# IJEGE-11_BS-George-&-Iverson

*Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza*

*DOI: 10.4408/IJEGE.2011-03.B-047*

**A TWO-PHASE DEBRIS-FLOW MODEL THAT INCLUDES COUPLED**

**EVOLUTION OF VOLUME FRACTIONS, GRANULAR DILATANCY,**

**AND PORE-FLUID PRESSURE**

**INTRODUCTION**

during initiation and subsequent rapid flow. Most nu-

merical simulations of debris-flow motion avoid this

challenge by specifying finite force imbalances in

static debris poised on slopes. In this way, modelers

artificially impose a static state before computations

begin. By contrast, initiation of natural debris flows

occurs when balanced forces are infinitesimally per-

turbed -- that is, when the factor of safety in static

debris becomes infinitesimally smaller than 1. As mo-

tion of the debris begins, however, the force imbal-

ance may significantly change because dilatancy and

pore-pressure feedback modify frictional resistance.

This feedback commonly determines whether motion

evolves to produce a rapid debris flow or a different

phenomenon, such as a landslide that creeps imper-

ceptibly downslope (i

*et alii,*2000; w

because it can cause frictional resistance to evolve in

response to changing stresses and deformation rates.

from initiation to post-depositional consolidation

by including the effects of coupled evolution of di-

latancy, solid and fluid volume fractions, and pore-

fluid pressure. This formulation results in a hyper-

bolic system of four simultaneous partial differential

**ABSTRACT**

contacts and thereby reduces intergranular friction.

Pore-pressure feedback accompanying debris deforma-

tion is particularly important during the onset of debris-

flow motion, when it can dramatically influence the

balance of forces governing downslope acceleration.

We consider further effects of this feedback by formu-

lating a new, depth-averaged mathematical model that

simulates coupled evolution of granular dilatancy, solid

and fluid volume fractions, pore-fluid pressure, and

flow depth and velocity during all stages of debris-flow

motion. To illustrate implications of the model, we use

a finite-volume method to compute one-dimensional

motion of a debris flow descending a rigid, uniformly

inclined slope, and we compare model predictions with

data obtained in large-scale experiments at the USGS

debris-flow flume. Predictions for the first 1 s of motion

show that increasing pore pressures (due to debris con-

traction) cause liquefaction that enhances flow accelera-

tion. As acceleration continues, however, debris dilation

causes dissipation of pore pressures, and this dissipa-

tion helps stabilize debris-flow motion. Our numerical

predictions of this process match experimental data

reasonably well, but predictions might be improved by

accounting for the effects of grain-size segregation.

*K*

*ey*

*worDS*

*: debris flow, dilatancy, pore pressure*

*D.L. GEORGE & R.M. IVERSON*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

*h*(

*x*,

*t*), because

in some instances rocks may protrude through a debris

flow's free surface, but in other instances all solid grains

may be submerged. To avoid this ambiguity we define

*h*(

*x*,

*t*) as the height of a

*virtual*free surface, such that the

debris-flow mass per unit basal area is

*ρh*(

*x*,

*t*).

*EVOLUTION OF MASS DISTRIBUTION*

mixture mass-conservation equation in the standard form

*q*must be balanced by a compensating

divergence of the grain velocity

*v*

*s*

*(i*

*v*

*s*

*implies that*

*m*evolves, leading

tion rate

*D*:

*m*=

*m*, we can recast (6) as

depth-averaged conservation of the granular phase, is the

first of the four evolution equations solved by DIGCLAW.

ume wave-propagation method similar to that used

in CLAWPACK and GEOCLAW (l

*et alii*, in press?). We implement the compu-

and we compare solutions generated by DIGCLAW

with aggregated data obtained in large-scale experi-

ments at the USGS debris-flow flume.

**MODEL FORMULATION**

sional motion of a two-dimensional debris flow

descending a rigid, impermeable plane uniformly

inclined at the angle

*θ*(Figure 1). The flow moves

downslope as an evolving surge that has a character-

istic length,

*L*, characteristic thickness,

*H*, and charac-

teristic grain diameter,

*δ*, such that

*L*>>

*H*>>

*δ*. The

disparity of these length scales justifies our use of a

depth-averaged continuum model.

*ρ*

*s*

*m*and incompressible fluid of mass

density

*ρ*

*f*

*m*, such

*ρ = ρ*

*s*

*m + ρ*

*f*

*(1-*

*m*)

*m*is a dependent variable that evolves as a func-

tion of the downslope coordinate, x, and time,

*t*, imply-

ing that the depth-averaged bulk density,

*ρ*, also evolves.

The other dependent variables are the depth-averaged

downslope flow velocity,

*v*(

*x*,

*t*), the flow thickness,

*h*(

*x*,

*t*),

and the basal pore-fluid pressure,

*p*

*bed*

*x*,

*t*) (Figure 1).

ence that moves with the solids. This approach utilizes

an apparent fluid velocity

*q*(

*i.e.*, fluid volume flux per

unit area) relative to the solids, defined as

*v*

*f*

*v*

*s*

tude of

*q*is sufficiently small that ||

*q||<<||v*

*s*

*||*(1-

*m*),

*e.g.*, when

*v*

*s*

*=*

tion of

*q*

*Fig. 1 - Schematic illustrating a debris flow of charac-*

*teristic length L, characteristic thickness H, and*

*characteristic local grain diameter δ descending*

*a uniform slope inclined at the angle θ. Magnified*

*slice illustrates the dependent variables h, v, m,*

*and p*

*bed*

*used in DIGCLAw*

**A TWO-PHASE DEBRIS-FLOW MODEL THAT INCLUDES COUPLED EVOLUTION OF VOLUME FRACTIONS,**

**GRANULAR DILATANCY, AND PORE-FLUID PRESSURE**

*Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza*

*ψ*cannot be a material constant. Rather,

*ψ*must evolve

and ultimately become zero during steady shearing. In

classical soil mechanics, where

*γ*

*<*

*0.01*

*ψ*

*=*

*γ*

*>>*

*0.01*

*ψ*we adopt a rationale

with those of dense grain-flow mechanics (f

*ψ*de-

*m*

*m*

*eq*

*m*

*eq*

*m*equil-

To gauge the effects of the stress state and shear rate on

*m*

*eq*

*N*that can be

is the characteristic time for local grain rearrangement

(mediated by pore-fluid viscosity,

*μ*), and the denomina-

tor is the characteristic time for bulk shear deformation,

1

*γ*(

*cf*. C

*et alii*, 2003; C

*et alii*,

cale as

*μ*

*ρ*

*-*

*ρ*

*f*

*gh*

*θ*, where (

*ρ*

*-*

*ρ*

*f*

*gh*

*θ*is the

this timescale with the depth-averaged bulk shearing

timescale 1

*γ = h*

*v*to express

*N*as

*N*is essentially the

ly to describe the stress state in debris flows (i

*ψ*by using the linear P

*N*:

*C*

*C*

*m*

*crit*

*m*

*eq*

drostatic, and

*N*

*N*increases from 0 to ∞, tanh

*N*increases almost linearly from 0 until it smoothly

asymptotes to 1, implying that the equilibrium volume

fraction

*m*

*eq*

increasing shear rates. Through its dependence on

*m*

*eq*

*N*,

*ψ*evolves in response to evolution of all of the

dependent variables in DIGCLAW:

*m*,

*v*,

*h*, and

*p*

*bed*

the flow depth to find

*dρ*/

*dt*in (8) accounts for changes

*ρ*due to changes in

*m*. Use of (1) and (7) in conjunc-

tion with the chain rule

shows that these changes can be expressed by

solved by DIGCLAW. If

*D*= 0, then (9) reduces to the

standard depth-averaged mass conservation equation

for incompressible materials, and (7) reduces to the

trivial relation

*dm*/

*dt*= 0.

*EVOLUTION OF DILATANCY*

describe the mechanical causes of dilation. For this

purpose we use an equation modified slightly from

one proposed by i

*γ*is the macroscopic shear rate,

*ψ*is the shear-

induced dilatancy (a property of granular materials

that is commonly expressed as an angle, -

*π*/2

*≤*

*ψ*≤

*/2),*

π

π

*α*is the mixture compressibility (a property that

commonly declines as m increases), and

*σ*

*e*

*σ*

*e*

*σ*

*p*, where

*σ*is

*p*is the pore-fluid

pressure). Positive dilatancy indicates that densely

packed grains move apart as they shear past one an-

other, whereas negative dilatancy implies that grains

converge during shearing, provided that

*σ*

*e*

*γ*

soil consolidation theories; but if shearing occurs in

a closed container that enforces ∆

*v*

*s*

*=*

*dσ*

*e*

*/*

*dt*

*=*

*γ*

*ψ*/

*α*. This equation erroneously

*σ*

*e*

*ψ>0*and shearing

*γ*, thereby demonstrating that

*D.L. GEORGE & R.M. IVERSON*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

*H*

*L*

*∂*

*p*

*∂y*

*∂*

*p*

*∂x*

*∂*

*∂y*

*H*

*∂*

*∂x*

*L*

*∂*

*p*

*∂x*

*v*

*y*

*y*

*h*)

*dh*

*dt*and

*v*

*x*

*=*

*v*to recast the total time de-

(s

surface boundary conditions

*p*

*h*)

*σ*

*h*)

*p*denotes the depth average of

*p*, and

*v*/

*h*is

used to approximate the depth-averaged shear rate.

The term

*ρ*

*f*

*g*

*θ*arises in (20) from depth integra-

application of a zero-flux basal boundary condition

that requires the pore-pressure gradient at the bed to

remain hydrostatic: [

*∂p*

*∂y*]

*ρ*

*f*

*g*

*θ*. The term

*σ*

*-*

*p*)

*dh*

*dt*arises from depth-integrating the term

*∂*

*σ*

*-*

*p*)

*∂y*in (19) by parts. This term

*κ*

is a longitudinal normal-stress coefficient that equals 1

if the stress state is hydrostatic (i

dimensional pore-pressure evolution equation, but it

retains two pore-pressure variables,

*p*and

*p*, rather

than the desired variable,

*p*

*bed*

*p*

*bed*

*p*and [

*∂p*

*∂y*]

*y=h*

*m*=

*m*at

all depths. This assumption implies that Λ

*v*

*s*

*q*are

*y*, further implying that

*∂*

*2*

*p*

*∂y*

*2*

*y*in (18) and (19). With this stipulation,

we solve

*∂*

*2*

*p*

*∂y*

*2*

*=*

*constant*and employ the hydro-

*∂p*

*∂y*

*|*

*ρ*

*f*

*g*

*θ*

*EVOLUTION OF PORE-FLUID PRESSURE*

*p*

*bed*

*q*to

the gradient of excess pore-fluid pressure,

*p*

*e*

*p*

*e*

*p*

*ρ*

*f*

*g*

*h*

*y*)

*θ*, where

*p*is the total fluid

*k*is the intrinsic hydraulic permeability of

the granular debris (b

phase-interaction forces in debris flows, but detailed

investigations of similar mixtures indicate that it prob-

ably provides a suitable first approximation (e.g., J

*et alii*, 1990; s

is related to the granular dilation rate, Δ

*v*

*s*

*v*

*s*

*k*

*μ*is assumed constant (an assump-

sulting equation reduces to

*k*

*αμ*plays the role of a pore-pressure diffusivity.

*p*

*e*

to find that

*dσ*

*e*

*dt*

*=*

*dσ*

*dt*

*-*

*dp*

*e*

*dt*

*-*

*d*

*ρ*

*f*

*g*

*h*

*y*)

*θ*]

*dt*.

tion rate

*γ*

*ψ*, the mean total stress

*σ*, and the hydro-

*ρ*

*f*

*g*

*h*

*y*)

*θ*. Note

*γ*

*ψ*is constant, the equation reduces to the steady-

is balanced by a steady influx of fluid that fills the

enlarging pores.

involve recasting (17) in terms of the total pore-fluid

pressure,

*p*

*p*

*e*

*ρ*

*f*

*g*

*h*

*y*)

*θ*, and invoking shallow-

**A TWO-PHASE DEBRIS-FLOW MODEL THAT INCLUDES COUPLED EVOLUTION OF VOLUME FRACTIONS,**

**GRANULAR DILATANCY, AND PORE-FLUID PRESSURE**

*Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza*

*p*(

*h*)

*p*(

*y*) satisfies the quadratic equation

*p*is represented entirely by the evolving values of the

basal pressure

*p*

*bed*

*t*) and flow thickness

*h*(

*t*). Equation

*d*(

*ρh*)

*dt*and

*dh*

*dt*can be elimi-

mass-conservation equation (9), yielding the final

form of the evolution equation for

*p*

*bed*

integrated effects of pressure diffusion, and the second

term accounts for the forcing effects of the evolving

gravitational load and dilation rate.

*EVOLUTION OF MOMENTUM DISTRIBUTION*

*ρ*

*=*

*ρ*, depth integration

*x*component of the mo-

mentum-conservation equation

yields a result that can be written in several forms,

including

*x*momentum as

a function of

*y*is negligible, and therefore omit the

integral containing

*v*

*x*

*v*

vanishes through application of (8), leaving only the

term

*ρh*(

*dv*/

*dt*). This term is mathematically correct

but physically "non-conservative" because it does not

*ρh*. The

final line of (27), which is used in DIGCLAW, dif-

fers from a conventional, conservative shallow-water

formulation owing to presence of the term -

*ρ*

*ρ*

*f*

*D*

*v*

*hv*

*dρ*

*dt*) and thereby accounts for

*D*on evolution of

*ρ*.

ing effects due to gravity and resisting effects due to

internal and boundary stresses, as derived by i

the fourth evolution equation solved by DIGCLAW,

*τ*

*s*

*τ*

*f*

*τ*

*s*

pressure and dilatancy,

*f*

*bed*

*f*

*bed*

*+*

*ψ*is the

present (i

*ρgh*

*θ*

*-*

*p*

*bed*

*σ*

*e bed*

*τ*

*f*

*m*is the fluid volume fraction and

*v*

*h*is, again,

*γ*.

*MATHEMATICAL CLOSURE*

*D*and

*κ*, and thereby complete the mathematical

model. We obtain an equation for

*D*by combining (5),

(6) and (15) to find that

*k*

*μ*is

obeys the quadratic relationship specified in (23).

It also shows that

*p*

*bed*

*D*as both quantities evolve. This be-

havior is a logical consequence of mass conserva-

*D.L. GEORGE & R.M. IVERSON*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

*et alii*, in press). The results presented here

results required about 49 s of CPU time on a standard

desktop computer with a 2.4 GHz processor.

*INITIAL CONDITIONS AND SHORT-TERM*

BEHAVIOR

BEHAVIOR

*h*,

*v*,

*m*and

*p*

*bed*

*x*= -5 m (behind the

headgate) to

*x*= 5 m, just downslope of the head-

gate (Figure 2). At

*t*= 0, the debris impounded be-

hind the headgate is static and the basal pore-fluid

pressure is hydrostatic (

*p*

*bed*

*=*

*ρ*

*f*

*gh*

*θ*). The solid

*m*

*et alii*(2010), and

of Figure 2 show these initial conditions as well as

the evolving longitudinal profiles of all dependent

variables at

*t*

resulting in progressive longitudinal extension and de-

creasing depth of the entire debris mass (Fig. 2 a and

b). At the same time, the solid volume fraction increas-

es everywhere except at the leading edge of the ad-

vancing flow (Fig. 2c), and this increase causes a com-

mensurately widespread increase in basal pore-fluid

pressure,

*p*

*bed*

*t*= 1 s, for example, much

*p*

*bed*

*h*~

*i.e*.,

*p*

*bed*

*≈*

*ρ*

*gh*

*θ*). An excep-

snout, which begins to dilate almost immediately be-

cause h→0 at the snout tip, and implied shear rates

assumptions of Darcian fluid flow and

*m*

*m*.

*κ*generally can vary from about 0.3 to 3, depending

on whether flowing debris undergoes longitudinal

extension or compression (s

potentially quite complicated. To avoid such compli-

cations while focusing our computations on coevolu-

tion of

*m*,

*v*,

*h*, and

*p*

*bed*

*κ*

*h∂p*

*bed*

*/*

*∂x*vanishes from (28), implying that pore-fluid

through its influence on basal Coulomb friction.

**MODEL PREDICTIONS AND TESTS**

measured in a series of eight replicate experiments

performed in the USGS debris-flow flume. In each

experiment 10 m

vertical headgate and travelled more than 70 m down

the uniformly sloping (31º), 2-m wide flume before

encountering flatter slopes and debouching from the

flume mouth. i

*et alii*(2010) presented de-

processing, and debris and flume properties. Because

the flume's sidewalls were vertical and much smooth-

er than its bumpy flume bed (1 mm vs. 16 mm char-

acteristic roughness amplitudes), a one-dimensional

model was appropriate for simulating flow within the

flume. Parameter values used to generate simulation

results generally matched values measured in labora-

tory tests (Tab. 1). A notable exception was the value

of

*k*, which we discuss below.

from

*x*= -10.0 m to

*x*= 90.0 m, where

*x*= 0 denoted

the flume headgate location. The code used explicit

computational time steps that were modified adaptive-

ly to satisfy a Courant-Friedrichs-Lewy (CFL) con-

dition. At each time step the numerical solution was

updated by using a finite-volume wave-propagation

method to solve Riemann problems at grid cell in-

terfaces, as detailed elsewhere (

*e.g*., l

*Tab. 1 - Comparison of parameter values used in DIG-*

*CLAw with values measured in "SGM" debris-*

*flow flume experiments of i*

*verSoN*

*et alii (2010)*

**A TWO-PHASE DEBRIS-FLOW MODEL THAT INCLUDES COUPLED EVOLUTION OF VOLUME FRACTIONS,**

**GRANULAR DILATANCY, AND PORE-FLUID PRESSURE**

values, and effective basal friction increases. Mean-

while, the flow elongates greatly while maintaining

a steep leading edge, or snout, as observed in experi-

ments (i

*et alii*, 2010). The peak flow depth

until it stabilizes at about

*h*= 0.1 m (Figure 3a),

while the snout speed stabilizes at between 8 and 10

m/s (Figure 3b), similar to behavior observed in the

experiments (i

*et alii*, 2010). The high speed

rates, dilation rates, and rates pore-pressure deple-

tion, thereby reinforcing the frictional resistance of

the snout. As a consequence, classic head-and-tail

debris-flow architecture develops and persists.

the snout declines in response to high shear rates, pore

pressures there also decline. Flow resistance therefore

begins to grow in the snout while most of the trailing

debris maintains a liquefied state that allows it to push

the snout from behind. As a consequence, the snout ini-

tially moves downslope faster than an ideal frictionless

body, despite its relatively high flow resistance.

*DOwNSLOPE BEHAVIOR*

term behavior because flow velocities become high

enough that the dominant debris response becomes

dilative (Figure 3). In conjunction with dilation, ba-

*Fig. 2 - Simulated short-term evolution of dependent vari-*

*ables over the interval from x = −5 m to 5 m. Longi-*

*tudinal profiles of variables are shown for the initial*

*condition (t =0 ) and for t= 0.2, 0.4, 0.6, 0.8, and 1.0 s*

*Fig. 3 - Simulated long-term evolution of dependent vari-*

*ables over the interval from x = -10 m to 75 m.*

*Longitudinal profiles are shown for the initial con-*

*dition (t =0 ) and for t= 2, 4, 6, 8, and 10 s*

*D.L. GEORGE & R.M. IVERSON*

*COMPARISON OF COMPUTED AND MEASU-*

RED TIME SERIES

RED TIME SERIES

*h*(

*t*), basal pore-fluid pressure,

*p*

*bed*

*t*), and

*σ*

*bed*

*t*) measured in the eight

*et alii*

(2010). Aggregated data from replicate experiments

provide a better basis for model tests than do data from

individual experiments or field observations, because

data aggregation minimizes the effects of idiosyncrasies

and reveals the effects of inherent variability. Therefore,

Figure 4 superposes model predictions (black lines)

on gray shaded envelopes that depict mean values ±1

standard deviation measured in eight experiments.

one 32 m downslope from the flume headgate and

one 66 m downslope from the headgate. All panels of

the figure show that model predictions of flow-front

arrival times differ from measured arrival times by

*h(*t) and basal total normal stress

*σ*

*bed*

*t*) at

*x*

pressure

*p*

*bed*

*t*) is somewhat lower than measured val-

*x*= 66 m, the predictions of flow

depth and basal normal stress remain relatively good,

but the predicted pore-fluid pressure is considerably

smaller than that observed in experiments. The next

section discusses some shortcomings of the model that

might account for these discrepancies.

**DISCUSSION**

cisely constrained by independent measurements.

When we adjusted the values of these parameters

within the range of physically plausible values, it

affected our predictions, but we did not make such

adjustments blindly.

*Fig. 4 - Comparison of model predictions (black lines) and aggregated time series data (gray shaded areas) measured at two*

*instrumented cross sections in the SGM debris-flow flume experiments of i*

*verSoN*

*et alii (2010). The shaded areas*

*depict the mean values +/- one standard deviation of measurements made in eight replicate experiments*

**A TWO-PHASE DEBRIS-FLOW MODEL THAT INCLUDES COUPLED EVOLUTION OF VOLUME FRACTIONS,**

**GRANULAR DILATANCY, AND PORE-FLUID PRESSURE**

enhance this development and lead to attendant

regulation of flow speeds. Recent work on grain-size

segregation in dry granular avalanches (e.g., G

**CONCLUSION**

tion of dilatancy, solid and fluid volume fractions,

pore-fluid pressure, and flow depth and velocity. This

formulation enables use of realistic initial conditions

with infinitesimal force imbalances, and it therefore

permits seamless simulation of debris-flow initiation

and subsequent flow. A key feature of our model is

non-monotonic evolution of computed pore pressures

during initiation (when pressures tend to increase and

promote liquefaction) and subsequent flow (when

pressures tend to relax diffusively). This non-monot-

onic evolution results from evolution of the dilatancy

angle and volume fractions in response to changes in

the debris' stress state and shear rate.

tions than most alternative models, and it therefore fa-

cilitates more stringent testing. Our model predictions

of evolving flow depths and velocities match experi-

mental data quite well, but to attain these matches, our

computations require use of a relatively high debris

permeability that leads to overprediction of pore-pres-

sure relaxation. This shortcoming might be remedied

by including grain-size segregation effects that lead to

evolving permeability distributions and sharper differ-

ences in debris-flow head and tail friction. At present,

however, we are encouraged that our model predic-

tions match many aspects of debris-flow behavior

measured in large-scale experiments.

**ACKNOWLEDGEMENTS**

contributed to successful experiments at the USGS

debris-flow flume.

velocities almost twice as large as those observed at

*x*

that friction remained close to zero. By increasing

the ratio

*k*

*μ*to values about five times larger than

with water-saturated SGM, we caused pore pres-

sures to relax more rapidly toward hydrostatic lev-

els. This adjustment increased basal friction enough

to produce reasonably accurate predictions of flow

speeds. We believe the adjustment is defensible

from a physical perspective, because rapidly shear-

ing, dilated debris is likely to be more permeable

than debris at rest in a permeameter.

*C*

*C*

model differ from those of the P

*C*

*C*

*C*

*C*

(Note that in the P

*k*

*C*

*k*

*C*

*C*

*C*

*ψ*(i.e., equations (12) and (13).

volves relaxation of basal pore-fluid pressure. In

our predictions the pressure relaxes more rapidly

than the measured pressure, but as noted above, this

rapid relaxation is necessary to obtain realistic flow

speeds. We believe the lack of grain-size segregation

in our model is responsible for this problem. In ex-

perimental debris flows, and in most natural debris

flows, large grains rapidly concentrate at flow fronts,

focussing more flow resistance there while leaving

finer-grained tails that remain largely liquefied (i

*k*can increase significantly in coarse-grained

flow fronts, leading to a zone of almost completely

depleted basal pore-fluid pressure. This heterogene-

ous architecture develops to some degree in our com-

*D.L. GEORGE & R.M. IVERSON*

**REFERENCES**

*Dynamics of fluids in porous media.*Dover, New York.

*The GEOCLAw software for depth-averaged flows with*

*adaptive refinement, Adv*. Water Resources.

*Submarine granular flows down inclined planes*. Physics of Fluids,

**17**: 103301.

*Granular avalanches in fluids.*Physical Rev. Let.,

**90**: 044301.

*Flows of dense granular media*. Annual Rev. Fluid Mech.,

**40**: 1-24.

*Segregation, recirculation and deposition of coarse particles near two-dimensional*

*avalanche fronts*. J. Fluid Mech.,

**629**: 387-423 doi:10.1017/S0022112009006466.

*The physics of debris flows*. Reviews of Geophysics,

**35**: 245-296.

*The debris-flow rheology myth*. In R

*Debris-flow Hazards Mitigation:*

*Mechanics, Prediction, and Assessment,*v.

**1**:303-314, Millpress, Rotterdam,.

*Regulation of landslide motion by dilatancy and pore-pressure feedback*. J. Geophys. Res.,

**110**: F02015

*Elements of an improved model of debris-flow motion*. Powders and Grains 2009 (Proc. Sixth Intl. Conf.

*Flow of variably fluidized granular masses across three-dimensional terrain: 1.*

*Coulomb mixture theory*. J. Geophys. Res.,

**106**: 537-552.

*Friction in debris flows: inferences from large-scale flume experiments*. Hydraulic

**2**: 1604-1609.

*The perfect debris flow? aggregated results from 28 large-scale*

*experiments.*J. Geophys. Res.,

**115**: F03005 doi:10.1029/2009JF001514.

*Acute sensitivity of*

*landslide rates to initial soil porosity.*Science,

**290**: 513-516.

*A review of interaction mechanisms in fluid-solid flow*s, Tech. Rep. DOE/

*Finite volume methods for hyperbolic problems*. Cambridge Univ. Press, Cambridge, U.K.

*A two-phase flow description of the initiation of underwater granular avalanches*. J. Fluid

**633**: 115-135 doi:10.1017/S0022112009007460.

*The motion of a finite mass of granular material down a rough incline.*J. Fluid Mech.,

**199**: 177-215.

*Surge dynamics coupled to pore-pressure evolution in debris flows*. In: R

*Debris-flow Hazards Mitigation: Mechanics, Prediction, and Assessment*. vol.

**1**: 503-514, Millpress, Rotterdam,

*Critical State Soil Mechanics*. London, McGraw-Hill.

*Coupled continuum-discrete model for saturated granular materials*. J. Engrg. Mech.,

**131**: 413-426.

*Pore-pressure generation and movement of rainfall-induced landslides: effects of grain size and*

*fine-particle content*. Eng. Geology,

**69**: 109-125.