## 3 Methodology
The initial gas pore volume \(V_{\rm i}\) [ft3] is expressed as  
\[V_{\rm i}=G_{\rm i}B_{\rm gi}=G_{\rm i}\left(\frac{p_{\rm sc}}{Z_{\rm sc}T_{ \rm sc}}\right)/\left(\frac{p_{\rm i}}{Z_{\rm i}T_{\rm i}}\right) \tag{5}\]  
where \(G_{\rm i}\) is the initial gas in place [SCF], and \(B_{\rm gi}\) is the gas formation volume factor at the reservoir's initial conditions [ft3/SCF]. If the initial gas in place \(G_{\rm i}\) [SCF] is unknown; the initial gas pore volume \(V_{\rm i}\) [ft3] can be calculated by  
\[V_{\rm i}=A\ h\ \varnothing\ (1-S_{\rm wi}) \tag{6}\]  
where \(A\) is the reservoir area [ft2], \(h\) is the reservoir thickness [ft], \(\varnothing\) is porosity [fraction], and \(S_{\rm wi}\) is the initial water saturation [fraction]. The effective compressibility \(c_{\rm eff}\) [1/psi], accounting for both rock and gas expansion with pressure change as defined by [PERSON] et al. [5], is expressed as  
\[c_{\rm eff}=\frac{c_{\rm f}+S_{\rm w}c_{\rm w}}{1-S_{\rm w}} \tag{7}\]  
where \(c_{\rm f}\) is the formation compressibility [1/psi] and \(c_{\rm w}\) is the water compressibility [1/psi]. Hydrogen mixes with methane after injection; therefore, we can express the total gas moles in the pore space with  
\[n_{\rm gt}=\frac{p_{\rm stor}V_{\rm g,\rm stor}}{Z_{\rm g,\rm stor}RT_{\rm stor}} \tag{8}\]  
where \(p_{\rm stor}\) is the storage pressure [psi], \(V_{\rm g,\rm stor}\) is the total gas volume in the pore space [ft3], \(Z_{\rm g,\rm stor}\) is the gas mixture compressibility factor, \(T_{\rm stor}\) is the reservoir temperature [\({}^{\circ}\)R], and \(R\) is the universal gas constant 10.73 [psi:ft3\(\cdot\)lbmol\({}^{-1}\cdot\cdot\)R\({}^{-1}\)]. The total moles of gas at storage conditions can also be expressed as  
\[n_{\rm gt}=n_{\rm CH_{4}\rm stor}+n_{\rm H_{2}\rm,\rm stor} \tag{9}\]  
where \(n_{\rm CH_{4}\rm stor}\) is the moles of methane at storage conditions [lbmol], and \(n_{\rm H_{2}\rm,\rm stor}\) is the moles of hydrogen at storage conditions [lbmol]. Based on the conservation of mass relation formulated in Equation (1), the molar quantity of methane is constant, i.e., \(n_{\rm CH_{4}\rm stor}=n_{\rm CH_{4}\rm,\rm dep}\), which was estimated using Equation (3). Therefore, we can solve for the molar quantity of hydrogen, \(n_{\rm H_{2}\rm,\rm stor}\), in two ways:  
1. By rearranging Equation (9): \[n_{\text{H}_{2}\text{,sc}}=n_{\text{H}_{2}\text{,stor}}\]0 (10)
2. By introducing the molar fraction \(x_{\rm H_{2}}\): \[n_{\text{H}_{2}\text{,sc}}=n_{\text{H}_{2}\text{,stor}}\]1 (11) Rearranging the equation: \[n_{\text{H}_{2}\text{,sc}}=n_{\text{H}_{2}\text{,stor}}\]2 (12) Finally, the surface volume of hydrogen can be computed from the moles of hydrogen using the real gas law as follows: \[n_{\text{H}_{2}\text{,sc}}=n_{\text{H}_{2}\text{,stor}}\]3 (13)The following steps demonstrate the iterative approach to estimating the injected hydrogen volume for storage \(V_{\text{H}_{2,\text{g}\text{c}}}\):  
1. Determine the pore volume occupied by methane, \(V_{\text{CH}_{4}\text{dep}}\), using Equation (4) at depletion conditions. 2. Compute the molar quantity of methane, \(n_{\text{CH}_{4}\text{dep}}\), using Equation (3). 3. Start with an initial guess for the hydrogen mole fraction, \(x_{\text{H}_{2}}\), such that \(0\leq x_{\text{H}_{2}}<1\). 4. Using the initial (or updated) \(x_{\text{H}_{2}}\) guess, calculate the compressibility factor, \(Z_{\text{g,stor}}\), for the gas mixture. 5. Determine the pore volume occupied by the total gas mixture, \(V_{\text{g,stor}}\), using Equation (4) at storage condition. 6. Calculate the total gas moles, \(n_{\text{gt}}\), using Equation (8). 7. Calculate the hydrogen moles, \(n_{\text{H}_{2,\text{g}\text{tor}}}\), using Equations (10) and (12). 8. Check for convergence (\(n_{\text{H}_{2}}\) from Equation (10) = \(n_{\text{H}_{2}}\) from Equation (12)). 9. If convergence is not reached, update \(x_{\text{H}_{2}}\) guess with the hydrogen moles from Equation (10): \[n_{\text{H}_{2}\text{,sc}}=n_{\text{H}_{2}\text{,stor}}\]4
10. Return to Step 4 and repeat. 11.