## 4 Result and Discussion
### Calculation Equation of Diffused Gas Volume  
Due to Equation (11) being a typical partial differential equation, only numerical solutions for \(P\)-\(t\) can be obtained through mathematical calculations, forming a numerical sequence about \(P\)-\(t\), and it is impossible to obtain an exact analytical solution \(f(x)=P(t)\). Similarly, the free gas concentration calculated based on the \(P\)-\(t\) sequence is also a numerical sequence about \(C\)-\(t\), and the exact analytical solution \(f(x)=C(t)\) cannot be obtained; however, geological history is a lengthy process, and geological parameters change extremely slowly during this time. The evolution of free gas concentration is a continuous function rather than independent scattered points that vary over time. When calculating the diffusion gas volume using Equation (2) and the free gas concentration at the top of the cap layer using Equation (5), it is necessary to obtain a continuous function for the free gas concentration of the reservoir over time, i.e., \(C(t)=C_{t}^{0}\). Therefore, when setting boundary conditions, it is necessary to approximate the complex sequence of free gas concentration in geological history as a combination of \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Well**} & **Numerical Simulation Results of** & **Numerical Simulation Results of** & **Actual Exploration** \\  & **Free Gas Content** & **Total Gas Content** & **Results [5; 6]** \\ \hline Wei 201 & 1.8 m\({}^{3}\)/t & 2.3–3.0 m\({}^{3}\)/t & 2.1–4.8 m\({}^{3}\)/t & 76.8\% \\ Wei 204 & 3.0 m\({}^{3}\)/t & 3.8–5.0 m\({}^{3}\)/t & 3.8–7.3 m\({}^{3}\)/t & 79.3\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of numerical simulation results and actual exploration results. multiple sets of quadratic polynomial changes (Figure 7). The change in diffusion coefficient in geological history is approximated as a combination of multiple sets of constant values, and the weighted average of diffusion coefficient at multiple time points is used to represent the diffusion coefficient in the short period of time. The mathematical model of shale gas diffusion in well Wei201 and well Wei204 is:  
\[\frac{\partial C}{\partial t}=D\frac{\partial^{2}C}{\partial x^{2}};\ D= \begin{cases} 343.4\ \ \text{m}^{2}/\text{Ma}&(0<t\leq 72)\\ 230.7\ \ \text{m}^{2}/\text{Ma}&(72<t\leq 83)\\ 120.3\ \ \text{m}^{2}/\text{Ma}&(83<t\leq 147)\\ 120.3\ \ \text{m}^{2}/\text{Ma}&(147<t\leq 244)\end{cases}  \tag{13}\]  
The initial condition of time variation of gas content at the top boundary of Wei201 L\({}_{1}\) in positive construction is:  
\[C_{0}^{x}=0;\ C_{t}^{0}= \begin{cases}-0.0001t\ ^{2}+0.0265t+0.1079&(0\leq t\leq 72) \\ -0.1091t\ ^{2}+16.869t-647.37&(72<t\leq 83)\\ -0.0001t\ ^{2}+0.0528t+6.5066&(83<t\leq 147)\\ 0.0012t\ ^{2}-0.5168t+63.651&(147<t\leq 215)\\ 0.0003t\ ^{2}-0.2644t+49.624&(215<t\leq 244)\end{cases}  \tag{14}\]  
The initial condition of time variation of gas content at the top boundary of Wei204 L\({}_{1}\) in negative construction is:  
\[C_{0}^{x}=0;\ C_{t}^{0}= \begin{cases}-0.0001t\ ^{2}+0.0265t+0.1079&(0\leq t\leq 72 )\\ -0.1091t\ ^{2}+16.869t-647.37&(72<t\leq 83)\\ -0.0001t\ ^{2}+0.0528t+6.5066&(83<t\leq 147)\\ 0.0021t\ ^{2}-0.8679t+95.694&(147<t\leq 215)\\ 0.0004t\ ^{2}-0.2313t+34.658&(215<t\leq 244)\end{cases}  \tag{15}\]  
The evolution of free gas content in the cover layer was obtained by solving the partial differential equation of the function using the forward difference approach, as shown in Figure 8. Figure 8 shows that prior to the shale gas diffusion time \(t=0\) (approximately 244 Ma years ago), the organic matter had not yet reached the hydrocarbon generation threshold, resulting in zero gas content in the 250 m thick cover layer that covered the L\({}_{1}\). The gas in the shale reservoir is gradually escaping upward due to the continuously strengthened ability of the formation subsidence to generate organic gas, causing the diffusion front to approach the interface of the direct cap top.