## 4 Verification with Numerical Simulation of a Tank Model  
In this section, we verify the proposed methodology by constructing a simple tank model consisting of a single grid block with a single well, as shown in Figure 3. It is important to note that the tank model approach is widely used in petroleum engineering, and there is extensive literature comparing its performance with numerical simulation, demonstrating its usefulness, limitations, and how it can be modified for various purposes; example references are provided [15; 33; 36; 37; 38; 39; 40; 41]. The verification is conducted for two cases: depletion/volumetric drive gas reservoir and water drive gas reservoir using the commercial compositional reservoir simulator CMG GEM [42]. The governing equations  
Figure 2: Flowchart illustrating the proposed methodology for estimating hydrogen storage capacity. for compositional simulators are comprised of a complex system of partial differential equations (associated with the material balance of each component or pseudo-component in the system), along with thermodynamic relationships and correlations to model fluid properties and phase equilibrium [43]. [PERSON] and [PERSON] [33] demonstrated how the zero-dimensional MBE can be derived from the governing partial differential equation used in numerical simulation, which is expressed as follows:  
\[-\
abla\cdot(\rho_{\text{G}}v_{\text{G}}y_{\text{i}}+\rho_{\text{L}}v_{\text{L }}x_{\text{i}})=\frac{\partial}{\partial t}[\phi(\rho_{\text{G}}S_{\text{G}}y_ {\text{i}}+\rho_{\text{L}}S_{\text{L}}x_{\text{i}})]-q_{i} \tag{14}\]  
where \(\rho_{\text{G}}\) and \(\rho_{\text{L}}\) are the densities of the gas and liquid phase, \(v_{\text{G}}\) and \(v_{\text{L}}\) are the gas and liquid velocities, and \(y_{\text{i}}\) and \(x_{\text{i}}\) are the mole fraction of component i in the gas phase and liquid phase, respectively. Finally, \(q_{i}\) is the injection/production rate per unit volume of component \(i\). When a single gridblock is considered and assuming no flow through the boundaries, the term on the left-hand side of Equation (14) becomes zero, and the material balance becomes a relationship between injection/production and the fluid in place. [PERSON] [44] describes the tank model as the fundamental component of reservoir simulators, where the rock and fluid properties do not exhibit any spatial variation. This makes it an ideal verification tool for the zero-order material balance equation (MBE). The verification is performed by comparing the resulting gas mixture compressibility factor, hydrogen molar fraction, and the volume of injected hydrogen at surface conditions. The parameters used to design this model were selected based on typical values encountered in the field. It is important to mention that the purpose of this verification exercise is to ensure that the MBE proposed in this work is mathematically accurate. A benchmark of the proposed MBE and a numerical solution considering a reservoir model with typical multidimensional discretization is presented later to investigate how the proposed MBE performs when mass and heat transfer within the porous formation is considered. Table 1 summarizes the model properties and indicates which ones are required as inputs for the Python code. The initial pore volume can be calculated using Equation (5) or Equation (6). Hence, the (if applicable) note in Table 1. In this exercise, we opted to calculate it using Equation (6). One of the most important considerations in this study is ensuring representative modeling of the phase behavior of the hydrogen-methane mixture. Accurately capturing fluid behavior is critical because it influences numerous factors such as density, compressibility, and the overall mixture behavior under varying pressure and temperature conditions. [PERSON] et al. [45] conducted a comparison of the performance of three equation-of-state (EOS) models in matching experimental data of the binary hydrogen-methane gas mixture density at different mole fractions, which were obtained from [PERSON] et al.