A Model for the Formation of Ice Lenses in an Unconfined, Watersaturated, Porous Medium consisting of Spherical Particles
Kunio Watanabe & Yoshiko Muto
Faculty of Bioresources, Mie University, Japan

Masaru Mizoguchi
Department of Biological and Environmental Engineering, The University of Tokyo, Japan

ABSTRACT: A model for simulating the formation of ice lenses during freezing of unconfined uniform
porous media is presented. The main notions of the model are that generation and jump are dependent on
the freezing rate, and that growth is dependent on supercooling. The critical freezing rate is assumed to varywith changes in the number of particles near the ice lens due to the growth of the ice lens. The proposedmodel was computed for the unidirectional freezing of a porous medium consisting of 10 µm-diameter glass
particles. The numerical results show that this model can represent the formation of intermittent layers of
ice lenses in the porous medium. This model can be applied to ice formation in unconfined water-soaked
porous media.

NOMENCLATUREa maverage molecular distance in liquid
A m-2s-1
coefficient in equation (2)B m s-1˚C-1 empirical constant in equation (3)C m-5
coefficient in equation (8)d mdiameter of the particle
D m2s-1
self-diffusion coefficient of water
G specific gravity
k J K-1
Boltzmann factor
k Wm-1˚C-1 thermal conductivity
K m2s-1J-1
hydraulic conductivity near the ice lens
L J g-1
Latent heat of ice
N number of particles per unit volume
Qm J
the molecular heat of melting of ice
tp s
elapsed time
T ˚Ctemperaturev m3
molecular volume
Vc m s-1
critical freezing rate for isolated particleVc‘ m s-1
critical freezing rate in porous media
Vf m s-1
rate of advancing freezing front
Vs m s-1
constant rate equivalent to the mean ofrate of advancing freezing frontw %water content by weightX mwidth of considered rectangularY mheight of considered rectangularz mcoordinate
Z mlength of the rectangular (freezing direction)
α m2 s-1
thermal diffusivity∆γ N m-1
difference of surface energy∆p Pa
pressure difference∆Til ˚Csupercooling degree
η Pa scoefficient of viscosity
ρ g m-3
density of iceSubscripts
0 initialiice
u unfrozen
f frozen
il ice lens
w water
H highL low

1 INTRODUCTION
When soil is frozen unidirectionally, the soil water
sometimes forms regions of ice that are almost
devoid of soil particles. These regions are called
ice lenses, and repetition of the process results in theformation of intermittent layers of ice lenses (Figure1a). The formation of ice lenses is recognized as anecessary condition for frost heave and is also
reported to occur in other porous media (e.g. Dash etal., 1995; Mutou et al., 1998). This paper focuseson the process of ice formation in porous media
consisting of fine particles.
Various models have been developed to describe
frost heave (Kujala, 1997). Presently, the model
most often used to describe the mechanism of frost
heave is the secondary frost heave theory proposedby Miller (1972), since it can explain the formationof intermittent layers of ice lenses. The modelemphasizes the importance of ice growing in thefrozen fringe, which was assumed to consist of anetwork of pore ice extending from the base of thewarmest ice lens to a position at or near the 0 °C
isotherm. However, the configuration of ice in aporous medium depends greatly upon the conditionsof freezing and the characteristics of the medium.
Watanabe and Mizoguchi (2000) observed theformation of intermittent layers of ice lenses in an
unconfined water-saturated uniform porous medium
consisting of micro glass particles 9.7 µm in
diameter. They found the ice in the medium does
not penetrate the warmer pore than the growing icelens.