4 Advanced Features
4.3 Implementation of permafrost development in
SHEMAT
Darius Mottaghy
4.3.1 Introduction
Frozen soils mainly occur in the polar regions and in the higher reaches of the
mountains. The associated thermal regime is called "permafrost" (Muller, 1945).
According to the general definition by (Lunardini, 1981) the term permafrost is
used, if the soil shows a temperature at or below 0 °C continuously for a significantly long time, but not necessarily for an entire geological period. However,
there is no defined time period during which the temperature of the material must
remain in the mentioned range. Soils freezing in an exceptional cold winter and
persisting over one or two years are not classified as permafrost.
The existence of permafrost is a result of the history and the present state of the
energy balance at the Earth’s surface - measured by the surface temperature - and
the deep Earth heat flow. The dominant physical processes in frozen soils are
thawing and freezing of water with the liberation of latent heat. Furthermore the
hydrology is greatly influenced by the frozen soil as the infiltration decreases and
long-range runoffs are caused. The quantitative understanding of these mechanism
is of paramount importance in order to forecast the results of a climate change, as
well as to improve the parameterization of models describing the soil-atmosphere
interactions.
4.3.2 Numerical modeling of permafrost
A detailed analysis of freezing processes including coupled heat- and mass transport in soils is very complex and the theory is not yet fully understood. Because of
the nonlinearity of the heat transport equation with phase change, and particularly
because of the complex coupling between thermal and hydrological processes on
one hand and climatological conditions on the other hand, a simplified model for
SHEMAT is presented.
4.3.2.1 Frozen soil physics
The heat transfer equation is in SHEMAT
Advanced Features
∇ ( λ ∇ T − ρf c f T v ) =
∂T
( φ ρf cf + (1 − φ) ρm cm ) − H ,
∂t
4.3_2
(4.3.1)
where λ is thermal conductivity tensor (W m-1 K-1), ρ is density (kg m-3), c is heat
capacity (J K-1), and H is volumetric heat production (W m-3). The subscripts f
and m account for the two-phase mixture between solid rock (m) and fluid-filled
pore space (f). This mixture is characterized by porosity φ. When modeling the
thermal effects of freezing and thawing, obviously Eq. (4.3.1) has to include three
phases: matrix, fluid, and ice. To achieve this, the following volume fractions are
defined:
φm = 1 − φ, φf = φ ⋅ Θ, φi = φ − φf ,
(4.3.2)
where φ denotes the fraction of fluid in the rock volume and an additional phase
marked by index i is introduced. The constraint φm + φI + φf = 1 implies that pore
space is saturated.
As an result of the complicated processes in the porous medium, thawing can not
be considered as a simple discontinuity. Θ is generally accounted a continuous
function of temperature in a specified interval, e. g.,
⎧
⎡ ⎛ T - T ⎞2 ⎤
L
⎪⎪exp ⎢ − ⎜
⎟ ⎥ ; T ≤ TL .
Θ=⎨
⎢⎣ ⎝ w ⎠ ⎥⎦
⎪
1
; T > TL
⎪⎩
(4.3.3)
This function is shown in Fig. 4.3.1, and is characterized by a thawing temperature
TL (liquidus, usually 0 °C) and a parameter w (usually 1 K). This corresponds to a
freezing interval ΔT = TL - TS = 2 K where TS is the freezing temperature
(solidus).
However, this range is a user specified parameter in SHEMAT, making it possible
to analyze a variety of ground conditions. The enthalpy during phase change is determined by latent heat effects. Through this both the cooling and the warming are
strongly decelerated. Fig. 4.3.2 is an example for this. It is realized with the presented model, monitoring the time dependence of the temperature of a point in 50
cm depth. For this the surface of a vertical model is exposed to a fixed temperature
of –3 °C, resulting in a cooling of the soil being initially at 0 °C. Due the finite
temperature interval the phase transition is not isothermal.
4.3_3
Darius Mottaghy
Fig. 4.3.1. Smooth Partition function Θ for several values of the thawing interval w, describing the unfrozen water content.
Fig. 4.3.2. Frozen soil properties: a qualitative example for a non isothermal cooling curve
(red).
Advanced Features
4.3_4
4.3.2.2 Apparent heat capacity
Usually the apparent or effective heat capacity concept is invoked by adding a
term to Eq. (4.3.1) (Kukkonen, 2001). Its right hand side can more generally be
written as the time derivative of enthalpy H. During thawing, this fluid enthalpy
per unit volume changes according to
ΔH = ∫ ( φf ρf cf + φiρi ci ) dT + ∫ ρf Ldφf .
(4.3.4)
L is the specific latent heat (for water ≈ 333.6 kJ kg-1). Obviously a volumetric apparent specific heat capacity (ρc)app can be defined, which includes additional energy sources or sinks due to latent heat effects and replaces the fluid contribution
of the term in the brackets on the right hand side of Eq. (4.3.1) and is written as
follows:
(ρc)app = φf ρf cf + φiρi ci dT +
ρf Ldφf
.
dT
(4.3.5)
The total derivative in the last term of Eq. (4.3.5) is usually approximated by a ratio of finite differences, resulting in a constant apparent specific heat:
L
φ − φf ,S
Δφf
= L f ,L
= L' .
ΔT
ΔT
(4.3.6)
The freezing range is thus described by the temperature interval ΔT = TL - TS with
a fixed temperature TS (solidus) at which all fluid is frozen (see Section 4.3.2.1).
This choice leads to an apparent specific heat capacity of
(ρc)app
T>TL
⎧ φ f ρf c f
⎪
ρ
L
'
⎪
= ⎨ φ f ρ f c f + φi ρ i c i + f
TS ≤ T ≤ TL
ΔT
⎪
T ≤ TS
.
⎪⎩φi ρi ci
(4.3.7)
ρ and c in this equation are functions of temperature. For the fully melted state the
variation with temperature of these properties are as described in Chapter 2 and
for the fully frozen state properties of ice at different temperatures are taken from
Landolt (1982). Fehler! Verweisquelle konnte nicht gefunden werden. shows
qualitatively, how (ρc)app varies with temperature. This approximation of a constant apparent heat capacity is consistent with assuming a ramp function for Θ. As
we have chosen a smoother function, we can simply differentiate Eq. (4.3.3)
⎧ 2(T − T )
⎡ ⎛ T - T ⎞2 ⎤
L
L
⎢− ⎜
exp
dΘ ⎪⎪−
⎟ ⎥ ; T ≤ TL .
=⎨
w2
⎢⎣ ⎝ w ⎠ ⎥⎦
dT ⎪
0
; T > TL
⎪⎩
(4.3.8)
4.3_5
Darius Mottaghy
Fig. 4.3.3. Constant apparent heat capacity, schematic
and use this in Eq. (4.3.5) instead of the approximation in (4.3.6). Eq. (4.3.8) is
plotted in Fig. 4.3.4
Fig. 4.3.4. Derivative of Θ for several values of the thawing interval w. The area below the
curves in the bottom figure are equal, and thus guaranteeing the latent heat condition
Advanced Features
4.3_6
According to Bonacina (1973) the actual shape of this curve is not important with
regard to the temperature fields calculated, but it must satisfy the latent heat condition,
TL
L = ∫ (ρc)app dT ,
TS
(4.3.9)
which is clearly fulfilled for both choices.
The apparent specific heat capacity is a function of temperature, however. Therefore this approach requires a nonlinear solution scheme. For this reason smoother
functions for θ generally improves convergence.
4.3.2.3 Thermal conductivity
In case of a phase change at a single temperature, thermal conductivity is not continuous in terms of temperature. However, considering the freezing range in rocks,
we use Eq. (4.3.2) for taking fluid and ice contributions into account. For temperatures below TS thermal conductivity of ice is assumed to vary linearly with temperature. All other temperature dependencies are given by the original code
SHEMAT. Since the materials are randomly distributed, the weighting between
them is realized by the square-root-mean, which has a greater physical basis than
the geometric mean (Roy et al., 1981).
λ (φm,f ,i , T) = ( λ m (T)φm + λ f (T)φf + λ i (T)φi ) 2 .
(4.3.10)
4.3.3 Model Verification
Analytical Solution
The solutions to conductive heat transfer problems with solidification phase
change - often referred to as "Stefan problems" (Stefan, 1891) - are inherently
nonlinear and thus, solution methods are very restricted. A classical solution for a
semi-infinite medium with constant temperature undergoing a step change of surface temperature was given by Neumann (1860) and has been expanded by
Carslaw and Jaeger (1959); it is called the Neumann solution. Accordingly at time
t=0 the surface x =0 is exposed to a temperature lower than TS, and it is T = TL for
x > 0. Hence it results for the temporal change of the phase front and with it the
isotherm T = TS (see Fig. 4.3.5)
4.3_7
Darius Mottaghy
Fig. 4.3.5. Phase front propagation for the Neumann problem
X(t) = 2 γ α i t .
(4.3.11)
αi,f indicates the thermal diffusivity of ice and water, respectively. The parameter
γ must be determined from the following equation (Carslaw and Jaeger, 1959) that
results from the boundary conditions of the associated differential equation (with
the thermal conductivities λ i,f of both materials):
exp[(αi − α f ) γ 2 / α f ]erfc[ γ α i / α f ] (TL − TS )λ f α i
.
−
erfγ
( TS − T0 ) λi α f
(4.3.12)
The solution is found by the null of the expression on the left hand side.
The latent heat effect is considered approximately in Eq. (4.3.12) by adding
the expression of Eq. (4.3.6) to the thermal diffusivity of the liquid:
αf =
λf
ρ L'
ρf c f + f
ΔT
.
(4.3.13)
Table (4.3.1) notes the computed values for several temperature differences ΔT =
TL - TS
Advanced Features
4.3_8
Table 4.3.1. The parameter γ at different temperatures.
γ
0.039
0.041
0.043
ΔT(°C)
2
3
4
Model properties
A horizontal field with 20 × 100 nodes and a mesh size of 1 cm is chosen as a
model for the semi infinite half space. The model is purely conductive. Since the
above illustrated analytical solution is only valid for a homogeneous fluid, the porosity must be chosen as large as possible. Hence in a first run a value of 0.95 is
chosen. In order to consider additionally the heterogeneous structure, the analytical solution is modified and a porosity of 0.05 is used. The initial temperature of
the half space is the top of the freezing range (TL = 0 °C); at t=0 the surface x=0 is
exposed to a temperature of T0=-3 °C < TS. The used parameters are summarized
in Table (4.3.2)
Table 4.3.2. Verification model parameters.
Parameter
Grid Size/Resolution
Temperature
Porosity
Matrix thermal capacity
Matrix thermal conductivity
Time step/Total simulation time
Value
20 x 100 m / 1 cm
0 °C (-3 °C at x=0)
0.95 and 0.05
2.06 MJ m-3 K-3
2.9 W m-1 K-1
864 s/100 days and 1.8 day<
Results and discussion
Fig. 4.3.6 shows the propagation of the phase front X(t). The cross symbols note
the numerical values and the line represent the analytical solution.
At the beginning of the run the error is rather large, but it decreases with time, being lower than 5 per cent after ten days. At longer times the difference between
analytical and numerical solution is even less. The initially significant deviation is
due to the rough discretization of the grid and boundary effects which decrease as
the phase front propagates. They also decrease when choosing smaller time steps.
Additional error sources arise from the non realizable porosity of 1.0 and the approximation by the choice of the function Θ shown in Fig. 4.3.4. Table (4.3.3)
notes some values of the analytical and numerical solution, as well as the percentage deviation.
4.3_9
Darius Mottaghy
Table 4.3.3. The isotherm T = TS at different times for φ = 0.95.
Time [d]
5
20
50
80
99
Xanalyt (cm)
5.4
12.6
19.6
24.6
26.9
Xnum (cm)
6
13
20
25
27
% Deviation
10
3.5
2.2
1.5
3.5
Fig. 4.3.6. Phase front propagation for the Neumann problem at φ = 0.95
Advanced Features
4.3_10
In a next step the heterogeneous soil structure is taken into account. The thermal
conductivity is weighted by the square-root-mean (see Sec. 4.3.2.3) and the mixing law for thermal capacity is the arithmetic mean. Since the properties in Eq.
(4.3.12) refer only to the fully melted (f) or the fully frozen (i) state, it is sufficient
to use the porosity φ as in Eq. (4.3.1). Thus, the thermal diffusivity of the fluid
from Eq. (4.3.11) becomes
α f → α f ,m =
(φ
)
λ f + (1 − φ) λ m
φρf cf + (1 − φ)ρm c m +
2
ρf L '
ΔT
(4.3.14)
And the ice thermal diffusivity changes to
α i → α i,m
(φ
=
λ i + (1 − φ) λ m
)
2
(4.3.15)
φρi ci + (1 − φ)ρm c m
Fig. 4.3.7 shows the devolution of the phase front for a porosity of 0.05 using the
modified analytical solution. Because of the lower water content an therefore the
less released amount of latent heat, the front propagates faster. Table(4.3.4) shows
some values of the analytical and numerical solution for comparison.
Table 4.3.4. The isotherm T = TS at different times for φ = 0.95.
Time (d)
8
18
22
27
32
Xanalyt (cm)
5.9
8.9
9.9
10.9
11.8
Xnum (cm)
6
9
10
11
12
% Deviation
1.5
1.4
1.4
1.4
1.4
4.3_11
Darius Mottaghy
Fig. 4.3.7. Phase front propagation for the Neumann problem at φ = 0.05
Permafrost and ground water flow
Due to the change in the ice content φi the fluid flow is affected, yielding lower
permeability when freezing. To account for the reduced flow, Jame (1980) introduced an impedance factor of the form
k frozen = k unfrozen ⋅10− Eφi
(4.3.16)
Here, k is permeability (m s-1) and E is an empirical constant, ranging from about
8 to 20-30. We use E=10 as in Corapcioglu (1988). By means of a 2-D vertical
model with an freezing area we could verify that the mass balance is assured except an error of 0.5 %.
4.3.4 Conclusions
The presented model includes latent heat effects due to freezing and melting processes in a heat transport model for porous media. In order to facilitate the complex
behavior of freezing processes, various assumptions have been made. First simulations of heat transport using this method reveal a considerable influence on thermal properties of soil. Fig. 4.3.8 shows a synthetic example of a temperature log,
Advanced Features
4.3_12
both the initial state and the situation after two step changes in surface temperature
(upper panel on the right), including and excluding latent heat effects. The maximum temperature shift between included and excluded latent heat effects is plotted in the lower panel on the right It becomes evident that the influence of permafrost development plays a significant role, which increases with porosity.
Further development of this model has to deal with unfrozen water relations, considering particular rock systems. Influences on the phase change temperature like
salinity and pressure can also be incorporated.
Fig. 4.3.8. Influence of permafrost formation on ground temperatures
4.3_13
Darius Mottaghy
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