###### Abstract  
Composite geomembrane cut-off walls (CGCW) have been widely used for the remediation of polluted sites, especially where the environmental conditions are complex. Accurate predictions of the GMB hole leakage and CGCW performance are essential for engineering design and cost control. This paper establishes empirical equations to predict the leakages through the CGCWs based on the numerical models. Additionally, an analytical solution for contaminant migration through the CGCW is proposed considering the effects of GMB holes. The accuracy of the established equations and analytical solution is verified by the numerical models. The key effects of the GMB thickness (\(T_{G}\)), head loss (\(H_{G}\)), cut-off wall hydraulic conductivity (\(k_{G}\)), hole radius (\(r_{G}\)) and shape on the leakage and CGCW performance are investigated. The results show that compared with other hole shapes, the leakage through the circular hole is lowest. This is mainly because the shape factor for the circular hole is 1.15-1.3 times lower than that for other shapes of holes with the same area. Additionally, the effects of the hole geometric properties and head loss on the CGCW performance can be more significant when the cut-off wall hydraulic coefficient is small. For example, the breakthrough time differences between the cases with \(r_{G}=0.005\) m and 0.05 m are 0.8 and 5.0 years when \(k_{G}=10^{-10}\) and \(10^{-9}\) m/s, respectively. This is because the impermeability of the CGCW is good when \(k_{G}\) is small. This will weaken the impacts of the hole geometric properties on the leakage. The proposed empirical equations and analytical solution can provide effective suggestions for the design of the CGCW in different GMB hole cases. empirical equations; analytical solution; GMB holes; composite geomembrane cut-off wall +
Footnote †: journal: Applied Sciences  
**Citation:** [PERSON]; [PERSON]; [PERSON]; [PERSON] Analytical Model for  
test to evaluate the hydraulic performance of the CGCWs based on the flexible wall permeameter. However, though experiments can permit direct observation, incorrect operations and low-precision apparatuses will lead to errors in experimental results [15]. Additionally, the modeling and computational steps of numerical simulations are always time-costly and software-dependent [16; 17]. Unlike the above methods, analytical solutions have been frequently used in recent years due to their ease of calculation [18; 19]. For example, [PERSON] et al. [20] considered the effects of diffusion, adsorption and degradation of organic contaminants on the performance of the CGCW and established a 2D analytical solution. GMB defects are the main cause of GMB failure and will affect the performance of the CGCW. It is found that a GMB hole (even if it is smaller than 10 mm) will lead to a significant leakage [21]. The increase in the hole diameter can also cause a large leakage increase [22]. There are several empirical equations for the hole leakage calculation [23]. For example, [PERSON] et al. [24] established an empirical equation to predict leakages of different wrinkle heights and widths and tested it with landfills observed data in North America. [PERSON] and [PERSON] [25] proposed new empirical equations to evaluate GMB hole leakage covered by saturated tailings based on experiments. However, all mentioned analytical studies on the performance of the CGCW didn't take into consideration the impact of GMB holes on contaminant transport. This will overestimate the service time and hydraulic performance of the CGCW [26]. Additionally, the field investigations and experiments show that the shapes of GMB holes are not always circular and water flow rates vary with GMB hole shapes [27; 28]. For example, [PERSON] and [PERSON] [29] found that the leakage of a triangular hole is 5% higher than that of a circular hole with the same area. There is no simple method to calculate the leakage of different hole shapes in the CGCW and aquifer system, currently.