420
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
2 GOVERNING EQUATIONS
2.1. General assumptions and mass balance equations
The governing equations for porous materials under freezing
action were assumed the following to define deformation
characteristics associated with THM phenomena.
First, the void of soils is fully saturated with water or water/ice.
That is, ice and unfrozen water fill the pore under frozen
condition, and the void is fully saturated with liquid water above
freezing temperature.
Second, porous material consisted of soil particles, water, and ice
is under local thermal equilibrium conditions.
Finally, a freezing porous medium, in the context of theory of
mixtures, is viewed as a mixed continuum of three independent
overlapping phases of solid and liquid (Bear & Bachmat, 1991).
For every phase, its mass conservation equations can be obtained
according to the principles of continuum mechanics and mixture
theory.
Figure 1. Phase diagram for frozen soil
The macroscopic balance of any species or property per unit
volume in a continuum can be expressed by the following
generalized partial differential equations (Lewis & Schrefler,
1998).
0
~~
j
t
(1)
where
is a species in porous material (e.g. soil, water), and
is mass per unit volume of each species.
is mass flux of
each species which can include advective and non-advective
components.
~
j
2.2. Soil mass balance equation
Soil particles only exist in solid-state, so mass balance equation
can be summarized as follows: mass density
, mass
flux
in Equation 1.
) 1(
s
~
~
~
) 1(
u
u
j
s
(2)
where
는
is mass density of soil particles,
s
is soil
porosity, and is velocity of soil particles.
~
u
2.3. Water mass balance equation
Since water can exist in liquid water or solid ice, the governing
equation was derived from generalized law of mass conservation
(
l
l
l
i
S
S
). Solid ice is assumed to be an
immobile substance which can make phase change from solid to
fluid.
w
w
1
(3)
where
l
is mass density of liquid water,
l
is degree of liquid
saturation in the void of material. is flow rate of liquid water
from Darcy’s law.
w
S
l
q
~
2.4. Energy balance equation
Although the energy balance is expressed by enthalpy balance
in most cases, it is preferable to express it in terms of internal
energy (Olivella, et al., 1996; Lewis, et al., 1998). If thermal
equilibrium between phases is assumed, the temperature is the
same in all phases and only one equation of total energy is
required. Adding the internal energy of each phase, the total
internal energy per unit volume of the porous medium becomes,
0
)
1(
)
1(
) 1(
)
1(
)
1(
) 1(
~
~
~
l
w
l
w
l
c
f
w
i
l
w
l
w
l l
w
i
w
i
l
s s
f
w
i
l
w
l
w
l l
w
i
w
i
l
s s
qE
i
u L S
E S E S
E
L S
E S E S
E
t
(4)
where
s
is internal energy of soil per unit mass, and
is
internal energy of water in solid phase per unit mass. Energy
transfer by heat conduction in porous materials was estimated
using from the Fourier's law (
c
~
~
). Last term
f
in first partial derivative represents internal
energy loss due to water phase change (Thomas, 2009; Tan,
2011; Jane, 1980).
E
w
i
l
)
w
i
E
T
i
L S
1(
2.5. Static equilibrium equation
Neglecting the inertial effects over all phases, the momentum
conservation equation reduces to the static stress equilibrium
based on the total stress.
0
~
~~
g
m
(5)
where
is total stress, average mass density is
, and gravity direction is
)3,1 ,(
~
ji
ij
w
i
l
s
S
)
1(
)
w
i l
m
S
1(
1,
0,0
i
g
.
2.6. Numerical implementation
Substituting Eq. 2 into Eq. 3 and 4, the differential equation
governing non-isothermal liquid flow through frozen-nonfrozen
porous material is obtained. The primary variables are
displacement components
~
u
, liquid pressure
l
P
, and
temperature
from fully coupled governing equations. The
material derivative with respect to the solid velocity field will be
very useful to obtain the final expressions for balance equations
and equilibrium equation. The material derivative relative to the
a phase is given by
T
(7)
(6)
Generalized trapezoidal rule (Eq. 7) is used to perform time
integration between
and
of coupled governing
equations, and they use discrete approximations to take
advantage of Newton's iterative process.
)(
n
t
)1 (
n
t
where
is an integration parameter to govern stability and
accuracy of the solution, and the solution is unconditionally
stable if
2/1
.
A volume integration of all governing equations then leads to a
weighted residual approximation to the governing equations,
based on the Galerkin method. After all governing equations are
discredited, the final system of algebraic equations can be
expressed in matrix form with respect to primary variables
.
) , , ,(
~
TPPu
l g
D
)
(
)
(
),(
)1 (
)1 ,(
),(
in
n
in
in
d
D F
F
D K
INT
EXT
(8)
0
)
1(
)
1(
~
~
~
~
l
w
l
l
w
l
l
w
i
l
w
l
l
w
i
q uS
uS
S
S
t
) 1(
) 1(
u
0
~
~
t
s
s
~ ~
u
t
dt
d
l
l
n
l
n
l
in
l
n
l
n
l
n
n
t
t
l
dP P
Pt
P
P t
P
P t
t
dt P
n
n
) (
) (
)1 , (
) (
)1 (
) (
)1 (
) 1(
) 1(
)
(
)1 (
) (
