2. Hydrofracture Vulnerability in Greenlandβs Ice Slab Areas
Riley Culberg, Yue Meng, Ching-Yao Lai
Department of Geosciences, Princeton University
Motivation Poromechanical Model
Application to the Greenland Ice Sheet
Will crevasses in ice slabs fill with water?
Are water-induced stresses sufficient to hydrofracture firn?
Non-Dimensional Analysis
Comparison of the rate of water infiltration into the
firn from the crevasse tip versus the rate at which
surface streams may feed water into a crevasse.
Blue bars show the plausible range of firn water
infiltration rates. Yellow bars show small stream
discharge values measured in the ablation zone of
Southwest Greenland. Red bars show large stream
discharges from the same region. Discharge from
the smallest streams is similar to the rate of leak-
off from the crevasse tip into the firn, so the
crevasse will not fill, preventing hydrofracture.
However, larger streams can inject water fast
enough to fill crevasses. Therefore, we also need to
understand whether the resulting water pressure in
the crevasse is sufficient to cause hydrofracture.
Firn Mechanical Properties
πΏππ₯π₯ πππ₯
β²
π½ππΏπ β
π
β π
ππ€π π»π€ β π»π
Constant Pressure
πΏπ ππ€ππ»π€
Constant Injection Velocity
πΏπ
π
ππ€πππππΏππππ£
π0
l
ππ€πππππΏππππ£
πππ€ππ§0π0
On the Greenland Ice Sheet, hydrofracture connects the supraglacial and subglacial
hydrologic systems, coupling surface runoff dynamics and ice velocity. Over the last two
decades, the growth of low-permeability ice slabs in the firn above the equilibrium line
has expanded Greenlandβs runoff zone, but the vulnerability of these regions to
hydrofracture is still poorly understood. Observations from Northwest Greenland
suggest that when meltwater drains through crevasses in ice slabs, it is often stored in
the underlying relict firn layer and does not reach the ice sheet bed. However, there is
also evidence for the drainage of buried supraglacial lakes in this same region,
suggesting some eventual transition from infiltration to fracture.
Motivating Questions:
βͺ What prevents water-filled crevasses in ice slabs from propagating unstably through
the underlying relict firn layer?
βͺ What drives the observed transition to full ice thickness hydrofracture once all pore
space directly beneath a lake has been filled by refreezing?
Parameter Sweep
To apply the analytical model, we must define reasonable values for the physical, mechanical, and
hydraulic properties of ice slab-firn systems in Greenland. Unfortunately, given the sparse and
uncertain observations available, it is hard to choose a single representative value for any of these
parameters. Therefore, we take a Monte Carlo simulation approach. For each variable, we define
an empirical distribution of reasonable values using a compilation of in situ, laboratory, and remote
sensing measurements reported in the literature. For the hydraulic and mechanical properties, we
use various empirical relations to define these properties as a function of firn density.
Analytical model to calculate the maximum effective stress at the crack tip for ice slab-firn systems
and solid ice.
We use a two-phase poromechanics
model to simulate water injection into
a firn layer with constant pressure and
constant injection velocity boundary
conditions. We run a suite of
simulations with different mechanical
and hydraulic properties to develop an
analytical estimate of the maximum
effective stress in the firn.
Distributions of Effective Stress
Key Conclusions
β’ The firn layer beneath ice slabs imparts significant
resilience to hydrofracture because:
1) Leak-off into the firn may prevent crevasses from filling
with water
2) When crevasses do fill, much of the hydrostatic stress
is accommodated by a change in pore pressure, rather
than a being transmitted to the solid skeleton
β’ Surface-to-bed drainage connections are unlikely to form
until all local pore space has been filled with refrozen ice.
Non-dimensional maximum effective stress as a function of firn porosity and non-dimensional
water height in the crevasse. a) Water-filled crevasses. Effective stress increases with firn porosity
and water height due to the increasing water pressure, stronger fluid-solid coupling, and reduced
lithostatic stress. b) Supraglacial lake over a crevasse. Effective stress becomes more compressive
as the water level increases, due to the added lithostatic stress. As water level increases, firn
porosity plays a great role in determining the stress, since it modulates both the hydrostatic stress
transmitted to the solid skeleton, and the portion of the lithostatic stress transmitted horizontally.
Contact:
rtculberg@princeton.edu
Physically plausible distributions of maximum effective stress in firn (purple bars) and solid ice
(blue bars). a) Partially water-filled crevasse. The ice slab-firn and solid ice systems are similar, as
reduced overburden in the ice slab-firn system balances the complete transmission of hydrostatic
stress in the solid ice system. b) Mostly water-filled crevasse. Effective stress in the solid ice
system is tensile, but remains compressive in the ice slab-firn system, as pore pressure
accommodates much of the hydrostatic stress. c) Supraglacial lake overtop a crevasse. In the ice
slab-firn system, the effective stress becomes more compressive, because lithostatic stress
increases faster with lake depth than the portion of hydrostatic stress felt by the solid skeleton.
Biot Coefficient:
portion of stress felt by
the solid skeleton
Poissonβs Ratio:
portion of vertical stress
transmitted horizontally
4. Surprising surface similitude to bed topography in Greenland
1. Interpreting subglacial geology; and 2
1. Interpreting subglacial geology; and 2
1. Interpreting subglacial geology; and 2
Joseph A. MacGregor (joseph.a.macgregor@nasa.gov), Liam Colgan + GreenValley team
Weβve long known that prominent subglacial topographic features beneath the ice sheets can generate observable surface expressions. Recent advances in digital
elevation models (e.g., GrIMP) and bed-to-surface transfer theory now permit widespread observation of this phenomenon and easier interpretation. Hillshading a digital
elevation model across the direction of ice flow highlights major surface features nicely. For Greenland, comparison against NASA/KU/CReSIS airborne
radar-sounding data confirms that most features are due to subglacial topography and are typically valleys. This suggests a better path toward: 1. Interpolating subglacial
topography between sparse radar observations by developing methods that also require fidelity to observed surface relief; 2. Interpreting subglacial geology.
Bumps in the night
Sun valley slopes
GrIMP mosaic hillshaded
across the local direction
of ice flow (explained
below).
(A) Map of whole island
with manually traced
lineations overlain
(BβG) Zoom-ins of
selected regions with bed
elevation anomaly Ξzb
(bed elevation minus 5-km
running mean) from
NASA/KU/CReSIS
radar-sounding tracks
overlain. Bed high / low.
(right column) Selected
radar-sounding tracks
from panels BβG with
along-track surface
elevation anomaly.
Ng et al. (2018)
Howβd they do that?
1. Filter both the GrIMP DEM and MEaSUREs surface velocity using a 5H
thickness-dependent triangular filter and resample to a 5 km grid.
2. For the slower interior (< 100 m yrβ1
), weight the flow direction toward filtered GrIMP
gradient direction.
3. Illuminate using a standard hillshade algorithm but allow illumination azimuth to vary for
each pixel, selecting the azimuth 90ΒΊ counter-clockwise from the filtered ice-flow
direction. This direction consistently highlights coherent surface textures / lineations.
Next season
1. Invert for ice thickness and sliding rate across
the interior using a mono-layer model.
2. Better resolve subglacial geology using this
improved ice thickness and seismic, gravity
and magnetic data.
3. Hiring a new post-doc! Could be you!
v
5. β’ Current fracture mechanics (i.e., LEFM) assumes that the stored elastic energy in an
impermeable solid matrix is instantaneously dissipated by creating new crack
surfaces, which only holds for impermeable solid media. Firn is porous material that
violates such assumption;
β’ We extend Biotβs poroelastic theory to two-phase immiscible flow to capture the
feedback between fluid flow and matrix deformation in the firn. We show that the
presence of a permeable firn layer prevents fracture propagation because a
significant portion of the hydrostatic stress is accommodated by changes in pore
pressure (~78% of total stress change), rather than being transmitted to the solid
skeleton (~22% of total stress change);
β’ To couple poromechanics, including thermoporoelasticity, thermoporoplasticity,
thermoporoviscoelasticity, with suitable glacial hydrology, rheology and fracture
models, to better understanding glacier dynamics.
Vulnerability of Firn to Hydrofracture: Poromechanics Modeling
Yue Meng, Riley Culberg, Ching-Yao Lai
Department of Geosciences, Princeton University
Motivation Poromechanics: The Concept of Effective Stress
Modeling Results Are water-induced stresses sufficient to hydrofracture firn?
On the Greenland Ice Sheet, hydrofracture connects the supraglacial and subglacial
hydrologic systems, coupling surface runoff dynamics and ice velocity. Over the last two
decades, the growth of low-permeability ice slabs in the firn above the equilibrium line
has expanded Greenlandβs runoff zone, but the vulnerability of these regions to
hydrofracture is still poorly understood. Observations from Northwest Greenland
suggest that when meltwater drains through crevasses in ice slabs, it is often stored in
the underlying relict firn layer and does not reach the ice sheet bed. However, there is
also evidence for the drainage of buried supraglacial lakes in this same region,
suggesting some kind of transition point from infiltration to fracture.
Motivating Questions:
βͺ What prevents water-filled crevasses in ice slabs from unstably propagating through
the underlying relict firn layer?
βͺ What drives the observed transition to full ice thickness hydrofracture once all pore
space directly beneath a lake has been filled by refreezing?
Analytical model to calculate the maximum effective stress at the crack tip for ice slab/firn systems and
solid ice. The poromechanical model predicts π½ 0.22.
When stress is applied to porous media, part of the stress is transmitted through the pore
fluid and part of the stress is transmitted through the solid skeleton. Effective stressβthe
fraction of the total stress that is transmitted through the solid skeletonβcontrols the
mechanical behavior of porous media.
Contact:
om3193@princeton.edu
Ξ΄π πΏπβ² β ππΏππ°
pore fluid (πΏπ)
solid skeleton (πΏπβ²)
total stress (πΏπ)
π β
πΎ
πΎπ
β [0 ]
What is the fracture criterion for the porous firn?
0
0
2
Water injection into the firn induces a
tensile effective stress change at the
crevasse tip ( πΏππ₯π₯
β² ). When the
horizontal effective stress exceeds the
firn tensile strength ( ππ‘
β²
), vertical
fractures are generated. The fracture
criterion at the crevasse tip is written
as follows:
ππ₯π₯
β²
ππ₯π₯ 0
β²
πΏππ₯π₯
β²
β₯ ππ‘
β²
calculated from
lithostatic stress
calculated from
poromechanics
The 2D, Two-Phase Poroelastic Continuum Model
We use a 2D, two-phase poroelastic continuum model to solve the infiltration-induced stress
and pressure changes. The model has four governing equations, two derived from
conservation of fluid mass and two derived from conservation of linear momentum. The
model solves the time evolution of four unknowns: (1) pore pressure field π π₯ π§ π‘ ; (2) water
saturation field π π₯ π§ π‘ ; (3) horizontal displacement field π’ π₯ π§ π‘ , and (4) vertical
displacement field π€ π₯ π§ π‘ of the porous firn layer. The governing equations are summarized
and written in the x, z coordinates as follows:
Model set-up
π. π
ππ
ππ‘
π π
ππππ
ππ‘ π
ππ
ππ‘
β
π0
ππ€
π
ππ₯
πππ€
ππ
ππ₯
β
π0
ππ€
π
ππ§
πππ€
ππ
ππ§
β ππ€π 0;
π. π
ππππ
ππ‘ π
ππ
ππ‘
β π0
π
ππ₯
πππ€
ππ€
πππ
ππ
ππ
ππ₯
βπ0
π
ππ§
πππ€
ππ€
πππ
ππ
ππ
ππ§
β
πππ€
ππ€
ππ€
πππ
ππ
ππ π 0;
π.
πππ₯π₯
ππ₯
πππ§π₯
ππ§
0;
π.
πππ₯π§
ππ₯
πππ§π§
ππ§
β π ππ π ππ β π ππ€π π 0.
Ξ΄π πΏπβ² β ππΏππ°
πΏπβ²
πΏπβ²
3πΎπ
π
ππππ°
3πΎ β 2π
π
π
Fluid continuity equations (for water and air phases):
Force balance equations (in x and z directions):
β
π
ππππ€ π β π ππ π β π ππ ; πππ€ π3 πππ β π 2.
Here, we consider two scenarios of water infiltration into the porous firn layer:
βͺ A constant water height (π»π€) in the surface crevasse;
βͺ A constant water injection velocity (ππππ) at the crevasse tip.
How does the pore pressure or the skeleton stress
evolves during meltwater infiltration?
0
0.22
πΏππ₯π₯ πππ₯
β²
π½ππΏπ β
π
β π
ππ€π π»π€ β π»π
The poromechanical model predicts π½ 0.22.
Constant Pressure
πΏπ ππ€ππ»π€
Constant Injection Velocity
πΏπ
2
π
ππ€πππππΏππππ£
π0
l
2ππ€πππππΏππππ£
πππ€ππ§0π0
How does πΉπππ πππ
β² depend on modeling parameters?
Analytical Expressions of πΉπ and πΉπππ πππ
β²
Key Conclusions
Future Work