## 6 Conclusions  
This paper proposes the empirical equations and analytical model to evaluate the performance of the CGCW while considering the effects of GMB holes. The accuracy of the established equations and analytical solution is confirmed by the numerical models. The key effects of the GMB thickness, head loss, cut-off wall hydraulic conductivity, hole radius and shape on the leakage and CGCW performance are investigated.  
1. The proposed solution can offer more accurate suggestions for the design of the CGCW than the pure diffusion model, especially when the leakage is large (e.g.,  
Figure 10: Influence of the head loss on the breakthrough curve.  
\(>\)10\({}^{-10}\) m/s). For example, the contaminant concentrations for \(v_{2}=10^{-9}\) m/s can be 7.43 and 1.79 times larger than that for \(v_{2}=10^{-10}\) m/s and \(v_{2}=5\times 10^{-10}\) m/s when \(t=50\) years, respectively.
2. The values of the leakage and head loss are linearly related. For example, increasing the head loss from 1 m to 10 m leads to a 10-time increase in the leakage. Additionally, changing the GMB thickness has an insignificant influence on the leakage when the \(k_{\rm G}\) is small. Thus, the pumping method is suggested to be used to control the head loss and improve the CGCW performance in practical engineering.
3. Various hole shapes can lead to different performances of the CGCW. Additionally, the performance of the CGCW is better when the GMB hole is circular. For example, the breakthrough time for the circular hole is 1.1 times larger than that for the rectangular hole when \(k_{\rm G}=10^{-9}\) m/s. This is mainly because the leakage through the circular hole is low as the shape factor for the circular hole is 1.15-1.3 times lower than that for other shapes of holes.
4. The influences of the hole radius, shape and head loss on the breakthrough curve of the CGCW seem to be more significant when hydraulic conductivity is large. For example, although the head loss increases by 33-fold (from 0.3 m to 10 m), the breakthrough time of the CGCW is only 1.2 times larger than before when \(k_{\rm G}=10^{-10}\) m/s. Thus, the hydraulic conductivity of the cut-off wall is suggested to be controlled lower than \(10^{-9}\) m/s.  
L.R.: Conceptualization, Methodology, Validation, Visualization, Writing--original draft, Writing--review & editing. G.W.: Validation, Investigation, Writing--review & editing. H.D.: Conceptualization, Project administration, Resources, Supervision, Writing--original draft, Writing--review & editing. H.X.: Funding acquisition, Investigation, Writing--review & editing. All authors have read and agreed to the published version of the manuscript.  
The financial supports from the National Key R&D research and development Program of China (Grant No. 2023 YFC3709602), National Natural Science Foundation of China (Grant Nos. 52278375, 41931289, and 41977223), and Zhejiang Provincial Natural Science Foundation (Gran No. LR20E080002) are greatly acknowledged.  
Not applicable.  
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.  
Data Availability Statement: Data are available from the authors upon reasonable request.  
Authors [PERSON], [PERSON] and [PERSON] were employed by the company The Architectural Design and Research Institute of Zhejiang University Co., Ltd.; the remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.  
References