Introduction
The detailed microstructure of sea ice is critical in both
governing the movement of fluid between the ocean and the sea ice surface and
controlling processes such as ice growth and decay
. Its complex pore structure influences many of
the bulk thermal and electric properties of sea ice. The permeability is of
primary interest to a wide range of disciplines (e.g., biology and
atmospheric chemistry) as it controls fluid flow through sea ice. The “Rule
of Fives” provides a guideline for describing the percolation threshold in
first-year columnar sea ice. Specifically, the ice becomes permeable to fluid
transport at brine volume fractions greater than 5 %, which are found in
ice at about -5 ∘C with a salinity of about five parts per thousand
. Although this rule of thumb is helpful in describing and
modeling basic phenomenon, it does not fully capture the spatially and
temporally evolving details of the sea ice microstructure. Here we provide a
more topologically complete characterization of sea ice pore structure.
Previous research has recognized the importance of thermally activated
percolation thresholds
e.g.,.
studied single-crystal laboratory-grown ice using X-ray micro-computed
tomography (μCT) to examine the thermal evolution of brine inclusions.
They found that brine volume fraction and pore space structure depend upon
temperature, with a percolation threshold observed at 4.6±0.7 %.
However, one expects natural polycrystalline ice to have a higher threshold
as pathways are sensitive to grain boundaries, flaws, and a certain degree of
horizontal transport . Since different growth rates in
natural sea ice produce different average spacing between brine layers, there
is also the potential for varying percolation thresholds and degree of
connectivity with depth .
Network models have successfully been employed in a variety of fields to
describe complex phenomena and predict future behavior, in particular fluid
flow in porous media e.g.,.
Specific to sea ice, utilized a Lattice–Boltzmann model
to emulate fluid flow through sea ice. Meanwhile, examined
critical percolation thresholds in their network model of sea ice. More
recently, used a two-dimensional pipe network model to
simulate fluid flow through sea ice using a fast multigrid method. This
network compared well with lab data for porosity above 0.15, but
overestimated permeability at lower porosities . The
majority of these models generate connectivity networks based on bulk brine
properties. Here we derive finer-grained statistics empirically, allowing for
models to more closely align with the physical properties of sea ice.
In this paper, we develop a methodology for describing the morphology
and variability of brine networks in a vertical column of first-year sea ice.
We construct a network model of the pore structure of sea ice and use
topological techniques to characterize this brine network. This yields a set
of network statistics that characterizes channels from different depths and
temperature, which we can later use to inform more sophisticated models of
sea ice. Future applications include refining under what conditions the
“Rule of Fives” applies, predicting bulk physical properties such as heat
transfer and fluid permeability, and improving the ability to describe
processes such as brine drainage and desalination. This approach provides
advances in quantifying the brine connectivity in sea ice, which we can then
incorporate into both global and regional sea ice models.
Methods
This work will focus on two of the ice cores extracted from different
locations in the Ross Sea, Antarctica during a October–November 2012 field
campaign. The 1.78 m Butter Point ice core was collected at
77∘35.133′ S and 164∘48.222′ E and had a temperature
gradient ranging from -16.1 ∘C at the top to -2.5 ∘C at
the bottom. For this core, the top 14 cm was frazil ice, the columnar ice
region was from 14 to 65 cm, and platelet ice formed the bottom 64 %
. The 1.89 m Iceberg Site ice core was located at
77∘7.131′ S and 164∘6.031′ E and had a temperature
gradient ranging from -17.7 ∘C at the top to -2.3 ∘C at
the bottom. Relative to the Butter Point core, the Iceberg Site core had more
frazil ice (0 to 30 cm), more columnar ice (30 to 137 cm), and less
platelet ice (137 to 189 cm) . Immediately following core
extraction, we recorded the temperature profile at 10 cm intervals, and
stored the cores in a -20 ∘C freezer at McMurdo station prior to
shipping. We then transported the cores at a constant temperature of
-20 ∘C back to Thayer School of Engineering's Ice Research
Laboratory at Dartmouth College, and stored them in a -33 ∘C cold
room prior to analysis. Cubic samples measuring 1 cm on edge were taken from
each core at 10 cm intervals. We will use the term “sample” throughout
this paper to refer to a particular 1 cm cube from a specified depth. We
used μCT to image each sample following the protocol developed by
. We scanned each sample from the two cores at in situ
temperatures using a Peltier cooling stage attached to our Skyscan
1172 μCT scanner, and analyzed the three-dimensional morphological data.
We build on the methodology of Lieb-Lappen et al. (2017) to convert the
binarized images of the brine phase to a simplified representation as a
network. Network models are now ubiquitous across many fields as they provide
mathematical descriptions of complex phenomena that are amenable to detailed
analysis. Abstractly, a network is a collection of nodes, N, and
edges, E. Most generally, an edge is an ordered pair of nodes,
e=(i,j), signifying a connection flowing from node i to node j. The
network we build for our application captures the structure of the brine
channels, but without all of the finest details. This network retains salient
permeability properties of the brine network while discarding less relevant
but complex fine-grained features. Our motivation is two-fold. First, the
network presentation allows us to easily analyze permeability of the ice core
samples in terms of network features. Second, the network structure is rich
enough to provide empirical estimates for the brine pore structure in
first-year sea ice at different temperatures. These estimates give a
statistical description of brine channel structure which we can use to inform
percolation and flow models.
Our model is, in a sense, a refinement of other network models used to study
transport within sea ice. Percolation models, such as those developed to
justify the “Rule of Fives” , rest on underlying network
connectivity models of brine pockets within sections of ice. While most of
these models use bulk brine properties to form a statistical basis for
generating the connectivity networks, our work derives finer-grained
statistics empirically, allowing for models more closely aligned with the
physical properties of sea ice. This approach occurs in other related fields:
soil scientists and geologists used network models of pore space to study
similar problems of connectivity and permeability in a different porous
medium e.g.,.
To create our network, we must identify nodes and edges between them that
reflect the structure of the brine channels in the sample. To begin, we think
of a sample as a cube in three dimensional space, R3, using
standard Cartesian coordinates denoted by (x,y,z). We fix the vertical
coordinate, z=C, to identify a horizontal slice of the sample at height
C. For such a slice, we associate a node to each distinct brine pocket.
Figure shows four horizontal slices associated to
consecutive z values, that we use to describe this process. In those
images, ice is grey, air is black, and brine is shown in different colors. In
the topmost image in the stack, where z=z0, we see 10 colored regions
showing 10 different brine channels within that slice. To define a node
associated to the ith region, we consider the collection of points,
{(xik,yik,z0)}k, that make up that region and calculate their
centroid, pi=(xi*,yi*,zi*), which labels the node. Centroids are
reasonable approximations of the positions of the regions since the brine
inclusions in each horizontal slice are primarily convex polygons with the
centroid located inside the connected component . While
this label records the position of the brine region, it contains no
additional information, so we record an approximation of the size of the
brine region along with each node. We fit an ellipse to the region and we
define the throat size of the region, denoted ri, by the length of its
semi-minor axis. The red ellipses in Fig. are examples of
ellipses that allow us to calculate these throat sizes. Repeating this
procedure for all values of z gives a complete list of nodes, {pi}, in
the network, with the associated throat sizes, {ri}. We note that this
process is a version of the maximal ball method
for creating networks from image data. While
other methods exist – e.g., random pipe, medial axis, and flow velocity
methods – the various methods
would yield similar results since the brine channels are mostly convex, and
our choice of method is the least computationally intensive.
Our collection of node–throat size pairs gives an approximation of the brine
structure in a sea ice sample but lacks an important feature – it does not
specify how the brine regions are connected to one another from one
horizontal slice to another. Including edges between nodes allows us to
encode this feature. We define an edge from node pi to node pj if the
pair meet two conditions: first, that node pj appears on the horizontal
slice just below that of node pi, and second, that the brine regions
represented by nodes pi and pj overlap when projected onto the same
image. For example, this connects the four green-colored regions of different
slices in Fig. into a single brine channel. We can formalize
these conditions as follows. We define an edge from node pi to node pj
if
zj*=zi*-1,
{(xik,yik)}k∩{(xjk,yjk)}k≠∅.
In the language of networks , this is a directed
edge as it points in a particular direction which, in this case, is
vertically downward. In some of our calculations, we will ignore the
direction, treating the edge as signaling the bidirectional connection
between the two brine regions. Figure shows several edges
between nodes, including one from node pi1 to pi2, denoted by
dashed black lines. In that figure, we can see that as we move from pi1
to pi2 and further down the vertical axis, the throat size of the brine
channel shrinks. This is not the only behavior – some channels shrink and
eventually disappear, new brine pockets sometimes appear below areas of ice.
Others split into multiple distinct regions, while others still join
together.
Sketch illustrating how the brine channel network is defined. Four
horizontal two-dimensional slices are shown with lines connecting adjacent
nodes (not all lines are drawn). Different colors represent different brine
channels in this sample.
With the definition of the network in place, we next introduce terminology
which helps describe the evolution of a brine channel as it progresses
downwards through an ice sample. For a fixed node pi, if the intersections
in the second part of the definition of an edge above are empty for all other
nodes, we say that node has died moving from slice z=zi* to
slice z=zi*-1. If there is only one non-empty intersection, we say the
node remains. Alternatively, if there are multiple non-empty
intersections, we say pi splits. Last, if more than one node at
height z overlaps with a single node at height z-1, we say that those
nodes join. This terminology allows us to depict the vertical
connectivity of the brine phase. Our definition precludes horizontal
connectivity as the horizontal extent of a brine region is captured in the
definition of the associated node. Since brine channels are primarily
vertically oriented with branches splitting both upwards and downwards, this
network definition yields a good model for depicting brine movement through
the sample.
With this mathematical description of the brine structure, we can calculate
statistics about its evolution. For a fixed throat size, r, we compute the
probabilities of a node of this size remaining, dying, splitting, or joining.
The probability of a node of throat size r remaining in the next step but
changing to throat size s is the number of connections xi*,yi*,zi*,r→xj*,yj*,zj*=zi*-1,s divided by the number of nodes of
throat size r:
Premainr→s=#xi*,yi*,zi*,r→xj*,yj*,zj*=zi*-1,s#i|ri=r.
For our discussions in this paper, we aggregate these fine statistics into a
single statistic recording the probability that a node of throat size r
remains in the next level:
Premain(r)=∑s>0#xi*,yi*,zi*,r→xj*,yj*,zj*=zi*-1,s#i|ri=r.
Similarly, we can calculate the probabilities that nodes of size r die,
split, or join. Taken together, these probability distributions summarize the
evolution of the brine channels in the ice sample statistically.
In summary, the set of probabilities defines a discrete-time probabilistic
process. Given a node at a particular height z and throat size r, we know
the possible outcomes for this node at height z-1 – remaining, dying,
splitting, or joining – and the chance that they occur by looking at the
approximations of the probabilities of their occurrence. In other words,
taking the collection of {pi,ri} as states and the probabilities as
transition probabilities, we have defined a Markov chain modeling the
evolution of brine channels within the sample.
Results: μCT 3-D imaging
We first used standard morphological metrics as defined in previous work to
describe the brine network shape and size .
Figure shows the brine volume fraction, definition, and
shape of the brine phase for the Butter Point and the Iceberg Site ice cores.
The trends in the top half of the core are similar to what we expect since
the temperature in the top half of the core is relatively cold and the
expected brine volume fraction is small. However, at around 100–120 cm the
brine volume fraction begins to increase and the expected C-shape profile
begins to appear. Although this trend persists for a few samples, it does not
continue as we would expect into the bottom of the core for the warmest
temperature samples. This suggests that perhaps the Peltier cooling stage was
not warming the temperatures of those samples above approximately
-7 ∘C. Since the average temperature of the cold room housing the
μCT scanner was -8 ∘C, either the cooling stage warming mode
was not functional or was overcome by the ambient temperature. This may
highlight that the cooling stage is not sufficiently warming the ice, but
instead producing a slush in the pore space that has X-ray attenuating
properties between ice and brine. Segmenting all the slush with the brine
phase (assuming it is possible to isolate only the slush from signal noise)
leads to an overestimate of the brine phase and an inaccurate depiction of
brine channel size and connectivity. Conversely, segmenting the slush with
the ice phase leads to an underestimate of the brine phase and also an
inaccurate depiction of the brine channels. We used segmentation thresholds
that split the difference with a threshold halfway between the peak of the
brine phase and the peak of the ice phase, recognizing that there was indeed
error in segmentation for these warmer samples. Thus, we will treat data
points at depths below roughly 120 cm with caution. Unfortunately, this
encompasses the region where the brine volume fraction crosses the 5 %
critical threshold, limiting our ability to examine the “Rule of Fives” in
this paper.
Definition and shape of brine channels scanned with a μCT
scanner and a Peltier cooling stage set at in situ temperatures. The red
squares and blue circles represent ice cores from Butter Point and the
Iceberg Site, respectively.
Since the cooling stage did not significantly warm samples beyond
-7 ∘C, we were not surprised that general trends shown in
Fig. for all metrics did not differ significantly from the
same samples scanned isothermally and presented in as
the percolation threshold was not crossed. As in , we
used the structure model index (SMI =6S′⋅VS2,
where S′ is the derivative of the change in surface area after a one pixel
dilation, V is the initial volume, and S2 is the initial surface area)
to quantify the similarity of the brine phase to plates, rods, or spheres. To
quantify size, we calculated a structure thickness by first identifying the
medial axes of all brine structures and then fit the largest possible sphere
at all points along said axes. The structure thickness is defined as the mean
diameter of all spheres over the entire volume. The structure separation is
the inverse metric, providing a measurement on the spacing between individual
objects. We then calculated the degree of anisotropy by finding the mean
intercept length for a large number of line directions, and forming an
ellipsoid with boundaries defined by these lengths. The eigenvalues for the
matrix defining this ellipsoid are calculated, and correspond to the lengths
of the semi-major and semi-minor axes. The ratio of the largest to smallest
eigenvalues then provides a metric for the degree of anisotropy, with zero
representing a perfectly isotropic object and one representing a completely
anisotropic object. We observed that the brine phase specific surface area
increased with depth, structure model index was roughly three (indicative of
cylindrical objects), structure thickness decreased, structure separation
increased, and the degree of anisotropy increased throughout the middle of
the core (Fig. ). From the metrics above, we conclude that
brine channels are primarily cylindrical in shape with more branches at lower
depths, consistent with previous observations .
We compared the μCT-measured brine volume fraction to expected values
derived from the Frankenstein and Garner relationship relating temperature,
salinity, and brine volume fraction in
Fig. . For this analysis, we used the core temperatures
measured in the field and salinity values estimated from ion chromatography
measured chloride concentrations presented in . From 0
to 120 cm, the measured brine volume fractions match the expected values
remarkably well. However, below a depth of 120 cm in both cores, the
expected brine volume fraction is greater than that measured using μCT.
For example, at a depth of 160 cm the expected brine volume fraction is 4.5
and 4.2 times greater than the values measured for samples from the Butter
Point and the Iceberg Site ice cores, respectively. This provides an estimate
for the degree by which the cooling stage failed to heat the warmer samples
in the μCT.
Comparing μCT-measured brine volume fraction to the expected
values derived from the Frankenstein and Garner relationship
. Results from the Butter Point (red) and the
Iceberg Site (blue) cores, where the μCT-measured values are the filled
circles and the expected values are the open circles.
Average throat size {er‾i} for the five largest
brine channels of each sample for the Butter Point ice core. For each
channel, at a given depth we calculated the average throat size of all nodes
and color-coded accordingly. Note that there are only six channels that connect
the top to the bottom of the sample (at depths of 0, 10, 70, and 120 cm).
Average throat size {r‾i} for the five largest
brine channels of each sample for the Iceberg Site ice core. For each
channel, at a given depth we calculated the average throat size of all nodes
and color-coded accordingly. Note that there are only six channels that connect
the top to the bottom of the sample (at depths of 0, 10, 30, 50, and
170 cm).
Results: brine network model
Model definitions
From the binarized images of the brine phase for the Butter Point and the
Iceberg Site ice cores, we created a mathematical network. We will use the
term network to refer to the entire brine phase of a given sample
and/or the entire brine phase of all samples in a given core. For a given
sample, we define each brine channel to be a single connected region
in the brine network. The number of brine channels per sample ranged from 830
to 4800, with maximum numbers occurring in samples from the top and bottom of
the cores. Previous work showed that these brine channels often appear in
layers or sheets spaced approximately 0.5–1.0 mm apart due to the ice
growth mechanism and original skeletal structure . A single
brine channel is a complex web containing many different parts, which we will
call the branches of the brine channel. As defined previously, a
join point is the node where two branches come together and a
split point is the node where a single branch splits into multiple
branches. We note that flipping the perspective of movement from downwards to
upwards changes a split point into a join point and vice
versa. This is an important observation since we did not record the vertical
orientation of the samples during cutting. Using this terminology, we use
techniques from network theory to topologically characterize the brine
network, gaining further insight into the connectivity and implications for
permeability.
In the analysis below, we use several metrics to describe the topology of the
brine network that have important fluid flow implications. We first look at
the throat sizes of the brine channels to gain an insight into the quantity
of fluid that can move through different regions of a given brine channel. We
then look at specific branches, both in terms of the number of branches and
the size of each branch to learn more about the specific pathways available
for fluid movement. As part of this analysis, we incorporate our various
transition probabilities to understand how the likelihood of a given branch
to split into multiple branches depends upon the throat size of the parent
branch. Finally, we will examine the size distribution of particular paths,
looking for “pinch points” that may restrict flow and large regions that
can provide maximum flow through the network. Together, these metrics will
provide a detailed description of the micro-scale complexity of the brine
network.
Throat sizes of channels and branches
For each brine channel we calculated the average throat size {r‾i} for all nodes {pi} at a given depth in the
sample. Figures and show the
collection of these throat sizes for the five largest (by vertical extent)
brine channels at each depth for the Butter Point and the Iceberg Site cores,
respectively. We note that since the entire length of the cores were not
scanned, there is no correlation between the five brine channels selected
from one sample (e.g., 20 cm depth) to the next (e.g., 30 cm depth). There
were six brine channels in each core that connected from the top to the
bottom of their respective sample, with the majority of these channels found
in samples from the top of each core. Generally speaking, the lengths of the
vertical extent of the longest channel decreased with depth from the top of
each core to around 60 cm, increased from 60 cm to roughly 120 cm, and
then decreased from 120 cm to the bottom of the core. The trend between the
top of the core and roughly 120 cm is consistent with what we would expect
due to brine volume fraction, temperature, and the expected C-shape profile.
The channels from the lower depths, which had even warmer temperatures, did
not reach in situ temperatures during scanning as described previously. We
also note that both the lengths of the vertical extent and average throat
size quickly diminished for channels beyond the few largest ones, as can be
seen in the bottom panel of Figs.
and . Thus, we learn that fluid flow is most controlled
by the behavior of the largest brine channel for a given section of sea ice.
The number of branches for a particular brine channel has potentially
significant implications for fluid flow and permeability, such as influencing
the rate at which chemical species may pass through the sea ice
. By increasing the number of
branches, split points can increase the number of potential paths through the
sample. A higher number of paths increases the probability of finding a path
connecting the top and bottom of a sample, thereby crossing the percolation
threshold . Alternatively, split points can represent
bottle-necks if the resulting child branches have smaller throat sizes than
the parent throat, measured either as a minimum or as an aggregate. We
observed that the largest channel in a cubic sample had by far the largest
number of branches, with the quantity dependent upon the ice type. For
example, in a columnar ice sample (70 cm depth of the Butter Point ice core)
the largest channel had a maximum of 20 branches at a given depth within the
sample. For comparison, the maximum number of branches for a sample in the
frazil ice region at the top of the core was 124 nodes at a single depth. As
expected, we find that brine channels in frazil ice have many more branches
than brine channels in columnar ice, providing more distinct pathways for
brine to move through the sample.
Throat size ri of each node for the largest channel of
representative samples in the Butter Point core. The top, middle, and bottom
rows show the largest brine channel from the sample at 0, 70, and 170 cm,
respectively. The left panels show the throat sizes at each depth in the
sample with nodes sorted by location, not by size. The right panel sorts the
nodes by throat size in descending order.
Throat size ri of each node for the largest channel of
representative samples in the Iceberg Site core. The top, middle, and bottom
rows show the largest brine channel from the sample at 0, 50, and 170 cm,
respectively. The left panels show the throat sizes at each depth in the
sample with nodes sorted by location, not by size. The right panel sorts the
nodes by throat size in descending order.
The figure shows {ri} probability distributions showing the likelihood
a node dies, remains, splits, or joins as a function of the throat size r.
The top four figures show probability distributions for frazil ice while the
bottom four figures show probability distributions for columnar ice. Note
that to complete these figures, in addition to the Butter Point and the
Iceberg Site cores, we used two additional first year sea ice cores from
previous work . Butter Point is shown in black, the
Iceberg Site is shown in blue, and the two additional first year ice cores
are shown in red and black.
To gain insight into the behavior of a channel, we visualized the number of
branches and distribution of throat sizes by plotting the throat size ri
of each node pi for the largest brine channel. Figures
and show the throat sizes as a function of depth in the sample
for three different representative sample depths: top, middle, and bottom of
the Butter Point and the Iceberg Site cores, respectively. For each channel
shown, there is a plot of {ri} at each depth sorted by physical
location in a two-dimensional grid (working line by line), not by size. A
second corresponding plot shows node sizes sorted by descending {ri}
for a given depth in the channel. The first set of plots illustrate the
connectivity of given branches, while the second set provide a visualization
of the distribution of ri. The sample taken from the top of each core is
from a region of frazil ice, which we would expect to have brine channels
that are not well connected and have a distribution of throat sizes
independent of depth in the sample. In both Figs.
and , panel (a) confirms this while panel (b) shows that there
was an even distribution of throat sizes. The two plots for mid-depth
networks (70 cm) are quite similar, illustrating less tortuosity and easier
ability to track particular branches in the brine channel. The bottom sample
of the Iceberg Site core had much larger throat sizes, although this sample
was an anomaly in Fig. . We did not observe a direct
correlation between the number of branches and the throat size of those
branches, as the distribution of throat sizes appeared to be more dependent
upon the particular depth of the sample in the ice core, and consequently,
the ice type of that sample. From this analysis, we learn that although there
may be more branches for a given brine channel in frazil ice, the branches
have better vertical connectivity in columnar ice. This means that fluid can
more readily move upwards or downwards through the larger well-connected
brine channels in columnar ice.
Probability distribution of branching nodes
Next we examined the branching of particular nodes to understand the behavior
of particular fluid flow paths. Following a branch of a channel downwards, at
an individual node the branch may end, continue onwards, or split into
multiple branches. Conversely, by looking upwards, a node can be considered
to be the first in a new branch, the continuation of a branch, or the joining
point of multiple branches. Thus, for each node in a brine channel, we can
count the number of edges above (incoming) and below (outgoing) to determine
the degree of splitting or joining of branches in the channel
. The large majority of nodes do not display branching, and
the number of two-way splits/joins was roughly the same as the number of
times a branch started/ended. We observed decay in frequency with increasing
quantity of splits/joins. The Iceberg Site core had a larger number of higher
order branching with a significant number of seven-way or eight-way
splits/joins. A branch that splits is most likely to split into only two
child branches, and thus for example, a contaminant introduced at a point
source is likely restricted to a small horizontal region, following only a
few separate paths through the ice. When a split occurred, we compared
ri for the parent node to the collection of ri for the children nodes
with similar behavior observed in both cores. 84 % of the time for the
Butter Point core and 86 % of the time for the Iceberg Site, the sum of
the throat sizes for the children node were greater than that of the parent
node. However, the parent node was still larger than the largest child node
67 % of the time for the Butter Point core and 68 % of the time for
the Iceberg Site core. Thus, we
learn that larger brine channels are more likely to split than smaller
channels, and after the split, the fluid can access a larger region of the
sea ice.
With knowledge of the total number of split points and join points, we then
investigated the likelihood that branching was dependent upon the throat
size. Figure shows the probability distributions
for pockets dying, remaining, joining, and splitting for two regions of each
of the four samples, frazil ice and columnar ice. Note that for these plots, we
used two additional first-year sea ice cores from previous work in addition
to the Butter Point and the Iceberg Site cores . The
most basic difference between the two regions is that larger pocket sizes do
not appear in columnar ice. However, for ranges of r occurring in both
regions, the shapes of the plots are similar. While small pockets can
disappear, larger ones generally do not – the probabilities tend to zero (as
marked by arrows in the plots in the first column). For pockets that remain
but do not split or join with others, smaller pockets remain with lower
probability (because more of them vanish) but then the probabilities follow
an inverted parabolic trajectory, peaking around r=130 µm. An
interesting difference appears as throat size grows. In columnar ice, for
throat size around r=500 µm, we see two distinct behaviors.
Some sizes remain with probability one (indicated by the top arrow in the
second plot of the second row), while others remain with probability zero
(bottom arrow). For the latter, looking at the last plot in the bottom row,
we see these pockets are splitting into two or more pockets (indicated by the
top arrow in that plot). This gives a signature for columnar ice – most
brine channels simply continue on with slightly varying throat size but the
ones that change generally split, creating a fork in the channel.
In frazil ice, the story for pockets that vanish is the same, as throat sizes become larger, they do not
vanish in the next level. For remaining, splitting, and joining, however,
there are new wrinkles in frazil ice relative to columnar ice. For pockets
that remain, for larger throat sizes we see three types of behavior, two of
which are similar to the behaviors in columnar ice (indicated by the top and
bottom arrows of the second plot in the first row); however, a third
behavior, where fifty percent of pockets remain, is new for frazil ice
(middle arrow). This new behavior is echoed in the probabilities of splitting
and joining (indicated by the middle arrows in those plots) which shows that
in this regime, brine channels have a complex behavior, remaining, splitting,
and joining with high frequency. This third category of behavior for large
throat sizes is a signature of frazil ice.
In addition, we summed the total number of edges leaving (splits) and
entering (joins) each node over all nodes for the five largest brine channels
of each sample. Figure plots these raw counts and the
difference between the two are given for the Butter Point and the Iceberg
Site cores. When we consider split points and join points separately, we are
considering the network as a graph with directed edges. The difference
between the number of splits and joins (i.e., difference between number of
incoming and outgoing edges) is a metric for the topological complexity of a
network. The raw counts for number of splits and joins had roughly C-shape
profiles for both cores, with largest values and variability observed towards
the bottom of the core. This is to be expected because the warmer part of the
core allows for greater interconnectivity of branches in the brine network.
For all brine channels, the number of splits was quite similar to the number
of joins, and hence the differences between the two were quite small.
However, there was still a general C-shape profile between 0 and 140 cm,
indicating that topological complexity is greatest near the top and bottom of
the core. This is consistent with frazil ice in the top of the core and
increased branching in the warmer ice. Interestingly, both cores showed a
decrease in topological complexity for the lowest samples below 140 cm. This
could either be an artifact of not achieving actual in situ temperatures with
the cooling stage, or potentially an indication of a thought-provoking trend.
If samples were not reaching in situ temperatures, isolated channels may not
have rejoined upon warming from storage temperatures, thereby reducing the
number of split points and join points. Alternatively, a possible explanation
of a real trend could be that as brine channels widen for the warmest
samples, branches join together, reducing the topological complexity. A
consequence of reducing the number of branches is a reduction in the number
of split points and join points.
Topological complexity of the five largest brine channels in each
sample for both the Butter Point (red) and the Iceberg Site (blue) cores. The
panel (a) shows the total number of splits (open circles) and joins
(filled squares) over all nodes in a given channel. The panel (b)
shows the absolute value of the difference between the number of splits and
joins. The dashed line highlights the depth below which there is concern
regarding the effectiveness of the cooling stage and whether samples were
scanned at actual in situ temperatures.
Cumulative distribution functions for number of brine channels as
functions of the total number of pixels in the channel. The
panels (a, b) are for the Butter Point and the Iceberg Site ice
cores, respectively. In both panels, each line represents a different sample
depth where the lines are colored on a gradient from red representing the top
of the core to blue for the bottom of the core. Note that pixels
in the original μCT images are 15 µm on each edge. Note
that all values for depth are in centimeters.
Capacity for fluid flow
We next examined the fluid flow capacity of each channel by both summing the
number of pixels associated with all nodes for each channel and summing the
total throat sizes of all nodes in each channel. We note that this represents
a region larger than the pathways used for current fluid flow since many
branches do not connect the top of a sample to the bottom. However, when the
ice begins to warm and the branches become more interconnected, the process
will likely start from the existing regions containing brine. Thus, this
metric offers a starting place for comparing the capacity for fluid flow
across different samples. Figure shows cumulative
distribution functions for the number of brine channels as functions of the
total number of pixels in the channel, with each line representing a different
sample depth. The lines are colored on a gradient from red representing the
top of the core to blue for the bottom of the core. The distribution
functions for all depths on both cores were remarkably similar, and pairwise
Kolmogorov–Smirnov tests did not detect that any two curves were from
different probability distributions (p≥0.1)
. Both cores did show a trend of increased
probability of brine channels with more pixels occurring at shallower depths,
with a more robust trend observed in the Iceberg Site core. This trend could
be due to samples at lower depths having an increased number of isolated
small channels that have yet to connect to larger channels. Since there is
doubt as to whether the samples below 120 cm were scanned at their in situ
temperatures, perhaps these small isolated channels would have connected to
larger channels under warmer conditions. Figure
presents similar cumulative distribution functions for the number of brine
channels as functions of the summed throat size of all nodes in the channel.
The curves yield the same observations as before, with the Iceberg Site core
again having a stronger correlation of increased probability of larger
channels occurring at shallower depths. Likewise, Kolmogorov–Smirnov tests
did not detect any two curves representing different probability
distributions (p≥0.1). Any noticeable changes to the relative shape
of the curves represent disproportionate changes in the shape of the brine
channel with size of the channel, however, these variations were quite minor.
In general, the shape of the curves in Fig. are similar
to those in Fig. . Thus, we conclude that brine channels in
samples near the top of the core provide fluid with multiple distinct
pathways to move through the sample, while deeper in the core there are only
a few large channels with many small isolated paths that may connect under
warmer ice conditions.
Cumulative distribution functions for number of brine channels as
functions of the summed throat size of all nodes in the channel. The
panels (a, b) are for the Butter Point and the Iceberg Site ice
cores, respectively. In both panels, each line represents a different sample
depth where the lines are colored on a gradient from red representing the top
of the core to blue for the bottom of the core. Note that throat
sizes in the original μCT images are 15 µm on each edge. Note
that all values for depth are in centimeters.
Largest brine channel at 70 cm in the Butter Point ice core.
Although this brine channel connects from top to bottom, there is not a
directed path that does so. Any connecting path involves movements both
upwards and downwards. One such path is highlighted in red.
Probability distributions of paths connecting the top to the bottom
for all brine channels in the Butter Point ice core. Only paths greater than
50 steps, or 750 µm were considered. Left, middle, and right
panels represent channels where rf<1500 µm,
1500≤rf<5250 µm, and
rf≥52500 µm, respectively. Top, middle, and
bottom rows represent probability distributions for rmin, rmax,
and summed throat size, respectively. For all panels, the colors red, blue,
and green represent channels where r1<1500 µm, 1500≤r1<5250 µm, and r1≥52500 µm,
respectively.
Following individual fluid flow paths
To further assess fluid flow capabilities, we analyzed individual branches of
brine channels to isolate particular paths through the network. By
construction, moving from pi to pj along an edge must either
increase or decrease the height in the sample by one step
(15 µm). First, we allowed only downward flow along the edges,
considering paths starting from the first node. Since the network does not
allow lateral movement, each step along an edge corresponds to a
15 µm step downwards. Although previously six brine channels were
found that connected the top to the bottom of a given sample, no such paths
were found in the directed graphs. This is because all paths connecting the
top to the bottom of a sample required some movement upwards along a branch
in order to reach the bottom. Figure shows an example of a brine
channel where although the network is connected, any connecting path involves
both upward and downward flow, such as the path highlighted in red. Thus, we
selected the longest downward directed path from each brine channel, as well
as any additional paths of the same length. This mimics a natural process
such as gravity drainage, allowing us to study its influence on brine
movement in the absence of pressure forces that aid upwards transport.
Summing over all brine channels in the Butter Point core resulted in 63 763
directed paths, of which 15 316 paths had a length of at least 50 steps
(750 µm). We then used this smaller subset for statistical
analysis of minimum throat size (rmin), maximum throat size
(rmax), and summed throat size. Future work will use this model to
statistically recreate brine channels that have this same distribution of
brine channel sizes.
We completed a similar analysis on the brine channel network, however this
time allowing for both upward and downward flow. Allowing for upward flow can
present a challenge in tracking various pathways if there is a repeating
loop. Thus, we only considered paths that reached every node but had no
loops. In the language of networks, we avoided complexities arising from
cycles by only considering different spanning trees. We used a depth-first
search algorithm to find all paths reaching the maximum vertical extent of
each channel . We checked results through
comparison of the distance obtained using Dijkstra's algorithm for finding
the shortest-path tree . This resulted in
36 449 paths over all the brine channels in the Butter Point core, of which
1753 were of length 50 steps (750 µm). We note that we can use
the adjacency matrix to calculate the number of different walks (paths
including cycles) that connected the top and bottom of a sample
. However, due to the size of the adjacency matrix, this
became computationally too expensive for large brine channels.
We used the 1753 paths of length greater than 50 steps to develop probability
distributions for basic network statistics important for fluid flow such as
rmin, rmax, and summed throat size of the path. These statistics
can yield valuable information regarding the location and distribution of
“pinch points,” large channels, and total fluid flow through a brine
channel. First, the paths were split into three categories based upon the
ending node (pf) throat size: rf<1500 µm, 1500≤rf<5250 µm, and
rf≥52500 µm. Then, we split each category into
three groups based upon the throat size of the starting node (p1), using
bins of the same size (r1<1500 µm, 1500≤r1<5250 µm, and r1≥52500 µm). This division
resulted in splitting the paths into nine separate sub-categories. For each
sub-category, we calculated the probability distributions for rmin,
rmax, and summed throat size of the path and we present these
histograms in Fig. . All plots are color-coded by r1,
with red, blue, and green histograms representing small, medium, and large
throats, respectively. All histograms show the similar trend that both
r1 and rf have a strong influence on resulting metrics, with
larger r1 and/or rf having larger rmin, rmax,
and total volumes. The magnitude of rf had slightly more of an impact, particularly
on rmin. There were no clear general trends in regards to the shape of
the distributions. However, we note that for the smallest rf, all
three histograms for rmin had a large peak around 30 µm
(Fig. , top row). This peak corresponds to the smallest
measurable branches, and we could potentially remove these paths from current
fluid flow analysis. However, as the ice begins to warm, these “pinch
points” are likely to have a significant impact on crossing percolation
thresholds.
Conclusions
The primary objective of this work has been to improve our
characterization of brine channel topology, morphology, and connectivity, in
order to provide sea ice modelers with a greater level of detail on the
factors that affect microstructural transport properties. While most
percolation models use coarse microstructural properties to form a
statistical basis for predicting connectivity, ours derives finer-grained
statistics empirically, allowing for better representation of the range of
physical properties found in sea ice of different types and conditions. We
can statistically model the evolution of brine channels as we move downwards
through the sea ice cover. Beginning with an initial brine pocket, our
estimates of the evolution probability distributions from μCT scans of
sea ice samples tell us how the channel changes as we progress downward
through the sample – Does it grow? Shrink? Split into more than one branch?
Join up with more than one branch? Close off entirely?
Overall, we observed similar morphological profiles for both first-year sea
ice cores. Topological complexity had the expected C-shape profile that is
consistent with complex frazil ice in the top of the core, relatively cold
columnar ice below it, and increasingly warmer columnar ice at lower depths.
However, we did not have good success in imaging and thresholding ice with
the warmest in situ temperatures at the bottom of the core.
Our estimates of the evolution probability distributions provides a
stochastic model of brine channels within sea ice at different temperatures,
extending the percolation models described above. Their structural features
reveal the onset of transitions between different types of ice: in our
analysis, we see different statistical features that delineate frazil and
columnar ice. Further, the level of detail inherent to this technique allows
us to quantify some of the finer details of brine channel structure and
development. In addition to estimating the expected brine volume and
permeability for ice at a fixed temperature, we can see when and why
permeability arises by analyzing the probabilistic structures. For example,
Fig. shows a stark structural difference between
frazil and columnar ice which points to the onset of percolation: brine
pockets in frazil ice larger than about 1 mm are extremely likely to join or
split while the largest brine pockets in columnar ice are more likely to
persist. We observe that brine channels in columnar ice simply continue
downwards with little change in size. For those in which there is a
significant change in size, there is a likely fork leading to a split in the
channel. A higher probability of interconnections between brine channels
translate directly into a higher probability of permeability of the ice.
When examining the branching in brine channels, we observe that the largest
channels have the greatest number of branches, but overall brine channel size
does not appear to have a direct correlation with the number of branches.
Brine channel size is most dependent upon the depth and consequently ice
type. When a split in a brine channel does occur, it is most likely to split
into two child branches, and after the split, the brine generally has access
to a larger region of the sea ice than before. Starting and ending brine
pocket sizes are strongly correlated with the flow capacity with larger
initial/final sizes strongly indicative of increased flow. We detected pinch
points in the brine channels that are critical points when determining
whether the sea ice cover has crossed the percolation threshold. However,
further work is needed in examining warmer ice with greater brine volume
fractions.
Our framework enables us to statistically replicate the pore structure of sea
ice at different depths and temperatures. The next step for this work is to
create a brine channel network from the probability distributions presented
here. For a sample at a given depth/temperature, first an initial region at
the top of the ice would be selected with size consistent with the statistics
shown here. The brine channel could grow or shrink, split into multiple
branches, join with other branches, remain constant, or stop, all with
probabilities dependent upon the thickness, depth/temperature, and proximity
to other brine channels. The model described herein can help address
questions such as how microstructural changes may be path dependent (e.g.,
whether to consider both upwards and downwards flow), how fluid flow may vary
with depth, and what are the percolation implications of temperature
fluctuations in an ice core. In summary, we successfully developed a method
using μCT imaging to characterize the geometry of brine channels, whereby
we can parameterize the pore networks using topological techniques that can
be adjusted for depth and temperature, correlated with physical properties,
and used in dynamical models of sea ice.