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Thesis Proposal
Computational aspects of direct and inverse problems in permafrost geophysics
Dmitry J. Nicolsky
Abstract.
One of the goals is to develop of a numerical model of diﬀerential frost heave dynamics, based
on physically realistic assumptions. The frost heave model has a numerous set of input parameters. Some
values of these parameters can not be measured directly in a laboratory. The way to obtain them is to
solve inverse problems. The second part of the PhD dissertation is devoted to the study of the so-called
non-sorted circles developed in the Arctic tundra.
1. Introduction
The research related to my Ph.D. thesis is focused on the development of a general thermo-mechanical
model of soil freezing/thawing and its applications to certain areas of the subsurface science: engineering,
subsurface hydrology, geocryology, and arctic biology and ecology. The area of possible applications of
the model includes the simulation of frost heave and thaw settlement and diﬀusion of water or solutes in
seasonally freezing and perennially frozen ground. The numerical implementation of the model realizes its
predictive capabilities to simulate complex systems and to gain better understanding of interactions between
physical and biological processes in the arctic ecosystems.
Currently, after a year of development, the model is based on thermo-mechanic equilibrium for a homo-
geneous fully saturated mixture of ice, water and soil particles. The elasticity laws govern the deformations
of the mixture. These deformations are developed by forces due to the pore water migration towards the
freezing zone and its consequent freezing. The ﬁnite element implementation of the model was used to cal-
culate the temperature, moisture regime, and frost heave dynamics at one of our ﬁeld research sites on the
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DMITRY J. NICOLSKY
North Slope of Alaska. Comparison between calculated and measured temperature and moisture dynamics
show a very good agreement.
The current numerical implementation of the model is also capable to realistically predict a frost heave.
We believe that the model could be very instrumental in studying the interaction between physical and
biological processes in the arctic ecosystems. Therefore, one of the applications of this model will focus
on cryoturbation processes in the Arctic tundra and on mechanisms that cause diﬀerential frost heave in
the active layer. The project will be concerned with the modeling of non-sorted circles along the Arctic
climatic gradient and with the reaction of these ecosystems to changes in climate, in the active layer, and
in vegetation cover. The main question to be addressed is, ”How can changes in surface conditions such as
vegetation, snow cover, and climate aﬀect the seasonal dynamics of water and heat ﬂuxes within non-sorted
circles in the Alaskan Arctic?”
2. Brief description of the general thermo-mechanical model of ground freezing
To describe the processes in the freezing soil, we use the theory of mixtures. The theory is aimed to
represent the phenomena of heat transfer, diﬀusion of liquids and solutes and mechanical deformation of the
ground.
Let us consider a mixture of
m
constituents occupying the region Ω
t
in space at the time
t.
Let
x
k
be the
coordinate vector of a particle of the
k-th
constituent. We mark all quantities related to the
k-th
constituent
with subscript
k.
Denote by
β
k
the volume fraction deﬁned as
β
k
=
dv
k
dv
,
where
dv
k
is the volume element
of the constituent of the mixture occupying the volume element
dv.
The volume fractions
β
k
k
= 1,
. . . , m
depend of the coordinate vector
x
k
and satisfy the following constrain
k
β
k
(x
k
, t)
= 1,
∀x
k
∈
Ω
t
.
Let
ρ
k
be the apparent mass density related to the intrinsic density
ρ
k
as
ρ
k
(x
k
, t)
=
β
k
(x
k
, t)¯
k
.
¯
ρ
At any point in Ω, we consider the actions on the part of the constituent
k
occupying Ω and calculate
the rates of growth of mass
c
k
, linear momentum
m
k
and energy
e
k
. It is shown in [1,
2]
that
∂ρ
k
+
∂t
dx
k
dt
(1)
c
k
=
·
ρ
k
(2)
dx
k
d
2
x
k
m
k
=
c
k
+
ρ
k
2
−
dt
dt
·
T
k
−
ρ
k
f
k
THESIS PROPOSALCOMPUTATIONAL
ASPECTS OF DIRECT AND INVERSE PROBLEMS IN PERMAFROST GEOPHYSICS
3
(3)
where
T
k
,
e
k
=
m
k
·
k
, f
k
,
h
k
dx
k
1
dx
k
dx
k
d
k
+
c
k
(
k
−
·
) +
ρ
k
−
T
k
:
dt
2
dt
dt
dt
dx
k
−
dt
·
h
k
−
ρ
k
r
k
and
r
k
stand for the stress tensor, speciﬁc energy, body force, heat ﬂux and body heating,
respectively. We impose the following constrains
c
k
= 0,
m
k
= 0,
e
k
= 0,
k
k
k
which are derived from the conservation principle of the mass, linear momentum and energy for the whole
mixture. The internal dissipation in the mixture is introduced by considering the speciﬁc entropy
s
k
. While
the entropy of each constituent may change, the entropy of the whole mixture cannot decrease. We regard
this statement as expressing the content of Clausius’ postulate:
Φ
≡
T
η
k
≥
0,
η
k
=
c
k
s
k
+
ρ
k
ds
k
−
dt
·
ρ
k
r
k
h
k
−
,
T
T
k
where
η
k
is the growth of entropy and
T
is the common temperature for the whole mixture. The so-called
dissipation function Φ = Φ(T,
h
i
, c
+
, β
i
,
i
x
k
,
x
k
) and the set of the speciﬁc free energies
ψ
k
≡
k
−
T s
k
,
k
= 1,
. . . , m
deﬁne physical processes in the mixture. The speciﬁc free energy
ψ
k
is given by
˜
ψ
k
=
ψ
k
(T,
β
i
,
x
k
,
T
k
) +
T
I(β
1
, . . . , β
m
),
ρ
k
¯
k
where function
I
has the following properties:
I(β
1
, . . . , β
m
) = 0 if
β
k
= 1 and
β
k
≥0;
or +∞ otherwise.
An appropriate choice of the speciﬁc free energy for each constituent (ice, liquid water and soil particles)
and the dissipation function sets up the realistic model of soil freezing. More details can be found in [3].
3. Modelling of the non-sorted circles.
Non-sorted circles are deﬁned as circular patterned ground features without a border of stones, usually
with barren or sparsely vegetated central area 0.5 to 3.0 m in diameter. Figure 1, left shows that the soil
proﬁle beneath the non-sorted circle and its complement area (inter-circle) diﬀer markedly. The non-sorter
circles occur on Turbic Cryosols, i.e. mineral soils that have permafrost within 2.0 m of the surface and show
marked evidence of cryoturbation laterally within the active layer, as indicated by disrupted mix of broken
horizons, or displaced material, or a combination of both. Turbic Cryosols generally have an organic-rich
mineral horizon (Ah) and (Bmy) horizon, in the sequel, referred as the vegetation cover. Underneath the
4
DMITRY J. NICOLSKY
Figure 1: Schematic description of physical processes in the non-sorted circles.
Soil horizons and layers:
A-Mineral
horizon forms at or near the surface in the zone of leaching
or eluviation of materials in solution or suspension, or of maximum
in situ accumulation of organic matter or both.
B-Mineral
horizon is characterized by enrichment in organic matter,
sesquioxides, or clay; or by the development of soil structure; or by a
change of color denoting hydrolysis, reduction, or oxidation.
C-Mineral
horizon is comparatively unaﬀected by the pedogenic pro-
cesses operating in A and B horizons.
h-A
horizon enriched with organic matter
m-A
horizon slightly altered by hydrolysis, oxidation, or solution, or
all three to give a change in color or structure, or both.
y-A
horizon aﬀected by cryoturbation as manifested by disrupted and
broken horizons, incorporation of materials from other horizons, and
mechanical sorting in at least half of the cross section of the pedon.
z-A
frozen layer.
vegetation cover the mineral horizon (Cy/Cz) is located. This horizon is comparatively unaﬀected by the
pedogenic processes operating in A and B horizons. Ah and Bmy horizons are less than 0.1 m thick.
The major hypothesis and assumptions regarding to the non-sorted circles that we are using in our
research are:
•
The vegetation and organic layers have thermo-rheological properties that are diﬀerent from ones
corresponding to the layer of mineral soil lying underneath the vegetation cover. However, the
mineral soil inside the circle is equivalent to the mineral soil inside the inter-circle.
•
To model the non-sorted circle dynamics, we hypothesize that the soil is always fully saturated,
i.e. there is enough liquid water to supply the processes that cause frost heave. This hypothesis
THESIS PROPOSALCOMPUTATIONAL
ASPECTS OF DIRECT AND INVERSE PROBLEMS IN PERMAFROST GEOPHYSICS
5
is modeled by prescribing the constant water pressure on the outer boundary of the non-sorted
circles.
•
Water freezing and mechanical deformation produces ice lenses and micro cracks in the near-surface
portion of the mineral soil that is seasonally thawed. Therefore, it is reasonable to assume that
rheological properties of the mineral soil are distinct above and below the depth of the active layer
thickness. This hypothesis is modelled by assuming that the soil in the permafrost has higher values
of the Young’s modulus, or the stiﬀness coeﬃcient.
•
Among the other assumptions are the isotropy of the material, the small strains, the piecewise
linear elasticity of the skeleton. Also we assume that the ﬂow of ice with respect to the skeleton
is negligible, the phase change that occurs between water and ice is reversible, the kinetic energy
terms and terms due to deformations in the expression of the balance of energy can be neglected.
Based on the above assumption we can formulate expressions for the speciﬁc free energies
T
1
T
) +
W
i
+
I(β
i
, β
w
, β
s
)
T
0
ρ
i
¯
ρ
i
¯
(4)
Ψ
i
=
−C
i
T ln(
(5)
Ψ
s
=
−C
s
T ln(
1
T
T
) +
W
s
+
I(β
i
, β
w
, β
s
)
T
0
ρ
s
¯
ρ
s
¯
(6)
Ψ
w
=
−C
w
T ln(
T
T
−
T
0
LT
T
)
−
[(C
i
−
C
w
)T
0
+
L]
+
f
(β
i
, β
w
, β
s
) +
I(β
i
, β
w
, β
s
)
T
0
T
0
T
0
ρ
s
¯
for ice, skeleton and liquid water, respectively. In the above,
C
k
is the speciﬁc heat capacity,
T
0
is the
temperature of fusion,
L
is the latent heat of fusion,
W
k
is the strain energy function and
f
is the so-called
the adsorption function of the skeleton. The form of function
f
is veriﬁed by experiments [4]. A sketch of
processes occurring inside the non-sorted circles are shown on ﬁgure 1, right.
In order to apply the model, numerous geophysical, thermal and hydraulic properties of the soils are
required. The required soil properties include the following: density, thermal conductivity (frozen and un-
frozen), heat capacity (frozen and unfrozen), water content, and porosity. We also specify the hydraulic
conductivity and freezing temperature of the soils as a function of unfrozen water content. Standard tech-
niques are available to determine these properties and parameters.
6
DMITRY J. NICOLSKY
The initial and boundary conditions can be derived from ﬁeld measurements. The instrumentation at
our ﬁeld sites allows monitoring of soil temperatures, soil moisture, frost heave, and cryoturbation activity.
4. Results up to date
For one non-sorted circle temperature and moisture distributions were measured at the Franklin Bluﬀs
site over the period of one year. The value of the maximum frost heave was also estimated at the same site.
The maximum frost heave equals to the maximum displacement of the ground surface relative to its position
by the end of summer when the active layer depth is maximal. For this speciﬁc circle, soil properties were
evaluated and used to simulate freezing of the non-sorted circle. As the result, the calculated temperature
and moisture at certain depths were compared to the measured ones and are shown in Figure 2. The
calculated value of the maximal frost heave is within the uncertainty of measurements. This example gives
us assurance that the physical model represents the reality objectively and our developed numerical model
can be used to investigate the process of frost heave and the development of non-sorted circles.
One of the questions that we will explore in our research
is ”How can changes in surface conditions such as vegeta-
tion, snow cover and climate aﬀect the seasonal dynamics
of water and heat within non-sorted circles?” We have to
note that the modelling of the non-sorted circles as a whole
is complicated, since processes interact with each other and
produce a non-linear behavior of the entire system. There-
fore, to obtain the answer to this posted question, we have to
Figure 2: Measured and calculated temperature
and volumetric water content in the center of the
circle at the depth of 0.35cm.
understand how each process inﬂuence the dynamics of the
system as a whole. We plan to apply a sensitivity analysis
where we will investigate this by changing crucial parame-
ters that characterize the certain process one by one and observe changes in the model’s output.
As an example, Figure 3 shows several graphs of the maximal frost heave (the result of computations),
corresponding to the sets of input parameters, where all elements but
B
were the same. The constant
B
parameterizes the unfrozen water content in the freezing soil as the function of temperature, shown in ﬁgure
THESIS PROPOSALCOMPUTATIONAL
ASPECTS OF DIRECT AND INVERSE PROBLEMS IN PERMAFROST GEOPHYSICS
7
Figure 3: Higher values of B correspond to higher amount of liquid water in partially frozen pores, and more
signiﬁcant body forces caused by the so-called ”cryogenic suction”.
.
3, right. The lower values of
B
correspond to soil that have more ﬁne texture such as silt and clay. The
bigger values relate to sand or organic matter. In the above calculations the diﬀerent values of constant
B
corresponded to the diﬀerent textures of soil inside the circle. The modeled situations correspond to observed
facts that the ﬁner the soil texture inside the circle the bigger values of the frost heave.
5. Improvements of the non-sorted circle model
As major improvements to the existing model we plan:
1.
to include the thaw settlement of frozen ground in the model. This would allow us to model the
non-sorted circle during the entire year and observe its evolution in time. The self-organization
between vegetation cover and deferential frost heave might be considered,
2.
to model insulating eﬀects of the snow cover explicitly, by introducing an additional layer of a
variable thickness above the ground,
3.
to include the diﬀusion of solutes in the frozen ground that can be modeled by introducing other
constituents in the mixture, and by modifying the dissipation function to include the irreversible
mixing of constituents,
8
DMITRY J. NICOLSKY
4.
to introduce in the model a more realistic thermo-rheological parameterization of the soil component
by modifying the strain energy
W
k
in (4)-(6), modiﬁcation of the dissipation function Φ to describe
hysteresis during the process of the water freezing.
References
[1]
C. Truesdell.
Rational Thermodynamics.
Modern applied mathematics. McGraw-Hill, 1969.
[2]
G. D.C Kuiken.
Thermodynamics of irreversible processes, Applications to Diﬀusion and Rheology.
Willey tutorial series in
Theoretical chemistry. Willey, 1994.
[3]
J. Hartikainen M. Mikkola. Mathematical model of soil frezzing and its numerical application.
Int. J. Numer. Meth. in
Engng,
52:543–557, 2001. DOI: 10.1002/nme.300.
[4]
M. Mikkola M. Fremond. Thermomechanical modelling of frezing soil.
In Ground Freezing 91, Proceedings of the 6th
International Symposium on Ground Freezing,
pages 17–24, 1991. Yu X, Wang C (eds), Balkema: Rotterdam, 1991.
Geophysical Institute, University of Alaska Fairbanks, PO Box 757320, Fairbanks, AK 99775
E-mail address:
[email protected]
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