# Analytical Solutions of Water Inflow for Foundation Pit in Confined Water Stratum
## 2 Water Inflow Solutions of Circular and Strip Foundation Pits
### The Influence Coefficient a of Seepage Path in Confined Water Stratum  
To investigate characteristics of the seepage path influence coefficient \(\alpha\), a seepage numerical model for the circular foundation pits is established. Based on the symmetry, a quarter of the model is set to increase calculation efficiency, which is shown in Figure 2. The stratum conditions of the numerical model are sandy-sticky mixed stratum, and the aquifer is mainly fine sand, where the permeability coefficient is \(k_{r}=k_{v}=5\times 10^{-6}\) m/s, the hydraulic head for the confined water aquifer is \(H=30\) m, the hydraulic head in the foundation pit is \(H_{d}=30\) m, and the thickness of the aquifer is \(M=10\) m. The radial boundary of the model is a fixed hydraulic head, the excavation surface of the foundation pit is the free boundary, and the pore water pressure \(0\) is fixed. The water-proof curtain and the impermeable layer are impermeable boundaries; the other boundaries are fixed by the normal seepage velocity. \(r_{0}/M=1\), \(3\), \(5\), \(7\), \(9\) are selected as the test cases and to explore the variation in the coefficient \(\alpha\). The remaining conditions are selected as the verification cases. Figure 3 shows the variation in water inflow with \(r_{0}/M\) and \(L/M\). It can be seen from Figure 3a that \(L/M\) has a greater impact on water inflow than \(r_{0}/M\). The water inflow in the foundation pit decreases as \(L/M\) increases, and when the scale of the foundation pit is small (\(r_{0}/M=1\)), the water inflow has the linear relationship with \(L/M\). However, in other cases, the water inflow initially decreases linearly as \(L/M\) increases, then decreases exponentially. Under the same value of \(L/M\), as \(r_{0}/M\) increases, the water inflow continues to increase, but the increase rate decreases. The variation in the influence coefficient \(\alpha\) with \(r_{0}/M\) and \(L/M\) are shown in Figure 4. It can be seen from Figure 3a that both the \(r_{0}/M\) and \(L/M\) have a  
Figure 2: Numerical model for the pit with suspended water-proof curtain in confined water stratum. significantly effect on \(\alpha\). What is more, \(L/M\) has a greater influence. If the effect of \(L/M\) on the equivalent seepage path is ignored, that is, taking \(\alpha\) equal to 1, the analytical solution of water inflow is consistent with the numerical results when \(L/M\) is 0.7-0.8 in Figure 4a. As \(L/M\) increases, \(\alpha\) firstly increases linearly, and then increases exponentially, approximately with \(L/M\) = 0.7 as the critical point. As \(r_{0}/M\) increases, \(\alpha\) firstly increases and then decreases, and the inflection point is \(r_{0}/M\) = 3. It can be seen from Figure 4b that \(r_{0}/M\) = 3 is the dividing point of the trend of influence coefficient. This is because, when \(r_{0}/M\) < 3, the seepage direction of water in the foundation pit is close to the vertical direction, and the equivalent seepage path is close to the shortest seepage path \(L_{\text{min}}\). Under the same \(L/M\), the smaller the \(r_{0}/M\), the smaller the coefficient \(\alpha\). According to the characteristics of \(\alpha\) in Figure 4a, the functional relationship between \(\alpha\) and \(L/M\) is established as follows:  
\[\alpha= \begin{cases}A+K\Big{(}\frac{L}{M}\Big{)}&(0<L/M<0.7)\\ I+B_{1}\Big{(}\frac{L}{M}\Big{)}+B_{2}\Big{(}\frac{L}{M}\Big{)}^{2}&(0.7\leq L /M<1)\end{cases}  \tag{15}\]  
where \(A\) is the intercept of the linear function; \(K\) is the slope of the linear function; \(I\) is the intercept of the quadratic polynomial; \(B_{1}\) is the coefficient of the linear term; \(B_{2}\) is the coefficient of a quadratic term. Fitting Equation (15) with the curve in Figure 4a obtains parameters \(A\), \(K\), \(I\), \(B_{1}\), and \(B_{2}\) under different \(r_{0}/M\). Then, the functional relationship between \(r_{0}/M\) and those parameters is established as follows:  
\[\left\{