## 5 Case Study  
In this section, we illustrate the practical application of the validated MBE analytical framework on a large-scale reservoir. The numerical simulation is performed on a synthetic model representing a heterogeneous volumetric gas reservoir (Figure 10). This methodology was applied only to volumetric gas reservoirs since modeling water-drive gas reservoirs is more challenging. There are numerous options for modeling the aquifer, leading to a nonunique solution. Without history-matched field data, it is challenging to determine the appropriate aquifer modeling approach. Therefore, the focus on volumetric gas reservoirs ensures a more controlled and reliable application of the proposed methodology. Table 7 summarizes the model properties. To model this case, we set the minimum bottom hole pressure to 500 psi. After 25 years of production, the abandonment pressure was reached, and 85% of the original gas in place was recovered. Next, the field was shut in for 5 years before commencing hydrogen injection. The maximum bottom hole pressure was equal to the initial reservoir to simulate the maximum capacity for hydrogen storage. Hydrogen was injected through two wells located in each of the domes in the model. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**P [psi]** & \(V_{\text{H}_{2,\infty}}\)**(MBE) [SCF]** & \(V_{\text{H}_{2,\infty}}\)**(simulation) [SCF]** & **\% Error** \\ \hline
3500 & 3.24 \(\times\) 10\({}^{7}\) & 3.27 \(\times\) 10\({}^{7}\) & 1.05\% \\
3300 & 2.83 \(\times\) 10\({}^{7}\) & 2.86 \(\times\) 10\({}^{7}\) & 0.98\% \\
3000 & 2.21 \(\times\) 10\({}^{7}\) & 2.23 \(\times\) 10\({}^{7}\) & 0.87\% \\
2750 & 1.68 \(\times\) 10\({}^{7}\) & 1.69 \(\times\) 10\({}^{7}\) & 0.76\% \\
2500 & 1.13 \(\times\) 10\({}^{7}\) & 1.14 \(\times\) 10\({}^{7}\) & 0.62\% \\
2300 & 6.84 \(\times\) 10\({}^{6}\) & 6.87 \(\times\) 10\({}^{6}\) & 0.42\% \\
2000 & 0.00 & 0.00 & 0.00\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Injected hydrogen volume from the numerical simulation and the proposed MBE (water-drive reservoir case). Figure 9: Molar fraction of hydrogen (\(x_{H_{2}}\)) computed by the simulator and the MBE code for the water-drive reservoir. We input the required data into the MBE Framework code and compared the obtained results with the simulation output, as reported in Table 8.  
With an average error of 0.82%, the close match between the MBE and the simulator results supports using the proposed framework for further analysis and practical applications in underground hydrogen storage. The slight discrepancies between the MBE and simulation results can be attributed to minor numerical approximations inherent in both methods. However, the low percentage error suggests that these discrepancies are negligible, reinforcing the reliability of this analytical framework for practical applications. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Number of active grid blocks & 47,151 \\ Grid block dimensions [ft] & \(300\times 300\) \\ Model dimensions & \(27\times 45\times 4\) \\ Grid average thickness [ft] & 31 \\ Initial pressure, \(p_{\mathrm{i}}\) [psi] & 3345 \\ Depletion pressure, \(p_{\mathrm{dep}}\) [psi] & 500 \\ Reservoir temperature, \(T_{\mathrm{i}}\) [\({}^{\circ}\)F] & 180 \\ Average reservoir porosity [frac] & 0.13 \\ Average reservoir permeability [md] & 67 \\ Original gas in place, \(G_{\mathrm{i}}\) [MMSCF] & \(2.27\times 10^{5}\) \\ Initial water saturation, \(S_{\mathrm{w}}\) [frac] & 0.25 \\ Formation rock compressibility, \(C_{\mathrm{f}}\) [psi\({}^{-1}\)] & \(1.0\times 10^{-6}\) \\ Water compressibility, \(C_{\mathrm{w}}\) [psi\({}^{-1}\)] & \(3.0\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Complex reservoir simulation model properties. Figure 10: Three-dimensional complex reservoir model used for the case study.