ρ3 = ρ30 , ρ1 = ν(T )ρ30 ≤ ρ10 , ρ1(0+) = ρ10
(35)
f ≡ f1 = -K1P′ - K2T ′ ,
K1(0+) = K0
(36)
K2 (T ) / K1(T ) = γ
0 > T ≥ T = -σ / γ
(37)
for
σ
P(δ-) = P = σ + P , σ ≥ 0
(38)
a
n
P′(δ) ≥ 0,
Vo ≥ 0
VoP′(σ) = 0
and
(39)
K2 (T ) = 0, K2 (T ) / K1(T ) = 0 for T ≤ Tm < Tσ
(40)
where γ is a constant (1.12 MPa/C), P (ξ) is the pressure of water, Pa is the applied confin-
ing pressure (uniaxial stress), σ is the effective confining pressure, and Ki (i = 1,2) is the
transport property of a given soil that generally depends on the temperature and the com-
position of the soil. Since ρ3 is a constant, we will assume that Ki is an increasing function of
T alone. This assumption implies the homogeneity of soils in a microscopic scale that corre-
sponds to the thickness of the frozen fringe, which is clearly an approximation. We will
assume that K1 (T) has a continuous first derivative. Because of eq 37, K2 (T) may be discon-
tinuous at T = Tσ. We will assume that the first derivative of K2 is continuous except at
T = Tσ. It is known that the mobility of unfrozen water tends to diminish as T decreases. We
will assume that there exists a negative number Tm < Tσ such that eq 40 holds true and that
K2 (T) > 0 and K2 (T)/ K1 (T) > 0 for T > Tm. According to M1, f is given by eq 36 in R1 while
Darcy's law holds true in Ro. Hence, f and P are continuous but P′ may be discontinuous at
no.
^
The M1 model is a generalization of somewhat simpler but more restrictive models, M1
(Derjaguin and Churaev 1978, Ratkje et al. 1982, Horiguchi 1987), in which the ratio K2/K1
^
is equal to γ regardless of T. In M1 the coupling mechanism for mass and heat transport is
based on irreversible thermodynamics in which local equilibrium is assumed under a tem-
where Tσ ≤ T < 0, but not in the part R11 where T (δ) < T < Tσ (Fig. 1). This generalization is
^
needed because M1 is too restrictive to accurately describe the behavior of porous media
(Nakano 1997). Equation 39, often referred to as the Signorini-type free boundary condi-
tion (Friedmann and Jiang 1984), is needed for the uniqueness proof of solutions when Vo
is positive. It is not certain that such a condition holds true because of the paucity of exper-
imental data. In addition to the above equations, we will assume that the thermal conduc-
tivities ko and k1 are given constants for the sake of simplicity.
When eq 35 holds true, u3 vanishes and eq 21 is reduced to
f(ξ) = fo + s2(ρ10 ρ30ν)Vo,
0 < ξ < δ.
(41)
The Λ (ξ) is given as
-1
Λ(ξ) = d1 d2 (ρ10 - ρ30 ν)Vο .
(42)
According to M1 the properties of a given soil are described by three empirically deter-
mined functions of T : K1, K2 and ν that are assumed to be functions of T alone for T < 0C.
The hydraulic field is specified by Pn, δo and Pa while the thermal field is specified by αo
and α1. Our problem is to find constants Vo ≥ 0, δ ≥ 0 and functions f (ξ), T (ξ) ≤ 0, P (ξ) so
that the following equations (P1 through P7) are satisfied:
7
Contents
Previous Page
