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Investigation of steady water inflow into a subsea grouted tunnel
Tunnelling and Underground Space Technology 80 (2018) 92–102
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Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
Investigation of steady water inflow into a subsea grouted tunnel Pengfei Li, Fan Wang, Yingying Long, Xu Zhao
⁎
T
The Key Laboratory of Urban Security and Disaster Engineering, Ministry of Education, Beijing University of Technology, Beijing 100124, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Subsea grouted tunnel Steady seepage field Analytical solution Numerical simulation Permeability of grouted zone
This paper investigates analytical solutions for steady water inflow into a subsea grouted tunnel. The relevant parameters include the pore pressure distribution in the aquifer, external water pressure on the grouted zone and water inflow. Analytical solutions are obtained by the complex variable method (CVM), mirror image method (MIM) and axisymmetric modeling method (AMM). A series of numerical simulations are performed to validate the analytical solutions. The calculation accuracy and application conditions of each method are stated by comparisons of the pore pressure distribution and water inflow. Finally, the influencing factors, such as the boundary conditions, permeability of grouted zone and water depth, are discussed. It is suggested that the width of the lateral boundary should be at least nine times the tunnel diameter when conducting a numerical simulation for the steady seepage field. The relative permeability (i.e., kr/kg) has influences on comparisons between the analytical solutions and numerical solution, but the results remain in an acceptable range. It is also found that both the external water pressure on the grouted zone and water inflow increase linearly with the rising of the water table.
1. Introduction Since the development of modern tunnel engineering, the subsea tunnel has become widely used for crossing rivers or seas. There are many advantages to a subsea tunnel, such as being unaffected by bad weather, requiring few land resources, and operating simultaneously with shipping operations that use the waterway (Fang et al., 2015). A subsea tunnel has a potentially infinite supply of water which cannot naturally drain off using a V-type longitudinal slope design. If full sealing is adopted to achieve a waterproof tunnel, the tunnel lining would have to sustain high water pressure for a long duration, which would result in water leakage and subsequent tunnel lining failure. If full drainage is implemented in the tunnel construction, a large-scale pumping system is needed, requiring a large amount of electric energy at high cost. Moreover, the water flowing into a subsea tunnel has strong corrosive capabilities, threatening the durability of the tunnel lining. If blocking with limited drainage is used, the water discharged into the tunnel is controlled and external water pressure on the tunnel lining is significantly reduced (Wang et al., 2008), an approach considered economical and safe. The key to the design of blocking with limited discharge is to obtain the distribution characteristics of the seepage field, including the water inflow and pore pressure of the soil and external water pressure of tunnel lining, which can be analyzed by field monitoring, numerical simulation and theoretical analysis. Using monitoring data, Farhadian
⁎
Corresponding author. E-mail address: [email protected] (X. Zhao).
https://doi.org/10.1016/j.tust.2018.06.003 Received 12 December 2017; Received in revised form 16 April 2018; Accepted 6 June 2018 0886-7798/ © 2018 Elsevier Ltd. All rights reserved.
and Katibeh (2017) presented a new empirical model to evaluate groundwater flow into a circular tunnel using multiple regression analysis. The method of field monitoring is important for tunnels in construction, to avoid potential deformations and failures. Also, in designing tunnels in similar ground conditions and lining types, former monitoring data could be significant references for the new design in similar projects. Compared with field monitoring, numerical simulation can obtain the spatial distribution characteristics of the seepage field against the complex formation conditions in practical engineering. Many researchers have applied numerical simulation to calculate the external water pressure on a tunnel lining and predict the water inflow (Lee and Nam, 2001, 2004; Shin et al., 2002; Ivars, 2006; Lee et al., 2007; Arjnoi et al., 2009; Butscher, 2012). In addition, complexities such as highly pervious geological features and complex drainage systems, which affect groundwater flow into a tunnel, can be investigated using numerical simulation (Moon and Jeong, 2011; Chen et al., 2008). Theoretical analysis is an efficient and convenient method for calculating the pore pressure distribution and predicting the water inflow into a tunnel. Harr (1962) presented the pore pressure distribution of a circular tunnel using the mirror image method. Goodman et al. (1965) predicted the analytical solutions of the water inflow for steady water inflow into undersea tunnels. Assuming a constant hydraulic head at the tunnel perimeter, Lei (1999) derived the analytical solutions to the hydraulic head distribution and water inflow for two-dimensional, steady water flow into a horizontal tunnel in a fully saturated,
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the water table to the ground surface is hw. The permeabilities of the aquifer and grouted zone are expressed as kr and kg, respectively. To obtain the analytical solutions for the steady seepage field of a subsea grouted tunnel, several simplifying assumptions were made:
homogeneous, isotropic and semi-infinite aquifer. Based on the Mobius transformation and Fourier series, El Tani (2003) gave the exact analytical solutions to the water inflow, pressure, leakage and recharging infiltrations for a circular tunnel in a semi-infinite, homogeneous and isotropic aquifer. Kolymbas and Wagner (2007) obtained the analytical expressions of the steady state groundwater ingress into a circular tunnel using conformal mapping. The derived equations can be generally applied to deep and shallow tunnels. For deep tunnels, Wang et al. (2008) studied the influence of controlled drainage on the external water pressure on a tunnel lining by theoretical and experimental methods. The results drainage measures were necessary to reduce the external water pressure on the tunnel lining. Moreover, considering the constant hydraulic head and constant water pressure boundary conditions at the tunnel perimeter, some researchers (Park et al., 2008; Huangfu et al., 2010) derived the analytical solutions to the total hydraulic head (or pore pressure) distribution and water inflow for the steady seepage into a drained circular tunnel in a semi-infinite aquifer. Zhang et al. (2017) investigated the analytical solutions to the seepage field for a lined tunnel considering the grouting effect and validated the analytical solutions using a numerical simulation. Besides, other researches were performed to analyze the effects of the hydraulic conductivity gradient and nonlinear consolidation on the analytical solutions of the water inflow and pore pressure distribution (Zhang and Franklin, 1993; Cao et al., 2014). As discussed, the design of tunnels below the water table has been systematically investigated by many researchers, many numerical simulations have been performed and several significant analytical solutions for the steady seepage field of tunnels have been obtained. However, most of these analytical solutions were obtained either neglecting the impact of the grouting permeability (without considering the grouted zone) or significantly simplifying the boundary conditions, and most of the numerical simulations didn’t clarify the influences of the boundary conditions, either. This paper focuses on the analytical solutions for the steady seepage field of a subsea grouted tunnel, validated by a series of numerical simulations and discusses influencing factors, such as the boundary conditions, water depth and relative permeability between the grouted zone and natural ground.
(1) The tunnel has a circular cross-section and is located in a fully saturated, homogeneous, isotropic and semi-infinite aquifer; (2) The grouting material is homogeneous and isotropic; (3) A state of steady flow is assumed; (4) The fluid is incompressible; (5) The water table is horizontal and remains unchanged. According to Darcy’s law and conservation of mass, in fully saturated, homogeneous, isotropic media, the differential equation for twodimensional steady-state groundwater flow is given by the Laplace equation:
∂ 2H ∂ 2H + =0 2 ∂x ∂y 2
(1)
where H is the total hydraulic head and is equal to the sum of the pressure head and elevation head, that is
H=
p +y γw
(2)
where p is the water pressure, γw is the unit weight of water and y is the elevation head. 3. Analytical solutions Considering the ground surface as the elevation reference datum in Fig. 1, the boundary conditions of the steady seepage field are given as follows: (1) The total hydraulic head at the ground surface remains constant and is defined as hw; (2) A constant total hydraulic headhrg at the outer boundary of grouted zone is assumed; (3) Constant water pressure pr0 at the inner boundary of grouted zone is assumed. Because of the complex boundary conditions, it is difficult to obtain analytical solutions to the steady seepage field by solving Eq. (1) directly. Consequently, the complex variable method (CVM), mirror image method (MIM) and axisymmetric modelling method (AMM) are introduced to obtain the analytical solutions for the steady seepage field of a subsea grouted tunnel.
2. Problem description Fig. 1 shows the model for analyzing the steady seepage field in a subsea grouted tunnel. In this model, r0 and rg represent the internal and external radii of the grouted zone, respectively. The tunnel depth from the tunnel center to the ground surface is h. The water depth from
3.1. Analytical solutions by the complex variable method (CVM) The complex boundary conditions can be mapped conformally to simple boundary conditions by adopting the conformal mapping technique (Fang et al., 2015; Huangfu et al., 2010; Zhang et al., 2017; Bobet and Yu, 2015; Bobet, 2016), which allows one to obtain the analytical solutions to the steady seepage field. The subsea grouted tunnel is located in a semi-infinite aquifer. The previous analytic model in Fig. 1 is separated into two sections: the aquifer region (Fig. 2(a)) and the grouted zone (Fig. 2(b)). An x-y coordinate system is applied, as shown in Fig. 2. 3.1.1. Steady seepage field analysis in the aquifer region. For the aquifer region, the appropriate conformal mapping function (Fang et al., 2015; Verruijt and Booker, 2000) is given by
z = ω (ζ ) = −ih
1−α 2 1 + ζ 1 + α 2 1−ζ
(3)
where α is a ratio determined by rg and h, namely
rg h
=
2α or α = h/ rg− (h/ rg )2−1 1 + α2
(4)
With the conformal mapping technique, the ground surface and outer boundary of the grouted zone in the z-plane can be mapped
Fig. 1. Analytic model for the seepage field of a subsea grouted tunnel. 93
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where r represents the seepage radius in the ζ-plane. The coordinate transformation between the z-plane (z = x + iy) and ζ-plane (ζ = ξ + iη) is determined by the following equations:
x 2 + y 2 −(h2−rg2)
ξ=
x 2 + (y− h2−rg2 )2
(5)
and
2x h2−rg2
η=
x 2 + (y− h2−rg2 )2
(6)
Therefore, the seepage radius r can be expressed as
r=
ξ 2 + η2 =
x 2 + (y +
h2−rg 2 )2
x 2 + (y− h2−rg 2 )2
(7)
According to Bear (1979), in the annulus region of the ζ-plane, the water inflow for a circular tunnel of a unit length is calculated as
(a) Analytic model for the aquifer region
(8)
Qr = 2πr |vr|
where vr is the radial seepage velocity. According to Darcy’s law, vr can be denoted as
vr = −kr
∂Hr ∂r
(9)
where Hr is the total hydraulic head in the aquifer. By substituting Eq. (9) into Eq. (8) and integrating by separation of variables, Hr can be expressed as
Hr =
Qr ln r + C1 2πkr
(10)
Qr and C1 are determined by the boundary conditions at the ground surface and outer boundary of the grouted zone: Hr (r = 1) = hw and Hr (r=α) =hrg . Substituting the boundary conditions above into Eq. (10) and considering Eq. (7), the water inflow Qr and total hydraulic head Hr in the aquifer region can be obtained as
Qr =
2πkr (hrg−h w ) (11)
ln α
and
(b) Analytic model for the grouted zone Fig. 2. The separated analytic models.
Hr (x , y ) =
conformally onto two circles with radii of 1 and α in the ζ-plane. In other words, the aquifer region in the z-plane is transformed conformally onto an annulus region in the ζ-plane, as indicated in Fig. 3,
hrg−h w ln α
ln
x 2 + (y +
h2−rg 2 )2
x 2 + (y− h2−rg 2 )2
+ hw (12)
According to Eq. (2), Hr can be denoted as Hr = pr/γw + y, where pr is the pore pressure in the aquifer region and y is the elevation head. The pore pressure distribution pr in the aquifer region can be written as
Fig. 3. Conformal mapping for the aquifer region. 94
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account. The pore pressure is related to the unit weight γ and elevation head y. Thus, the real expression of the pore pressure pg should be assumed to be (Huangfu et al., 2010)
pg = γw hrg + X (x , y ) +
pr0 −γw hrg + Y (x , y ) ln(r0/ rg )
ln
x 2 + (y + h)2 rg
(20)
where X(x,y) and Y(x,y) are determined by the boundary conditions at the outer and inner boundaries of the grouted zone. The boundary conditions considering the unit weight and elevation head are illustrated as follows: At the outer boundary of the grouted zone (i.e., x2 + (y + h)2 =rg2 ), pg (x,y) = γwhrg − yhw; At the inner boundary of the grouted zone (i.e., x2 + (y + h)2 =r02 ), pg (x,y) = pr0 . Substituting the boundary conditions above into Eq. (20), the solution to X(x,y) and Y(x,y) can be calculated as
X (x , y ) = −yγw
(21)
and
Y (x , y ) = γw ⎡r0 sin ⎛arctan ⎢ ⎝ ⎣
Fig. 4. Polar coordinates for the grouted zone.
pr (x , y ) =
(hrg−h w ) γw ln α
ln
x 2 + (y +
h2−rg 2 )2
x 2 + (y− h2−rg 2 )2
ρ=
+ (y +
(22)
Substituting Eqs. (21) and (22) into Eq. (20), the pore pressure pg in the grouted zone is denoted as
+ (h w−y )·γw
pg (x , y ) = (hrg−y )·γw
(13)
h)2
(
y+h x
pr0 + γw r0 sin arctan
+
3.1.2. Steady seepage field analysis of the grouted zone Fig. 4 illustrates the analytic model for the grouted zone. A ρ-θ polar coordinate system is applied with the origin at the tunnel center. The xy Cartesian coordinate system and ρ-θ polar coordinate system can be transformed to each other using the following equations:
x2
y + h⎞ ⎤ −h x ⎠ ⎥ ⎦
)−γ h
w rg −γw h
ln(r0/ rg )
(23)
pr 0 γw
(
−hrg−h + r0 sin arctan
y+h x
) ln
x 2 + (y + h)2
(14)
Hg (x , y ) = hrg +
(15)
By substituting Eqs. (14) and (15) into Eq. (24), the total hydraulic head Hg in the grouted zone can be rewritten in the polar coordinate system of Fig. 4 as
ln(r0/ rg )
rg (24)
y+h θ = arctan x
The grouted zone is axisymmetric relative to the polar coordinate system. Therefore, under the conditions in which gravity is neglected and the elevation head is not taken into account, the pore pressure pg in the grouted zone coincides with the Laplace equation under the axisymmetric condition in the polar coordinate system, namely
∂ρ2
rg
The total hydraulic head Hg in the grouted zone is
and
∂2pg
x 2 + (y + h)2
ln
1 ∂pg + =0 ρ ∂ρ
pr 0
Qg = k g
The analytical solution to Eq. (16) is
pg = γw hrg +
ln(r0/ rg )
pg = γw hrg +
ln(r0/ rg )
ln
ρ rg
(25)
∫0
2π
∂Hg ∂ρ
·ρdθ =
2πk g
(
pr0
ln(r0/ rg ) γw
−hrg−h )
(26)
3.1.3. Analytical solutions for the steady seepage field Because the water inflow in the aquifer is equal to the water inflow in the grouted zone (i.e., Qr = Qg), the total hydraulic head hrg at the outer boundary of grouted zone can be derived by equating Eq. (11)– (26) as follows:
ln
ρ rg
pr
hrg =
ln
Q=
x 2 + (y + h)2 rg
(kr / k g ) ln(rg / r0) h w + (h− γ 0 ) lnα w
(kr / k g ) ln(rg / r0)−ln α
(27)
Substituting Eq. (27) into Eq. (11), the analytical solution to the water inflow Q for the subsea grouted tunnel is obtained as
(18)
Considering Eq. (14), Eq. (18) can be rewritten as
pr0 −γw hrg
ln(r0/ rg )
(17)
where C2 and C3 are constants that can be determined by the boundary conditions at the outer and inner boundary of the grouted zone. Under the conditions in which gravity is neglected and the elevation head is not taken into account, the boundary conditions in the polar coordinate system are described as follows: pg (ρ = rg) = γwhrg and pg (ρ = r0) = pr0 . Substituting the boundary conditions above into Eq. (17), the solution to Eq. (17) can be calculated as
pr0 −γw hrg
−hrg−h + r0 sin θ
According to Darcy’s law, the water inflow Qg in the grouted zone can be obtained by
(16)
pg = C2 + C3 ln ρ
γw
Hg = hrg +
(19)
(
pr
2πkr h + h w− γ 0 w
)
(kr / k g ) ln(rg / r0)−ln α
(28)
Substituting Eq. (27) into Eq. (13), the analytical solution to the pore pressure pr in the aquifer region for the subsea grouted tunnel is denoted as
Note that, Eqs. (18) and (19) are obtained under the conditions in which gravity is neglected and the elevation head is not taken into 95
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P. Li et al. pr
pr (x , y ) =
(h + h w− γ 0 ) γw w
(kr / k g ) ln(rg / r0)−ln α
ln
x 2 + (y +
h2−rg 2 )2
x 2 + (y− h2−rg 2 )2
H2 =
+ (h w−y )·γw
where α is determined by Eq. (4). 3.2. Analytical solutions by the mirror image method (MIM) The separated analytic models in Fig. 2 is adopted to analyze the steady seepage field for a subsea grouted tunnel, again. The analytical solutions for the aquifer region are solved by using MIM, while the solutions for the grouted zone are obtained by the method used in Section 3.1.2.
r2 = rg2−4hy , when r1 = rg. Here, y represents the ordinate value at the outer boundary of the grouted zone. As shown in Fig. 5, y follows the relationship -h-rg ≤ y ≤ -h + rg and the external radius of the grouted zone rg is small enough relative to the distance from the tunnel center to the ground surface h. Therefore, y can be approximately denoted by y = −h, and r2 can be rewritten as r2 = rg2 + 4h2 . According to the boundary conditions above, the water inflow Qr and total hydraulic head Hr in the aquifer region can be obtained as Qr =
2πkr (hrg−h w ) ln(rg / rg2 + 4h2 )
(31)
hrg−h w
ln
ln(rg / rg2 + 4h2 )
x 2 + (y + h)2 + hw x 2 + (y−h)2
(36)
Similar to Eq. (13), the pore pressure distribution pr in the aquifer region can be written as
The water inflow and total hydraulic head for the proto steady seepage field are denoted as Q1 and H1, while those for the virtual steady seepage field are expressed as Q2 and H2. According to Eq. (10), the relationships between H1, H2 and Q1, Q2 are
Q1 H1 = ln r1 + D1 2πkr
(35)
and
(30)
Hr (x , y ) =
x 2 + (y−h)2
(34)
When the point M(x,y) is on the ground surface (i.e., r1 = r2), the total hydraulic head for the infinite steady seepage field is equal to hw. When M(x,y) is at the outer boundary of the grouted zone (i.e., r1 = rg), the total hydraulic head for the infinite steady seepage field is equal to hrg . In addition, according to Eqs. (30) and (31) , r2 can be obtained as
and
r2 =
Qr r ln 1 + D1 + D2 2πkr r2
Hr = H1 + H2 =
3.2.1. Steady seepage field analysis in the aquifer region. Considering the ground surface as the mirror face, the proto semiinfinite steady seepage field can be imaged into a virtual one using MIM. With the superposition of the two semi-infinite seepage fields, an infinite steady seepage field is obtained. The proto semi-infinite steady seepage field corresponds to a steady seepage field generated by a pumping well, while the virtual steady seepage field corresponds to a steady seepage field formed by an injection well. The seepage velocity and water inflow for the proto seepage field are equal to those for the virtual seepage field. Fig. 5 describes the analytic model for the aquifer region using MIM. A Cartesian coordinate system is applied. r1 represents the distance from a certain point M(x,y) in the aquifer to the proto tunnel center (0,h), while r2 represents the distance from the point M(x,y) to the virtual tunnel center (0,h). According to geometric properties, r1 and r2 can be given by
x 2 + (y + h)2
(33)
where D1 and D2 are constants that can be determined by the boundary conditions. Suppose the value of the water inflow for the infinite seepage field is denoted as Qr, water inflow is positive and water outflow is negative, Q1 and Q2 can be denoted as Q1 = −Q2 = Qr. On the basis of the superposition principle (Bear, 1979), the total hydraulic head Hr for the infinite steady seepage field is written as
(29)
r1 =
Q2 ln r2 + D2 2πkr
pr (x , y ) =
(hrg−h w ) γw ln(rg /
rg2
+ 4h2 )
ln
x 2 + (y + h)2 + (h w−y )·γw x 2 + (y−h)2
(37)
(32) 3.2.2. Steady seepage field analysis of the grouted zone. The analytical solutions to the steady seepage field in the grouted zone are obtained by the method described in Section 3.1.2.
and
3.2.3. Analytical solutions for the steady seepage field. Similar to Eq. (27), on the basis of the relation Qr = Qg, the total hydraulic head hrg at the outer boundary of the grouted zone can be derived as follows:
hrg =
(
pr
(kr / k g ) ln(rg / r0) h w + ln(rg / rg2 + 4h2 ) h− γ 0 (kr / k g ) ln(rg / r0)−ln(rg /
rg2
+
4h2 )
w
) (38)
Substituting Eq. (38) into Eq. (35), the analytical solution to the water inflow Q for the subsea grouted tunnel is obtained as
Q=
(
pr
2πkr h + h w− γ 0 w
)
(kr / k g ) ln(rg / r0)−ln(rg / rg2 + 4h2 )
(39)
Substituting Eq. (38) into Eq. (37) , the analytical solution to the pore pressure pr in the aquifer region for the subsea grouted tunnel is denoted as
Fig. 5. Analytic model for the aquifer region using MIM. 96
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P. Li et al.
can be obtained by pr
Qg = k g
∫0
2π
dHg dρ
·ρdθ = 2πk g
hrg− γ 0 w
ln(rg / r0)
(44)
3.3.3. Analytical solutions for the steady seepage field Similar to Eqs. (27) and (38), on the basis of the relationship Qr = Qg, the total hydraulic head hrg at the outer boundary of the grouted zone can be derived as follows:
(h + h w ) ln(rg / r0) + (k g / kr ) hrg =
pr 0 γw
ln(h/ rg )
(k g / kr ) ln(h/ rg ) + ln(rg / r0)
(45)
Substituting Eq. (45) into Eq. (42), the analytical solution to the water inflow Q for the subsea grouted tunnel is obtained as
Q=
pr
pr (x , y ) =
w
(kr / k g ) ln(rg / r0)−ln(rg /
rg2
+
4h2 )
+ (h w−y )·γw
ln
x 2 + (y + h)2 x 2 + (y−h)2
pr
⎜
(48)
ln(ρ / rg )
h + h w−hrg dHr ·ρdθ = 2πkr dρ ln(h/ rg )
(41)
4. Comparisons of the analytical solutions and numerical solutions 4.1. The numerical model
(42) To validate the analytical solutions, the program FLAC3D was used to establish a three-dimensional model to analyze the steady seepage field for a subsea grouted tunnel. Taking the F4 weathered slot of Xiamen Xiang’an subsea tunnel in China as an example, the analytical solution of steady seepage field is compared with the numerical solution. Based on the area equivalence principle, the four-centered circular section with excavation area of 170 m2 is equivalent to a circular section with diameter of 7.4 m (Zhang et al., 2014; Li and Zhou, 2015). The thickness of grouted zone is 6 m. The related parameter values are presented in Table 1. The numerical model measured 148 m in width, 1 m in length and 126.4 m in height. The aquifer and grouted zone were
3.3.2. Steady seepage field analysis of the grouted zone. For the grouted zone, the boundary conditions are described as follows: Hg (ρ = rg)=hrg and Hg(ρ=r0)= pr0 /γw. Taking the boundary conditions into the general solution of the Laplace equation, the total hydraulic head Hg in the grouted zone is calculated as
Hg =
pr0 γw
pr ln(ρ / r0) + ⎜⎛hrg− 0 ⎟⎞ γw ⎠ ln(rg / r0) ⎝
⎟
where y represents the ordinate of a certain point at the outer boundary of the grouted zone. Note that, there is a limitation that AMM cannot be applied to analyze the pore pressure distribution in the aquifer region, but can only be used to obtain the water inflow and external water pressure on the grouted zone.
According to Darcy’s law, the water inflow Qr in the aquifer region can be obtained by 2π
(47)
0 ⎡ (h + h w ) ln(rg / r0)−(k g / kr ) γ ln(rg / h) ⎤ y ⎞ w ⎥ ⎛1− p (ρ = rg ) = γw ⎢ (k g / kr ) ln(h/ rg ) + ln(rg / r0) ⎢ ⎥ ⎝ h + hw ⎠ ⎣ ⎦
3.3.1. Steady seepage field analysis in the aquifer region. For the aquifer region, the boundary conditions are described as follows: Hr (ρ = rg) = hrg and Hr(ρ=h)=h + hw. Taking the boundary conditions into the general solution of the Laplace equation, the total hydraulic head Hr in the aquifer region is calculated as
∫0
(46)
It should be mentioned that the external water pressure on the grouted zone obtained by Eq. (47) has the same value around the outer boundary of the grouted zone because both the unit weight and elevation head have been neglected. However, a constant total hydraulic head hrg at the outer boundary of grouted zone is assumed. According to Eq. (2), the external water pressure on the grouted zone varies with the elevation head. Therefore, the external water pressure on the grouted zone obtained by Eq. (47) should be modified considering the elevation head and initial hydrostatic pressure of the tunnel center. The modified external water pressure on the grouted zone is expressed as
In this section, a simplified model using AMM is shown in Fig. 6 to analyze the water inflow and external water pressure on the grouted zone for a subsea grouted tunnel. The far field boundary for the axisymmetric model is regarded as the distance from the tunnel center to the ground surface h. As shown in Fig. 6, an x-y Cartesian coordinate system and a ρ-θ polar coordinate system are applied. The total hydraulic head H(ρ) is given by the Laplace equation under axisymmetric conditions. The general solution to the Laplace equation is H(ρ) =E1 + E2lnρ, where E1 and E2 are constants that can be derived from the boundary conditions.
Qr = kr
)
pr
(40)
ln(h/ rg )
w
(kr / k g ) ln(rg / r0) + ln(h/ rg )
0 ⎡ (h + h w ) ln(rg / r0)−(k g / kr ) γ ln(rg / h) ⎤ w ⎥ p (ρ = rg ) = γw ⎢ (k g / kr ) ln(h/ rg ) + ln(rg / r0) ⎢ ⎥ ⎣ ⎦
3.3. Analytical solutions by the axisymmetric modeling method (AMM)
Hr = hrg + (h + h w−hrg )
pr
For AMM, both the unit weight and elevation head are neglected. Substituting Eq. (45) into Eq. (41) and considering H(ρ = rg) = p (ρ = rg)/γw, the analytical solution to the external water pressure on the grouted zone p(ρ = rg) for the subsea grouted tunnel is denoted as
Fig. 6. Analytic model using AMM.
(h + h w− γ 0 ) γw
(
2πkr h + h w− γ 0
(43)
According to Darcy’s law, the water inflow Qg in the grouted zone 97
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0
Table 1 Parameter values.
Analytical solution obtained by CVM Analytical solution obtained by MIM Numerical solution obtained by FLAC3D
-5
r0 (m)
rg (m)
h (m)
hw (m)
kr (m/s)
kg (m/s)
pr 0 (Pa)
7.4
13.4
52.4
20
5.0 × 10−6
5 × 10−8
0
-10
Depth (m)
-15
simulated as solid elements. The lateral displacement boundaries were fixed in the normal direction and displacement boundaries at the bottom were fixed in both the horizontal and vertical directions. The ground surface boundaries were free and permeable.
-20 -25 -30 -35
4.2. Pore pressure distribution
-40 200
Fig. 7 compares the pore pressure between the analytical solutions (CVM and MIM) and numerical solution. The analytical solution obtained by CVM almost coincides with that obtained by MIM. In Fig. 7(a) and (b), the analytical solutions match well with the numerical solution. However, as shown in Fig. 7(c), the analytical solutions and numerical solution differ at line 5–6, while the tendencies of the curves obtained by the analytical solutions and the numerical solution are coincident. The analysis suggests that the far field boundary influences the accuracy of the numerical solution, which will be illustrated in a later section. The external water pressure on the grouted zone obtained by the analytical solutions (CVM, MIM and AMM) is compared with the numerical solution in Fig. 8. The analytical solutions match quite well with the numerical solution. Moreover, the analytical solutions obtained by CVM and MIM are slightly less than the numerical solution. However, the formulas derivation of AMM is simpler than that of CVM and MIM. The external water pressure on the grouted zone obtained by AMM is slightly higher than the numerical solution, which provides a safe prediction for an engineering application. Overall, the analytical solutions obtained by CVM, MIM and AMM can accurately predict the external water pressure on the grouted zone for a subsea grouted tunnel (see Fig. 8(d)). The relative permeability (i.e., kr/kg) has a significant influence on the deviation between the analytical solutions and numerical solution (Shin et al., 2002; Zhang et al., 2017). Fig. 9 describes the relative deviation between the analytical solutions and numerical solution for water pressure above the tunnel crown (at point 2). As relative permeability increases, the deviation between the analytical solutions and numerical solution decreases gradually and the rate of decrease slows, regardless of which method is used. That is to say, the lower the permeability of the grouted zone is, the smaller the deviation between the analytical solutions and numerical solution becomes. As shown in Fig. 9, when the relative permeability is more than 40, the deviation between the analytical solutions and numerical solution is less than 4%, which implies that the precision of the analytical solutions is adequate and in an acceptable range. In addition, it can also be found from Fig. 9 that the water pressure at point 2 obtained by MIM and AMM is closer to the numerical solution than that obtained by CVM.
250
300
350
400
450
500
550
600
Pore pressure (kPa) (a) Above the tunnel crown (1-2) -65 Analytical solution obtained by CVM Analytical solution obtained by MIM Numerical solution obtained by FLAC3D
Depth (m)
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-91
-104
-117
-130 800
900
1000
1100
1200
1300
1400
1500
70
80
Pore pressure (kPa) (b) Underneath the tunnel invert (3-4) 730
Pore pressure (kPa)
725
Analytical solution obtained by CVM Analytical solution obtained by MIM Numerical solution obtained by FLAC3D
720 715 710 705 700 695 10
4.3. Water inflow
20
30
40
50
60
Distance from tunnel axis (m)
Fig. 10 shows the comparisons of the water inflow between the analytical solutions (CVM, MIM and AMM) and numerical solution. The water inflow obtained by the analytical solutions is very close to that obtained by the numerical solution. In addition, the water inflow predicted by CVM and MIM is almost equivalent, just 0.7% less than that predicted by the numerical solution. The water inflow predicted by AMM is 0.4% more than that predicted by the numerical solution. In summary, the analytical solutions obtained by CVM, MIM and AMM can accurately predict the water inflow for a subsea grouted tunnel.
(c) At line 5-6 Fig. 7. Comparisons of pore pressure between analytical solutions and numerical solution.
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(Numerical solution obtained by FLAC3D) Analytical solution obtained by CVM
(572) 566
(611) 605
(611) 605
Tunnel
(704) 700
(611) 615
(704) 700
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(704) 708
(798) 800
(798) 800 (836) 839
(a) Comparison between CVM and FLAC3D
(c) Comparison between AMM and FLAC3D 3D
(Numerical solution obtained by FLAC ) Analytical solution obtained by MIM
(572) 567
(Numerical solution obtained by FLAC3D) Analytical solution obtained by CVM Analytical solution obtained by MIM Analytical solution obtained by AMM
(611) 606
Tunnel
(704) 700
(611) 615
Units:kPa
(836) 834
(611) 606
(Numerical solution obtained by FLAC3D) Analytical solution obtained by AMM
Tunnel
(704) 708
(798) 795
(798) 795
(572) 577
(704) 700
Units:kPa
Tunnel
(798) 793
(798) 793 (836) 832
(b) Comparison between MIM and FLAC3D
(d) Comparison between the analytical solutions and numerical solution
Fig. 8. Comparisons of external water pressure on the grouted zone between analytical solutions and numerical solution.
4
18 16
Relative deviation (%)
14
Water inflow (m3/(d·m))
Analytical solution obtained by CVM Analytical solution obtained by MIM Analytical solution obtained by AMM
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3.222
CVM
MIM
AMM
FLAC3D
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2
1
2 0
0
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40
60
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100
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180
0
200
Relative permeability kr/kg
Fig. 10. Comparison of water inflow between analytical solutions and numerical solution.
Fig. 9. Relative deviation for water pressure at point 2 between analytical solutions and numerical solution.
However, the boundaries of the numerical simulation are finite. Therefore, it is necessary to discuss the influence of the far field boundary on the numerical solution. A case in which the relative permeability kr/kg is equal to 100 is chosen to analyze the influence of the far field boundary. The influence of the lateral boundary on the pore pressure distribution at line 5–6 is described in Fig. 11, where B represents the width of the lateral boundary and D represents the internal diameter of the grouted zone.
5. Analysis of influencing factors 5.1. Influence of the far field boundary For the numerical simulation, the far field boundary should be large enough to obtain more accurate solutions. In this paper, the analytical solutions by CVM and MIM are obtained on the basis of the semi-infinite boundary. The lateral and bottom boundaries are infinite. 99
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0
725
Analytical solution obtained by CVM Analytical solution obtained by MIM Numerical solution obtained by FLAC3D
-5
720
710 705
-15
Depth (m)
Pore pressure (kPa)
-10 Analytical solution obtained by CVM Analytical solution obtained by MIM Numerical solution by FLAC3D (B=5D) Numerical solution by FLAC3D (B=7D) Numerical solution by FLAC3D (B=9D) Numerical solution by FLAC3D (B=11D) Numerical solution by FLAC3D (B=13D)
715
-20 kr/kg=20
-25 kr/kg=50
-30
kr/kg=100
700
-35
B
695
0
20
40
60
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100
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180
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200
Distance from tunnel axis (m)
kr/kg=5
250
kr/kg=10
300
350
400
450
500
550
600
Pore pressure (kPa)
Fig. 11. The influence of boundary conditions on pore pressure (5–6).
(a) Pore pressure above the tunnel crown (1-2)
External water pressure at point 2 (kPa)
Five cases with different lateral boundaries (B = 5D, 7D, 9D, 11D, 13D) are discussed. From Fig. 11, as the width of the lateral boundary B increases from 5D to 13D, the numerical solution gradually approaches the analytical solutions obtained by CVM and MIM. However, when B increases to 9D ∼ 11D, the change in the numerical solution is very small. So, in the numerical simulation, the range of B = 3D ∼ 5D (which is generally adopted in a dry condition) is not sufficient to analyze the steady seepage field. Stated another way, the range of influence for the seepage field is larger than that for the stress field. It is suggested that the width of the lateral boundary should be at least 9D when conducting a numerical simulation for the steady seepage field. In addition, the enlargement of the bottom boundary has little influence on the numerical solution after analysis. 5.2. The permeability of the grouted zone
600 550 Analytical solution obtained by CVM Analytical solution obtained by MIM Analytical solution obtained by AMM Numerical solution obtained by FLAC3D
500 450 400 350 300 250
0
20
40
60
80
100
120
140
160
180
200
Relative permeability kr/kg
The permeability of the grouted zone has a significant influence on the steady seepage field for a subsea grouted tunnel. Fig. 12 shows the influence of the relative permeability (i.e., kr/kg) on the steady seepage field for a subsea grouted tunnel. The influence of the relative permeability on the pore pressure above the tunnel crown is described in Fig. 12(a). For various relative permeabilities, the regularities of the pore pressure distribution above the tunnel crown are almost coincident. The pore pressure increases gradually with the increase of the relative permeability, regardless of the analytical solutions and numerical solution. The pore pressure increases more remarkably when kr/kg < 50 than when kr/kg > 50. In addition, Fig. 12(a) shows the greater the relative permeability is, the better the analytical solutions and numerical solution match. The influences of the relative permeability on the external water pressure on the grouted zone above the tunnel crown (at point 2) and water inflow are shown in Fig. 12(b) and (c), respectively. The analytical solutions for the external water pressure are closer to the numerical solution when the relative permeability is greater. However, the analytical solutions for the water inflow always match quite well with the numerical solution no matter how the relative permeability changes. As relative permeability increases, the external water pressure on the grouted zone at point 2 increases and the water inflow is gradually reduced, regardless of the analytical solutions and numerical solution. Moreover, the patterns described are stronger when kr/ kg < 50 than when kr/kg > 50. Based on the previous analysis, some significant conclusions can be drawn. Decreasing the permeability of the grouted zone contributes to a decrease of the water inflow. However, at the same time, the grouted zone must support a greater water pressure. When the permeability of the grouted zone decreases to a certain value, the external water
(b) External water pressure on the grouted zone at point 2
Water inflow (m3/(d·m))
50
Analytical solution obtained by CVM Analytical solution obtained by MIM Analytical solution obtained by AMM Numerical solution obtained by FLAC3D
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30
20
10
0
0
20
40
60
80
100
120
140
160
180
200
Relative permeability kr/kg (c) Water inflow Fig. 12. The influence of relative permeability on the steady seepage field.
pressure on the grouted zone is no longer increased significantly and the water inflow is no longer decreased significantly. Therefore, choosing a reasonable permeability for the grouted zone is crucial to ensure the economy (i.e., the lower discharge) and stability (i.e., the 100
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External water pressure at point 2 (kPa)
900
lower water pressure on grouted zone) for a subsea grouted tunnel.
Analytical solution obtained by CVM Analytical solution obtained by MIM Analytical solution obtained by AMM Numerical solution obtained by FLAC3D
800
5.3. Water depth
4.0
The influences of water depth on the external water pressure on the grouted zone above the tunnel crown (at point 2) and water inflow are shown in Fig. 13(a) and (b), respectively. With the rising of the water table, both the external water pressure and water inflow increase linearly, regardless of the analytical solutions and numerical solution. The influence coefficient of the external water pressure on the grouted zone is defined as λ = p/p0, where p represents the external water pressure on the grouted zone above the tunnel crown (at point 2) under steady seepage state and p0 represents the initial hydrostatic pressure at point 2. The influence of the water depth on the influence coefficient λ is shown in Fig. 13(c). The influence coefficient λ obtained by AMM is constant and not affected by the water depth, while the influence coefficients λ obtained by CVM, MIM and the numerical simulation depend on the water depth. As the water table rises, the influence coefficients λ obtained by CVM, MIM and the numerical simulation increase gradually, while the amount of increase is not significant if the water depth is greater than 20 m. In summary, when the permeabilities of the aquifer and the grouted zone are of certain values, the absolute value of the external water pressure on the grouted zone and influence coefficient are affected by the water depth. Moreover, the water depth has a more direct and significant impact on the absolute value than the influence coefficient.
3.5
6. Conclusions
700 600 500 400 300
0
10
20
30
40
50
Water depth hw (m) (a) External water pressure on the grouted zone at point 2 5.0 Analytical solution obtained by CVM Analytical solution obtained by MIM Analytical solution obtained by AMM Numerical solution obtained by FLAC3D
Water inflow (m3/(d·m))
4.5
To investigate the steady seepage field for a subsea grouted tunnel, analytical solutions were obtained by the complex variable method, mirror image method and axisymmetric modeling method. A series of numerical simulations were performed to validate the analytical solutions and influencing factors, such as the boundary conditions, permeability of grouted zone and water depth for a subsea tunnel, were discussed. The water discharge into the subsea tunnel and distribution of the pore pressure in the aquifer and external water pressure on the grouted zone are the main concerns. The findings of this investigation are summarized as follows.
3.0 2.5 2.0
0
10
20
30
40
50
Water depth hw (m) (b) Water inflow
Influence coefficient λ
1.00
(1) For the steady seepage problem of a subsea grouted tunnel, the analytical solutions by each of the three methods can be applied to analyze the external water pressure on the grouted zone and water inflow into the tunnel with sufficient accuracy. The formula derivation of AMM is simpler than that of CVM and MIM, but AMM cannot be applied to analyze the pore pressure distribution in the aquifer region especially for shallow tunnels. (2) The range of the far field boundary has an important impact on the precision of the pore pressure distribution when the numerical simulation is adopted to analyze the seepage problem of a subsea grouted tunnel. Three to five times the tunnel diameter for the range of the far field boundary, which is generally adopted for dry conditions, is not sufficient to analyze the steady seepage field. It is suggested that the width of the lateral boundary should be at least nine times the tunnel diameter when conducting a numerical simulation for the steady seepage field. (3) The permeability of grouted zone influences comparisons between the analytical solutions and numerical solution, especially for the pore pressure distribution and external water pressure on the grouted zone, but the results remain in an acceptable range. With a decrease of the permeability of grouted zone, both the pore pressure in the aquifer and external water pressure on the grouted zone increase gradually, while the water inflow decreases gradually. (4) When the permeabilities of the aquifer and grouted zone are of certain values, both the external water pressure on the grouted zone and water inflow increase linearly with the rising of the water table.
0.98
0.96
0.94
Analytical solution obtained by CVM Analytical solution obtained by MIM Analytical solution obtained by AMM Numerical solution obtained by FLAC3D
0.92
0.90
0
10
20
30
40
50
Water depth hw (m) (c) Influence coefficient of the external water pressure on the grouted zone at point 2 Fig. 13. The influence of water depth on the steady seepage field.
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In addition, along with the rising of water table, the influence coefficients of the external water pressure on the grouted zone obtained by CVM, MIM and the numerical simulation increase gradually, but the amount of increase is not significant if the water depth is greater than 20 m.
tunnel driving. Eng. Geol. 2 (1), 39–56. Harr, M.E., 1962. Groundwater and Seepage. McGraw-Hill, New York. Huangfu, M., Wang, M.S., Tan, Z., Wang, X.Y., 2010. Analytical solutions for steady seepage into an underwater circular tunnel. Tunn. Undergr. Space Technol. 25 (4), 391–396. Ivars, D.M., 2006. Water inflow into excavations in fractured rock—a three-dimensional hydro-mechanical numerical study. Int. J. Rock Mech. Min. Sci. 43 (5), 705–725. Kolymbas, D., Wagner, P., 2007. Groundwater ingress to tunnels – the exact analytical solution. Tunn. Undergr. Space Technol. 22, 23–27. Lee, I.M., Nam, S.W., 2001. The study of seepage forces acting on the tunnel lining and tunnel face in shallow tunnels. Tunn. Undergr. Space Technol. 16 (1), 31–40. Lee, I.M., Nam, S.W., 2004. Effect of tunnel advance rate on seepage forces acting on the underwater tunnel face. Tunn. Undergr. Space Technol. 19 (2), 273–281. Lee, S.W., Jung, J.W., Nam, S.W., Lee, I.M., 2007. The influence of seepage forces on ground reaction curve of circular opening. Tunn. Undergr. Space Technol. 22 (1), 28–38. Lei, S., 1999. An analytical solution for steady flow into a tunnel. Ground Water 37, 23–26. Li, P.F., Zhou, X.J., 2015. Mechanical behavior and shape optimization of lining structure for subsea tunnel excavated in weathered slot. China Ocean Eng. 29 (6), 875–890. Moon, J., Jeong, S., 2011. Effect of highly pervious geological features on ground-water flow into a tunnel. Eng. Geol. 117, 207–216. 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. Space Technol. 23 (2), 206–209. Shin, J., Addenbrooke, T., Potts, D., 2002. A numerical study of the effect of groundwater movement on long-term tunnel behaviour. Geotechnique 52 (6), 391–403. Verruijt, A., Booker, J.R., 2000. Complex variable analysis of mindlin’s tunnel problem. Development of Theoretical Geomechanics, Sydney, 3–22, Balkema. Wang, X.Y., Tan, Z.S., Wang, M.S., Zhang, M., Huangfu, M., 2008. Theoretical and experimental study of external water pressure on tunnel lining in controlled drainage under high water level. Tunn. Undergr. Space Technol. 23, 552–560. Zhang, D.L., Fang, Q., Lou, H.C., 2014. Grouting techniques for the unfavorable geological conditions of Xiang’an subsea tunnel in China. J. Rock Mech. Geotech. Eng. 6, 438–446. Zhang, D.M., Huang, Z.K., Yin, Z.Y., Ran, L.Z., Huang, H.W., 2017. Predicting the grouting effect on leakage-induced tunnels and ground response in saturated soils. Tunn. Undergr. Space Technol. 65, 76–90. Zhang, L., Franklin, J.A., 1993. Prediction of water flow into rock tunnels: an analytical solution assuming a hydraulic conductivity gradient. Int. J. Rock Mech. Min. Sci. 30 (1), 37–46.
Acknowledgement This study is supported by the National Natural Science Foundation of China (No. 51778025) and the State Key Laboratory of Disaster Reduction in Civil Engineering, China (No. SLDRCE16-02). References Arjnoi, P., Jeong, J.H., Kim, C.Y., Park, K.H., 2009. Effect of drainage conditions on porewater pressure distributions and lining stresses in drained tunnels. Tunn. Undergr. Space Technol. 24 (4), 376–389. Bear, J., 1979. Hydraulics of Groundwater. McGraw-Hill, New York. Bobet, A., 2016. Deep tunnel in transversely anisotropic rock with groundwater flow. Rock Mech. Rock Eng. 49, 4817–4832. Bobet, A., Yu, H., 2015. Stress field near the tip of a crack in a poroelastic transversely anisotropic saturated rock. Eng. Fract. Mech. 141, 1–18. Butscher, C., 2012. Steady-state groundwater inflow into a circular tunnel. Tunn. Undergr. Space Technol. 32, 158–167. Cao, Y., Jiang, J., Xie, K.H., Huang, W.M., 2014. Analytical solutions for nonlinear consolidation of soft soil around a shield tunnel with idealized sealing linings. Comput. Geotech. 61, 144–152. Chen, Y., Zhou, C., Zheng, H., 2008. A numerical solution to seepage problems with complex drainage systems. Comput. Geotech. 35 (3), 383–393. El Tani, M., 2003. Circular tunnel in a semi-infinite aquifer. Tunn. Undergr. Space Technol. 18, 49–55. Fang, Q., Song, H.R., Zhang, D.L., 2015. Complex variable analysis for stress distribution of an underwater tunnel in an elastic half plane. Int. J. Numer. Anal. Meth. Geomech. 39, 1821–1835. Farhadian, H., Katibeh, H., 2017. New empirical model to evaluate groundwater flow into circular tunnel using multiple regression analysis. Int. J. Rock Mech. Min. Sci. 27 (3), 415–421. Goodman, R., Moye, D., Schalkwyk, A., Javandel, I., 1965. Ground-water inflow during
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Investigation of steady water inflow into a subsea grouted tunnel Pengfei Li, Fan Wang, Yingying Long, Xu Zhao
⁎
T
The Key Laboratory of Urban Security and Disaster Engineering, Ministry of Education, Beijing University of Technology, Beijing 100124, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Subsea grouted tunnel Steady seepage field Analytical solution Numerical simulation Permeability of grouted zone
This paper investigates analytical solutions for steady water inflow into a subsea grouted tunnel. The relevant parameters include the pore pressure distribution in the aquifer, external water pressure on the grouted zone and water inflow. Analytical solutions are obtained by the complex variable method (CVM), mirror image method (MIM) and axisymmetric modeling method (AMM). A series of numerical simulations are performed to validate the analytical solutions. The calculation accuracy and application conditions of each method are stated by comparisons of the pore pressure distribution and water inflow. Finally, the influencing factors, such as the boundary conditions, permeability of grouted zone and water depth, are discussed. It is suggested that the width of the lateral boundary should be at least nine times the tunnel diameter when conducting a numerical simulation for the steady seepage field. The relative permeability (i.e., kr/kg) has influences on comparisons between the analytical solutions and numerical solution, but the results remain in an acceptable range. It is also found that both the external water pressure on the grouted zone and water inflow increase linearly with the rising of the water table.
1. Introduction Since the development of modern tunnel engineering, the subsea tunnel has become widely used for crossing rivers or seas. There are many advantages to a subsea tunnel, such as being unaffected by bad weather, requiring few land resources, and operating simultaneously with shipping operations that use the waterway (Fang et al., 2015). A subsea tunnel has a potentially infinite supply of water which cannot naturally drain off using a V-type longitudinal slope design. If full sealing is adopted to achieve a waterproof tunnel, the tunnel lining would have to sustain high water pressure for a long duration, which would result in water leakage and subsequent tunnel lining failure. If full drainage is implemented in the tunnel construction, a large-scale pumping system is needed, requiring a large amount of electric energy at high cost. Moreover, the water flowing into a subsea tunnel has strong corrosive capabilities, threatening the durability of the tunnel lining. If blocking with limited drainage is used, the water discharged into the tunnel is controlled and external water pressure on the tunnel lining is significantly reduced (Wang et al., 2008), an approach considered economical and safe. The key to the design of blocking with limited discharge is to obtain the distribution characteristics of the seepage field, including the water inflow and pore pressure of the soil and external water pressure of tunnel lining, which can be analyzed by field monitoring, numerical simulation and theoretical analysis. Using monitoring data, Farhadian
⁎
Corresponding author. E-mail address: [email protected] (X. Zhao).
https://doi.org/10.1016/j.tust.2018.06.003 Received 12 December 2017; Received in revised form 16 April 2018; Accepted 6 June 2018 0886-7798/ © 2018 Elsevier Ltd. All rights reserved.
and Katibeh (2017) presented a new empirical model to evaluate groundwater flow into a circular tunnel using multiple regression analysis. The method of field monitoring is important for tunnels in construction, to avoid potential deformations and failures. Also, in designing tunnels in similar ground conditions and lining types, former monitoring data could be significant references for the new design in similar projects. Compared with field monitoring, numerical simulation can obtain the spatial distribution characteristics of the seepage field against the complex formation conditions in practical engineering. Many researchers have applied numerical simulation to calculate the external water pressure on a tunnel lining and predict the water inflow (Lee and Nam, 2001, 2004; Shin et al., 2002; Ivars, 2006; Lee et al., 2007; Arjnoi et al., 2009; Butscher, 2012). In addition, complexities such as highly pervious geological features and complex drainage systems, which affect groundwater flow into a tunnel, can be investigated using numerical simulation (Moon and Jeong, 2011; Chen et al., 2008). Theoretical analysis is an efficient and convenient method for calculating the pore pressure distribution and predicting the water inflow into a tunnel. Harr (1962) presented the pore pressure distribution of a circular tunnel using the mirror image method. Goodman et al. (1965) predicted the analytical solutions of the water inflow for steady water inflow into undersea tunnels. Assuming a constant hydraulic head at the tunnel perimeter, Lei (1999) derived the analytical solutions to the hydraulic head distribution and water inflow for two-dimensional, steady water flow into a horizontal tunnel in a fully saturated,
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the water table to the ground surface is hw. The permeabilities of the aquifer and grouted zone are expressed as kr and kg, respectively. To obtain the analytical solutions for the steady seepage field of a subsea grouted tunnel, several simplifying assumptions were made:
homogeneous, isotropic and semi-infinite aquifer. Based on the Mobius transformation and Fourier series, El Tani (2003) gave the exact analytical solutions to the water inflow, pressure, leakage and recharging infiltrations for a circular tunnel in a semi-infinite, homogeneous and isotropic aquifer. Kolymbas and Wagner (2007) obtained the analytical expressions of the steady state groundwater ingress into a circular tunnel using conformal mapping. The derived equations can be generally applied to deep and shallow tunnels. For deep tunnels, Wang et al. (2008) studied the influence of controlled drainage on the external water pressure on a tunnel lining by theoretical and experimental methods. The results drainage measures were necessary to reduce the external water pressure on the tunnel lining. Moreover, considering the constant hydraulic head and constant water pressure boundary conditions at the tunnel perimeter, some researchers (Park et al., 2008; Huangfu et al., 2010) derived the analytical solutions to the total hydraulic head (or pore pressure) distribution and water inflow for the steady seepage into a drained circular tunnel in a semi-infinite aquifer. Zhang et al. (2017) investigated the analytical solutions to the seepage field for a lined tunnel considering the grouting effect and validated the analytical solutions using a numerical simulation. Besides, other researches were performed to analyze the effects of the hydraulic conductivity gradient and nonlinear consolidation on the analytical solutions of the water inflow and pore pressure distribution (Zhang and Franklin, 1993; Cao et al., 2014). As discussed, the design of tunnels below the water table has been systematically investigated by many researchers, many numerical simulations have been performed and several significant analytical solutions for the steady seepage field of tunnels have been obtained. However, most of these analytical solutions were obtained either neglecting the impact of the grouting permeability (without considering the grouted zone) or significantly simplifying the boundary conditions, and most of the numerical simulations didn’t clarify the influences of the boundary conditions, either. This paper focuses on the analytical solutions for the steady seepage field of a subsea grouted tunnel, validated by a series of numerical simulations and discusses influencing factors, such as the boundary conditions, water depth and relative permeability between the grouted zone and natural ground.
(1) The tunnel has a circular cross-section and is located in a fully saturated, homogeneous, isotropic and semi-infinite aquifer; (2) The grouting material is homogeneous and isotropic; (3) A state of steady flow is assumed; (4) The fluid is incompressible; (5) The water table is horizontal and remains unchanged. According to Darcy’s law and conservation of mass, in fully saturated, homogeneous, isotropic media, the differential equation for twodimensional steady-state groundwater flow is given by the Laplace equation:
∂ 2H ∂ 2H + =0 2 ∂x ∂y 2
(1)
where H is the total hydraulic head and is equal to the sum of the pressure head and elevation head, that is
H=
p +y γw
(2)
where p is the water pressure, γw is the unit weight of water and y is the elevation head. 3. Analytical solutions Considering the ground surface as the elevation reference datum in Fig. 1, the boundary conditions of the steady seepage field are given as follows: (1) The total hydraulic head at the ground surface remains constant and is defined as hw; (2) A constant total hydraulic headhrg at the outer boundary of grouted zone is assumed; (3) Constant water pressure pr0 at the inner boundary of grouted zone is assumed. Because of the complex boundary conditions, it is difficult to obtain analytical solutions to the steady seepage field by solving Eq. (1) directly. Consequently, the complex variable method (CVM), mirror image method (MIM) and axisymmetric modelling method (AMM) are introduced to obtain the analytical solutions for the steady seepage field of a subsea grouted tunnel.
2. Problem description Fig. 1 shows the model for analyzing the steady seepage field in a subsea grouted tunnel. In this model, r0 and rg represent the internal and external radii of the grouted zone, respectively. The tunnel depth from the tunnel center to the ground surface is h. The water depth from
3.1. Analytical solutions by the complex variable method (CVM) The complex boundary conditions can be mapped conformally to simple boundary conditions by adopting the conformal mapping technique (Fang et al., 2015; Huangfu et al., 2010; Zhang et al., 2017; Bobet and Yu, 2015; Bobet, 2016), which allows one to obtain the analytical solutions to the steady seepage field. The subsea grouted tunnel is located in a semi-infinite aquifer. The previous analytic model in Fig. 1 is separated into two sections: the aquifer region (Fig. 2(a)) and the grouted zone (Fig. 2(b)). An x-y coordinate system is applied, as shown in Fig. 2. 3.1.1. Steady seepage field analysis in the aquifer region. For the aquifer region, the appropriate conformal mapping function (Fang et al., 2015; Verruijt and Booker, 2000) is given by
z = ω (ζ ) = −ih
1−α 2 1 + ζ 1 + α 2 1−ζ
(3)
where α is a ratio determined by rg and h, namely
rg h
=
2α or α = h/ rg− (h/ rg )2−1 1 + α2
(4)
With the conformal mapping technique, the ground surface and outer boundary of the grouted zone in the z-plane can be mapped
Fig. 1. Analytic model for the seepage field of a subsea grouted tunnel. 93
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where r represents the seepage radius in the ζ-plane. The coordinate transformation between the z-plane (z = x + iy) and ζ-plane (ζ = ξ + iη) is determined by the following equations:
x 2 + y 2 −(h2−rg2)
ξ=
x 2 + (y− h2−rg2 )2
(5)
and
2x h2−rg2
η=
x 2 + (y− h2−rg2 )2
(6)
Therefore, the seepage radius r can be expressed as
r=
ξ 2 + η2 =
x 2 + (y +
h2−rg 2 )2
x 2 + (y− h2−rg 2 )2
(7)
According to Bear (1979), in the annulus region of the ζ-plane, the water inflow for a circular tunnel of a unit length is calculated as
(a) Analytic model for the aquifer region
(8)
Qr = 2πr |vr|
where vr is the radial seepage velocity. According to Darcy’s law, vr can be denoted as
vr = −kr
∂Hr ∂r
(9)
where Hr is the total hydraulic head in the aquifer. By substituting Eq. (9) into Eq. (8) and integrating by separation of variables, Hr can be expressed as
Hr =
Qr ln r + C1 2πkr
(10)
Qr and C1 are determined by the boundary conditions at the ground surface and outer boundary of the grouted zone: Hr (r = 1) = hw and Hr (r=α) =hrg . Substituting the boundary conditions above into Eq. (10) and considering Eq. (7), the water inflow Qr and total hydraulic head Hr in the aquifer region can be obtained as
Qr =
2πkr (hrg−h w ) (11)
ln α
and
(b) Analytic model for the grouted zone Fig. 2. The separated analytic models.
Hr (x , y ) =
conformally onto two circles with radii of 1 and α in the ζ-plane. In other words, the aquifer region in the z-plane is transformed conformally onto an annulus region in the ζ-plane, as indicated in Fig. 3,
hrg−h w ln α
ln
x 2 + (y +
h2−rg 2 )2
x 2 + (y− h2−rg 2 )2
+ hw (12)
According to Eq. (2), Hr can be denoted as Hr = pr/γw + y, where pr is the pore pressure in the aquifer region and y is the elevation head. The pore pressure distribution pr in the aquifer region can be written as
Fig. 3. Conformal mapping for the aquifer region. 94
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P. Li et al.
account. The pore pressure is related to the unit weight γ and elevation head y. Thus, the real expression of the pore pressure pg should be assumed to be (Huangfu et al., 2010)
pg = γw hrg + X (x , y ) +
pr0 −γw hrg + Y (x , y ) ln(r0/ rg )
ln
x 2 + (y + h)2 rg
(20)
where X(x,y) and Y(x,y) are determined by the boundary conditions at the outer and inner boundaries of the grouted zone. The boundary conditions considering the unit weight and elevation head are illustrated as follows: At the outer boundary of the grouted zone (i.e., x2 + (y + h)2 =rg2 ), pg (x,y) = γwhrg − yhw; At the inner boundary of the grouted zone (i.e., x2 + (y + h)2 =r02 ), pg (x,y) = pr0 . Substituting the boundary conditions above into Eq. (20), the solution to X(x,y) and Y(x,y) can be calculated as
X (x , y ) = −yγw
(21)
and
Y (x , y ) = γw ⎡r0 sin ⎛arctan ⎢ ⎝ ⎣
Fig. 4. Polar coordinates for the grouted zone.
pr (x , y ) =
(hrg−h w ) γw ln α
ln
x 2 + (y +
h2−rg 2 )2
x 2 + (y− h2−rg 2 )2
ρ=
+ (y +
(22)
Substituting Eqs. (21) and (22) into Eq. (20), the pore pressure pg in the grouted zone is denoted as
+ (h w−y )·γw
pg (x , y ) = (hrg−y )·γw
(13)
h)2
(
y+h x
pr0 + γw r0 sin arctan
+
3.1.2. Steady seepage field analysis of the grouted zone Fig. 4 illustrates the analytic model for the grouted zone. A ρ-θ polar coordinate system is applied with the origin at the tunnel center. The xy Cartesian coordinate system and ρ-θ polar coordinate system can be transformed to each other using the following equations:
x2
y + h⎞ ⎤ −h x ⎠ ⎥ ⎦
)−γ h
w rg −γw h
ln(r0/ rg )
(23)
pr 0 γw
(
−hrg−h + r0 sin arctan
y+h x
) ln
x 2 + (y + h)2
(14)
Hg (x , y ) = hrg +
(15)
By substituting Eqs. (14) and (15) into Eq. (24), the total hydraulic head Hg in the grouted zone can be rewritten in the polar coordinate system of Fig. 4 as
ln(r0/ rg )
rg (24)
y+h θ = arctan x
The grouted zone is axisymmetric relative to the polar coordinate system. Therefore, under the conditions in which gravity is neglected and the elevation head is not taken into account, the pore pressure pg in the grouted zone coincides with the Laplace equation under the axisymmetric condition in the polar coordinate system, namely
∂ρ2
rg
The total hydraulic head Hg in the grouted zone is
and
∂2pg
x 2 + (y + h)2
ln
1 ∂pg + =0 ρ ∂ρ
pr 0
Qg = k g
The analytical solution to Eq. (16) is
pg = γw hrg +
ln(r0/ rg )
pg = γw hrg +
ln(r0/ rg )
ln
ρ rg
(25)
∫0
2π
∂Hg ∂ρ
·ρdθ =
2πk g
(
pr0
ln(r0/ rg ) γw
−hrg−h )
(26)
3.1.3. Analytical solutions for the steady seepage field Because the water inflow in the aquifer is equal to the water inflow in the grouted zone (i.e., Qr = Qg), the total hydraulic head hrg at the outer boundary of grouted zone can be derived by equating Eq. (11)– (26) as follows:
ln
ρ rg
pr
hrg =
ln
Q=
x 2 + (y + h)2 rg
(kr / k g ) ln(rg / r0) h w + (h− γ 0 ) lnα w
(kr / k g ) ln(rg / r0)−ln α
(27)
Substituting Eq. (27) into Eq. (11), the analytical solution to the water inflow Q for the subsea grouted tunnel is obtained as
(18)
Considering Eq. (14), Eq. (18) can be rewritten as
pr0 −γw hrg
ln(r0/ rg )
(17)
where C2 and C3 are constants that can be determined by the boundary conditions at the outer and inner boundary of the grouted zone. Under the conditions in which gravity is neglected and the elevation head is not taken into account, the boundary conditions in the polar coordinate system are described as follows: pg (ρ = rg) = γwhrg and pg (ρ = r0) = pr0 . Substituting the boundary conditions above into Eq. (17), the solution to Eq. (17) can be calculated as
pr0 −γw hrg
−hrg−h + r0 sin θ
According to Darcy’s law, the water inflow Qg in the grouted zone can be obtained by
(16)
pg = C2 + C3 ln ρ
γw
Hg = hrg +
(19)
(
pr
2πkr h + h w− γ 0 w
)
(kr / k g ) ln(rg / r0)−ln α
(28)
Substituting Eq. (27) into Eq. (13), the analytical solution to the pore pressure pr in the aquifer region for the subsea grouted tunnel is denoted as
Note that, Eqs. (18) and (19) are obtained under the conditions in which gravity is neglected and the elevation head is not taken into 95
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pr (x , y ) =
(h + h w− γ 0 ) γw w
(kr / k g ) ln(rg / r0)−ln α
ln
x 2 + (y +
h2−rg 2 )2
x 2 + (y− h2−rg 2 )2
H2 =
+ (h w−y )·γw
where α is determined by Eq. (4). 3.2. Analytical solutions by the mirror image method (MIM) The separated analytic models in Fig. 2 is adopted to analyze the steady seepage field for a subsea grouted tunnel, again. The analytical solutions for the aquifer region are solved by using MIM, while the solutions for the grouted zone are obtained by the method used in Section 3.1.2.
r2 = rg2−4hy , when r1 = rg. Here, y represents the ordinate value at the outer boundary of the grouted zone. As shown in Fig. 5, y follows the relationship -h-rg ≤ y ≤ -h + rg and the external radius of the grouted zone rg is small enough relative to the distance from the tunnel center to the ground surface h. Therefore, y can be approximately denoted by y = −h, and r2 can be rewritten as r2 = rg2 + 4h2 . According to the boundary conditions above, the water inflow Qr and total hydraulic head Hr in the aquifer region can be obtained as Qr =
2πkr (hrg−h w ) ln(rg / rg2 + 4h2 )
(31)
hrg−h w
ln
ln(rg / rg2 + 4h2 )
x 2 + (y + h)2 + hw x 2 + (y−h)2
(36)
Similar to Eq. (13), the pore pressure distribution pr in the aquifer region can be written as
The water inflow and total hydraulic head for the proto steady seepage field are denoted as Q1 and H1, while those for the virtual steady seepage field are expressed as Q2 and H2. According to Eq. (10), the relationships between H1, H2 and Q1, Q2 are
Q1 H1 = ln r1 + D1 2πkr
(35)
and
(30)
Hr (x , y ) =
x 2 + (y−h)2
(34)
When the point M(x,y) is on the ground surface (i.e., r1 = r2), the total hydraulic head for the infinite steady seepage field is equal to hw. When M(x,y) is at the outer boundary of the grouted zone (i.e., r1 = rg), the total hydraulic head for the infinite steady seepage field is equal to hrg . In addition, according to Eqs. (30) and (31) , r2 can be obtained as
and
r2 =
Qr r ln 1 + D1 + D2 2πkr r2
Hr = H1 + H2 =
3.2.1. Steady seepage field analysis in the aquifer region. Considering the ground surface as the mirror face, the proto semiinfinite steady seepage field can be imaged into a virtual one using MIM. With the superposition of the two semi-infinite seepage fields, an infinite steady seepage field is obtained. The proto semi-infinite steady seepage field corresponds to a steady seepage field generated by a pumping well, while the virtual steady seepage field corresponds to a steady seepage field formed by an injection well. The seepage velocity and water inflow for the proto seepage field are equal to those for the virtual seepage field. Fig. 5 describes the analytic model for the aquifer region using MIM. A Cartesian coordinate system is applied. r1 represents the distance from a certain point M(x,y) in the aquifer to the proto tunnel center (0,h), while r2 represents the distance from the point M(x,y) to the virtual tunnel center (0,h). According to geometric properties, r1 and r2 can be given by
x 2 + (y + h)2
(33)
where D1 and D2 are constants that can be determined by the boundary conditions. Suppose the value of the water inflow for the infinite seepage field is denoted as Qr, water inflow is positive and water outflow is negative, Q1 and Q2 can be denoted as Q1 = −Q2 = Qr. On the basis of the superposition principle (Bear, 1979), the total hydraulic head Hr for the infinite steady seepage field is written as
(29)
r1 =
Q2 ln r2 + D2 2πkr
pr (x , y ) =
(hrg−h w ) γw ln(rg /
rg2
+ 4h2 )
ln
x 2 + (y + h)2 + (h w−y )·γw x 2 + (y−h)2
(37)
(32) 3.2.2. Steady seepage field analysis of the grouted zone. The analytical solutions to the steady seepage field in the grouted zone are obtained by the method described in Section 3.1.2.
and
3.2.3. Analytical solutions for the steady seepage field. Similar to Eq. (27), on the basis of the relation Qr = Qg, the total hydraulic head hrg at the outer boundary of the grouted zone can be derived as follows:
hrg =
(
pr
(kr / k g ) ln(rg / r0) h w + ln(rg / rg2 + 4h2 ) h− γ 0 (kr / k g ) ln(rg / r0)−ln(rg /
rg2
+
4h2 )
w
) (38)
Substituting Eq. (38) into Eq. (35), the analytical solution to the water inflow Q for the subsea grouted tunnel is obtained as
Q=
(
pr
2πkr h + h w− γ 0 w
)
(kr / k g ) ln(rg / r0)−ln(rg / rg2 + 4h2 )
(39)
Substituting Eq. (38) into Eq. (37) , the analytical solution to the pore pressure pr in the aquifer region for the subsea grouted tunnel is denoted as
Fig. 5. Analytic model for the aquifer region using MIM. 96
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can be obtained by pr
Qg = k g
∫0
2π
dHg dρ
·ρdθ = 2πk g
hrg− γ 0 w
ln(rg / r0)
(44)
3.3.3. Analytical solutions for the steady seepage field Similar to Eqs. (27) and (38), on the basis of the relationship Qr = Qg, the total hydraulic head hrg at the outer boundary of the grouted zone can be derived as follows:
(h + h w ) ln(rg / r0) + (k g / kr ) hrg =
pr 0 γw
ln(h/ rg )
(k g / kr ) ln(h/ rg ) + ln(rg / r0)
(45)
Substituting Eq. (45) into Eq. (42), the analytical solution to the water inflow Q for the subsea grouted tunnel is obtained as
Q=
pr
pr (x , y ) =
w
(kr / k g ) ln(rg / r0)−ln(rg /
rg2
+
4h2 )
+ (h w−y )·γw
ln
x 2 + (y + h)2 x 2 + (y−h)2
pr
⎜
(48)
ln(ρ / rg )
h + h w−hrg dHr ·ρdθ = 2πkr dρ ln(h/ rg )
(41)
4. Comparisons of the analytical solutions and numerical solutions 4.1. The numerical model
(42) To validate the analytical solutions, the program FLAC3D was used to establish a three-dimensional model to analyze the steady seepage field for a subsea grouted tunnel. Taking the F4 weathered slot of Xiamen Xiang’an subsea tunnel in China as an example, the analytical solution of steady seepage field is compared with the numerical solution. Based on the area equivalence principle, the four-centered circular section with excavation area of 170 m2 is equivalent to a circular section with diameter of 7.4 m (Zhang et al., 2014; Li and Zhou, 2015). The thickness of grouted zone is 6 m. The related parameter values are presented in Table 1. The numerical model measured 148 m in width, 1 m in length and 126.4 m in height. The aquifer and grouted zone were
3.3.2. Steady seepage field analysis of the grouted zone. For the grouted zone, the boundary conditions are described as follows: Hg (ρ = rg)=hrg and Hg(ρ=r0)= pr0 /γw. Taking the boundary conditions into the general solution of the Laplace equation, the total hydraulic head Hg in the grouted zone is calculated as
Hg =
pr0 γw
pr ln(ρ / r0) + ⎜⎛hrg− 0 ⎟⎞ γw ⎠ ln(rg / r0) ⎝
⎟
where y represents the ordinate of a certain point at the outer boundary of the grouted zone. Note that, there is a limitation that AMM cannot be applied to analyze the pore pressure distribution in the aquifer region, but can only be used to obtain the water inflow and external water pressure on the grouted zone.
According to Darcy’s law, the water inflow Qr in the aquifer region can be obtained by 2π
(47)
0 ⎡ (h + h w ) ln(rg / r0)−(k g / kr ) γ ln(rg / h) ⎤ y ⎞ w ⎥ ⎛1− p (ρ = rg ) = γw ⎢ (k g / kr ) ln(h/ rg ) + ln(rg / r0) ⎢ ⎥ ⎝ h + hw ⎠ ⎣ ⎦
3.3.1. Steady seepage field analysis in the aquifer region. For the aquifer region, the boundary conditions are described as follows: Hr (ρ = rg) = hrg and Hr(ρ=h)=h + hw. Taking the boundary conditions into the general solution of the Laplace equation, the total hydraulic head Hr in the aquifer region is calculated as
∫0
(46)
It should be mentioned that the external water pressure on the grouted zone obtained by Eq. (47) has the same value around the outer boundary of the grouted zone because both the unit weight and elevation head have been neglected. However, a constant total hydraulic head hrg at the outer boundary of grouted zone is assumed. According to Eq. (2), the external water pressure on the grouted zone varies with the elevation head. Therefore, the external water pressure on the grouted zone obtained by Eq. (47) should be modified considering the elevation head and initial hydrostatic pressure of the tunnel center. The modified external water pressure on the grouted zone is expressed as
In this section, a simplified model using AMM is shown in Fig. 6 to analyze the water inflow and external water pressure on the grouted zone for a subsea grouted tunnel. The far field boundary for the axisymmetric model is regarded as the distance from the tunnel center to the ground surface h. As shown in Fig. 6, an x-y Cartesian coordinate system and a ρ-θ polar coordinate system are applied. The total hydraulic head H(ρ) is given by the Laplace equation under axisymmetric conditions. The general solution to the Laplace equation is H(ρ) =E1 + E2lnρ, where E1 and E2 are constants that can be derived from the boundary conditions.
Qr = kr
)
pr
(40)
ln(h/ rg )
w
(kr / k g ) ln(rg / r0) + ln(h/ rg )
0 ⎡ (h + h w ) ln(rg / r0)−(k g / kr ) γ ln(rg / h) ⎤ w ⎥ p (ρ = rg ) = γw ⎢ (k g / kr ) ln(h/ rg ) + ln(rg / r0) ⎢ ⎥ ⎣ ⎦
3.3. Analytical solutions by the axisymmetric modeling method (AMM)
Hr = hrg + (h + h w−hrg )
pr
For AMM, both the unit weight and elevation head are neglected. Substituting Eq. (45) into Eq. (41) and considering H(ρ = rg) = p (ρ = rg)/γw, the analytical solution to the external water pressure on the grouted zone p(ρ = rg) for the subsea grouted tunnel is denoted as
Fig. 6. Analytic model using AMM.
(h + h w− γ 0 ) γw
(
2πkr h + h w− γ 0
(43)
According to Darcy’s law, the water inflow Qg in the grouted zone 97
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0
Table 1 Parameter values.
Analytical solution obtained by CVM Analytical solution obtained by MIM Numerical solution obtained by FLAC3D
-5
r0 (m)
rg (m)
h (m)
hw (m)
kr (m/s)
kg (m/s)
pr 0 (Pa)
7.4
13.4
52.4
20
5.0 × 10−6
5 × 10−8
0
-10
Depth (m)
-15
simulated as solid elements. The lateral displacement boundaries were fixed in the normal direction and displacement boundaries at the bottom were fixed in both the horizontal and vertical directions. The ground surface boundaries were free and permeable.
-20 -25 -30 -35
4.2. Pore pressure distribution
-40 200
Fig. 7 compares the pore pressure between the analytical solutions (CVM and MIM) and numerical solution. The analytical solution obtained by CVM almost coincides with that obtained by MIM. In Fig. 7(a) and (b), the analytical solutions match well with the numerical solution. However, as shown in Fig. 7(c), the analytical solutions and numerical solution differ at line 5–6, while the tendencies of the curves obtained by the analytical solutions and the numerical solution are coincident. The analysis suggests that the far field boundary influences the accuracy of the numerical solution, which will be illustrated in a later section. The external water pressure on the grouted zone obtained by the analytical solutions (CVM, MIM and AMM) is compared with the numerical solution in Fig. 8. The analytical solutions match quite well with the numerical solution. Moreover, the analytical solutions obtained by CVM and MIM are slightly less than the numerical solution. However, the formulas derivation of AMM is simpler than that of CVM and MIM. The external water pressure on the grouted zone obtained by AMM is slightly higher than the numerical solution, which provides a safe prediction for an engineering application. Overall, the analytical solutions obtained by CVM, MIM and AMM can accurately predict the external water pressure on the grouted zone for a subsea grouted tunnel (see Fig. 8(d)). The relative permeability (i.e., kr/kg) has a significant influence on the deviation between the analytical solutions and numerical solution (Shin et al., 2002; Zhang et al., 2017). Fig. 9 describes the relative deviation between the analytical solutions and numerical solution for water pressure above the tunnel crown (at point 2). As relative permeability increases, the deviation between the analytical solutions and numerical solution decreases gradually and the rate of decrease slows, regardless of which method is used. That is to say, the lower the permeability of the grouted zone is, the smaller the deviation between the analytical solutions and numerical solution becomes. As shown in Fig. 9, when the relative permeability is more than 40, the deviation between the analytical solutions and numerical solution is less than 4%, which implies that the precision of the analytical solutions is adequate and in an acceptable range. In addition, it can also be found from Fig. 9 that the water pressure at point 2 obtained by MIM and AMM is closer to the numerical solution than that obtained by CVM.
250
300
350
400
450
500
550
600
Pore pressure (kPa) (a) Above the tunnel crown (1-2) -65 Analytical solution obtained by CVM Analytical solution obtained by MIM Numerical solution obtained by FLAC3D
Depth (m)
-78
-91
-104
-117
-130 800
900
1000
1100
1200
1300
1400
1500
70
80
Pore pressure (kPa) (b) Underneath the tunnel invert (3-4) 730
Pore pressure (kPa)
725
Analytical solution obtained by CVM Analytical solution obtained by MIM Numerical solution obtained by FLAC3D
720 715 710 705 700 695 10
4.3. Water inflow
20
30
40
50
60
Distance from tunnel axis (m)
Fig. 10 shows the comparisons of the water inflow between the analytical solutions (CVM, MIM and AMM) and numerical solution. The water inflow obtained by the analytical solutions is very close to that obtained by the numerical solution. In addition, the water inflow predicted by CVM and MIM is almost equivalent, just 0.7% less than that predicted by the numerical solution. The water inflow predicted by AMM is 0.4% more than that predicted by the numerical solution. In summary, the analytical solutions obtained by CVM, MIM and AMM can accurately predict the water inflow for a subsea grouted tunnel.
(c) At line 5-6 Fig. 7. Comparisons of pore pressure between analytical solutions and numerical solution.
98
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P. Li et al.
(Numerical solution obtained by FLAC3D) Analytical solution obtained by CVM
(572) 566
(611) 605
(611) 605
Tunnel
(704) 700
(611) 615
(704) 700
Units:kPa
(704) 708
(798) 800
(798) 800 (836) 839
(a) Comparison between CVM and FLAC3D
(c) Comparison between AMM and FLAC3D 3D
(Numerical solution obtained by FLAC ) Analytical solution obtained by MIM
(572) 567
(Numerical solution obtained by FLAC3D) Analytical solution obtained by CVM Analytical solution obtained by MIM Analytical solution obtained by AMM
(611) 606
Tunnel
(704) 700
(611) 615
Units:kPa
(836) 834
(611) 606
(Numerical solution obtained by FLAC3D) Analytical solution obtained by AMM
Tunnel
(704) 708
(798) 795
(798) 795
(572) 577
(704) 700
Units:kPa
Tunnel
(798) 793
(798) 793 (836) 832
(b) Comparison between MIM and FLAC3D
(d) Comparison between the analytical solutions and numerical solution
Fig. 8. Comparisons of external water pressure on the grouted zone between analytical solutions and numerical solution.
4
18 16
Relative deviation (%)
14
Water inflow (m3/(d·m))
Analytical solution obtained by CVM Analytical solution obtained by MIM Analytical solution obtained by AMM
12 10 8 6 4
3.2
3.198
3.235
3.222
CVM
MIM
AMM
FLAC3D
3
2
1
2 0
0
20
40
60
80
100
120
140
160
180
0
200
Relative permeability kr/kg
Fig. 10. Comparison of water inflow between analytical solutions and numerical solution.
Fig. 9. Relative deviation for water pressure at point 2 between analytical solutions and numerical solution.
However, the boundaries of the numerical simulation are finite. Therefore, it is necessary to discuss the influence of the far field boundary on the numerical solution. A case in which the relative permeability kr/kg is equal to 100 is chosen to analyze the influence of the far field boundary. The influence of the lateral boundary on the pore pressure distribution at line 5–6 is described in Fig. 11, where B represents the width of the lateral boundary and D represents the internal diameter of the grouted zone.
5. Analysis of influencing factors 5.1. Influence of the far field boundary For the numerical simulation, the far field boundary should be large enough to obtain more accurate solutions. In this paper, the analytical solutions by CVM and MIM are obtained on the basis of the semi-infinite boundary. The lateral and bottom boundaries are infinite. 99
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0
725
Analytical solution obtained by CVM Analytical solution obtained by MIM Numerical solution obtained by FLAC3D
-5
720
710 705
-15
Depth (m)
Pore pressure (kPa)
-10 Analytical solution obtained by CVM Analytical solution obtained by MIM Numerical solution by FLAC3D (B=5D) Numerical solution by FLAC3D (B=7D) Numerical solution by FLAC3D (B=9D) Numerical solution by FLAC3D (B=11D) Numerical solution by FLAC3D (B=13D)
715
-20 kr/kg=20
-25 kr/kg=50
-30
kr/kg=100
700
-35
B
695
0
20
40
60
80
100
120
140
160
180
-40 200
200
Distance from tunnel axis (m)
kr/kg=5
250
kr/kg=10
300
350
400
450
500
550
600
Pore pressure (kPa)
Fig. 11. The influence of boundary conditions on pore pressure (5–6).
(a) Pore pressure above the tunnel crown (1-2)
External water pressure at point 2 (kPa)
Five cases with different lateral boundaries (B = 5D, 7D, 9D, 11D, 13D) are discussed. From Fig. 11, as the width of the lateral boundary B increases from 5D to 13D, the numerical solution gradually approaches the analytical solutions obtained by CVM and MIM. However, when B increases to 9D ∼ 11D, the change in the numerical solution is very small. So, in the numerical simulation, the range of B = 3D ∼ 5D (which is generally adopted in a dry condition) is not sufficient to analyze the steady seepage field. Stated another way, the range of influence for the seepage field is larger than that for the stress field. It is suggested that the width of the lateral boundary should be at least 9D when conducting a numerical simulation for the steady seepage field. In addition, the enlargement of the bottom boundary has little influence on the numerical solution after analysis. 5.2. The permeability of the grouted zone
600 550 Analytical solution obtained by CVM Analytical solution obtained by MIM Analytical solution obtained by AMM Numerical solution obtained by FLAC3D
500 450 400 350 300 250
0
20
40
60
80
100
120
140
160
180
200
Relative permeability kr/kg
The permeability of the grouted zone has a significant influence on the steady seepage field for a subsea grouted tunnel. Fig. 12 shows the influence of the relative permeability (i.e., kr/kg) on the steady seepage field for a subsea grouted tunnel. The influence of the relative permeability on the pore pressure above the tunnel crown is described in Fig. 12(a). For various relative permeabilities, the regularities of the pore pressure distribution above the tunnel crown are almost coincident. The pore pressure increases gradually with the increase of the relative permeability, regardless of the analytical solutions and numerical solution. The pore pressure increases more remarkably when kr/kg < 50 than when kr/kg > 50. In addition, Fig. 12(a) shows the greater the relative permeability is, the better the analytical solutions and numerical solution match. The influences of the relative permeability on the external water pressure on the grouted zone above the tunnel crown (at point 2) and water inflow are shown in Fig. 12(b) and (c), respectively. The analytical solutions for the external water pressure are closer to the numerical solution when the relative permeability is greater. However, the analytical solutions for the water inflow always match quite well with the numerical solution no matter how the relative permeability changes. As relative permeability increases, the external water pressure on the grouted zone at point 2 increases and the water inflow is gradually reduced, regardless of the analytical solutions and numerical solution. Moreover, the patterns described are stronger when kr/ kg < 50 than when kr/kg > 50. Based on the previous analysis, some significant conclusions can be drawn. Decreasing the permeability of the grouted zone contributes to a decrease of the water inflow. However, at the same time, the grouted zone must support a greater water pressure. When the permeability of the grouted zone decreases to a certain value, the external water
(b) External water pressure on the grouted zone at point 2
Water inflow (m3/(d·m))
50
Analytical solution obtained by CVM Analytical solution obtained by MIM Analytical solution obtained by AMM Numerical solution obtained by FLAC3D
40
30
20
10
0
0
20
40
60
80
100
120
140
160
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Relative permeability kr/kg (c) Water inflow Fig. 12. The influence of relative permeability on the steady seepage field.
pressure on the grouted zone is no longer increased significantly and the water inflow is no longer decreased significantly. Therefore, choosing a reasonable permeability for the grouted zone is crucial to ensure the economy (i.e., the lower discharge) and stability (i.e., the 100
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External water pressure at point 2 (kPa)
900
lower water pressure on grouted zone) for a subsea grouted tunnel.
Analytical solution obtained by CVM Analytical solution obtained by MIM Analytical solution obtained by AMM Numerical solution obtained by FLAC3D
800
5.3. Water depth
4.0
The influences of water depth on the external water pressure on the grouted zone above the tunnel crown (at point 2) and water inflow are shown in Fig. 13(a) and (b), respectively. With the rising of the water table, both the external water pressure and water inflow increase linearly, regardless of the analytical solutions and numerical solution. The influence coefficient of the external water pressure on the grouted zone is defined as λ = p/p0, where p represents the external water pressure on the grouted zone above the tunnel crown (at point 2) under steady seepage state and p0 represents the initial hydrostatic pressure at point 2. The influence of the water depth on the influence coefficient λ is shown in Fig. 13(c). The influence coefficient λ obtained by AMM is constant and not affected by the water depth, while the influence coefficients λ obtained by CVM, MIM and the numerical simulation depend on the water depth. As the water table rises, the influence coefficients λ obtained by CVM, MIM and the numerical simulation increase gradually, while the amount of increase is not significant if the water depth is greater than 20 m. In summary, when the permeabilities of the aquifer and the grouted zone are of certain values, the absolute value of the external water pressure on the grouted zone and influence coefficient are affected by the water depth. Moreover, the water depth has a more direct and significant impact on the absolute value than the influence coefficient.
3.5
6. Conclusions
700 600 500 400 300
0
10
20
30
40
50
Water depth hw (m) (a) External water pressure on the grouted zone at point 2 5.0 Analytical solution obtained by CVM Analytical solution obtained by MIM Analytical solution obtained by AMM Numerical solution obtained by FLAC3D
Water inflow (m3/(d·m))
4.5
To investigate the steady seepage field for a subsea grouted tunnel, analytical solutions were obtained by the complex variable method, mirror image method and axisymmetric modeling method. A series of numerical simulations were performed to validate the analytical solutions and influencing factors, such as the boundary conditions, permeability of grouted zone and water depth for a subsea tunnel, were discussed. The water discharge into the subsea tunnel and distribution of the pore pressure in the aquifer and external water pressure on the grouted zone are the main concerns. The findings of this investigation are summarized as follows.
3.0 2.5 2.0
0
10
20
30
40
50
Water depth hw (m) (b) Water inflow
Influence coefficient λ
1.00
(1) For the steady seepage problem of a subsea grouted tunnel, the analytical solutions by each of the three methods can be applied to analyze the external water pressure on the grouted zone and water inflow into the tunnel with sufficient accuracy. The formula derivation of AMM is simpler than that of CVM and MIM, but AMM cannot be applied to analyze the pore pressure distribution in the aquifer region especially for shallow tunnels. (2) The range of the far field boundary has an important impact on the precision of the pore pressure distribution when the numerical simulation is adopted to analyze the seepage problem of a subsea grouted tunnel. Three to five times the tunnel diameter for the range of the far field boundary, which is generally adopted for dry conditions, is not sufficient to analyze the steady seepage field. It is suggested that the width of the lateral boundary should be at least nine times the tunnel diameter when conducting a numerical simulation for the steady seepage field. (3) The permeability of grouted zone influences comparisons between the analytical solutions and numerical solution, especially for the pore pressure distribution and external water pressure on the grouted zone, but the results remain in an acceptable range. With a decrease of the permeability of grouted zone, both the pore pressure in the aquifer and external water pressure on the grouted zone increase gradually, while the water inflow decreases gradually. (4) When the permeabilities of the aquifer and grouted zone are of certain values, both the external water pressure on the grouted zone and water inflow increase linearly with the rising of the water table.
0.98
0.96
0.94
Analytical solution obtained by CVM Analytical solution obtained by MIM Analytical solution obtained by AMM Numerical solution obtained by FLAC3D
0.92
0.90
0
10
20
30
40
50
Water depth hw (m) (c) Influence coefficient of the external water pressure on the grouted zone at point 2 Fig. 13. The influence of water depth on the steady seepage field.
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In addition, along with the rising of water table, the influence coefficients of the external water pressure on the grouted zone obtained by CVM, MIM and the numerical simulation increase gradually, but the amount of increase is not significant if the water depth is greater than 20 m.
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