Simulating particle settling in inclined narrow channels with the unresolved CFD-DEM method

Acceleration of particle settling in inclined containers, known as the Boycott effect, is a common natural phenomenon, which can significantly affect the particle conveying process. Particularly for proppant transport constrained in narrow channels, physical mechanisms related to this effect remain undetermined. In this study, an Eulerian-Lagrangian method, i.e., the unresolved computational fluid dynamics–discrete element method, has been adopted to simulate the particle settling process in inclined narrow channels, with an aim to deepen the understanding of the Boycott effect on proppant settling. The simulation results illustrate that two types of granular-induced instabilities dominate the sediment settling process in inclined fractures, which are analogous to Kelvin-Helmholtz instability and Rayleigh-Taylor instability, respectively. Due to the instabilities, spatial inhomogeneity of the particle phase, i.e., particle clustering, is clearly observed. Based on parametric studies, it is proven that there exists a critical inclination angle where the acceleration ratio reaches a maximum, which ranges from 2 to 6 in this work. Moreover, the critical angle can shift between 60 ° and 75 ° as physical properties change, such as fluid viscosity, particle size, and particle density.

Figure 1

Illustration of particle settling in an inclined vessel (a) and in an inclined fracture or narrow channel (b). $L, W$, and $H$ denote horizontal spreading length, transverse width, and longitudinal height of vessel or fracture in the $x, y$, and $z$ directions, respectively. $\vec{g}$ is the gravity in the vertical direction, and $\theta$ is the inclination angle. Note that $\theta =\pi /2$ indicates a rigorously vertical state.

Figure 2

Flowchart of the unresolved CFD-DEM framework.

Figure 3

Illustration of particle-packed porous medium (a) and comparison results of friction factor $f^{*}$ versus modified particle Reynolds number ${\mathrm{Re}}_p^{*}$ between the Ergun equation and the present work (b).

Figure 4

Images of sediment transport of four cases. (a), (b): bed-load regime; (c), (d): suspended-load regime. The direction of gravity is $-y$, and the downstream direction is $+z$.

Figure 5

Nondimensional transport rate versus shield stress in sediment transport tests. The red solid line is Wong and Parker's [49] empirical curve describing the bed-load regime, and the blue solid line is Nielsen's [50] empirical curve describing the suspended-load regime. The black square dots indicate numerical results of the present work, and the red circle and blue diamond are the numerical results of CFD-DEM methods in previous studies [31, 32].

Figure 6

Illustration of the relationship between field-scale (left) and mesoscale (right) study for proppant transport. The blue shaded region indicates the packed sand bed, and the gray shaded region indicates the suspended slurry.

Figure 7

Images of proppant settling in an inclined fracture of the case $\theta ={75}^{\circ }$ (plotted on the $x\text{−}z$ plane). Proppant particles are scattered as black points.

Figure 8

Phase-averaged settling velocity curve in an inclined fracture of the case $\theta ={75}^{\circ }$. The curve can be divided into three stages over time: stratified acceleration stage (stage I); resuspended stage (stage II); and secondary instability stage (stage III).

Figure 9

Different stages of development for granular-induced instability. (a), (b) stratified acceleration stage (I); (c), (d) resuspended stage (II); (e) secondary instability stage (III). (a)–(c) are presented on the $y\text{−}z$ plane, and (d), (e) are presented on the $x\text{−}z$ plane. Red and black arrows indicate velocity vectors of the fluid phase and the particle phase, respectively.

Figure 10

Images of proppant settling in inclined fractures of various inclination angles (base case).

Figure 11

Correlation curve between ensemble-averaged settling velocities and inclination angles.

Figure 12

Particle distributions of $y$-direction coordinates (a) and settling velocity (b).

Figure 13

Images of proppant settling in inclined fractures of various inclination angles: (a)–(d) for cases $\mathrm{Ga}=145.6$; (e)–(h) for cases $\mathrm{Sr}=0.0667$; (i)–(l) for cases $\mathrm{Sg}=1.5$.

Figure 14

Curves of ensemble-averaged settling velocity (a) and acceleration ratio (b) against inclination angles for four groups of cases. For the base case, the dimensionless numbers are $\mathrm{Ga}=1456, \mathrm{Sr}=0.1$, and $\mathrm{Sg}=2.65$.