Generalized Shields criterion for weakly cohesive granular materials

The erosion of natural sediments by a superficial fluid flow is a generic situation in many usual geological or industrial contexts. However, there is still a lack of fundamental knowledge about erosional processes, especially concerning the role of internal cohesion and adhesive stresses on issues such as the critical flow conditions for the erosion onset or the kinetics of soil mass loss. This contribution investigates the influence of cohesion on the surface erosion by an impinging jet flow based on laboratory tests with artificially bonded granular materials. The model samples are made of spherical glass beads bonded either by solid bridges made of resin or by liquid bridges made of a highly viscous oil. To quantify the intergranular cohesion, the capillary forces of the liquid bridges are here estimated by measuring their main geometrical parameters with image-processing techniques and using well-known analytical expressions. For the solid bonds, the adhesive strength of the materials is estimated by direct measurement of the yield tensile forces and stresses at the particle and sample scales, respectively, with specific traction tests developed for this purpose. The proper erosion tests are then carried out in an optically adapted device that permits a direct visualization of the scouring process at the jet apex by means of the refractive index matching technique. On this basis, the article examines qualitatively the kinetics of the scour crater excavation for both scenarios, namely, for an intergranular cohesion induced by either liquid or solid bonds. From a quantitative perspective, the critical condition for the erosion onset is discussed specifically for the case of the solid bond cohesion. In this respect, we propose here a generalized form of the Shields criterion based on a common definition of a cohesion number from yield tensile values, derived at both micro- and macroscales. The article finally shows that the proposed form manages to reconcile the experimental data for cohesive and cohesionless materials, the latter in the form of the so-called Shields curve along with some previous results of the authors which have been appropriately revisited.

Figure 1

(a) Capillary bridge of Ucon oil between two borosilicate glass beads of diameter $d=3$ mm with a Ucon oil mass content ${\zeta }_u=0.5\%$. (b) Solid bridge of dried polyurethane resin between two borosilicate glass beads of diameter $d=3$ mm with an initial resin mass content ${\zeta }_r=0.2\%$.

Figure 2

Drying process at ambient conditions from liquid to solid bond made of a commercial resin (Syntilor), with initial resin mass content ${\zeta }_r=0.1\%$, between two beads of diameter 7 mm at successive times $t=0$, 4, and 8 min shown from left to right.

Figure 3

Typical capillary bridge profile deduced from a picture (similar to the left one in Fig. 1) after some image processing using Image J. This figure defines the radius of curvature of the meniscus $r_1$, the radius of the capillary bridge $r_2$, the half-filling angle $\varphi$, and the contact angle $\theta$.

Figure 4

Mean capillary force due to a liquid bridge of Ucon oil between two 3 mm borosilicate glass spheres as a function of the Ucon oil mass content ${\zeta }_u$. The mean capillary force weakly decreases with ${\zeta }_u$ but remains roughly constant in this range of Ucon oil mass content, with a median value $F_{\text{cap}}=0.425\pm 0.013$ mN. The solid line stands for the median value, while the light gray rectangle depicts the overall error domain.

Figure 5

(a) Sketch of the traction test setup between two bonded beads. (b) Pictures of the breakage of a solid resin bond between two borosilicate beads, featuring larger beads than those used in the actual tests for an improved visibility of the bond rupture (here 7 mm in diameter, just as in Fig. 2).

Figure 6

Typical force measurement recorded during a test and enabling the determination of the bond tensile yield force $F_t^{*}$ as depicted on the curve.

Figure 7

Yield tensile force $F_t^{*}$ measured from the microscale traction test as a function of the initial resin mass content ${\zeta }_r$ for silicate beads with $2.85<d<3.30$ mm. The dashed line is a proportional relation $F_t^{*}=A^F{\zeta }_r$ with $A^F=467$ mN (and ${\zeta }_r$ in $\%$) while the solid line stands for the stretched exponential law in Eq. (10) with $B^F=85.6$ mN and ${\zeta }_r^F=9.0\times {10}^{-2}$ (in percent by mass).

Figure 8

Sketch of the traction device at the sample scale.

Figure 9

Yield tensile stress ${\tau }_t^{*}$ measured from the macroscale traction test as a function of the initial resin mass content ${\zeta }_r$ for silicate beads with $0.75<d<1.0$ mm. The dashed line is a proportional relation ${\sigma }_t^{*}=A^{\sigma }{\zeta }_r$ with $A^{\sigma }=4.95$ kPa (and ${\zeta }_r$ in percent) while the solid line stands for the stretched exponential relation in Eq. (12) with $B^{\sigma }=879$ Pa and ${\zeta }_r^{\sigma }=9.8\times {10}^{-2}$ (in percent by mass).

Figure 10

Sketch of the Jet Erosion Test device adapted for refractive-index-matching visualization.

Figure 11

Time evolution of the nondimensional crater depth $\frac{\delta x}{\delta x_f}$ during a JET test with an injection velocity $U_J=8.2\text{m}{\text{s}}^{-1}$ for borosilicate glass beads $d=3\pm 0.02$ mm mixed with an initial resin mass content ${\zeta }_r=0.1\%$ (in black) and $U_J=1.5\text{m}{\text{s}}^{-1}$ for the same beads mixed with Ucon oil at the same liquid mass content ${\zeta }_u=0.1\%$ (in blue).

Figure 12

Evolution of the crater depth $\delta x$ in a logarithmic timescale for different samples of borosilicate beads ($d=3\pm 0.02$ mm), with five varying Ucon oil mass contents ${\zeta }_u$, tested by RIM-JET with an injection velocity $U_J=1.5\text{m}{\text{s}}^{-1}$. A $t+1$ timescale is used in order to plot the initial state $\delta x=0$ on the graph. The mean final crater depth obtained for a similar sample without cohesion ($\delta x_f=53.6$ mm) is also represented with a dashed line, while the gray area marks the uncertainty range due to fluctuations. The picture is an illustration obtained for ${\zeta }_u=0.2\%$.

Figure 13

(a) Final crater depth $\delta x_f$ normalized by the bead diameter $d$ and plotted as a function of the Ucon oil mass content ${\zeta }_u$. The error bar of the test without cohesion represents the dispersion of $\delta x_f$, and the one of the cohesive samples represents one bead diameter. The dashed line stands for the mean value of $\delta x_f$ with viscous bonds, $\delta x_e=54.5$ mm. (b) Characteristic scouring time $t_{sc}$ vs Ucon oil mass content ${\zeta }_u$, where $t_{sc}$ is defined as the time needed to reach $60\%$ of the final crater depth. The dashed line is only a guide for the eye emphasizing the increase of $t_{sc}$ with ${\zeta }_u$.

Figure 14

Critical injection velocity $U_J^{*}$ as a function of the initial resin mass content ${\zeta }_r$ for borosilicate beads with $d=3\pm 0.02$ mm ($•$) and silicate beads with $2.85<d<3.30$ mm ($\blacksquare$) and $0.75<d<1.0$ mm ($▴$). Error bars are not shown here since they are smaller than the markers' size.

Figure 15

Critical shear stress ${\tau }_m^{*}$ exerted by the impinging flow at erosion onset as a function of (a) the initial resin mass content ${\zeta }_r$ and (b) the combination ${{\zeta }_r}^{3/2}{d}^2$, for borosilicate beads with $d=3\pm 0.02$ mm ($•$) and silicate beads with $2.85<d<3.30$ mm ($\blacksquare$) and $0.75<d<1.0$ mm ($▴$). The dashed lines stand for power-law relationships: ${\tau }_m^{*}\propto {{\zeta }_r}^{3/2}$ (a) and ${\tau }_m^{*}\propto {{\zeta }_r}^{3/2}{d}^2$ (b).

Figure 16

Time evolution of the crater depth $\delta x$ within a sample of borosilicate glass beads (diameter $3\pm 0.02$ mm) with an initial resin mass content ${\zeta }_r=0.2\%$. Successive incremental steps of injection velocity $U_J$ are implemented, from 9.45 $\text{m}{\text{s}}^{-1}$ to 10.16 $\text{m}{\text{s}}^{-1}$.

Figure 17

JET test on the sample of borosilicate glass beads (diameter $3\pm 0.02$ mm) with an initial resin mass content ${\zeta }_r=0.2\%$, with an injection velocity $U_J=10.16\text{m}{\text{s}}^{-1}$. The images in the sequence are taken 7 s (a), 8 s (b), 14 s (c), and 41 s (d) after the increment of velocity.

Figure 18

Evolution of the crater depth $\delta x$ during a time-linear increase of the jet velocity and plotted versus the instantaneous jet Reynolds number ${\text{Re}}_j$. The tests were carried out with samples of borosilicate glass beads (diameter $3\pm 0.02$ mm) without cohesion (black curve), ${\zeta }_r=0\%$, and with initial resin mass contents ${\zeta }_r=0.05\%$ (light blue curve), ${\zeta }_r=0.10\%$ (blue curve), and ${\zeta }_r=0.20\%$ (dark blue curve).

Figure 19

(a) Critical values at erosion onset of the classical Shields number ${\text{Sh}}_0^{*}$ (closed symbols) and the generalized Shields number ${\text{Sh}}^{*}$ (open symbols), with $\alpha =2.26$ [see Eq. (17)], obtained for borosilicate beads with $d=3\pm 0.02$ mm (blue circles) and silicate beads with $2.85<d<3.30$ mm (black squares) and $0.75<d<1.0$ mm (red triangles). The cohesionless values extracted from Ref. [48] (dark yellow stars) and the implicit formulation of the Shields curve by Guo [18] (solid line) are also shown for comparison. (b) Plot of $({\text{Sh}}_0^{*}/{\text{Sh}}^{*}-1)$ versus the cohesion number Co for all the data with cohesion. The line stands for a linear regression with a slope $\alpha =2.26$ ($R^2=0.813$).

Figure 20

Critical Shields number ${\text{Sh}}_0^{*}$ versus shear particle Reynolds number ${\text{Re}}_{\tau }$ for $a=0.30$ (squares), $a=0.37$ (up triangles), and $a=0.23$ (down triangles). The solid line stands for the Shields curve given by the explicit formulation from Guo [18].