Modeling Flow-Regime TransitionsUsing an Extension of the

Norem–Irgens–Schieldrop Rheological Model

Dieter Issler1, Arne S. Moe1,3, Peter Gauer1,3 and Fridtjov Irgens2,3

1Norwegian Geotechnical Institute, Oslo, Norway2Norwegian Univ. of Science and Technology, Trondheim, Norway

3International Centre for Geohazards, c/o NGI, Oslo, Norway

Work funded by the EU 5th Framework Programme (project SATSIE,EVG1-CT2002-00059) and the Norwegian Research Council

Content

1. What do we observe in large avalanches? — The case for a fluidized flow regime

2. What could be responsible for fluidization?

3. The NIS model — virtues and shortcomings

4. How to model flow-regime transitions?

5. Towards a practical avalanche model

Typical deposit profile from Ryggfonn , Norway:

● Often (not always) clear distinction between “thick” and relatively “thin” deposits.

● Flow part forming thin deposits easily surmounts obstacles, “thick” flow is stopped or deflected.

● Scattered snowballs on and in “thin” deposits.

Example: 1995 Albristhorn avalanche, Switzerland

Deposit area of dense flow

Approximate deposit area of fluidized layer

Powder-snow avalanche de-posits extend ap-prox. 500 m to the left (uphill).

Separation of fluidized layer from dense flow:

Deposit of a small mixed avalanche, photo taken in a region not reached by dense flow (after sharp bend of gully).Largest snowballs approx. 40 cm in diameter!

1999 measurements at Vallée de la Sionne

3.0 m

3.9 m

7.0 m

19 m

Load cell measurements

FMCW radar profile

~ 10 s> 300 m

suspension layerfluidized layerdense layer

Content

1. What do we observe in large avalanches? — The case for a fluidized flow regime

2. What could be responsible for fluidization?3. The NIS model — virtues and shortcomings

4. How to model flow-regime transitions?

5. Towards a practical avalanche model

Present view of avalanche structure

0 10

2

3

4

q0,1q1,0

q2,0 q0,2

q2,3

q3,2

4 ambient air3 suspension layer2 fluidized layer1 dense flow0 snow cover qi,j: mass flow rates

q4,3

Order-of-magnitude estimates:

Flow type Density Concentr. Mean free path Granular (kg/m3) (—) (particle diam.) regime

Dense 100–500 0.1–0.5 0–1 frictional / collisional

Fluidized 10–100 0.01–0.10 1–4 grain-inertial

Suspension 1–10 < 10–2 > 4 macro-viscous (turbulent)

Physical properties and transport processes differ substantially between flow types!

Two causes for fluidization to consider:

A) Purely granular mechanism:Dispersive pressure from collisions between particles overcomes normal load.Conditions: high shear rates,

sufficiently elastic collisions dispersive shear stresses small

B) Pneumatic mechanism:Air flow over avalanche creates stagnation pressure at snout, underpressure on the head.Conditions: high velocity

? small cohesion in avalanche

The granular view:

• At macroscopic scale, frictional and collisional regimes can coexist at same location.

• Frictional regime:Mean free path → 0, continuous contact between particles, Coulombian friction

• Collisional regime:Short-duration collisional contacts,dispersive pressure ~ (shear rate)2, but alsodispersive shear stress ~ (shear rate)2

• Fluidization occurs when and where the dispersive pressure supports the avalanche weight.Seems to require slopes with tan θ ~ 1, however!

Content

1. What do we observe in large avalanches? — The case for a fluidized flow regime

2. What could be responsible for fluidization?

3. The NIS model — virtues and shortcomings4. How to model flow-regime transitions?

5. Towards a practical avalanche model

Original formulation of NIS rheology:

For simplicity, consider plane shear flow here.

pe effective pressure pu pore pressurec cohesion [Pa] μ dry friction [–]ν1, ν2 viscometric coeff. [m2] m shear viscosity [m2]

xx=−pe− pu−2−12

yy=−pe− pu

zz=−pe− pu−2 2

xz=c pem 2

Solution for steady gravity-driven shear flow:

Neglect cohesion and pore pressure. Then

Obtain

⇒ Steady-state flow only for μ < tan θ < m/ν2Implies the condition m > μν2

g h cos= pe2 2

g hsin= pem 2

pe= g h−z cosm /2−tan

m/2−

2 z =g h−z cos tan−

m−2

Problems with original formulation of NIS:

Unphysical behavior:• Why no dispersive pressure in transversal direction?

• With ν1 = 10 ν2 as suggested by Norem et al., longitudinal stress becomes contractive before fluidization threshold!

Formulation is incomplete:• What happens at the fluidization threshold?• How do the stresses depend on the density and

particle properties?

Problem of negative pressures can be remedied with an isotropic dispersive pressure term and restrictions on parameter choice:

Requirements: ν1 < ν2 + ν3ν2 > 0, ν3 > 0m > μ (ν2 + ν3)

N.B. Longitudinal and lateral pressure still diminish with growing shear rate! No experiments available...

xx=−pe− pu−32−1 2

yy=−pe− pu−3 2

zz=−pe− pu−32 2

xz=c pem 2

Content

1. What do we observe in large avalanches? — The case for a fluidized flow regime

2. What could be responsible for fluidization?

3. The NIS model — virtues and shortcomings

4. How to model flow-regime transitions?5. Towards a practical avalanche model

What happens at fluidization (according to NIS)?

● If tan θ > m/ν2, dispersive pressure supports entire overburden, and pe = 0 ⇒ expansion.

● Where pe = 0, fluidization takes place throughout entire depth simultaneously.

● NIS model parameters must depend on density and particle properties, but model does not specify how.

● Assume flow-regime transitions to be rapid⇒ Use algebraic instead of differential equation

to determine local depth-averaged density.

How do we determine the saltation-layer density?

● Cheapest way out:Assume ρ2 constant in the range 20–50 kg/m3.

● Simple solution (SL-1D):Ballistic trajectories, h2 ∝ u2

α , 1 ≤ α ≤ 2.

● Physical approach:Dispersive pressure depends on collision rate and thus on density.⇒ Adjust density in each time step so that

dispersive pressure matches normal load.

(Details to be worked out.)

However, we expect all shear-rate dependent terms to have the same density dependence to first order.

Resulting problem:

Ratio of shear stress to normal stress remains constant during expansion (= tan θ).

⇒ Fluidized avalanche finds no new equilibrium density, but expands forever!

⇒ Include additional effects that increase shear stress more strongly than normal stress.

Stresses in the saltation layer and interaction with other layers

● Saltation layer must be very “granular”.● Must collapse to dense-flow layer if the dispersive

pressure is unable to sustain the weight of the layer.

● Preliminary suggestion: Use CEF rheology with cohesion and dry-friction terms switched off.

● Is excessive pore pressure due to air flow from the front important?

● Search for simple yet physically reasonable expressions for exchange terms between layers.

The effect of ambient air:

Suction

Stagnationpressure

Simulation of subaqueous laboratory debris flow by Peter Gauer, NGI

Content

1. What do we observe in large avalanches? — The case for a fluidized flow regime

2. What could be responsible for fluidization?

3. The NIS model — virtues and shortcomings

4. How to model flow-regime transitions?

5. Towards a practical avalanche model

+Flow physics is mod-eled more realistically.

+Range of friction pa-rameters may become narrower.

+Potentially smaller zones of destructive pressures in hazard maps.

Advantages/disadvantages of including the fluidized layer in the model:

− Physics of fluidizedlayer is incompletely understood.

−Model becomes more complex.

− 2-layer version:computation time~ doubled.

Why was the bunker at Vallée de la Sionnenot destroyed on 1999/02/25?

PSA

up to 5 m

Velocity at bunker:U ≈ 40–50 m/s

Pressure:p ≈ 70 kPa

Saltation density:≈ p

U2≈35kg /m3

Dense-flow part deflectedor stopped at opposite slope.

Density of saltation layer<< density of dense flow.