深埋隧道排水系统非对称堵塞后渗流场的解析研究

Analytical study on seepage field of the deep tunnel with asymmetric blocked drainage system

基于目前未见对隧道排水系统非对称堵塞渗流场进行解析研究,通过对研究区域进行简化假设,求解代表渗流场的Laplace方程。对研究区域内未知边界条件,采用PDE工具进行数值解后,得到近似的非齐次边界条件。基于此,再运用复平面内的保角变换、坐标变换、微分方程定解问题的分解等方法,将边界条件齐次化,在Hilbert空间中将最后唯一的非齐次边界条件表达为正交函数族的级数和,从而得到了此情况下的隧道周边渗流场解析级数,其后考查了级数项的多少对计算结果的影响。研究结果表明:(1)隧道排水系统非对称堵塞后渗流场可以表达为解析后级数,可简化将来针对这一情况下渗流场的求解难度;(2)隧道排水系统非对称堵塞后渗流场大致呈漏斗状,但在堵塞侧水头较高;(3)求和项数对计算精度的影响至多到100项,多数情况下只需50项;(4)考查了隧道可能受到的水压力分布和渗流量,分别与相同条件下的数值模拟结果进行对比,两者相差仅4.28%和4.05%。

Analytical study on seepage field of the deep tunnel with asymmetric blocked drainage system has not been carried out at present. An analytical solution of the Laplace equation was derived based on the simplification of the studying area. With the PDE tools, the unknown but necessary boundary conditions were converted to the approximate inhomogeneous ones. These boundary conditions were homogenized with methods like conformal transformation in complex plane, transformation of coordinates, separation of problem for determining the solution of differential equation. The last inhomogeneous boundary condition was expressed as the sum of a family of orthogonal functions in Hilbert space. Finally, a solution of series for the seepage field were derived. The seepage flow field is similar to the funnel shape. The hydraulic head is higher in blocked side. And the further study shows that in most situations the sum of series converges quickly when the number of term in series greater than 50. The differences between the analytical and numerical results of the distribution of hydraulic pressure and seepage flux are only 4.28% and 4.05%.