## 3 Theoretical Model of Fluid Flow in Porous Media
To accurately simulate these complex flow mechanisms, several modified theoretical models have been proposed, taking into account the multi-scale nature of the flow and the impact of displacement mechanisms. The MD equation characterizes the overall transport properties of fluids by studying the motion laws of all fluid particles in the statistical model, which can precisely describe the distribution of fluid particles in the model, thereby obtaining properties such as adsorption and diffusion of the fluid in the model. Figure 4 illustrates the molecular dynamics model for fluid migration in nanopores [45]. The MD equation, which solves the motion equations for each particle to obtain the changes in particle positions, differs significantly from the disordered pore structure of real porous media, and the basic mathematical governing equation is shown in Equation (3):  
\[m_{i}\frac{d^{2}r_{i}(t)}{dt^{2}}=-\sum_{i<j}\frac{\partial V(r_{ij})}{ \partial r_{ij}} \tag{3}\]  
where \(m_{i}\) represents the mass of particle \(i\), g/mol; \(r_{i}\) is the position vector of particle \(i\); \(V(r_{ij})\) is the interaction potential function between particles \(i\) and \(j\); \(r_{ij}\) is the relative position between particle \(i\) and \(j\); \(t\) is time, in picoseconds (ps). The NS equation is based on the continuum hypothesis, which treats the fluid as a continuous medium filling the entire flow domain. This allows the definition of properties  
Figure 4: Schematic diagram of the migration mode of fluid particles inside a single nanotube [45]. such as fluid density, velocity, temperature, and pressure at each point in the flow domain. The continuum hypothesis is a fundamental assumption in fluid mechanics and can be seen as an approximation of fluid behavior. The motion of the fluid satisfies the laws of conservation of mass, momentum, and energy, and can be mathematically described by a set of partial differential equations [46], as shown in Equation (4):  
\[\left\{ \begin{array}{c}\frac{\partial p}{\partial t}+\
abla\cdot(\rho u)=0\\ \frac{\partial(\rho u)}{\partial t}+\
abla\cdot(\rho uu)=\
abla\cdot\sigma\\ \frac{\partial(\rho e)}{\partial t}+\
abla\cdot(\rho ue)=\sigma:\
abla u- \
abla\cdot q\end{array} \right. \tag{4}\]  
where \(u\) and \(e\) represent the velocity vector and the specific internal energy, respectively; \(\sigma\) is the stress tensor, Pa; \(q\) is the heat flux caused by thermal conduction and radiation, J/s. To better reveal the inherent mechanism of the two-phase flow process under various scenarios, researchers have developed robust theoretical models to describe the flow process, which not only improve our understanding of the underlying fluid dynamics at the pore scale but also aid in optimizing the displacement efficiency of two-phase systems [47]. Flow in porous media can be described using different methods at various scales. At the pore scale, the flow behavior is typically analyzed by directly solving flow equations based on physical mechanisms. For single-phase flow, the Navier-Stokes (NS) equations are commonly used to describe the balance between fluid inertia and viscous forces. In the case of two-phase flow, methods such as the phase field or level-set methods are employed to solve the interface evolution equations, while considering microscopic forces like capillary pressure and surface tension. Molecular dynamics (MD) simulations are primarily used at the molecular scale, focusing on phenomena like wettability, surface chemistry, and nanoscale confinement effects. MD describes fluid flow through molecular interactions, such as collisions and transfer, while NS models flow as a continuous medium, applicable to the entire fluid domain. At the macroscopic scale, flow behavior is typically described using the continuum assumption and macroscopic equivalent equations. Common models include Darcy's law, which is used for single-phase saturated flow, establishing the relationship between permeability and pressure gradient to determine flow velocity.