Abstract
This paper provides an analytical solution and solving procedure for the seepage field around two noncircular tunnels in anisotropic permeable ground. The anisotropic problem is first made equivalent to an isotropic problem by coordinate transformation. Then, by using Schwartz alternate method and conformal mapping, a precise analytical solution is obtained based on the anisotropic problem of single tunnel with redundant hydraulic heads at tunnel boundary. The noncircular tunnel interaction and the permeable anisotropy are considered accurately. The iterative procedure is simple and efficient in its calculations, and achieves good convergence. The results of the analytical model are compared with the finite element results, and show good agreement. Finally, some parametric studies are present to research the influences of tunnel shapes and the anisotropic permeability ratio on the seepage field, and the effect of the location of the pilot tunnel on the seepage field of the primary tunnel is also investigated.
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Abbreviations
- a :
-
Width of the quasi-rectangular tunnel
- b :
-
Height of the quasi-rectangular tunnel
- d j :
-
Tunnel height
- H :
-
Total hydraulic head
- H 1 (H 2):
-
Total hydraulic head at Tunnel 1 (2) boundary
- H w :
-
Depth of water above the ground surface
- \(H_{kj}^0\) :
-
Hydraulic head in the jth step of the kth iteration
- \(H_{11}^R\) :
-
Redundant hydraulic head around Tunnel 2 boundary in the first step of the first iteration
- \(H_{12}^R\) :
-
Redundant hydraulic head around Tunnel 1 boundary in the second step of the first iteration
- \(H_{k1}^R\,\,\,\left( {H_{\left( {k - 1} \right)2}^R} \right)\) :
-
Redundant hydraulic head around the Tunnel 2 (1) boundary in the first (second) step of the kth ((k-1)th) iteration
- h 1 (h 2):
-
Buried depths of Tunnel 1 (2)
- i :
-
Imaginary unit
- K x :
-
Permeability along the horizontal direction
- K y :
-
Permeability along the vertical direction
- L :
-
Tunnel spacing
- N :
-
Total number of iterations
- n :
-
Anisotropic permeability ratio
- p :
-
Pore pressure
- P j :
-
ater pressure around boundary of Tunnel j
- r :
-
Radius of the pilot tunnel
- r * :
-
Equivalent circle radius
- S j :
-
Area of original tunnel j
- x :
-
Horizontal coordinate in global coordinate system
- x 0 :
-
Horizontal coordinate after coordinate transformation (global)
- x 0j :
-
Horizontal coordinate after coordinate transformation (local)
- x j :
-
Horizontal coordinate in local coordinate system (Tunnel j)
- y :
-
Vertical coordinate in global coordinate system
- y 0 :
-
Vertical coordinate after coordinate transformation (local)
- y 0j :
-
Vertical coordinate after coordinate transformation (local)
- y j :
-
Vertical coordinate in local coordinate system (Tunnel j)
- z j :
-
A point in the physical plane
- γ w :
-
Volumetric weight of water
- ζ j :
-
A point in the image plane
- θ j :
-
Polar angle in the mapped plane ρi = Polar radius in the mapped plane
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Acknowledgments
This study was supported by the National Natural Science Foundation of China (Grant Nos.11872281, 51890911) and the State Key Laboratory of Disaster Reduction in Civil Engineering (SLDRCE19-A-06). This support was greatly appreciated.
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Wei, F., Wang, H., Zeng, G. et al. Analytical Solution to the Seepage Field of Two Parallel Noncircular Tunnels in Permeable Anisotropic Ground. KSCE J Civ Eng 26, 5328–5341 (2022). https://doi.org/10.1007/s12205-022-0054-0
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DOI: https://doi.org/10.1007/s12205-022-0054-0