## 2 Physical and Mathematical Model
### Mathematical Model
#### 2.2.2 Leakage Diffusion Modelling  
The dispersion of HBNG leakage in utility tunnels is determined by three fundamental equations: the continuity equation, the energy equation, and the momentum equation. It is important to follow both the component transport equation and the turbulence equation when studying the diffusion process of HBNG, which has many parts and behaves erratically as it leaks from high-pressure jets.  
1. Continuity equations: \[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}+\frac{ \partial(\rho v)}{\partial y}=0\] (1)
2. Energy equation: \[\frac{\partial(\rho h)}{\partial t}+\frac{\partial(\rho u_{x}h)}{\partial x}+ \frac{\partial(\rho u_{y}h)}{\partial y}=pdivU+div(\lambda gradT)+\phi+E_{k}\] (2)
3. Momentum equation: \[\left\{\begin{array}{l}\frac{\partial(\rho u_{x})}{\partial t}+\frac{ \partial(\rho u_{x}u_{y})}{\partial x}+\frac{\partial(\rho u_{x}u_{y})}{ \partial y}=-\frac{\partial p}{\partial x}+\mu\left(\frac{\partial^{2}u_{x}} {\partial x^{2}}+\frac{\partial^{2}u_{y}}{\partial y^{2}}\right)+F_{x}\\ \frac{\partial(\rho u_{y})}{\partial t}+\frac{\partial(\rho u_{x}u_{y})}{ \partial x}+\frac{\partial(\rho u_{y}u_{y})}{\partial y}=-\frac{\partial p}{ \partial y}+\mu\left(\frac{\partial^{2}u_{y}}{\partial x^{2}}+\frac{\partial ^{2}u_{y}}{\partial y^{2}}\right)+F_{y}\end{array}\right.\] (3)
4. Component transport equation: \[\frac{\partial(\rho Y_{i})}{\partial t}+\
abla(\rho V_{i}Y_{i})=-\
abla\cdot I _{i}+R_{i}+S_{i}i\] (4)  
\[J_{i}=-\rho\frac{1-X_{i}}{\sum\limits_{j,j\
eq i}{(X_{j}/D_{ij})}}\
abla\cdot Y _{i} \tag{5}\]
5. Turbulence equation: This study utilizes the standard \(k\)-\(\varepsilon\) turbulence model. This model was initially introduced by Launder and Spalding in 1991 Launder and Spalding in 1991 Launder and Spalding in 1991 Launder and Spalding in 1991 Launder and Spalding in 1991 Launder and Spalding in 1991 Launder and Spalding in 1991 Launder and Spalding in 1991 Launder and Spalding in 1991 Launder. It is founded on the transport equations for turbulent kinetic energy \(k\) and its dissipation rate. The conventional \(k\)-\(\varepsilon\) model is widely employed in industrial fluid dynamics including gas diffusion because of its robustness and cost-effectiveness. Below are the formulae for turbulent kinetic energy \(k\) and its dissipation rate \(\varepsilon\).  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho\mu_{i}k)}{\partial x_ {i}}=\frac{\partial\left[\left(\mu_{i}+\frac{\mu_{i}}{\varepsilon_{k}} \right)\frac{\partial k}{\partial x_{j}}\right]}{\partial x_{j}}+G_{k}+G_{b}- \rho\varepsilon-Y_{M}+S_{K} \tag{6}\]  
\[\frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial(\rho\mu_{i} \varepsilon)}{\partial x_{i}}=\frac{\partial\left[\left(\mu_{i}+\frac{\mu_{ i}}{\varepsilon_{k}}\right)\frac{\partial k}{\partial x_{j}}\right]}{\partial x_{j}}+ C_{\varepsilon 1}\frac{\varepsilon}{k}(G_{k}+C_{\varepsilon 3}G_{b})-C_{ \varepsilon 2}\rho\frac{\varepsilon^{2}}{k}+S_{\varepsilon} \tag{7}\]  
The model constants \(C_{\varepsilon 1}\) is 1.44, \(C_{\varepsilon 2}\) is 1.92, \(\sigma_{k}\) is 1.0, and \(\sigma_{\varepsilon}\) is 1.3.  
To close the equations, the transport equation and the mixed gas density equation need to be augmented as follows [PERSON] and Spalding (1991):  
\[\frac{\partial(\rho\varsigma)}{\partial t}+\frac{\partial}{\partial x_{j}}( \rho u_{j}\varsigma)=\frac{\partial}{\partial x_{j}}\left(\mu_{i}\frac{ \partial\varsigma}{\partial x_{j}}\right) \tag{8}\]  
\[\rho=\frac{P}{RT}\frac{M_{m}M_{a}}{[\varsigma M_{a}+(1-\varsigma)M_{m}]} \tag{9}\]