# Analytical Solutions of Water Inflow for Foundation Pit in Confined Water Stratum
## 1 Introduction  
In recent years, with the continuous improvements in national economies and urbanization levels, more and more high-rise buildings and large-scale underground structures have appeared in cities, resulting in the number of deep foundation pits sharply increasing [1, 2, 3, 4]. However, the deep foundation pit often encounters the confined water stratum during the excavation process, and dewatering measures need to be taken to avoid problems such as groundwater intrushing and overall floating when the hydraulic head is high [5, 6]. Among the existing dewatering measures, the \"wall-well\" dewatering system [7, 8, 9], which is composed of pumping wells and suspended water-proof curtains, has been widely used because of the advantages of outstanding water control effect and economic benefits. During the design of the \"wall-well\" system, it is necessary to accurately calculate the water inflow to reasonably determine the dewatering capacity [10, 11, 12]. Extensive existing research has studied the water inflow of the foundation pits with suspended water-proof curtains from analytical solutions, numerical simulation, and site measurements. [PERSON] [13] considered the barrier effect of the water-proof curtain from two cases and proposed a series of simple equations to calculate the head difference at the two sides of the water-proof curtain. [PERSON] [14] carried out the numerical simulation of the large deep excavation by 3D finite difference method (FDM) and compared it with the observation results; the analysis results show that the water-proof curtain can minimize seepage and uplift in large excavation pits, although settlement outside the pit may need treatment measures. [PERSON] [15] used numerical and analytical methods to derive semi-empirical equations to quantify the two barrier effects for semi-permeable, partially penetrating (or fully penetrating but finite in length), and barriers with a by-pass in confined aquifers, obtaining excellent agreementbetween the calculated and observed barrier effects at two construction sites. [PERSON] [16] investigated the mechanism of the blocking effect, which is affected by dewatering through the finite difference method; the results show that drawdown varies along the depth of the confined aquifer, and the influence factors of drawdown inside and outside the pit include insertion depth of retaining walls, anisotropy of a confined aquifer, and screen length of pumping wells. [PERSON] [17] conducted a three-dimensional finite element simulation to analyze the dewatering process of the Dongjiadu subway tunnel repair foundation pit in Shanghai. [PERSON] [18] regarded the foundation pits with suspended water-proof curtains as a large-diameter vertical well, and a simplified formula for water inflow calculation of foundation pit dewatering was derived; the simplified formula had a higher accuracy when the curtain insertion ratio was greater than 0.6 and the value of the foundation pit radius to confined aquifer thickness was less than 2.0.  
In conclusion, the existing analytical solutions do not have a good consistency with numerical solutions and site-measured values. They only have good accuracy in some specific \(r_{0}/M\) and \(L/M\) intervals (where \(r_{0}\) is the radius of the circular foundation pit, \(L\) is the insertion depth of the water-proof curtain, and \(M\) is the thickness of the confined aquifer). The reason is that the equivalent length of the seepage path applied in water inflow calculation is unreasonable. Numerical methods are based on specific engineering projects, and the accuracy of the results depends on the selection of parameters and boundary conditions [8; 9; 10; 11; 14], which is not universal. This paper obtained the analytical solutions of water inflow for multi-shape foundation pits, introduced the influence coefficient \(\alpha\) of seepage path length, and proposed the expressed formula. The results show that the analytical solutions in this paper are in good agreement with the numerical results and measured data.