Heat Flow Analysis for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 496

equations can be solved for situations where the bound-
ary conditions are constant as well as situations where the
boundary conditions change with time. These two situations
are referred to as steady-state conditions and transient (or
unsteady-state) conditions, respectively. The divergence of
heat flow ( ∂/∂x

q hy + ∂ q hy
∂

∂ y

y

y

x = plane and
z = plane insulated

∂/∂z ) is equated to the change in
heat storage in the elemental volume with respect to time.
The PDEs for heat flow are solved in a manner similar to the
procedures used for water flow and air flow in unsaturated soils.
Heat flow formulations have the appearance of being relatively
simple to solve. Heat flow problems can also be complex when
the water undergoes a phase change. The heat flow formula-
tions are presented in a hierarchical manner from the simplest
situation to more complex situations. Each situation is consid-
ered in terms of one, two, and three dimensions and also in
terms of steady-state conditions and transient conditions.
The first set of formulations is for the analysis of heat flow
when the temperature conditions in the soil range from just
above freezing conditions to near the boiling point of water.
The formulation of the partial differential equation does not
need to take any phase changes into consideration. The sec-
ond set of formulations is for the case where the soil may
experience freezing and thawing conditions, and as a result,
the latent heat of fusion needs to be taken into consideration.
The third set of formulations is for the case where water may
move into the vapor phase and as a result the latent heat of
vaporization needs to be taken into consideration. Finally,
some special cases are also presented.

+ ∂/∂y +

z

dy

dz

x

q hy

q hy = rate of flow of mass

dx

Figure 10.12 Representative elemental volume showing heat

fluxes for one-dimensional heat flow, other sides insulated.

The net heat flow can be written as follows:
dq hy
dy =

0 (10.12)

Let us assume that the thermal conductivity λ of the soil
might vary from one location to another in a soil column
due to a variation in the degree of saturation of the soil.
Substituting Fourier's law for heat flow into the net heat
flow equation yields

d {− λ dT /dy }

=

10.4.1 Steady-State Heat Flow Partial Differential
Equation for Unfrozen Soils
One-Dimensional Heat Flow: Consider the one-dimensional
flow of heat in the y- direction through a REV of unsaturated
soil (Fig. 10.12). The element has infinitesimal dx, dy, and dz
dimensions. The dy and dz sides of the element are assumed
to be perfectly insulated, and therefore, heat flow occurs only
in the y- direction. One-dimensional heat flow could also occur
when the soil on the dy and dz sides is identical to the soil
in the column. Boundary conditions on the soil adjacent to
the element being analyzed must also be the same as for the
element being analyzed.
The heat flow rate in the y- direction is designated as q hy .
The conservation of thermal energy requires that the amount
of heat flowing in and out of the element must be equal for
steady-state conditions:
q hy +

0 (10.13)

dy

where:

λ = thermal conductivity of the soil, W/m/K,

dT / dy = temperature gradient in the y -direction, and

temperature, ◦ C. T =

The above equation can be solved for temperature T pro-
vided there are independent, known values for thermal con-
ductivity. For example, the variations in the degree of satura-
tion in the soil might he known as a result of an independent
unsaturated water seepage analysis. The variation in thermal
conductivity of the soil from one location to another would
need to be taken into consideration when solving Eq. 10.13.
The above equation can be rewritten in the following non-
linear differential form:

dy

dq hy
dy

λ d 2 T
dy 2

dλ
dy

dT
dy

− q hy = 0 (10.11)

+ =

0 (10.14)

where: The above equation is nonlinear since the thermal conduc-

tivity is, in general, variable throughout the soil column. If
the unsaturated soil varied in soil suction in the y- direction,
then it would be necessary to take into account the variation
of thermal conductivity with matric suction, water content,
or degree of saturation. It is also possible for the thermal

q hy = heat flow rate across a unit area of the soil

in the y -direction and

dx , dy , dz = dimensions in the x -, y -, and z -directions,

respectively.