Sand Particle Transport Mechanisms in Rough-Walled Fractures: A CFD-DEM Coupling Investigation

Abstract

Utilizing a coupled Computational Fluid Dynamics and Discrete Element Method (CFD-DEM) approach, this study constructs a comprehensive three-dimensional numerical model to simulate particle migration dynamics within rough artificial fractures subjected to the high-energy impact of water inrush. The model explicitly incorporates key governing factors, including intricate fracture wall geometry characterized by the joint roughness coefficient (JRC) and aperture variation, hydraulic pressure gradients representative of inrush events, and polydisperse sand particle sizes. Sophisticated simulations track the complete mobilization, subsequent acceleration, and sustained transport of sand particles driven by the powerful high-pressure flow. The results demonstrate that particle migration trajectories undergo a distinct three-phase kinetic evolution: initial acceleration, intermediate coordination, and final attenuation. This evolution is critically governed by the complex interplay of hydrodynamic shear stress exerted by the fluid flow, frictional resistance at the fracture walls, and dynamic interactions (collisions, contacts) between individual particles. Sensitivity analyses reveal that parameters like fracture roughness exert significant nonlinear control on transport efficiency, with an identified optimal JRC range (14–16) promoting the most effective particle transit. Hydraulic pressure and mean aperture size also exhibit strong, nonlinear regulatory influences. Particle transport manifests through characteristic collective migration patterns, including “overall bulk progression”, processes of “fragmentation followed by reaggregation”, and distinctive “center-stretch-edge-retention” formation. Simultaneously, specific behaviors for individual particles are categorized as navigating the “main shear channel”, experiencing “boundary-disturbance drift”, or becoming trapped as “wall-adhered obstructed” particles. Crucially, a robust multivariate regression model is formulated, integrating these key parameter effects, to quantitatively predict the critical migration time required for 80% of the total particle mass to transit the fracture. This investigation provides fundamental mechanistic insights into the particle–fluid dynamics underpinning hazardous water–sand inrush phenomena, offering valuable theoretical underpinnings for risk assessment and mitigation strategies in deep underground engineering operations.

1. Introduction

In fracture systems involving fluid flow, such as those encountered in geotechnical engineering (e.g., hydraulic fracturing, oil/gas production, geothermal energy extraction, groundwater extraction), subsurface seepage, and construction materials, understanding the transport and migration behavior of particles driven by fluid flow is critical. In recent years, numerical simulations—particularly the coupled Computational Fluid Dynamics–Discrete Element Method (CFD-DEM)—and physical experiments have emerged as the primary tools for investigating these complex processes. These approaches have revealed key mechanisms and influencing factors governing particle migration.

Research on Migration Mechanisms and Patterns:

Particle migration in porous media or fractures exhibits diverse mechanisms. Beyond fundamental mechanisms, like sedimentation, fluidization, and suspension, numerical simulations reveal that the vortex mechanism also constitutes a significant mode of particle transport [1]. Fluid–particle flow patterns become considerably more complex within non-planar tortuous fractures, where phenomena such as particle vortices and re-suspension frequently occur near bends and corners. As migration proceeds, particles ultimately form particle beds with irregular geometries, such as structures featuring dual depressions or triple terraces on their surfaces [2,3]. The overall migration pattern of particles can be categorized into distinct stages, typically encompassing phases of particle acceleration, deceleration, accumulation, and the development of clogging layers (or particle dams) [1,4]. Furthermore, the migration of cohesive particle aggregates within fractures manifests in three primary modes: collective motion, fragmentation of large aggregates into smaller ones, and the detachment of particles from the aggregate boundaries. Aggregate migration velocity shows a negative correlation with the degree of aggregation (aggregate size/density) but a positive correlation with the erosion rate [5]. Influence of Particle and Fluid Properties: Fluid velocity serves as a critical factor governing particle migration. Elevated velocities enhance efficient suspension and deep transport of low-density particles [6], extend deposition distances of proppants and increase particle migration distances [7]. However, they concurrently reduce the final equilibrium height of particle beds [8], while the flow-impeding effect of particle sedimentation becomes significantly pronounced under low velocities [9]. Increased fluid viscosity likewise aids particle suspension [10] but exerts a relatively minor influence on the ultimate coverage of proppants [10]. For specialized fluids such as viscoelastic fluids, particle migration kinetics are modulated by wall slip conditions and the degree of shear thinning, potentially driving particles toward either the walls or the channel center [10]. In contrast, supercritical CO₂ establishes distinct preferential flow channels within porous media, with particle migration primarily confined to these paths and clogging occurring along their flanks [10]. Particle physical attributes—size, density, concentration, morphology, and stiffness—exert decisive control over migration and clogging dynamics. Higher particle concentrations elevate fluid–solid velocities, intensify mechanical obstruction at walls [11], and promote particle accumulation near walls—particularly on rough surfaces—forming elongated aggregates [11] while also amplifying particle erosion in interfacial seepage flows [12]. Enhanced particle angularity (morphology) facilitates the formation of more stable clogging layers [13]. Greater particle density and larger size tend to impede migration. Particle diameter critically governs migration distance—which increases with growing particle-to-pore size ratios [14] or decreases with larger pore/particle diameter ratios [7]—and dictates clogging patterns. Large particles (relative to flow paths) predominantly cause superficial layer clogging (e.g., bridging clogs, pinhole clogs) [1]. Medium-sized particles (0.3–0.6 mm) induce deep-seated clogging. Small particles typically traverse systems rapidly [15]. Increased particle density and Young’s modulus contribute to more stable clogging zones [13]. The critical bridging concentration exhibits sensitivity to the inlet/outlet size ratio of fractures and the outlet/particle diameter ratio [13].

Effect of Fracture Geometry and Roughness:

Fracture geometries profoundly alter flow fields and particle behaviors. Compared to straight fractures, flow and particle patterns within non-planar configurations (e.g., tortuous fractures with bend segments) exhibit greater complexity [2]. Crucially, wall roughness—typically quantified by the joint roughness coefficient (JRC) or fractal dimension—intensifies flow instability [6,11], inducing more frequent particle–particle and particle–wall collisions that augment energy dissipation [11]. Rough surfaces promote particle settlement at high-roughness zones [8] yet paradoxically drive migration of additional particles toward central fracture regions, thereby modifying deposition patterns [11]. Rough fractures with high fractal dimensions often demonstrate enhanced proppant coverage [8]. The aperture-to-particle diameter ratio stands as a pivotal parameter governing both flow characteristics and clogging susceptibility [5]. Macroscopic fracture geometry further modulates gravitational effects. When a fracture aligns parallel to gravity, gravitational resistance exerts a more pronounced influence on particle trajectories [13]. In interfacial seepage flows at fracture boundaries (e.g., between soil layers), higher initial porosity and flow velocity at the interface trigger earlier and more severe particle erosion, with the critical hydraulic gradient increasing nonlinearly with soil compactness [10].

Clogging, Bridging, and Seepage Effects:

The terminal stage of particle migration often culminates in clogging, with studies identifying multiple scenarios: superficial clogging, internal/deep-seated clogging, and hybrid clogging [1]. Among these, hybrid clogging exhibits the most severe degradation of porosity and permeability [1]. Pinhole clogging and bridging clogging represent two primary modes [7]. Research on bridge plugging technology (Leak Control Material, LCM) in fractured formations reveals that the initial structure of clogging layers typically follows a binary sphere bridging model. The formation location of such bridges is dominated by particle diameter, while their development rate depends on particle concentration. Optimizing particle combinations enhances the stress-bearing capacity of these structures, though excessive hydraulic pressure can compromise their stability [12]. Under seepage conditions, the size ratio (intruding particle to porous media grain diameter) serves as a dominant factor governing fine particle aggregation and migration. At lower size ratios, most fine particles accumulate at the media surface (enhancing surface clogging). At higher size ratios, it becomes a key determinant of particle migration distance [16]. Additionally, porosity directly modulates flow behavior. Increased porosity initially reduces and then extends particle retention time. Conversely, granular temperature displays an opposing trend [17].

In summary, fluid-driven particle migration through fractured and porous media constitutes a highly complex multiphysics process, governed by the coupled effects of particle properties, fluid characteristics, fracture geometry/roughness, and interfacial conditions. These factors collectively dictate ultimate migration distance, deposition morphology, and clogging patterns. A thorough investigation of these variables and their interaction mechanisms holds significant implications for predicting and controlling particle-related challenges in practical engineering applications—particularly in petroleum extraction, geothermal development, subsurface storage, and concrete construction [18,19,20,21,22].

Mondal [23] employed the CFD-DEM method to investigate the minimum particle concentration required for bridge formation during temporary plugging, while Shahri [24] conducted CFD and DEM simulations using the open-source software OpenFOAM(5.x) and LIGGGHTS(3.8), respectively, deeply exploring the bridging and sealing behaviors of temporary plugging agent particles at perforation or fracture entrances and detailedly analyzing the effects of parameters including particle size distribution, shape, displacement, and fluid viscosity before validating the simulation results against experimental data. Kloss [25] further developed the CFDEM open-source software based on these platforms, enriching the toolbox for particle migration simulation. However, in the coupled CFD-DEM method, the continuous phase is influenced by the discrete phase (e.g., the volume occupied by particles), and since DEM necessitates solving the motion equations independently for each particle, an increase in particle number (particularly during field-scale simulations) leads to extremely high computational costs. To address engineering demands, improved methodologies, such as the MP-PIC method [26] and the Representative Particle Model (RPM) [27], have been developed; these partition the discrete phase into parcels containing multiple particles with identical properties, represented by a single parcel particle, significantly reducing computational effort by neglecting particle–particle interactions within parcels. The parcel particle is treated as a virtual soft sphere, effectively simulating collision behavior by calculating the elastic and damping forces arising from collision overlap. Additionally, the Dense Discrete Phase Model (DDPM), as an Eulerian–Lagrangian particle flow model [28], integrates the advantages of the TFM (Two-Fluid Model) and MP-PIC approaches, solving the fluid phase within an Eulerian framework while tracking the motion of parcel particles using a Lagrangian method. DDPM treats groups of proximate particles as a single virtual entity for collective solution, further reducing computational overhead, and calculates particle concentration by mapping the particle volume fraction within parcels to the Eulerian grid [29]. Zhang Feng [30] achieved successful simulation of solid particle migration trajectories in horizontal wellbores and analyzed the sealing behavior of temporary plugging balls on perforations employing this DDPM model.

While existing studies have systematically examined particle transport mechanisms in fractures, persistent limitations include (1) geometric oversimplification through the neglect of three-dimensional complex structures in realistic rough-walled fractures due to inadequate characterization of the joint roughness coefficient (JRC); (2) the scarcity of simulations addressing transient hydrodynamic behavior under high-energy water inrush conditions (1–4 MPa) characterized by intense shear turbulence; and (3) insufficient investigation into coupled effects governing roughness, hydraulic pressure, and aperture (

Table 1. Summary of knowledge, limitations, and gaps in particle migration through fractures.

Dimension	Current Knowledge Summary (Literature Review)	Research Limitations	Knowledge Gaps (Present Study Focus)
Fracture Geometry	(1) Non-planar tortuous fractures increase flow/particle complexity [1] (2) JRC/fractal dimension quantifies roughness, enhancing flow instability and collision dissipation [6,11] (3) Aperture-to-particle diameter ratio controls clogging susceptibility [5]	Geometric oversimplification: Neglect of 3D complex structures in realistic rough-walled fractures due to inadequate JRC characterization	Establish 3D rough fracture models via Barton curves (JRC = 6–20), quantify nonlinear roughness effects on transport pathways [1,2,3,8]
Fluid Conditions	(1) Fluid velocity dominates migration efficiency (high velocity promotes suspension) [6,7] (2) Supercritical CO2 forms preferential flow channels; viscoelastic fluids modulated by wall slip [10] (3) Particle sedimentation impedes flow under low velocities [9]	Scarcity of transient simulations under high-energy water inrush (1–4 MPa) with intense shear turbulence	Reveal three-phase kinetic evolution (acceleration–coordination–attenuation) driven by high-energy inrush (1–4 MPa) [6,7,8,9,10]
Parameter Coupling	(1) Roughness drives particles toward the center but enhances wall deposition [8,11] (2) Particle size dictates clogging patterns (large: bridging; small: rapid penetration) [1,7,15] (3) Hydraulic gradient positively correlates with migration distance [10,16]	Insufficient investigation into coupled effects of roughness, hydraulic pressure, and aperture	Quantify nonlinear synergy of JRC–pressure–aperture, identifying optimum transport range (e.g., JRC = 14–16) [5,8,9,10,11,12,13]
Particle Transport Modes	(1) Collective migration: acceleration–deceleration–accumulation–clogging [1,4] (2) Individual behaviors: main shear flow, vortex disturbance, wall adhesion [2,5] (3) Cohesive aggregates: collective motion/fragmentation/detachment [5]	_	Identify novel patterns: “bulk progression”, “fragmentation-reagglomeration”, “central stretching with edge retention” [1,4,5]
Numerical Methods	(1) CFD-DEM as a primary tool; MP-PIC/RPM/DDPM improves computational efficiency [23,24,25,26,27,28,29,30] (2) Validations focused on laminar flows; scarce data for turbulence [26,27,28,29]	High computational cost for large-scale simulations; lack of experimental validation under high-pressure turbulence	Develop a high-pressure turbulent CFD-DEM framework, validated via pipe pressure-drop benchmark (error 0.45%) [23,24,25,26,27,28,29,30]
Engineering Predictive Models	(1) LCM bridging depends on particle size/concentration [12] (2) Pore/throat ratio controls critical bridging concentration [13,16]	Absence of quantitative models for migration time in high-pressure rough fractures	Formulate a multivariate regression model (Equation (1)) predicting t80 (80% mass breakthrough, R2 = 0.782) and integrating JRC/P/aperture/particle size [12,13]

). This study innovatively establishes a fully coupled CFD-DEM model, achieving the first quantitative characterization of three-phase particle migration evolution under synergistic high-pressure/high-roughness conditions (JRC = 6–20), thereby addressing critical knowledge gaps.

2. CFD-DEM Coupling Framework

2.1. CFD-DEM Workflow

Maintaining technical precision and logical flow involves the CFD-DEM coupling simulation workflow and the CFD phase (Figure 1). First, define the fluid properties and problem boundary conditions. Mesh the fluid domain. Solve the governing equations (e.g., Navier–Stokes equations for the fluid flow, potentially including additional fields, like electric fields). Compute the fluid motion using the PISO solver. Transfer the computed fluid field data (velocity, pressure, etc.) to the DEM particles. In the DEM phase, establish the DEM domain and generate/disperse particles according to the simulation setup. Calculate particle–particle interaction forces (e.g., using contact models like Hertz–Mindlin). Calculate the particle–fluid interaction forces acting on each particle (e.g., drag, buoyancy) based on the received fluid data. Compute the net force acting on each particle. Update particle motion states (position, velocity, rotation) by integrating Newton’s laws of motion using the net force. Transfer the updated particle state data (position, velocity, radius, temperature, etc.) back to the CFD solver. For the coupling interface (e.g., CFDEM), exchange data between the CFD and DEM solvers (transfer fluid data to particles, particle data to fluid). Based on the exchanged information, calculate the interphase coupling forces (fluid acting on particles and vice versa). Calculate the local fluid volume fraction (void fraction) occupied by particles in CFD cells. Optionally track and process phase-averaged particle field information (e.g., mean particle concentration distribution in space over time).

CFDEM Coupling Calculation Steps:

(1)

CFD Setup: Define the computational domain, generate the mesh, and set fluid boundary conditions.

(2)

DEM Initialization: Import particles into the domain.

(3)

Force Calculation: Compute particle–particle contact forces and particle–fluid interaction forces and then determine the net force on each particle.

(4)

Particle Motion Update: Solve particle motion equations to update particle velocities and positions.

(5)

Fluid Flow Solution: Solve CFD governing equations to update the fluid velocity/pressure fields.

(6)

Data Transfer: Pass updated particle data (position, velocity, etc.) to the fluid solver.

(7)

Loop: Repeat steps (3)–(6) iteratively for each time step.

2.2. Fluid Flow Solution

As stipulated in the paper requirements, here is the formal translation for the governing equations of the fluid phase in an incompressible liquid–solid two-phase flow model with constant fluid density and conserved liquid–phase mass, expressed in standard academic notation as follows:

$${\displaystyle \frac{\partial ({\rho }_f{\epsilon }_f)}{\partial t}}+\nabla \cdot ({\rho }_f{\epsilon }_f{V}_f)=0$$

(1)

$${\displaystyle \frac{\partial ({\rho }_f{\epsilon }_f{V}_f)}{\partial t}}+\nabla \cdot (\rho f{\epsilon }_f{V}_f{V}_f)=-{\epsilon }_f\nabla P-\nabla \cdot ({\epsilon }_f{\tau }_f)+{\epsilon }_f{\rho }_{f}g+F_p$$

(2)

In the equations, g denotes the gravitational acceleration, ρ_f represents the fluid density, p and ε_f indicate the fluid-phase pressure and volume fraction, respectively, u_f is the fluid velocity vector, fp→f designates the fluid–particle interaction force, and τ_f stands for the viscous stress tensor.

The high-energy water and sand inrush process is intrinsically characterized as turbulent flow. To meet the requirements of high accuracy, simple application, computational efficiency, and generalizability, this study employs the standard k-ε model for calculating the turbulent viscosity of the drilling fluid. Here, k represents the turbulent kinetic energy and ε denotes the dissipation rate. The transport equations for k and ε are given by the following:

$${\displaystyle \frac{\partial ({\rho }_{f}k)}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_i}}({\rho }_{f}k{V}_i)={\displaystyle \frac{\partial }{\partial x_j}}((\mu +{\displaystyle \frac{{\mu }_t}{{\delta }_k}}){\displaystyle \frac{{\delta }_k}{\partial x_j}})+Gk-{\rho }_f\epsilon -Y_M$$

(3)

$${\displaystyle \frac{\partial ({\rho }_f\epsilon )}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_i}}({\rho }_f\epsilon {\epsilon }_f{V}_i)={\displaystyle \frac{\partial }{\partial x_j}}((\mu +{\displaystyle \frac{{\mu }_t}{{\delta }_{\epsilon }}}){\displaystyle \frac{{\delta }_{\epsilon }}{\partial x_j}})+C_{1\epsilon }{\displaystyle \frac{\epsilon }{k}}(G_k+G_{3\epsilon }{G}_b)-C_{2\epsilon }{\rho }_f{\displaystyle \frac{{\epsilon }^2}{k}}$$

(4)

In these equations, ρ_f denotes the liquid-phase density (unit: kg/m³); u_f represents the liquid-phase velocity vector (unit: m/s); C_1ε, C_2ε, and C_3ε are empirical constants, with C_1ε = 1.44 and C_2ε = 1.92; σ_k and σ_ε are the turbulent Prandtl numbers for k and ε, respectively, assigned values of σ_k = 1.0 and σ_ε = 1.3; G_k designates the turbulence kinetic energy generation due to mean velocity gradients; Y_M signifies the contribution of fluctuating dilatation to the overall dissipation rate in compressible turbulence; and μ_t is the turbulent viscosity coefficient.

2.3. Solid Phase Solution Procedure

Granular flow within the narrow slit exhibits highly complex non-planar behavior, characterized by frequent fluid–particle interactions, particle–particle collisions, and particle–wall contacts. In simulations, each particle possesses both translational and rotational degrees of freedom [31,32]. The governing equations for particle momentum are described by the following:

$$m{\displaystyle \frac{d{\upsilon }_p}{dt}}=m_{p}g+{\displaystyle \sum }{F}_c+F_d+F{S}_f+F_{Mf}+F_m+F_p+F_b$$

(5)

$$I{\displaystyle \frac{d{\omega }_p}{dt}}={\displaystyle \sum T_c+T_f}$$

(6)

where m_p is particle mass; v_p is particle translational velocity vector; ω_p is particle angular velocity vector; F_c is contact forces (particle–particle and particle–wall interactions); F_d is fluid drag force (DiFelice drag model); F_p is pressure gradient force (imposed by the fluid); F_f is buoyancy force; I is particle moment of inertia; T_c is contact torque; and T_f is hydrodynamic torque.

Interaction between particles, as well as between particles and walls, is simulated using the soft-sphere approach, which allows deformation during contact. The contact model employs the nonslip Hertz–Mindlin–Deresiewicz formulation. Within this model, the normal force component is calculated based on Hertzian contact theory; the tangential force component is calculated based on Mindlin–Deresiewicz theory; and the normal and tangential damping components are incorporated. The damping coefficients are determined via a relationship established with the coefficient of restitution using a collision energy dissipation model. Tangential contact behavior is governed by the Coulomb friction law, which is characterized by a static friction coefficient determining sliding friction between particles and between particles and walls. Rolling friction is modeled by applying a constant-magnitude resistance moment opposing rolling motion at the contact point. This rolling resistance moment acts in the direction opposite to the relative angular velocity.

When the particle volume fraction is sufficiently high, the particle cloud exerts a significant disturbance on the fluid flow field. In such cases, the forces exerted by the particulate phase on the fluid phase must be accounted for, necessitating a two-way coupling approach. The force density applied by particles within a CFD grid control volume to the fluid phase is calculated using Equation (7).

$$F_{pf}={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n(F_d^i+F_{Sl}^i+F_{Ml}^i+F_m^i+F_p^i+F_B^i)}}{V_{cell}}}$$

(7)

where N is the number of particles within the cell. According to the fundamental principle of Newton’s third law, when the fluid exerts a force on the particles, the particles simultaneously exert a force with an equal magnitude and an opposite direction on the fluid.

2.4. Model Verification

The effectiveness of the CFD-DEM coupling method has been extensively validated through multiple classic models and benchmark cases in prior studies. These investigations demonstrate its broad applicability in accurately and stably simulating fluid–solid interactions. Building on this foundation, this study extends the commonly studied laminar flow conditions in existing research to turbulent regimes, addressing the demands of high-velocity flows and complex flow structures encountered in practical engineering scenarios. Although direct experimental validation data for CFD-DEM coupling under high-pressure turbulent conditions remain scarce, the particle–fluid interaction in numerical solutions critically depends on fluid flow structures. Specifically, turbulent boundary layers and velocity distributions significantly influence particle dynamics. Consequently, independent verification of the CFD turbulence model is imperative. To this end, a classic benchmark case for pressure drop in turbulent pipe flow was selected. The accuracy of the SST k-ω turbulence model was verified by comparing the simulation results with theoretical analytical values under the current mesh resolution and boundary conditions. This verification not only establishes a reliable fluid dynamics foundation for subsequent fluid–solid coupling simulations but also enhances the physical realism and numerical credibility of the overall CFD-DEM framework.

This study validates the rationality of the employed turbulence model using a classic experimental benchmark for turbulent pipe pressure drop, ensuring the accuracy and robustness of numerical simulations under turbulent conditions.

The simulation models water flowing through a smooth horizontal pipe with the following specifications: pipe length, 2 m; pipe radius, 0.002 m; water density, 1000 kg/m³; dynamic viscosity, 1.003 × 10⁻³ Pa·s; inlet velocity, 50 m/s; and outlet pressure, 0 Pa. The pressure drop within the pipe is calculated. Under identical conditions, the k-ε turbulence model is adopted to compute the pressure drop. In this scenario, Equation (9) is applied to determine the friction factor, which subsequently yields the pressure drop.

$$f=0.316\times {\mathrm{Re}}^{-1/4}$$

(8)

$$\Delta p=\rho f{\displaystyle \frac{L}{D}}{\displaystyle \frac{v^2}{2}}$$

(9)

Re represents Reynolds number (Re), which is calculated using Equations (2)–(4) as follows:

$$\mathrm{Re}={\displaystyle \frac{\rho \mu D}{\mu }}$$

(10)

The pressure drop obtained using the model is 22,263.97 Pa, while the value calculated with Equation (9) is 22,366.31 Pa, showing an error of 0.45%. The minor deviation between the two results confirms the correctness of the turbulence model adopted in this study.

2.5. Grid Independence Verification

A structured hexahedral mesh was employed to ensure computational accuracy and numerical convergence. To examine the influence of mesh size on the simulation results, three mesh resolutions (length × height × width) were tested: 3 mm × 3 mm × 3 mm, 2.5 mm × 2.5 mm × 2.5 mm, and 1.5 mm × 1.5 mm × 1.5 mm. The corresponding mesh counts for each resolution are listed in

Table 2. Mesh counts for different mesh sizes.

Mesh Size	Mesh Count
3 mm × 3 mm × 3 mm	NODES = 6390, QUADS = 3076, HEXAS = 4760
2.5 mm × 2.5 mm × 2.5 mm	NODES = 10458, QUADS = 4300, HEXAS = 8200
1.5 mm × 1.5 mm × 1.5 mm	NODES = 38360, QUADS = 11628 HEXAS = 32368

.

As shown in Figure 2a, a comparison of the particle migration results under different mesh sizes indicates that variations in mesh resolution have only a minor effect on particle migration velocity. Figure 2b presents the pressure field distribution curves at the midsection of the fracture, revealing that changes in mesh size likewise exert a limited influence on the overall pressure distribution pattern. Considering both computational accuracy and efficiency, the medium mesh size (3 mm ×3 mm ×3 mm) was, therefore, selected as the final mesh configuration for subsequent simulations.

2.6. Model Construction

This experiment conducts coupled simulations using OpenFOAM (fluid solver) and LIGGGHTS (particle solver). The fracture geometry was constructed in Rhinoceros to build a 3D model, and the flow field was generated by ICEM. The roughness control strategy adopts Barton curves to define the joint roughness coefficient (JRC) values, with each original curve segment measuring 10 cm in length.

Three-Dimensional Fracture Modeling Workflow

Baseline construction: Repeat and splice single Barton curves in Rhinoceros → form a 20 cm horizontal baseline, extending a 5 cm particle storage zone at the left end. Surface generation: Stretch the profile along the baseline to 5 cm width → translate upward by 10 mm (setting fracture aperture) to form the upper surface. Three-dimensional processing: Duplicate the one-dimensional curve twice → construct a 5 cm-long 3D rough fracture structure (Figure 3). The geometric model of the fracture was imported into the DEM model, and approximately 9589 particles were generated in the storage zone. The model was then initiated to eliminate large unbalanced forces between particles and reach an equilibrium state.

ICEM generates the fluid mesh, which is then imported into the DEM model (Figure 4). Subsequently, a water pressure of 1–4 MPa is applied at the left boundary of the fracture to drive fluid flow and initiate particle migration; concurrently, the right boundary water pressure is set to 0 MPa, remaining unchanged throughout the simulation. The top, bottom, and side boundaries of the computational domain are configured as impermeable and fixed in all three spatial directions.

Figure 5 displays the initial particle migration field in a fracture with JRC = 16–18.

(a) The void fraction contour plot indicates particles accumulating in the left-side injection zone (void fraction 0.36–0.68), while the rest of the area is a pure water region (void fraction 1.0); at this stage, particles are stationary. (b) A 2 MPa water pressure is applied at the model inlet, decreasing gradiently along the flow direction (x-direction) to zero at the outlet, forming a pressure-driven condition. (c) and (d) show the side/top views of the initial velocity field. A perturbed continuous flow channel forms within the fracture passage (maximum velocity 1.1 m/s, concentrated near the inlet zone); velocity sharply decreases near the wall regions due to shear effects; and the overall flow pattern exhibits turbulent characteristics.

2.7. Model Parameters and Simulation Scheme

To control contact behavior, a rolling resistance contact model was adopted in the simulation. This model applies a torque at contact points to resist interparticle rolling, thereby simulating the motion characteristics of angular particles. The parameters used in the DEM and CFD models are summarized in

Table 3. Model parameters in the CF-DEM model.

Parameter	Value
Particle diameter	0.0005 m
Particle density	2650 kg/m3
Coefficient of interparticle friction	0.3
Rolling resistance coefficient	0.1
Normal stiffness of the particles	3.9 × 105 N/m
Shear stiffness of the wall	1.95 × 105 N/m
Normal stiffness of the wall	3.9 × 105 N/m
Shear stiffness of the wall	1.95 × 105 N/m
Coefficient of wall friction	0.3
Fluid density	1000 kg/m3
Fluid viscosity	1.0 × 10−3 Pa·s

[12]. For sand particles, referencing prior studies, the following parameters were employed: density, 2650 kg/m³; friction coefficient, 0.3; and rolling resistance coefficient, 0.1. Both normal and tangential stiffness remained constant throughout the simulation, with values consistent with typical sand stiffness properties.

It is assumed that fracture walls share identical mineral composition and surface roughness with particles. Consequently, wall stiffness and friction coefficients were set equal to particle properties. In the CFD model, pure water was adopted as the working fluid, with hydraulic parameters assigned values corresponding to standard atmospheric pressure at 20 °C. According to Equation (10), Reynolds numbers throughout the domain significantly exceeded 2300, confirming that the flow exists in a turbulent flow regime, indicating high-velocity fluid flows.

To investigate particle transport mechanisms within rough fractures under different influencing factors, a series of numerical experiments defined in

Table 4. Numerical simulation parameters.

Hydraulic Pressure (MPa)	Fracture Aperture (mm)	JRC	Average Particle Diameter (mm)
1, 2, 3, 4	10	14~16	0.0005
2	5, 10, 15, 20	14~16	0.0005
2	10	6~8, 10~12, 14~16, 18~20	0.0005
2	10	14~16	0.0005, 0.001, 0.0015, 0.002

were conducted. Twenty distinct simulation sets were performed, with experimental groupings detailed in Table 4.

3. Particle Migration in Fractures

Based on the CFD-DEM coupled simulation results, this chapter systematically analyzes the migration dynamics behavior, typical patterns, and regulatory effects of major influencing factors for sand particles in rough fractures (aperture, water pressure, particle diameter) under high-water-pressure driving.

3.1. Migration Processes

Particle migration exhibits a pronounced three-stage temporal evolution pattern (Figure 6).

(1)

Initial Acceleration Stage (t < 0.0015 s): The particle cluster rapidly initiates motion from a static state (average coordination number ~3.5–4) under intense shear from high-pressure fluid. Average velocity and drag force surge dramatically, with drag and pressure gradient forces dominating (>95% combined contribution) (Figure 7 and Figure 8).

(2)

Coordinated Advancement Stage (t = 0.0015–0.005 s): Particle dispersion intensifies (coordination number declines rapidly), exhibiting near-constant-velocity collective migration. Fluid driving forces and wall/particle friction/collision resistance reach dynamic equilibrium. Average velocity increases linearly, while drag force peaks and maintains a high magnitude (Figure 8).

(3)

Tail Attenuation Stage (Stage III, t > 0.005 s): Tail and near-wall particles re-accelerate under diminishing hydrodynamic forces and gradually discharge. Intra-system particles become sparse (coordination number < 1), drag force decays rapidly, and migration terminates.

The spatial distribution evolution of the particle cluster manifests as follows: initially dense low-velocity accumulation → formation of a high-velocity front with cluster elongation in the midsection → emergence of a highly sparse, high-speed “floating body” near the exit (Figure 8 and Figure 9). Concurrently, the velocity distribution shifts from an initial right-skewed pattern to approximate normality (peak ~40 m/s), ultimately concentrating within the high-velocity range (~55–60 m/s) (Figure 10).

3.2. Collective Particle Behavior

The particle cluster (as distinct from fully discrete particles) constitutes the fundamental migration unit, governed by coupled fluid shear, wall friction, and interparticle interactions. It manifests three characteristic evolutionary patterns (Figure 11, Figure 12 and Figure 13).

(1)

Bulk-Style Cohesive Advancement: During the initial acceleration stage (high coordination number), the cluster maintains tight packing with dense, high-strength force chains, moving as a unified rigid block-like structure propelled within the mainstream flow region. Distinctive features include stable morphology and robust collective coordination.

(2)

Fragmentation–Reagglomeration: With increasing fluid percolation efficacy, macro-agglomerates disintegrate at high-velocity zones or structural discontinuities. The resulting sub-clusters either migrate independently or undergo reagglomeration in regions of localized deceleration/complex geometry, accompanied by declining coordination numbers with measurable stochastic fluctuations.

(3)

Midzone Tensile Stretching—Boundary Retention: Particles within the principal shear zone undergo high-velocity migration (predominantly compressive high-strength force chains), while marginal particles experience wall-induced frictional retardation (weak force chains with sparse tensile dominance), resulting in an asymmetric “bulging central core—trailing edge” topological conformation. At the terminal phase (coordination number approaching critical zero), inter-granular contacts become negligible, transitioning to discrete particle-dominated flow, with boundary particles being ultimately evacuated due to residual frictional hysteresis.

3.3. Individual Particle Behavior

Based on three-dimensional trajectory and velocity evolution behavior, the motion patterns of individual particles are dominated by local flow field structures and can be classified into three categories.

(1)

Primary Shear Channel Type (e.g., ID-4537): It is located in the core region of the main flow and is driven by the following intense shear forces: maximum velocity peak (70 m/s), shortest transit time, significant fluctuations in Z/Y directions, and slightly tortuous trajectory but highly efficient advancement (Figure 14).

(2)

Boundary Disturbance Drift Type (e.g., ID-21257): It is located in boundary layers or vortex regions and exhibits medium-high velocity, with an overall straight trajectory; yet, it develops “slow-then-rapid” acceleration characteristics and moderate lateral deviation due to disturbance/wall undulations (Figure 15).

(3)

Wall-Adhesion Hindered Type (e.g., ID-15264): It closely adheres to rough walls and exhibits the lowest velocity, with trajectories dictated by wall geometry, where speed variations precisely mirror wall undulations, resulting in prolonged duration and high susceptibility to detaining or accumulation (

Table 5. Particle basic parameters.

Particle ID	Total Path Length	Trajectory Curvature	Z-Direction Undulation Root Mean Square	Maximum Offset in the Y Direction	Time-Consuming
21257	189.6	1.0109	2.54	2.75	0.00575
4537	191.21	1.0124	2.65	4.84	0.00540
15264	191.96	1.0121	2.59	1.39	0.00595

).

4. Influence of Key Factors on Particle Migration

4.1. Effect of Fracture Roughness

4.1.1. Particle Transit Times and Mass Transfer

Previous studies have demonstrated that fracture roughness significantly impacts fluid flow and particle migration behaviors. To elucidate the influence of fracture roughness, four fracture models with distinct joint roughness coefficients (JRCs) were constructed. Crucially, all parameters other than JRCs were maintained consistently across the four models. These JRC-varied fracture models and their baseline flow fields are presented in Figure 16. As observable in Figure 16a–d, fractures with higher JRCs exhibit increased surface irregularity and more abrupt flow deflections. The corresponding flow fields within different JRC fractures are shown in Figure 16e–h, where the velocity distributions closely mirror the geometric morphology of the fractures.

Figure 17a demonstrates that when the JRC increases from 6–8 to 14–16, the curve shifts leftward with particle transit time decreasing from 8.6 ms to 7.3 ms, indicating significantly enhanced migration efficiency. However, further elevating the JRC to 18–20 shifts the curve markedly rightward (transit time increases to 9.9 ms) and reduces particle passage efficiency. As revealed in Figure 17b, particle passage efficiency exhibits a V-shaped relationship with fracture roughness, reaching its optimal level at JRC = 14–16 (minimum transit time and peak efficiency). This identifies JRC ≈ 14–16 as the transitional zone yielding maximal particle migration efficiency. Both excessively low and high fracture roughness impair fluid-borne particle transport capacity, while moderate roughness facilitates optimal flow paths for particle migration.

4.1.2. Variation Pattern of Particle Transport Velocity

Figure 18 illustrates the spatial distribution patterns of particle transport within fractures of different roughness. From the figure, it is observed that as the joint roughness coefficient (JRC) increases from 4–8 to 14–16, the migration velocity of the particle group increases significantly, and the overall migration pattern becomes more concentrated and uniform. However, when the fracture roughness further increases (e.g., JRC = 18–20), the particle group advances slowly, exhibiting greater spatial dispersion and the emergence of distinct lag zones. This indicates that the wall disturbance effects within high-roughness fractures significantly impede the free migration of particles, exhibiting a strong hindering effect.

Figure 19 presents velocity variation curves of particles in fractures with different roughness coefficients. As observed in Figure 18a,b, as the joint roughness coefficient (JRC) increases from 6–8 to 14–16, the average migration velocity of particles rises in both temporal and spatial dimensions, reaching its peak value at JRC = 14–16. When the JRC further increases from 14–16 to 18–20, the particle velocity decreases substantially. Figure 18c further illustrates the influence of the JRC variation on the average particle velocity. The graph shows that as the JRC increases from 6–8 to 14–16, the coefficient exhibits a positive correlation with particle velocity. The average migration velocity increases with increasing roughness. Conversely, when the JRC increases from 14–16 to 18–20, the coefficient demonstrates a negative correlation with particle velocity. The average migration velocity decreases with increasing roughness.

4.1.3. Variation Patterns of Coordination Number

Figure 20 shows the evolution of the average particle coordination number over time in fractures with different joint roughness coefficients (JRCs). As shown in Figure 19a, with increasing fracture roughness, all coordination number curves exhibit a rapid decay from high to low values. This reflects the transition of the granular assembly from an initially dense configuration to a loose, disordered structure under high-potential-energy, high-velocity fluid perturbation. At the initial stage (t = 0 s), the average coordination number ranges between 3.5 and 4.2, indicating tight interparticle contacts. Over time, it rapidly decreases to below 1 and approaches zero at t ≈ 0.01 s, signifying the near-complete loss of particle contacts and completion of granular decoupling and rearrangement. Distinct evolutionary paths and decay rates emerge under different JRC conditions. For low-roughness fractures (JRC = 6–8 and 10–12), the coordination number drops most rapidly, falling below 1 within ca. 0.003 s. This indicates insufficient fracture–wall constraints, enabling easier particle dispersion and flow under hydrodynamic shear. For high-roughness fractures (JRC = 14–16 and 18–20), the coordination number decays more gradually. Temporary recovery phases and “steady-state plateaus” are observed, demonstrating the restraining capacity of rough fracture surfaces in maintaining particle structures, thereby exhibiting stronger resistance to fluid-driven mobilization.

4.1.4. Variation Patterns of Drag Force

Figure 21 shows the variation trend of the average drag force on particles within fractures under different joint roughness coefficients (JRCs). From Figure 21a, it can be concluded that regardless of how the fracture roughness coefficient changes, the drag force experienced by particles generally shows a trend of increasing first and then decreasing over time, differing only in the peak values among different groups. From Figure 21b, it can be concluded that fracture roughness has a significant nonlinear regulatory effect on the force characteristics of particles. With the increase in the JRC, the average drag force does not show a monotonic upward trend; instead, it reaches a peak under the condition of JRC = 14–16 and then significantly decreases under higher roughness (JRC = 18–20). Specifically, under low fracture roughness conditions (JRC = 6–8, 10–12), the fracture interface is relatively smooth, particles are prone to slip and rapid decoupling, the relative motion between fluid and particles is relatively stable, and the resulting shear disturbance is limited, leading to an overall low level of drag force. Under medium-to-high roughness (JRC = 14–16), the geometric protrusions on the fracture surface significantly enhance the friction and collision effects between particles, causing unstable motion and frequent contact of particles during migration, thereby amplifying local flow field disturbances and shear stress transmission, resulting in a significant increase in the average drag force. For JRC = 14–16, the higher peak drag force is primarily attributed to the pronounced surface asperities at this roughness level, which significantly enhance particle–wall friction and collision effects, leading to unstable particle motion, intensified local flow disturbances, and increased shear stress transmission. Under low roughness conditions (JRC = 6–8, 10–12), the fracture surface is relatively smooth, allowing particles to slip and rapidly decouple from the wall, thereby maintaining a lower drag force. In contrast, under higher roughness conditions (JRC = 18–20), excessive geometric protrusions generate low-shear recirculation zones behind the asperities, making particles more prone to entrapment and reducing the overall drag force.

The observed decline in drag force over time is due to the initial collision-dominated phase, in which frequent particle–particle and particle–wall contacts amplify shear transfer from fluid to particles; in the later stage, particle clusters gradually loosen and their motion becomes more streamlined, reducing relative velocity differences and weakening effective shear transfer, which leads to a gradual decrease in the average drag force.

4.2. Effect of Hydraulic Gradient

4.2.1. Particle Passing Mass and Time

Figure 22 illustrates the variation pattern of the proportion of particles passing through fractures over time under different water pressure conditions. From Figure 22a, it can be concluded that the cumulative migration proportion curves of particles under different water pressures all exhibit an “S-shaped” growth trend. As hydraulic pressure increases, the curves shift significantly leftward along the time axis, indicating a substantial increase in migration speed and a significant reduction in the time required for particles to completely traverse the fracture. Particularly, under high hydraulic pressures (3–4 MPa), sand particle groups complete fracture traversal within a very short time. This demonstrates that under the combined action of strong shear stress from substantial water flow and intense fluid driving forces, the initial contact structures between particles rapidly disintegrate, disperse quickly, and are efficiently transported by the fluid.

Figure 22b further quantifies the relationship between the time t required for particles to fully traverse fractures and the water pressure. As depicted, the particle transit time gradually decreases with increasing hydraulic pressure at the fracture inlet, exhibiting a nearly linear declining trend. This indicates that elevated water pressure at the fracture entrance significantly enhances the sediment transport capacity of the flow, thereby effectively improving particle transport efficiency.

4.2.2. Particle Transport Velocity

Figure 23 shows the variation characteristics of particle velocity under different hydraulic pressures. As observed in Figure 23a,b, the slopes of the velocity curves continuously increase with rising water pressure, leading to rapid acceleration of particle transport in both spatial and temporal dimensions. At high hydraulic pressures, particles continuously gain momentum input during transport, resulting in swift velocity increments where the acceleration effect manifests more prominently along the spatial path, demonstrating enhanced driving capacity and kinetic energy transmission efficiency.

Figure 23 illustrates particle velocity characteristics under varying hydraulic pressures. As shown in (a) and (b), the slopes of velocity curves progressively increase with rising water pressure, accelerating particle transport in both spatial and temporal dimensions. Under high pressures, particles continuously gain momentum, exhibiting rapid velocity growth that manifests most prominently along spatial trajectories—evidencing enhanced driving capacity and kinetic energy transmission efficiency. Furthermore, (c) quantifies the regulatory effect of hydraulic pressure on average particle velocity, revealing an approximately linear positive correlation. This highlights hydrodynamic conditions as the key driver of particle migration. Collectively, hydraulic pressure not only dictates the initiation intensity of particle transport but also governs sustained acceleration behavior and kinetic energy levels during migration, thereby serving as a critical control parameter for sand particle transport efficiency and dynamic behavior.

4.2.3. Coordination Number

Figure 24 and Figure 25 presents the evolution of particle coordination numbers under varying hydraulic pressures. As depicted in (a), the average coordination number exhibits a decreasing trend over time across all pressure conditions. During the initial phase, particles maintain high coordination numbers (approximately 3–4), indicating strong interparticle constraints and closely packed configurations. Subsequently, the coordination number decreases markedly, signifying progressive separation between particles. Higher hydraulic pressures accelerate this decline, demonstrating enhanced particle detachment rates.

4.2.4. Drag Force

Figure 26 illustrates the time-dependent variation of the average drag force on particles under different water pressure conditions. As shown in Figure 26a, higher water pressure significantly increases the drag force acting on particles. Notably, a distinct peak force is observed, with both its magnitude and time of occurrence directly correlating with pressure levels. This indicates stronger fluid–particle interactions and more rapid dynamic responses under elevated pressures. Figure 26b quantitatively analyzes the relationship between average particle drag force and inlet water pressure, revealing a significant positive correlation. These results demonstrate that increasing the inlet water pressure effectively enhances the hydrodynamic forces on particles, intensifies momentum exchange and fluid–solid coupling, and further promotes particle initiation and transport processes.

4.3. Effect of Fracture Aperture

4.3.1. Particle Transit Times and Mass Transfer

Figure 27 illustrates the variation in the proportion of particles passing through fractures over time under different fracture aperture conditions. From Figure 27a, it can be concluded that as the fracture aperture gradually increases from 10 mm to 25 mm, the cumulative passing rate curve of particles crossing the fracture outlet shifts rightward overall, indicating a significant increase in the time required for particles to traverse the fracture. Figure 27b further quantifies the relationship between the critical time for complete particle penetration and the fracture aperture. When the aperture increases from 10 mm to 15 mm, the penetration time remains relatively stable; however, as the aperture exceeds 20 mm, the required time exhibits a sharp increase followed by a plateau. This suggests a marked decline in particle migration efficiency under larger aperture conditions.

4.3.2. Variation Pattern of Particle Transport Velocity

As shown in Figure 28a,b, the velocity of sand particles progressively increases over time during migration. However, distinct differences exist in velocity evolution curves across aperture conditions. At an aperture of 10 mm, the average particle velocity consistently exceeds other aperture groups, exhibiting the maximum growth rate and highest terminal velocity. As the aperture increases—particularly beyond 20 mm—the velocity increment markedly decelerates, ultimately plateauing.

Figure 28c quantitatively compares the average migration velocities of particle populations. It reveals that as the aperture expands from 10 mm to 25 mm, particle velocity rapidly declines. Increased aperture fails to enhance particle migration rates; instead, dispersion of fluid kinetic energy and weakened directional propulsion collectively reduce the average particle velocity.

4.3.3. Variation Patterns of Coordination Number

Figure 29 shows the evolution of sand particle coordination numbers under varying fracture apertures. As shown in Figure 29a, the average coordination number of the particle population continuously decreases over time during migration. Initially, higher coordination numbers rapidly decline as particles migrate under fluid forces, causing gradual loosening of the granular structure. Crucially, larger apertures correspond to lower overall coordination levels and accelerated decline rates. At apertures ≥20 mm, coordination numbers remain persistently low, indicating significantly weakened interparticle contacts and cooperative interactions. Figure 29b quantifies the total coordination number across apertures. The results reveal that total coordination numbers decrease markedly with increasing aperture, with the most pronounced reduction occurring when apertures expand from 15 mm to 20 mm, followed by a subsequent plateau.

4.3.4. Variation Patterns of Drag Force

Figure 30 illustrates the evolution of the average drag force on particles during migration under different fracture apertures. As shown in Figure 30a, the average drag force for all aperture groups initially increases with time, reaching a peak before declining sharply. This pattern reflects the kinetic process of particles traversing the fracture outlet driven by fluid flow. Notably, as fracture aperture increases, both the overall magnitude and the peak intensity of the average drag force decrease significantly. The drag force observed for the 10 mm aperture group substantially exceeds that for groups with apertures of 20 mm or larger. Figure 30b further quantifies the average drag force across different apertures. The results reveal a rapid decline in drag force with increasing aperture; this trend plateaus at larger apertures. This behavior indicates that enlarged apertures disperse the fluid flow field, weakening the hydrodynamic force acting on an individual particle and consequently reducing the collective migration momentum and velocity of the particles. In summary, fracture aperture exerts a pronounced suppressing effect on particle drag force. Smaller apertures concentrate fluid power, enhancing the drag force on particles and improving migration efficiency. Conversely, larger apertures substantially reduce drag force due to fluid power dispersion, leading to diminished migration efficiency.

4.4. Effect of Particle Diameter

4.4.1. Particle Transit Times and Mass Transfer

Particle size significantly influences the migration of sand particles through fractures. Figure 31a depicts the cumulative percentage of particles traversing the fracture outlet as a function of time for different particle sizes. The figure clearly shows that the slope of the curve steepens with increasing particle size, while the transit time decreases substantially. Figure 31b further quantifies the relationship between transit time and particle size. The transit time exhibits a decreasing trend with larger particle sizes, indicating an inverse correlation between the two parameters. This phenomenon primarily stems from the greater fluid forces imposed on larger particles compared to smaller ones, resulting in swifter and smoother migration through the fracture channel. Consequently, larger particles demonstrate enhanced migration capacity.

4.4.2. Variation Pattern of Particle Transport Velocity

The experimental results demonstrate that particle size significantly influences particle velocity, exhibiting position-dependent patterns. Figure 32a reveals that within an extremely short temporal scale, particle velocity rapidly escalates to its peak. Both peak velocity and acceleration increase markedly with larger particle sizes, while larger particles exhibit a temporal lag in acceleration initiation, reflecting inertial effects on the acceleration process. Figure 32b delineates velocity variations at different X-positions. As particles migrate along the X-direction, their velocities display an overall increasing trend, with larger particles maintaining consistently higher velocities throughout the migration path. This indicates that hydrodynamic effects coupled with particle inertia confer dynamic advantages to larger particles. Concurrently, velocity fluctuations intensify with increasing particle size, manifesting complex fluid disturbance and interparticle interactions during migration. Figure 32c quantifies that mean particle velocity exhibits a nearly linear positive correlation with particle size, confirming that larger particles attain higher time-averaged velocities. Collectively, these findings validate the critical control of particle size on migration capability under constant hydraulic pressure. Larger particles achieve faster acceleration, higher velocities, and demonstrate enhanced dynamic persistence alongside more complex flow responses within fractures.

4.4.3. Variation Patterns of Drag Force

Figure 33a shows the temporal evolution of instantaneous time-averaged drag force for three particle sizes, demonstrating progressive force augmentation followed by post-peak fluctuations. Peak drag force for larger particles (0.00075 mm) exceeds 1.5 N—over 30 times greater than that of minimal-sized particles (0.00025 mm, <0.05 N)—confirming exceptional size sensitivity. (b) Nonlinear scaling of time-averaged drag force with particle radius results in larger radii inducing disproportionately amplified resistance due to enhanced interfacial shear and kinetic dominance.

4.5. Granular Navigability Quantification in Fractured Media

Particle migration through fractures is influenced by JRC, aperture width, hydraulic pressure, fracture opening, and particle diameter. We therefore develop a regression equation to assess granular traversability in rough fractures. The penetration time (denoted t_p) for 80% of particles negotiating fractures is adopted as the navigability metric. A multivariate polynomial expression is proposed to quantify penetration time, formulated through regression analysis using hydraulic pressure, fracture aperture, particle diameter, fracture length, and JRC. The following equation demonstrates universal applicability across all simulated scenarios, with coefficients consolidated in

Table 6. Summary of the numerical simulation results.

JCR	Hydraulic Pressure (MPa)	Fracture Aperture (mm)	Particle Diameter (mm)	Through Time (s)
7	2	10	0.0005	0.00725
11	2	10	0.0005	0.007
15	2	10	0.0005	0.00635
19	2	10	0.0005	0.0079
15	2	10	0.00025	0.00675
15	2	10	0.0005	0.00635
15	2	10	0.00075	0.0052
15	2	10	0.0005	0.0063
15	2	15	0.0005	0.0063
15	2	20	0.0005	0.00685
15	2	25	0.0005	0.0069
15	1	10	0.0005	0.00895
15	2	10	0.0005	0.0063
15	3	10	0.0005	0.00515
15	4	10	0.0005	0.00455

:

$$t_p=0.0272\cdot JR{C}^{-0.095}\cdot P^{-0.496}\cdot d^{0.069}\cdot r^{-0.186}\cdot l^{1.5}$$

(11)

Penetration time t (unit: seconds) quantifies the duration for particle groups to traverse fractures. JRC denotes the roughness coefficient of fractures. P represents hydraulic pressure (MPa). d is the aperture of fractures (m). r signifies the radius of particles (mm). l defines the length of fractures (m). The correlation coefficient of this formula is 0.782, indicating its effective capacity to reflect the penetration ability of particle groups through fractures. This moderate determination coefficient primarily arises from intrinsic uncertainties in particulate flows, including (1) stochastic turbulence effects that introduce randomness into particle–fluid interactions; (2) natural particle heterogeneity in shape, roughness, and density; and (3) multi-scale coupling limitations due to unresolved sub-grid phenomena and collective particle behaviors. For given fractures and particle groups, larger tp values correspond to lower penetration ability of particle groups, and vice versa. This equation finds broad application in geological and geotechnical fields, such as assessing the migration capacity of sand grains or drilling cuttings in fractures during oil–gas extraction; calculating grouting efficiency; and predicting transport capacity of fracturing proppants.

5. Conclusions

Based on the CFD-DEM coupled simulation method, this study systematically reveals the essential characteristics and core control mechanisms of sand particle transport dynamics in rough fractures under high-potential-energy water inrush conditions.

(1)

The research identifies a typical three-phase temporal evolution of sand particles driven by high-pressure turbulent flow: initial acceleration, linear stable progression, and terminal re-acceleration. Both the average particle velocity and predominant forces (drag force and pressure gradient, collectively contributing > 95%) exhibit a “rapid rise—plateau sustainment—rapid decay” pattern. The strong coupling among particle–fluid–wall (P–F–W) interactions is confirmed as the physical core driving collective particle transport.

(2)

Fracture structure (roughness JRC, aperture b), hydrodynamic conditions (water pressure P), and particle properties (radius r_p) demonstrate complex nonlinearly synergistic control over transport efficiency. An optimal JRC range (14–16) enhances transport velocity. A critical fracture aperture threshold (experimental value ≈8rp) exists—exceeding it disperses fluid kinetic energy and suppresses transport. Water pressure (P) serves as the key master control parameter for transport initiation and maintenance, with migration speed exhibiting a strong positive correlation (R > 0.93). Increasing particle size amplifies inertial thrust but significantly reduces coordination number (>40% decline) and weakens particle structural synergy, forming a competition mechanism.

(3)

Particle migration demonstrates multi-scale characteristics. At the collective level, it exhibits three dominant patterns—“bulk-advancing mode” (low JRC regions), “splitting-reaggregation cycles” (intermediate JRC regions), and “central stretching with boundary retention” (high JRC regions); at the individual level, particle trajectories governed by local flow structures are categorized into “major shear channelization” (high-flow zones), “boundary-vortex drifting” (eddy zones), and “wall-adhesion obstruction” (rough asperity areas). The coordination number dynamically decays during migration (over 60% decrease from peak values), clearly unveiling the transition of the particle system from “tightly packed” to “dispersed transport” and highlighting the intense spatiotemporal coupling effects of fluid–solid interactions.

(4)

A predictive model based on 80% particle breakthrough time (t₈₀, goodness-of-fit R² = 0.782) identifies critical impact factors. Increased water pressure substantially reduces t₈₀ (enhancing breakthrough efficiency); an optimal JRC range (14–16) exists; and fracture aperture (b) and particle radius (rp) cooperatively regulate breakthrough capacity. This model provides essential quantitative assessment tools and theoretical foundations for practical engineering applications, including water-inrush sand disaster prevention, precision grouting parameter design, and optimized proppant selection in hydraulic fracturing.

In conclusion, this research elucidates the complex physical mechanisms governing high-energy-water-inrush-induced sand migration in rough fractures and establishes a predictive framework for breakthrough capacity. The outcomes deliver significant theoretical guidance and engineering value for advancing the understanding of groundwater hydrodynamic hazards, optimizing engineering protection measures, and developing coupled multiphysics numerical models.