Particle size segregation and diffusion in saturated granular flows: implications for grain sorting in debris flows

. Sorting of rocks, boulders, and silt/sand-sized particles according to their size is a characteristic feature of debris flow deposits and is an active process during flow which significantly affects the mobility. The degree at which size sorting occurs in debris flows depends on the relative magnitudes of granular processes such as particle size segregation and diffusion. Since debris flows are fluid-saturated phenomena, accurate modelling of size sorting requires the understanding of the influence of fluids on these processes, which have not been systematically studied. Here, we present simulation results and the associated empirical expressions for particle size segregation and diffusion which take into account fluid effects due to buoyant and drag forces. These expressions are developed through scaling analysis of data obtained from coupled granular-fluid simulations of saturated bi-disperse mixtures under simple shear. We further show that using these scaling relationships, an existing segregation-diffusion continuum equation can be extended to model particle sorting in debris flows with various types of fluids.


Particle size sorting in debris flows
Vertical size sorting is an important process in debris flows [1] wherein differently-sized particles separate into distinct layers [2]. This process is active during the flow and influences its mobility [3] and deposit morphology. In dry cases, this particle re-arrangement depends on the relative magnitude of granular processes such as particle size segregation and diffusion [4]. Size segregation drives smaller particles to rapidly percolate downwards towards the base and larger particles to slowly rise upwards due to persistent frictional contacts between particles [5]. On the other hand, diffusion due to random collisions in the granular flow drives the segregated layers to re-mix.
However, the segregation-diffusion behaviour is more complicated in debris flows, which are fluidsaturated phenomena involving complex solid-fluid interactions [6,7]. Fig. 1a demonstrates that debris flow experiments having water as the interstitial fluid exhibits a clear separation of large (free surface) and fine grains (base), whereas Fig. 1b shows that size sorting in muddy debris flow deposits in the Jiangjia Gully, China is not evident. Although the difference in these scenarios may also be attributed to other processes, better understanding of how fluids affect size segregation and diffusion will elucidate this problem and enhance the modelling of size sorting in debris flows.
Here, we investigate the effects of different types of fluids that are relevant to debris flows, characterized by * Corresponding author: gordon@imde.ac.cn density and viscosity, on particle size segregation and diffusion. Expressions incorporating these fluid effects are developed based on scaling analysis of granularfluid simulation results of sheared bi-disperse mixtures saturated in an ambient fluid. We further demonstrate that by substituting these empirical formulas into an existing segregation-diffusion continuum equation, it is possible to predict the temporal evolution of particle size distributions in saturated granular flow simulations.

Numerical simulation of saturated shear flows
Size sorting in debris flows (or the lack thereof) is studied using numerical simulations of granular shear flows saturated in various fluids. This is done by coupling the computational fluid dynamics (CFD) and discrete element method (DEM). The CFD and DEM are implemented using two independent open-source solvers which communicate with each other after a set number of timesteps through a message passing algorithm. Particles and fluid interact with each other through buoyant and drag forces. A detailed description of the CFD-DEM can be found in [8]. Fig. 1c shows the simulation setup with its dimensions. The DEM phase is a granular bed composed of large and small spherical particles, with average diameters = 0.01m and = 0.005m (size ratio of 2) respectively. The particles are bounded by rough walls in the vertical direction. The Hertz contact model is used to simulate particle interactions with Young's modulus 5 × 10 7 Pa, Poisson's ratio 0.4, friction coefficient 0.5, and the coefficient of restitution 0.8. The particles are initially normallygraded, i.e., large particles are initially positioned at the base while small particles are at the top. The CFD phase is an incompressible, Newtonian fluid evenly discretized into cells wherein 5 large particles would fit. Periodic boundaries are along the streamwise ( ) and spanwise directions ( ) in both the CFD and DEM.
The top particle wall imposes a constant confining pressure 0 and at the same time drives the particles to flow by translating at a constant velocity = , where ̇ is an imposed shear rate and is the vertical position of the wall which slightly changes with time due to the dilation and contraction of the granular bed. As the granular bed is sheared by the top wall, the ambient fluid becomes sheared along with it. Drag forces acting opposite to the streamwise flow can result in non-linear flow velocity profiles. To ensure that a constant ̇ is maintained along the flow height and reduce the localized shear effects [9], a small stabilizing force = (̇− ( )), is applied to each particle at a height having a streamwise velocity at each timestep [10].

Fluids are varied by setting different fluid viscosities
and densities for the fluid domain. Changes in and represent different concentrations of suspended fine particles in the fluid phase [1]. Note that although highly viscous pore fluids in debris flows are usually non-Newtonian, the fluids simulated here are assumed to have exceeded their yield strength. Density effects are expressed as the density difference between the particles and the fluid ̂= ( − )⁄ [11]. The ̂ values used in this study are summarized in Table  1 and are denoted by different symbols. The fluids are also varied for different flow conditions: 0 , , , , and the mean diameter ̅ = ( + )/( + ), where and are volume fractions of the large and small particles, respectively. These test cases are represented by different colors as listed in Table 2. Dry granular flows are also simulated to serve as reference cases (denoted as cross marks).

Measuring segregation and diffusion
The segregation velocity is calculated from DEM data as , = − where is the bulk vertical velocity and is the averaged velocity of a size species at a specific height. Fig. 2a shows that , increases when plotted against the local concentration = /( + ) of the opposing size species [12]. This tendency can be defined by the linear equation [13]: indicated by the dashed lines, where is the segregation velocity magnitude encoding the dependence on the inertial properties of the flow, as well as on the fluid material properties. The positive and negative signs denote the direction of the large and small particle motion. Diffusion is quantified by the diffusion coefficient = 〈∆ (∆ ) 2 〉/2∆ which is a function of the mean squared displacement of particles measured along the -direction over a time interval ∆ , and 〈… 〉 denotes an   Fig. 2b shows that the mean squared displacement of all particles changes linearly with time, having a slope that is twice . Diffusion is only measured in the lateral direction ( -axis) because particle motion is independent of segregation ( -axis) and streamwise flow ( -axis) along this direction.
It can be seen in Fig. 3a that decreases with which is obvious only after a certain threshold value (between 0.05-0.1 Pa s). This transitional behavior is also observed for other flow conditions (Fig. 3b), where the effects of flow conditions and ̂ primarily shift the curves along the vertical direction. Similar to , also exhibits a tendency to decrease with after exceeding a limit value. However, unlike the effect of density ratio on , does not seem to vary significantly with ̂ (Fig.  3c). Fig. 3d shows that the diffusion coefficient is strongly dependent on ̇ and ̅ but less so with the other flow parameters. in the right-hand side captures the influence of viscous stresses in such a way that it reduces to approximately 1 in the absence of a viscous fluid but diminishes segregation or diffusion when the viscosity is high. In the low viscosity limit, the segregation velocity depends only on the flow inertia and buoyancy. When ̂= 0, no segregation occurs as particles are neutrally buoyant. In this limit, the equation for diffusion in dry granular flows is also recovered, i.e., =̇̅ 2 , with a value of = 0.047 that falls within the range of previous results of dry granular flow experiments and simulations [15,16].   for different ̂ and flow conditions. The downward pointing arrow represents decreasing ̂. Refer to Tables 1 and 2 for the representation of the different symbols and colors. Gray & Chugunov (2006) previously developed a continuum framework for the evolution of particle size concentrations of bi-disperse mixtures resulting from the competition of size segregation and diffusion. In a coordinate system similar to that illustrated in Fig. 1c the trajectory of particle species in a steady flow is governed by a transport equation [13]:

Continuum modelling
Eq. 4 is developed for dense dry granular flows. Here, it is extended for fluid-saturated flows by incorporating the expressions for , and (i.e., Eqs. 1-3) to account for fluid effects. Fig. 5 shows the trajectory of the large particle centers of mass obtained from the simulations (dotted lines) and predicted by the model above (solid lines) for flows with different ̂ and . It can be observed that good agreement is obtained between simulation data and theory. This shows that by accounting for the fluid effects on size segregation and diffusion, it is possible to predict the evolution of size distributions in saturated bidisperse granular flows, or even more realistic debris flows, with different types of interstitial fluids.

Conclusions
Particle size distributions resulting from size sorting in debris flows is controlled by the competing influence of size segregation and diffusion. Here, expressions are proposed for these two granular processes which incorporate the influence of interstitial fluids. This is accomplished through scaling analysis of segregation and diffusion data obtained from saturated bi-disperse granular flow simulations. We show that both processes are weakened in viscous fluids but only after exceeding certain threshold values (i.e., in the viscous regime).
Below the threshold (in the inertial regime), segregation and diffusion depend primarily on the inertial conditions of the flow. Using these new relationships, we extend an existing segregation-diffusion equation previously developed for dry granular flows, to describe the segregation of saturated granular flows spanning over a wide range of fluid densities and viscosities. Theoretical predictions are validated by simulation measurements with reasonable agreement. Results here are expected to enhance prediction and modeling of grading patterns, and its effects on the dynamics, in debris flows over a wide range of material and flow parameters.