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Predicting external water pressure and cracking of a tunnel lining by measuring water inflow rate
Tunnelling and Underground Space Technology 71 (2018) 115–125
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Predicting external water pressure and cracking of a tunnel lining by measuring water inflow rate
MARK
⁎
Yuqi Tan, John V. Smith, Chun-Qing Li , Matthew Currell, Yufei Wu School of Engineering, RMIT University, Melbourne 3001, Australia
A R T I C L E I N F O
A B S T R A C T
Keywords: Tunnel Groundwater inflow Lining Crack Inhomogeneity Hydraulic conductivity
The flow of water into a tunnel, through the surrounding rock mass and the tunnel lining, has important implications for tunnel design and tunnel condition assessment. The hydraulic conductivity of the lining is a major controlling factor on the water inflow rate. Previous work had been mainly based on the assumption of constant hydraulic conductivity of the lining. In this work, the existing analytical models of water flow through a tunnel lining under steady-state, saturated conditions are extended to incorporate a linear variation of hydraulic conductivity with distance from the tunnel wall. The inhomogeneity of a lining is shown to have a significant impact on water inflow rate and water pressure distribution according to the model. The relation between lining inhomogeneity and other hydraulic parameters was established. This model can be used to predict the hydraulic pressure and crack condition at the outer face of the lining based on measured water inflow rate and the crack condition at the inner face, with significantly increased accuracy compared with the existing models based on constant hydraulic conductivity. Design charts are also developed for engineering applications.
1. Introduction Groundwater inflow is a problem for tunnels worldwide, and numerous cases of tunnel lining failure due to water inflow have been reported (ITA, 1991). Chemically aggressive water inflow can cause degradation of concrete lining (Gérard et al., 2002) and excessive hydraulic pressure in the lining can trigger concrete spalling (Jansson and Boström, 2010) and affect the structural stability of the tunnel (Fang et al., 2016). A numerical approach was developed to assess the hydraulic conductivity of the tunnel lining according to the water inflow rate when the other parameters are given (Bagnoli et al., 2015). Tunnels can be classified into three types with respect to interaction with water: unlined tunnels, drained lined tunnels and water-sealed lined tunnels (Butscher, 2012). The inflow water problem has been extensively studied for all the three cases (Polubarinova-Kochina, 1963; Goodman et al., 1965; Heuer, 1995; Zhang and Franklin, 1993; El Tani, 2003; Hwang and Lu, 2007; Lei, 1999; Kolymbas and Wagner, 2007; Park et al., 2008; Fernandez and Moon, 2010a, 2010b; Huang et al., 2013; Fang et al., 2016). These previous approaches for solving tunnel water inflow problems assume that the lining has a homogeneous hydraulic conductivity. The surrounding rock mass is also commonly assumed to have homogeneous hydraulic conductivity, although two studies with inhomogeneous properties have been conducted (Table 1). Schematic
⁎
Corresponding author. E-mail address: [email protected] (C.-Q. Li).
http://dx.doi.org/10.1016/j.tust.2017.08.015 Received 30 January 2017; Received in revised form 17 July 2017; Accepted 16 August 2017 0886-7798/ © 2017 Published by Elsevier Ltd.
illustration of the hydraulic conductivity distribution of the three types of tunnel under homogeneous conditions is shown in Fig. 1. The hydraulic conductivity of a homogeneous lining, which primarily depends on the crack width and crack density, can be estimated from the crack features of the lining (Fernandez and Moon, 2010a). The water flow rate through a crack has a cubic relationship with crack width, and a linear relationship with crack density (Snow, 1965). A reduction factor for crack roughness has been developed for application to water flow through cracked concrete (Reinhardt, 1997). Calculations of water inflow are very sensitive to the hydraulic conductivity of the lining and inhomogeneity of the lining can potentially cause significant differences in the results. The hydraulic conductivity of a lining is typically deduced from observations of joint and crack features on the inner face of the lining. Such features can vary through the lining which would affect the hydraulic conductivity along the water flow path. However such variation within the lining cannot be directly observed, and thus there is a need for methods to predict and account for this. This paper presents an analytical solution for water inflow rate and hydraulic pressure distribution in tunnel linings with inhomogeneous hydraulic conductivity, assuming saturated conditions and steady-state flow. The solution uses the hydraulic conductivity of the surface of the lining inside the tunnel as the reference value. Variation in hydraulic conductivity is considered linear within the lining. In combination with
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Therefore, the water head reduction caused by a lining, and the water head at the spring line at the outer surface, houter can be calculated as (Fernández, 1994)
Table 1 Framework for water inflow prediction models. Type of tunnel
Rock mass
Unlined
Homogenous
Lining/lining-like zone
Lined water sealing
Homogenous
Homogenous
Homogenous Inhomogeneous Inhomogeneous
Inhomogeneous Homogenous Inhomogeneous
Homogenous
Homogeneous
Homogenous Inhomogeneous
Inhomogeneous Homogenous
Inhomogeneous
Inhomogeneous
1+γ
Kolymbas and Wagner (2007) Park et al. (2008) NONE NONE NONE Fernandez and Moon (2010a) (unlined tunnel with a lining-like zone) Addressed in this paper Fernandez and Moon (2010b) (unlined tunnel with a lining-like zone) NONE
ql = dθrK (r )
where dh is the change in total hydraulic head, which comprises both hydraulic pressure head and elevation head, and dr is the change in radius. According to Darcy’s law, the water discharge Q per unit length of the tunnel for a homogenous lining (Fetter, 2001) is (2)
2πK (houter −hinner ) ln
(
(a + d) a
)
a + d⎞ a ⎠
(5)
dh dr
(6)
where dθr is the flow area of unit arc length of a unit length tunnel and K (r ) is the hydraulic conductivity of the lining, which varies with distance from the tunnel centre. The hydraulic head variation caused by the lining inhomogeneity is shown in Fig. 3. The hydraulic head decreases linearly through the homogeneous lining as shown in the right hand side of Fig. 3. The hydraulic conductivity through the lining is a constant under the homogeneous assumption. The hydraulic head decrease will be affected by the variation of the hydraulic conductivity through the lining in an inhomogeneous case. The left hand side of Fig. 3 shows the condition when hydraulic conductivity of the lining increases from the inner side of lining surface to the outer side of the lining surface. An inhomogeneous hydraulic conductivity coefficient can be defined as
where K is the hydraulic conductivity of material. For a permeable lining zone of a tunnel, the hydraulic head at the outer surface and inner surface of lining will affect the water inflow rate. The outer surface and inner surface of lining were defined using the term extrados and intrados in a published document (ITA, 1991). The water inflow rate was discussed by introducing the boundary conditions of the hydraulic head at the outer lining surface and the inner lining surface (Fernández, 1994) as
Q=
ln ⎛ ⎝
Previous research only considered the condition when the lining is homogeneous, i.e. K is a constant. Assuming a homogeneous lining is a practical simplification for a thin lining where the lining inhomogeneity would not be expected to have a significant effect on water inflow. Inhomogeneity of a lining needs to be considered when the lining is thick. For a lining which is inhomogeneous, the hydraulic conductivity K(r) is a function of r. A study of crack spacing in reinforced concrete showed that differences in strain in reinforcing elements would result a linear variation of crack spacing (Bazant and Oh, 1983). A linear form of variation of K with respect to r is assumed in this work since the crack spacing has been shown to have a linear relationship with hydraulic conductivity (Reinhardt, 1997). This study only analyses a linear variation of hydraulic conductivity, however other types of variation are also possible. The method applied in this study could also be applied to other types of lining inhomogeneity as long as the lining hydraulic conductivity varies continuously. An equation is established to describe the water inflow rate (ql ) for a unit arc length of a unit length of tunnel lining (dθr ) based on Darcy’s law as shown in Eq. (6). The hydraulic conductivity of a tunnel lining is considered to change continuously with the radial direction (K (r ) ).
(1)
dh dr
(4)
3. Water inflow through an inhomogeneous lining
To calculate the water inflow rate of a tunnel lining, a tunnel can be defined in a cylindrical coordinate system. The origin of the coordinate system is the centre point of the tunnel. All points are defined by an angle θ, and the distance r (Fig. 2). For a tunnel under the water table (Fernandez and Moon, 2010a), the hydraulic gradient i can be described as
Q = 2πKr
K Km
2H ⎞ γ = ln ⎛ ⎝a + d⎠
2. Water inflow through a homogeneous lining
dh dr
( )
where H is the vertical distance from the spring line of the tunnel to the water table (or equivalent phreatic pressure head); K is the equivalent hydraulic conductivity of the lining; Km is the hydraulic conductivity of the surrounding rock mass and γ is a coefficient related to tunnel radius and lining thickness, which is defined by
the observations of cracking at inner surface and the water inflow rate, this model can be used to estimate the level of inhomogeneity within the lining including the water pressure and as the crack condition at the outer face of the lining.
i=
H
houter = Goodman et al. (1965) Lei (1999) El Tani (2003) Zhang and Franklin (1993)
Inhomogeneous Lined drained
Reference
Cl =
(3)
where water head at the outer surface at the spring line of the lining is defined as houter ; hinner is the water head at the inner surface at the spring line of the lining; a is the tunnel inner radius (from the centre to the inner surface) and d is the lining thickness. For a non-pressurised tunnel, the pressure head at the inner surface is taken to be zero, so the hydraulic head loss across the lining is equal to the pressure head at the outer surface of the lining (at the spring line). The datum level is defined as the spring line of the tunnel.
K outer Kinner
(7)
where Kinner and K outer are the hydraulic conductivity of the inner surface of the lining and outer surface of the lining, respectively. The value of Kinner can potentially be determined directly by observation and measurement of crack density and crack width according to Reinhardt (1997).
Kinner = ζ 116
w 3ρg 12μΔ
(8)
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Fig. 1. Schematic illustration of hydraulic conductivity of three types of homogeneous tunnel.
where w is the crack width; 1 is the crack density; ζ is the crack Δ roughness factor; w is the crack width; ρ is the water density; g is the standard gravity; μ is the viscosity of water. Eq. (8) is a simplified approach to estimate the hydraulic conductivity of a bulk media along the radial direction of a tunnel. This method has been validated experimentally in water inflow tests through cracked concrete samples (Reinhardt and Jooss, 2003) and applied by Fernandez and Moon (2010a) to the hydraulic conductivity estimation of a tunnel lining. The hydraulic conductivity of the outer surface of the lining (K outer ) typically cannot be determined directly by observation. The hydraulic conductivity of any position in the lining can be represented as
K (r ) = Kinner + (Cl−1)
Kinner (r −a) d
The function that contains radius r can be rearranged as
1 d 1 = 2 =d rK (r ) r Kinner (Cl−1) + rKinner (d−aCl + a) r (αr + β )
(10)
where α = Kinner (Cl−1) and β = Kinner (d−aCl + a) . Based on Eq. (6), the hydraulic head through the lining can be written as
dh =
ql 1 dr dθ rK (r )
(11)
Integration of Eq. (11) gives
h (r ) =
(9) 117
ql d ⎡ 1 αr + β ⎤ − ln +C ⎥ β r dθ ⎢ ⎣ ⎦
(12)
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Fig. 2. Illustration of tunnel geometry parameters.
where C is a constant. To determine Eq. (12), the boundary conditions need to be considered. For a lined tunnel with relatively low water seepage velocity, the loss of velocity head around the outer lining is negligible compared with the total head. Similarly, it is assumed that any frictional loss at the rock mass-lining interface is very small and can be neglected. When the water head loss along the rock tunnel interface is negligible, the decrease of elevation is equal to the increase of pressure head along the interface. Therefore, the hydraulic head (total head) which is equal to the summation of the pressure head and elevation head at the boundary of the rock and lining is considered a constant. Since there is no water pressure at the inner surface of the lining, the total head at the inner surface is equal to the elevation head when the datum level is the spring line of the tunnel. Therefore the boundary conditions (Kolymbas and Wagner, 2007) are as follows
h|r = a = asinθ
(13)
h|r = a + d = houter
(14)
C = asinθ +
ql d ⎡ 1 αa + β ⎤ ln ⎥ dθ ⎢ a ⎣β ⎦
(15)
Substituting the boundary condition Eq. (14) into Eq. (12), the water head can be further developed as
houter =
ql d ⎡ 1 α (a + d ) + β ⎤ − ln +C ⎥ a+d dθ ⎢ ⎣ β ⎦
(16)
Eq. (16) can be rewritten as
houter =
ql d ⎡ 1 α (a + b) + β αa + β ⎤ ⎤ 1 + ⎡ ln − ln + asinθ ⎥ ⎢ ⎥ dθ ⎢ β a + b β a ⎣ ⎦⎦ ⎣ (17)
Considering Kinner (Cl−1) = α and Kinner (d−aCl + a) = β , Eq. (17) can be further developed as dC
d
⎡ln Kinner a + ld −ln Kinner a ⎤ dθ(houter −asinθ) ⎦ = −⎣ ql d Kinner (d−aCl + a)
(18)
The equation of water flow rate across a unit arc length of the lining can be expressed by simple transformation of Eq. (18) as
The boundary conditions Eqs. (13) and (14) are used to calculate the function of water head with respect to the cylindrical coordinate system. Substituting the boundary condition Eq. (13) into Eq. (12), C can be obtained as
ql =
(houter −asinθ) Kinner (aCl−a−d ) d ln
aCl a+d
dθ (19)
Fig. 3. Head loss under both homogeneous (right) and inhomogeneous (left) conditions.
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lining. The primary lining also included application of shotcrete to the tunnel wall prior to the installation of the secondary lining. The primary and secondary lining elements are together considered to comprise the lining. The hydraulic conductivity can be variable through the lining due to the different allocation of reinforcement as in Fig. 4. The tunnel was designed as a water-sealed tunnel to prevent water table draw down. Even though the concrete lining was designed to minimise the water infiltration, cracks in the concrete lining have been observed. The mean crack width (winner ) is 0.3 mm, and the crack density is 0.5 m/m2. The hydraulic conductivity of the inner surface of the lining is approximately 1 × 10−8 m/s based on the observed crack density and crack width, according to the model outlined earlier (Eq. (8)). The mean value of hydraulic conductivity of the surrounding rock mass is approximately 1 × 10−7 m/s based on previous geological surveys (Robinson and Kenna, 1992). The value of the hydraulic parameters used in calculation is summarized in Table 2. The hydraulic parameter for this case varies in a range as shown in Table 3. The variation of the hydraulic parameter will be used for calibration purpose. It has also been reported generally, that the type of reinforcement in a concrete lining can result in variations of crack width and crack density in different parts of the concrete (Bazant and Oh, 1983). While observations of crack density and crack width exposed on the inner surface of the lining of the tunnel were used to estimate the hydraulic conductivity, the crack density and crack width within the lining (and therefore potential variation in hydraulic conductivity within the lining), is not known. Based on the distribution of support elements within the concrete lining, inhomogeneous hydraulic conductivity is considered a likely condition for the lining of the example tunnel. In this study we incorporate the potential variability of hydraulic conductivity through the lining into the model, as described in Section 3. We consider the linear inhomogeneous hydraulic conductivity coefficient (Cl ) across a range from 0.01 to 100. The concept of Cl is proposed as a practical way to frame the problem as the condition of the inner surface of a tunnel lining can be observed by direct inspection or by use of devices such as televiewers.
The total water inflow rate of a unit length of tunnel Ql with a cylindrical lining can be obtained by integrating θ, to give
2πhouter Kinner (aCl−a−d )
Ql =
d ln
aCl a+d
(20)
If the tunnel has a constant total head at the interface of rock and lining, previous works (Lei, 1999; Kolymbas and Wagner, 2007; Park et al., 2008) have shown that the water flow rate through the rock mass can be written as
2πKm (H −houter )
Qm =
D ln ⎛ a + d + ⎝
D2 −1 ⎞ (a + d)2
(21)
⎠
where D is the distance from tunnel spring line to the water table or the ground surface, depending on which one is smaller; the hydraulic conductivity of the surrounding rock mass is Km . Since water is considered to be an incompressible fluid in this study, water flow rate through a lining under saturated conditions is equal to the water flow rate through the rock mass, based on mass conservation. Combining Eqs. (20) and (21), the following equation can be obtained
2πhouter Kinner (aCl−a−d ) d ln
aCl a+d
2πKm (H −houter )
=
D2 −1 ⎞ (a + d)2
D
ln ⎛ a + d + ⎝
⎠
(22)
Transforming (22), the water head at the outer surface (houter ) of the lining can be expressed as
H
houter =
⎛ D + Kinner (aCl − a − d)ln ⎜ a+d ⎝ Km dln
D2 (a + d)2
⎞ − 1⎟ ⎠
aCl a+d
+1 (23)
Substituting Eq. (23) into Eq. (20), the water inflow rate per unit length of tunnel Q can be expressed as
2πKinner (aCl−a−d ) H
Q= d ln
aCl a+d
⎡ Kinner (aCl − a − d)ln ⎛⎜ D + a+d ⎢ ⎝ aCl ⎢ Km dln a+d ⎢ ⎣
D2 (a + d)2
⎞ − 1⎟ ⎠
⎤ + 1⎥ ⎥ ⎥ ⎦
4.2. Water head variation (24)
The result of the analytical solution (Eq. (24)) shows the water head behaviour at the spring line through the lining for Cl values of 0.01, 0.1, 1, 10 and 100 (Fig. 5). For the homogeneous case (Cl = 1) it can be seen that the water head distribution is slightly non-linear due to the effect of flow convergence described above (Fig. 5). For cases of inhomogeneous hydraulic conductivity coefficient (Cl ) the water head variation through the lining is distinctly non-linear (Fig. 5(a)). When Cl > 1, the hydraulic gradient decreases from the inner surface to the outer surface. This relationship becomes more pronounced as Cl increases (Fig. 5(b)). The relationship is due to the relatively minor influence of the flow area change and relatively major influence of the increase in average hydraulic conductivity, represented by Cl > 1. In contrast, the hydraulic gradient decreases through the lining when Cl < 1. Increasing Cl represents an increase in hydraulic conductivity at the outer surface while the hydraulic conductivity at the inner surface is fixed. Therefore, as Cl increases, the average hydraulic conductivity of the lining would also increase. The average or equivalent hydraulic conductivity of the lining has not been defined or used in this study because it cannot be independently verified. Rather, the hydraulic conductivity at the inner surface has been used as the reference (as this can be estimated using direct observation) and hydraulic conductivity at the outer surface is a variable factor defined according to Cl . The variation of hydraulic conductivity in an inhomogeneous lining will cause hydraulic head redistribution at the outer surface. The water head at the outer surface (x axis = 0.56, Fig. 5) will decrease if the average hydraulic conductivity of lining decreases based on Eq. (4). In general,
All the parameters have been defined above. To calculate the hydraulic head in the lining, Eqs. (15) and (19) are substituted into the water head distribution function Eq. (12) and combined with the water head at the outer surface of the lining (Eq. (23)). The water head through the lining can be calculated as
h (r ) =
(houter −asinθ) ln
aCl a+d
ln
a [r (Cl−1) + a + d−aCl] + asinθ rd
(25)
For any position within the lining with given r and θ , the water head in the lining can be calculated accordingly. 4. Application 4.1. Parameters used in application An example tunnel (Fig. 4) was selected for the purpose of illustrating the influence of variable hydraulic conductivity through a lining. The example tunnel is cylindrical and approximately 35 m underground. The water table is approximately 15 m above the spring line of the tunnel. The radius (a ) of the tunnel from the tunnel centre to the inner surface of the lining is 3000 mm and the thickness of the cast in situ concrete lining (d ) is 560 mm. The tunnel lining comprises steel beams as primary support embedded in the cast-in-situ lining near its outer surface. Steel reinforcement bars were cast in the concrete close to the inner surface of the 119
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Fig. 4. Illustration of the example concrete lining with steel reinforcement elements (a) transverse section and (b) longitudinal section.
Fig. 6 shows the water head at the outer surface for various values of Cl . When Cl increases from 0.01 to 1, the water head at the outer surface decreases. When Cl increases from 1 to 100, the water head at the outer surface of the lining drops from 6.7 m to less than 1 m. The homogeneous case represents an inflection point on the curve representing the range of inhomogeneous conditions (Fig. 6).
Table 2 Parameters from the example tunnel. Parameter
Km (m/s) −7
Value
1 × 10
Kinner (m/s) 1 × 10
−8
a (m)
d (m)
H (m)
3
0.56
15
Table 3 Parameter ranges used for validation of the water inflow factor chart.
Km (m/s) −7
Low Middle High
1 × 10 1 × 10−6 1 × 10−5
Kinner (m/s) −9
1 × 10 1 × 10−7 1 × 10−5
4.3. Water inflow rate variation
a (m)
d (m)
H (m)
Cl
2.8 3 3.2
0.5 0.56 0.6
5 15 20
0.01 1 100
In addition to influencing water head distribution, the Cl value will also influence the water inflow rate. When Cl increases from 0.01 to 1, the water inflow rate increases (Fig. 7). The variation of water inflow rate is controlled by both hydraulic conductivity and hydraulic head. The homogeneous condition again represents an inflection point in the interplay of these parameters (Fig. 7). For the homogeneous condition the calculated water inflow rate for the example tunnel is 2.46 × 10−6 m3/s. When Cl increases from 0.01 to 100, the water inflow increases, ranging from 9.7 × 10−7 m3/s to 4.3 × 10−6 m3/s,
lower average hydraulic conductivity of the lining will allow greater hydraulic pressure at the outer surface, if all other parameters are constant.
15
120
100
60
40
Outer lining surface
Cl=10
80
Inner lining surface
6
Hydraulic gradient
9
Outer lining surface
Inner lining surface
Water head (m)
12
3
20 Cl=100
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0
Distance outward from the inner lining surface (m)
Cl=10
Cl=100
0
0.1
0.2
0.3
0.4
0.5
0.6
Distance outward from the inner lining surface (m)
(a)
(b)
Fig. 5. Relation of the water head (a) and the hydraulic gradient (b) at different lining position r-a with different Cl value when Km/ Kinner = 10. The horizontal axis is the distance outward from the inner surface of the lining, equal to r-a. Parameters derived from example tunnel.
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Fig. 6. The relationship of water head and inhomogeneous hydraulic conductivity coefficient at outer surface of the lining. Parameters derived from example tunnel.
4.4. Design chart and crack prediction
respectively. By using Fig. 7, the overall condition of the lining can be predicted based on the observed water inflow rate. An example of water inflow rate data collected in the tunnel is plotted in Fig. 7. The measured water inflow rate is 1.64 × 10−6 m3/s. Based on Fig. 7, the Cl value is 0.16, indicating the hydraulic conductivity increases in the direction of flow, from the outer surface to the inner surface of the tunnel lining. It is noted that an increase of Cl would increase the water inflow rate even beyond the discussed range. The Cl variation discussed in this section aims to illustrate a range of two orders of magnitude each side of the homogeneous case. The observed variation within that range indicates that beyond the Cl range from 0.01 to 100 there would be less significant water inflow variation.
In an engineering application such as a tunnel, the parameters are expected to vary, for example, along the length of the tunnel. Parameters such as depth to water table and hydraulic conductivity of the rock mass and the tunnel lining would have a range of values. To further demonstrate the application of the analytical equation, a water inflow factor has been defined. This is a dimensionless parameter comprising a four main variables, water inflow rate per length of tunnel (Q ), lining thickness (d ), total water head above the spring line of the tunnel (H ) and the rock mass hydraulic conductivity (Km ). One dependent variable (λ ) is also included in the water inflow factor. The dependent variable (λ ) is a simplification of the intersection of the main variables in (Eq. (24)). The dependent variable (λ ) is generated and calibrated based on a selected range of hydraulic parameter as shown in Fig. 7. The relationship of water inflow rate and inhomogeneous hydraulic conductivity coefficient Cl at the outer surface of the lining when Cl varies from 0.01 to 100. Parameters derived from the example tunnel.
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Fig. 8. The water inflow factor chart for parameters of the example tunnel. An example of collected data is shown.
Table 3. The other term used in the chart is the relation between rock mass hydraulic conductivity (Km ) and hydraulic conductivity at the inner surface (Kinner ). The relationship between water inflow factor and the inhomogeneity of a tunnel lining is shown in a chart that can cover a range of hydraulic conductivity conditions (Fig. 8). Since the dependent variable (λ ) is a simplification of the intersection of basic variables, the chart is not perfectly accurate; to quantify the inaccuracy a comparison of the calculated water flow value and the chart-derived value is presented (Fig. 9). The input parameters used in the comparison were taken from an arbitrary range shown in Table 3. The result shows a strong correlation between calculated values and the simplified chart (Fig. 9). The design chart is generated based on the analytical solution of water inflow as in Eq. (24). The design chart is valid as long as the analytical solution holds. The dependent variable (λ ) is generated based on the hydraulic parameters presented in Table 3. A larger error value is possible for the hydraulic parameters beyond the calibrated range. A measurement of water flow from the inner surface of the example tunnel is plotted in Fig. 8 to illustrate the application of this chart. For this specific location, the Cl value has been obtained from Fig. 8. This value represents the relation between inner and outer lining hydraulic conductivity. The variation of hydraulic conductivity can be attributed to crack density variation or crack width variation according to the cubic law (Fig. 10). The procedure for using such data (Tan et al., 2016) is shown in Table 4.
Fig. 9. Water flow rate is calculated compared with the water inflow rate according to the water inflow factor chart.
5. Discussions 5.1. Models of hydraulic conductivity of tunnels The typical conditions of hydraulic conductivity variation in a tunnel lining can be divided into three general types. The first type is homogeneous lining (Cl = 1), where the hydraulic conductivity through the lining is constant. This type of lining has been studied 122
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Fig. 10. Relation between crack features and inhomogeneous hydraulic conductivity coefficient (Cl ). Parameters for the example tunnel and collected data are marked.
the inner surface of the lining. The expansive water flow could induce unsaturated conditions, which would be more complex compared to the homogeneous and constrictive conditions. The hydraulic conductivity of the surrounding rock mass is usually higher than that of the concrete lining, as in our case study tunnel. The condition when the rock mass is less permeable than the concrete lining is not considered in this case. Three conditions with respect to the hydraulic conductivity of the tunnel lining are illustrated as graphs in Fig. 11. The approach in this study analysed all the three lining hydraulic conductivity conditions. In a practical engineering application, it is often difficult to identify which of the three cases a lining belong to. This information can be obtained by using the design chart as shown in Fig. 8. When the identified Cl > 1, the cracking condition inside the lining is inferred to be worse (more cracks or wider cracks) than it appears on the lining inner surface. When the identified Cl < 1, the cracking condition inside the lining is inferred to be better than it
previously (Fernandez and Moon, 2010a). The water flow rate through the homogenous lining (QCl= 1) is also used as a reference value to contrast the flow rates when the hydraulic conductivity is inhomogeneous in this study. The hydraulic head and water inflow rate under homogeneous condition is marked to compare with the inhomogeneous condition in Figs. 6 and 7. The second type is the constrictive lining (Cl > 1). In this type of lining the hydraulic conductivity decreases from the outer surface to the inner surface of the lining. The decreasing trend can be continuous or non-continuous. In this study, we have discussed a circumstance when the decreasing trend of hydraulic conductivity is continuous and linear. We showed that the hydraulic head difference through the lining decreased from approximately 7 m to less than 1 m when Cl increased from 1 to 100 as in Fig. 6. The flow rate is almost doubled when Cl increase from 1 to 100 as in Fig. 7. The third type is the expansive lining (Cl < 1). In this type of lining, the hydraulic conductivity increases from the outer surface to Table 4 Procedure for using water inflow factor chart. Steps
General explanation
Application in example tunnel
1 2
Measure the water inflow rate (Q ) per length of the tunnel Calculate the equivalent hydraulic conductivity of the inner surface (Kinner ) based on observed crack width and crack density
3
Calculate water inflow factor based on the water head (H ), lining thickness (d ), dependent variable (λ ) and the hydraulic conductivity of the rock mass (Km )
4
Determine the ratio between hydraulic conductivity of the rock mass (Km ) and the equivalent hydraulic conductivity of the inner surface (Kinner ) Identify the intersecting point of y axis value and Km /Kinner curve, and then find Cl value Predict the crack features through concrete lining according to the calculated Cl value. The relation between hydraulic conductivity of cracks features and hydraulic conductivity is quantified by Reinhardt (1997)
Measured local water inflow rate is 1.64 × 10−6 m3/s (0.099 L/min) Inner surface crack width (winner ) is 0.3 mm and inner surface crack density (Dinner ) is 0.5 m/m2, the calculated hydraulic conductivity of inner surface is approximately 1 × 10−8 m/s Water head is approximately 15 m, lining thickness is 0.56 m, and the hydraulic conductivity of rock mass is approximately 1 × 10−7 m/s based on literature. Water inflow factor is calculated to be 0.62 Based on step (2) and (3), the ratio (Km /Kinner ) is 1 × 101
5 6
123
The intersecting point is shown in Fig. 8, the Cl value is found to be 0.16 Inner surface crack width (winner ) is 0.3 mm, and Cl value is 0.16 in this example. For a constant crack density wouter is 0.16 mm. For a constant crack width Douter is approximately 0.35 m−1 (Fig. 10)
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Fig. 11. Illustrations of types of lining and rock mass hydraulic conductivity conditions. (a) Lining hydraulic conductivity homogeneous; (b) lining hydraulic conductivity constrictive; (c) lining hydraulic conductivity expansive.
5.3. Water pressure in an inhomogeneous lining
appears on the lining inner surface. The distribution of cracking along and within a lining could be interpreted with a range of Cl value readings along a tunnel. For example, a tunnel with differential settlement could have different Cl values along its length. The collected water inflow data from the case study tunnel represents a Cl value smaller than 1 (Fig. 11(c)). Hence, the hydraulic conductivity of the inner surface is predicted to be higher than the outer surface of the lining, according to our model. The case studied is a tunnel designed to be water-sealed with no drainage layer. However, in modern tunnels, a drainage layer is typically incorporated to drain away the potential inflow water and relieve the hydraulic pressure from the lining. In a drained tunnel, the primary and secondary lining is separated by a drainage layer. In such cases, the secondary lining is considered effectively impermeable as most of the water pressure has been relieved by the drainage layer. The hydraulic conductivity of the drainage layer is designed to be higher than that of both the lining and the rock mass. Impermeability of the lining is an idealised situation even where a drainage layer is present. The flow models of tunnels with a drainage layer for comparison with the simpler examples are shown in Fig. 12.
The water pressure distribution in a concrete lining can also be affected by the inhomogeneity of the lining. This effect has not been considered in previous research. Based on our study, if the Cl value is greater than 1, the water pressure at the outer surface will be lower than for the homogeneous condition (e.g., Fig. 5). However, the water pressure close to the inner surface could be higher than for the homogenous condition. The hydraulic pressure in the lining is an important parameter for structure stability analysis. Higher than expected water pressure close to the inner surface could trigger concrete spalling and then lead to structural problems (Jansson and Boström, 2010). For a selected location (Table 4), the Cl value is calculated to be 0.16, based on the collected water inflow rate and other hydraulic parameters. According to that Cl value and Fig. 6, the hydraulic head at the outer face of the lining is approximately 9.8 m in this method. In contrast, the hydraulic head at the outer face of the lining would be approximately 7.5 m for a homogeneous lining, which would underestimate the hydraulic pressure by approximately 30%. 5.4. Outer face crack prediction
5.2. Water flow rate in an inhomogeneous lining The degree of cracking of the outer face of a lining can be different from that of the inner face of the lining as discussed above. Variation of the degree of cracking will affect lining hydraulic conductivity causing water inflow variation. The relation between water inflow variation and the degree of cracking is quantified in this study. Based on our study, when the Cl value is greater than 1, cracks at the outer face of the lining are wider or more abundant than at the inner face of the lining. Cracks at the outer lining are narrower or less abundant than at the inner face of the lining when the Cl value is less than 1. The degree of cracking at the outer lining is predicted for one selected location based on the water inflow rate and other hydraulic parameters (Table 4). The crack width and crack density of the inner face of the lining at the selected location is 0.3 mm and 0.5 m/m2 respectively. Based on a homogeneous lining, the crack formation of the outer face of the lining would be assumed to be the same as the inner face. In the method presented in this paper, the cracking at the outer face of the lining is predicted to have narrower
The water flow rate through an inhomogeneous lining can be affected by the inhomogeneity of the lining as demonstrated in Section 4.3. The overall inflow rate (flux) through a lining depends on both hydraulic gradient and hydraulic conductivity of the lining; under saturated conditions, the flux is a combination of hydraulic gradient and hydraulic conductivity, according to Darcy’s law. For an underground, lined tunnel under steady state flow, variation of hydraulic conductivity will cause corresponding variation of the hydraulic gradient through the lining. An increase in Cl value will decrease the overall hydraulic conductivity of the tunnel lining, and the inflow rate will also decrease. Variation of basic variables like water table and rock mass hydraulic conductivity will also cause water flow rate variation. This study developed a multiple parameter chart to quantify the effect of these different parameters on water inflow rate (Fig. 8). The parameter values (e.g., Cl ) can also be inferred if the water flow rate is measured.
Fig. 12. Illustrations of types of hydraulic conductivity conditions of a tunnel with a drainage layer. (a) Homogeneous lining hydraulic conductivity; (b) constrictive lining hydraulic conductivity; (c) expansive lining hydraulic conductivity.
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crack width (0.16 mm) or less abundant crack density (0.35 m/m2) when other hydraulic parameters are fixed.
References Bagnoli, P., Bonfanti, M., Della Vecchia, G., Lualdi, M., Sgambi, L., 2015. A method to estimate concrete hydraulic conductivity of underground tunnel to assess lining degradation. Tunn. Undergr. Sp. Technol. 50, 415–423. http://dx.doi.org/10.1016/j. tust.2015.08.008. Bazant, Z.P., Oh, B.H., 1983. Spacing of cracks in reinforced concrete. J. Struct. Eng. 109, 2066–2085. Butscher, C., 2012. Steady-state groundwater inflow into a circular tunnel. Tunn. Undergr. Sp. Technol. 32, 158–167. http://dx.doi.org/10.1016/j.tust.2012.06.007. El Tani, M., 2003. Circular tunnel in a semi-infinite aquifer. Tunn. Undergr. Sp. Technol. 18, 49–55. Fang, Y., Guo, J., Grasmick, J., Mooney, M., 2016. The effect of external water pressure on the liner behavior of large cross-section tunnels. Tunn. Undergr. Sp. Technol. 60, 80–95. http://dx.doi.org/10.1016/j.tust.2016.07.009. Fernández, G., 1994. Behavior of pressure tunnels and guidelines for liner design. J. Geotech. Eng. 120, 1768–1791. Fernandez, G., Moon, J., 2010a. Excavation-induced hydraulic conductivity reduction around a tunnel – Part 1: guideline for estimate of ground water inflow rate. Tunn. Undergr. Sp. Technol. 25, 560–566. http://dx.doi.org/10.1016/j.tust.2010.03.006. Fernandez, G., Moon, J., 2010b. Excavation-induced hydraulic conductivity reduction around a tunnel–Part 2: verification of proposed method using numerical modeling. Tunn. Undergr. Sp. Technol. 25, 567–574. Fetter, C.W., 2001. Applied Hydrogeology. Prentice Hall. Gérard, B., Le Bellego, C., Bernard, O., 2002. Simplified modelling of calcium leaching of concrete in various environments. Mater. Struct. 35, 632–640. Goodman, R.E., Moye, D.G., Van Schalkwyk, A., Javandel, I., 1965. Ground Water Inflows during Tunnel Driving. College of Engineering, University of California. Heuer, R.E., 1995. Estimating rock tunnel water inflow. In: Proceedings of the Rapid Excavation and Tunneling Conference. Society for Mining, Metallurgy & Exploration, INC, pp. 41–60. Huang, Y., Yu, Z., Zhou, Z., 2013. Simulating groundwater inflow in the underground tunnel with a coupled fracture-matrix model. J. Hydrol. Eng. 18, 1557–1561. http:// dx.doi.org/10.1061/(ASCE)HE.1943-5584.0000455. Hwang, J.-H., Lu, C.-C., 2007. A semi-analytical method for analyzing the tunnel water inflow. Tunn. Undergr. Sp. Technol. 22, 39–46. ITA, 1991. Report on the damaging effects of water on tunnels during their working life. Tunn. Undergr. Sp. Technol. 6, 11–76. Jansson, R., Boström, L., 2010. The influence of pressure in the pore system on fire spalling of concrete. Fire Technol. 46, 217–230. Kolymbas, D., Wagner, P., 2007. Groundwater ingress to tunnels–the exact analytical solution. Tunn. Undergr. Sp. Technol. 22, 23–27. Lei, S., 1999. An analytical solution for steady flow into a tunnel. Groundwater 37, 23–26. Park, K.-H., Owatsiriwong, A., Lee, J.-G., 2008. Analytical solution for steady-state groundwater inflow into a drained circular tunnel in a semi-infinite aquifer: a revisit. Tunn. Undergr. Sp. Technol. 23, 206–209. Polubarinova-Kochina, P.Y., 1963. Theory of ground water movement. SIAM Rev. 5, 297. http://dx.doi.org/10.1137/1005089. Reinhardt, H.W., 1997. Penetration and Permeability of Concrete, Barriers to Organic and Contaminating Liquids: State-of-the-art Report Prepared by Members of the RILEM Technical Committee 146-TCF. RILEM Report, first ed. E & FN Spon, London; New York. Reinhardt, H.-W., Jooss, M., 2003. Permeability and self-healing of cracked concrete as a function of temperature and crack width. Cem. Concr. Res. 33, 981–985. http://dx. doi.org/10.1016/S0008-8846(02)01099-2. Robinson, M., Kenna, M., 1992. Report on Pumping Tests at Two Sites in the King’s Domain Garden. Rural Water Corporation, Melbourne. Snow, D.T., 1965. A parallel Plate Model of Fractured Permeable Media (Thesis). University of California, Berkeley. Tan, Y.Q., Smith, J.V., Li, C.Q., Dauth, J., 2016. Water infiltration estimation in tunnels: with reference to the Melbourne underground. In: Transforming the Future of Infrastructure through Smarter Information: Proceedings of the International Conference on Smart Infrastructure and Construction, Cambridge, pp. 597–602. http://dx.doi.org/10.1680/tfitsi.61279.597. Zhang, L., Franklin, J.A., 1993. Prediction of water flow into rock tunnels an analytical solution assuming an hydraulic conductivity gradient. Int. J. Rock Mech. Min. Sci. Geomech. 37–46.
6. Conclusions The analytical model of water inflow rate into a cylindrical tunnel under steady state, saturated conditions has been extended to the case of tunnel lining with linearly inhomogeneous hydraulic conductivity. The linear change of hydraulic conductivity of the lining is represented by an inhomogeneous hydraulic conductivity coefficient, Cl . This coefficient is used to quantify the ratio of the hydraulic conductivity at the outer surface and inner surface of the lining. Features such as cracks and joints which may allow water to flow through a concrete lining can be observed on the inner surface of a tunnel lining. The hydraulic conductivity of the inner surface of the lining can be calculated from the observed crack density and crack width. The approach proposed in this paper is to project the hydraulic conductivity from the inner surface of the lining into the body of the lining with a continuous linear distribution. Using the parameters of an example tunnel, the water head distribution through the lining was calculated for a range of Cl values. The tunnel lining is considered to be homogeneous when the Cl value equal to 1. Water inflow rate and the hydraulic head of the homogeneous condition are used as the reference to make comparison to the inhomogeneous condition. As the Cl value increases, the hydraulic conductivity of the lining also increases. Correspondingly, the water head near the outer surface of the lining decreases. In general, the assumption of homogeneous hydraulic conductivity of a tunnel lining can result in water pressure at the rock-lining interface being under-estimated by as much as 90% if the Cl value is less than 1. Conversely, the assumption of homogeneous hydraulic conductivity of a tunnel lining can result in water pressure near the inner surface of the lining being underestimated by as much as 30% if the Cl value is greater than 1. This finding has implications for stability analyses as the water pressure in and behind the tunnel lining is an important contribution to stability of a tunnel lining. A water inflow factor chart relating several different parameters has been developed to enable these parameters to be assessed for a given case. A method to interpret the cracking conditions within a concrete lining using the water inflow factor chart has been developed. The cracked concrete within the lining is described by the Cl value. The Cl value can be determined once the other hydraulic parameters including, water table elevation and water inflow rate are known. An example has been demonstrated to show the procedure for using the water inflow factor chart and the degree of inhomogeneity of a tunnel lining, due to cracking, predicted by measuring water inflow. Acknowledgments Financial support from Metro Trains Melbourne, Australia and Australian Research Council under DP140101547, LP150100413 and DP170102211 is gratefully acknowledged.
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Predicting external water pressure and cracking of a tunnel lining by measuring water inflow rate
MARK
⁎
Yuqi Tan, John V. Smith, Chun-Qing Li , Matthew Currell, Yufei Wu School of Engineering, RMIT University, Melbourne 3001, Australia
A R T I C L E I N F O
A B S T R A C T
Keywords: Tunnel Groundwater inflow Lining Crack Inhomogeneity Hydraulic conductivity
The flow of water into a tunnel, through the surrounding rock mass and the tunnel lining, has important implications for tunnel design and tunnel condition assessment. The hydraulic conductivity of the lining is a major controlling factor on the water inflow rate. Previous work had been mainly based on the assumption of constant hydraulic conductivity of the lining. In this work, the existing analytical models of water flow through a tunnel lining under steady-state, saturated conditions are extended to incorporate a linear variation of hydraulic conductivity with distance from the tunnel wall. The inhomogeneity of a lining is shown to have a significant impact on water inflow rate and water pressure distribution according to the model. The relation between lining inhomogeneity and other hydraulic parameters was established. This model can be used to predict the hydraulic pressure and crack condition at the outer face of the lining based on measured water inflow rate and the crack condition at the inner face, with significantly increased accuracy compared with the existing models based on constant hydraulic conductivity. Design charts are also developed for engineering applications.
1. Introduction Groundwater inflow is a problem for tunnels worldwide, and numerous cases of tunnel lining failure due to water inflow have been reported (ITA, 1991). Chemically aggressive water inflow can cause degradation of concrete lining (Gérard et al., 2002) and excessive hydraulic pressure in the lining can trigger concrete spalling (Jansson and Boström, 2010) and affect the structural stability of the tunnel (Fang et al., 2016). A numerical approach was developed to assess the hydraulic conductivity of the tunnel lining according to the water inflow rate when the other parameters are given (Bagnoli et al., 2015). Tunnels can be classified into three types with respect to interaction with water: unlined tunnels, drained lined tunnels and water-sealed lined tunnels (Butscher, 2012). The inflow water problem has been extensively studied for all the three cases (Polubarinova-Kochina, 1963; Goodman et al., 1965; Heuer, 1995; Zhang and Franklin, 1993; El Tani, 2003; Hwang and Lu, 2007; Lei, 1999; Kolymbas and Wagner, 2007; Park et al., 2008; Fernandez and Moon, 2010a, 2010b; Huang et al., 2013; Fang et al., 2016). These previous approaches for solving tunnel water inflow problems assume that the lining has a homogeneous hydraulic conductivity. The surrounding rock mass is also commonly assumed to have homogeneous hydraulic conductivity, although two studies with inhomogeneous properties have been conducted (Table 1). Schematic
⁎
Corresponding author. E-mail address: [email protected] (C.-Q. Li).
http://dx.doi.org/10.1016/j.tust.2017.08.015 Received 30 January 2017; Received in revised form 17 July 2017; Accepted 16 August 2017 0886-7798/ © 2017 Published by Elsevier Ltd.
illustration of the hydraulic conductivity distribution of the three types of tunnel under homogeneous conditions is shown in Fig. 1. The hydraulic conductivity of a homogeneous lining, which primarily depends on the crack width and crack density, can be estimated from the crack features of the lining (Fernandez and Moon, 2010a). The water flow rate through a crack has a cubic relationship with crack width, and a linear relationship with crack density (Snow, 1965). A reduction factor for crack roughness has been developed for application to water flow through cracked concrete (Reinhardt, 1997). Calculations of water inflow are very sensitive to the hydraulic conductivity of the lining and inhomogeneity of the lining can potentially cause significant differences in the results. The hydraulic conductivity of a lining is typically deduced from observations of joint and crack features on the inner face of the lining. Such features can vary through the lining which would affect the hydraulic conductivity along the water flow path. However such variation within the lining cannot be directly observed, and thus there is a need for methods to predict and account for this. This paper presents an analytical solution for water inflow rate and hydraulic pressure distribution in tunnel linings with inhomogeneous hydraulic conductivity, assuming saturated conditions and steady-state flow. The solution uses the hydraulic conductivity of the surface of the lining inside the tunnel as the reference value. Variation in hydraulic conductivity is considered linear within the lining. In combination with
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Therefore, the water head reduction caused by a lining, and the water head at the spring line at the outer surface, houter can be calculated as (Fernández, 1994)
Table 1 Framework for water inflow prediction models. Type of tunnel
Rock mass
Unlined
Homogenous
Lining/lining-like zone
Lined water sealing
Homogenous
Homogenous
Homogenous Inhomogeneous Inhomogeneous
Inhomogeneous Homogenous Inhomogeneous
Homogenous
Homogeneous
Homogenous Inhomogeneous
Inhomogeneous Homogenous
Inhomogeneous
Inhomogeneous
1+γ
Kolymbas and Wagner (2007) Park et al. (2008) NONE NONE NONE Fernandez and Moon (2010a) (unlined tunnel with a lining-like zone) Addressed in this paper Fernandez and Moon (2010b) (unlined tunnel with a lining-like zone) NONE
ql = dθrK (r )
where dh is the change in total hydraulic head, which comprises both hydraulic pressure head and elevation head, and dr is the change in radius. According to Darcy’s law, the water discharge Q per unit length of the tunnel for a homogenous lining (Fetter, 2001) is (2)
2πK (houter −hinner ) ln
(
(a + d) a
)
a + d⎞ a ⎠
(5)
dh dr
(6)
where dθr is the flow area of unit arc length of a unit length tunnel and K (r ) is the hydraulic conductivity of the lining, which varies with distance from the tunnel centre. The hydraulic head variation caused by the lining inhomogeneity is shown in Fig. 3. The hydraulic head decreases linearly through the homogeneous lining as shown in the right hand side of Fig. 3. The hydraulic conductivity through the lining is a constant under the homogeneous assumption. The hydraulic head decrease will be affected by the variation of the hydraulic conductivity through the lining in an inhomogeneous case. The left hand side of Fig. 3 shows the condition when hydraulic conductivity of the lining increases from the inner side of lining surface to the outer side of the lining surface. An inhomogeneous hydraulic conductivity coefficient can be defined as
where K is the hydraulic conductivity of material. For a permeable lining zone of a tunnel, the hydraulic head at the outer surface and inner surface of lining will affect the water inflow rate. The outer surface and inner surface of lining were defined using the term extrados and intrados in a published document (ITA, 1991). The water inflow rate was discussed by introducing the boundary conditions of the hydraulic head at the outer lining surface and the inner lining surface (Fernández, 1994) as
Q=
ln ⎛ ⎝
Previous research only considered the condition when the lining is homogeneous, i.e. K is a constant. Assuming a homogeneous lining is a practical simplification for a thin lining where the lining inhomogeneity would not be expected to have a significant effect on water inflow. Inhomogeneity of a lining needs to be considered when the lining is thick. For a lining which is inhomogeneous, the hydraulic conductivity K(r) is a function of r. A study of crack spacing in reinforced concrete showed that differences in strain in reinforcing elements would result a linear variation of crack spacing (Bazant and Oh, 1983). A linear form of variation of K with respect to r is assumed in this work since the crack spacing has been shown to have a linear relationship with hydraulic conductivity (Reinhardt, 1997). This study only analyses a linear variation of hydraulic conductivity, however other types of variation are also possible. The method applied in this study could also be applied to other types of lining inhomogeneity as long as the lining hydraulic conductivity varies continuously. An equation is established to describe the water inflow rate (ql ) for a unit arc length of a unit length of tunnel lining (dθr ) based on Darcy’s law as shown in Eq. (6). The hydraulic conductivity of a tunnel lining is considered to change continuously with the radial direction (K (r ) ).
(1)
dh dr
(4)
3. Water inflow through an inhomogeneous lining
To calculate the water inflow rate of a tunnel lining, a tunnel can be defined in a cylindrical coordinate system. The origin of the coordinate system is the centre point of the tunnel. All points are defined by an angle θ, and the distance r (Fig. 2). For a tunnel under the water table (Fernandez and Moon, 2010a), the hydraulic gradient i can be described as
Q = 2πKr
K Km
2H ⎞ γ = ln ⎛ ⎝a + d⎠
2. Water inflow through a homogeneous lining
dh dr
( )
where H is the vertical distance from the spring line of the tunnel to the water table (or equivalent phreatic pressure head); K is the equivalent hydraulic conductivity of the lining; Km is the hydraulic conductivity of the surrounding rock mass and γ is a coefficient related to tunnel radius and lining thickness, which is defined by
the observations of cracking at inner surface and the water inflow rate, this model can be used to estimate the level of inhomogeneity within the lining including the water pressure and as the crack condition at the outer face of the lining.
i=
H
houter = Goodman et al. (1965) Lei (1999) El Tani (2003) Zhang and Franklin (1993)
Inhomogeneous Lined drained
Reference
Cl =
(3)
where water head at the outer surface at the spring line of the lining is defined as houter ; hinner is the water head at the inner surface at the spring line of the lining; a is the tunnel inner radius (from the centre to the inner surface) and d is the lining thickness. For a non-pressurised tunnel, the pressure head at the inner surface is taken to be zero, so the hydraulic head loss across the lining is equal to the pressure head at the outer surface of the lining (at the spring line). The datum level is defined as the spring line of the tunnel.
K outer Kinner
(7)
where Kinner and K outer are the hydraulic conductivity of the inner surface of the lining and outer surface of the lining, respectively. The value of Kinner can potentially be determined directly by observation and measurement of crack density and crack width according to Reinhardt (1997).
Kinner = ζ 116
w 3ρg 12μΔ
(8)
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Fig. 1. Schematic illustration of hydraulic conductivity of three types of homogeneous tunnel.
where w is the crack width; 1 is the crack density; ζ is the crack Δ roughness factor; w is the crack width; ρ is the water density; g is the standard gravity; μ is the viscosity of water. Eq. (8) is a simplified approach to estimate the hydraulic conductivity of a bulk media along the radial direction of a tunnel. This method has been validated experimentally in water inflow tests through cracked concrete samples (Reinhardt and Jooss, 2003) and applied by Fernandez and Moon (2010a) to the hydraulic conductivity estimation of a tunnel lining. The hydraulic conductivity of the outer surface of the lining (K outer ) typically cannot be determined directly by observation. The hydraulic conductivity of any position in the lining can be represented as
K (r ) = Kinner + (Cl−1)
Kinner (r −a) d
The function that contains radius r can be rearranged as
1 d 1 = 2 =d rK (r ) r Kinner (Cl−1) + rKinner (d−aCl + a) r (αr + β )
(10)
where α = Kinner (Cl−1) and β = Kinner (d−aCl + a) . Based on Eq. (6), the hydraulic head through the lining can be written as
dh =
ql 1 dr dθ rK (r )
(11)
Integration of Eq. (11) gives
h (r ) =
(9) 117
ql d ⎡ 1 αr + β ⎤ − ln +C ⎥ β r dθ ⎢ ⎣ ⎦
(12)
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Fig. 2. Illustration of tunnel geometry parameters.
where C is a constant. To determine Eq. (12), the boundary conditions need to be considered. For a lined tunnel with relatively low water seepage velocity, the loss of velocity head around the outer lining is negligible compared with the total head. Similarly, it is assumed that any frictional loss at the rock mass-lining interface is very small and can be neglected. When the water head loss along the rock tunnel interface is negligible, the decrease of elevation is equal to the increase of pressure head along the interface. Therefore, the hydraulic head (total head) which is equal to the summation of the pressure head and elevation head at the boundary of the rock and lining is considered a constant. Since there is no water pressure at the inner surface of the lining, the total head at the inner surface is equal to the elevation head when the datum level is the spring line of the tunnel. Therefore the boundary conditions (Kolymbas and Wagner, 2007) are as follows
h|r = a = asinθ
(13)
h|r = a + d = houter
(14)
C = asinθ +
ql d ⎡ 1 αa + β ⎤ ln ⎥ dθ ⎢ a ⎣β ⎦
(15)
Substituting the boundary condition Eq. (14) into Eq. (12), the water head can be further developed as
houter =
ql d ⎡ 1 α (a + d ) + β ⎤ − ln +C ⎥ a+d dθ ⎢ ⎣ β ⎦
(16)
Eq. (16) can be rewritten as
houter =
ql d ⎡ 1 α (a + b) + β αa + β ⎤ ⎤ 1 + ⎡ ln − ln + asinθ ⎥ ⎢ ⎥ dθ ⎢ β a + b β a ⎣ ⎦⎦ ⎣ (17)
Considering Kinner (Cl−1) = α and Kinner (d−aCl + a) = β , Eq. (17) can be further developed as dC
d
⎡ln Kinner a + ld −ln Kinner a ⎤ dθ(houter −asinθ) ⎦ = −⎣ ql d Kinner (d−aCl + a)
(18)
The equation of water flow rate across a unit arc length of the lining can be expressed by simple transformation of Eq. (18) as
The boundary conditions Eqs. (13) and (14) are used to calculate the function of water head with respect to the cylindrical coordinate system. Substituting the boundary condition Eq. (13) into Eq. (12), C can be obtained as
ql =
(houter −asinθ) Kinner (aCl−a−d ) d ln
aCl a+d
dθ (19)
Fig. 3. Head loss under both homogeneous (right) and inhomogeneous (left) conditions.
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lining. The primary lining also included application of shotcrete to the tunnel wall prior to the installation of the secondary lining. The primary and secondary lining elements are together considered to comprise the lining. The hydraulic conductivity can be variable through the lining due to the different allocation of reinforcement as in Fig. 4. The tunnel was designed as a water-sealed tunnel to prevent water table draw down. Even though the concrete lining was designed to minimise the water infiltration, cracks in the concrete lining have been observed. The mean crack width (winner ) is 0.3 mm, and the crack density is 0.5 m/m2. The hydraulic conductivity of the inner surface of the lining is approximately 1 × 10−8 m/s based on the observed crack density and crack width, according to the model outlined earlier (Eq. (8)). The mean value of hydraulic conductivity of the surrounding rock mass is approximately 1 × 10−7 m/s based on previous geological surveys (Robinson and Kenna, 1992). The value of the hydraulic parameters used in calculation is summarized in Table 2. The hydraulic parameter for this case varies in a range as shown in Table 3. The variation of the hydraulic parameter will be used for calibration purpose. It has also been reported generally, that the type of reinforcement in a concrete lining can result in variations of crack width and crack density in different parts of the concrete (Bazant and Oh, 1983). While observations of crack density and crack width exposed on the inner surface of the lining of the tunnel were used to estimate the hydraulic conductivity, the crack density and crack width within the lining (and therefore potential variation in hydraulic conductivity within the lining), is not known. Based on the distribution of support elements within the concrete lining, inhomogeneous hydraulic conductivity is considered a likely condition for the lining of the example tunnel. In this study we incorporate the potential variability of hydraulic conductivity through the lining into the model, as described in Section 3. We consider the linear inhomogeneous hydraulic conductivity coefficient (Cl ) across a range from 0.01 to 100. The concept of Cl is proposed as a practical way to frame the problem as the condition of the inner surface of a tunnel lining can be observed by direct inspection or by use of devices such as televiewers.
The total water inflow rate of a unit length of tunnel Ql with a cylindrical lining can be obtained by integrating θ, to give
2πhouter Kinner (aCl−a−d )
Ql =
d ln
aCl a+d
(20)
If the tunnel has a constant total head at the interface of rock and lining, previous works (Lei, 1999; Kolymbas and Wagner, 2007; Park et al., 2008) have shown that the water flow rate through the rock mass can be written as
2πKm (H −houter )
Qm =
D ln ⎛ a + d + ⎝
D2 −1 ⎞ (a + d)2
(21)
⎠
where D is the distance from tunnel spring line to the water table or the ground surface, depending on which one is smaller; the hydraulic conductivity of the surrounding rock mass is Km . Since water is considered to be an incompressible fluid in this study, water flow rate through a lining under saturated conditions is equal to the water flow rate through the rock mass, based on mass conservation. Combining Eqs. (20) and (21), the following equation can be obtained
2πhouter Kinner (aCl−a−d ) d ln
aCl a+d
2πKm (H −houter )
=
D2 −1 ⎞ (a + d)2
D
ln ⎛ a + d + ⎝
⎠
(22)
Transforming (22), the water head at the outer surface (houter ) of the lining can be expressed as
H
houter =
⎛ D + Kinner (aCl − a − d)ln ⎜ a+d ⎝ Km dln
D2 (a + d)2
⎞ − 1⎟ ⎠
aCl a+d
+1 (23)
Substituting Eq. (23) into Eq. (20), the water inflow rate per unit length of tunnel Q can be expressed as
2πKinner (aCl−a−d ) H
Q= d ln
aCl a+d
⎡ Kinner (aCl − a − d)ln ⎛⎜ D + a+d ⎢ ⎝ aCl ⎢ Km dln a+d ⎢ ⎣
D2 (a + d)2
⎞ − 1⎟ ⎠
⎤ + 1⎥ ⎥ ⎥ ⎦
4.2. Water head variation (24)
The result of the analytical solution (Eq. (24)) shows the water head behaviour at the spring line through the lining for Cl values of 0.01, 0.1, 1, 10 and 100 (Fig. 5). For the homogeneous case (Cl = 1) it can be seen that the water head distribution is slightly non-linear due to the effect of flow convergence described above (Fig. 5). For cases of inhomogeneous hydraulic conductivity coefficient (Cl ) the water head variation through the lining is distinctly non-linear (Fig. 5(a)). When Cl > 1, the hydraulic gradient decreases from the inner surface to the outer surface. This relationship becomes more pronounced as Cl increases (Fig. 5(b)). The relationship is due to the relatively minor influence of the flow area change and relatively major influence of the increase in average hydraulic conductivity, represented by Cl > 1. In contrast, the hydraulic gradient decreases through the lining when Cl < 1. Increasing Cl represents an increase in hydraulic conductivity at the outer surface while the hydraulic conductivity at the inner surface is fixed. Therefore, as Cl increases, the average hydraulic conductivity of the lining would also increase. The average or equivalent hydraulic conductivity of the lining has not been defined or used in this study because it cannot be independently verified. Rather, the hydraulic conductivity at the inner surface has been used as the reference (as this can be estimated using direct observation) and hydraulic conductivity at the outer surface is a variable factor defined according to Cl . The variation of hydraulic conductivity in an inhomogeneous lining will cause hydraulic head redistribution at the outer surface. The water head at the outer surface (x axis = 0.56, Fig. 5) will decrease if the average hydraulic conductivity of lining decreases based on Eq. (4). In general,
All the parameters have been defined above. To calculate the hydraulic head in the lining, Eqs. (15) and (19) are substituted into the water head distribution function Eq. (12) and combined with the water head at the outer surface of the lining (Eq. (23)). The water head through the lining can be calculated as
h (r ) =
(houter −asinθ) ln
aCl a+d
ln
a [r (Cl−1) + a + d−aCl] + asinθ rd
(25)
For any position within the lining with given r and θ , the water head in the lining can be calculated accordingly. 4. Application 4.1. Parameters used in application An example tunnel (Fig. 4) was selected for the purpose of illustrating the influence of variable hydraulic conductivity through a lining. The example tunnel is cylindrical and approximately 35 m underground. The water table is approximately 15 m above the spring line of the tunnel. The radius (a ) of the tunnel from the tunnel centre to the inner surface of the lining is 3000 mm and the thickness of the cast in situ concrete lining (d ) is 560 mm. The tunnel lining comprises steel beams as primary support embedded in the cast-in-situ lining near its outer surface. Steel reinforcement bars were cast in the concrete close to the inner surface of the 119
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Fig. 4. Illustration of the example concrete lining with steel reinforcement elements (a) transverse section and (b) longitudinal section.
Fig. 6 shows the water head at the outer surface for various values of Cl . When Cl increases from 0.01 to 1, the water head at the outer surface decreases. When Cl increases from 1 to 100, the water head at the outer surface of the lining drops from 6.7 m to less than 1 m. The homogeneous case represents an inflection point on the curve representing the range of inhomogeneous conditions (Fig. 6).
Table 2 Parameters from the example tunnel. Parameter
Km (m/s) −7
Value
1 × 10
Kinner (m/s) 1 × 10
−8
a (m)
d (m)
H (m)
3
0.56
15
Table 3 Parameter ranges used for validation of the water inflow factor chart.
Km (m/s) −7
Low Middle High
1 × 10 1 × 10−6 1 × 10−5
Kinner (m/s) −9
1 × 10 1 × 10−7 1 × 10−5
4.3. Water inflow rate variation
a (m)
d (m)
H (m)
Cl
2.8 3 3.2
0.5 0.56 0.6
5 15 20
0.01 1 100
In addition to influencing water head distribution, the Cl value will also influence the water inflow rate. When Cl increases from 0.01 to 1, the water inflow rate increases (Fig. 7). The variation of water inflow rate is controlled by both hydraulic conductivity and hydraulic head. The homogeneous condition again represents an inflection point in the interplay of these parameters (Fig. 7). For the homogeneous condition the calculated water inflow rate for the example tunnel is 2.46 × 10−6 m3/s. When Cl increases from 0.01 to 100, the water inflow increases, ranging from 9.7 × 10−7 m3/s to 4.3 × 10−6 m3/s,
lower average hydraulic conductivity of the lining will allow greater hydraulic pressure at the outer surface, if all other parameters are constant.
15
120
100
60
40
Outer lining surface
Cl=10
80
Inner lining surface
6
Hydraulic gradient
9
Outer lining surface
Inner lining surface
Water head (m)
12
3
20 Cl=100
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0
Distance outward from the inner lining surface (m)
Cl=10
Cl=100
0
0.1
0.2
0.3
0.4
0.5
0.6
Distance outward from the inner lining surface (m)
(a)
(b)
Fig. 5. Relation of the water head (a) and the hydraulic gradient (b) at different lining position r-a with different Cl value when Km/ Kinner = 10. The horizontal axis is the distance outward from the inner surface of the lining, equal to r-a. Parameters derived from example tunnel.
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Fig. 6. The relationship of water head and inhomogeneous hydraulic conductivity coefficient at outer surface of the lining. Parameters derived from example tunnel.
4.4. Design chart and crack prediction
respectively. By using Fig. 7, the overall condition of the lining can be predicted based on the observed water inflow rate. An example of water inflow rate data collected in the tunnel is plotted in Fig. 7. The measured water inflow rate is 1.64 × 10−6 m3/s. Based on Fig. 7, the Cl value is 0.16, indicating the hydraulic conductivity increases in the direction of flow, from the outer surface to the inner surface of the tunnel lining. It is noted that an increase of Cl would increase the water inflow rate even beyond the discussed range. The Cl variation discussed in this section aims to illustrate a range of two orders of magnitude each side of the homogeneous case. The observed variation within that range indicates that beyond the Cl range from 0.01 to 100 there would be less significant water inflow variation.
In an engineering application such as a tunnel, the parameters are expected to vary, for example, along the length of the tunnel. Parameters such as depth to water table and hydraulic conductivity of the rock mass and the tunnel lining would have a range of values. To further demonstrate the application of the analytical equation, a water inflow factor has been defined. This is a dimensionless parameter comprising a four main variables, water inflow rate per length of tunnel (Q ), lining thickness (d ), total water head above the spring line of the tunnel (H ) and the rock mass hydraulic conductivity (Km ). One dependent variable (λ ) is also included in the water inflow factor. The dependent variable (λ ) is a simplification of the intersection of the main variables in (Eq. (24)). The dependent variable (λ ) is generated and calibrated based on a selected range of hydraulic parameter as shown in Fig. 7. The relationship of water inflow rate and inhomogeneous hydraulic conductivity coefficient Cl at the outer surface of the lining when Cl varies from 0.01 to 100. Parameters derived from the example tunnel.
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Fig. 8. The water inflow factor chart for parameters of the example tunnel. An example of collected data is shown.
Table 3. The other term used in the chart is the relation between rock mass hydraulic conductivity (Km ) and hydraulic conductivity at the inner surface (Kinner ). The relationship between water inflow factor and the inhomogeneity of a tunnel lining is shown in a chart that can cover a range of hydraulic conductivity conditions (Fig. 8). Since the dependent variable (λ ) is a simplification of the intersection of basic variables, the chart is not perfectly accurate; to quantify the inaccuracy a comparison of the calculated water flow value and the chart-derived value is presented (Fig. 9). The input parameters used in the comparison were taken from an arbitrary range shown in Table 3. The result shows a strong correlation between calculated values and the simplified chart (Fig. 9). The design chart is generated based on the analytical solution of water inflow as in Eq. (24). The design chart is valid as long as the analytical solution holds. The dependent variable (λ ) is generated based on the hydraulic parameters presented in Table 3. A larger error value is possible for the hydraulic parameters beyond the calibrated range. A measurement of water flow from the inner surface of the example tunnel is plotted in Fig. 8 to illustrate the application of this chart. For this specific location, the Cl value has been obtained from Fig. 8. This value represents the relation between inner and outer lining hydraulic conductivity. The variation of hydraulic conductivity can be attributed to crack density variation or crack width variation according to the cubic law (Fig. 10). The procedure for using such data (Tan et al., 2016) is shown in Table 4.
Fig. 9. Water flow rate is calculated compared with the water inflow rate according to the water inflow factor chart.
5. Discussions 5.1. Models of hydraulic conductivity of tunnels The typical conditions of hydraulic conductivity variation in a tunnel lining can be divided into three general types. The first type is homogeneous lining (Cl = 1), where the hydraulic conductivity through the lining is constant. This type of lining has been studied 122
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Fig. 10. Relation between crack features and inhomogeneous hydraulic conductivity coefficient (Cl ). Parameters for the example tunnel and collected data are marked.
the inner surface of the lining. The expansive water flow could induce unsaturated conditions, which would be more complex compared to the homogeneous and constrictive conditions. The hydraulic conductivity of the surrounding rock mass is usually higher than that of the concrete lining, as in our case study tunnel. The condition when the rock mass is less permeable than the concrete lining is not considered in this case. Three conditions with respect to the hydraulic conductivity of the tunnel lining are illustrated as graphs in Fig. 11. The approach in this study analysed all the three lining hydraulic conductivity conditions. In a practical engineering application, it is often difficult to identify which of the three cases a lining belong to. This information can be obtained by using the design chart as shown in Fig. 8. When the identified Cl > 1, the cracking condition inside the lining is inferred to be worse (more cracks or wider cracks) than it appears on the lining inner surface. When the identified Cl < 1, the cracking condition inside the lining is inferred to be better than it
previously (Fernandez and Moon, 2010a). The water flow rate through the homogenous lining (QCl= 1) is also used as a reference value to contrast the flow rates when the hydraulic conductivity is inhomogeneous in this study. The hydraulic head and water inflow rate under homogeneous condition is marked to compare with the inhomogeneous condition in Figs. 6 and 7. The second type is the constrictive lining (Cl > 1). In this type of lining the hydraulic conductivity decreases from the outer surface to the inner surface of the lining. The decreasing trend can be continuous or non-continuous. In this study, we have discussed a circumstance when the decreasing trend of hydraulic conductivity is continuous and linear. We showed that the hydraulic head difference through the lining decreased from approximately 7 m to less than 1 m when Cl increased from 1 to 100 as in Fig. 6. The flow rate is almost doubled when Cl increase from 1 to 100 as in Fig. 7. The third type is the expansive lining (Cl < 1). In this type of lining, the hydraulic conductivity increases from the outer surface to Table 4 Procedure for using water inflow factor chart. Steps
General explanation
Application in example tunnel
1 2
Measure the water inflow rate (Q ) per length of the tunnel Calculate the equivalent hydraulic conductivity of the inner surface (Kinner ) based on observed crack width and crack density
3
Calculate water inflow factor based on the water head (H ), lining thickness (d ), dependent variable (λ ) and the hydraulic conductivity of the rock mass (Km )
4
Determine the ratio between hydraulic conductivity of the rock mass (Km ) and the equivalent hydraulic conductivity of the inner surface (Kinner ) Identify the intersecting point of y axis value and Km /Kinner curve, and then find Cl value Predict the crack features through concrete lining according to the calculated Cl value. The relation between hydraulic conductivity of cracks features and hydraulic conductivity is quantified by Reinhardt (1997)
Measured local water inflow rate is 1.64 × 10−6 m3/s (0.099 L/min) Inner surface crack width (winner ) is 0.3 mm and inner surface crack density (Dinner ) is 0.5 m/m2, the calculated hydraulic conductivity of inner surface is approximately 1 × 10−8 m/s Water head is approximately 15 m, lining thickness is 0.56 m, and the hydraulic conductivity of rock mass is approximately 1 × 10−7 m/s based on literature. Water inflow factor is calculated to be 0.62 Based on step (2) and (3), the ratio (Km /Kinner ) is 1 × 101
5 6
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The intersecting point is shown in Fig. 8, the Cl value is found to be 0.16 Inner surface crack width (winner ) is 0.3 mm, and Cl value is 0.16 in this example. For a constant crack density wouter is 0.16 mm. For a constant crack width Douter is approximately 0.35 m−1 (Fig. 10)
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Fig. 11. Illustrations of types of lining and rock mass hydraulic conductivity conditions. (a) Lining hydraulic conductivity homogeneous; (b) lining hydraulic conductivity constrictive; (c) lining hydraulic conductivity expansive.
5.3. Water pressure in an inhomogeneous lining
appears on the lining inner surface. The distribution of cracking along and within a lining could be interpreted with a range of Cl value readings along a tunnel. For example, a tunnel with differential settlement could have different Cl values along its length. The collected water inflow data from the case study tunnel represents a Cl value smaller than 1 (Fig. 11(c)). Hence, the hydraulic conductivity of the inner surface is predicted to be higher than the outer surface of the lining, according to our model. The case studied is a tunnel designed to be water-sealed with no drainage layer. However, in modern tunnels, a drainage layer is typically incorporated to drain away the potential inflow water and relieve the hydraulic pressure from the lining. In a drained tunnel, the primary and secondary lining is separated by a drainage layer. In such cases, the secondary lining is considered effectively impermeable as most of the water pressure has been relieved by the drainage layer. The hydraulic conductivity of the drainage layer is designed to be higher than that of both the lining and the rock mass. Impermeability of the lining is an idealised situation even where a drainage layer is present. The flow models of tunnels with a drainage layer for comparison with the simpler examples are shown in Fig. 12.
The water pressure distribution in a concrete lining can also be affected by the inhomogeneity of the lining. This effect has not been considered in previous research. Based on our study, if the Cl value is greater than 1, the water pressure at the outer surface will be lower than for the homogeneous condition (e.g., Fig. 5). However, the water pressure close to the inner surface could be higher than for the homogenous condition. The hydraulic pressure in the lining is an important parameter for structure stability analysis. Higher than expected water pressure close to the inner surface could trigger concrete spalling and then lead to structural problems (Jansson and Boström, 2010). For a selected location (Table 4), the Cl value is calculated to be 0.16, based on the collected water inflow rate and other hydraulic parameters. According to that Cl value and Fig. 6, the hydraulic head at the outer face of the lining is approximately 9.8 m in this method. In contrast, the hydraulic head at the outer face of the lining would be approximately 7.5 m for a homogeneous lining, which would underestimate the hydraulic pressure by approximately 30%. 5.4. Outer face crack prediction
5.2. Water flow rate in an inhomogeneous lining The degree of cracking of the outer face of a lining can be different from that of the inner face of the lining as discussed above. Variation of the degree of cracking will affect lining hydraulic conductivity causing water inflow variation. The relation between water inflow variation and the degree of cracking is quantified in this study. Based on our study, when the Cl value is greater than 1, cracks at the outer face of the lining are wider or more abundant than at the inner face of the lining. Cracks at the outer lining are narrower or less abundant than at the inner face of the lining when the Cl value is less than 1. The degree of cracking at the outer lining is predicted for one selected location based on the water inflow rate and other hydraulic parameters (Table 4). The crack width and crack density of the inner face of the lining at the selected location is 0.3 mm and 0.5 m/m2 respectively. Based on a homogeneous lining, the crack formation of the outer face of the lining would be assumed to be the same as the inner face. In the method presented in this paper, the cracking at the outer face of the lining is predicted to have narrower
The water flow rate through an inhomogeneous lining can be affected by the inhomogeneity of the lining as demonstrated in Section 4.3. The overall inflow rate (flux) through a lining depends on both hydraulic gradient and hydraulic conductivity of the lining; under saturated conditions, the flux is a combination of hydraulic gradient and hydraulic conductivity, according to Darcy’s law. For an underground, lined tunnel under steady state flow, variation of hydraulic conductivity will cause corresponding variation of the hydraulic gradient through the lining. An increase in Cl value will decrease the overall hydraulic conductivity of the tunnel lining, and the inflow rate will also decrease. Variation of basic variables like water table and rock mass hydraulic conductivity will also cause water flow rate variation. This study developed a multiple parameter chart to quantify the effect of these different parameters on water inflow rate (Fig. 8). The parameter values (e.g., Cl ) can also be inferred if the water flow rate is measured.
Fig. 12. Illustrations of types of hydraulic conductivity conditions of a tunnel with a drainage layer. (a) Homogeneous lining hydraulic conductivity; (b) constrictive lining hydraulic conductivity; (c) expansive lining hydraulic conductivity.
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crack width (0.16 mm) or less abundant crack density (0.35 m/m2) when other hydraulic parameters are fixed.
References Bagnoli, P., Bonfanti, M., Della Vecchia, G., Lualdi, M., Sgambi, L., 2015. A method to estimate concrete hydraulic conductivity of underground tunnel to assess lining degradation. Tunn. Undergr. Sp. Technol. 50, 415–423. http://dx.doi.org/10.1016/j. tust.2015.08.008. Bazant, Z.P., Oh, B.H., 1983. Spacing of cracks in reinforced concrete. J. Struct. Eng. 109, 2066–2085. Butscher, C., 2012. Steady-state groundwater inflow into a circular tunnel. Tunn. Undergr. Sp. Technol. 32, 158–167. http://dx.doi.org/10.1016/j.tust.2012.06.007. El Tani, M., 2003. Circular tunnel in a semi-infinite aquifer. Tunn. Undergr. Sp. Technol. 18, 49–55. Fang, Y., Guo, J., Grasmick, J., Mooney, M., 2016. The effect of external water pressure on the liner behavior of large cross-section tunnels. Tunn. Undergr. Sp. Technol. 60, 80–95. http://dx.doi.org/10.1016/j.tust.2016.07.009. Fernández, G., 1994. Behavior of pressure tunnels and guidelines for liner design. J. Geotech. Eng. 120, 1768–1791. Fernandez, G., Moon, J., 2010a. Excavation-induced hydraulic conductivity reduction around a tunnel – Part 1: guideline for estimate of ground water inflow rate. Tunn. Undergr. Sp. Technol. 25, 560–566. http://dx.doi.org/10.1016/j.tust.2010.03.006. Fernandez, G., Moon, J., 2010b. Excavation-induced hydraulic conductivity reduction around a tunnel–Part 2: verification of proposed method using numerical modeling. Tunn. Undergr. Sp. Technol. 25, 567–574. Fetter, C.W., 2001. Applied Hydrogeology. Prentice Hall. Gérard, B., Le Bellego, C., Bernard, O., 2002. Simplified modelling of calcium leaching of concrete in various environments. Mater. Struct. 35, 632–640. Goodman, R.E., Moye, D.G., Van Schalkwyk, A., Javandel, I., 1965. Ground Water Inflows during Tunnel Driving. College of Engineering, University of California. Heuer, R.E., 1995. Estimating rock tunnel water inflow. In: Proceedings of the Rapid Excavation and Tunneling Conference. Society for Mining, Metallurgy & Exploration, INC, pp. 41–60. Huang, Y., Yu, Z., Zhou, Z., 2013. Simulating groundwater inflow in the underground tunnel with a coupled fracture-matrix model. J. Hydrol. Eng. 18, 1557–1561. http:// dx.doi.org/10.1061/(ASCE)HE.1943-5584.0000455. Hwang, J.-H., Lu, C.-C., 2007. A semi-analytical method for analyzing the tunnel water inflow. Tunn. Undergr. Sp. Technol. 22, 39–46. ITA, 1991. Report on the damaging effects of water on tunnels during their working life. Tunn. Undergr. Sp. Technol. 6, 11–76. Jansson, R., Boström, L., 2010. The influence of pressure in the pore system on fire spalling of concrete. Fire Technol. 46, 217–230. Kolymbas, D., Wagner, P., 2007. Groundwater ingress to tunnels–the exact analytical solution. Tunn. Undergr. Sp. Technol. 22, 23–27. Lei, S., 1999. An analytical solution for steady flow into a tunnel. Groundwater 37, 23–26. Park, K.-H., Owatsiriwong, A., Lee, J.-G., 2008. Analytical solution for steady-state groundwater inflow into a drained circular tunnel in a semi-infinite aquifer: a revisit. Tunn. Undergr. Sp. Technol. 23, 206–209. Polubarinova-Kochina, P.Y., 1963. Theory of ground water movement. SIAM Rev. 5, 297. http://dx.doi.org/10.1137/1005089. Reinhardt, H.W., 1997. Penetration and Permeability of Concrete, Barriers to Organic and Contaminating Liquids: State-of-the-art Report Prepared by Members of the RILEM Technical Committee 146-TCF. RILEM Report, first ed. E & FN Spon, London; New York. Reinhardt, H.-W., Jooss, M., 2003. Permeability and self-healing of cracked concrete as a function of temperature and crack width. Cem. Concr. Res. 33, 981–985. http://dx. doi.org/10.1016/S0008-8846(02)01099-2. Robinson, M., Kenna, M., 1992. Report on Pumping Tests at Two Sites in the King’s Domain Garden. Rural Water Corporation, Melbourne. Snow, D.T., 1965. A parallel Plate Model of Fractured Permeable Media (Thesis). University of California, Berkeley. Tan, Y.Q., Smith, J.V., Li, C.Q., Dauth, J., 2016. Water infiltration estimation in tunnels: with reference to the Melbourne underground. In: Transforming the Future of Infrastructure through Smarter Information: Proceedings of the International Conference on Smart Infrastructure and Construction, Cambridge, pp. 597–602. http://dx.doi.org/10.1680/tfitsi.61279.597. Zhang, L., Franklin, J.A., 1993. Prediction of water flow into rock tunnels an analytical solution assuming an hydraulic conductivity gradient. Int. J. Rock Mech. Min. Sci. Geomech. 37–46.
6. Conclusions The analytical model of water inflow rate into a cylindrical tunnel under steady state, saturated conditions has been extended to the case of tunnel lining with linearly inhomogeneous hydraulic conductivity. The linear change of hydraulic conductivity of the lining is represented by an inhomogeneous hydraulic conductivity coefficient, Cl . This coefficient is used to quantify the ratio of the hydraulic conductivity at the outer surface and inner surface of the lining. Features such as cracks and joints which may allow water to flow through a concrete lining can be observed on the inner surface of a tunnel lining. The hydraulic conductivity of the inner surface of the lining can be calculated from the observed crack density and crack width. The approach proposed in this paper is to project the hydraulic conductivity from the inner surface of the lining into the body of the lining with a continuous linear distribution. Using the parameters of an example tunnel, the water head distribution through the lining was calculated for a range of Cl values. The tunnel lining is considered to be homogeneous when the Cl value equal to 1. Water inflow rate and the hydraulic head of the homogeneous condition are used as the reference to make comparison to the inhomogeneous condition. As the Cl value increases, the hydraulic conductivity of the lining also increases. Correspondingly, the water head near the outer surface of the lining decreases. In general, the assumption of homogeneous hydraulic conductivity of a tunnel lining can result in water pressure at the rock-lining interface being under-estimated by as much as 90% if the Cl value is less than 1. Conversely, the assumption of homogeneous hydraulic conductivity of a tunnel lining can result in water pressure near the inner surface of the lining being underestimated by as much as 30% if the Cl value is greater than 1. This finding has implications for stability analyses as the water pressure in and behind the tunnel lining is an important contribution to stability of a tunnel lining. A water inflow factor chart relating several different parameters has been developed to enable these parameters to be assessed for a given case. A method to interpret the cracking conditions within a concrete lining using the water inflow factor chart has been developed. The cracked concrete within the lining is described by the Cl value. The Cl value can be determined once the other hydraulic parameters including, water table elevation and water inflow rate are known. An example has been demonstrated to show the procedure for using the water inflow factor chart and the degree of inhomogeneity of a tunnel lining, due to cracking, predicted by measuring water inflow. Acknowledgments Financial support from Metro Trains Melbourne, Australia and Australian Research Council under DP140101547, LP150100413 and DP170102211 is gratefully acknowledged.
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