Andrew, Ines and Seth
Consider a control volume of soil, where
where
Let
Let
We assume we are dealing with a closed system with respect to water - so there is no inflow or outflow. The continuity equation is:
Therefore we have
We know that
On a per unit volume basis we have
Since we are talking about changes in time, we can write
and since the densities are constant,
and
This is the mass balance equation for the situation where the water doesn't move.
Let
and
or
We now have to deal with the left hand side, that is the changes in internal energy, and the right hand side, that is the net addition, or removal, of heat. These can be treated separately.
Since we are neglecting the movement of water, we can ignore advection and have heat transport only by diffusion. Diffusive transport is described by Fick's first law
And the right hand side of the continuity equation is simply Fick's second law, which is
The internal energy takes an arbitrary value - we are only interested in relative changes in energy. Changes in energy are due to sensible and latent heat. Internal energy associated with sensible heat,
or
or
where
Internal energy associated with latent heat,
or
Since the soil has four components, and three of these are dynamic, the change in internal energy can be written:
and further, the internal energy of a single component,
meaning we end up with the following eight terms:
Now, since we have no phase change associated with the soil solids or the gas (in this case), we can ignore the last two terms. We can also make the important assumption that the components of our soil are all in thermal equilibrium, hence
So, considering changes with time we can write
and
Since there is no change in mass of soil solids, and the change in air mass is negligible, we have
or
(Note, do the last two terms above explain why the apparent latent heat capacity is a function of temperature? Maybe.)
At this point, if we examine the equation above, we are left with the question, how can we know how a certain amount of net energy exchange (i.e. from the diffusion equation above) is partitioned between changes in sensible and latent heat. I'm not sure that there is a completely rigorous solution beyond this point for the general case. However, we will now make a simplifying assumption - that is, assume that the soil is completely saturated.
The change in internal energy for a saturated (with liquid or ice) soil is now given by
Recall from the water balance equation we have
so
and
Now, we also know that due to freezing point depression, we have a relationship between the liquid water content,
Hence
where
Or in otherwords:
Note, for
Also note, we have not here defined how
van Genuchten:
It can be shown (ask me for my derivation) that the slope of this is given by
Start from this
where
Therefore
and
where
Substituting the above into the van Genuchten model we get the SFC
and the CFC (which is the slope of the SFC)
Which is