Research Progress in Microscopic Mechanisms and Cross-Scale Simulation of Seepage Behavior in Porous Media

Abstract

With the advancement of aerospace equipment toward high-speed and heavy-duty applications, conventional forced lubrication systems are facing significant challenges in terms of reliability and adaptability to complex operating conditions. Porous medium materials, owing to their unique self-lubricating and oil-retention capabilities, are regarded as an ideal lubrication solution. However, their seepage behavior is governed by the strong coupling effects of microscopic pore structures and fluid physicochemical properties, the mechanisms of which remain inadequately understood, thereby severely constraining the design and application of high-performance lubricating materials. To address this, this paper systematically reviews recent research progress on seepage behavior in porous media, with the aim of establishing a correlation between microstructural characteristics and macroscopic performance. Starting from the characterization of porous media, this work comprehensively analyzes the structure–seepage relationships in porous polymers, metal foams, and porous ceramics, and constructs a multi-scale theoretical framework encompassing macroscopic continuum theories, mesoscopic lattice Boltzmann methods (LBM), pore network models, and microscopic molecular dynamics. The advantages and limitations of experimental measurements and numerical simulation approaches are also compared. In particular, this study critically highlights the current neglect of key interfacial parameters such as surface wettability and pore roughness, and proposes an in-depth investigation into the seepage mechanisms of polyimide porous cage materials based on LBM. Furthermore, the potential application of emerging research paradigms such as data-driven approaches and intelligent computing in seepage studies is discussed. Finally, it is emphasized that future efforts should focus on developing deeply integrated cross-scale simulation methodologies, strengthening multi-physics coupling and artificial intelligence-assisted research, and advancing the development of intelligent porous lubricating materials with gradient structures or stimulus-responsive characteristics. This is expected to provide a solid theoretical foundation and technical pathway for the rational design and optimization of high-performance lubrication systems.

1. Introduction

With the continuous advancement of aerospace equipment toward high-speed, heavy-duty, and long-life operation, the reliability of lubrication systems for core components such as high-speed bearings under extreme working conditions faces severe challenges. Conventional forced lubrication systems, due to their structural complexity, reliance on external pumping, and difficulty in maintaining a stable oil film under high centrifugal forces, no longer meet the performance requirements of next-generation equipment. In this context, self-lubricating porous materials have demonstrated unique advantages. These materials store lubricants within their three-dimensional pore networks and utilize intrinsic mechanisms such as capillary forces to achieve self-driven, on-demand oil supply, thereby significantly enhancing the reliability and adaptability of lubrication systems under complex operating conditions [1].

However, the successful application of porous media in high-performance lubrication systems still faces a fundamental scientific challenge: the mechanism governing their internal seepage behavior remains unclear. Seepage in porous media is a typical cross-scale physical process, controlled by the strong coupling of microscopic pore structure, fluid physicochemical properties (particularly surface wettability), and external force fields such as centrifugal and mechanical stresses [2]. The inherent randomness and heterogeneity of pore structures, combined with an insufficient understanding of key interfacial parameters such as surface wettability and pore roughness, make it extremely difficult to accurately describe and predict the “absorption–storage–transport” behavior of lubricants within the pore network. This knowledge bottleneck confines the design of high-performance porous lubricating materials to a largely trial-and-error approach, lacking theoretical guidance and significantly hindering their practical engineering application [3].

Moreover, a root cause of this bottleneck lies in the limited understanding of the nature of the interaction between porous media and lubricants. This interaction arises from physicochemical forces at the solid–liquid interface—such as van der Waals forces and electrostatic interactions—which macroscopically govern surface wettability, contact angle hysteresis, and lubricant adsorption/desorption behavior. These ultimately determine the magnitude and direction of capillary driving forces, the uniformity of lubricant distribution, and its retention stability. Neglecting this fundamental aspect impedes the establishment of accurate structure–performance relationships and rational material design. Therefore, this article systematically reviews recent advances in seepage behavior in porous media, aiming to build a bridge from microstructure to macroscopic performance.

From a systemic perspective, the lubrication behavior of porous media in bearings constitutes a dynamic cycle: lubricant is absorbed and stored in the pore network via capillary action; during operation, centrifugal and capillary forces compete and jointly regulate lubricant transport and release to the friction interface to maintain the oil film; and part of the lubricant is expected to be recaptured and re-infiltrate into the porous structure after lubrication. The core mechanism driving this process lies in the coupling of capillary effects, seepage dynamics, and external force fields. However, the effectiveness of this system is closely dependent on the physicochemical properties of the lubricant. Inherent limitations—such as centrifugal loss at high speeds, viscosity oxidation and degradation at high temperatures, and interfacial compatibility with the porous medium surface—collectively define the key boundary conditions determining the reliability of the lubrication system.

Although recent advances in materials science and computational methods have promoted the study of seepage in porous media—for instance, high-resolution imaging techniques such as micro/nano-CT and FIB-SEM provide powerful tools for characterizing real pore structures [4], and multiscale theoretical frameworks have extended from macroscopic continuum theories to mesoscopic lattice Boltzmann methods (LBM) [5], with experimental and numerical approaches becoming increasingly complementary—a systematic review that integrates perspectives from “microscopic mechanisms” to “cross-scale simulations,” critically identifies current limitations, and clarifies innovative interdisciplinary directions is still lacking. Existing studies often focus either on the material itself or on specific scale-specific simulation methods, failing to effectively establish a complete chain linking material structure–interfacial properties–multiscale models–lubrication application. In addition, most current modeling and experimental analyses treat porous media as rigid structures and assume constant lubricant properties. This overlooks the elastic deformation, damage, and even fatigue failure of porous media (e.g., polymer cages) under mechanical stress and centrifugal forces in real bearing operation, as well as the time-varying evolution of lubricant properties such as viscosity and surface tension due to high-temperature shear and oxidation. These intrinsic material behaviors and their interactions with the environment dynamically alter pore structure and seepage pathways, profoundly affecting the long-term reliability of the lubrication system. Therefore, integrating the mechanical behavior of porous media, the evolving properties of the lubricant, and their coupling effects into the research framework is essential for achieving accurate prediction and design.

Based on this, the present review aims to systematically summarize research progress on seepage behavior in porous media, focusing on three aspects: material characterization, theoretical models, and research methodologies, with a specific emphasis on proposing a future research pathway based on LBM. It particularly highlights the influence of surface wettability and pore roughness on seepage behavior—interfacial properties often overlooked yet critical to seepage performance. Finally, the article adopts a critical perspective to examine the limitations of current research paradigms and outlines innovative directions for interdisciplinary integration, aiming to provide a solid theoretical foundation and forward-looking technical pathway for promoting the practical application of porous media lubrication technology in high-end equipment.

This review strives to build a “bridge” connecting fundamental theory and engineering practice. By systematically clarifying the intrinsic relationships among material structure, interfacial properties, and multiscale models, it not only aims to elucidate underlying mechanisms but also to provide a clear methodology and actionable research roadmap for material selection, performance prediction, and optimization of porous retainers in aerospace bearings. It should be noted that seepage research in porous media has a strong foundation in fields such as oil, gas, and geotechnics (“subsurface utilization”), whose theoretical models and experimental methods offer valuable references for this study. Therefore, in reviewing cross-domain results, the discussion will consistently be grounded in the specific conditions of aerospace lubrication—such as high-speed centrifugal fields, extreme temperatures, and specific lubricant–material pairs—critically assessing their applicability and translating them from a lubrication perspective, thereby bridging the cognitive gap between general seepage theory and specific lubrication applications.

Specifically, this study aims to achieve the following objectives:

(1) Elucidate quantitative structure–seepage relationships in porous media: Systematically summarize the microstructural characteristics of porous polymers, metal foams, and porous ceramics, clarify the quantitative influence of key geometric parameters such as porosity, tortuosity, and pore size distribution on the “absorption–storage–transport” behavior of lubricants, and review the regulatory role of surface physicochemical properties such as wettability.

(2) Build bridges across multiscale simulation methods: Organize the multiscale theoretical framework spanning macroscopic continuum theories, mesoscopic LBM, pore network models, and microscopic molecular dynamics, compare their applicability and limitations, and provide a theoretical basis for model selection and coupling tailored to specific bearing conditions—such as evaluating lubricant supply efficiency under centrifugal fields and screening surface modification strategies.

(3) Explore the application potential of emerging research paradigms: Prospectively discuss integration pathways and development prospects for data-driven approaches, AI-assisted computing, and smart material design in seepage research—for instance, using AI surrogate models to rapidly optimize retainer pore parameters, laying a methodological foundation for the development of next-generation intelligent lubricating materials.

By systematically achieving these objectives and integrating the various theoretical, experimental, and simulation tools reviewed herein from a lubrication-oriented perspective, this study aims to deepen the mechanistic understanding of seepage behavior in porous media and provide comprehensive support—from theory to application—for the design and optimization of high-performance aerospace lubrication systems.

2. Characterization of Porous Media

The macroscopic properties of porous media, including oil storage capacity, permeability, and mechanical strength, are fundamentally determined by their microscopic structural characteristics. According to the standards set by the International Union of Pure and Applied Chemistry (IUPAC), porous materials can be classified into three categories based on their pore size: microporous (pore diameter < 2 nm), mesoporous (pore diameter between 2 and 50 nm), and macroporous materials (pore diameter > 50 nm) [6]. Inorganic porous materials primarily include metallic porous materials (e.g., metal foams) and non-metallic porous materials, as well as organic-inorganic hybrid materials such as Metal-Organic Frameworks (MOFs) [7,8,9]. This classification not only reflects differences in scale but is also intrinsically linked to distinct surface effects and fluid transport mechanisms. In the field of lubrication applications, porous organic polymers, metal foams, and porous ceramics have garnered significant attention due to their unique performance advantages. Their properties essentially result from the combined effects of their microstructure and surface physicochemical characteristics.

In the context of lubrication, porous organic polymers, metal foams, and porous ceramics have attracted extensive interest due to their unique pore structures and seepage characteristics. Key microstructural features of these materials—such as porosity, pore size distribution, tortuosity, and connectivity—directly determine critical seepage parameters, including oil storage capacity, permeability, and capillary-driven behavior.

2.1. Porous Organic Polymers

The seepage behavior of porous organic polymers (POPs) is closely linked to their precisely designable hierarchical pore structures and surface chemical properties, which affords significant potential for regulating the “absorption–storage–transport” behavior of lubricants. The pore architecture in POPs typically exhibits a continuous distribution spanning micropores (<2 nm), mesopores (2–50 nm), and macropores (>50 nm) [10]. Their precisely tunable pore structure and functionality—particularly the powerful capability of covalent organic frameworks (COFs) enabled by topological design—have made them a focus of cutting-edge research [11]. This multiscale pore network is essential to seepage behavior: micropores and mesopores provide selective adsorption and storage of lubricant molecules through strong capillary forces, while macropores serve as pathways for rapid macroscopic transport.

Seepage performance can be quantitatively described by the classical Kozeny–Carman equation (($k\propto {\phi }^3/\left({\tau }^2{S}^2\right)$), where $k$ is permeability, $\phi$ is porosity, $\tau$ is tortuosity, and $S$ is specific surface area). POPs generally exhibit high porosity (often >50%), which underpins their high oil storage capacity and potential permeability. However, their large specific surface area, while providing abundant adsorption sites, also significantly increases frictional resistance to fluid flow. Therefore, optimizing the seepage performance of POPs hinges on balancing porosity, specific surface area, and tortuosity.

In a review on porous polymer molding, S. Das et al. [12] outlined the molecular structural features of various types of porous polymers—including hyper-crosslinked polymers (HCPs), polymers of intrinsic microporosity (PIMs), COFs, conjugated microporous polymers (CMPs), covalent triazine frameworks (CTFs), and porous aromatic frameworks (PAFs)—noting their distinct characteristics in topology, specific surface area, and crystallinity. For instance, COFs possess well-defined periodic pore channels and highly ordered pore arrangements, facilitating directional molecular transport and selective separation. In contrast, CMPs exhibit unique optoelectronic properties and charge transport behavior due to their extended π-conjugated systems. HCPs are formed via post-crosslinking reactions and exhibit high specific surface area and good thermal stability, while PIMs rely on rigid and contorted molecular backbones to generate intrinsic microporosity. Critically, the topological structure of the material directly governs the tortuosity of the seepage path. As summarized by Lu Yanqiu et al. [13] based on studies by Wang [4], Yanqiu L [14], and Jingyi W [15], COFs with one-dimensional straight channels (e.g., COF-5) exhibit low tortuosity, favoring rapid lubricant transport. In comparison, three-dimensional interpenetrating pore structures offer higher specific surface area and stronger oil retention capacity but may increase flow resistance. This structure–seepage relationship provides clear guidance for material design. In recent years, rational design strategies based on reticular chemistry—which employ specific molecular building blocks to predetermine network topology and pore environment—have provided a powerful theoretical toolkit for constructing a new generation of functional porous polymers [16]. In their seminal review on COFs, Feng et al. [11] systematically elucidated how different topological structures decisively influence physicochemical properties, including mass transport capability.

Surface wettability, as a key physicochemical parameter, regulates the initial infiltration and spontaneous seepage dynamics of lubricants through capillary pressure (($P_c=\left(2\mathsf{\gamma }\cos \theta \right)/r$), where $\mathsf{\gamma }$ is surface tension, $\theta$ is contact angle, and $r$ is pore throat radius). It macroscopically reflects the interactions between the chemical composition of the polymer framework and lubricant molecules. For example, the Banerjee group [17] introduced alkyl chains into the framework via post-synthetic modification, enhancing van der Waals interactions between the framework and lubricant molecules. This effectively improved oleophilicity (increasing $\cos \theta$), reduced capillary resistance, and promoted lubricant seepage and replenishment. This synergistic “structure–surface” effect enables porous polymers to achieve adaptive lubricant regulation: under low-speed conditions, capillary action maintains oil film integrity, while under high-speed conditions, macrochannels enable rapid oil replenishment.

In lubrication applications, the mechanical properties of porous polymer cages—such as elastic modulus and compressive strength—are crucial. Under assembly and operational stresses, the pore structure may undergo recoverable elastic deformation or even irreversible collapse, dynamically altering key seepage parameters such as porosity and tortuosity, thereby influencing oil supply behavior. Therefore, evaluating the coupled “structure–seepage–mechanics” relationship is essential for predicting the long-term performance of these materials under real service conditions.

2.2. Metallic Foams

The seepage performance of metal foams is highly dependent on their pore connectivity and structural morphology. These materials typically exhibit two characteristic forms: reticulated (open-cell) and cellular (closed-cell/through-pore) structures [18], which lead to distinctly different seepage behaviors and application scenarios. As described by Liu Peisheng, reticulated metal foams possess a highly interconnected three-dimensional pore-ligament structure that forms fluid pathways with low tortuosity, resulting in exceptionally high permeability and enabling efficient, rapid lubricant transport under centrifugal forces. In contrast, the closed-cell structures reported by Liu Hong et al. [19] exhibit almost no permeability due to their isolated pores. Although through-pore structures mentioned by Liu Peisheng show some connectivity, their highly tortuous fluid paths and significant flow resistance can lead to permeability values several orders of magnitude lower than those of reticulated foams. This highly connected three-dimensional network represents an ideal seepage skeleton; its well-defined pore throats and cavities provide an ideal model for analyzing single-phase fluid permeability and calculating tortuosity. The difference between closed-cell and through-pore structures fundamentally reflects how connectivity governs seepage characteristics: closed-cell structures, with their isolated pores, are suited for structural applications, whereas through-pore structures, despite possessing basic connectivity, exhibit significantly higher flow resistance than reticulated foams due to their greater tortuosity.

The macroscopic seepage behavior of metal foams is quantitatively determined by their microstructural parameters. High porosity (typically >90%) provides substantial oil storage capacity [20,21]. The average pore size and its distribution directly influence the uniformity of capillary pressure, while tortuosity ($\tau$) is a key factor in predicting permeability ($k$). The Kozeny–Carman equation clearly describes the coupling among these parameters: high porosity, low tortuosity, and large pore size collectively contribute to high permeability. As Chen Xiang and Li Yanxiang noted in their review [22], excessively high tortuosity significantly increases flow resistance—this is the fundamental reason why the permeability of through-pore metal foams is generally lower than that of reticulated foams.

In the context of high-speed bearing lubrication, seepage behavior is governed by the dynamic competition between centrifugal and capillary forces. A promising future direction is the design of metal foam components with graded pore structures, featuring a continuous variation in pore size along the radial direction of the bearing. Such structures can establish capillary pressure gradients that either cooperate or compete with the centrifugal field, enabling intelligent storage and on-demand release of lubricant under different operating conditions. For example, during low-speed startup, small inner pores can provide prioritized oil supply via strong capillary action; during high-speed operation, larger outer pores facilitate rapid oil replenishment dominated by centrifugal forces, thereby maintaining optimal lubrication throughout the operational cycle.

The key advantage of metal foams lies in their high specific strength, which helps maintain structural stability under high centrifugal forces. Nevertheless, the coupling between seepage and mechanical behavior requires careful attention: under high-cycle fatigue loading, the initiation and propagation of cracks in pore ligaments may lead to sudden changes in pore connectivity and even generate metallic debris that contaminates the lubrication system. Therefore, mechanical reliability is a prerequisite for ensuring long-term stable seepage performance.

2.3. Porous Ceramics

The effectiveness of porous ceramics in lubrication under extreme conditions relies not only on their excellent thermal stability but also, more critically, on their seepage characteristics dictated by microstructure. Taking porous alumina ceramics as an example, their typical structure consists of sintered necks connecting randomly stacked ceramic particles formed at high temperatures, resulting in a complex, interconnected, yet non-uniform three-dimensional pore network [23]. Such structures often exhibit a wide pore size distribution, which in turn leads to complex seepage behavior.

A profound quantitative relationship exists between the lubricating functionality of porous ceramics and their microstructural parameters. Among these, the open-cell porosity (which can be as high as 95% [24]) forms the foundation for efficient lubrication. More critically, the pore size distribution directly determines the range of capillary pressure. In high-speed bearings, lubricant seepage involves a dynamic competition between centrifugal and capillary forces. The lubricant tends to be preferentially expelled from larger pores due to centrifugal force, while oil in smaller pores can become trapped due to excessively strong capillary forces, leading to uneven lubricant distribution or even supply failure [25]. Therefore, precise control over the slurry composition and sintering process to achieve a more focused pore size distribution is crucial for maintaining stable seepage. Furthermore, porous ceramics (e.g., alumina) typically possess high surface energy and abundant surface hydroxyl groups, which can induce strong polar interactions and even hydrogen bonds with polar components in mineral base oils. These specific interfacial interactions fundamentally influence the spreading behavior and retention stability of the lubricant at elevated temperatures, representing a key factor for maintaining lubrication efficacy under extreme conditions.

To overcome the inherent randomness of traditional sintered ceramic structures and their adverse effects on seepage uniformity, advanced fabrication techniques like ice-templating have been introduced to fabricate porous ceramics with highly aligned, lamellar pores. Waschkies et al. [26] successfully prepared porous alumina using this method, demonstrating significant permeability anisotropy—the permeability along the freezing direction was much higher than in the perpendicular direction. These controllable and directional seepage pathways provide an ideal material prototype for the precise transport of lubricants in bearing systems. Under high-temperature conditions, the aligned structure also enables efficient coupling between thermal and flow fields: heat transfer along the directional channels is more effective, helping to avoid localized overheating and the consequent sharp decrease in lubricant viscosity, thereby maintaining seepage stability. Studies indicate that at 300 °C, the lubricant consumption rate for aligned porous ceramics can be approximately 30% lower than that for structures with random porosity, fully highlighting their structural advantage.

However, the inherent brittleness of porous ceramics remains a major challenge for their application in moving components. Under impact loads or thermal stress, the propagation of microcracks can significantly alter the pore structure and permeability, potentially leading to catastrophic failure. Therefore, in-depth investigation into the relationship between damage evolution and seepage performance is crucial for assessing the suitability of these materials under extreme operating conditions.

2.4. Controlled Fabrication Methods for Porous Structures

The precise control of microstructure is central to realizing porous media with desired seepage performance. As mentioned earlier, key geometric parameters—including porosity, pore size distribution, tortuosity, and connectivity—govern seepage behavior. To date, various fabrication techniques have been developed for different material types to achieve targeted pore structures:

For porous organic polymers (POPs), methods such as templating, hyper-crosslinking, and reversible covalent bond assembly enable precise design of hierarchical pore architectures spanning micro- to macropores. Pore size and topology can be controlled through monomer geometry, template dimensions, and reaction kinetics [27].

In the case of metal foams, pore structure is highly dependent on the manufacturing route. Powder metallurgy and foaming allow control over pore size by adjusting the size and distribution of pore-forming agents. Electrodeposition techniques can replicate highly interconnected three-dimensional network structures using sacrificial templates (e.g., polymer foams) [20]. Directional solidification, moreover, can be employed to fabricate graded structures with aligned porosity.

For porous ceramics, conventional sintering regulates porosity by controlling powder particle size and sintering conditions. Advanced methods such as ice-templating (freeze-casting) enable the fabrication of highly aligned, lamellar channel structures by governing freezing direction and rate, significantly reducing seepage tortuosity [28]. Furthermore, 3D printing offers unprecedented control in constructing complex, pre-designed 3D pore networks [29]. This technology has evolved beyond mere structural replication to the coordinated programming of pore size, tortuosity, and material chemistry [30].

These fabrication routes provide a practical pathway for structure–performance-oriented material design. Nevertheless, a key challenge bridging materials science and lubrication engineering remains: how to reverse-translate specific lubrication requirements—such as the optimal balance between permeability and capillary forces—into well-defined pore structure parameters and realize them through processing.

3. Theoretical Frameworks and Models for Seepage

A profound understanding of seepage behavior in porous media requires the establishment of multi-scale theoretical frameworks, spanning from the macroscopic to the microscopic level. These frameworks, each with its applicable scale and physical basis, collectively form a complete knowledge system for interpreting seepage phenomena. However, the selection of an appropriate theoretical model is not arbitrary; it is strongly influenced by the spatial and temporal scales of the problem under investigation, computational resource constraints, and the complexity of the target physical phenomena. Therefore, a thorough grasp of the applicable scope and inherent limitations of each theoretical model is a prerequisite for achieving accurate simulations.

3.1. Macroscopic Continuum Theory

The macroscopic continuum theory serves as the historical cornerstone and primary framework for engineering applications in the study of seepage through porous media. Since its introduction in 1856, Darcy’s law has become the foundational principle describing macroscopic seepage behavior and has been maturely applied for over a century and a half in traditional fields such as civil engineering and hydrogeology. Its mathematical expression, as shown in Equation (1), establishes a linear relationship between the volumetrically averaged seepage velocity and the pressure gradient [1]. This formulation remains valid under conditions of low Reynolds number flow and fully saturated, homogeneous media.

$$v=-{\displaystyle \frac{K}{\mu }}\nabla p$$

(1)

where $v$ is the seepage velocity; $K$ is the permeability tensor; $\mu$ is the dynamic viscosity of the fluid; $\nabla p$ is the pressure gradient.

However, Darcy’s Law has limited applicability. When the flow velocity is sufficiently high for inertial effects to become non-negligible, the flow deviates from this linear relationship, entering the non-Darcy flow regime. Under these conditions, the Forchheimer equation is typically employed to provide a more accurate description [31]:

$$-\nabla p={\displaystyle \frac{\mu }{K}}v+\beta \rho v^2$$

(2)

where $\rho$ is the fluid density; $\beta$ is the non-Darcy flow coefficient.

When the effects of viscous shear stresses within the pore spaces must be considered, such as in regions near solid walls, the Brinkman equation must be incorporated [32]:

$${\displaystyle \frac{\mu }{K}}v+\beta \rho \left|v\right|v=-\nabla p+{\mu }_{eff}{\nabla }^{2}v$$

(3)

where ${\mu }_{eff}$ is the effective viscosity.

The complete set of macroscopic governing equations comprises the mass conservation equation and the momentum conservation equation. For the steady-state flow of an incompressible fluid, the mass conservation equation (continuity equation) is given by:

$$\nabla \cdot \left(\rho v\right)=0$$

(4)

The momentum conservation equation represents a generalized form of the Navier–Stokes (N-S) equations within porous media, and its specific expression can be written as:

$${\displaystyle \frac{\rho }{\varphi }}\left({\displaystyle \frac{\partial v}{\partial t}}+\left(v\cdot \nabla \right)v\right)=-\nabla p+v_{eff}{\nabla }^{2}v-{\displaystyle \frac{\mu }{K}}v-\beta \rho \left|v\right|v+F$$

(5)

where $F$ represents other body forces (e.g., gravity); $\varphi$ denotes the porosity.

The key dimensionless number for determining the applicability of the continuum hypothesis is the Knudsen number (Kn), defined as the ratio of the molecular mean free path to the characteristic flow length scale (such as the pore diameter):

$$Kn={\displaystyle \frac{\lambda }{d}}$$

(6)

The continuum hypothesis remains valid when Kn ≤ 0.001, allowing macroscopic models based on the Navier–Stokes (N-S) equations to be appropriately employed [33]. This condition is generally satisfied for most engineering-scale seepage problems in porous media.

While macroscopic continuum theory is well-established for engineering-scale applications, it faces two fundamental and interrelated limitations when applied to novel porous materials such as those used in aerospace lubrication. First, the validity of this approach strongly depends on the accuracy of equivalent averaged parameters—such as permeability and tortuosity. However, for newly developed porous materials with highly complex microstructures, these parameters are often difficult to determine a priori, becoming a major bottleneck for predictive modeling. Second, this theoretical framework cannot reveal pore-scale physical mechanisms—such as the instability of microscopic displacement fronts or the formation of trapping effects—which are essential for understanding and optimizing seepage performance.

In scenarios involving deformation of the porous skeleton, the macroscopic theory must be extended to poroelastic or poro-elastoplastic models. In such cases, the governing equations couple Darcy’s law (or its non-Darcy extensions) for seepage with mechanical constitutive relations (e.g., linear elasticity or elastoplastic models) describing solid matrix deformation. This coupling is typically achieved by treating porosity and permeability as stress-dependent variables—for instance, through modified forms of the Kozeny–Carman equation to dynamically capture the influence of stress on permeability. Such developments provide a theoretical basis for simulating the evolution of seepage behavior in loaded porous retainers under actual bearing operating conditions.

Therefore, in modern research paradigms, the role of Darcy’s law and macroscopic continuum theory has undergone a significant shift: they no longer serve merely as computational tools but have become a bridge and an endpoint connecting microscopic mechanisms with macroscopic performance. Macroscopic models, such as CFD-based system-level simulations, are suitable for performance prediction and structural design of entire bearing lubrication systems incorporating porous retainers. Meanwhile, a key task of meso- and microscale simulations—such as LBM and MD—is to uncover underlying physical mechanisms at the representative elementary volume scale and compute effective transport parameters that can be accurately incorporated into Darcy’s law and its generalizations. In other words, Darcy’s law serves as the macroscopic boundary and final performance descriptor in a multiscale research framework. Integrating the engineering applicability of macroscopic theory with the mechanistic insights provided by meso- and microscale simulations represents both a key objective and a central challenge in contemporary cross-scale modeling research.

3.2. Mesoscopic Kinetic Theory

In the study of fluid flow within micron to submicron pore scales, where the characteristic size approaches the applicability limit of the continuum assumption (Kn > 0.001), the lattice Boltzmann method (LBM)—rooted in mesoscopic kinetic theory—demonstrates distinct advantages. It is important to note that LBM is primarily applicable to mesoporous and macroporous scales (tens of nanometers to micrometers). Its “mesoscopic” nature refers to its theoretical foundation and does not restrict its use to nanoscale pores. This scale range—particularly the micrometer scale—is critically important in lubrication studies: it constitutes the primary pathway for macroscopic lubricant transport within porous retainers and represents the typical domain where capillary and centrifugal forces compete. Therefore, accurate modeling of pore structures at the micrometer scale is essential for understanding the “absorption–storage–transport” behavior of lubricants.

For flows at true nanoscale dimensions (<2 nm), where nanoconfinement effects become significant and intermolecular interactions dominate, conventional LBM faces challenges. In such scenarios, either more sophisticated modified LBM approaches or a transition to molecular dynamics (MD) simulations is generally required.

Unlike methods that directly solve the macroscopic Navier–Stokes equations, LBM tracks the evolution of fluid particle distribution functions, from which macroscopic quantities are obtained through statistical averaging. This characteristic makes LBM particularly suitable for simulating flow in porous media with complex microscale pore structures. Specifically, LBM enables flow simulations directly on pore geometries reconstructed from techniques such as micro-CT imaging. This capability allows the capture of flow details that are difficult to resolve with traditional macroscopic models, including: the critical capillary entry pressure at micron-sized pore throats, the formation of preferential flow paths resulting from the competition between centrifugal and capillary forces within the pore network, and the influence of surface wettability on local flow structures at the microscale.

Consequently, LBM serves as an effective tool for modeling seepage in porous structures at the micrometer scale and has been widely applied in related studies involving multiphase flow, complex boundaries, and microscopic interfacial effects [34,35].

The core of the LBM is the discrete Boltzmann equation [36]:

$$f_i\left(x+e_i\delta t,t+\delta t\right)-f_i\left(x,t\right)={\Omega }_i$$

(7)

where $f_i\left(x,t\right)$ represents the particle distribution function moving with discrete velocity $e_i$ at position $x$ and time $t$; $\delta t$ is the time step size; ${\Omega }_i$ is the collision operator, describing the changes in the distribution function due to interparticle collisions.

In practical applications, the BGK (Bhatnagar-Gross-Krook) single relaxation time approximation is widely adopted to simplify the expression of the collision term:

$${\Omega }_i=-{\displaystyle \frac{1}{\tau }}\left[f_i\left(x,t\right)-f_i^{eq}\left(x,t\right)\right]$$

(8)

where $\tau$ is the dimensionless relaxation time, which governs the rate at which the distribution function approaches its equilibrium state; $f_i^{eq}$ is the equilibrium distribution function determined by the local macroscopic physical quantities.

The specific form of the equilibrium distribution function depends on the adopted discrete velocity model. Taking the classical D2Q9 model as an example, its expression is given by:

$$f_i^{eq}={\omega }_i\rho \left[1+{\displaystyle \frac{c_i\cdot u}{c_s^2}}+{\displaystyle \frac{{\left(c_i\cdot u\right)}^2}{2c_s^4}}-{\displaystyle \frac{u^2}{2c_s^2}}\right]$$

(9)

where ${\omega }_i$ is the weight coefficient associated with the discrete direction $i$; $\rho$ and $u$ are the macroscopic density and velocity, respectively; $c_s$ is the lattice speed of sound.

A significant feature of this method is the well-defined correspondence between its mesoscopic parameters and macroscopic fluid properties. The fluid kinematic viscosity, ν, is explicitly given by:

$$v=c_s^2\left(\tau -{\displaystyle \frac{1}{2}}\right)\Delta t$$

(10)

To simulate multiphase flow phenomena, it is necessary to introduce methods that describe intermolecular interactions. The Shan-Chen model achieves phase separation through an interaction force term [37]:

$$F\left(x,t\right)=-G\psi \left(x,t\right){\displaystyle \sum _i{\omega }_i}\psi \left(x+c_i\Delta t,t\right)c_i$$

(11)

where $G$ is the interaction strength controlling the phase separation intensity; $\psi \left(x\right)$ is the effective density [37].

The unique advantage of LBM lies in its mesoscopic nature, which makes it almost “naturally” suited for handling complex boundaries and multiphase flow problems. Compared to conventional CFD methods that directly solve the Navier–Stokes equations, LBM offers simpler boundary treatment (particularly for complex geometries), higher parallel efficiency, and inherent capability for simulating multiphase/multicomponent flows [38]. It demonstrates superior flexibility and stability when dealing with high-curvature pore interfaces and dynamic contact line movement, successfully bridging the gap between microscopic molecular dynamics and macroscopic continuum mechanics. As such, LBM has become a powerful tool for investigating pore-scale flow phenomena. However, LBM also faces specific challenges: for instance, the single relaxation time (BGK) model may exhibit numerical instability in high Reynolds number flows; the balance between physical rigor (such as the implementation of equations of state) and computational cost in multiphase flow models requires careful consideration. Furthermore, the computational results of LBM are highly dependent on the fidelity of digital rock core reconstruction—the principle of “garbage in, garbage out” is particularly evident in this context.

3.3. Pore Network Modeling Theory

Pore network models (PNMs) serve as a powerful topology-reduction tool, bridging macroscopic continuum theories and microscopic resolution methods. The core concept involves abstracting complex porous media into a network of pore bodies connected by pore throats. Fluid flow is described by the conductivity of the throats, while mass conservation is enforced at each pore body node.

The governing equation for PNM is based on flow conservation at each node:

$${\displaystyle \sum _j{g}_{ij}}\left(p_i-p_j\right)=0$$

(12)

where $g_{ij}$ is the conductivity of the throat connecting nodes $i$ and $j$, typically calculated using semi-empirical formulas (e.g., modified forms of the Hagen–Poiseuille equation) based on throat geometry (such as radius and length) and fluid properties.

Compared to direct numerical simulation methods like LBM, PNM offers significantly higher computational efficiency, enabling simulations at the core scale (centimeters or even meters) and rapid calculation of macroscopic transport properties (e.g., absolute permeability, relative permeability curves). However, this efficiency comes at the cost of lost physical detail. PNM cannot resolve the precise flow field within pores and has inherent limitations in predicting phenomena involving complex fluid dynamics (e.g., high-inertia flows) or interface behaviors strongly dependent on local geometry (e.g., film flow, snap-off mechanisms).

Therefore, PNM and LBM represent complementary extremes in mesoscale research: LBM prioritizes physical fidelity and is suitable for mechanism exploration and high-resolution simulation of small samples; PNM emphasizes computational efficiency and is ideal for large-scale statistical analysis and sensitivity studies. The integration of PNM with image-based pore structure extraction techniques has become a standard practice in the oil and gas industry for rapid reservoir characterization.

3.4. Microscopic Molecular Interaction Theory

At the molecular scale (nanoscale pores), fluid behavior is governed by both the intermolecular interaction potentials and the interactions between fluid molecules and solid wall atoms. Molecular Dynamics (MD) simulation serves as the definitive tool for investigating phenomena at this scale [39]. MD simulates the temporal evolution of a system by numerically solving Newton’s equations of motion for all constituent particles. The fundamental equation is:

$$m_i{\displaystyle \frac{d^2{r}_i}{d{t}^2}}=F_i=-{\nabla }_i{\displaystyle \sum _{i\ne j}U\left(r_{ij}\right)}$$

(13)

where $m_i$ is the mass of particle $i$; $r_i$ is its position vector; $F_i$ is the total force acting on particle $i$; $U\left(r_{ij}\right)$ is the interaction potential energy between particles $i$ and $j$ (e.g., the Lennard–Jones potential).

Among these, the Lennard–Jones potential describes generic van der Waals interactions, effectively balancing short-range repulsion and long-range attraction effects:

$$U_{LJ}\left(r_{ij}\right)=4ϵ\left[{\left({\displaystyle \frac{\sigma }{r_{ij}}}\right)}^{12}-{\left({\displaystyle \frac{\sigma }{r_{ij}}}\right)}^6\right]$$

(14)

where $U_{LJ}\left(r_{ij}\right)$ is the interaction potential energy between particles $i$ and $j$; $r_{ij}$ is the distance between particles $i$ and $j$; $ϵ$ represents the strength of the intermolecular interaction; $\sigma$ is the van der Waals diameter, defined as the intermolecular distance where the potential energy is zero.

For systems involving polar molecules or charged particles, the Coulomb potential governs the long-range electrostatic interactions:

$$U_{Coul}\left(r_{ij}\right)={\displaystyle \frac{1}{4\pi {ϵ}_0{ϵ}_r}}{\displaystyle \frac{q_i{q}_j}{r_{ij}}}$$

(15)

where $U_{Coul}\left(r_{ij}\right)$ is the electrostatic potential energy between two charged particles $i$ and $j$; $q_i$ and $q_j$ are the respective charges of particles $i$ and $j$; ${ϵ}_0$ is the vacuum permittivity, a fundamental physical constant with a value of approximately 8.854 × 10⁻¹² $C^2\cdot J^{-1}\cdot m^{-1}$; ${ϵ}_r$ is the relative permittivity, a dimensionless number characterizing the dielectric material.

Molecular dynamics (MD) simulations serve as a fundamental tool for elucidating seepage mechanisms at the molecular level. Their core value lies in the ability to directly reveal the nature of interactions between lubricant molecules and pore wall atoms from first principles [40,41]. By employing appropriate force fields (e.g., COMPASS, OPLS-AA), MD simulations can quantitatively evaluate the contributions of van der Waals forces, electrostatic interactions, and other factors to interfacial phenomena. This approach provides microscopic insights that are difficult to obtain through direct experimental measurements, such as: Specific adsorption configurations, orientations, and binding energies of lubricant molecules on pore surfaces; Structural ordering transitions of fluids under nanoconfinement effects; Dynamic slip boundary conditions at solid–liquid interfaces; Specific interaction energies between surface chemical groups (e.g., imide rings in polyimide, hydroxyl groups on ceramic surfaces) and lubricant molecules (e.g., alkanes, esters).

Crucially, the nanoscale physical information obtained from MD simulations—including potential energy surfaces, slip lengths, and intrinsic contact angles—can be transferred as key constitutive relationships and boundary conditions to mesoscale lattice Boltzmann method (LBM) simulations. This integration significantly enhances the predictive accuracy of LBM in modeling near-wall behaviors for submicron to micron-scale flows. Therefore, MD and LBM form a coupled research chain linking molecular interactions to pore-scale flow phenomena, serving as an indispensable approach for clarifying the fundamental interactions between porous media and lubricants.

3.5. Fractal Theory

The pore structures of natural and many artificial porous media exhibit statistical self-similarity, known as fractal characteristics. Fractal theory provides a powerful mathematical framework for describing such complex, irregular structures [42]. By introducing parameters like the fractal dimension ($D_f$), it enables the quantitative characterization of structural complexity and spatial filling capacity.

Based on fractal theory, researchers have established analytical relationships between transport parameters (such as permeability and porosity) and fractal dimensions as well as pore size distributions. For instance, a classic expression relating permeability ($K$), porosity ($\varphi$), and fractal dimension ($D_f$) can be formulated as:

$$K=C\cdot {\displaystyle \frac{{\varphi }^{3+D_f}}{{\left(1+\varphi \right)}^{2{\tau }^2}}}\cdot d_{\max }^2$$

(16)

where $C$ is a dimensionless proportionality constant; $\tau$ is the tortuosity, dimensionless; $d_{\max }$ is the maximum pore diameter.

The fractal dimension ($D_f$) can be determined from microscopic images using various methods such as the box-counting method and the area-perimeter method. For different porous media, $D_f$ typically ranges between 2 and 3, with higher values indicating more complex structures. Fractal theory provides significant value by employing a concise mathematical framework—the fractal dimension—to quantify the essential characteristics of complex, disordered structures, thereby establishing an analytical bridge between microscopic structure and macroscopic properties. This offers important implications for rapidly predicting material permeability and guiding material design. However, the theory faces a fundamental challenge: the self-similarity observed in actual porous media is typically limited and exists in a statistical sense rather than representing a mathematically strict fractal. Consequently, the predictive accuracy of fractal models strongly depends on the reliability of the fractal dimension estimation method and its universality across the relevant scale range under investigation.

4. Research on Seepage Behavior in Porous Media

This chapter systematically reviews experimental and numerical simulation methodologies for studying seepage behavior. It should be noted that a substantial body of mature research in this field originates from “subsurface utilization” disciplines such as petroleum extraction and geomechanics. While drawing upon these established works, this review aims to extract their universal methodological principles and explore their potential implications for lubrication research. Particular attention will be given to advances directly relevant to lubrication applications—for instance, the characterization of pore structures in polyimide retainers. When evaluating various techniques and findings, we will consistently adopt a “lubrication perspective,” interpreting how these methods can be adapted to address the specific requirements of aerospace bearings. These include critical issues such as oil film maintenance under high-speed centrifugal fields and lubrication failure mechanisms in high-temperature environments.

Within this methodological framework, the accurate reconstruction of pore structures forms the physical foundation of seepage research. The quality of these reconstructions directly determines the reliability of both numerical simulations and experimental results, making it a long-standing focus of extensive academic investigation. Current mainstream pore-scale reconstruction approaches can be categorized into two primary types [43]: The first is physical reconstruction based on image tomography, which utilizes microscopic imaging techniques like CT and SEM to directly acquire the three-dimensional morphology of real pore structures. The second category comprises numerical reconstruction methods, including particle deposition algorithms that simulate packing processes, statistical reconstruction techniques that replicate structures based on statistical features like porosity and pore size distribution, and the four-parameter stochastic growth method that generates pore networks by controlling skeleton growth parameters.

With the continuous development and refinement of pore-scale reconstruction techniques, research on seepage in porous media has progressively evolved into two major methodological systems: experimental investigation and numerical simulation. These approaches are distinct in their fundamental principles yet functionally complementary. Through systematic integration and mutual validation, they collectively establish a comprehensive methodological framework for unraveling complex seepage mechanisms.

4.1. Experimental Investigations

Experimental research represents the most direct and reliable approach for elucidating seepage mechanisms in porous media. The research paradigm has now evolved from single-scale macroscopic parameter measurements toward integrated, multiscale investigations that combine macroscopic testing with microscopic visualization and in situ dynamic characterization.

In accurate measurement of macroscopic seepage parameters, precision permeation experimental systems are widely employed to investigate the influence of stress fields and interfacial physicochemical effects. Studies by Tang Jupeng et al. [44] and Wang Jianmei et al. [45] have demonstrated that external stress fields can significantly regulate the permeability of porous media. Research by Jia Zhenqi et al. [46] on ultra-low permeability cores further revealed that strong solid–liquid interfacial effects can cause sharp declines in effective permeability and generate significant threshold pressure gradients, resulting in substantial deviations from classical Darcy’s law. These findings not only experimentally confirm that interfacial effects replace viscous forces as the dominant mechanism in micro-nano channels, but also drive theoretical models to incorporate non-equilibrium effects such as threshold pressure gradients, providing crucial guidance for both low-permeability hydrocarbon development and micro-nano scale lubrication design.

In visualization of microscopic flow behavior, advanced imaging techniques have enabled direct observation of pore-scale flow mechanisms. Liu Jianjun et al. [47] utilized transparent rock analogs combined with high-speed microscopic imaging systems, enabling direct observation of interface dynamics and trapping behaviors during oil-water two-phase seepage, clearly demonstrating the controlling role of wettability and capillary forces on displacement fronts. Niu Gang et al. [48] systematically revealed through laboratory experiments the combined influence of displacement pressure differentials and crude oil viscosity on water-flooding seepage characteristics in conventional heavy oils.

In in situ and dynamic three-dimensional characterization, advanced techniques like synchrotron radiation X-ray CT have enabled “transparent” observation of seepage processes. The “in-situ CT seepage experiments” pioneered by Blunt’s team [49] allow real-time, non-destructive tracking of fluid saturation evolution in three-dimensional pore spaces under simulated mechanical loading conditions, providing an unprecedented window for understanding multiphase seepage mechanisms. This approach has transformed seepage research from a “black box” to a “transparent chamber,” representing the current state-of-the-art in experimental technology, despite inherent compromises between equipment cost, technical barriers, and spatiotemporal resolution. Utilizing this technology, Bo Zhao et al. [50] systematically investigated the evolution of mechanical and seepage characteristics in sandstones under different saturation states following liquid nitrogen treatment.

In coupled mechanical-seepage behavior research, Sheng-Qi Yang et al. [51] employed rock mechanics servo-testing systems to reveal the intrinsic relationship between gas permeability and damage evolution in sandstones throughout complete stress–strain processes. Their research demonstrated that permeability undergoes orders-of-magnitude changes as rock damage accumulates—from micro-crack compaction and initiation to propagation and eventual macroscopic failure—elucidating the controlling mechanisms and evolutionary patterns of how mechanical damage governs seepage capacity. While such “stress-damage-seepage” coupling relationships originate from geomechanics, their physical mechanisms possess universal applicability. When translating these findings to lubrication applications, the key lies in substituting geological stresses with bearing mechanical loads and assembly stresses, and replacing rock damage models with fatigue damage models for polymer or metal-ceramic retainers, thereby providing theoretical references and experimental paradigms for predicting performance evolution of lubrication components under complex operating conditions.

4.2. Numerical Simulation

Numerical simulation recreates and predicts seepage processes by solving mathematical equations governing fluid motion, offering unique advantages including cost efficiency, parameter controllability, and comprehensive data acquisition. However, it is essential to maintain a clear perspective that, as George Box famously stated, “all models are wrong, but some are useful.” The reliability of simulation results depends not only on the governing equations themselves but also on the accuracy of input parameters (e.g., pore structure, fluid properties), the rationality of boundary conditions, and the model’s capacity to capture targeted physical processes. To address different spatial scales and physical problems, researchers have developed distinct simulation methodologies. Currently prevalent numerical approaches include commercial software based on the finite element method (e.g., Fluent Version R1 of 2020, COMSOL Multiphysics Version 6.1) and microscopic particle-based methods such as Molecular Dynamics (MD), Smoothed Particle Hydrodynamics (SPH), and the Lattice Boltzmann Method (LBM). MD, SPH, and LBM describe the states and interactions of microscopic particles through discretization; after convergence of discrete factors, statistical averaging of these microscopic states characterizes the macroscopic motion of fluids. This bottom-up approach is particularly suitable for investigating fluid behavior within pore structures [43].

At the macroscopic scale, with advancements in theoretical models and computational power, Computational Fluid Dynamics (CFD) has become an important tool for studying multiphase flow in porous media [52,53,54,55,56]. CFD software like Fluent solves volume-averaged Navier–Stokes equations, simplifying complex porous structures into continuous media characterized by equivalent parameters such as permeability and porosity. Its simulation accuracy is influenced by scale, flow characteristics, model design, and parameter selection, making it more suitable for macro- to meso-scale problems. Dung A. Pham et al. [57] developed a porous media model within a two-phase flow framework, accounting for liquid dispersion, mass transfer, and chemical reactions in packed beds. Ju Qiming et al. [58] systematically simulated the influence of porosity and viscous resistance coefficients on the pressure field during heavy oil seepage, providing theoretical guidance for heavy oil recovery. Li Guodong et al. [33] proposed a novel seepage simulation method based on CFD software FLUENT, introducing Darcy resistance as a source term into the fluid governing equations and employing the VOF method to track free infiltration surfaces, achieving precise simulation of seepage processes. Validation through two typical cases—an underground river channel and a rectangular earth dam—demonstrated close agreement with theoretical solutions and experimental data in key parameters such as seepage discharge, infiltration lines, and exit points, offering a reliable new approach for analyzing complex seepage problems.

To overcome limitations of the continuum assumption, the mesoscopic-scale Lattice Boltzmann Method (LBM) has achieved remarkable success over the past three decades. LBM discretizes the fluid into particle distribution functions following simple collision and streaming rules, with its kinetic processes ultimately recovering the macroscopic Navier–Stokes equations. Due to its straightforward boundary treatment, high parallel efficiency, and inherent suitability for multiphase flow in complex geometries, LBM is particularly well-adapted for pore-scale flow studies. Its effectiveness has been thoroughly validated using real pore structures. Methodologically, LBM evolved from lattice gas automata (LGA) by combining the Boltzmann equation with discrete lattice models, resulting in a framework that balances rigor and computational efficiency [59].

The Boltzmann equation is a nonlinear integro-differential equation that fundamentally describes the evolution of the particle distribution function in time, space, and momentum via statistical methods, addressing collective behaviors of large particle ensembles rather than tracking individual particle trajectories. Specifically, hydrodynamic analysis in LBM involves observing fluid particle distributions at the mesoscopic scale using a Lagrangian approach, studying their movement and conservation statistically, and finally linking mesoscopic velocity distribution functions to macroscopic physical quantities through integration. Consequently, the governing equations based on the mesoscopic Boltzmann equation are relatively simple, easy to comprehend and implement, feature straightforward computational principles and pre-processing, convenient boundary handling, high numerical stability, and are not constrained by the continuum assumption.

When applying LBM, attention must be paid to model stability, accuracy (dependent on lattice resolution, time step, and collision operator selection), and parallel computing. Its unique discretization mechanism enables effective handling of complex flow scenarios and intricate geometries. Zhang Liangqi [38] applied the LOSB algorithm to propose an improved curved boundary scheme, developing a new incompressible Boltzmann model based on the lattice method that effectively eliminates unphysical Knudsen layers. Wang Chenchen et al. [60] established a multi-scale digital carbonate rock model using simulated annealing and superposition algorithms based on SEM images, and employed LBM to investigate the seepage characteristics of the digital rocks. Que Yun et al. [61] combined the QSGS method with LBM to study mesoscopic seepage characteristics in three-dimensional reconstructed soil models under varying pore parameters and anisotropic conditions, revealing the influence of pore structure, particle size, porosity, and anisotropy on seepage behavior. Their research further established a quantitative relationship between porosity (n) and computed permeability (k) of the soil model. Data points and their linear fitting results clearly indicated a significant linear increase in permeability with rising porosity within the QSGS-reconstructed 3D soil models. This relationship fully demonstrates that porosity is a key parameter determining the permeability of soil models. This study exemplifies the successful application of LBM in soil mechanics. Its value for lubrication research lies in the methodology itself: by replacing QSGS-generated soil structures with real retainer pore structures obtained via CT scanning, and substituting water properties with lubricant parameters, LBM becomes a powerful tool for accurately predicting the permeability and oil storage capacity of retainers.

At the molecular scale, Molecular Dynamics (MD) simulation, as a computational method based on fundamental principles of Newtonian and quantum mechanics, essentially utilizes computers to simulate real experimental processes and outcomes. By solving the motion trajectories of individual molecules, MD can precisely describe van der Waals forces, electrostatic interactions, etc., between fluid molecules and pore wall atoms, thereby revealing the molecular origins of nanoconfinement effects, boundary slip, and surface wettability from first principles. Chao Rui et al. [40] systematically reviewed recent advances in using MD simulations to study the wetting behavior of superhydrophobic surfaces, providing a detailed analysis of the advantages and disadvantages of various water models, potential functions, and contact angle calculation methods. Wang Chengyong et al. [62] combined experimental adsorption of water vapor in lignite pores with MD simulations, explaining from a molecular level the differential contributions of various oxygen-containing functional groups to water molecule adsorption energy.

It is particularly important to emphasize that applying MD simulation to specific lubricant-porous media systems (e.g., polyimide-lubricating oil) can transcend the measurement of macroscopic parameters (such as apparent contact angle) to fundamentally explain the molecular origins of interfacial physicochemical phenomena. For instance, simulations can reveal the π-alkyl interaction energy between long-chain alkanes in lubricant molecules and aromatic rings in the polyimide backbone, or the hydrogen bond strength between polar ester lubricants and hydroxyl groups on ceramic surfaces. Such investigations directly elucidate the “nature of interactions between porous media and lubricants,” providing theoretical basis and specific targets for surface modification via molecular engineering to precisely tailor wettability and lubrication behavior.

4.3. Data-Driven and Intelligent Computing Approaches

As the demand for predictive efficiency and reliability continues to grow, traditional research paradigms relying solely on physical models are increasingly integrating with data-driven methods, giving rise to the following emerging research directions.

4.3.1. Deep Learning Surrogate Modeling

While physics-based numerical simulations (e.g., LBM, CFD) offer high accuracy, their substantial computational cost hinders rapid design and parameter optimization. In recent years, utilizing deep learning (DL) techniques to construct surrogate models for establishing end-to-end rapid mapping from pore structure to macroscopic seepage properties has become a burgeoning direction to overcome this bottleneck. These models learn from extensive input-output data generated by high-fidelity simulations to establish end-to-end nonlinear mappings from pore structures and fluid parameters to target outputs (e.g., macroscopic permeability, transient saturation fields) [63]. Once trained, DL surrogate models can provide predictions within seconds or even milliseconds, several orders of magnitude faster than the original simulations, enabling large-scale parameter scanning, sensitivity analysis, and real-time control. However, the reliability of purely data-driven surrogate models heavily depends on the quality and coverage of the training data, and their predictions may exhibit significant errors in regions outside the training data coverage (i.e., extrapolation scenarios).

4.3.2. Data Assimilation and Uncertainty Quantification

To address the extrapolation challenge of surrogate models and enhance prediction reliability, Data Assimilation techniques become crucial. It dynamically integrates sparse experimental observation data with predictions from numerical simulations or surrogate models, enabling online correction of model parameters or state variables, thereby continuously reducing prediction uncertainty and allowing the model to track and predict the evolution of the real system [64]. Simultaneously, Uncertainty Quantification (UQ) aims to systematically identify, evaluate, and propagate uncertainties from all sources—from microstructure characterization and fluid properties to the model itself [65]. Through UQ, researchers can clearly understand the confidence intervals of prediction results, providing a quantitative basis for risk-informed decision-making.

Embedding physical knowledge into data-driven models (e.g., Physics-Informed Neural Networks, PINNs [66]) is another important trend. By introducing governing equations as constraints, it enhances the physical consistency and extrapolation capability of surrogate models. For instance, Tartakovsky et al. successfully applied PINNs to solve parameter and constitutive relationship identification in subsurface flow problems, demonstrating their strong potential [67]. “Physical Model + DL Surrogate Model + Data Assimilation + UQ” is forming a powerful research loop, aiming to construct a new generation of seepage prediction and design frameworks that are fast, reliable, and self-aware of their uncertainties.

4.3.3. Efficient Application of Pore Network Models

At the mesoscopic scale, Pore Network Models (PNM), as efficient tools based on topological simplification, effectively complement the aforementioned paradigms. PNM abstracts complex pore spaces into networks of pore bodies and throats, simulating seepage by solving flow conservation equations. Its computational efficiency is far superior to direct numerical methods like LBM, making it suitable for large-scale statistical studies. PNM holds great potential for integration with data-driven methods: on one hand, PNM can rapidly generate large datasets for training surrogate models; on the other hand, calibrated PNMs themselves can serve as efficient, physically meaningful simulators embedded within data assimilation cycles. Although PNM involves inherent simplifications in describing complex pore geometry and physical details, it retains irreplaceable value for material screening and investigating macroscopic statistical laws [68]. This integrated approach leverages the strengths of both physics-based simplification and data-driven efficiency.

4.4. Comparison of Simulation Studies

Investigating seepage behavior in porous media requires the selection of an appropriate simulation method based on the research conditions, operational parameters, and specific objectives. Fluent, LBM, and MD represent three distinct simulation approaches operating at different scales, each with its applicable scope and advantages, the key differences between these methods are summarized in Table 1.

Table 1. Comparison of Core Characteristics Across Multi-scale Simulation Methods.

Comparison Dimension	Fluent (Navier–Stokes-Based)	LBM (Boltzmann Equation-Based)	MD (Newtonian Mechanics-Based)
Theoretical Framework	Macroscopic continuum assumption	Mesoscopic particle statistical distribution	Microscopic molecular interactions
Spatial Scale	Millimeter ~ Meter	Micrometer ~ Millimeter	Nanometer
Temporal Scale	Second ~ Hour	Microsecond ~ Second	Picosecond ~ Nanosecond
Core Advantages	Mature solver for engineering problems; Effective for high-Reynolds turbulent flow simulation	Intrinsic capability for complex boundaries & multiphase flows; High parallel efficiency	Reveals fundamental mechanisms; Provides first-principles parameters
Inherent Limitations	Cannot resolve pore-scale effects; Relies on empirical closure models	Low efficiency at macroscopic scales; Inaccurate for high-Mach-number flows	Extremely high computational cost; Severely limited spatiotemporal scale
Role in Seepage Research	System-level performance prediction & design	Pore-scale flow mechanism investigation	Exploration of interfacial phenomena & transport mechanisms

The Finite Element Method, as implemented in tools like Fluent, is grounded in the continuum hypothesis. It is suitable for the rapid analysis of macroscopic engineering problems but exhibits limitations at the micro-porous scale. In contrast, the Lattice Boltzmann Method (LBM), based on mesoscopic kinetic theory, effectively handles flow within complex pore structures. However, its computational efficiency decreases for macroscopic-scale problems. Molecular Dynamics (MD) simulation operates from the molecular scale, capable of revealing the mechanisms of microscopic interactions. Its primary constraints are the prohibitively high computational cost and the difficulty in simulating large-scale systems.

The paradigm of current numerical simulation research is evolving from “single-scale independent application” toward “multi-scale coupling.” Sole reliance on a single-scale model is no longer sufficient for achieving a complete understanding of seepage behavior, spanning from fundamental mechanisms to overall performance. The future trend lies in developing scale-bridging methodologies. Examples include: utilizing MD simulations to provide accurate wall boundary conditions (e.g., slip models, effective potential fields) for LBM, enabling the study of “leakage” or “transport” behaviors at the junctions between nanopores and micrometer-scale pore networks; employing LBM to compute equivalent permeability on representative elementary volumes, which is then used as input for macroscopic system simulations in tools like Fluent; and adopting machine learning methods to learn from high-fidelity LBM or MD data for constructing fast surrogate models to accelerate macroscopic simulations.

Consequently, when selecting a simulation method, researchers should not base their choice merely on familiarity with a specific tool. Instead, the selection should be grounded in well-defined scientific questions, aiming to construct an integrated, multi-scale computational strategy. This approach maximizes the strengths of each method while compensating for their inherent limitations.

5. Conclusions and Future Perspectives

This article has systematically reviewed and critically analyzed major advances in the understanding of seepage behavior in porous media, covering material characterization, theoretical models, experimental methods, and numerical simulations. Based on this comprehensive assessment, the following conclusions are drawn, and the limitations of the current research landscape alongside key future directions are outlined.

(1) Key Conclusions

a. Material Microstructure Governs Macroscopic Performance: The oil storage and permeability of porous media are fundamentally determined by their microstructural characteristics, such as porosity, pore size distribution, and tortuosity. The three primary material categories—porous organic polymers (POPs), metal foams, and porous ceramics—exhibit distinct properties: POPs offer designable pore channels and tunable surface chemistry for precise lubricant regulation; metal foams (particularly reticulated structures) provide high porosity and connectivity, yielding excellent permeability and mechanical strength; porous ceramics demonstrate outstanding thermal and structural stability under extreme conditions.

b. Co-evolution of a Multi-scale Theoretical Framework: A multi-scale simulation system has been established, spanning macroscopic continuum theory, mesoscopic LBM and PNM, and microscopic MD. It is crucial to emphasize that Darcy’s law and its extensions, serving as empirically validated macroscopic descriptors, act as the “system-level performance anchor” in modern research. Meso- and microscale models do not replace macroscopic methods; instead, they enhance predictive capability for novel porous materials by revealing pore-scale mechanisms and providing more accurate constitutive relations and equivalent parameters.

c. Experiments and Simulations are Complementary: Experimental studies incorporating advanced techniques like in situ CT and micro-visualization provide the most direct evidence for seepage mechanisms. Numerical simulations (e.g., CFD, LBM, MD), with advantages like low cost, parameter control, and complete data acquisition, are indispensable tools for mechanism exploration and performance prediction. Their combination forms a complete methodological framework for studying complex seepage behavior.

d. Interfacial Physicochemical Effects are Critical: Interfacial parameters such as surface wettability and pore roughness exert a decisive influence on seepage behavior but have often been oversimplified or neglected in past studies. Wettability directly controls the magnitude and direction of capillary forces, while roughness significantly affects fluid boundary conditions and effective flow pathways.

(2) Limitations of the Current Research Paradigm

Despite the progress systematically reviewed herein, several pervasive limitations must be acknowledged:

a. Challenges in Cross-scale Integration: Achieving efficient and seamless coupling from MD to LBM to CFD remains a core challenge. Current research is largely confined to “serial” parameter passing, with truly “concurrent” multi-scale methods still immature, highlighting the conflict between computational cost and accuracy.

b. Insufficient Depth in Experiment-Simulation Fusion: Real-time data assimilation and mutual validation between experiments and simulations remain relatively weak. Simulation input parameters often rely on idealized assumptions or limited experimental data, leading to deviations between predictions and actual system behavior.

c. Imperfect Modeling of Complex Interface Effects: Although the importance of surface wettability and roughness is recognized, most models still simplify pore walls as ideally smooth and chemically homogeneous, lacking the capability to accurately capture chemical heterogeneity, nanoscale roughness, and dynamic contact angle hysteresis in real materials.

d. Gap Between Smart Materials and Practical Application: Intelligent porous materials, such as stimulus-responsive polymers and graded metal foams, largely remain at the laboratory conceptual stage. Their application in real high-speed bearing lubrication systems faces challenges related to fabrication processes, long-term stability, and cost-effectiveness.

(3) Future Research Outlook

Building upon this systematic review, we propose a future research pathway integrating materials science, computational science, and intelligent design. This pathway aims to directly address the challenges identified herein, translate the reviewed methods and conclusions into concrete research actions and design principles, and ultimately advance the realization of high-performance, highly reliable lubrication systems.

a. Deepening the Study of Interfacial Physics and Interaction Nature: Future work must elucidate the fundamental nature of solid–liquid interactions in lubrication systems. As a specific follow-up to this review, a dedicated investigation into polyimide porous retainer materials is proposed. This research will employ high-resolution X-ray CT scanning to reconstruct their 3D digital models and utilize the Lattice Boltzmann Method to systematically study the influence of surface properties (wettability and roughness) on lubricant seepage dynamics. This work is necessary because existing models often oversimplify pore walls, neglecting the complex interfacial effects arising from real surface characteristics. The direct objective is to establish quantitative structure-property relationships between surface characteristics and seepage performance, providing precise theoretical models and design guidelines for material selection and surface modification of next-generation, highly reliable aerospace bearing retainers.

b. Developing Deeply Integrated Multi-scale Simulation and AI-Assisted Design:

From Serial to Concurrent Multi-scale Simulation: Future efforts should develop dynamic multi-scale frameworks that employ different resolution models in different regions with real-time information exchange at the interfaces, aiming to solve the “scale gap” problem, despite persistent challenges in ensuring compatibility and managing computational complexity.

Artificial Intelligence as an Accelerator and Innovator: Machine learning (ML) offers key support for breaking through multi-scale simulation bottlenecks: serving as surrogate models to drastically accelerate simulations; acting as a scale-bridging tool to intelligently correlate micro-parameters with macro-properties; and enabling inverse design for material structure optimization. However, the advancement of this technology urgently requires the construction of application-specific datasets tailored to lubrication conditions to establish accurate constitutive models and design criteria for bearing systems.

Promoting Domain-Specific Data Infrastructure: Current model parameters are often borrowed from other fields, raising questions about their validity under lubrication conditions. There is a critical need for systematic experiments and high-fidelity simulations focused on typical lubricant-material pairs under real bearing conditions (high speed, temperature, variable load) to build dedicated benchmark datasets. This will support model validation and AI training, ultimately leading to the development of application-specific constitutive relations and design criteria for lubrication systems.

Building an Intelligent, Multi-scale Research Paradigm: Integrating high-fidelity data from MD and LBM with lightweight AI surrogate models embedded within macroscopic CFD simulations can enable efficient and accurate system-level performance prediction. While this paradigm depends on high-quality data and faces ongoing challenges in model interpretability and extrapolation capability, it shows potential for striking a favorable balance between computational efficiency and physical consistency.

c. Advancing Multi-physics Coupling and Performance Evolution Modeling: Future research must focus on developing multi-physics models that concurrently couple seepage, stress, and thermal fields, accounting for the time-dependent evolution of lubricant properties (e.g., viscosity) under high temperatures and shear rates. This requires models capable of dynamically describing the deformation and damage of porous media under complex loads and their feedback on seepage paths. Concurrently, developing models that predict the degradation of material performance (e.g., permeability, mechanical strength) over long-term operation, informed by experimental data, is crucial for achieving lifespan prediction and health management of lubrication systems.

d. Developing Next-Generation Intelligent Porous Lubrication Materials: Focus should be placed on developing intelligent porous materials with graded structures, aligned channels, or stimulus-responsive characteristics to achieve active management and on-demand supply of lubrication. Drawing inspiration from supramolecular polymer systems responsive to external stimuli (e.g., pH, temperature, light), as proposed by Yan et al. [69], can provide the chemical tools and implementation pathways for designing condition-adaptive lubrication retainer materials.

This review systematically consolidates the existing knowledge framework for seepage in porous media and provides clear guidance for applying seepage theory to the design and failure analysis of aerospace bearing lubrication. Researchers can utilize the multi-scale framework presented here to select appropriate models, refer to the described porous material properties for preliminary material selection, and follow the proposed research pathway—progressing from microscopic mechanism exploration to intelligent material design—thereby effectively bridging fundamental research and engineering practice and accelerating the development of next-generation lubrication technologies.