Introduction
The coupled mechanical response of ice, water, and sediment can control
the flow of glaciers residing on deformable sediment
e.g.,.
This is clearly expressed by ice streams in Greenland and Antarctica,
where low levels of basal friction enable high flow rates. These ice
streams are of particular interest, since they are large contributors
to the polar ice sheet mass balance e.g.,.
Although most flow-limiting friction of ice streams
terminating into ice shelves is likely provided by ice shelf
buttressing , the
disintegration of these ice shelves leaves lateral
and basal friction
as the main
mechanical components resisting the flow. A fundamental understanding
of subglacial dynamics is a requirement for our ability to predict
future response of the ice sheets to climate change.
The pressure and flow of pore water in the subglacial bed can
influence subglacial deformation in a number of ways. Assuming
a Mohr–Coulomb constitutive relation of the basal till strength, an
increase in pore-water pressure weakens the bed by reducing the
effective stress, and this may facilitate basal movement if the
driving shear stresses become sufficient to overcome the sediment
yield strength
.
If the hydraulic diffusivity of the bed is sufficiently low relative
to the rate of deformation, a modulation of the pore-water pressure
at the ice–bed interface is, over time, carried into the subglacial
bed, resulting in variable internal yield strength and ultimately
variable shear strain rates with depth
. Owing to local
volumetric changes, variations from the hydrostatic fluid pressure
distribution can be created inside the sediment by the onset and halt
of granular deformation. This influences the local effective pressure
and, in turn, the sediment yield strength e.g.,.
In the case of non-planar ice–bed geometry, excess pore-water pressures
can develop on the stoss side of objects ploughing through
a subglacial bed . The
elevated pore-water pressure weakens the sediment by lowering the
effective stress, resulting in a net strain-rate weakening rheology
e.g.,,
which has been associated with the stick-slip behavior of Whillans Ice
Stream .
Early field studies suggested a strain-rate strengthening Bingham or
slightly non-linear viscous rheology of till ,
which has been used to simplify analytical and numerical modeling of
till mechanics e.g.,.
Laboratory studies have, however, strongly favored the notion of till having
a plastic, Mohr–Coulomb rheology, with a very small rate dependence in the case
of critical-state deformation . This Mohr–Coulomb rheology is also supported by field
investigations . However,
a rate-weakening rheology is expected in the case where obstacles plough through
a soft and deformable bed .
Schematic representation of body and surface forces of two
non-rotating and interacting particles submerged in a fluid with
a pressure gradient.
The low viscosity of water facilitates flow at low stress
and can only be expected to influence the
overall strength of subglacial materials in a few select scenarios, e.g., when
pore-water flow due to porosity change in early stages of shear deformation is limited by low hydraulic diffusivities
e.g.,. The mechanics of coupled granular-fluid
mixtures have previously been numerically investigated for studies of
fluidized beds e.g.,, the stability of
inclined, fluid-immersed granular materials e.g.,, mechanics during confined deformation
e.g.,, debris flow
e.g.,, and for the design
of industrial components, e.g., hydrocyclones e.g.,, or silos and hoppers .
In this study, we explore the interaction between the fluid and granular
phases in water-saturated consolidated particle assemblages undergoing
slow shear deformation. A dry granular assemblage deforms
rate-independently in a pseudo-static manner when deformational rates
are sufficiently low . Critical-state
deformation of water-saturated granular materials is rate independent
during slow shear, but transient changes in pore pressure may cause
initiation of liquefaction at higher shear velocities . The
particle–fluid mixture is in this study sheared at velocities and
stresses comparable to those found in subglacial settings. The
computational approach allows for investigating the internal granular
mechanics and feedbacks during progressive shear deformation.
In the following section, we present the details of the numerical
implementation of particle–fluid flow and describe the experimental
setup. We then present and discuss the modeled deformational
properties of the particle–fluid mixture. Finally, we analyze how the
fluid influences formation of shear zones and under which conditions
deformation is rate dependent.
Methods
The granular model
We use the discrete element method (DEM) to
simulate the granular deformation. With the DEM, particles are treated
as separate, cohesionless entities, which interact by soft-body
deformation defined by a prescribed contact law. The contact mechanics
are micromechanically parameterized. The temporal evolution is
handled by integration of the momentum equations of translation:
mi∂2xi∂t2=mig︸Gravity+∑jfnij+ftij︷Contact forces+fii,
and rotation:
Ii∂2Ωi∂t2=∑j-ri+δnij2nij×ftij︸Contact torques.
i and j are particle indexes, m is the particle mass, I is the
particle rotational inertia, and x and Ω are linear
and rotational particle positions, respectively.
The model considers the transient kinematics during time t. Gravitational
acceleration is included as the vector g.
The inter-particle contact force
vectors are denoted fn and ft
in the normal and tangential direction relative to the contact interface. The
particle radius is included as r. The fluid-particle interaction force is denoted fi
(Fig. ), and the inter-particle
normal vector is termed n. Here, δn is the virtual inter-particle overlap
at the contact, scaling the compressive strain of the particle. The inter-particle contact forces are
determined by a linear-elastic contact model causing repulsion between particle
pairs and providing frictional contact strength. The elastic stiffness is
kn in the normal direction to the contact and kt
parallel (tangential) to the contact interface. The magnitude of the
tangential force ft is limited by the Coulomb
frictional coefficient μ
. The bulk Coulomb
friction coefficient of a granular assemblage depends on the elastic properties,
the inter-particle frictional coefficient, and the granular packing arrangement
e.g.,.
fnij=-knδnijnijandftij=-maxkt||δtij||,μ||fnij||δtij||δtij||
The vector δt is the tangential displacement on
the inter-particle interface when corrected for contact rotation. In
the case of slip, the length of δt is adjusted
to a length consistent with Coulomb's condition
(||δt||=-μ||fn||/kt) .
The linear elasticity allows temporal integration with a constant time
step length Δt.
The fluid model
The inter-particle fluid is handled by using conventional continuum mechanics.
The implementation follows the compressible Darcian flow model presented by
. This
approach was favored over a full Navier–Stokes solution of fluid flow
because it allows for
convenient parameterization of the hydrological permeabilities. The
model assumes insignificant fluid inertia, which is appropriate for
subglacial settings. Fluid compressibility is only important under undrained
or very low-permeability conditions e.g.,. We include fluid
compressibility since the numerical method does not require us to impose the
common simplification of incompressibility.
The volumetric fraction of the fluid phase (the porosity, ϕ) is
incorporated in the Eulerian formulations of the compressible
continuity equation and momentum equation using the local average
method . The Darcy constitutive equation
is used for conserving momentum (Eq. )
:
∂pf∂t=1βfϕμf∇⋅k∇2pf+∇pf⋅∇k︸Spatial diffusion+1βfϕ(1-ϕ)∂ϕ∂t+v‾p⋅∇ϕ︸Particle
forcing(vf-vp)ϕ=-kμf∇pf,
where vf is the fluid velocity, vp
is the particle velocity, and k is the hydraulic permeability estimated from the
local porosity.
The adiabatic fluid compressibility is denoted βf, and
μf is the dynamic fluid viscosity. The continuity
equation (Eq. ) is in the form of a transient
diffusion equation with the forcing term acting as a source/sink for
the fluid pressure. The pressure, pf, is the pressure
deviation from the hydrostatic pressure distribution. This pressure
deviation is sometimes referred to as the excess pressure. We
refrain from using this term, as it may be misleading for pressures
that are smaller than the hydrostatic value.
The simulation domain is discretized in a regular rectilinear
orthogonal grid. The pressure is found using the Crank–Nicolson
method of mixed explicit and implicit temporal integration, which is
unconditionally stable and second-order accurate in time and space
e.g.,. The implicit
solution is obtained using the iterative Jacobi relaxation method
e.g.,, which is light on
memory requirements and ideal in terms of parallelism for our graphics
processing unit (GPU) implementation, although not optimal in terms of
convergence. The numerical solution is continuously checked by the
Courant–Friedrichs–Lewy condition . The partial
derivatives are approximated by finite differences.
The granular-fluid coupling
The particle and fluid algorithms interact by direct forcings
(Eqs. and ) and through
measures of porosity and permeability .
Left: a cell in the fluid grid. The node for pressure
(pf), the gradient of fluid pressure (∇pf), porosity (ϕ), permeability (k), and average
grain velocity (v‾) are calculated at the cell
center. Right: the weight function (Eq. ) at
various distances.
Porosity
The local porosity is determined at the fluid cell center. For a cell
with a set of N grains in its vicinity, it is determined by
inverse-distance weighting the grains using a bilinear interpolation
scheme . The weight function s has
the value 1 at the cell center and decreases linearly to 0 at
a distance equal to the cell width (Δx,
Fig. ):
ϕ(xf)=1-∑i∈NsiVgiΔx3si=Πd=131-|xdi-xf,d|Δx
if|x1i-xf,1|,|x2i-xf,2|,|x3i-xf,3|<Δx0 otherwise.
Vgi is the volume of grain i.
Δx3 is the fluid cell volume, and xf is the
cell center position. Π is the product operator. The average grain
velocity at the cell center (v‾) is found using the same
weighting function described above (Eq. ). Additionally, large
grains with velocity vi contribute to the cell-average velocity with
a greater magnitude:
v‾(xf)=∑i∈NsiVgivi∑i∈Nsi.
The change in porosity is the main forcing the particles exert onto
the fluid (Eq. ). At time step n it is
estimated by central differences for second-order accuracy:
∂ϕ∂tn≈ϕn+1-ϕn-12Δt.
The porosity at n+1 is found by estimating the upcoming particle
positions from temporal integration of their current positions and
velocities.
Permeability
Relations between grain size and hydraulic properties of
sediments supported by empirical evidence e.g.,. The Kozeny–Carman
estimation of permeability k is commonly used and is of the form
k=d2180ϕ3(1-ϕ)2,
where d is the representative grain diameter. Due to constraints on
the computational time step we are unable to simulate fine grain sizes
with realistic elastic properties within a reasonable time frame. In
order to give a first-order estimate of the deformational behavior of
fine-grained sediments, we therefore use a modified version of the
above relationship, where the permeability varies as a function of the
porosity and a predefined permeability pre-factor kc:
k=kcϕ3(1-ϕ)2.
Using this approach we can simulate large particles with the
hydrological properties of fine-grained materials, while retaining the
effect of local porosity variations on the intrinsic permeability. We
do note, however, that the dilative magnitude during deformation is
likely different for clay materials due to their plate-like
shape. This is because sediments with a considerable amount of arbitrarily oriented
clay minerals are likely to compact during deformation as the clay
particles align to accommodate shear strain.
Particle–fluid interaction
The dynamic coupling from the pore fluid to the solid particles acts
through the particle–fluid force (fi) in
Eq. (). Our implementation of this coupling
follows the procedure outlined by ,
,
and (scheme 3).
In a complete formulation, the interaction force between particles and the
pore fluid consists of several components e.g.,.
The most well-known interaction force is the drag force, which is evident in
Stokes' experiments of
particle settling. This force is typically incorporated in particle–fluid models
by semi-empirical relationships. It scales linearly with the relative velocity
difference between particle and fluid and the particle-surface drag coefficient
.
The pressure-gradient force is another important fluid-interaction force
. It results when there is a difference in pressure
across a surface and is well known from buoyancy. In liquids with hydrostatic
pressure distributions the pressure grows linearly with depth as the weight of
the overburden fluid increases. The resultant surface pressure on submerged body
increases correspondingly with depth and causes a buoyant force
upwards. In fluids without hydrostatic pressure distribution, the
pressure-gradient force can be a highly variable vector field.
Several weaker interaction forces have been identified ,
including forces due to particle rotation Magnus force;,
lift forces on the particles caused by fluid velocity gradients
Saffman force;, and interaction forces caused by particle
acceleration virtual mass force;.
Initial tests using a full Navier–Stokes solution for the
fluid phase showed us that the pressure-gradient force was by far the
dominant interaction force in our pseudo-static shear experiments. The
drag force was the second-most important force but 2 orders of
magnitude weaker than the pressure-gradient force due to very slow fluid and
particle flow. Since we neglect fluid inertia in our fluid description, we
included only the pressure-gradient force. The fluid pressure in our model
records the pressure difference from the hydrostatic pressure. For this reason
we add a term to the pressure-gradient force which describes the buoyancy of
a fully submerged particle as the weight of the displaced fluid:
fi=-Vg∇pf-ρfVgg,
where Vg is the volume of the particle, ρf is the
fluid density, and g is the vector of gravitational
acceleration. The particle–fluid interaction force is added to the sum
of linear forces per particle (Eq. ). The particle
force is not added to the fluid pressure equation
(Eq. ) since fluid inertia is ignored. The
fluid is instead forced by variations in porosity.
Computational experiments
The computational fluid dynamics algorithm is implemented in
CUDA C in order to allow a direct integration
with the GPU-based particle solver. The coupled particle–fluid code
is free software (source code available at
https://github.com/anders-dc/sphere), licensed under the GNU
Public License v.3. The simulations were performed on a GNU/Linux
system with a pair of NVIDIA Tesla K20c GPUs. The experimental
results are visualized using ParaView and
Matplotlib .
Experimental setup for the shear experiments. The fluid
cells containing the mobile top wall are given a prescribed
fixed-pressure boundary condition (pftop,
Dirichlet). The bottom boundary is impermeable (no flow, free slip
Neumann). The fluid grid is extended upwards to allow for dilation
and movement of the upper wall. The granular phase is compressed
between a fixed wall at the bottom and a dynamic top wall, which
exerts a normal stress (σ0) downwards. The material is
sheared by moving the topmost particles parallel to the
x axis.
Parameter values used for the shear experiments.
Parameter
Symbol
Value
Particle count
Np
9600
Particle radius
r
0.01 m
Particle normal stiffness
kn
1.16×109 Nm-1
Particle tangential stiffness
kt
1.16×109 Nm-1
Particle friction coefficient
μ
0.5
Particle density
ρ
2600 kgm-3
Fluid density
ρf
1000 kgm-3
Fluid dynamic viscosity
μf
1.797×10-8 to 1.797×10-6 Pas
Fluid adiabatic compressibility
βf
1.426×10-8 Pa-1
Hydraulic permeability prefactor
kc
[3.5×10-15, 3.5×10-13] m2
Normal stress
σ0
20 kPa
Top wall mass
mw
280 kg
Gravitational acceleration
g
9.81 ms-2
Spatial domain dimensions
L
[0.52, 0.26, 0.55] m
Fluid grid size
nf
[12, 6, 12]
Shear velocity
vp, topx
2.32×10-2 ms-1
Inertia parameter value
I
1.7×10-4
Time step length
Δt
2.14×10-7 s
Simulation length
ttotal
20 s
The experimental setup is a rectangular volume
(Fig. ) where a fluid-saturated particle
assemblage deforms due to forcings imposed at the outer boundaries.
We deform the consolidated material by a constant-rate shearing motion
in order to explore the macro-mechanical shear strength under
different conditions.
To determine the effects of the pore water, we perform experiments
with and without fluids, and for the experiments with fluids present,
the permeability pre-factor kc is varied to constrain the
effect of the hydraulic conductivity and diffusivity on the overall
deformation style. The low value used for kc (3.5×10-15 m2) results in an intrinsic permeability of k=1.9×10-16 m2 for a porosity of 0.3
(Eq. ). The highest value (kc=3.5×10-13 m2) matches a permeability of 1.9×10-14 m2. These two end-member permeabilities roughly
correspond to what and
estimated for the clay-rich Two Rivers till and the clay-poor
Storglaciären till, respectively.
Shear experiments with different shearing rates. (Top)
Unsmoothed and smoothed shear friction values, (center) dilation
in number of grain diameters, and (bottom) minimum, mean, and
maximum fluid pressures. The permeability prefactor value is
kc=3.5×10-15 m2. The shear
friction values (top) are smoothed with a moving Hanning window
function to approximate the strength of larger particle
assemblages. The material peak friction increases with strain rate
due to reductions of internal fluid pressure. This strengthening
is taking place when the dilation rate exceeds the dissipation
rate of the fluid. The diagonal hatched pattern marks the earliest state of
shear characterized by rapid dilation.
Temporal evolution
(x axis) of the horizontally averaged fluid pressures (color bar) as a
function of the height above the base (y axis). At
fast shear rates (top) there are large internal pressure decreases
and slow recovery due to a large dilation rate and an insufficient
pressure dissipation. When the shearing velocity is decreased
(middle) and (bottom) the dissipation rate becomes increasingly
capable of keeping internal pressures close to the hydrostatic
pressure (0 kPa).
The lower boundary is impermeable, and a fixed fluid pressure is
specified for the top boundary. These boundary conditions imply that
the simulated ice–bed interface is a relatively permeable zone with
rapid diffusion of hydrological pressure, which is likely for
subglacial beds with low permeability
e.g.,. In
coarse-grained tills the subglacial till diffusivity
may exceed the hydraulic diffusivity at the ice–bed interface, unless a network
of linked cavities causes rapid diffusion of pore-water pressures here
e.g.,. The lateral boundaries are periodic (wrap-around).
When
a particle moves
outside the grid on the right side it immediately reappears on the
left side. Likewise, particle pairs can be in mechanical contact
although placed on opposite sides of the grid.
The particle size distribution is equidimensional, a stark contrast relative to
subglacial tills which often display a fractal size distribution
e.g.,. Fractal size distributions minimize
internal stress heterogeneities , but, in
the absence of grain crushing, an assemblage with a wide particle size
distribution dilates from a consolidated state with the same magnitude
as assemblages with a narrow particle size distribution
and displays the same frictional strength
. The comparable dilation
magnitude justifies the computationally efficient narrow particle size
distribution used here. However, as previously noted, shear zones in clay-rich
materials can compact during shear due to preferential parallel
alignment, which is not possible to capture with the methodology
presented here.
Peak frictional strength before the critical state of the
low-permeability granular bed (kc=3.5×10-15 m2) at different shear velocities. The
frictional strength is constant and rate independent at velocities
lower than 101 ma-1 as pore-pressure diffusion
rates far exceed rates in volumetric change.
(Top) Shear strength, (center)
dilation in number of grain diameters, and (bottom) minimum, mean,
and maximum fluid pressures in shear experiments with different
permeability properties.
Horizontal particle displacement
with depth (shear strain profiles) for the dry and fluid-saturated
shear experiments. Top: displacement profiles from experiments
with different shear velocities. Bottom: displacement profiles
from experiments with different permeabilities.
The simulated particle size falls in the gravel category of grain
size (Table ). The large size allows us to perform the temporal integration
with larger time steps . The
frictional force between two bodies is independent of their size
(Amontons' second law) but is proportional to the normal force on the
contact interface , as reflected in the contact
law in the discrete element method (Eq. ). We
prescribe the normal forcing at the boundary as a normal stress, which
implies that the normal force exerted onto a particle assemblage at
the boundaries scales with domain size. For a number of total
particles in a given packing configuration the ratio between particle
size and inter-particle force is constant, which causes the shear
strength to be independent of simulated particle size. This
scale independence is verified in laboratory experiments, where the
granular shear strength of non-clay materials is known to be mainly
governed by grain shape and surface roughness instead of grain size
.
Experiment preparation and procedure
The particles are initially positioned in a dry, tall volume, from where
gravity allows them to settle into a dense state at the bottom of the bounding
box. The particle assemblage is then consolidated by moving the
top wall downwards until the desired level of consolidation stress is reached
for an extended amount of time. The same top wall is thereafter used
to shear the material in a fluid-saturated state
(Fig. ).
For the shear experiments, the uppermost particles are forced to move
with the top wall at a prescribed horizontal velocity
(Fig. ). The particles just above the bottom wall
are prescribed to be neither moving or rotating. The physical size of the
domain is 0.52 m long in the direction of shear (x), 0.26 m in the transverse
horizontal direction (y), and 0.55 m tall (z). We use the exact same
material for all shear experiments, allowing for perfect
reproducibility and effectively removing inter-experimental variability
caused by differences in grain packing.
The micromechanical properties and geometrical values used are listed in
Table .
Particle displacement and fluid forces for different
permeabilities at a shear strain of 0.25. (Left) Particles colored
by their original position; (center) particles colored by their
displacement along the x axis; (right) vertical (z) forces
from the fluid onto the particles. In the permeable material
(top), the internal volumetric
changes are accommodated by porous (Darcian) flow. This keeps the fluid
pressures close to hydrostatic values and causes deep deformation
(top center). In materials which are impermeable (bottom), the volumetric
changes cause drastic pore-pressure reductions, effectively strengthening
the material (bottom right) and focusing deformation at the top
(bottom left and center).
Horizontally averaged fluid and particle behavior with
progressive shear strain. (left) Vertical particle displacement,
(center left) mean permeability, (center right) mean fluid
pressure, and (right) vertical component of the mean fluid stress,
calculated as fii/Ai, where
fii is the fluid pressure force on particle i
from Eq. () and Ai is its surface
area. Note that the limits of the horizontal axes differ between the
experiments.
Temporal evolution (x axis) of horizontally averaged
fluid pressures (y axis). The permeable material (top) is able
to quickly respond to internal volumetric changes, which are
short-lived and of small magnitude. The low-permeability material
(bottom) is dominated by large pressure reductions and relatively
slow relaxation.
Particle–fluid interaction during deformation of a consolidated sediment (after
).
Micromechanical cause of dilatant
hardening. A consolidated sediment (top) is deformed with
a vertical gradient in velocity. The grains are forced past each
other in order to accommodate the shear strain. The deformation
causes dilation, which increases porosity locally and decreases
fluid pressure (bottom). The gradient in fluid
pressure pulls particles together
(Eq. ), which increases the load on
inter-particle contacts. The larger inter-particle normal stress
increases the shear strength of the contact (Eq. )
resulting in a stronger sediment.
Scaling of the shear velocity
The heavy computational requirements of the discrete element method
necessitates upscaling of the shearing velocity in order to reach
a considerable shear strain within a manageable length of
time. Temporal upscaling does not influence the mechanical behavior of
dry granular materials, as long as the velocity is below a certain
limiting velocity . The
shearing velocity used here (2.32×10-2 ms-1),
although roughly 3 orders of magnitude greater than the velocities
observed in subglacial environments (e.g., 316ma-1=10-5 ms-1), guarantees quasi-static, rate-independent
deformation in the granular phase, identical to the behavior at lower
strain rates. The particle inertia parameter, I, quantifies the
influence of grain inertia in dry granular materials
. Values of I below 10-3 correspond to
pseudo-static and rate-independent shear deformation in dry granular
materials. I has a value of 1.7×10-4 in the shear
experiments of this study.
The fluid phase needs separate treatment in order to correctly resolve
slow shear behavior at faster shearing velocities. This behavior is
achieved by linearly scaling the fluid dynamic viscosity with the
relationship between actual shearing velocity and the reference
glacial sliding velocity. By decreasing the viscosity, the fluid is
allowed to more quickly adjust to external and internal forcings. The
velocity scaling adjusts the time-dependent parameters of hydraulic
conductivity and diffusivity correctly. The intrinsic permeability k
is time independent, and the values produced here are directly
comparable with real geological materials. The fluid viscosity is
scaled to a lower value of 1.797×10-6 Pa s, consistent
with the scaling factor used for the shearing velocity. We test the
influence of shearing rate by varying this parameter.
Results
First we investigate the strain-rate dependence of the sediment
strength and dilation by shearing a relatively impermeable sediment
(kc=3.5×10-15 m2) at different shear
velocities. The shear velocity directly influences the magnitude of
the peak shear strength, dilation, and internal fluid pressure
(Figs.
and ).
At the reference shearing velocity the peak shear frictional strength
is 0.71, which corresponds to a shear stress of 14 kPa at an effective
stress of 20 kPa (Figs. , top left, and
). When sheared 100 times
slower, the peak shear friction has decreased to 0.62, corresponding
to a shear stress of 12 kPa (Figs. , top
right, and ). The peak values are measured
during the transition from the dense and consolidated pre-failure
state to the critical state where a shear zone is fully
established. This transition is characterized by rapid dilation due to
porosity increases in the shear zone
(Fig. , middle). During fast shearing
velocities the volumetric change outpaces the diffusion of fluid
pressure, causing the internal pore-water pressure in the shear zone
to decline (Figs. , bottom, and
). Dilatant hardening causes
the peak shear strength to increase at large shear velocities
(Fig. ), while the strength reduces to
values corresponding to the rate-independent granular strength for lower
velocities.
In this model framework, adjusting the hydraulic permeability of the
same coarse sediment leads to similar conditional strengthening as
shearing the sediment at different rates
(Fig. ). Without fluids (the dry experiment),
the peak shear friction (Fig. , left) is
relatively low and is dominated by high-frequency
fluctuations. The fluid-saturated experiment with the relatively high
permeability (kc=3.5×10-13 m2) has
similar shear strength, but the high-frequency oscillations in shear
friction are reduced by the fluid presence. The dilation is similar to
the dry experiment but with slightly decreased magnitude. The mean
fluid pressure deviation from hydrostatic values
(Fig. , bottom left) is close to 0. The
low-permeability experiment (Fig. , right) is
characterized by the largest initial peak strength and lowest
magnitude of dilation. Compared to the other experiments, the dilation
reaches its maximum values at lower shear strain. The fluid pressure
decreases almost instantaneously at first, whereafter it equilibrates
towards the hydrostatic value (0 Pa).
At constant shearing rate with different permeabilities
(Fig. , top) or at variable shearing rates with
constant permeability (Fig. , bottom), we
observe that pore-water dynamics have a significant effect on the
distribution of strain.
The effects of the fluid are visible at different depths within the
deforming material (Figs.
and ). The deformation is pervasive with
depth for the relatively permeable experiment
(Fig. top), and the fluid pressures deviate
only slightly from the hydrostatic values. The relatively small
pressure gradients cause only weak fluid forces on the particles in
this experiment. Contrasting these results, deformation in the
impermeable experiment is primarily governed by shallow deformation beneath the
top wall (Figs.
and , bottom).
Differences in hydraulic permeability influence the dynamics of the
fluid over time, as illustrated in
Fig. . The fluid pressures in the
permeable material (top) are initially predominantly negative,
reflecting the increasing dilation (Fig. ,
middle). In the critical state (after a shear strain value of 0.1),
the fluid pressures fluctuate around the hydrostatic value
(0 Pa).
Discussion
Strain-rate dependency
Several studies have highlighted the importance of feedbacks between
the solid and fluid phases during granular deformation
e.g.,. A shear-rate dependence
in a grain–fluid mixture can only originate from the fluid phase,
since dry granular materials deform rate-independently under
pseudo-static shear deformation
. Rate dependence emerges, however,
as soon as the flow of viscous pore fluids starts to influence the
solid phase.
Water has a relatively low viscosity, which implies that the shear
stress required to deform the fluid phase alone is extremely low. However, the
fluid phase influences the bulk rheology if diffusion of
fluid pressures is limited relative to volumetric forcing rates, as in
a rapidly deforming but relatively impermeable porous material. The
coupled particle–fluid interactions cause the material to respond as
a low-pass filter when forced with changes in volume and porosity. The
reequilibration of pressure anomalies depends on the volumetric strain
rate, water viscosity, and material permeability. Any forcing that
affects local porosity causes the material to respond in part like
a viscous dashpot due to internal fluid flow.
The pore-fluid viscosity likely increases if suspended clay-particles or
ice crystals are present. The exact rheology of such mixtures may very
well be non-Newtonian, and we have therefore not included such effects in this
study.
Dilatant hardening: effects on sediment strength and deformation depth
Transient porosity changes take place when granular materials deform before
attaining the critical state
e.g.,. These
changes can cause deviation from the hydrostatic pressure
distribution, which affects the magnitude of the local effective normal stress.
Considering the Mohr–Coulomb constitutive relation for till
rheology, a reduction in pore-water pressure increases the effective
stress, which in turn strengthens the material in the shear zone
(Fig. ) e.g.,. In our results we obtain insight into the
micromechanical mechanisms of strengthening without explicitly specifying
Mohr–Coulomb constitutive behavior.
Deviations from the hydrostatic pressure distribution
cause pressure-gradient forces, which during dilation resist the
development by pushing the
grains together towards the low pressure of the shear zone
(Fig. ). The tangential strength of inter-particle
contacts in the DEM is determined by Coulomb friction (Eq. ),
which implies a linear correlation between contact normal force and tangential
contact strength. Heavily loaded particle contacts are thus less likely to
slip, and chains of particles with strong contacts cause increased resistance to
deformation . The fluid force on the particles
strengthens the inter-particle contacts and increases the shear friction until
hydrostatic pressure conditions are reestablished.
The dilatant hardening causes a shallower
deformational profile in these experiments
(Figs. , left,
and ).
The hardening vanishes at higher strains as the hydrostatic pressure distribution
is recovered at all depths of the material, and the deformational profiles
progressively deepen until they match the dry experiment at higher shear-strain values (≳2).
This progressive widening of the shear zone causes dilation to slowly
increase while shear strength only displays very slight decrease
(Figs. and ).
The shallow deformation at low strains is consistent with the laboratory results
by , where the shear zone in the coarse-grained
Storglaciären till in all cases was wider than the shear zone of the
fine-grained Two Rivers till.
The velocity profile of the shear zone determines the material
flux. A shallower deformation depth and a transiently lower subglacial sediment
transport rate is thus to be expected from subglacial shearing of compacted,
low-permeable sediments, relative to permeable counterparts. These results are
consistent with observations of very shallow deformation in subglacial tills
with a relatively low permeability ,
although shallow deformation is also consistent with fine grain sizes
, low effective stresses , and
rate-weakening ploughing at the ice–till interface .
Owing to the granularity
of the material, the dilation displays small fluctuations
around the critical-state value. The small
volumetric oscillations create new fluid-pressure deviations from the
hydrostatic value, which alternately slightly weaken or strengthen the sediment
(Fig. , top). In cases
where the shear stress is close to the sediment shear strength, the
hardening may be sufficient to stabilize patches of the bed
, or the weakening may trigger slip .
Shear deformation can cause compaction instead of dilation if the material
porosity exceeds the critical-state porosity, if elongated grains rotate
into a more compact arrangement with high fabric strength, or if relatively soft
granular components such as aggregates disintegrate.
Compaction-induced weakening is a primary mechanism for
debris-flow mobilization in landslides, as demonstrated experimentally
at small scales and at field scales .
The granular model applied here is not able to reproduce the
shear-induced compaction that, e.g., clay-rich materials can display during
early shear due to clay-fabric development e.g.,.
Compaction causes increased pore-water pressure in the shear zone in cases
where the volumetric change exceeds the timescale of pore-water pressure
diffusion. Some of the effective normal stress on the shear zone
is consequentially reduced, which in turn decreases the material strength. The
reduction of strength due to compaction is rate dependent like the dilative
hardening.
Our results confirm that the interplay between the solid and fluid
phases can influence the sediment strength in a transient manner e.g.,. Pore-water pressures
decrease during deformation, and shear strength increases until
deformation ceases or the critical state is reached. Once the local
and regional hydraulic system recovers from the pore-pressure
reduction, the sediment strength is once again reduced and a new
deformation phase may be initiated. The
magnitude of strengthening is dictated by consolidation at times between slip
events as well
as the ability of the subglacial hydrological system to accommodate reductions
in pressure at the ice–bed interface.
A variable shear strength of the till influences ice flow when the basal
shear stress is in the range of till strength variability. Since surface
slopes of ice streams are low, driving stresses tend to be low as
well. Inferred values of driving stresses at the Northeast Greenland
ice stream , Whillans Ice Stream and ice plain
, and Pine Island Glacier lie
within the range of 2 to 23 kPa
and are thus potentially sensitive to the variability in till
strength. If the glacier moves with variable velocities in
a stick-slip or surging manner, periods of stagnation may
consolidate and strengthen the sediment, in effect delaying the
following slip event . Future studies will focus on
investigating the mechanical consequences of transiently variable stresses,
caused by changes in pore-water pressure or shear stress.
Conclusions
We numerically simulate a two-way coupled particle–fluid mixture under
pseudo-static shear deformation and confirm results on transient strengthening
from previous physical experiments e.g., and calculations
e.g.,.
The grains are simulated individually
by the discrete element method, while the fluid phase is treated as
a compressible and slowly flowing fluid adhering to Darcy's law. The
fluid influences the particles through local deviations from the
hydrostatic pressure distribution. Due to the extremely low viscosity
of water, the deformational behavior of dense granular material is
governed by inter-grain contact mechanics. The porosity of a granular
packing evolves asymptotically towards a critical-state value with increasing
shear strain.
Changes in porosity cause deviations from the hydrostatic pressure if
the rate of porosity change exceeds the rate of pressure
diffusion. The rate of pressure diffusion is governed by the fluid
viscosity, the local porosity, and the hydraulic permeability. Low
fluid pressures developing due to sediment dilation increase the frictional
strength of inter-grain contacts by increasing the loading between grain pairs.
The magnitude of the strengthening effect is
rate dependent and increases with shear velocity and decreases with
increasing hydraulic permeability. The rheology is
plastic but
rate-dependent dilative strengthening can contribute to the material
strength during early stages of fast deformation.
If the till is very porous or the deformation is accompanied by
strong fabric development or grain crushing, compaction
in the shear zone is expected to weaken the
sediment, causing a rate weakening with increased shear rate until the
excess pressures are reduced by hydraulic diffusion.
We furthermore show that for fast shear velocities
permeable sediments are only weakly influenced
by the fluid phase, resulting in little shear strengthening and a deep
decimeter-scale deformation dictated by the normal stress and grain
sizes. Impermeable and consolidated sediments display slight dilatant
strengthening at high shear velocity. The strengthening causes
deformation to focus at the ice–bed interface if pore-water
pressures are higher there. The resultant depth of deformation
is on the millimeter-to-centimeter scale. Actively deforming
patches in the subglacial mosaic of deforming and stable spots act as
sinks for meltwater. Additionally, sediment dilation can cause substantial thinning of a water film
at the ice–bed interface. If the subglacial shearing movement halts,
the sediment gradually weakens as the fluid pressure readjusts to the
hydrostatic value. These temporal changes in sediment strength may
explain observed variability in glacier movement.