water are of concern. At this scale we refer to the

(B21)

porous medium behaving as a continuum display-

in which the reduced volumetric flux of water

ing uniform material properties, and we are inter-

ested in the flow into and then out of a volume

of material. This suggests that a different length

ζη

scale is required to describe the gradient, since λ

(B22)

λγ IA

is applicable only at the pore scale. The additional

macroscopic length scale is ζ. Using this new

The continuity equation for pore constituents

length scale to reduce the gradient operator re-

is

sults in the reduced body force:

∇ (vW + *Y*IWvI )

ζλ

(B18)

γ AW

[ W (uW , *u*I ) + *Y*IW I (uW , *u*I ) ]

=-

(B23)

The proportionality factor in Darcy's equation

is the source of much confusion. In eq B17 it is

in which *v*I (m3/m2s) is the volumetric flux of

ice and *t *is time. The operator ∇ is the diver-

entists tend to use hydraulic conductivity *K *(m/

gence, which simply scales by multiplying by ζ.

s), and geologists and petroleum engineers use

intrinsic permeability *k *(m2). These three param-

The volumetric flux of ice must scale like the

eters are related by

volumetric flux of water to maintain dimensional

equality on the left side of eq B23. Since the volu-

=

(B19)

metric water and ice contents are already dimen-

ρ*g * η

sionless, the reduced time *T *is

in which ρ (kg/m3) is the density of water, *g *is

λγ IA

(B24)

the acceleration due to gravity (or other body

ζ2η

force) and η (N/m2s) is the viscosity of water.

They are all functions of water and ice content,

The conservation of thermal energy is

which are functions of the water and ice pres-

[ CH (uW , *u*I ) θ

∇ (vH + *h*IWvW ) = -

sures.

The process of scaling the capillary conduc-

tivity consists of first realizing that it is already

]

θ + *h*IWW (uW , *u*I )

(B25)

inversely proportional to viscosity (i.e., eq B19),

so it must therefore be scaled by multiplying by

in which *v*H (W/m2) is the macroscopic flux of sen-

the viscosity. It then becomes the intrinsic perme-

sible heat and *C*H(*u*W, *u*r) (J/m3 K) is the volumet-

ability, which is in dimensions of length squared.

To complete the scaling we must decide on the

ric heat capacity excluding heat resulting from

correct length scale to use. Again relying on the

phase change. The volumetric heat capacity is as-

continuum approximation, the conductivity

sumed to be a material property that is constant

should not depend on the size of the sample once

in the continuum assumption for a given state of

it attains the minimum size required for the con-

ice and water contents. Starting with the left side

tinuum approximation. This requires the use of

of eq B25 the reduced macroscopic flux of sen-

the microscopic length scale λ to scale the con-

sible heat *V*H is obtained

ductivity. Miller and Miller (1955) deduced the

ζη

same length scale choice by considering the

(B26)

γ IA

2

NavierStokes equation applied to flow through

the pore. The reduced capillary conductivity is

The reduced volumetric heat capacity *C*H must

then

therefore be

η

(B20)

λθo

γ IA H ( W I )

λ2

(B27)

Substituting eq B20, B19, B6 and the scaling fac-

tor ζ to reduce the gradient operator into eq B17

The macroscopic flux of sensible heat is given

gives the scaled form of Darcy's equation:

by Fourier 's law:

16

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