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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
415
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
d
avid
L. GEORGE
(*)
& R
iCHaRd
M. IVERSON
(*)
(*)
U.S. Geological Survey, 1300 SE Cardinal Ct., Vancouver - WA 98683 USA
Email: dgeorge@usgs.gov - riverson@usgs.gov
INTRODUCTION
One of the greatest challenges in debris-flow
modeling involves seamlessly simulating behavior
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
veRson
et alii, 2000; w
anG
&
s
assa
, 2003; i
veRson
, 2005). Similar feedback is also
important during later stages of debris-flow motion,
because it can cause frictional resistance to evolve in
response to changing stresses and deformation rates.
Here we summarize a new, depth-averaged com-
putational model that simulates debris-flow motion
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
Pore-fluid pressure plays a crucial role in debris
flows because it counteracts normal stresses at grain
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
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D.L. GEORGE & R.M. IVERSON
416
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
Relative motion of solid and fluid phases can lead
to possible ambiguity in the definition of 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
By employing the mixture bulk density defined in
(1) and mixture velocity defined in (3), we can utilize a
mixture mass-conservation equation in the standard form
Mass conservation additionally requires that the
divergence of q must be balanced by a compensating
divergence of the grain velocity v
s
(i
veRson
, 1997):
Divergence of v
s
implies that m evolves, leading
to our definition of the depth-averaged granular dila-
tion rate D:
Here
denotes a material time
derivative that follows motion of the granular phase.
Because we assume that m = m, we can recast (6) as
where
denotes a depth-averaged
material time derivative. Equation (7), which expresses
depth-averaged conservation of the granular phase, is the
first of the four evolution equations solved by DIGCLAW.
equations we solve numerically by using a finite-vol-
ume wave-propagation method similar to that used
in CLAWPACK and GEOCLAW (l
e
v
eQue
, 2002;
b
eRGeR
et alii, in press?). We implement the compu-
tations in a new FORTRAN code we call DIGCLAW,
and we compare solutions generated by DIGCLAW
with aggregated data obtained in large-scale experi-
ments at the USGS debris-flow flume.
MODEL FORMULATION
To emphasize physical concepts and minimize
mathematical complexity, we focus on one-dimen-
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.
Our model treats debris as a mixture of incom-
pressible solid grains of mass density ρ
s
occupying the
volume fraction m and incompressible fluid of mass
density ρ
f
occupying the volume fraction 1-m, such
that the mixture bulk density is
ρ = ρ
s
m + ρ
f
(1-m)
In DIGCLAW the depth-averaged solid volume
fraction 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).
Our model emphasizes motion of the granular
solid phase, and treats fluid flow in a frame of refer-
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
where v
f
and v
s
are the velocities of the fluid and sol-
ids, respectively, in a fixed frame of reference (b
eaR
,
1972). Formally, the model assumes that the magni-
tude of q is sufficiently small that ||q||<<||v
s
||(1-m),
although violation of this assumption (e.g., when v
s
=
0) poses no significant problem, provided that varia-
tion of q
has negligible effect on the mass-weighted
mixture velocity, defined as
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
(1)
(2)
(3)
(4)
(5)
(6)
(7)
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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
417
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
s
-1
is typical,
such steady states are called critical states (s
CHofield
&
w
RotH
, 1968). In DIGCLAW quasi-steady states with
ψ
=
0 can develop even if γ
>>
0.01
s
-1
.
To calculate evolution of ψ we adopt a rationale
similar to that of P
ailHa
& P
ouliQuen
(2009), who
combined the principles of critical-state soil mechanics
with those of dense grain-flow mechanics (f
oRteRRe
&
P
ouliQuen
, 2008) to postulate that the value of tan
ψ de-
pends linearly on m
-
m
eq
, where m
eq
is a value of m equil-
ibrated with the ambient state of stress and shear rate.
To gauge the effects of the stress state and shear rate on
m
eq
, they used a dimensionless parameter N that can be
interpreted as a timescale ratio in which the numerator
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
ouRReCH
du
P
ont
et alii, 2003; C
assaR
et alii,
2005). Here we identify the grain-rearrangement times-
cale as μ
/
(ρ
-
ρ
f
)gh
cos
θ, where (ρ
-
ρ
f
)
gh
cos
θ is the
characteristic effective normal stress, and we combine
this timescale with the depth-averaged bulk shearing
timescale 1
/
γ = h
/
v to express N as
This relationship shows that N is essentially the
reciprocal of the friction number introduced previous-
ly to describe the stress state in debris flows (i
veRson
& l
a
H
usen
, 1993; i
veRson
, 1997).
Next we define ψ by using the linear P
ailHa
-
P
ouliQuen
(2009) formula
but we include nonlinear dependence of on N:
Here C
1
and C
2
are positive coefficients that re-
quire calibration (cf. P
ailHa
& P
ouliQuen
, 2009), and
m
crit
is the static, critical-state value of m
eq
that applies
when the stress is lithostatic, the pore pressure is hy-
drostatic, and N
=
0. As 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
decreases monotonically but not indefi-
nitely in response to decreasing normal stresses and
increasing shear rates. Through its dependence on m
eq
and N, ψ evolves in response to evolution of all of the
dependent variables in DIGCLAW: m, v, h, and p
bed
.
We evaluate depth-averaged mass conservation
for the two-phase mixture by integrating (4) through
the flow depth to find
The term including /dt in (8) accounts for changes
in 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
, and substitution of this
equation in the second line of (8) leads to
This depth-averaged mass-conservation equa-
tion for the mixture is the second evolution equation
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
Although (6), (7), and (9) summarize the kin-
ematic effects of the granular dilation rate, they do not
describe the mechanical causes of dilation. For this
purpose we use an equation modified slightly from
one proposed by i
veRson
(2009),
where 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
π
/2), α is the mixture compressibility (a property that
commonly declines as m increases), and σ
e
is the ef-
fective normal stress (defined as σ
e
=
σ
-
p, where σ is
the mean total normal stress and 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
is constant.
If no macroscopic shearing occurs (i.e., γ
=
0), then
(10) reduces to a standard equation used in quasi-static
soil consolidation theories; but if shearing occurs in
a closed container that enforces ∆v
s
=
0, then (10) re-
duces to
e
/
dt
=
γ
tan
ψ/α. This equation erroneously
predicts that σ
e
increases with time if ψ>0 and shearing
proceeds at a constant rate γ, thereby demonstrating that
(8)
(9)
(10)
(11)
(12)
(13)
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D.L. GEORGE & R.M. IVERSON
418
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
flow scaling that applies if H
/
L
<<
1. This scaling in-
dicates that
2
p
/
∂y
2
>>
2
p
/
∂x
2
because
2
/
∂y
2
scales
with 1
/
H
2
, whereas
2
/
∂x
2
scales with 1
/
L
2
. Conse-
quent neglect of
2
p
/
∂x
2
reduces (17) to
Another step involves use of the approximations
v
y
=
(y
/
h)
dh
/
dt and v
x
=
v to recast the total time de-
rivatives in (18) as
(s
avaGe
& i
veRson
, 2003). Then (18) can be rewritten as
Depth integration of (19) is accomplished term-by-
term by using Leibniz' rule and applying the stress-free
surface boundary conditions p
(h)
=
σ
(h)
=
0, yielding
where p denotes the depth average of p, and v/h is
used to approximate the depth-averaged shear rate.
The term ρ
f
g
cos
θ arises in (20) from depth integra-
tion of the pore-pressure diffusion term in (19) and
application of a zero-flux basal boundary condition
that requires the pore-pressure gradient at the bed to
remain hydrostatic: [∂p
/
∂y]
y=0
=
- ρ
f
g
cos
θ . The term
-
(σ
-
p)
dh
/
dt arises from depth-integrating the term
that includes
(σ
-
p)
/
∂y in (19) by parts. This term
cancels some other terms and thereby reduces (20) to
where
is the depth-averaged mean total normal stress, and κ
is a longitudinal normal-stress coefficient that equals 1
if the stress state is hydrostatic (i
veRson
& d
enlinGeR
,
2001). Equation (21) is a valid, depth-integrated, one-
dimensional pore-pressure evolution equation, but it
retains two pore-pressure variables, p and p, rather
than the desired variable, p
bed
.
To express (21) in terms of p
bed
, approximations of
p and [∂p
/
∂y]
y=h
are necessary, and we obtain these ap-
proximations by utilizing the assumption that m=m at
all depths. This assumption implies that Λv
s
and Λq are
not functions of y, further implying that
2
p
/
∂y
2
is not
a function of y in (18) and (19). With this stipulation,
we solve
2
p
/
∂y
2
=
constant and employ the hydro-
static basal boundary condition ∂p
/
∂y
|
y=0
=
-
ρ
f
g
cos
θ
EVOLUTION OF PORE-FLUID PRESSURE
Development of a depth-averaged evolution
equation for p
bed
involves several steps. The first en-
tails use of a linear, Darcian drag formula to relate q to
the gradient of excess pore-fluid pressure, p
e
:
Here p
e
=
p
-
ρ
f
g
(h
-
y)
cos
θ, where p is the total fluid
pressure, and k is the intrinsic hydraulic permeability of
the granular debris (b
eaR
, 1972). A linear drag formula
such as (14) may oversimplify the effects of complex
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
oHn
-
son
et alii, 1990; s
Hamy
& z
eGHal
, 2005).
Substitution of (14) into (5) yields a fundamental
equation that shows how the divergence of
is related to the granular dilation rate, Δv
s
:
Next, Δv
s
can be eliminated from (15) through
use of (10). If k
/
μ is assumed constant (an assump-
tion that is easily relaxed computationally), the re-
sulting equation reduces to
where k
/
αμ plays the role of a pore-pressure diffusivity.
From (16) we obtain a forced, advection-diffusion
equation governing evolution of p
e
by first using the defi-
nitions of effective stress and excess pore-fluid pressure
to find that
e
/
dt
=
/
dt
-
dp
e
/
dt
-
d
[ρ
f
g
(h
-
y)
cos
θ]
/
dt.
Substitution of this equation into (16) yields
The forcing terms on the right-hand side of (17)
express the evolving effects of the shear-induced dila-
tion rate γ
tan
ψ, the mean total stress σ, and the hydro-
static pore-pressure component ρ
f
g
(h
-
y)
cos
θ. Note
that if all of the time derivatives in (17) are zero and
γ
tan
ψ is constant, the equation reduces to the steady-
state balance
which can alterna-
tively be expressed as .
This result
shows that porosity creation during steady dilation
is balanced by a steady influx of fluid that fills the
enlarging pores.
The next step in obtaining our pore-pressure evo-
lution equation is depth integration. Preliminary steps
involve recasting (17) in terms of the total pore-fluid
pressure, p
=
p
e
+
ρ
f
g
(h
-
y)
cos
θ, and invoking shallow-
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
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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
419
and pressure-free surface boundary condition p(h)
=
0
to find that p(y) satisfies the quadratic equation
This equation indicates that temporal evolution of
p is represented entirely by the evolving values of the
basal pressure p
bed
(t) and flow thickness h(t). Equation
(23) also implies that
Substitution of (22) and (24) into (21) then yields
The derivatives d(ρh)
/
dt and dh
/
dt can be elimi-
nated from the right-hand side of (25) by using the
mass-conservation equation (9), yielding the final
form of the evolution equation for p
bed
:
The first term on the right-hand side of (26) ac-
counts for pore-pressure relaxation due to the depth-
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
For a debris mixture with ρ
=
ρ, depth integration
of the left-hand side of the x component of the mo-
mentum-conservation equation
yields a result that can be written in several forms,
including
The first line of (27) is exact, but subsequent lines
assume that differential advection of x momentum as
a function of y is negligible, and therefore omit the
integral containing v
x
-
v
. In the second line of (27),
the term in brackets expresses mass conservation and
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
explicitly represent the effects of evolving 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
.
This term equals hv
(
/
dt) and thereby accounts for
the effects of D on evolution of ρ.
The right-hand side of the depth-averaged momen-
tum-conservation equation expresses the sum of forc-
ing effects due to gravity and resisting effects due to
internal and boundary stresses, as derived by i
veRson
(1997). Addition of the right-hand side to (27) yields
the fourth evolution equation solved by DIGCLAW,
Here τ
s
and τ
f
are the basal shear tractions exerted
by the solid and fluid phases, respectively.
To estimate τ
s
we use the Coulomb-Terzaghi
equation for granular friction influenced by pore
pressure and dilatancy,
where f
bed
is the steady-state (zero-dilatancy) friction
angle of grains in contact with the bed, f
bed
+
ψ is the
effective basal friction angle when nonzero dilatancy is
present (i
veRson
, 2005), and ρgh
cos
θ
-
p
bed
is an estimate
of the basal effective stress, σ
e bed
. To estimate τ
f
we use
where 1
-
m is the fluid volume fraction and v
/
h is, again,
a depth-averaged approximation of the shear rate γ.
MATHEMATICAL CLOSURE
Two additional relationships are needed to evalu-
ate D and κ, and thereby complete the mathematical
model. We obtain an equation for D by combining (5),
(6) and (15) to find that
The second line of (31) assumes that k
/
μ is
constant and that the pore-pressure distribution
obeys the quadratic relationship specified in (23).
It also shows that p
bed
remains equilibrated to the
dilation rate D as both quantities evolve. This be-
havior is a logical consequence of mass conserva-
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
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D.L. GEORGE & R.M. IVERSON
420
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
b
eRGeR
et alii, in press). The results presented here
depict 10 s of debris-flow motion; generation of the
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
We focus first on predicted short-term evolution
of the dependent variables h, v, m and p
bed
in the
upper part of the flume, from 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
cos
θ). The solid
volume fraction is uniformly m
=
0.61, matching the
mean value measured by i
veRson
et alii (2010), and
implying a loosely packed initial state. The panels
of Figure 2 show these initial conditions as well as
the evolving longitudinal profiles of all dependent
variables at t
=
0.2, 0.4, 0.6, 0.8 and 1.0 s.
The model predicts that as downslope motion be-
gins, the highest velocities occur near the flow front,
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
(Figure 2d). At t = 1 s, for example, much
of the mass has p
bed
~
10 kPa and h ~
0.5 m, implying a
mostly liquefied state (i.e., p
bed
ρ
gh
cos
θ). An excep-
tion to this behavior develops in the advancing flow
snout, which begins to dilate almost immediately be-
cause h→0 at the snout tip, and implied shear rates
tion in a fully saturated mixture together with our
assumptions of Darcian fluid flow and m
=
m.
The value of the longitudinal stress coefficient
κ generally can vary from about 0.3 to 3, depending
on whether flowing debris undergoes longitudinal
extension or compression (s
avaGe
& H
utteR
, 1989;
i
veRson
, 1997, 2009; i
veRson
& d
enlinGeR
, 2001).
The effects of such variation are relatively subtle but
potentially quite complicated. To avoid such compli-
cations while focusing our computations on coevolu-
tion of m, v, h, and p
bed
, we have used the traditional
shallow-flow assumption, κ
=
1. The most important
consequence of this assumption is that the term (1
-
κ)
h∂p
bed
/
∂x vanishes from (28), implying that pore-fluid
pressure exerts direct effects on flow momentum only
through its influence on basal Coulomb friction.
MODEL PREDICTIONS AND TESTS
As a demonstration and test of our model predic-
tions, we have used DIGCLAW to simulate behaviour
measured in a series of eight replicate experiments
performed in the USGS debris-flow flume. In each
experiment 10 m
3
of water-saturated sand, gravel
and mud ("SGM") discharged abruptly from behind a
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
veRson
et alii (2010) presented de-
tails of experimental protocols, data acquisition and
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.
Our DIGCLAW simulations used 1000 fixed, uni-
formly spaced Eulerian grid cells on a domain ranging
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
e
v
eQue
, 2002;
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)
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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
421
sal pore pressures gradually relax from their elevated
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
veRson
et alii, 2010). The peak flow depth
occurs just behind the snout and gradually declines
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
veRson
et alii, 2010). The high speed
of the snout leads to commensurately large shear
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.
are accordingly high. As the solid volume fraction in
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
Behavior computed as the simulated flow moves
further downslope differs qualitatively from short-
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
background image
D.L. GEORGE & R.M. IVERSON
422
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
COMPARISON OF COMPUTED AND MEASU-
RED TIME SERIES
Next we compare model predictions with values of
flow depth, h(t), basal pore-fluid pressure, p
bed
(t), and
total basal normal stress, σ
bed
(t) measured in the eight
replicate SGM experiments described by Iverson 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.
Figure 4 compares model predictions with time
series measured at two instrumented cross sections:
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
<
1
s. Viewed in more detail, the evolving values of
predicted flow depth h(t) and basal total normal stress
σ
bed
(t) at x
=
32 m match those of data relatively well
(Figure 4 a and b), but the predicted basal pore-fluid
pressure p
bed
(t) is somewhat lower than measured val-
ues (Figure 4c). At 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
In our simulations of debris-flow flume experi-
ments, values of some parameters were not pre-
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
background image
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
423
putations, but we believe that size segregation would
enhance this development and lead to attendant
regulation of flow speeds. Recent work on grain-size
segregation in dry granular avalanches (e.g., G
Ray
&
a
nCey
, 2009) may point the way toward including
segregation effects in our debris-flow model.
CONCLUSION
Our model differs from previous models of
debris-flow dynamics by describing coupled evolu-
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.
By computing simultaneous evolution of several
variables, our model provides more detailed predic-
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
We thank Roger Denlinger and Mark Reid for
manuscript reviews, and we thank all those who have
contributed to successful experiments at the USGS
debris-flow flume.
Initially, when we began simulating the flume
experiments, we found that the model predicted flow
velocities almost twice as large as those observed at
x
=
66 m. The high velocities were due to widespread
persistence of very high pore-fluid pressures, such
that friction remained close to zero. By increasing
the ratio k
/
μ to values about five times larger than
those measured in quasistatic permeameter tests
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.
Additionally, we followed the precedent es-
tablished by P
ailHa
& P
ouliQuen
(2009) and se-
lected values of the dilatancy coefficients C
1
and
C
2
to optimize model fits to the data. Interestingly,
however, despite the fact that many aspects of our
model differ from those of the P
ailHa
-P
ouliQuen
(2009) model, our values C
1
=
0.5 and C
2
=
20 are
comparable to the values C
1
=
4.09 and C
2
=
25 used
by P
ailHa
& P
ouliQuen
to optimize their fits to data
from small-scale, underwater granular avalanches.
(Note that in the P
ailHa
-P
ouliQuen
formulation, k
3
is analogous to our C
1
, and k
2
is analogous to our
C
2
.) This consistency lends some credibility to the
formulae that employ C
1
and C
2
for calculating the
dilatancy angle ψ (i.e., equations (12) and (13).
The most significant discrepancy between our
model predictions and measured flow behavior in-
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
veR
-
son
, 2003). Thus, as segregation occurs, the perme-
ability 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-
background image
D.L. GEORGE & R.M. IVERSON
424
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
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