The seepage field around twin noncircular tunnels in anisotropic sloping

Abstract

A new and efficient theoretical model for studying the seepage field around two noncircular tunnels in sloping ground is proposed through analytical derivation. The model considers the exact interaction between two tunnels with arbitrary shapes and locations, as well as the permeability anisotropy, whose principal orientation is along any direction. By applying the Schwartz alternating method and conformal mapping, the analytical solutions of the hydraulic head and pore pressure can be obtained. These solutions strictly satisfy all governing equations and conditions. The correctness and accuracy of the analytical solutions are verified by the good agreement between the theoretical and numerical results. The proposed theoretical model can be applied to the cases with any tunnel shape, slope angle, anisotropic direction and degree. The quantitative influences of the anisotropy ratio, slope angle and tunnel shape on the seepage field are obtained, and some useful charts are provided. The proposed model can efficiently and quickly predict the seepage flow around tunnels in anisotropic sloping ground.

Résumé

Un nouveau modèle théorique efficace pour étudier le champ d’infiltration autour de deux tunnels non circulaires dans un terrain en pente est proposé par dérivation analytique. Le modèle prend en compte l’interaction exacte entre deux tunnels de formes et d’emplacements arbitraires, ainsi que l’anisotropie de perméabilité, dont l’orientation principale est dans n’importe quelle direction. En appliquant la méthode alternative de Schwartz et la cartographie conforme, les solutions analytiques de la charge hydraulique et de la pression interstitielle peuvent être obtenues. Ces solutions satisfont strictement à toutes les équations et conditions régissantes. L’exactitude et la précision des solutions analytiques sont vérifiées par le bon accord entre les résultats théoriques et numériques. Le modèle théorique proposé peut être appliqué aux cas avec n’importe quelle forme de tunnel, angle de pente, direction et degré d’anisotropie. Les influences quantitatives du rapport d’anisotropie, de l’angle de pente et de la forme du tunnel sur le champ d’infiltration sont obtenues et quelques graphiques utiles sont fournis. Le modèle proposé permet de prédire efficacement et rapidement l’écoulement des eaux d’infiltration autour des tunnels dans des terrains en pente anisotropes.

Resumen

Se propone un modelo teórico nuevo y eficiente para estudiar mediante derivación analítica la infiltración en el entorno de dos túneles no circulares en un terreno inclinado. El modelo considera la interacción exacta entre dos túneles con formas y ubicaciones arbitrarias, así como la anisotropía de la permeabilidad, cuya orientación principal es a lo largo de cualquier dirección. Al aplicar el método alterno de Schwartz y el mapeo conforme, se pueden obtener las soluciones analíticas de la carga hidráulica y la presión de poros. Estas soluciones satisfacen estrictamente todas las ecuaciones y condiciones determinantes. La corrección y precisión de las soluciones analíticas se verifican por la buena concordancia entre los resultados teóricos y numéricos. El modelo teórico propuesto puede aplicarse a los casos con cualquier forma de túnel, ángulo de inclinación, dirección y grado de anisotropía. Se obtienen las influencias cuantitativas de la relación de anisotropía, el ángulo de inclinación y la forma del túnel en el campo de infiltración, y se proporcionan algunos gráficos útiles. El modelo propuesto puede predecir de forma eficiente y rápida la infiltración que se produce alrededor de los túneles en un terreno con pendiente anisótropa.

摘要

提出了一种新的高效理论模型, 用于通过分析推导研究坡地两个非圆形隧道周边的渗流场。该模型考虑了两个具有任意形状和位置的隧道之间的准确相互作用, 以及渗透率的各向异性, 其主导方向沿任意方向。通过应用Schwartz交替法和共形映射, 可以得到水头和孔隙压力的解析解。这些解析解严格满足所有主导方程和条件。通过理论和数值结果之间的良好一致性验证了解析解的正确性和准确性。提出的理论模型可应用于具有任意隧道形状、坡度角、各向异性方向和程度的情况。得到了各向异性比、坡度角和隧道形状对渗流场的定量影响, 并提供了一些有用的图表。提出的模型能够高效快速地预测各向异性坡地隧道周边的渗流流动。

Resumo

Um novo e eficiente modelo teórico para estudar o campo de infiltração em torno de dois túneis não circulares em terreno inclinado é proposto através de derivação analítica. O modelo considera a interação exata entre dois túneis com formas e localizações arbitrárias, bem como a anisotropia de permeabilidade, cuja orientação principal é em qualquer direção. Aplicando o método alternado de Schwartz e o mapeamento conformal, as soluções analíticas da carga hidráulica e da pressão dos poros podem ser obtidas. Essas soluções satisfazem estritamente todas as equações e condições governantes. A correção e precisão das soluções analíticas são verificadas pela boa concordância entre os resultados teóricos e numéricos. O modelo teórico proposto pode ser aplicado a casos com qualquer formato de túnel, ângulo de inclinação, direção e grau anisotrópico. As influências quantitativas da razão de anisotropia, ângulo de inclinação e formato do túnel no campo de infiltração são obtidas e alguns gráficos úteis são fornecidos. O modelo proposto pode prever com eficiência e rapidez o fluxo de infiltração em torno de túneis em terrenos inclinados anisotrópicos.

Appendix: Nomenclature

a:

Length of the major half-axis of the elliptical tunnel

b:

Length of the short semi-axis of the elliptical tunnel

c_j:

Vertical projection of ${o}_j{o}_j^{\prime }$

d_j:

Height of the equivalent hole

e_j:

Horizontal projection of ${o}_j{o}_j^{\prime }$

H₁:

Buried depths of tunnel 1

H₂:

Buried depths of tunnel 2

H_w:

Depth of water above tunnel 1

i:

Imaginary unit

K_x:

Permeability along the principal anisotropy direction x₀

K_y:

Permeability along the principal anisotropy direction y₀

L:

Horizontal distance between the centres of the two tunnels

m_j,λ:

Real part of the coefficient δ_{j, λ}

n_j,λ:

Imaginary part of the coefficient δ_{j, λ}

N:

Total number of the iterations

n:

Anisotropic permeability ratio

p:

Pore pressure

p₁:

Water pressure at the boundary of tunnel 1

p₂:

Water pressure at the boundary of tunnel 2

p_j:

Water pressure at the boundary of tunnel j

q_j:

Specific flow at an arbitrary point on the tunnel j boundary

r:

Radius of the pilot tunnel

r^*:

Equivalent circle radius

z_j:

A point in the physical plane z

β:

Slope angle

β′:

Slope angle in the transformed plane

β₁:

Angle between anisotropy direction x₀ and horizontal direction

γ_w:

Volumetric weight of water

ζ_j:

A point in the image plane

θ_j:

Polar angle in the mapped plane

ρ_j:

Polar radius in the mapped plane

δ_j,λ:

Coefficient determined by the specific boundary shape

Φ:

Total hydraulic head

Φ₁:

Total hydraulic head at the tunnel 1 boundary

Φ₂:

Total hydraulic head at the tunnel 2 boundary

Φ₁₁:

Hydraulic head in the first step of the first iteration

Φ₁₂:

Hydraulic head in the second step of the first iteration

Φ₂₁:

Hydraulic head in the first step of the second iteration

Φ₂₂:

Hydraulic head in the second step of the second iteration

Φ_kj:

Hydraulic head in the jth step of the kth iteration

${\varPhi}_{11}^R$ :

Redundant hydraulic head at the tunnel 2 boundary in the first step of the first iteration

${\varPhi}_{12}^R$ :

Redundant hydraulic head at the tunnel 1 boundary in the second step of the first iteration

${\varPhi}_{21}^R$:

Redundant hydraulic head at the tunnel 2 boundary in the first step of the second iteration

${\varPhi}_{22}^R$ :

Redundant hydraulic head at the tunnel 1 boundary in the second step of the second iteration

${\varPhi}_{kj}^R$ :

Redundant hydraulic head at the fictitious boundary of another tunnel boundary in the jth step of the kth iteration

${\varPhi}_{\left(k-1\right)2}^R$:

Redundant hydraulic head at the tunnel 1 boundary in the second step of the (k-1)th iteration