# Water flow into tunnels in discontinuous rock: a short critical review of the analytical solution of the art

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## Abstract

Indeed, determination of groundwater inflow into tunnels, especially in discontinuous rock masses, is still one of the most challenging issues faced by tunneling experts, which may have destructive consequences and result in impediments to tunneling operations. Hence, it is necessary to predict the location and amount of inflow in the tunneling designation phase. In order to quantify the groundwater inflow rate into tunnels, various analytical, empirical, and numerical methods exist in the literature. Analytical solutions are of great interest since they provide desirable approximations quickly and without the necessity for advanced computers. However, there are no comprehensive and unique databases concerning analytical methods. Therefore, the attempt of this study was to present the various analytical methods that exist in the literature, their application domain, and their validation.

## Keywords

Water inflow Tunnel Analytical equation Rock media## Notations

## Latin

*A*Cross-section area (m

^{2})*a*Constant parameter

*a*_{b}Distance from the axis of the tunnel to the bottom of the aquifer (m)

*a*_{F( J)}Aperture of joints in the field (mm)

*a*_{i}Corner of normal direction of fracture and coordinate axis (°)

*a*_{J}Aperture surface area of joints (m

^{2})*b*Aquifer thickness (m)

**b**Constant parameter

- C
A coefficient only related to the shape and depth

*C*_{j}Inter-connectivity coefficient

*C*_{J}Obtained circumference of shape from intersection of joint strike with tunnel axis (m)

*C*_{m}Median of the specific capacity of the rock mass (m

^{2}/s)*D*Hydraulic load above land surface (m)

*D*_{ij}Discontinuity (

*i*th discontinuity for*j*th set) of diameter (discontinuity with circular shape) (m)*d*_{i}Average aperture of the

*i*-set of discontinuities (m)*e*_{i}Average frequency of the

*i*-set of discontinuities (m^{− 1})*g*Earth gravity (m/s

^{2})*H*Hydraulic head into the tunnel (m)

*h*Depth of the tunnel center from the water table (m)

**h**Water depth above the ground (m)

*h*_{e}Energy head (m)

*h*_{gs}Distance of the tunnel center to the ground surface (m)

*h*_{gssf}Distance of the tunnel center to the ground surface or sea floor (m)

*h*_{wt}Distance of the tunnel center to the water table (m)

*I*Hydraulic gradient (m)

**I**Identity matrix

- I
_{0} Modified Bessel function of the first kind of order zero

*i*Cluster number of fracture

*K*Permeability coefficient (m/s)

*K*_{epm}Equivalent hydraulic conductivity (m/s)

*K*_{max}Maximum components of the hydraulic conductivity (m/s)

*K*_{min}Minimum components of the hydraulic conductivity (m/s)

*K*_{sim}Empirical hydraulic conductivity (m/s)

- K
_{0} Modified Bessel function of the second kind of order zero

*k*_{gc}Permeability coefficient of grouting circle (m/s)

*k*_{l}Permeability coefficient of lining (m/s)

*k*_{sr}Permeability coefficient of surrounding rocks (m/s)

*k*_{sl}Permeability coefficient of secondary lining (m/s)

*L*Tunnel length (m)

*m*The number of joint sets

*N*Total number of discontinuities sets

*P*_{i}Crude seepage field force (kg/m.s

^{2})*P*_{r}Excess pressure at opening (kg/m

^{2})*Q*Inflow to tunnel water (m

^{2}/s)*Q*_{ap}Approximated water inflow rate (m

^{2}/s)*Q*_{f}The average flow rate (m

^{2}/s)*Q*_{x}Leakage into a certain section of a tunnel quantified according to the seismic velocity (m

^{3}/s)*R*Influence radius of the flow (m)

**R**Radius of the objective volume of the rock mass around the tunnel (m)

*R*_{x}The horizontal influence distance of groundwater level drawdown from the center of the tunnel (m)

*R*_{z}The vertical influence distance of groundwater level drawdown from the initial groundwater level (m)

*r*Tunnel radius (m)

**r**Borehole radius for which the capacity is estimated (m)

*r*_{e}External tunnel radius with lining (m)

*r*_{gc}Radius of grouting circle (m)

*r*_{i}Internal tunnel radius with lining (m)

*r*_{l}Radius of lining (m)

*r*_{sl}Radius of secondary lining (m)

*r*_{sr}Radius of surrounding rock (m)

*r*_{0}Influence radius of borehole (m)

**S**A coefficient only related to the tunnel’s shape and depth

*s*Drawdown (m)

*S*_{J}Spacing of joints (m)

*V*Mean value of seismic velocity (m/s) of the whole tunnel line

*Z*Depth of the tunnel’s centerline (m)

## Greek

*∆*Gradient operator

*γ*Unit weight of water

*γ*_{w}Water specific weight

*θ*_{ij}Acute angle between the axis of the tunnel and the normal vector of the

*i*th discontinuity of the*j*th set (°)*θ*_{min}Minimum joint dip (°)

*μ*Dynamic viscosity of water (kg/m/s)

*ϑ*Kinematic viscosity of water (3.20 ·10

^{− 6}m^{2}/s)- \( \overline{v} \)
Mean value of the seismic velocity (m/s) of the rock mass which has been penetrated by the boreholes

- π
A mathematical constant, commonly approximated as 3.14159

- ∅
_{min} Angle between direction minimum components of the hydraulic conductivity and the horizontal plane (°)

- ∅
_{0} Hydraulic head at the tunnel perimeter (m)

## Operator and symbol

- \( \overrightarrow{n_i} \) ⌊n
_{1}; n_{2}; n_{3}⌋ Dimensionless unitary vector normal to the average plane of the discontinuity set

## Notes

### Acknowledgements

The authors would like to thank the enthusiastic researchers in the field of groundwater in underground spaces and related topics. Moreover, the valuable and constructive comments of the anonymous reviewers are greatly appreciated.

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