JCT1 Benchmarking exercise

Depth-averaged Two-phase Debris-flow Model with a Shear-rate Dependent

Internal Friction

Xie Y.X., Cui, K.F., Zhou G.D., Song D.R., Zhou M.J.

a Key Laboratory of Mountain Hazards and Earth Surface Process, Chinese Academy of Sciences, Chengdu, China

b Institute of Mountain Hazards and Environment, Chinese Academy of Sciences & Ministry of Water Conservancy, Chengdu, China

Introduction

The interaction between the solid and fluid phases play an important role

in the dynamics of the system and on the flow mobility.

Fixed rheology debris flow models cannot accurately represent its

complex evolving behavior – from nearly static to liquefied.

Iverson, R.M., The Physics of Debris Flows, Reviews of Geophysics, 35, 3, August 1997Iverson, R.M., The Debris Flow Rheology Myth, Debris-Flow Hazards Mitigation: Mechanics, Prediction, and Assessment, Rickenmann & Chen, 2003

USGS Flume experiments

Natural debris flow

Granular

System

SlurryWater + Fines

Coarse debris

Pebbles

Boulders

Two-phase debris flow model

Mass and Momentum Balance Equations

Flow of a mixture composed of solids and fluids. Based on the dynamics of an assemblyof solid particles immersed in a Newtonian fluid.

𝜌𝑠 𝜕𝑡𝐶𝑠 + 𝛻 ∙ 𝐶𝑠𝑣𝑠 = 0

𝜌𝑓 𝜕𝑡(1 − 𝐶𝑠) + 𝛻 ∙ (1 − 𝐶𝑠)𝑣𝑓 = 0

𝜌𝑠𝐶𝑠 𝜕𝑡𝑣𝑠 + 𝛻(𝑣𝑠 ∙ 𝑣𝑠 = −𝛻 ∙ 𝜏𝑠 + 𝑓𝑖 + 𝜌𝑠𝐶𝑠g

𝜌𝑓(1 − 𝐶𝑠) 𝜕𝑡𝑣𝑓 + 𝛻(𝑣𝑓 ∙ 𝑣𝑓 ) = −𝛻 ∙ 𝜏𝑓 − 𝑓𝑖 + 𝜌𝑓(1 − 𝐶𝑠)g

Solid phase

Fluid phase

Solid-fluid interaction term

Shear stress in granular flows

Viscosity of Newtonian Fluid Bouchut F., Fernandez-Nieto E.D., Mangeney A., Narbona –Reina G. 2015. A Two-phase debris flow model with energy balance. ESAIM: M2AN 49:101-140, doi: 10.1051/m2an/2014026

𝐶𝑠 =ℎ𝑠

ℎ𝑓 + ℎ𝑠

𝜌𝑚 = 𝜌𝑠𝐶𝑠 + 𝜌𝑓(1 − 𝐶𝑠)

𝑣𝑚 =𝜌𝑠𝐶𝑠𝑣𝑠 + 𝜌𝑓(1 − 𝐶𝑠)𝑣𝑓

𝜌𝑠𝐶𝑠 + 𝜌𝑓(1 − 𝐶𝑠)

Mixture Properties

Two-phase debris flow model

Depth- averaged forms

𝜌𝑠𝜕ℎ𝑠𝜕𝑡

+𝜕(ℎ𝑠𝑢𝑠)

𝜕𝑥+𝜕(ℎ𝑠𝑣𝑠)

𝜕𝑦= 0

𝜌𝑓𝜕ℎ𝑓

𝜕𝑡+𝜕(ℎ𝑓𝑢𝑓)

𝜕𝑥+𝜕(ℎ𝑓𝑣𝑓)

𝜕𝑦= 0

𝜌𝑠𝜕(ℎ𝑠𝑢𝑠)

𝜕𝑡+𝜕 ℎ𝑠𝑢𝑠

2 + ൗ𝑘𝑠𝑔ℎ𝑠2

2𝜕𝑥

+𝜕(ℎ𝑠𝑢𝑠𝑣𝑠)

𝜕𝑦= 𝜌𝑠𝑔𝑧ℎ𝑠

𝜕𝑧

𝜕𝑥− 𝜏𝑠𝑥 + 𝑓𝑖𝑥

𝜌𝑠𝜕(ℎ𝑠𝑣𝑠)

𝜕𝑡+𝜕(ℎ𝑠𝑣𝑠𝑢𝑠)

𝜕𝑥+𝜕 ℎ𝑠𝑢𝑠

2 + ൗ𝑘𝑠𝑔ℎ𝑠2

2𝜕𝑦

= 𝜌𝑠𝑔𝑧ℎ𝑠𝜕𝑧

𝜕𝑦− 𝜏𝑠𝑦 + 𝑓𝑖𝑦

𝜌𝑓𝜕(ℎ𝑓𝑣𝑓)

𝜕𝑡+𝜕(ℎ𝑓𝑣𝑓𝑢𝑓)

𝜕𝑥+𝜕 ℎ𝑓𝑢𝑓

2 + ൘𝑘𝑓𝑔ℎ𝑓2

2𝜕𝑦

= 𝜌𝑓𝑔𝑧ℎ𝑓𝜕𝑧

𝜕𝑦− 𝜏𝑓𝑦 + 𝑓𝑖𝑦

𝜌𝑓𝜕(ℎ𝑓𝑢𝑓)

𝜕𝑡+𝜕 ℎ𝑓𝑢𝑓

2 + ൘𝑘𝑓𝑔ℎ𝑓2

2𝜕𝑥

+𝜕(ℎ𝑓𝑢𝑓𝑣𝑓)

𝜕𝑦= 𝜌𝑓𝑔𝑧ℎ𝑓

𝜕𝑧

𝜕𝑥− 𝜏𝑓𝑥 + 𝑓𝑖𝑥

Bouchut F., Fernandez-Nieto E.D., Mangeney A., Narbona –Reina G. 2015. A Two-phase debris flow model with energy balance. ESAIM: M2AN 49:101-140, doi: 10.1051/m2an/2014026

Solid phase

Fluid phase

Solid-fluid coupling

𝑓𝑖 = −𝑣𝑝𝑖𝛻𝑝 +1

2𝐶𝑑𝜌𝑓

𝜋𝑑2

4𝑼− 𝑽 𝑼− 𝑽 𝑛−𝜒+1 +

BuoyancyHydrostatic

Drag ForceHydrodynamic

Virtual Mass ForceBasset ForceStaffman ForceMagnus Force

Drag Coefficient

Porosity Correction Factor

Reynold’s Number (Particle Scale)

Zhao T., 2017. Coupled DEM-CFD Analyses of Landslide-Induced Debris Flows. Singapore, Springer Nature.Zhu H.P., Zhou Z.Y., Yang R.Y., Yu A.B., Discrete particle simulation of particulate systems: Theoretical developments, Chemical Engineering Science 62 (2007) 3378 – 3396

𝐶𝐷 =24

𝑅𝑒𝑝1 + 0.15𝑅𝑒𝑝

0.681 +0.407

1 +8710𝑅𝑒𝑝

𝜒 = 3.7 − 0.65exp −1.5 − log10 𝑅𝑒𝑝

2

2

𝑅𝑒𝑝 = Τ𝜌𝑓𝑑 𝑼 − 𝑽 𝜇

Brown & Lawler 2003

Internal Friction

Shear stress models for granular flows

𝜏𝑠 = 𝜇𝜌𝑔𝑐𝑜𝑠ѲℎCoulomb model

Voellmy model

𝜏𝑠 = 𝜇𝜌𝑔𝑐𝑜𝑠Ѳℎ + 𝑔ℎ𝑢2

𝜌ξℎ2

Non-local 𝝁 𝑰 rheology model

𝜏𝑠 = 𝜌𝑔𝑐𝑜𝑠Ѳℎ 𝜇𝑠 +𝜇2 − 𝜇𝑠 𝑢𝑑50

𝐼0ℎ32 𝐶𝑠𝑔 + 𝑢𝑑50

𝐼0 =5

2

𝑑

𝑙0

𝛽

𝜙𝑐𝑜𝑠𝜃

• Only apply to quasi-static conditions;unable to capture the effects of particle contacts within the granular body.

• Adds a dynamic shear resistance term;turbulence coefficient is not physically well-defined.

• The shear rate dependence enables the frictional model to transition from a quasi-static regime to a rapid shear regime.

Zhou G G D, Ng C W W, Sun Q C. 2014. A new theoretical method for analyzing confined dry granular flows, Landslides, 11(3), 369-384Jop P., Fortrerre Y., Poliquen O. 2006. A Constitutive law for dense granular flows. Nature, 441(8), 727-730

𝜇𝑠 function of bed friction angle 𝜇2 function of internal friction angle

𝛽 and 𝑙0 are material constants𝑑50 is the mean particle diameter𝐶𝑠 is the solid volume concentration

Experiments to determine material bed friction angle (𝜽𝒃𝒆𝒅) and internal friction angle (𝜽𝒊𝒏𝒕).

Numerical Scheme (FDM)

S1 S2

S3S4

Non-oscillatory Central Differencing Scheme

• Based on the NT scheme, natural extension of the Lax first order scheme.

• Scheme viscosity is treated desirably.

• Offers the advantage of avoiding the costly Riemann characteristic decomposition while retaining high

resolution.First-order Lax central

differencing scheme

Second-order central

differencing scheme

under staggered grid

Second or higher-order

central differencing under

non-staggered grid

Van-Leer

interpolation

Re-averaged,

reconstructed

value of staggered

form

Nessyahu H, Tadmor E. 1990. Non-oscillatory central differencing for hyperbolic conservation laws, Journal of computational physics, 87(2), 408-463

Model Validation

Solid Parameters Values

Solid Volume Concentration

𝐶𝑠 0.4

𝜇𝑠 0.25

𝜇2 0.32

𝐼0 0.28

Mean Particle Diameter

𝑑50 (𝑚) 0.0005

Fluid Parameters Values

Fluid Viscosity (𝑃𝑎 ⋅ 𝑠) 0.01

Two-phase flume experiment is compared with model.

Flume experiment using mono-sized glass beads mixed with glycerol

Mixture of glass beads and glycerol

solution

0.4

Model Validation

Numerical result

Flume test

Point B

Height

Shear and normal forces obtained from flume experiments are validated

against model calculated values.

Numerical result

Flume testPoint A

Height

𝑃 = 𝜌𝑠𝑔ℎ 𝑠 + 𝜌𝑓𝑔ℎ 𝑓

𝜏 = 𝐶𝑠𝜏𝑠 + (1 − 𝐶s)𝜏𝑓

Case study I

Source location

Disruption to traffic

A

B

C

D

E

General Features:The Yu Tung Debris Flow of Hong Kong is a series of 19 landslides that occurred at the hillside area adjacent the Yu Tung

road. The largest debris flow had a total detached mass of 2,350 m3 and a total run-out distance of 600 m. The velocity at 100

m from the source location was estimated to be about 12 m/s which reduced to about 10m/s at 400 m.

Yu Tung debris flow, Hong Kong Solid Parameters Values

Solid Volume Concentration

𝐶𝑠 0.2~ 0.5

𝜇𝑠 0.23

𝜇2 0.3

𝐼0 0.28

Mean Particle Diameter

𝑑50 (𝑚) 0.001

Fluid Parameters Values

Fluid Viscosity (𝑃𝑎 ⋅ 𝑠) 0.005

*Parameter values are based on Zhou et al. 2014

Zhou G G D, Ng C W W, Sun Q C. 2014. A new theoretical method for analyzing confined dry granular flows, Landslides, 11(3), 369-384

Flow Velocities

Velocities calculated two-phase model against field-estimated velocities.

• The higher the solid concentration the lower the velocity.

• Best predictions come at a relatively low solid concentration (i.e. 0.4)

• Highest velocities correspond to the sudden bend in the flow path at ~300 m.

• Total run-out distance is ~600 m.

Two - phase

Evolution of Solid Concentration

• The model shows a constant re-distribution of the solid concentration during the

course of the flow.

Initial state is assumed to be well-mixed.

Volume= 2350 m3

Mixture Cs= 0.3

Runout= 530 m

Grid: 325× 88

Space gridding: 2 m

Comp. time=10 min. Actual footage

Flow is highly saturated. Head

region is watery/highly fluidized

followed by a more solid-rich region.Time= 0 s Time= 20 s Time= 50 s Time= 100 s

Cs

Conclusions

• A two-phase depth-averaged model was presented which incorporates solid-

fluid interaction and shear-stress dependent internal friction for the flow of the

solid mixture.

• Validation results show over-prediction of shear and normal stresses, and flow

height.

• Benchmark results for the Yu Tung debris flow show reasonable agreement

with known velocities. Fluctuations in the velocities arise due to the re-

arrangement of phases during flow.

• The model features a separation of the solid and fluid phases during the course

of the flow as marked by sharp transitions in the solid volume fractions.

Thank you.That’s all for now

DEM of landInformation of

material in source area

Basal parameters

Initial condition

Begin

Data initializtion

Model of single phase

Shallow water equations

frictional

equations

Discrete equationsusing high resolution scheme

over

( ) ( ) ( ) ( )2 2 2

t dt t dt tU t t U t F t G t S t

dx dy

Current time=total time

( ) ( ) ( ) ( )2

t dt dtU t U t F t G t

dx dy

crt crt dt

2 2

2 2

1( )

( ) ( )2r

hu kghhu huv u Z

f ght x y xu v

( ) ( ) ( )h hu hvE

t x y

2 2

2 2

1( )

( ) ( ) 2r

hv kghhv huv v Z

f ght x y yu v

Grid mesh

Difinite function

Current time<total time

Aftertreatment(animation)

Data analysis

Erosion block

Pre processing: Mesh grid, read terrain info…What you need to do:

1. Get the Dem(digital elevation map) of terrain and source material2. Define the basal friction ( basal friction parameters)

Main processing: 1. Discrete the governing equations using specific math schemes

2. Iterate the variables

Post processing: 1. Data analysis2. Animation and figures (Tecplot, Gmesh, Google

earth, and Arcgis, surfer)

Appendix 1: Program setting