## 2 Simplified Theoretical Model  
The collaborative drainage system consists of a tunnel blind pipe network and parallel adit drain borehole. The tunnel is drained through the blind pipe network, and the parallel adit is drained using the drain boreholes (Figure 1).  
According to the equal permeable area principle, the sections of tunnel and parallel adit are equivalent to circular sections, and the crown position of the tunnel remains unchanged (Figure 2). In addition, the following assumptions are proposed in this paper. (1) Only the external water pressure of the lining (without considering the thickness of the lining) is considered, and the permeability coefficient of surrounding rock is isotropic and constant; (2) The fluid is incompressible and has a state of steady flow; (3) The water table is horizontal and remains constant; (4) The radial seepage velocities are equal at both the tunnel and parallel adit boundaries.  
A global Cartesian coordinate system is established with the water table line as the x-axis and the vertical symmetry axis of the parallel adit as the y-axis. The half-plane \(Z\) below the water table contains the half-plane \(Z_{\rm P}\) and half-plane \(Z_{\rm t}\), which correspond to the coordinate system \(x_{\rm P}\rho_{\rm P}y_{\rm P}\) above the parallel adit and \(x_{\rm t}\rho_{\rm t}y_{\rm t}\) above the tunnel, respectively; \(Q_{\rm P}\) and \(Q_{\rm t}\) are the drainage volume of the parallel adit and tunnel, respectively; \(r_{\rm P}\) and \(r_{\rm t}\) are the radius of the equivalent circle of the parallel adit and tunnel, respectively; \(d_{\rm P}\), \(d_{\rm t}\), and \(d_{\rm w}\) are the burial depths of the parallel adit, tunnel, and water table, respectively; \(H_{\rm t}\) is the pressure head at the tunnel boundary when the parallel adit and tunnel are drained; \(H_{\rm t}^{0}\) is the initial hydrostatic pressure head at the tunnel boundary; \(S_{\rm h}\) is the horizontal net distance between the tunnel and the parallel adit; and \(S_{\rm v}\) is the height difference between the parallel adit and the tunnel. The points A, B, C, and D represent the position of tunnel crown, left wall, inverted arch, and right wall respectively.  
Figure 1: Drainage system for the tunnel with its parallel adit.  
According to [PERSON]'s law, mass conservation law, and the above assumptions, the total head satisfies the Laplace equation:  
\[\frac{\partial^{2}h}{\partial x^{2}}+\frac{\partial^{2}h}{\partial y^{2}}=0 \tag{1}\]  
where \(h\) is the total head, given by the sum of the elevation head and pressure head (kinetic head is neglected because groundwater flow is assumed to be slow): \(h=u/\gamma_{\rm w}+y\), \(u\) is the pore water pressure, \(\gamma_{\rm w}\) is the unit weight of water, \(u/\gamma_{\rm w}\) is the pressure head, and \(y\) is the elevation head.  
Using the basic assumptions, the boundary conditions were determined as follows:  
At the water table:  
\[h\big{|}_{y=0}=0 \tag{2}\]  
At the parallel adit circumference:  
\[\frac{\partial h}{\partial n_{\rm p}}\Big{|}_{x^{2}+(y+c_{\rm p})^{2}=r_{\rm p }^{2}}=i_{\rm p}=\frac{Q_{\rm p}}{2K\pi r_{\rm p}} \tag{3}\]  
At the tunnel circumference:  
\[\frac{\partial h}{\partial n_{\rm t}}\Big{|}_{(x-l)^{2}+(y+c_{\rm t})^{2}=r_{ \rm t}^{2}}=i_{\rm t}=\frac{Q_{\rm t}}{2K\pi r_{\rm t}} \tag{4}\]  
where \(c_{\rm p}=d_{\rm p}+r_{\rm p}-d_{\rm w}\), \(c_{\rm t}=d_{\rm t}+r_{\rm t}-d_{\rm w}\), \(l=r_{\rm t}+r_{\rm p}+S_{\rm h}\); \(n_{\rm p}\) and \(n_{\rm t}\) are the radial direction of the circumference of the parallel adit and the tunnel, respectively (the outer normal is positive); \(K\) is rock permeability coefficient; and \(i_{\rm p}\) and \(i_{\rm t}\) are the radial hydraulic gradient of the parallel adit and tunnel circumference, respectively.