In the numerical simulation of water-sediment flow in fractures, three basic assumptions are proposed as follows: (1) water is incompressible, and its density is a constant; (2) the sediment particle is assumed to be a rigid sphere of fixed radius, without obvious damage; and (3) the flow rate is uniformly distributed around the cross section of the fracture inlet, and the fluid velocity of the discrete phase sediment particle is the same as that of the continuous phase particle.

Figure 1 shows the computational domains of the smooth and rough fractures. In Figure 1A, the water-sediment flow in the area is defined by two parallel smooth fracture surfaces. The distance between the two surfaces is h, and the length of the fracture is L. The projection of the upper and lower surfaces on the OX ₁ X ₂ section is a straight line. The boundary of the flow domain Ω comprises the inlet section O D 1 ¯ , the upper surface D 1 D 2 ¯ , the outlet surface D 2 D 3 ¯ , and the lower surface D 3 O ¯ . In Figure 1B, the water-sediment flow in the area is defined by two coincident surfaces. The projections of the upper and lower surfaces on the OX ₁ X section are broken lines, dividing the fracture surface into 50 equal broken line segments F L i u and F L i d , where i = 1 , ...50 . The boundary of the flow domain Ω consists of the inlet section O C 1 ¯ , upper surface ∪ i = 1 50 F L i u , outlet section C 2 C 3 ¯ , and lower surface ∪ i = 1 50 F L i d .

ANSYS Fluent 17.0 numerical simulation software was used to establish models of the water-sediment two-phase flow in smooth and rough fractures. Since the boundary of the smooth fracture model was relatively regular, structured grids were used for division in the ANSYS ICEM CFD. The evenly distributed grid nodes were arranged at the inlet to make the sediment particles uniformly distributed along the X ₂ direction. In addition, the Stress-Omega RSM turbulence model was used in this study. It was required that y ⁺ (the dimensionless distance to the wall) at the first layer of grid nodes near the wall was approximately 1; thus, fine grids were arranged. After calculation, y ⁺ was checked. To ensure the accuracy and efficiency of the calculation results, after the grid independence test, the boundary layer grid size was set as 0.005 mm, and the global grid size was 0.08 mm. Figure 2 shows the final division result.

Considering the severe bending of the wall boundary of the rough fracture model, hybrid grids were chosen for division in the meshing module on ANSYS Workbench, as shown in Figure 3. Multilayer structured grids were arranged in the boundary layer, and unstructured quadrilateral-dominant grids were used in other domains. After the division, y ⁺ was checked and a grid-independent test was conducted. The total number of the grids was 145,000, as shown in Figure 3A. Figure 3B shows the grids in the boundary layer.

The parameters were adjusted according to the results of the laboratory test, and the material properties were determined, as shown in Table 1. Similarly, the interaction parameters between the sediment particle and the wall were determined. The friction coefficient, normal restitution coefficient, and tangential restitution coefficient were 0.45, 0.2, and 0.9, respectively. The interaction parameters among sediment particles were fixed. The static friction coefficient, sliding friction coefficient, and restitution coefficient were 0.3, 0.2, and 0.05, respectively.

In this study, the sediment volume concentration, sediment particle size, and sediment mass density were taken as variables to investigate the evolution characteristics of the pressure gradient, mean velocity distribution, and turbulent kinetic energy distribution of the water-sediment two-phase flow under two fracture surface conditions. When a variable was used, the other two variables were fixed. Table 2 shows the specific values.

In particular, the inlet segment (X ₁ = 5 mm), the middle segment (X ₁ = 50 mm) of the smooth fracture surface, the bending segment (X ₁ = 5 mm), and the parallel segment (X ₁ = 50.5 mm) of the rough fracture surface were selected as the typical segments to comprehensively study the change laws of mean velocity distributions. Similarly, the inlet velocity was fixed as 0.869 m/s, the observation time node t was 0.3 s, and the number of variables was reduced.

Pressure gradient is one of the main parameters describing the seepage, which can reflect the pressure change along the flow direction. Figure 4 shows the absolute value of pressure gradient–sediment volume concentration (G _p –Ф) curves. In Figure 4A, when the water-sediment flowed in the smooth fracture, with a gradual increase of Ф, G _p decreased linearly from 6.51 kPa m⁻¹ to 6.39 kPa m⁻¹, decreasing by 1.72%. In Figure 4B, during the flow of water-sediment in the rough fracture, with a gradual increase of Ф, G _p first increased and then decreased. When Ф increased from 0 to 1.02%, G _p sharply decreased from 191.68 kPa m⁻¹ to 181.57 kPa m⁻¹, decreasing by 5.27%. When Ф increased from 1.02 to 4.06%, G _p gradually increased from 181.57 kPa m⁻¹ to 185.08 kPa m⁻¹, increasing by 1.93%. Through comparisons, it can be found that under the same conditions, the absolute value of the pressure gradient of the water-sediment flow in the rough fracture was about 40 times that in the smooth fracture. In addition, the change characteristics of the absolute value of the pressure gradient with the volume concentration of sediment particles were different under different fracture conditions. It indicates that the fracture surface morphology affects the influence of the volume concentration of sediment particles on the pressure gradient.

Figure 5 shows the absolute value of the pressure gradient–sediment particle size (G _p -D _p ) curves. In Figure 5A, during the water-sediment flow in the smooth fracture, G _p gradually decreased in a nonlinear form with the increase of D _p . As D _p increased from 0 to 0.12 mm, G _p quickly decreased from 6.51kPa m⁻¹ to 6.34 kPa m⁻¹, decreasing by 2.61%. In Figure 5B, during the water-sediment flow in the rough fracture, G _p first decreased, then increased, and decreased again. As D _p increased from 0 to 0.12 mm, G _p rapidly decreased from 191.68 kPa m⁻¹ to 171.76 kPa m⁻¹, decreasing by 10.39%. It can be found that during the water-sediment flow in the rough fracture, when the sediment particle size is small, the pressure loss increases with the increase of the particle size; when the sediment particle size is relatively large, the pressure loss decreases with the increase of the particle size. Through the comparisons, it can be found that the absolute value of the pressure gradient varies with the change of the sediment volume concentration under two types of fractures. It proves that the surface morphology of fractures affects the influence of sediment particle size on the pressure gradient.

Figure 6 shows the absolute value of the pressure gradient–sediment particle mass density (G _P -ρ _p ) curves. In Figure 6A, when the water sediment flowed in the smooth fracture, with the increase of ρ _p , G _P decreased in an approximately linear form. As ρ _p increased from 0 kg/m³ to 4,500 kg/m³, G _P rapidly decreased from 6.51kPa m⁻¹ to 6.33 kPa m⁻¹, decreasing by 2.84%. In Figure 6B, during the water-sediment flow in the rough fracture, G _P was smaller than that in the single-phase flow, and it first increased and then decreased with the increase of ρ _p . When ρ _p was 4,500 kg/m³, the minimum G _P was obtained. Through the comparison, it is found that the absolute value of the pressure gradient during the water-sediment flow in the rough fracture is about 30 times that in the smooth fracture under the same conditions. In addition, the absolute value of the pressure gradient varies with the sediment particle mass density. It indicates that the surface morphology of the fracture affects the impact of sediment particle mass density on the pressure gradient.

In summary, under the conditions of two types of surface fractures, the absolute value of the pressure gradient with different sediment particle volume concentrations, sediment particle sizes, and sediment particle mass density in the two-phase flow was smaller than that in the single-phase flow. Under the smooth fracture surface, the absolute value of the pressure gradient changed linearly with the change of sediment particle volume concentrations, sediment particle sizes, and sediment particle mass density, while this value has different changing trends under the rough fracture.

The mean velocity distribution is one of the important technical indicators for the study of seepage problems. Figure 7 and Figure 8 show the mean velocity distributions of the continuous-phase fluid on typical cross sections of smooth and rough fractures under various sediment volume concentrations. In Figure 7A, at a cross section of X ₁ = 5 mm, the water-sediment fluid was not fully developed on the smooth fracture surface. In Figure 7B, the flow became fully developed. Through the comparison, it can be found that the mean velocity distributions on the two sections rarely change with the sediment volume concentration Ф, and they were symmetrically distributed along the center line of X ₂ = 0.9 mm.

In Figure 8, when the water sediment flowed in the rough fractures, the mean velocity distributions of the continuous-phase fluid presented remarkable differences. In Figure 8A, v showed the asymmetric M-shaped distribution at the cross-section of X ₁ = 50 mm, with two extreme points, and the peak values were within 1.2 mm ≤ X ₂ ≤ 1.5 mm. In Figure 8B, there were multiple extreme points at the cross section of X ₁ = 50.5 mm, and the peak values were near the center line X ₂ = 1.4 mm. Compared with other positions, v has the greatest change with Ф near the wall.

As shown in Figure 7 and Figure 8, v of the continuous-phase fluid was dramatically different under two types of fracture conditions. The maximum v in the rough fracture was about three times that in the smooth fracture. In addition, the distribution curves are different. During the continuous-phase fluid flow in the smooth fracture, v-X ₂ curves are rectangular or semi-sine shaped, while they are M-shaped during the continuous-phase fluid flow in the rough fracture.

Figure 9 and Figure 10 show the mean velocity distribution of the continuous-phase fluid on the typical cross sections of the smooth and rough fractures under different sediment particle sizes. In Figure 9A, when the water sediment flowed in the smooth fracture, the fluid was not fully developed at the cross section of X ₁ = 5 mm. In Figure 9B, the fluid became fully developed at the cross section of X ₁ = 50 mm. It can be observed that v rarely changed with D _p under two types of cross sections and the distribution of v was symmetrical along the center line of X ₂ = 0.9 mm.

Figure 10 shows that when the water sediment flowed in the rough fracture, v of the continuous-phase fluid was greatly influenced by D _p . Different v-X ₂ curves varied significantly. In Figure 10A, on the cross section of X ₁ = 50 mm, v first increased and then decreased with the increase of X ₂, and the maximum value was between 1.2 and 1.5 mm. In Figure 10B, at the cross section of X ₁=50.5mm, v also first increased and then decreased with the increase of X ₂. In addition, there was an upward fluctuation at X ₂ = 2.1 mm, and the peak value was within 1.2 mm ≤ X ₂ ≤ 1.5 mm.

Figure 11 and Figure 12 show the mean velocity distribution of the continuous-phase fluid on the typical cross sections of the smooth and rough fractures under different sediment particle mass densities, respectively. In Figure 11, at the cross sections of X ₁ = 5 mm and X ₁ = 50 mm, v rarely changed with the sediment particle size, and v was symmetrically distributed along the center line of X ₂ = 0.9 mm.

In Figure 12, v was greatly affected by the sediment particle size during the flow in the rough fracture, and multiple extreme points can be observed. On the section of X ₂ = 50 mm, v of the fluid particle at each position was significantly affected by the sediment volume concentration, and the maximum v was in the range of 1.2 mm ≤ X ₂ ≤ 1.5 mm. On the section of X ₂ = 50.5 mm, the peak value was near the center line of X ₂ = 1.4 mm.

In summary, in the smooth fracture, the changes of the sediment particle volume concentration, sediment particle mass density, and sediment particle size rarely affect the mean velocity distribution of the continuous-phase fluid, while these influencing factors significantly affect the mean velocity distribution of the continuous-phase fluid in the rough fracture.

The turbulent kinetic energy is an indicator to measure the development and decline of turbulence. Figure 13 and Figure 14 show the distribution of turbulent kinetic energy of the continuous-phase fluid on typical cross sections of smooth and rough fractures at various sediment particle volume concentrations. As shown in Figure 13A, on the section of X ₁ = 5 mm in the smooth fracture, k _t first increased and then decreased with X ₂, and k _t reached the maximum at X ₂ = 0.9 mm. The distribution curves of k _t are symmetrical along with X ₂ = 0.9 mm under various Ф. In particular, within X ₂ = 0.15–0.45 mm and X ₂ = 1.35–1.65 mm, there was a significant negative correlation between Ф and k _t . It indicates that the movement of sediment particles can inhibit the turbulent kinetic energy of the continuous-phase fluid. However, on the section of X ₁ = 50 mm, the k _t –X ₂ curves are M-shaped, as shown in Figure 13B. Differing from the situation on the section of X ₁ = 5 mm, k _t reached a minimum value at X ₂ = 0.9 mm. At this point, Ф affected k _t in the entire X ₂ interval, namely, the greater the Ф, the smaller the k _t . It can be observed that the distribution of k _t –X ₂ curves was approximately symmetrical along X ₂ = 0.9 mm.

As presented in Figure 14, the distribution of k _t on the cross section of the rough fracture can be greatly affected by the sediment particle volume concentration Ф, which was manifested as the deviation of the extreme point position. In Figure 14A, on the section of X ₁ = 50 mm, k _t fluctuated in the interval of 0 mm ≤ X ₂ ≤ 2 mm and gradually decreased within 2 mm ≤ X ₂ ≤ 2.8 mm. In Figure 14B, on the section of X ₁ = 50.5 mm, k _t gradually increased in the interval of 0.5 mm ≤ X ₂ ≤ 0.8 mm and fluctuated in the interval of 0.8 mm ≤ X ₂ ≤ 2.3 mm. It can be observed that the maximum k _t was in the interval of 1.0 mm ≤ X ₂ ≤ 1.3 m on both sections. In addition, the fluctuations of k _t can be found on both sections, indicating that the turbulence intensity was higher in the rough fracture.

As shown in Figure 13 and Figure 14, the turbulent kinetic energy of the fluid in the rough fracture was an order of magnitude higher than that in the smooth fracture under the same conditions. The distribution of fluid turbulent kinetic energy in the X ₁ direction was numerically different, and the distribution curves were completely different during the water-sediment flow in the smooth and rough fractures. The impacts of the sediment volume concentration in the smooth fracture on the continuous-phase fluid turbulent kinetic energy had significant laws, while no obvious laws were observed in the rough fracture.

Figure 15 and Figure 17 show the distribution of turbulent kinetic energy of the continuous-phase fluid on typical cross sections of smooth and rough fractures under various sediment particle sizes. In Figure 15A, on the section of X ₁ = 5 mm of the smooth fracture, k _t first increased and then decreased with X ₂, and the maximum k _t was obtained at X ₂ = 0.9 mm. The sediment particles significantly reduced k _t in the intervals of 0.15 mm ≤ X ₂ ≤ 0.45 and 1.35 mm ≤ X ₂ ≤ 1.65 mm, but the influence laws were different. In the interval of 0.15 mm ≤ X ₂ ≤ 0.45, the distribution curves of k _t were basically consistent, and D _p had no obvious influence on k _t , while in the interval of 1.35 mm ≤ X ₂ ≤ 1.65 mm, k _t gradually decreased with the decrease of D _p . When D _p was 0.01 mm, the distribution curve of k _t was symmetrically distributed along X ₂ = 0.9 mm. When the values of D _p were 0.04, 0.08, and 0.12 mm, the distribution curves of k _t were asymmetrical along X ₂ = 0.9 mm. On the section of X ₁ = 5 mm, k _t was slightly affected by D _p in intervals other than X ₂ = 0.15–0.45 mm and X ₂ = 1.35–1.65 mm, and the curves were relatively consistent.

In Figure 15B, on the section of X ₁ = 50 mm, k _t changed with X ₂ in an M shape and reached the minimum at X ₂ = 0.9 mm. At this point, D _p had effects on k _t in the entire X ₂ interval, and k _t gradually decreased with the decrease of D _p . The influence laws of D _p on k _t were different in the intervals of X ₂ ≤ 0.9 mm and X ₂ ≥ 0.9 mm. In the interval of X ₂ ≤ 0.9 mm, the distribution curves of k _t were nearly consistent with various D _p , indicating that D _p had no obvious effect on k _t . However, in the interval of X ₂ ≥ 0.9 mm, k _t gradually reduced with the decrease of D _p . When D _p was 0.01 mm, the distribution curve of k _t was symmetrical along with X ₂ = 0.9 mm, and the curves are asymmetrical with a D _p of 0.04, 0.08, and 0.12 mm.

To illustrate the cause of the asymmetry in Figure 15, the distributions of sediment particles in fractures were given, as shown in Figure 16. The size of the sediment particle was doubled, and the concentration was diluted by one-tenth in order to clearly show the distribution of particles in the fracture. It can be observed that the sediment particles of 0.01 mm had no contact with the upper and lower walls of the fracture and were distributed symmetrically along X ₂ = 0.9 mm. The sediment particles of 0.04, 0.08, and 0.12 mm were gradually shifted downward under the action of gravity, and some of them settled on the bottom wall. This is because the sediment particles with smaller sizes have better flowability and are not easy to settle under the action of gravity, while the sediment particles with larger sizes have larger Stokes numbers and are easily affected by gravity. According to the aforementioned analysis, the sediment particles of 0.01 mm were symmetrically distributed along with X ₂ = 0.9 mm, and the turbulent kinetic energy of the fluid subjected to them was also symmetrically distributed along X ₂ = 0.9 mm. The sediment particles of 0.04, 0.08, and 0.12 mm were asymmetrical along with X ₂ = 0.9 mm and so was the turbulent kinetic energy.

In Figure 17, the turbulent kinetic energy distribution of the continuous-phase fluid on the cross section of the fracture was greatly affected by sediment particle size, which was manifested as the deviation of the extreme points. On the section of X ₁ = 50 mm, k _t was relatively large in the interval of X ₂ ≤ 2 mm, while it was smaller in the interval of X ₂ ≥ 2 mm. On the section of X ₁ = 50.5 mm, k _t was relatively small when X ₂ ≤ 0.8 mm, and it was larger when X ₂ ≥ 0.8. The distribution of the fluid turbulent kinetic energy was disordered on the aforementioned two cross sections, indicating that the fluid pulsation was severe in the rough fracture and the turbulence intensity was high. Through the comparison of Figure 15 and Figure 17, it is found that the impacts of the sediment particle size on the fluid turbulent kinetic energy are completely different in smooth and rough fractures, and the difference is nearly an order of magnitude.

Figure 18 and Figure 19 show the distribution of turbulent kinetic energy of the continuous-phase fluid on typical cross sections of the smooth fracture and rough fracture under various sediment mass densities.

In Figure 18A, on the section of X ₁ = 5 mm of the smooth fracture, k _t first increased and then decreased and reached the maximum when X ₂ = 0.9 mm. Obviously, the effects of the sediment volume concentration on k _t were very significant in the intervals of 0.15 mm ≤ X ₂ ≤ 0.45 and 1.35 mm ≤ X ₂ ≤ 1.65 mm, where k _t decreased with the increase of ρ _p . It was indicated that the movement of sediment particles can inhibit the turbulent kinetic energy with the increase of ρ _p . During the single-phase flow, when ρ _p was 1,500 kg/m³, k _t was distributed symmetrically along X ₂ = 0.9 mm. When the values of ρ _p were 2650 kg/m³, 3500 kg/m³, and 4,500 kg/m³, k _t in the interval of 0.15 mm ≤ X ₂ ≤ 0.45 mm was smaller than that in the interval of 1.35 mm ≤ X ₂ ≤ 1.65 mm, indicating that the sediment particle in the interval of 0.15 mm ≤ X ₂ ≤ 0.45 mm had a greater impact on the turbulent kinetic energy. This is because the sediment particles are more affected by gravity as the mass density increases, and they are easily deposited on the lower wall. In the intervals except for 0.15 mm ≤ X ₂ ≤ 0.45 and 1.35 mm ≤ X ₂ ≤ 1.65 mm, the distribution curves of k _t were basically consistent and were symmetrical along X ₂ = 0.9 mm. It indicated that k _t was slightly affected by the sediment particle mass density in these intervals. In Figure 18B, on the section of X ₁ = 50 mm, k _t - X ₂ curves were M-shaped, and ρ _p affected k _t in the whole X ₂ interval. The larger the ρ _p , the smaller the k _t . In the single-phase flow, when ρ _p was 1,500 kg/m³, k _t was symmetrically distributed along X ₂ = 0.9 mm. When the values of ρ _p were 2650 kg/m³, 3500 kg/m³, and 4,500 kg/m³, k _t under X ₂ ≤ 0.9 mm was smaller than that under X ₂ ≥ 0.9 mm. It indicates that the sediment particles with the aforementioned mass densities have a greater influence on the turbulent kinetic energy in the interval of X ₂ ≤ 0.9 mm. This is because the sediment particles are more affected by gravity and easily deposited on the lower wall, as the mass density of sediment particles increases.

In Figure 19, the distribution of k _t was greatly affected by ρ _p during the water-sediment flow in the rough fracture. The extreme points were shifted. On the section of X ₁ = 50 mm, k _t was larger in the interval of X ₂ ≤ 2 mm than that in the interval of X ₂ ≥ 2 mm. On the section of X ₁ = 50.5 mm, k _t was smaller when X ₂ ≤ 0.8 mm. It can be observed that the distribution of k _t was disordered on both the cross sections, suggesting that the fluid pulsation was violent in the rough fracture and the turbulence intensity was high.

In summary, under the conditions of the same sediment mass density, same sediment size, same sediment volume concentration, and same inlet velocity, the influence of sediment mass density on fluid turbulent kinetic energy exhibits completely different laws in smooth and rough fractures, with the difference of nearly an order of magnitude.