E3S Web of Conferences 415, 01003 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341501003

Particle size segregation and diffusion in saturated granular
flows: implications for grain sorting in debris flows
Kahlil Fredrick Cui1,2, Gordon G.D. Zhou1,2,3*, Lu Jing4, Yunxu Xie1,2, and Xueqiang Lu1,2
1Key Laboratory of Mountain Hazards and Earth Surface Process / Institute of Mountain Hazards and Environment, Chinese

Academy of Sciences, Chengdu 610000, China
2University of Chinese Academy of Sciences, Beijing 100000, China
3China-Pakistan Joint Research Center on Earth Sciences, CAS-HEC, Islamabad, Pakistan

4Institute for Ocean Engineering, Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, China

Abstract. Sorting of rocks, boulders, and silt/sand-sized particles according to their size is a characteristicfeature 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 empiricalexpressions 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.

1 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 theflow 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]. Sizesegregation drives smaller particles to rapidly percolate
downwards towards the base and larger particles toslowly rise upwards due to persistent frictional contacts
between particles [5]. On the other hand, diffusion due
to random collisions in the granular flow drives thesegregated layers to re-mix.
However, the segregation-diffusion behaviour ismore 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 fluidexhibits a clear separation of large (free surface) and finegrains (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 anddiffusion will elucidate this problem and enhance themodelling of size sorting in debris flows.
Here, we investigate the effects of different types offluids that are relevant to debris flows, characterized bydensity and viscosity, on particle size segregation anddiffusion. 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 demonstratethat by substituting these empirical formulas into an
existing segregation-diffusion continuum equation, it ispossible to predict the temporal evolution of particle size
distributions in saturated granular flow simulations.

2 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 bycoupling the computational fluid dynamics (CFD) anddiscrete 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 otherthrough buoyant and drag forces. A detailed descriptionof 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, withaverage diameters 𝑑𝐿 = 0.01m and 𝑑𝑆 = 0.005m (sizeratio of 2) respectively. The particles are bounded by* Corresponding author: gordon@imde.ac.cn© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0(https://creativecommons.org/licenses/by/4.0/).

E3S Web of Conferences 415, 01003 (2023)
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https://doi.org/10.1051/e3sconf/202341501003

Table 1. Relative density and fluid viscosity
Symbol𝜌̂𝜂𝐹 (Pa s)
0.60○
0.480.36Δ□◊⊲

0.001 ≤ 𝜂𝐹 ≤ 1

0.24
0.12
Table 2. Different test cases𝛾̇ (s-1)

𝑃0(Pa)

𝑔(m/s2)

𝜌𝐺(kg/m3)𝑑̅ (m)

1
1
1.5
5
10
1
1
1

100
500
100
100
100
100
100
100

9.8
9.8
9.8
9.8
9.8
19.6
9.8
9.8

2500
2500
2500
2500
2500
2500
3000
2500

0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.004

Color

●
●
●
●
●
●
●
●
Fluids are varied by setting different fluidviscosities 𝜂𝐹 and densities 𝜌𝐹 for the fluid domain.
Changes in 𝜂𝐹 and 𝜌𝐹 represent different concentrationsof suspended fine particles in the fluid phase [1]. Notethat although highly viscous pore fluids in debris flows
are usually non-Newtonian, the fluids simulated here areassumed to have exceeded their yield strength. Density
effects are expressed as the density difference between
the particles and the fluid 𝜌̂ = (𝜌𝐺 − 𝜌𝐹 )⁄𝜌𝐺 [11].
The 𝜌̂ values used in this study are summarized in Table
1 and are denoted by different symbols. The fluids arealso varied for different flow conditions: 𝑃0 , 𝛾̇ , 𝑔, 𝜌𝐺 ,and the mean diameter 𝑑̅ = (𝜙𝐿 𝑑𝐿 + 𝜙𝑆 𝑑𝑆 )/(𝜙𝐿 +𝜙𝑆 ), where 𝜙𝐿 and 𝜙𝑆 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).

Fig. 1 Deposits of (a) experimental debris flows where inverse
grading is evident and of (b) natural debris flows where it isnot obvious. (c) Snapshot of the numerical set-up. Images are
obtained from [7].

rough walls in the vertical direction. The Hertz contact
model is used to simulate particle interactions with
Young’s modulus 5 × 107 Pa, Poisson’s ratio 0.4,
friction coefficient 0.5, and the coefficient ofrestitution 0.8. The particles are initially normallygraded, i.e., large particles are initially positioned at thebase while small particles are at the top. The CFD phaseis an incompressible, Newtonian fluid evenly
discretized into cells wherein 5 large particles would fit.
Periodic boundaries are along the streamwise (𝑥) andspanwise directions (𝑦) in both the CFD and DEM.
The top particle wall imposes a constant confiningpressure 𝑃0 and at the same time drives the particles toflow by translating at a constant velocity 𝑢𝑤𝑎𝑙𝑙 =𝛾̇ 𝐻𝑤𝑎𝑙𝑙 , where 𝛾̇ is an imposed shear rate and 𝐻𝑤𝑎𝑙𝑙 isthe vertical position of the wall which slightly changes
with time due to the dilation and contraction of thegranular bed. As the granular bed is sheared by the topwall, the ambient fluid becomes sheared along with it.
Drag forces acting opposite to the streamwise flow canresult in non-linear flow velocity profiles. To ensure thata constant 𝛾̇ is maintained along the flow height andreduce the localized shear effects [9], a small stabilizing
force 𝐹𝑠𝑡 = 𝐴𝑠𝑡 (𝛾̇ 𝑧 − 𝑢(𝑧)), is applied to each particle
at a height 𝑧 having a streamwise velocity 𝑢 at eachtimestep [10].

3 Results
3.1 Measuring segregation and diffusion
The segregation velocity is calculated from DEM data
as 𝑤𝑝,𝑖 = 𝑤 − 𝑤𝑖 where 𝑤 is the bulk vertical velocity
and 𝑤𝑖 is the averaged velocity of a size species 𝑖 at aspecific height. Fig. 2a shows that 𝑤𝑝,𝑖 increases whenplotted against the local concentration 𝐶𝑖 = 𝜙𝑖 /(𝜙𝐿 +𝜙𝑆 ) of the opposing size species [12]. This tendency canbe defined by the linear equation [13]:𝑤𝑝,𝑖 = ±𝑄(1 − 𝐶𝑖 ),(1)indicated by the dashed lines, where 𝑄 is the segregation
velocity magnitude encoding the dependence on theinertial 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
𝐷 = 〈∆𝑦(∆𝑡)2 〉/2∆𝑡 which is a function of the meansquared displacement of particles measured along the 𝑦-direction over a time interval ∆𝑡, and 〈… 〉 denotes an2

E3S Web of Conferences 415, 01003 (2023)
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https://doi.org/10.1051/e3sconf/202341501003

Fig. 2 (a) Measurements of the segregation velocity 𝑤𝑝,𝑖 ofboth large and small particles vs. the concentration of theopponent species. Dashed inclined lines are best fits based on
Eq. 1. (b) Mean squared displacement of all particles for agiven timestep. The dashed line is a linear fit with slope 2𝐷.

ensemble average. Fig. 2b shows that the mean squared
displacement of all particles changes linearly with time,
having a slope that is twice 𝐷. Diffusion is only
measured in the lateral direction (𝑦-axis) because
particle motion is independent of segregation (𝑧-axis)
and streamwise flow (𝑥-axis) along this direction.
It can be seen in Fig. 3a that 𝑄 decreases with 𝜂𝐹
which is obvious only after a certain threshold value(between 0.05-0.1 Pa s). This transitional behavior isalso observed for other flow conditions (Fig. 3b), where
the effects of flow conditions and 𝜌̂ primarily shift thecurves along the vertical direction. Similar to 𝑄, 𝐷 also
exhibits a tendency to decrease with 𝜂𝐹 after exceedinga limit value. However, unlike the effect of density ratio
on 𝑄, 𝐷 does not seem to vary significantly with 𝜌̂ (Fig.
3c). Fig. 3d shows that the diffusion coefficient isstrongly dependent on 𝛾̇ and 𝑑̅ but less so with the otherflow parameters.

Fig. 4 Scaling relationships for (a) 𝑄 and (b) 𝐷. The dashedlines have a slope of 1. Refer to Tables 1 and 2 for therepresentation of the different symbols and colors. Cross marks
represent dry granular flow results.

3.2 Scaling relationships for segregation anddiffusion
Fig. 4 shows scaling relationships derived for thenormalized segregation velocity magnitude 𝑄 anddiffusion coefficient 𝐷:
St𝑄/√𝑔𝑑̅ = 𝐵𝐼 𝛼 𝜌̂𝛽 (𝜆,1 +St)

𝐷=St𝑘 𝛾̇ 𝑑̅ 2(𝜆2 + St) 𝑑

respectively, where St =𝜌𝐺 𝑑̅ 218𝜂𝐹(2)(3)̂𝑔𝜌√ ̅ is the Stokes number,2𝑑

𝐼 = 𝛾̇ 𝑑̅ √𝜌𝐺 /𝑃 is the inertial number, and 𝑃 = 𝑃0 +𝜙𝐺 (𝜌𝐺 − 𝜌𝐹 )𝑔𝐻 is the total granular pressure. 𝜆1 =0.44, 𝜆2 = 0.17, 𝐵 = 0.22, 𝛼 = 0.88 and 𝛽 = 0.65 are
fitting coefficients. A more detailed derivation of thescaling relationships can be found in Ref. [14]. Note that
St and 𝐼 are dimensionless numbers which define thecharacteristic motion of particles under viscous andfrictional-collisional interactions respectively.
StIn Eqs. 2 and 3, the termin the right-hand(𝜆1,2 +St)

side captures the influence of viscous stresses in such away that it reduces to approximately 1 in the absence ofa viscous fluid but diminishes segregation or diffusion
when the viscosity is high. In the low viscosity limit, thesegregation velocity depends only on the flow inertia
and buoyancy. When 𝜌̂ = 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.,𝐷 = 𝑘𝑑 𝛾̇ 𝑑̅ 2 , with a value of 𝑘𝑑 = 0.047 that falls within
the range of previous results of dry granular flow
experiments and simulations [15,16].

Fig. 3 Change of (a, b) 𝑄 and (c, d) 𝐷 with 𝜂𝐹 for different 𝜌̂and flow conditions. The downward pointing arrow represents
decreasing 𝜌̂. Refer to Tables 1 and 2 for the representation of
the different symbols and colors.

3

E3S Web of Conferences 415, 01003 (2023)
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https://doi.org/10.1051/e3sconf/202341501003

Below the threshold (in the inertial regime), segregationand diffusion depend primarily on the inertial conditionsof the flow. Using these new relationships, we extend anexisting segregation-diffusion equation previously
developed for dry granular flows, to describe thesegregation of saturated granular flows spanning over awide range of fluid densities and viscosities. Theoreticalpredictions are validated by simulation measurementswith reasonable agreement. Results here are expected toenhance prediction and modeling of grading patterns,
and its effects on the dynamics, in debris flows over awide range of material and flow parameters.
The authors acknowledge financial support from the National
Natural Science Foundation of China (grant no. 41941017),
the International Science & Technology Cooperation Programof China (No. 2018YFE0100100), CAS "Light of West China"
ProgramFig. 5 Simulated (dotted curve) and predicted (solid curve)
large particle centers of mass for immersed shear flows having(a) different 𝜌̂ but constant 𝜂𝐹 (0.001 Pa s) and (b) constant 𝜌̂(0.6) but different 𝜂𝐹 . The arrows denote decreasing 𝜌̂ andincreasing 𝜂𝐹 , respectively.

References
1.
2.

3.3 Continuum modelling

3.

Gray & Chugunov (2006) previously developed acontinuum framework for the evolution of particle size
concentrations of bi-disperse mixtures resulting fromthe competition of size segregation and diffusion. In acoordinate system similar to that illustrated in Fig. 1c
the trajectory of particle species 𝑖 in a steady flow isgoverned by a transport equation [13]:
𝜕𝐶𝑖
𝜕𝑡+

𝜕
𝜕𝑧(𝑤𝑝,𝑖 𝐶𝑖 ) = 𝐷

𝜕2 𝐶𝑖
𝜕𝑧 2.

4.
5.
6.

(4)
7.

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

8.

9.

10.
11.
12.
13.

4 Conclusions
Particle size distributions resulting from size sorting indebris flows is controlled by the competing influence ofsize segregation and diffusion. Here, expressions areproposed for these two granular processes which
incorporate the influence of interstitial fluids. This isaccomplished through scaling analysis of segregationand diffusion data obtained from saturated bi-disperse
granular flow simulations. We show that both processes
are weakened in viscous fluids but only after exceedingcertain threshold values (i.e., in the viscous regime).

14.
15.
16.

4

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