The specific surface area of fine-grained soils is on the order of 20200 m2/g, and unfro-
zen water is known to exist in the form of thin films. The thickness of such films depends
on temperature and pressure, and is estimated on the order of 10100 nm at the tempera-
tures around 0.1C under atmospheric conditions (Ishizaki et al. 1994). Unfrozen water in
frozen soils is one special case of a wide class of confined liquids. The key issue underlying
the transport of unfrozen water is deemed to be the dynamic collective behavior of water
confined to small spaces in frozen soils, which depends on complex solidliquid interac-
tions.
It is generally accepted that a thin transitional zone, often referred to as the frozen
fringe, exists between the 0C isotherm (frost front) and the growing surface of an ice layer.
The unfrozen water content in frozen soils under equilibrium conditions is routinely meas-
ured by nuclear magnetic resonance, differential scanning calorimetry, or time domain
reflectometry. However, since the unfrozen water content under dynamic conditions is dif-
ficult to measure, the phase composition of a frozen fringe is not known. Since the proper-
ties of all parts except the frozen fringe are understood, the dynamic behavior of the frozen
fringe has been one of the major subjects in the study of soil freezing in recent years. Since
the 1960s, many mathematical models (Talamucci 1977, Kay and Perfect 1988) of a frozen
fringe have been proposed on the basis of various hypotheses. With the widespread use of
computers, the methods of numerical analysis became very popular. However, because of
the paucity of basic knowledge and the complex nature of the problem, these numerical
studies have not been effective for the critical evaluation of the multiple hypotheses used.
Around 1980, two important semiempirical models of soil freezing were introduced for
engineering applications: the segregation potential (SP) model (Konrad and Morgenstern
1981) and the Takashi model (Takashi et al. 1978). Today the SP model is widely used for
engineering in Europe and North America, while the Takashi model is the standard of
engineering design in Japan. These two semiempirical models share a common approach
that the freezing characteristics of a given soil are determined empirically under certain
quasi-steady conditions, where a frost front moves with a constant speed. These models
also share a common weakness of requiring one or more empirically determined parame-
ters. These are known to depend on not only the properties of a given soil but also a partic-
ular quasi-steady condition specified by given thermal and hydraulic fields. The empirical
determination of such dependence is elaborate and costly. An accurate mathematical
model is needed that provides the functional dependence of parameters on pertinent vari-
ables specifying given thermal and hydraulic conditions in terms of well-defined functions
(or parameters) describing the properties of a given soil.
As the 1980s were ending, there were many mathematical models of soil freezing
(Gilpin 1980, O'Neill and Miller 1985, Fowler 1989, etc.), but they all suffer from the com-
mon fault of little or no experimental verification. Efforts were initiated to study the prob-
lem analytically and to verify the hypotheses used in the analysis by comparing the prop-
erty and the behavior of solutions with empirical findings. Adopting such an approach,
Nakano (1990) introduced a mathematical model called M1. This model was shown
(Nakano and Takeda 1991, 1994) to be consistent with experimental data on the growth
condition of an ice layer without overburden load (Takeda and Nakano 1990) and under
load (Takeda and Nakano 1993). The growth process of final ice lenses was accurately
described by M1 (Nakano 1992, Nakano and Takeda 1993). Nakano (1994b) has shown that
the functional dependence of SP on thermal and hydraulic conditions predicted by M1 is
consistent with empirical findings that were used to build the SP model.
According to the Takashi model the freezing characteristics of a given soil are described
by two empirical formulas that specify the dependence of the frost heave ratio and the
water intake ratio on given thermal and hydraulic conditions. Two theoretical equations
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