## 3 Theoretical Calculation and Result Verification
### Degradation Analysis  
Using Taylor series expansion theorem [32], the result is  
\[\frac{e^{nh}}{n^{2}}-\frac{h}{n}=\frac{1}{n^{2}}+\frac{1}{n}\overset{+\infty} {\underset{i=2}{\sum}}\frac{n^{i-1}h^{i}}{i!} \tag{22}\]  
Substituting Formula (22) into (19), the result is  
\[\frac{1}{n}\overset{+\infty}{\underset{i=2}{\sum}}\frac{n^{i-1}h^{i}}{i!}= \frac{q}{2m}x-\frac{w}{2m}x^{2}+\frac{1}{n}\overset{+\infty}{\underset{i=2}{ \sum}}\frac{n^{i-1}H^{i}}{i!}-\frac{q^{2}}{8 mw} \tag{23}\]Equation (23) is the head function expression when only considering the permeability coefficient of surrounding rock affected by in situ stress. When \(n=0\), \(m=k_{r}\) Formula (23) degenerates into  
\[h^{2}=-\frac{wx^{2}}{k_{r}}+\frac{qx}{k_{r}}+H^{2}-\frac{q^{2}}{4k_{r}w} \tag{24}\]  
Comparing Equation (24) with Equations (4) and (7), the difference between the three equations is only the permeability coefficient of each structure and the parabolic constant term. The structural permeability coefficient is determined by the properties of the three regions, and the constant term is determined by the boundary conditions. That is, Formula (23) can be degenerated into the head potential function expression without considering the permeability coefficient of the surrounding rock changing with the ground stress: (4) and (7), which verifies the rationality and correctness of the derivation to a certain extent.  
Substituting Formula (22) into (14), the result is  
\[\frac{1}{n}\sum_{i=2}^{+\infty}\frac{n^{i-1}h^{i}}{i!}=-\frac{w}{a}x+\frac{1}{n }\sum_{i=2}^{+\infty}\frac{n^{i-1}h_{\text{A}}{}^{i}}{i!}+\frac{w}{a}l+\frac{qa +2 wm_{0}}{2a^{2}}\ln\frac{ax+m_{0}}{al+m_{0}} \tag{25}\]  
By using [PERSON]'s formula, then the results obtained are  
\[\ln(x+1)=\sum_{i=0}^{\infty}\frac{\left(-1\right)^{i}}{i+1}x^{i+1} \tag{26}\]  
Substituting Equation (26) into Equation (25), the result is  
\[\frac{1}{n}\sum_{i=2}^{+\infty}\frac{n^{i-1}h^{i}}{i!}=-\frac{w}{a}x+\frac{1}{ n}\sum_{i=2}^{+\infty}\frac{n^{i-1}h_{\text{A}}{}^{i}}{i!}+\frac{w}{a}l+\frac{qa+2 wm_{0}}{2a^{2}}\sum_{i=0}^{\infty}\left[\frac{\left(-1\right)^{i}}{i+1}\left( \frac{ax-al}{al+m_{0}}\right)^{i+1}\right] \tag{27}\]  
Equation (27) is the head function expression considering the permeability coefficient of surrounding rock is simultaneously affected by the coupling of excavation disturbance and ground stress. When \(a=0\), \(m_{0}=m\), Formula (27) degenerates into  
\[\frac{1}{n}\sum_{i=2}^{+\infty}\frac{n^{i-1}h^{i}}{i!}=\frac{1}{n}\sum_{i=2}^ {+\infty}\frac{n^{i-1}h_{\text{A}}{}^{i}}{i!}-\frac{x^{2}}{2m}-\frac{l^{2}}{2 m}+\frac{xl}{m} \tag{28}\]  
Comparing Equations (28) and (23), the difference between the two equations is only the symmetry axis of the parabola and the constant line, which is affected by the abscissa of the boundary and the height of the water head. Therefore, Equation (13) can be degenerated into Equation (19) that does not consider the influence of the surrounding rock permeability coefficient on excavation disturbance, and Equation (19) is the special working condition derived from Equation (13).  
After degradation analysis, the head expression for the permeability coefficient of the surrounding rock as a fixed value is considered as the special case of this article considering the permeability coefficient of the surrounding rock is affected by the in-situ stress; The limit condition of the influence of ground stress and excavation disturbance on the permeability coefficient of surrounding rock verifies the correctness of the formula derivation in this paper to a certain extent.