## 3 Theoretical Model of Fluid Flow in Porous Media  
The prediction of fluid transport properties in the rock at the pore scale is the basis for analyzing macroscopic flow characteristics. Due to the complexity of the pore structure inside the rock, fluid flow within the rock often exhibits highly complex multi-scale features. At the nanoscale or mesoscopic scale, fluid flow can be modeled using fictive methods such as the lattice Boltzmann method, where the behavior of the fluid is described by the interactions of fluid molecules or pseudo-particles in random motion. This approach captures the random and discrete nature of fluid flow at small scales. However, at the macroscopic scale, the fluid flow process can be considered as continuous, with the flow characteristics described by classical fluid dynamics equations. The study of hydrogen flow in reservoirs is crucial for understanding its permeability in porous media. Due to the small size of hydrogen molecules, it exhibits higher diffusivity and greater permeability, which causes its flow behavior in low-permeability rocks to differ significantly from that of water or oil. Understanding the flow mechanisms of hydrogen in reservoirs is essential for the development of hydrogen storage and recovery technologies, particularly in the context of underground hydrogen storage and two-phase flow dynamics. The flow of fluids in porous media is influenced not only by the complexity of the pore structure but also by various displacement mechanisms. In the case of two-phase flow, fluid movement within the pores typically exhibits different displacement patterns. Common displacement mechanisms include piston-like displacement, breakthrough phenomena, and hysteretic displacement. Piston-like displacement generally occurs in porous media with larger pore sizes or uniform pore structures, where the fluid moves in a relatively stable manner, resulting in a well-defined interface [41]. In contrast, breakthrough phenomena arise in more complex pore structures, where the fluid interface is irregular and accompanied by a significant pressure gradient. These displacement mechanisms have a significant impact on the macroscopic behavior of fluid flow, including permeability, seepage velocity, and other related properties. Therefore, understanding these mechanisms and incorporating them into flow models is essential for describing two-phase flow behavior in porous media, particularly in the context of underground hydrogen storage [42]. Figure 3: Schematic diagram of the CT scanning workflow. [PERSON]'s law is the most fundamental and widely used theory for describing the macroscopic flow behavior of fluids in porous media, and it is derived from steady-state flow experiments, as shown in Equation (1). The expression for two-phase flow in porous media can be derived from Equation (1), as shown in Equation (2) [43]:  
\[k=\frac{q\mu L}{\
abla pA} \tag{1}\]  
\[q_{\alpha}=-\frac{kk_{\pi}}{\mu}(\
abla p) \tag{2}\]  
where \(q\) is the flow rate, m\({}^{3}\)/s; \(k\) and \(kr\) represent the absolute permeability and relative permeability respectively, m\({}^{2}\); \(\mu\) is the fluid's viscosity, Pa-s; \(\
abla p\) is the pressure gradient, Pa; \(\alpha\) equals 1 or 2, representing different fluid phases; \(A\) and \(L\) represent sample dimension, i.e., cross-sectional area and length, respectively. Although [PERSON]'s law is the fundamental theory for describing fluid flow in porous media, in certain special cases, particularly when the pore structure is complex, flow channels are narrow, or two-phase flow occurs, the fluid flow behavior often deviates from the predictions of [PERSON]'s law. In these situations, the flow no longer follows the linear behavior described by [PERSON]'s law, but instead exhibits typical non-Darcy flow characteristics. As a result, researchers have started using non-Darcy flow models to describe two-phase flow [44].