## 4 Verification with Numerical Simulation of a Tank Model
### Water-Drive Gas Reservoir  
Water-drive gas reservoirs are characterized by an active aquifer, causing water encroachment as the reservoir is depleted. Consequently, the pressure in such reservoirs does not decline as much as in volumetric gas reservoirs. Additionally, gas production is significantly hindered by water coning, resulting in a recovery factor ranging between 50 and 80% [5]. To test the performance of the proposed MBE methodology on this type of reservoir, we initialized the tank model as a gas reservoir that has been depleted with a 63% recovery factor and has been encroached upon by the aquifer. Therefore, the water saturation in this case is set at 50%, and the reservoir pressure is set at 2000 psi. Hydrogen is immediately injected at the start of the simulation run until the initial reservoir pressure of the tank model 3500 psi is reached. Figure 7 illustrates the increase in pressure during the storage period. The next step is translating the simulation conditions to inputs for the MBE. The produced water volume, \(W_{p}\), and the aquifer influx volume, \(W_{e}\), are set to arbitrary values such that their total equals the change in reservoir water volume after depletion. The change in reservoir water volume, \(\Delta V_{w}\), is calculated from the change in pore volume using Equation (6) at the initial water saturation in the original tank model and the final water saturation of the depletion case tank model as follows:  
\[\Delta V_{\rm w}=V_{\rm f}-V_{\rm i} \tag{16}\]  
Table 4 summarizes the resulting MBE inputs used for this case. The outputs for this case are reported in Table 5.  
Figure 7: Increase in average reservoir pressure with the injection of hydrogen. The first variable examined for the verification is the gas mixture compressibility factor. Figure 8 compares the compressibility factors calculated using the Python code and those obtained from the numerical simulator. Despite the complexity introduced by the two-phase flow, which includes gas and water in the system, the compressibility factor computation is not compromised, with results that are extremely close to those generated by the simulator. The presence of water typically complicates the calculations due to the interactions between the gas and liquid phases, as well as the changes in the overall mixture's properties. Nonetheless, there is an excellent agreement between the analytical framework and the numerical simulation output of hydrogen mole fraction, as shown in Figure 9. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Initial pressure, \(p_{\text{i}}\) [psi] & 3500 \\ Depletion pressure, \(p_{\text{dep}}\) [psi] & 2000 \\ Water Saturation, \(S_{\text{w}}\) [frac] & 0.50 \\ Reservoir temperature, \(T_{\text{i}}\) [\({}^{\circ}\)F] & 100 \\ Volume of produced water, \(W_{\text{p}}\) [STB] & 5419 \\ Volume of influx water, \(W_{\text{e}}\) [bbbl] & 53,428 \\ Water formation volume factor, \(B_{\text{w}}\) [bbl/STB] & 0.986 \\ \hline \end{tabular}
\end{table} 
Table 4: MBE inputs for the water-drive reservoir case. Figure 8: Gas mixture compressibility factor computed by the simulator and the MBE code for the water-drive reservoir. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**T** [\({}^{\circ}\)F] & **P [psi]** & \(Z_{\text{CH}_{4}}\) & \(Z_{\text{H}_{2}}\) & \(V\) [ft\({}^{3}\)] & \(x_{\text{H}_{2}}\) & \(Z_{\text{g}}\) & \(n_{\text{H}_{2}}\) \\ \hline
100 & 3500 & 0.91 & 1.14 & \(4.50\times 10^{5}\) & 0.33 & 1.03 & \(8.54\times 10^{4}\) \\
100 & 3300 & 0.90 & 1.13 & \(4.50\times 10^{5}\) & 0.30 & 1.01 & \(7.46\times 10^{4}\) \\
100 & 3000 & 0.89 & 1.12 & \(4.49\timesLastly, the injected hydrogen volume (\(V_{\text{H}_{2,\infty}}\)) is compared in Table 6. The results show good agreement between the analytical framework and the numerical simulator. The percentage error is higher than the volumetric reservoir case due to the added layer of complexity with the two-phase flow of gas/water.