groundwater discharge and runoff conditions are changed by tunnel construction. If the amount of groundwater discharge exceeds the infiltration replenishment of the groundwater, the groundwater table will draw down and form a drainage funnel. In general, an appropriate groundwater depth exists at which typical vegetation can thrive in a certain region, and this groundwater depth is named the ecological groundwater table (Wan et al., 2005). In the influence scope of the funnel, if the drawdown of the groundwater table exceeds the ecological groundwater table, the vegetation can be destroyed due to a lack of water. Maitre et al. (1999) studied the root system and root depth of shrubs and woody plants. The root depth can reach 60 m for some plants, and vegetation is dependent on groundwater. Plant growth and the species assemblage can be influenced by the availability of groundwater, which can be in the form of a water table declining at a rate faster than root growth or an alteration in the annual fluctuations of the water table. Groundwater extraction can also result in these changes. Horton et al. (2001) investigated the physiological response of three vegetation species to groundwater availability along different groundwater depth gradients. The result showed that to effectively conserve or restore forests, it is necessary to reduce human activities that lower the groundwater beyond the rooting depth of tree species, because deep groundwater has a greater negative physiological impact on vegetation. Newman et al. (2006) proposed that the relationship between groundwater and vegetation growth should be considered according to the viewpoint of ecohydrology. Jin et al. (2007), Yan et al. (2007), Zhao et al. (2008), Jin and Jin(2009), Jin (2010), Cheng et al. (2012) studied the relationship between the groundwater table and vegetation. They found that the groundwater controls the vegetation growth conditions and that vegetation has a suitable groundwater table depth. The above analysis shows that the vegetation is closely related to the groundwater table. If the groundwater table drawdown caused by tunnel seepage does not exceed the ecological groundwater table, the vegetation can grow normally. Thus, the ecological groundwater table in a certain region can be determined by in situ investigation. Based on the relationship between groundwater inflow and the drawdown of the groundwater table, the tunnel water inflow can be determined to ensures that the drawdown of the groundwater table does not exceed the depth of the ecological groundwater table to avoid the destruction of the ecological environment.
2. Analysis model and method From the theory and practice of groundwater inflow, in well flow theory (Xue, 1997; Chen and Lin, 1999), the groundwater table (GWT) will drop after water is pumped from a well, and the drawdown area will be funnel-shaped. The situation for a tunnel will be similar that for a well after groundwater drainage from the tunnel. The funnel-shaped area will form at the top of the tunnel area, as shown in Fig. 1. The initial groundwater table of the tunnel is horizontal. Then, the groundwater table drops after a large amount of water being discharged out of the tunnel, which forms a drainage funnel. Inside the funnel area, vegetation is abundant before tunnel excavation. The vegetation will decline and even die due to a lack of water when the groundwater table drops after the construction of the tunnel. Thus, if the minimum groundwater table that can ensure the normal growth of vegetation is
GWT after drainage
3. Method of calculating drawdown of groundwater table for a tunnel
Original GWT
3.1. Water inflow forecasting for a tunnel Many methods, such as groundwater dynamics methods, empirical formulas and hydrogeological analogue method, are available for estimating tunnel water inflow. The first two methods above have been widely applied in engineering practice because the physical meanings of each parameter are clear (Huang and Yang, 1999; Wang et al., 2004; Zhu and Li, 2000; Li et al., 2012). For partially penetrating tunnel in a phreatic aquifer, the Goodman formula and the Kuniaki Sato method (Zhu and Li, 2000) can be used to calculate the maximum water inflow q0 (m3·m−1·d−1) and the water inflow (less than the maximum flow) at a certain time t.
root system
aquifer
tunnel Fig. 1. Calculation model. 23
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q0
0
=
q0
1
=
2 KH ln(2H r0 )
(1)
2 mKH
{
ln tan
(2H r0) 4h c
r
cot 4h0
c
}
(2)
K 2t r0 Kr0 µB q0
qt = q0
(3)
where K is the hydraulic conductivity of the aquifer; H is the distance from the groundwater table to the centreline of the tunnel; r0 is the radius of the tunnel; m is the conversion coefficient, which is generally taken as 0.86; hc is the thickness of the aquifer; ε is an empirical factor, which is generally taken as 12.8; μ is the specific yield of the aquifer; and B is the width of the tunnel before being lined. For the tunnel buried deeply in a confined aquifer, the Hiroshi Oshima, Ochiai Toshiro and empirical formula recommended in the standard “Code for Hydrogeological Investigation of Railway Engineering” (National Railway Administration of the People's Republic of China, 2015) are used to predict water inflow.
q0
2
=
2 K (H r0 ) m ln[2(H r0 )/r0 ]
(4)
q0
3
= 0.0255 + 1.9224KH
(5)
Fig. 2. Sketch map of the vertical section for the tunnel.
well that is pumping from the well-group area. Water seepage in a tunnel is similar to pumping from a well. Thus, the calculation methods for water inflow and the drawdown of groundwater after drainage are generally based on well flow theory. However, the assumptions in well flow theory are that the diameter of the well is infinitely small and the well is vertical. In this case, a groundwater drawdown funnel appears near the mouth of the well. Because a tunnel is similar to a horizontal well, the drawdown funnel will appear at the top of the tunnel. However, the length of the tunnel is much larger than the width, when computing the drawdown of groundwater, it differs from that for a single well, and the tunnel wall can be assumed to be composed of an infinite number of pumping wells. Then, considering the superposition of drawdown for the infinite number of pumping wells, as shown in Fig. 2, a method of calculating groundwater drawdown for a tunnel can be obtained. Therefore, the area-well method is introduced to calculate the drawdown of groundwater in this study. After the drawdown of the groundwater table is calculated, based on the perspective of protecting the vegetation and ecology, the groundwater table is allowed to decrease to a certain depth that enables the vegetation to survive, and a reasonable drainage quantity is determined.
Because of the diversity and complexity of engineering geological and hydrogeological conditions for different tunnels, there is no general method of predicting the exact water inflow. Thus, it is better to use various methods to predict and obtain the average water inflow q0AVG based on the actual situation at a specific tunnel. 3.2. Relationship between water inflow and drawdown of groundwater table Calculating groundwater table drawdown is a classical problem in groundwater dynamics. For steady flow and unsteady flow, the equations for the drawdown funnel curve under confined water and phreatic water in well flow are established based on the Dupuit well-flow model (Xue, 1997; Chen and Lin, 1999). Generally, groundwater flow is unsteady after a tunnel is excavated. Thus, the unsteady flow model is applied in this paper. The following formulas have been widely applied to calculate the unsteady flow of groundwater (Xue, 1997; Chen and Lin, 1999):
s=
q W (u ) 4 T
(6)
r 2µ 4Tt
(7)
3.2.1. Assumptions (1) The surrounding tunnel rock is homogeneous and isotropic. (2) The tunnel cross section is circular, and the tunnel is cylindrical in shape. (3) The water inflow at each point on the tunnel wall is uniform, stable, and equal. (4) The fluid is incompressible and can be instantaneously released.
and
u=
W (u) =
x
e u
x
dx =
0.577216
( 1)n
ln u + u n= 2
3.2.2. Derivation process of groundwater table drawdown for the tunnel The tunnel wall is expanded to a plane, as shown in Fig. 3. The plane is assumed to be composed of an infinite number of tiny water seepage surfaces, and the amount of water inflow is equal everywhere on the surface. A single surface can be regarded as a well-point, and the influence of a single surface is equal to that of one well-point. Therefore,
un n ·n!
(8)
where s is the drawdown of the groundwater table (m); q is the tunnel water inflow per unit length (m3·m−1·d−1); T is the transmissibility coefficient of the aquifer (m3·m−1·d−1); a is the coefficient of pressure conductivity, where a = T/μ, and μ is the specific yield; t is time from the start of leakage (d); R is the radius of the tunnel (m); and W (u) is the well function. Theis formula can be directly applied to a single well (Chen and Lin, 1999). For multiple wells, a superposition method can be used to obtain reliable results. The superposition method applied in this paper is called the area-well method, and it can be explained as follows (Chen and Lin, 1999). There are many single wells constitute various geometric shapes, such as rectangular and circular. When the amount of the water inflow from a single well is similar to that of other wells, the well-group can be regarded as an entity, and the single well can be regarded as an area
Fig. 3. Sketch map of area well. 24
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by integrating all the water seepage surfaces in the plane, the drawdown at an arbitrary point P can be obtained. We assume that a tunnel exists with length L and radius r0 and transform the tunnel wall into a plane of length L and width 2πr0, assuming that 2 lx = L and 2ly = 2πr0. The amount of water inflow per linear meter in the tunnel is q, and total water inflow Qa for the plane can be expressed as follows.
2
4a
we can obtain the following equation.
d =d
1
x + lx 4a
a
x lx 4a
0 x lx 4a
a
2d
e
2d
e
+
0
x + lx 4a
2d
e
(17) x 2dx .
x + lx 4a
x
erf
lx (18)
4a
1
(y y )2 4a
e
ly
y + ly
dy = a erf
erf
4a
y
ly (19)
4a
Substituting Eqs. (18) and (19) into (15) and setting T = aμ yields the following equation.
d 2
=
x
d
0
dx =
q 4 T
s=
(11)
e
r2
4a
x
t
·x
d
=
t
e
4a
0
d
t
e
r2 4a
0
dx dy 4 T
t
=
e
0
d =
dx dy 4 T
t
e
(x x )2 4a
0
·
e
(y y )2 4a
s=
0
lx lx
e
(x x )2 4a
dx ·
1
lx lx
e
(y y )2 4a
ly
d
4a
(20)
4µ
1 0
{
t 4µ
erf
(
1 0
)
erf
(
x
lx 4at
)
ly + y
lx + x 4at
4at
1 0
ly + y
lx x 4at
( ) } td erf ( ) erf ( ) dt erf ( ) erf ( ) dt
× erf
( ) erf ( ) dt + erf ( ) erf ( ) dt +
1 erf 0
+
x + lx 4at
( ) y + ly 4at
lx + x 4at
1 0
4at
lx x 4at
y
erf
ly
4at
ly
y
4at ly
y
4at
(21) Then, a function is defined as follows.
S ( , )=
1 0
erf
erf
d
(22)
The drawdown equation for an arbitrary point in the rectangular well area can be expressed as follows.
d
S=
t Ar (l x , l y , x , y , at ) 4µ
(23)
where
The groundwater table drawdown at the point P (x, y) that is influenced by the total inflow through the water seepage surface can be obtained by taking the integral of the entire planar area.
1
y
erf
4a
lx 4a
t
=
(14)
t
y + ly
x
erf
and substitute variables into Eq. (20) to yield the following expression.
(13)
(x x )2 (y y )2 4a
x + lx 4a
erf
0
Next, we set
(12)
d
t
4µ
× erf
r2
For an arbitrary point P (x, y) in the tunnel area, when the amount of water inflow through the water seepage surface is Qa, the groundwater table drawdown s at point P that is influenced by the water seepage surface can be calculated as follows. Taking the arbitrary point (x', y') as the centre and selecting the differential surface dx'dy', the total amount of water inflow through the differential surface area is ωdx'dy'. The groundwater table drawdown ds at point P (x, y) that is influenced by the differential surface dx'dy' can be obtained based on Eq. (13), as follows.
4
2
· 4a d
In the same way, we can obtain Eq. (19).
For a well point with a specific water inflow q in the plane, the drawdown at the arbitrary point in the infinite plane that is influenced by the well-point can be expressed as follows.
s=
2
2
x + lx 4a
dx = a erf
ly
r2 4a
ds =
(x x )2 4a
e
lx lx
Therefore, the Eq. (8) can be transformed to following equation.
s=
x lx 4a e
1
e The Gaussian error function is expressed as erf ( ) = 0 Therefore, Eq. (17) can be transformed into the following formula.
where
x
dx =
2
r2 x= 4a
r2 4at
(x x )2 4a
lx e lx
=
Exchanging the variables in well functions Eq. (8) yields the following formula. Setting
x
(16)
(10)
r2 u= 4at
e
dx 4a
=
=
The coordinate system of the plane is shown in Fig. 3. The location of the differential surface in the seepage area is expressed by the coordinate system x'oy', and the location of the calculation point for drawdown of the groundwater table is expressed by the coordinate system xoy. Due to a = T/μ, Eq. (7) can be changed to
dx =
x) 4a
1
Therefore, the water inflow per unit area is given by the following equation.
qL Qa = = 2l x ·2l y 2l x ·2l y
(x
Therefore, the following formulas can be derived.
(9)
Qa = qL
x )2
(x
=
Ar (lx , l y , x , y, at ) = S +S
dy d
(
( lx
)+S ( )+S (
lx + x ly + y , 4at 4at
x ly + y , 4at 4at
lx
lx + x ly y , 4at 4at
x ly y , 4at 4at
)
) (24)
Here, Ar is the well function of the well in the rectangular area. The value of the function S*(α, β) can be obtained from reference (Chen and Lin, 1999) when the ranges of values are α = 0.02–3.0 and
(15)
By substituting variables and defining 25
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β = 0.02–3.0. However, the length of a tunnel is typically larger than the width; therefore, α and β often exceed the ranges given in the literature (Chen and Lin, 1999). Thus, a computational procedure in MATLAB is developed to determine S*(α, β) for arbitrary scopes of α, β. If we set (x = 0, y = 0) in Eq. (24), the drawdown equation of the centre of the tunnel is as follow.
sc =
t S µ
lx , 4at
(2) Geological surveys can be conducted to identify impervious boundaries, recharge boundaries, obvious catchment depressions, and other characteristics around the tunnel. If the scope of an impervious boundary or recharge boundary is smaller than that of the calculated influence scope of water inflow for the tunnel, the scope of the impervious boundary or recharge boundary is used to be the influence scope of the tunnel water inflow. If the scope of the impervious boundary or recharge boundary is larger than the calculated influence scope of water inflow for the tunnel, another method should be adopted. For example, if the tunnel passes through a catchment basin or depression, the area can be regarded as the catchment area of the tunnel, and the average width of the area can be considered the width of influence of tunnel water inflow.
ly (25)
4at
The drawdown is largest at the tunnel centre, and Eq. (25) is used to calculate the largest drawdown of the groundwater table. Based on Eq. (25), the t value can be achieved when sc and q are given. 4. The shape and expansion of the drawdown funnel
Therefore, if the tunnel corresponds to the above two situations, the influence scope of tunnel water inflow can be determined based on the above methods. Nevertheless, if the tunnel does not correspond these cases, e.g., if the boundary cannot be determined and there is no existing tunnel for reference, specific empirical formulas can be used to determine the influence scope. These formulas include the Kusakin formula, the Jihaerte formula and the formula recommended in the standard “Code for Hydrogeological Investigation of Railway Engineering” (National Railway Administration of the People's Republic of China, 2015). However, these formulas are typically associated with a single factor only and do not consider temporal changes or the expansion of the drawdown of the groundwater table. A drawdown funnel will form around the well after pumping for a long time. Based on the flow characteristic of the well, a drawdown funnel will form above the tunnel when the amount of drainage is too high. However, unlike the well, the tunnel length is much larger than the width; therefore, the drawdown funnel is shaped like an inverse elliptical cone. The elliptical scope of the drawdown funnel is the influence scope of the tunnel and the rainfall infiltration recharge area. Eq. (21) is the groundwater table drawdown formula, and the drawdown at the edge of the influence scope is 0, that is sc = 0, so Ar = 0. If x and y are known, the major and minor axes of the ellipse funnel can be determined, and the influence scope can then be obtained. The crosswise drawdown at the central of the tunnel can be expressed as follows when x = 0 and y = b.
Eq. (21) is the formula for groundwater drawdown, and the parameters for the tunnel are given as follows. The length of the water-bearing section of the tunnel is L = 1020 m; the equivalence circle radius of the tunnel is r0 = 6.85 m; the width of the tunnel is B = 12.8 m; the hydraulic conductivity of the surrounding rock is k = 0.0258 m/d; the specific yield is μ = 0.056; the groundwater hydraulic head is H = 450 m; and q0AVG = 9.12 m3·m−1·d−1 based on the Eqs. (1), (2), (4) and (5); T = 11.61 m2/d; a = T/μ = 207.3 m2/d. The drawdown of the groundwater table from 1 to 3 days is shown in Figs. 4–6. (1) Two-dimensional cross-sectional plot of the drawdown funnels for the tunnel (2) Three-dimensional plot of the drawdown funnel for the tunnel As shown in Figs. 4–6, the drawdown of the groundwater table increases, and the influence scope gradually expands as t increases. 5. Determination of the influence scope of tunnel water inflow There is no broadly used prediction method for the influence scope of tunnel water inflow. Hydrogeological analogue methods and geologic survey methods are recommended in the standard “Code for Hydrogeological Investigation of Railway Engineering” (National Railway Administration of the People's Republic of China, 2015). (1) If the engineering geological and hydrogeological conditions for a new tunnel are similar to those of an existing tunnel, the influence scope of the water inflow of the new tunnel can be based on that of the existing tunnel. This approach is called hydrogeological analogue method.
s =
t S 2µ
ly - b lx , 4at 4at
ly + b lx , +S 4at 4at
(26)
For s = 0, the minor axis b of the elliptical funnel is obtained. The lengthways drawdown at the central of the tunnel can be
0
Drawdown of GWT (m)
2
4
0 day 1st day 2nd day 3rd day
6
8
10
-60
-40
-20
0
Distance of cross direction (m)
20
40
Fig. 4. The plot of groundwater table (GWT) drawdown of the tunnel centre cross section. 26
60
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Fig. 5. Three-dimensional plot of the drawdown funnel for a tunnel on the 1st day.
expressed as follows when x = l and y = 0.
s =
t S 2µ
lx + l , 4at
ly 4at
+S
lx - l , 4at
portion of the total rainfall W. By dividing these values, we can obtain the rainfall infiltration coefficients p (Xiao et al., 2010), i.e., p = Wr/W. If p and W are known, the recharge amount can be obtained. In plain, hill and mountain areas, the rainfall infiltration coefficient p of an aquifer can be approximately determined from historical values, as shown in Table 1, when there is a lack of experimental data (Zhu and Li, 1995). Rainfall amount can be obtained according to local meteorological data. When the values of F, W and p are known, the recharge amount Wr of groundwater per day can be obtained as follows:
ly 4at
(27)
For s = 0, the major axis of the elliptical funnel l can be obtained. Strictly speaking, a steady flow is unable to form when tunnel seepage occurs without any recharge in an infinite aquifer. On the contrary, the groundwater hydraulic head decreases everywhere in the aquifer (Chen and Lin, 1999). However, if vertical recharge occurs in the aquifer, the amount of recharge increases with the expansion of the funnel area of the tunnel. When the amount of the vertical recharge equals the tunnel water inflow, the drawdown funnel will stop expanding. Then, the groundwater flow becomes a steady flow. Therefore, we take the distance from the central of the tunnel to the distance where groundwater table drawdown can be neglected as the influence scope of the tunnel water inflow. The influence scope can be obtained after the major and minor axes of the ellipse funnel are determined, and is expressed as follow:
Wr = p · W · F 365
Wr is the daily average amount of rainfall infiltration recharge (m3/d); p is the rainfall infiltration coefficient; W is the annual rainfall (m); and F is the rainfall infiltration area (m2). However, the amount of water inflow per unit time for the tunnel is q, and the total water inflow Q within t can be defined as follows.
Q = qLt
(28)
F = lb
(29)
F is the area of rainfall infiltration recharge.
(30)
When the total water inflow Q and the recharge amount Wt within t are known for specific groundwater table drawdown conditions, if Q = Wt, the discharge amount and recharge amount are balanced, and the groundwater table will no longer continue to fall. If Q > Wt, groundwater drainage can result in aquifer dewatering, which can negatively affect the ecological environment. Thus, the tunnel water inflow q should be properly reduced and to achieve a balance between recharge and discharge. If Q < Wt, the water inflow is sufficient and does not need to be addressed.
6. Analysis of the groundwater balance To keep the groundwater balance, the total rainfall infiltration recharge of groundwater must be equals to the total drainage groundwater. In general, the primary recharge source for mountain tunnel groundwater is rainfall. However, the recharge amount Wr is only a
Fig. 6. Three-dimensional plot of the drawdown funnel for a tunnel on the 3rd day. 27
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daily recharge is Wr = 339.1 m3/d according to Eq. (29), and the total recharge is Wt = 3.39 × 103 m3 for 10 d.
Table 1 Experimental data for the rainfall infiltration coefficient. Strata
p
Strata
p
Silty clay Silty soil Silty sand Fine sand Medium sand Coarse sand Round gravel (sand) Pebble (sand) Intact rock
0.01–0.02 0.02–0.05 0.05–0.08 0.08–0.12 0.12–0.18 0.18–0.24 0.24––0.30 0.30–0.35 0.01–0.10
Relatively intact rock Relatively broken rock Broken rock Extremely broken rock Very weak karst development Weak karst development Karst medium development Karst development
0.10–0.15 0.15–0.18 0.18–0.20 0.20–0.25 0.01–0.10 0.10–0.15 0.15–0.20 0.20–0.50
(6) Analysis of groundwater balance From above analysis, Q > Wt, which means that for a long time, the aquifer will be dewatered, and vegetation could be destroyed under these conditions. Therefore, the amount of drainage water should be reduced to achieve the balance between drainage and recharge. (7) Determination method of the amount of limiting drainage to maintaining a groundwater balance Due to the above analysis, the amount of water inflow drainage is too high to maintain a groundwater balance; therefore, the water inflow q0 should decrease until the amount of drainage water equals the amount of recharge water, i.e., Q = Wt, and the water inflow q0 is reasonable amount of drainage water. Because the scope of the q0 is unknown before the calculations, and the computational efficiency will seriously decrease if the searching scope is too large. Therefore, it is better to select several values of q0 and determine a preliminary scope, and then perform loop computing to obtain a reasonable q0. In this paper, values of q0 = 2.7 m3·m−1·d−1, 2.86 m3·m−1·d−1 and 3.0 m3·m−1·d−1 were separately used to assess the relationships between drainage and recharge, drainage and time, and drainage and influence scope of water inflow, as shown in Figs. 7–9. Fig. 7 illustrates that a q0 value between 2.86 and 2.88 m3·m−1·d−1 can make Q equal Wt. Thus, the search interval for loop computing is set from 2.86 m3·m−1·d−1 to 2.88 m3·m−1·d−1 to determine the exact q0 value. According to Fig. 8, when q0 = 2.86–2.88 m3·m−1·d−1, the t value is in the range of 120–125 d. Similarly, the major and minor axes of tunnel drawdown funnel are in the range of l = 1450–1480 m, and b = 1150–1200 m, respectively, according to Fig. 9. However, Fig. 7 cannot accurately reflect the q0 values, so the scopes of the q0, t, l and b values should be appropriately expanded to avoid the result exceeding the scope. As shown in Figs. 7–9, in the case of a given drawdown, the time required to achieve a given depth and influence scope of water inflow gradually decreases as the unit amount of water inflow increases. Moreover, total groundwater discharge and recharge also decrease. The amount of total discharge is related to the unit discharge and time, although the unit discharge increases, but the time required to reach a given depth is reduced, and the total drainage decreases. However, the recharge amount is related to the influence scope of water inflow, and the decreasing scope causes the recharge amount to decrease. Because the decreasing range of the total recharge amount is greater than the
7. Case study A case study was investigated for a tunnel with a total length of 850 m and an aquifer composed of mudstone and siltite. The groundwater hydraulic head is H = 180 m. The parameters of the aquifer are as follows: the aquifer thickness is hc = 380 m, the hydraulic conductivity K = 0.0386 m/d, the storage coefficient is μ = 0.048, and the transmissibility coefficient is T = 13.51 m2/d. The equivalent circle radius of the tunnel cross section r0 is 5.94 m, and the width of the tunnel is B = 11.6 m. Based on the investigation and analysis of the vegetation and ecological conditions at the tunnel site, the maximum allowable drawdown of the groundwater table is 20 m. (1) The amount of water inflow without plugging Based on Eqs. (1), (2), (4) and (5), the water inflow of the tunnel is q0-0 = 12.80 m3·m−1·d−1, q0-1 = 9.10 m3·m−1·d−1, q03 −1 −1 ·d , and q0-3 = 13.89 m3·m−1·d−1. Thus, the average 2 = 8.92 m ·m value q0AVG = 11.18 m3·m−1·d−1 is considered the water inflow. (2) The time t when the allowable drawdown is reached The drawdown of the groundwater table at the tunnel centre reaches 20 m after time t based on Eq. (3). Therefore, the following equation can be established: qt = 11.18–0.0014 t. Considering Eq. (25) and given that sc = 20 m, t can be determined. Taking the integral, a value of t = 10 d is obtained. Thus, the drawdown of groundwater table at the central tunnel will reach 20 m after 10 d. (3) Influence scope of tunnel water inflow
5
3
Drainage Q and discharge Wt (×10 m )
After 10 d, sc < 0.0001 m at the location 341 m away from the central of the crosswise tunnel. The drawdown of the groundwater table can be neglected at this distance; therefore, the minor axis of the influence scope of the tunnel is set to b = 341 m. Similarly, the minor axis is l = 772 m. Thus, the rainfall infiltration recharge area is F = 7.73 × 105 m2 based on Eq. (28). (4) The total amount of groundwater inflow The amount of water inflow after 10 d, q10d = 11.17 m3·m−1·d−1 is obtained using Eq. (3), and q = (q0 + q10d)/2 = 11.18 m3·m−1·d−1. Thus, the total groundwater inflow after 10 d is Q = 9.50 × 104 m3 according to Eq. (30). (5) The total amount of rainfall infiltration recharge The average annual rainfall at the tunnel site is 1000 mm according to local meteorological data, and the rainfall infiltration recharge area is 7.73 × 105 m2. The rock surrounding the tunnel is fractured, so the rainfall infiltration coefficient p is selected as 0.16 from Table 1. The
4.0 3.9 3.8 3.7 3.6 3.5 3.4 3.3 3.2 3.1 3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.65
Q
Wt 2.70
2.75
2.80
2.85
2.90
2.95 3
-1
3.00
-1
Unit water inflow of tunnel q0 (m .m .d ) Fig. 7. Relationship between total drainage and recharge. 28
3.05
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The method presented is used to calculate the limiting drainage of the Yuanliangshan Tunnel (Jiang, 2005), and the allowable drawdown is set to 20 m. Based on calculations, when q0 = 2.70 m3·m−1·d−1, QWt = 1.8 × 103 m3, which is able to maintain a sufficient groundwater balance. However, the limiting drainage standard of the Yuanliangshan Tunnel is 5.0 m3·m−1·d−1 (Wang et al., 2005) which is larger than the calculated value in practical engineering, and result in groundwater table drawdown, ground subsidence and soil erosion problems (Gan et al., 2008). If the limiting drainage is lower than 2.70 m3·m−1·d−1, the problems will be avoided, and the vegetation and ecological environment can be effectively protected. 8. Conclusions The ecological groundwater table of vegetation is introduced in this study, and the limiting drainage of a tunnel to maintain a balance between groundwater and the ecological environment is analysed based on the groundwater dynamic method that considers groundwater recharge by rainfall. The main conclusions of the study are as follows:
Fig. 8. Relationship between water inflow and t. 1550
(1) Because the length of a tunnel is much larger than the width, the superposition of the influence of water seepage should be considered. Based on the area-well theory, a groundwater seepage model of a tunnel is established. Due to the model and groundwater dynamics, the equation describing the relationship between the amount of water inflow and drawdown of the groundwater table for a tunnel is derived under unsteady flow conditions. Due to the formula, two-dimensional and three-dimensional plots of groundwater table drawdown are created, and the plots show the intuitional acquaintance of the characteristics of the drainage funnel. (2) The method for determining the limiting drainage value of groundwater is presented. This limit must be not being exceeded to maintain the balance between groundwater and the vegetation environment. When the allowable drawdown of the groundwater table sc, the time t when the maximum allowable drawdown is reached, and the rainfall infiltration recharge area F are determined by calculating, the total amount of rainfall infiltration recharge Wt is obtained. Based on the comparation of Wt and the total groundwater drainage Q, the analytical solution for the limiting drainage of groundwater q0 can be obtained. If Wt ≥ Q, the amount of recharge is larger than the discharge, and the groundwater table will not drop; therefore, the vegetation ecological environment will not be destroyed. In this case, the initial water inflow by forecasting is the limiting drainage mount of the tunnel. However, if Q > Wt, the drawdown of groundwater table will exceeds the ecological groundwater table of vegetation and causing vegetation destroy, and the water inflow q0 of the tunnel must be gradually decreased, and the calculation must be repeated until Wt ≥ Q, then the value of water inflow is the limited drainage of groundwater for the tunnel, and the local vegetation and ecological environment will be protected and preserved. (3) The results and case study show that the proposed method exhibits good agreement with actual conditions and is suitable for use in tunnel construction.
1500
Influence scope (m)
1450
l
1400 1350 1300 1250 1200 1150 1100 2.65
b 2.70
2.75
2.80
2.85
2.90
3
2.95 -1
-1
3.00
3.05
Unit water inflow of tunnel q0 (m .m .d )
Fig. 9. Relationship between water inflow and the influence scope.
decreasing range of the total discharge amount, a water inflow value exists that can equate the recharge amount and discharge amount. Based on calculations, when q0 = 2.872438 m3·m−1·d−1, t = 124 d, Q = 29361.0 m3, Wt = 29405.9 m3, and the recharge amount is slightly larger than the discharge amount, with a difference of only 442.6 m3. When the difference value is converted to daily discharge over the total length of the tunnel, there is only 3.6 m3/d, which can be considered negligible. Typically, when the resulting numbers of q0 are down to the 6th or even more decimal points, the difference between Q and Wt is approximately equal to zero. However, such exact q0 value have no engineering significance, and a resulting q0 value down to the 2nd decimal point is more reasonable. If the resulting numbers of q0 is down to the 2nd decimal point, the difference between Q and Wt will reach an order magnitude of 103, even though in this situation, the daily discharge amount is only 8.06 m3/d, which can be also ignored. Thus, related measures can be terminated under the condition that |Q Wt | < 1000 . In this case, when q0 = 2.87 m3·m−1·d−1, the total groundwater recharge is approximately equal to the total groundwater recharge, and a groundwater balance is achieved. Therefore, the drawdown of the groundwater table will not continually decrease when it reaches 20 m, and the local vegetation and ecological environment will be protected and preserved. Of course, if the limiting drainage is less than 2.87 m3·m−1·d−1, groundwater recharge will be larger than the groundwater discharge, and the groundwater table drawdown will be smaller than 20 m, which is more beneficial for vegetation and the ecological environment. Therefore, the limiting drainage for the present case can be determined as q0 < 2.87 m3·m−1·d−1.
Acknowledgements This research project was jointly supported by the National Natural Science Foundation of China (51809271, 61505259, 51478477, 11572349, 11602299) and the Scientific Research Program of National University of Defense Technology (ZK2017-03-40). Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.tust.2018.10.014. 29
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