Marine Geology 213 (2004) 439 – 455
www.elsevier.com/locate/margeo
Flow models of natural debris flows originating from
overconsolidated clay materials
F.V. De Blasioa,*, A. Elverhbia, D. Isslera, C.B. Harbitzb, P. Brync, R. Lienc
a
Institutt for Geofag, Universitetet i Oslo, Postboks 1047 Blindern, 0316 Oslo, Norway
b
Norges Geotekniske Institutt, Postboks 3930 Ullevål Stadion, 0806 Oslo, Norway
c
Norsk Hydro ASA, 0246 Oslo, Norway
Abstract
In this paper, slides and debris flows in overconsolidated clay materials are simulated numerically. As a case study, the
models are applied to the Storegga slide in the Norwegian Sea and in particular to the sub-region called Ormen Lange, where
the information available is the most precise for a subaqueous debris flow. Three different models for the rheology of clay are
used: a viscoplastic (Bingham) fluid model, a viscoplastic fluid with interspersed solid blocks, and a viscoplastic model with
yield strength increasing with depth.
The small-scale debris flows in the Ormen Lange area can be reasonably well understood in terms of a pure Bingham model
without granular effects and blocks. The presence of intact blocks in the region, however, indicates that at least the top layer of
the sliding sediments was not destroyed by the flow. It suggests that the flow occurred mainly at high shear rate in a lubricating
layer of mud deriving partly from the disintegration of the block’s own material, and possibly from the entrainment of
hemipelagic sediments along the flow path while the top part was left unsheared. The failure of the model with blocks probably
stems from the use of the Coulomb friction law to represent the interaction between the block and the seabed. The Bingham
model works better because during the flow of such fluids an unsheared plug region is formed naturally, even in unconsolidated
materials. Combining the simulations with these three models, a possible scenario for the Ormen Lange debris flows is deduced
according to which the lubricating layer supporting the blocks has a yield strength of about 10–15 kPa.
D 2004 Elsevier B.V. All rights reserved.
Keywords: consolidation; debris flows; modeling; submarine landslides; viscoplastic materials
1. Introduction
Submarine slides developing into debris flows and
turbidity currents represent the most effective process
* Corresponding author. Fax: +47 22856422.
E-mail address: [email protected]
(F.V. De Blasio).
0025-3227/$ - see front matter D 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.margeo.2004.10.018
of sediment transport from the shallow continental
margin to deeper parts of ocean basins. Slides and
debris flows are particularly common along glaciated
margins that have experienced high sediment fluxes to
the shelf break during and after glacial maxima
(Elverhøi et al., 2002; Vorren et al., 1998). During
one single event, typically lasting for a few hours or
less, enormous sediment volumes (up to several
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F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
thousand cubic kilometers) can be transported over
distances exceeding hundreds of kilometers (e.g.,
Locat and Lee, 2002). When information from
glacially influenced as well as non-glacial margins is
combined, it is seen that the flows with long runout
distances (more than 100 km) generally occur on
slopes of less than 28 (Booth et al., 1993; Vorren et al.,
1998) and that the largest volumes of displaced
sediments are associated with long runout distances
(Elverhøi et al., 2002). Based on information from
cable breaks, flow velocities in the range from 20 to
100 km h1 are calculated, even on slopes less than 18
(Heezen and Ewing, 1952; Bjerrum, 1971).
In order to understand the physics of these mass
flows, the process is divided into a release phase,
followed by break-up, flow and final deposition
(Elverhøi et al., 2002). The release phase involving
initial slide triggering and instability is typically
treated on the basis of geotechnical principles while
the flow phase is analyzed either from fluid dynamical
principles or as a granular flow, depending on the
composition of the moving masses. The phase of
break-up and disintegration represents an intermediate
stage. Moving masses are being disintegrated and
dispersed into smaller particles. The break-up phase
can be approached from a geotechnical perspective
while the movement of the masses follows the laws of
motion of a non-Newtonian fluid. Depending on the
sediment composition and energy available, the
duration of the break-up phase may vary. In turn,
the disintegration may alter run-out length, velocity
and other important quantities. In the case of soft and
Fig. 1. Overview map showing the location of the Storegga slide between the Vbring Plateau and the North Sea Fan. The debris-flow area has a
length of more than 400 km. The rectangle indicates the coverage of the detailed map of the Ormen Lange area shown in Fig. 2A.
F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
easily remolded sediments, the break-up phase represents a short episode. On the other hand, when more
overconsolidated sediments are involved, the break-up
phase may represent a significant part of the total flow
441
time. In particular, in small and medium size slides
with short run-out distances and in slides originating
from overconsolidated materials, the flowing period is
not long enough to cause complete remolding of the
Fig. 2. (A) Detail map of the Ormen Lange area (cf. Fig. 1 for location). The debris-flow lobes E4 and E5 and the corresponding profile lines
used for the numerical simulations are indicated. (B) Bathymetric profile of the slide path E4 used in the numerical simulations. The simulations
reported in Section 4.1 used the pre-Storegga glide plane, those in Section 4.2 were run on the more ragged present-day profile.
442
F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
Fig. 2 (continued).
masses. A situation with incomplete remolding is
found in the latest phase of the Storegga event on the
western margin of Norway.
The Storegga area shown in Fig. 1 (which includes
the Ormen Lange area, see detail map in Fig. 2A) has
been intensively investigated due to plans for gas field
development. Detailed sea floor surveying combined
with high-resolution seismic exploration and sediment
coring have provided an extensive database and
detailed maps of the various slide phases. In this
article we have used the interpretation by Haflidason
et al. (2002) on the depositional sequence in Storegga.
Although the sequences of slides and slide phases
proposed by them may be subject to some controversy, the extensive data base of all the slides shows a
rather consistent pattern of runout and drop height
with respect to sediment volume. Due to this
consistency of the field data, our basic approach is
to use these data as a background for our modeling
work. As long as no unified and general flow model
for sub-aqueous gravity mass flows exists, we cannot
test or validate the field data independently. Our
approach has been to use the extensive database in
order to develop and subsequently test various
numerical approaches.
2. Slide evolution, flow types and model selection
The Storegga area is characterized by periodic
sliding, the period being on the order of 2–300,000
years, with the latest slide event at about 8200
calendar years B. P. Thus we find ourselves at the
beginning of such a cycle, and are not able to sample
and investigate the type of sediments that were
actually involved in the last Storegga slide. The
glaciation has heavily influenced the sediments in
the present headwall area and further landward by
deposition and/or by ice loading. Due to this situation
the sediments included in the Storegga slide may have
been characterized by significant lateral variation and
strong horizontal gradients with regard to composition
and properties. After an initial phase of release of
normally consolidated sediments, the later phases of
sliding involved sediment overconsolidated due to ice
loading. The debris-flow lobes found in the Ormen
Lange area (Fig. 2) contain large blocks or slabs of
nearly intact sediment interspersed in remolded
material. The sea-floor images and the seismic profiles
strongly suggest that the break-up process did not
proceed beyond an early stage in which only material
from the shear zones was remolded. A rather delicate
feedback mechanism may determine whether a submarine landslide comes to a halt after a relatively short
distance or becomes liquid and covers enormous
distances. If the transition between consolidated and
liquid state is sufficiently rapid, the flow will travel
for a long time, during which the remolding and loss
of strength due to water entrainment (wetting) can
progress even further. If it is too slow, the friction
remains relatively high and the slide comes to a halt
before most of the mass had a chance to be remolded.
If the grains constituting the sliding material are
sufficiently large, the effects due to grain–grain
contacts (both frictional and collisional) become
important, resulting in deviations from the simple
hydrodynamical behavior.
Sandy flows are characterized by their granular
behavior where grain–grain interactions play a very
important role in the flow dynamics (see, e.g.,
Iverson, 1997). The physics of dry granular flows
(geological examples of which are some pyroclastic
flows or rock avalanches) has progressed more than
our knowledge of wet gravity flows, especially
because of the industrial use of granular materials
(Campbell, 1990). In the subaqueous environment,
grain–grain interaction is mediated by the presence
of the fluid, and this has several consequences
(Bagnold, 1954; Savage, 1984; Iverson, 1997). The
fluid alters both the single-particle motion and the
F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
grain–grain interaction. The fluid can also develop
excess pore pressure. An increase of pore water
pressure leads to local weakening of the grain
contacts, resulting in a dramatic reduction of the
shear strength of the material. The formation and
diffusion of pore water pressure through the body of
the debris flow is controlled by the clay–sand ratio,
among other factors.
Presently, two classes of models are used to
simulate subaqueous debris flows. A depth-integrated
granular model (Savage and Hutter, 1989, 1991) has
been extended to incorporate the effects of viscous
pore fluid (Iverson, 1997). The model by Norem et al.
(1987, 1989) belongs to the same class; it implements
Bagnold’s dispersive pressure caused by grain–grain
and grain–fluid interaction. Such models are specifically used if the flow contains a substantial amount of
large clasts (i.e., sand-sized or larger).
The second class is constituted by viscoplastic
models (e.g., Johnson, 1970; Huang and Garcı́a, 1998,
1999) treating the material as a fluid with nonNewtonian properties. The rheological properties of
the flowing mud would ideally be inferred from
geotechnical laboratory measurements on soil samples. The viscoplastic model works well for clay-rich
or muddy materials with cohesion and a very low
content of coarse particles capable of particle–particle
interactions. Both types of models have been widely
discussed (Johnson, 1970; Locat and Lee, 2002;
Norem et al., 1990; Iverson and Vallance, 2001).
A more general model taking into account both
effects should be employed in cases where both
particle–particle and particle–water (viscous) interactions are important. The dimensionless Bagnold
number characterizes the relative importance of
viscous and collisional forces; it is defined as
frictional or collisional forces between grains
viscous forces
mċck1=2
ð1Þ
¼
gd
Ba ¼
where ċ is the shear rate (Bu/Bz in the case of planar
shear), k is Bagnold’s dimensionless linear grain
concentration, m is the particle mass, g is the
viscosity, and d is the particle diameter. The dispersive pressure plays an important role if the
Bagnold number is substantially larger than about 40.
443
The grain-size distribution in the Storegga area is
strongly influenced by the input of fine-grained
glacial erosion products. A typical particle-size distribution in this kind of subaqueous debris flows is
30–40%, clay, 30–405% silt and 20–30% sand
(Vorren et al., 1998); gravel is almost completely
absent. We find that the Bagnold number for the
typical Storegga material is on the order of Bac105,
which implies that the dispersive pressure is negligible. Such a small value, mainly due to the lack of
large particles and correspondingly high clay content,
favors the viscoplastic model for explaining debris
flow dynamics along these margins. Recent modeling
of muddy debris flows using the viscoplastic approach
shows a high degree of correspondence between
observed and modeled run-out distances for small
debris flows (Huang and Garcı́a, 1999). The composition of the material used in the experiments by
Mohrig et al. (1998, 1999) was selected to reflect the
field data, supporting our choice of model for muddy,
almost clast-free debris flows.
In this work a numerical code for the flow of a
Bingham material is used. The model (called BING)
was developed by Imran et al. (2001). BING is
derived from a two-dimensional formulation of the
conservation equations (i.e., only the main flow
direction of the material and the direction perpendicular to the bed are included). Vertical integration of
the conservation equations reduces their information
content but even more so their complexity. The
flowing material is divided into several segments
along the flow direction, and the acceleration of each
node is calculated according to the forces acting on it.
Although in the present work only the Bingham
rheology has been used, the model is actually more
general, as it implements both a Herschel–Bulkley
fluid (which is a generalization of the Bingham fluid
allowing for non-linear dependence of the stress on
the strain rate) and a bi-viscous rheology (which does
not require the shear strength). The model is
Lagrangian (i.e., the acceleration of the nodes is
calculated in a frame moving with the slide).
As discussed earlier, the physical state of the
material during the flow defies precise analysis at
present. There is clear evidence that the debris flow
lobes in the Ormen Lange area were never completely
remolded: There are blocks in the final deposit and the
deposit thickness is approximately independent of the
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F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
local slope. Lacking a robust debris flow model for
only partially liquid soil, we apply a viscoplastic
model with different modifications and select those
that fit the field data best and also are physically
realistic. The yield strength will be varied and we will
study the deposit shape, runout and velocity as a
function of the yield strength. For reasons explained
in the paragraph dedicated to the debris flow models,
the viscosity plays a minor role and will be kept
constant.
We will consider the following scenarios:
(1)
A pure Bingham fluid with rheological properties
independent of the flow, depth and previous
history (BING). This will provide a reference
model. Note that the model works well in some
cases (see the extensive literature on the debris
flows of the Bear Island Fan, e.g., Marr et al.,
2002 and references therein), but may not be
fully appropriate for Storegga, at least for the
small to medium-size flows of the later stages.
The top panel of Fig. 3 shows a sketch of the
lowslide evolution in this approximation.
(2) One or more solid blocks embedded in a
Bingham fluid (B-BING): The persistence of
isolated solid blocks is clearly indicated by the
field observations of the final deposit. In
addition, the presence of blocks is a logical
consequence of the high yield strength of the
initial sediment pile. There is not enough shear
deformation and enough time to remould the
material to a liquid form. In order to account for
these blocks, a number of additional parameters
are introduced by this model: (i) N b, the number
of blocks. (ii) If the blocks shed mass at the
bottom, two material constants (quantifying the
hardness of the material and the abrasion
efficiency) describe the efficiency of block
reduction. (iii) The bed friction angle d, determining the force exerted on the block by the bed.
Fig. 3. A schematic view of the models used in this study. Pure Bingham fluid (upper panel), Bingham fluid with blocks (middle) and Bingham
fluid with a depth-dependent yield strength (bottom).
F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
(3)
These constants can only be crudely estimated.
The present model must be considered as
tentative. To our knowledge, no model coupling
liquid soil to solid blocks has been published
before. A possible flow configuration for this
case is shown schematically in the middle panel
of Fig. 3.
A modified Bingham model with an increase of
the yield strength with depth (C-BING, bottom
panel of Fig. 3): Static measurements of the
soils in Storegga and Ormen Lange show a
marked increase of shear strength with depth.
Sediments in the deeper parts have low water
content and are strongly consolidated as a
consequence. In this case, the yield strength is
split into a sum of two different terms: the
cohesion (i.e., a part independent of the depth)
and a part increasing linearly with depth, which
mathematically is equivalent to a friction
proportional to the stress generated by overburden mass. The effective friction angle also
incorporates the effects of excess pore pressure
and is therefore much smaller than for a pile of
dry grains. The model is characterized by three
instead of two parameters.
3. Mathematical formulation of the models
3.1. Pure Bingham-fluid model
The simplest rheological model of a non-Newtonian fluid is the Bingham (or viscoplastic) model. In
this model, the shear rate (c˙=Bu/Bz in the present
setting) where u is the velocity in the x-direction
parallel to the bed, grows linearly with the shear stress
s if a threshold s y, termed the yield strength, is
exceeded. This condition is translated into the
rheological equation
lB
Bu
¼ sgnðsÞ jsj sy Q jsj sy
Bz
ð2Þ
where l B is the dynamic (Bingham) viscosity and H
is the Heaviside function, which is unity if the
argument is positive and zero if it is zero or negative.
It follows that in a Bingham fluid a state of shearing
motion is possible only if the absolute value of the
shear stress s is larger than the yield strength. The
445
existence of a non-zero yield strength is generally due
to friction and interlocking of the microscopic
constituents of the material, as well as electrostatic
attraction between clay platelets.
It is not the purpose of this work to explain all the
details of the model, which can be found in the
literature (Huang and Garcı́a, 1998, 1999; Imran et al.,
2001)—we only sketch the essence of the BING
equations here. To be more specific, let us consider
the first of the basic equations of the fluid (which are
the two components of the Navier–Stokes equations
(NSE) and the continuity equation):
Bu Bu2
BðuwÞ
1 Brxx
1 Bsxz
þ
¼
þ
þ
Bt
Bz
qd Bx
qd Bz
Bx
þ RVgx :
ð3Þ
Here RV=(q dq w)/q d, g x =g sinb, where g is the
gravitational acceleration, q d and q w are the density of
the solid material and water, respectively, b is the
slope angle and w is the velocity in the z-direction
(perpendicular to the bed). By integrating this
equation in the direction perpendicular to the bed,
once in the region where |s xz (z)|bs y (the no-shear or
plug region) and once over the entire flow depth, one
can obtain workable relations for the plug layer and
for the whole material. The Leibniz rule is applied to
the non-linear terms to interchange the derivative and
the integral.
Note that all the models are two-dimensional, i.e.,
only the dimensions belonging to the same plane of
the gravity force have been considered. The volume
per unit width (in km2) is so defined as the volume
of the slide divided by its width, and it is a
substitute of the slide volume in our two-dimensional simulations.
The depth integrals of the advective terms can be
approximately evaluated using the velocity profile for
a steady uniform flow. The total flow depth, D, is the
sum of the depths of the shear layer, D s, and the plug
layer, D p. The velocity in the latter is uniform and is
denoted by U p. Assuming a steady-state velocity
profile in a transient flow implies neglecting the
inertial forces with regard to the velocity profile (but
not for the flow in its entirety). The normal stress in the
x-direction is approximated by the pressure due to the
overburden load, r xx (x, z,t)=(q dq w) (D(x, t)z)g
g z (x)=r zz (x, z, t). Upon integration of the shear stress
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F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
term, one obtains the difference between the shear
stresses at the upper and lower interfaces, i.e., the sum
of the hydrodynamic drag, s drag, and the viscous
friction, sy þ lB Bu
Bz jz¼0 . Details can be found in the
papers cited above and in (De Blasio et al., 2004). We
use a curvilinear coordinate system and neglect
curvature effects, which are small for large curvature
radii.
One can further simplify the equations by adopting a Lagrangian framework, i.e., a description in
which the reference frame is moving with the
material. The final result—after some algebra—are
two equations for the acceleration of the plug layer
and of the whole sediment. For the plug layer
(velocity U p) one finds
DUp
BD sgnðU Þ ¼ RVgz
sy þ sdrag
Bx
Dp qd
Dt
BUp
ð4Þ
þ RVgx þ U Up
Bx
where U is the average velocity over the entire flow
height (plug and shear layers), the material derivative
D/Dt is defined with respect to U as (B/Bt)+U(B/Bx),
and s drag is the drag shear stress due to the
interaction of the ambient marine water with the
moving debris. The terms on the right hand side are
the earth-pressure acceleration (which tends to
equilibrate the inhomogeneities in the thickness of
the deposit), the resistance at the boundaries, the
gravity acceleration along the bed and finally a term
which accounts for velocity gradients in the debris
flow.
For the depth-averaged acceleration one finds
DU
1 B
7
2 2
BD
2
¼
U D Up U D þ Up D RVgz
Dt
D Bx
5
5
Bx
Up
sgnðU Þ
sy þ 2lB
þ sdrag þ RVgx : ð5Þ
Dqd
Ds
In deriving this expression, a steady-state velocity
profile in the shear layer has been assumed.
The drag force is expressed as
sdrag ¼
CT
q U 2:
2 w p
ð6Þ
The total drag coefficient C T is approximated as
BD BD
Q
sgnðU Þ þ CF ;
CT cCP ð7Þ
Bx Bx
where the coefficients C P and C F correspond to the
pressure and the frictional drag, respectively. The part
proportional to C F is the (specific) drag force on the
upper face of a rough plate, while the part proportional to C P accounts for the finite thickness of the
sediment. |BD/Bx|Dx i is the equivalent area of element
i normal to the flow direction.
The coefficient C P can be estimated by measuring
the drag force on objects of given thickness. In our
study, the value of the coefficients was inferred from
the literature (Newman, 1977, pp. 17–20). The drag
forces turn out to be important for the front part of the
debris flow, which travels at least at twice the speed of
the rear part. In the model, we do not account for
added-mass effects, which arise from the inertial
resistance of the ambient water against accelerations.
In practice, the solution of the problem as a
function of time must be obtained iteratively, using
finite time steps and a space mesh. The entire flowing
mass is divided into N vertical elements, which at the
beginning of the calculation are equally spaced. The
equations of motion for the nodes are derived from
Eqs. (4) and (5) by temporal and spatial discretization.
When the acceleration of one node is calculated, its
new velocity and position after a finite (short) time
step can be found. Since adjacent nodes can have
different accelerations, their relative velocities and
positions will change as a function of time. The
conservation of the total amount of mud is ensured by
calculating the new height of the mud at node i, D i , so
that the area of the longitudinal section, A A i , of all
the elements i=1, . . . , N is constant during the
calculation:
Di ¼
Ai
;
Xiþ1 Xi
ð8Þ
where X i is the position of the ith node.
The thickness of the shear regions is calculated at
each time step and for each node from the equation
U
Ds ¼ 3D 1 ;
ð9Þ
Up
and for the plug region from
Dp ¼ D Ds :
ð10Þ
Fig. 4 shows the time evolution of a debris flow
calculated with the BING code.
F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
447
In addition, the block interacts with the bottom via
a Coulomb-like friction force
!
n
X
Di Dxi ðqd qw Þg cosb 4tan h
FCoul ¼ sgnðUÞ
i¼m
ð14Þ
Fig. 4. Example of flow evolution over time as computed with the
BING model for a yield strength of 8 kPa. The bathymetry
corresponds to the pre-Storegga profile of slide E4. The flow height
is exaggerated by the same factor as the bathymetric profile
(approximately 50-fold).
where h is the bottom friction angle and b* is the slope
angle averaged over the segments m to n. The
acceleration of the block is thus calculated from the
previous expressions for the forces acting on the block,
using an average value for the slope angle.
In order to account for block disintegration at the
bottom, a model for the abrasion of metals is adopted
(Rabinowicz, 1995). If we imagine the block-bed
interface as a region of interpenetration of the material
accompanied by the formation of grooves on the two
surfaces, we can write the volume of eroded material
as
DV c 3.2. Bingham fluid with interspersed solid blocks
The BING code has been extended to account for
solid blocks embedded in a viscoplastic fluid. Some of
the vertical segments are made bsolidQ by preventing
relative motion between them and shear within them.
The shear stress at the bottom is replaced by a
Coulomb-friction force. Suppose the segments
between m and n describe a solid block, while the
rest of the material is liquid.
Its mass per unit width is given by
M ¼ qd
n
X
Di Dxi :
ð11Þ
i¼m
The downslope gravitational and earth-pressure forces
per unit width acting on this block are
!
n
X
Di Dxi g sin b;
FG ¼ ðqd qw Þ
ð12Þ
D P tan c
DPD x
Dx ¼
kabr
pp
p
ð15Þ
where DP=(q dq w) Vgcos b is the load (i.e., the total
force exerted by the mass on the surface on which it
flows), c is the typical angle formed by the asperities,
p is the hardness of the material (to be measured in
pascal), V is the volume of the material and D x is the
length of relative movement between the block and
the bottom. The dimensionless coefficient k abr=tan c/
p takes into account the average size of asperities.
For cases in which the load is due to gravity one
readily finds (for a constant slope angle b)
dV
kabr
¼ V Dqgcosb;
dx
p
Dqgcosbkabr
x ;
V ð xÞ ¼ V0 exp p
ð16Þ
ð17Þ
ð13Þ
namely an exponential decay of the block volume as a
function of the distance. The disintegration length k is
given by
p
k¼
:
ð18Þ
Dqgcosbkabr
where D_=D m1 and D+=D n+1 are the heights of the
liquid at the left and right end, respectively, and
cosb _, are the corresponding angles.
Field evidence suggests kc3104 m for Storegga,
from which we find p/k abrc3108 kg m-1 s-2. With a
tentative value k abr=10-3 (of the order of the abrasive
coefficient of metal–metal surface with loose abrasive
i¼m
Fep ¼
1
ðqd qw Þg D 2 cosb D2þ cosbþ ;
2
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F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
grains) one finds pc300 kPa for the hardness of the
material.
It is also likely that some superficial material will
be scraped off the sea bottom by the moving slide and
set into motion. It is difficult at the present stage to
quantify the entity of this effect, even though one can
imagine that the thickness of the material involved
would be at most few metres, corresponding to the
weak veneer of recent hemipelagic sediments.
3.3. Bingham fluid with depth-dependent yield
strength
We now turn to the model where the yield strength
depends on the depth. This modification is directly
suggested by laboratory measurements of the static
strength of the material in the Storegga area, which
shows a considerable increase of the shear strength
with depth as a consequence of consolidation and
glacial loading. Here we show how the Bingham
model may be extended to account for such an
increase.
The simplest approximation is to consider the yield
strength as a function of the effective stress as for a
granular medium obeying the Mohr–Coulomb law:
sy ¼ c þ ðrn pw Þtan /;
ð19Þ
where r n is the stress normal to the bed, p w is the total
water pressure (hydrostatic water pressure plus
possibly excess pore pressure), / is the internal
friction angle of the material and c is the cohesion
due to cementation and electrostatic inter-particle
forces. In the absence of excess pore water pressure,
the pressure difference term simply becomes DqgD
and accounts for the buoyancy effect. For a pure
viscoplastic material tan /=0 and the yield strength is
only due to material cohesion, which is assumed
independent of the pressure. A finite internal friction
angle implies an increase in the shear strength as a
function of the depth. This makes the material much
stiffer in the regions where it is thicker (D is large). It
is interesting to observe that this effect compensates
for the tendency of a pure Bingham fluid to flow in
the parts where it is thicker. Intuitively, then, this
model of Bingham fluid plus granular behavior should
flow more independently of the local thickness of the
material. This corresponds to some extent to the
intuitive picture of sand pile behavior. From a formal
point of view, it is indifferent whether the increase of
shear strength with depth is due to granular effects or
to consolidation.
Writing the above expression explicitly one finds
sy ð zÞ ¼ c þ ð Dqgð D zÞ pu Þtan /Vcc
þ Dqg ð D zÞ tan /tot:
ð20Þ
where D is the local debris flow height, and the
coordinate z is measured perpendicular to the bed.
The depth of the shear layer is evaluated on the basis
of this modified yield stress. The total friction angle,
/ tot., includes effects due to excess pore pressure and
may therefore be rather small (0.5–5).
A simple extension of the BING code to account
for such effects was made. The stress at the bottom is
proportional to the thickness of the material at the top,
and this implies some rearrangements and extension
of the BING code where the equations of motion are
calculated. In particular, in the equation of motion for
the plug layer (4), the fixed yield strength s y is
replaced by an expression taking into account the
increase of the yield strength from the top to the
bottom of the plug region:
sy Yc þ DqgDp tan /tot: :
ð21Þ
In Eq. (5) for the averaged acceleration, s y is to be
replaced by an analogous expression, in which the
total flow depth D occurs in place of the plug-layer
depth D p, however.
Finally, it is well known that in some cases the
excess pore pressure may increase due to continuous
loading. In the regions where the excess pore pressure
is high, the grain interactions can be highly damped
with consequent decrease of the shear resistance and
loss of stability. In principle, one should consider the
complicated problem of water diffusion through the
material in which the diffusion coefficient depends on
the density itself. In a program like BING, one can
partially account for the excess pore pressures by
using a simplified solution to the diffusion equation in
the form of an error function. However, it does not
seem that these effects played an important role for
Storegga, as the material was well equilibrated and
the water in the sediment was probably at normal
hydrostatic pressure (except in the weak layer on
which the mass began to slide after the external
triggering influence).
F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
449
4. Results
Calculations were performed with the three
models described above. For a series of reasons, we
found it of special interest to consider the Ormen
Lange area in the Storegga region, and in particular
the lobes denoted E4 and E5 in Ormen Lange
(Haflidason et al., 2002). These lobes extend for
about 15–20 km from the present headwall of the
Ormen Lange area down into the basin. Not only are
they well bounded and relatively easy to track, but
their mass, geometrical characteristics and material
properties are known with some precision. The
reason for such good quality data is that these lobes
cover a large portion of the area destined to the future
exploitation of the natural gas field.
4.1. Model calibration
Fig. 5 shows a back-calculation of the slide E4
using the pure Bingham model, without blocks. The
best fit is provided by a yield strength of about 8–10
kPa. The pure Bingham model is clearly unable to
reproduce the deposit irregularities and the presence
of deposits at high slope. Fig. 6 shows the results with
blocks. The runout is now shorter. In particular, in
Fig. 5. Back-calculation of slide E4 with the BING model: Runout
distances and deposition profiles are shown for various values of the
yield stress. The bed topography corresponds to the reconstructed
sea floor just prior to the slide E4 (Haflidason et al., 2002), the
release mass was chosen to correspond to the reconstructed volume.
The present-day bathymetry along the profile is superimposed for
comparison.
Fig. 6. Back-calculation of slide E4 with the B-BING model with
four blocks. The yield strength in the liquefied parts of the flow is
chosen as 6 kPa while the bed friction angle d of the blocks is varied
between 1 and 58. In order to obtain a stopped block on the steeper
slope before 3 km, d has to be chosen so large that the runout
distance becomes much shorter than observed.
order to obtain a stopped block on the steeper slope
before 3 km, the Coulomb bed friction angle d has to
be chosen so large that the runout distance becomes
much shorter than observed.
Fig. 7 shows the results from C-BING, the model
to adopt if the yield strength in the dynamical
calculations should parallel the unremolded yield
strength, which increases with depth. The model
Fig. 7. Back-calculation of slide E4 with the C-BING model:
Runout distances and deposition profiles are shown for constant
cohesion c=0.1 kPa and various values of the total friction angle
/ tot. An extremely low total friction angle would have to be chosen
in order to reproduce the runout distance of the E4 slide.
450
F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
clearly predicts the occurrence of deposits only at the
toe of the headwall, resembling a rotational slumping—the observed runout distance cannot be reproduced with reasonable model parameters. Irrespective
of the increase of the shear strength with depth in the
pre-failure sediments, the remolded yield strength
relevant for the flow is the one of the thin shear layer
at the bottom and is probably determined primarily by
remolding and wetting (i.e., water entrainment) during
the flow and not so much by the original strength of
the soil.
4.2. Simulation of potential small slides in the Ormen
Lange area
In the previous section we studied the flow of
past debris flows with the purpose of comparing the
results with the observed deposits. We consider here
the flow of future possible debris flows on the
present topography. The Storegga slide reached all
the way up to the continental shelf in the Ormen
Lange area, and the headwall is therefore situated
behind the former shelf edge. During the last ice-age
this area was subjected to glacier loading, and later
was unloaded when the ice-age came to an end. Soil
profiles from geoborings and core samples show stiff
and hard overconsolidated clays with a significant
content of silt, sand and gravel, low liquidity index
and low sensitivity (s tb8). Undrained shear strengths
from 100 to 200 kPa are measured at a depth of 20
m below the seabed in cores collected behind the
headwall (Haflidason et al., 2002).
In order to make the simulations of future slides
from the Storegga headwall as realistic as possible, the
present-day bathymetry of the sea bottom is used for
generating the flow path profiles. Slides E4 and E5 are
used as reference slides for model verification and
calibration, giving a good indication whether initial
conditions and material parameter assumptions are
within reasonable values.
Assuming a slide release in the headwall slope,
the mean slope angle is of the order of 20–25 in
the uppermost part and about 5 in the lower part of
the starting zone. The slope becomes more gentle as
the debris flow runs away from the headwall, and
there is a mean slope angle of less than 2 in the
runout area downslope. A mean sediment density of
1800 kg m-3 is assumed. Table 1 summarizes the
Table 1
Summary of model parameters and initial conditions used in the
study of potential slides from the headwall in the Ormen Lange area
Parameter
Units
Value range
Yield strength
Viscosity
Density
Release volume per unit width
kPa
Pa s
kg m-3
km2
2, 6, 8, 10, 12
30.0
1800
0.059, 0.124
When varying one set of parameters, the other parameters were kept
at their main values indicated in bold face in the third column.
different parameter sets and initial conditions
explored in our simulations.
Fig. 8 (top) shows the final debris flow deposit
for a pure Bingham fluid from two simulations
obtained using different volumes but the same low
yield strength of 2 kPa. Note that this assumed yield
strength is very small considering that values at
least five times larger were necessary to reproduce
the observed runout for the old, quite massive slides
E4 and E5. The reason for this choice is that the
contouritic sediments that have accumulated since
the Storegga slide have a high liquidity index and
are correspondingly soft and sensitive. According to
the preliminary information available (H. Haflidason, personal communication, 2002), these sediments are rather unevenly distributed, with
accumulations at most 1 m deep in troughs and
no sediments at all on the crests. We do not know
how strongly they are able to reduce the yield
strength at the interface between the bed and the
flow, and their effect on the flow dynamics is not
exactly the same as that of assuming an equally low
yield strength in the bulk of the sliding material.
Nevertheless, such simulations will give an idea of
the runout distances and velocities to expect if bed
entrainment is an important factor in a potential
future slide.
The dependence on the yield strength is further
shown in the bottom plot of Fig. 8, where the larger
of the two volumes is adopted. A yield strength of
12 kPa, i.e., close to the value found from the backcalculations of the Storegga lobes E4 and E5, would
produce a debris flow stopping 7 km from the
release area.
Note also the rather strong dependence on both the
yield strength and the volume. The dependence on the
debris flow volume is due to the fact that the drag
force and the resistive force at the bottom are surface
F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
451
data can be compared with tsunami deposits, if they
can be found around the coasts facing the area. If
one is interested in the potential risk from a future
debris flow, the calculated velocities can give
valuable information on both the associated tsunami
and the impact of the debris flow on the sea
bottom. In fact, if a debris flow collides against an
installation, the resulting damage depends on both
the peak pressure during the first milliseconds of
the impact and the (somewhat lower) sustained
dynamic pressure afterwards. These pressures are
determined by the velocity, density and strength of
the flowing soil. Figs. 9 and 10 show the velocity
as a function of time and distance traveled,
respectively, for some selected volumes and yield
strengths. The velocity first increases while the
gravitational force exceeds the friction and drag
forces, reaches a peak value and then decreases as
the flowing sediment layer becomes thinner and the
slope angle decreases.
Estimating the impact forces acting on specific
structures is still a very difficult problem because the
shear-rate dependent properties of the flowing material play an important role. We therefore give only the
double stagnation pressure q du 2 in Fig. 10. The figure
shows maximal values between about 1 and 2 MPa,
according to the volume per unit width and the yield
stress. The actual pressure exerted on, e.g., a pipeline
Fig. 8. Deposition profiles and runout distances of potential slides
along flow path E5 in the Ormen Lange area. (A) Effect of varying
the release volume by a factor 2 at a low yield strength of 2 kPa. (B)
Effect of varying the yield strength from 2 to 12 kPa with the larger
of the two initial volumes in the upper plot. The position of the
future pipeline is also shown.
forces; therefore their relative importance compared to
gravity (body force) decreases with increasing flow
depth and volume.
4.3. Velocity
Another interesting parameter is the velocity of
the debris flow. Calculated velocities can be useful
in at least two respects. For past slides, a knowledge of the velocity can be used for estimates of
the tsunami produced by the debris flow, and these
Fig. 9. Time evolution of front velocity along flow path E5 in the
Ormen Lange area, showing the very strong dependence on the
initial mass and a marked decrease of flow duration and velocity in
the flatter part of the path as the yield strength is increased.
452
F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
Fig. 10. Variation of front velocity (A) and impact pressure (B)
along flow path E5 for the same initial volumes and yield stresses
as in Fig. 9.
is expected to be even larger because of the strength of
the material.
4.4. Sediments piled up at high slope
A feature of many small to medium-size slide
deposits in the Ormen Lange area is a bulge of piledup sediment just upslope of the lowest slope break at a
depth of about 550 m below sea level (Figs. 5 and 8).
The length of these deposits is about 1.5 km, and their
height exceeds that of the deposits immediately
downstream by a factor of 2 to 3. Similar deposits
can be seen at the first slope break at a depth of 430 m
below sea level. Their height is even more impressive,
but they are significantly shorter.
The presence of these sediments at high-slope
positions is not easily reproduced by our simulations,
and they deserve a separate discussion. There are two
distinct possibilities for interpreting these deposits:
The first considers the change of slope to be gradual
enough for the soil to follow the sea-bed topography
without any appreciable resistance and compression.
In that case, the deposits must be attributed to
secondary slides, which, due to their significantly
smaller mass, had a larger total friction angle and
lower speed; they would rapidly have come to a halt
when the slope became less than that angle. The same
effect may occur if we take into account that the tail of
a slide tends to be much slower than the head and thus
reacts more sensitively to changes of slope angle;
also, the release height is smaller in the uppermost
parts of the slab than in its middle. The difficulty with
this interpretation is the considerable fine-tuning
needed for the deposits to stop just at the bend rather
than, say, half a kilometer before or after it.
An alternative interpretation considers the change
of slope to be abrupt and to have a similar effect as a
retaining dam against avalanches. The flowing mass
needs to be accelerated to change its direction, leading
to compressive stresses in the direction perpendicular
to the sea floor. As the soil is very weak (i.e., the
internal friction angle is very low), it will fail along a
shear plane that is almost parallel to the sea floor.
Material is then deposited in the bend until the slope
transition has become so gradual that the soil flowing
over it no longer fails. If this interpretation were
correct, however, one would expect the extra deposits
to extend to roughly equal distances on both sides of
the bend and to be deepest at the bend. But the
bathymetric data suggests that the soil has accumulated mostly upslope of the bend.
In view of these considerations, we lean towards
the first interpretation of these bulges as blate-comersQ
too slow to progress into the much less inclined terrain
at 2 km from the slide scar. In passing we also note
that the BING model and its variants, using a
Lagrangian description of the flow, are not well suited
to capture bshocksQ of the kind described in connection with the second interpretation (see Tai et al.,
2002) for a discussion of this issue in the context of
granular flows). Neither can the development of shear
F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
planes be satisfactorily described in a depth-integrated, one-dimensional approach.
5. Discussion
The sliding of overconsolidated sediments differs
from the sliding of soft sediments including quick
clay. In the latter case the sediments may liquefy or
become completely remolded almost instantaneously
after slide release. Such conditions seem to prevail in
front of fast ice streams where sediments are
deposited at high rates, on the order of tens of
centimeters per year (Dimakis et al., 2000; Elverhøi
et al., 2002). In the case of overconsolidated
materials, the initial phase is characterized by
disintegration of the soil and lasts for a significant
part of the runout distance.
If we use the static value of the shear strength
measured in the laboratory as an input for the model
(Haflidason et al., 2002), there will be no movement
at all—these values are in fact far higher than those
inferred from the location and thickness of the
deposits. Evidently, some kind of remolding or
softening of the sediments must have occurred. This
is indeed very likely, as the material will develop
cracks during the flow, it will shear and water will
be incorporated, with consequent increase in the
liquidity index and decrease of the yield strength. It
might appear that these extremely complex and
unknown physical phenomena should be incorporated in the flow modeling. However, we argue here
that a great deal of phenomenology can be inferred
from simple flow models. A first approach is to
consider a simple Bingham model with a strongly
reduced yield strength as compared to the shear
strength of the original material. The yield strength,
which results from the complex phenomena mentioned earlier, cannot be calculated a priori (Locat
and Demers, 1988). One should simply use the value
that gives the best fit. This approach seems to give
reasonable results, but it does not take into account
the presence of blocks. However, a model incorporating blocks interacting with Coulomb friction (BBING) gives a poorer description than the simple
Bingham model. In fact, as also shown in a parallel
paper (Issler et al., 2005), the simulated results for
the runout as a function of the mass per unit width
453
fit fairly well a power-law function. Additionally,
one finds that the slope ~0.87 of the regression line
is rather insensitive to the value of the assumed yield
strength.
A possible explanation for these findings is that the
material fails along a weak layer (at a depth of about
100 m in the case of Ormen Lange). The material
properties around the weak layer are rapidly changed
due to the high shear and water entrainment (Elverhøi
et al., 2002). Both these effects dramatically reduce
the cohesion of clay along this layer, and the yield
strength plummets. Material is continuously eroded
and comminuted along this layer, and the block moves
in a self-lubricating fashion. This would explain
several facts: (i) the apparent lack of Coulomb friction
(the shear occurs in the self-lubricating layer, with
Bingham fluid properties, rather than Coulombian,
frictional ones), (ii) the presence of blocks, apparently
inconsistent with a pure Bingham model, (iii) the
superiority of a Bingham model. Indeed, if the selflubricating layer is thicker than the shear layer of a
Bingham fluid under the same conditions (same
density, velocity and so on), then the flow of a
Bingham liquid carrying a block at the top is
indistinguishable from the one of a completely
liquefied material. Therefore, blocks and Bingham
fluid can in principle coexist and give the same flow
as a pure Bingham fluid. However, if the selflubricating material is thinner than the equivalent
thickness of the shear layer, the two flows will not be
identical, but still similar. (iv) The fact that the blocks
are not so easily comminuted is due to the very high
shear strength (easily over 100 kPa); there is no
contradiction with the failure of C-BING, a program
that does not incorporate the weak layer. The
lubricating material acts like pore pressure, reducing
the effect of Coulomb friction.
Once we accept this picture, we see that, at least for
this kind of materials, we can by-pass the complex
problem of sediment disintegration in the body of the
debris flow and replace it with the problem of the loss
of strength due to abrasion and erosion along a single
plane (which is probably easier to address experimentally). We conclude that the flow of overconsolidated clay such as in Storegga is controlled more by
shearing in a weak layer than by the disintegration of
the whole body, which is an effect rather than a cause
of the flow.
454
F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
Having described the model emerging from the
present simulations, we can be more specific on the
values that should be attributed to the Bingham fluid
in the Ormen Lange area. The BING model based on
the Bingham rheology reproduces the runout distance
for a wide range of initial volumes per unit width,
with a value for the yield strength of about 8–12 kPa.
Both the slope of the runout curve and the value
seem to be well reproduced. The initial size of the
debris flow constitutes an important parameter for the
flow. As already stated, a statistical analysis of the
debris flow lobes in Storegga shows that the runout
scales like the volume per unit width to the power of
0.87 (Issler et al., 2005). The multiplicative factor in
the power-law relationship, instead, depends much
more strongly on the yield strength. For example, an
increase of the yield strength from 6 to 14 kPa in our
simulations, determines a decrease of the runout
distance approximately by a factor of 2. Clearly,
there is a rather strict empirical correspondence
between volume per unit width and runout. The use
of the empirical power law relation could be
particularly appealing for risk assessment and prediction in the Ormen Lange area. Possibly, similar
considerations could be extended to other clay-rich
areas of the continental shelf.
Peak velocities found from our simulations are on
the order of 30 m s-1 or less depending on the volume
and are reached after only about 1–2 min from start.
The velocity then decreases slowly and the debris
flow comes to rest after a total time of about half an
hour. Such data might turn out to be important for the
risk assessment in the area.
leads us to conclude that the flow dynamics was
dominated by a remolded clay layer at the base of the
slide. The best correspondence is obtained with a
yield strength of 10–15 kPa.
As mentioned previously, we consider the following two issues the most pressing for a significant
improvement of our capacity for predicting potential
flowslides:
6. Conclusions
Research on the topics delineated above will
require a combination of experiments, theoretical
work, and field observations.
In the present paper we addressed the physical
effects at play during the flow of a slide formed by
overconsolidated clay. Due to the valuable data of
unprecedented quality and abundance, the debris
flows in the Storegga–Ormen Lange region in the
Norwegian Sea was considered as a case study. We
found that the values for the yield strength that
produce results comparable to field data are substantially smaller than those measured statically in the
laboratory. This, together with the indications of
absence of granular and Coulomb frictional behavior,
(1)
Break-up and disintegration of overconsolidated
clay is rather poorly understood, despite its key
role in determining whether a slab of unstable
sediment will produce a mere slump, or has the
potential to become a debris flow with long runout. The disintegration process might turn out to
be important in affecting the further dynamics of
the debris flow especially for small slides, as
they presumably spend a higher time fraction in
the overconsolidated state.
(2) Both hydroplaning and shear-wetting of the
bottom layer of a debris flow with the water
penetrating underneath it during hydroplaning or
with the overflowed soft, water-rich marine
sediments may lead to a significant reduction
of bottom friction and thus to long runout
distances. In fact, the two wetting mechanisms
share many dynamical aspects. Neither the
dynamics of bed entrainment nor the diffusion
of the lubricating water layer under a hydroplaning mudflow have been studied to any
depth. In this context, the phenomenon of
(hydroplaning) outrunner blocks, observed both
in the field and in the laboratory, also deserves
further investigation.
Acknowledgments
The authors wish to thank Lars Engvik, Peter
Gauer, Haflidi Haflidason, Kaare Hbeg, Trygve Ilstad,
Tore J. Kvalstad, Jacques Locat, David Mohrig, Gary
Parker (who also provided us with the code of the
original BING model, which is now available on the
Internet) and Anders Solheim for discussions at
F.V. De Blasio et al. / Marine Geology 213 (2004) 439–455
different stages of the work and for helpful comments
on the manuscript. The work was carried out as part of
projects funded by the EU (project COSTA-Continental Slopes and Stability, project no. EVK3-CT-199900006), by Norsk Hydro ASA, and by the Research
Council of Norway. Support from the International
Centre for Geohazards (ICG) at the Norwegian Geotechnical Institute (Oslo, Norway) is gratefully
acknowledged. This is the ICG paper no. 69.
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