Soil Heat Flow and
Temperature
By R.L. Snyder and K.T. Paw U
Copyright  Regents of the University of
California
Created  June 22, 2000
Last Revision –June 13, 2001
Question: Why do
we want to know the soil heat flow and temperature?
Neglecting
small energy components, the energy balance on a surface can be
expressed using
the following equation, where R_{n} is net radiation, G
is the soil heat flux density, H is the sensible heat flux
density, and LE
is the latent heat flux density.
In
this document, the methodology to estimate G at the surface is
discussed.
1)
Temperature
(T) is a measure of the heat stored (^{o}C or K)
2)
Upper
temperature (T_{1}) is the
temperature at depth z_{1}
3)
Lower
temperature (T_{2}) is the
temperature at depth z_{2 }
4)
Volumetric
Heat Capacity is the amount of heat required to raise the temperature
of a unit
volume by one Kelvin (J m^{3} K^{1})
5)
Thermal
Conductivity (C_{1}) is the
ratio of the heat flux density to the temperature gradient in (W m^{1} K^{1})
1)
Soil
heat flux density (G) is the conduction of energy per unit area
in response to a
temperature gradient. For small depth
changes,
_{}
(1)
Here, the thermal
conductivity C_{1} = KA, where K is the heat conductivity of the material in W
m^{3 }K^{1} and A is the surface area in m^{2}, so for heat flow through a
unit
surface area C_{1} has the units W m^{1}K^{1}. In Eq. 1, _{}is
positive when the temperature decreases with depth in the
soil. The negative sign is included to
make G positive when heat is
transferring downward. Because of
instrument limitations, it is not possible to accurately measure the temperature gradient unless there is
sufficient distance between the sensors.
Consequently, G is estimated as
_{}
(2)
where z_{2}
is sufficiently far below z_{1}
to allow for a measurable difference between T_{2} and T_{1}. Equation 2
assumes that C_{1} is constant with depth in
the soil.
2)
Thermal
Diffusivity (k) is the ratio of the
thermal conductivity to the volumetric heat capacity.
_{}
(3)
C_{1}=the thermal conductivity (W m^{1 }K^{1})
r_{s} = the apparent soil density
(kg m^{3})
C_{p} = the apparent mass
specific heat capacity (J kg^{1} K^{1})
C_{V} =
volumetric heat capacity (J m^{3} K^{1})
k = thermal
diffusivity (m^{2 }s^{1})
where
_{}
(4)
So the thermal conductivity in terms of
diffusivity
and volumetric heat capacity is
_{}
(5)
Therefore, G can be expressed in terms of
diffusivity, volumetric heat capacity, and the temperature gradient as:
_{}
(6)
In terms of the temperature gradient, G is
_{}
(7)
The variable k is useful as a measure of how fast the
temperature
of a soil layer changes. The rate at
which the heat content of a layer of soil changes depends on the
volumetric
heat capacity (C_{V}) and the rate of temperature change of the
soil
volume per unit time. For a unit
surface area, the rate of change in heat storage within the soil layer
is
expressed as _{}. For a unit surface
area, the rate of change in heat storage within the soil is also equal
to the
change in heat flux density through the soil layer _{} . Assuming that the
physical properties of the
soil are constant with depth in the soil and equating these two
expressions, we
get _{} which simplifies to
_{}
(8)
Therefore, k is useful to determine the rate of
temperature
change of a soil layer. Recall that k is directly proportional to C_{1} and inversely proportional
to C_{v}. Both C_{1} and C_{v} increase as the water content of the soil
increases; however, C_{1} increases more rapidly with water content
when the
soil is dry and it slows as the soil becomes wet. C_{v} continues to increase even when the soil is
relatively wet. Consequently, for a dry
soil, k increases with water content,
but it slows and sometimes decreases when the soil nears saturation. As a result, the maximum change in
temperature with time will occur at a water content below saturation.
Question: How do
you determine G= G_{1} at the
surface?
The
change in heat storage of a layer of soil between the surface z_{1}=0
and some depth z_{2} is given by
_{}
(9)
where
G is positive downward into
the soil.
Note: Soil heat
flux density is positive downwards.
When DS = (G_{2}
– G_{1}) is positive,
then more heat is entering the top than leaving the bottom of the soil
layer
and the soil is warming. If DS =
(G_{2} – G_{1})_{ }is
negative, then more heat is leaving than entering the layer and the
soil is
cooling.
Accounting
for the change in soil heat storage is important for estimating G at the surface. This is
clearly indicated in Fig. 1, which
shows a plot of G_{2} + DS versus G_{2} for
half hour measurements
of G_{2} at 0.02 m depth
under a cover of about 0.1 m tall, coolseason grass.
The regression equation indicates that the soil heat flux
density
at the surface is underestimated by about 40% when not corrected for
the
change in soil heat storage.
The
net change of heat storage per unit time (DQ/Dt) in J
s^{1} within a soil
volume (V) is given by:
_{}
(10)
J s^{1}
= (J m^{3} K^{1}) (K s^{1})
(m^{3})
where
T_{f} is the final
temperature at time t_{f} and
T_{i} is the initial
temperature at time t_{i}.
Because
the change in stored soil heat (DS) per unit surface area is
equal to the net rate of heat storage divided by the soil surface area
and V/A=Dz, we can calculate DS in W
m^{2} as
_{}
(11)
If G_{2} is measured and DS is calculated using Eq.11,
then G (= G_{1} at the surface) is
calculated using:
_{}
(12)
Example: Given
a uniform soil with C_{V} = r_{s}C_{p} = 1.677x10^{6}
J m^{3} K^{1}, a
soil heat flux plate placed at an 0.08 m depth, and soilaveraging
temperature sensors
located at 0.02 and 0.06 m depth to represent the change in heat
storage
between the surface and 0.08 m depth.
If the heat flux plate measurement averages G_{2} =
20 W m^{2} and the average soil temperature from the two
sensors
increases by 1 K during a onehour period, what is the soil heat flux
density (G = G_{1}) at the surface?
Question: Using the same criteria as above but now with
the mean soil temperature dropping by 1 K, what is the soil heat flux
density (G = G_{1}) at the surface?
If the soil heat
flux density at 0.08 m depth is G = 20 W m^{2}
and the temperatures 20^{o}C
and 20.8^{o}C are measured on either side of the heat flux
plate. What
is the thermal conductivity (C_{1}) if the
temperature sensors are 0.02 m below and above the
heat flux plate? Which temperature is
recorded
above the heat flux plate?
T=20.8^{o}C is the upper
temperature because G>0 W m^{2}K^{1}
Heat
capacity depends on the mineral, organic matter, and water content of a
soil. The apparent heat capacity (C_{v}) on a volume basis has the
units J m^{3} K^{1} and on a mass basis (C_{p}) it has the units J kg^{1}
K^{1}. Equation 13 is used to
express heat capacity.
_{}
(13)
J m^{3}K^{1}
= (kg m^{3})(J kg^{1}K^{1})
where
r_{s} is the apparent density in
kg of moist soil per m^{3}, r_{b} is the bulk density of the soil in kg of dry
soil
per m^{3}, c_{pav}
is the average
heat capacity on a mass basis for the solid constituents of the soil in
J per
kg of dry soil per Kelvin, q_{m} is the water content on a mass basis in kg
of water
per kg of dry soil, and c_{pw} is the heat capacity of water on a mass
basis in J
per kg of H_{2}0 per Kelvin.
The
heat capacity of the solid constituents of the soil (C_{pav}) depends on the amount of
sand, clay, silt, and organic matter in the soil. For
most mineral soils,
_{ }
_{ }
J
kg^{1}K^{1}
_{ }
The heat capacity of
the water component of the soil
(q_{mcpw}) depends
on the water content and the heat capacity for water.
_{} J kg^{1}K^{1}^{}
The volumetric heat capacity is r_{w}c_{pw}, which is approximated as
_{}
J
m^{3}K^{1}
and
the volumetric water content q_{v} in m^{3} H_{2}O per m^{3}
of dry soil is
_{}
Therefore,
the heat capacity of the soil component is approximated as
_{}
and
the heat capacity of the water component is approximated as
_{}
Substitution
into Eq. 13 gives the following approximation for C_{v}.
_{}
J
m^{3}
K^{1}
(15)
Because
we are interested in heating and cooling the soil, it is useful to have
an
expression for the amount of energy (Q) needed to raise or lower the temperature of
a known volume (V) of soil from T_{i} to T_{f} . The equation
is
_{}
J
=
J m^{3 }K^{1 }(K) m^{3}
(16)
Problems:
1)
Given
a wet soil at T_{i} = 18^{o}C
with q_{v} = 0.23 and r_{b} = 1300 kg m^{3}.
Find the amount of heat required to raise a
1 m^{3} volume of soil to a temperature T_{f}
= 20^{o}C for a surface area of 1 m^{2}
to a depth of 1 m.
2)
Given
a wet soil with heat flux G = 25 W m^{2} into the soil surface (q_{v} = 0.18, r_{b} = 1300 kg m^{3}
and Q = G A t). Find the average temperature increase that occurs in
one day in
the top 1 m of soil assuming that the heat is evenly distributed in the
1 m
soil layer.
3)
Given
a wet soil with q_{v} = 0.18, r_{b} = 1300 kg m^{3}, Q = G A t,
and the average daily net radiation is R_{n} = 150 W m^{2}, find the ratio G/R_{n} if the increase in average
soil temperature to 1 m depth over a month’s time is 10^{o}C.
_{}
From problem 2, we know that Cv^{.}V
= 1.84´10^{6}
J m^{3}
K^{1}
_{}
_{}
4)
Given
a soil with the following temperatures and properties, find the
quantity of
heat stored in the soil per unit area (Q/A), for the period from 0500
to 1400 hrs.
Soil Depth 
T_{i} 
T_{f} 
r_{b} 
q_{v} 
m 
0500 h 
1400 h 
kg m^{3} 

0.000.05 
15 
35 
1.0 
0.05 
0.050.20 
20 
30 
1.1 
0.10 
0.200.60 
18 
25 
1.2 
0.20 
0.601.20 
17 
18 
1.3 
0.25 