Geoscience Reference
In-Depth Information
Isothermal flow in the absence of gravity: desorption
In a number of studies the formulation has been further simplified by considering the
second stage of drying as a problem of desorption. This formulation, which was first used
by Gardner (1959), is based on several additional assumptions, beside that of isothermal
liquid flow. First, it is assumed that the effect of gravity is negligible. In other words, it is
assumed that the drying rate of a vertical soil column is the same as that of a horizontal
column, so that Equations (9.3) or (9.6) govern the flow. Second, the boundary conditions
are taken to be same as given in (9.7), except that in desorption, instead of
θ
0
, let
θ
s
represent
the water content at the dry soil surface, so that
θ
i
>θ
s
; thus, when applied to this case,
in the first of (9.7) it is assumed that the initial water content is uniform, and in the second
that the water content at the surface is always very low. These conditions are equivalent with
the assumption that the energy-limiting drying rate is so large, that the duration of the first
stage of drying is negligibly short. Third, vapor transfer in the drier soil near the surface is
neglected.
As was the case for the sorption problem, by applying the Boltzmann transform (9.11),
Equation (9.6) can be reduced to the ordinary differential equation (9.13). Because
θ
i
>θ
s
,
in desorption the water content is normalized as
θ
−
θ
s
θ
i
−
θ
s
S
n
=
(9.106)
so that, instead of (9.14), the boundary conditions are
θ
=
θ
i
(or
S
n
=
1)
φ
→∞
(9.107)
θ
=
θ
s
(or
S
n
=
0)
φ
=
0
To date no general exact solution has been obtained for this problem, but only approximate
solutions or exact solutions for certain types of diffusivity functions. Gardner (1959) made
use of two solutions. One was the linearized solution obtained by means of a weighted
mean diffusivity calculated by Crank's method; his second solution, which was presented
graphically, was obtained by iteration for the exponential-type diffusivity (8.39). Parlange
et al.
(1985) proposed an approximate, but quite accurate method of solution for arbitrary
diffusivity functions
D
w
(
θ
), in a manner similar to the techniques used for sorption. Finally,
it has been shown by Brutsaert (1982) that there is a large class of
D
w
(
) functions that
admit exact solutions for desorption as formulated by Equations (9.13) and (9.107); one
such function, in particular appeared to have practical relevance to flow in soils and other
porous materials. A detailed discussion of these methods of solution is beyond the scope of
the present treatment.
In the present context, however, the most interesting feature of any solution by means of
the Boltzmann transform, regardless of the solution method and regardless of the assumed
diffusivity function
D
w
(
θ
), is that the total water volume lost from the soil profile is pro-
portional to the square root of time. Actually, by analogy with (9.19) and (9.17), for any
solution
x
=
φ
(
θ
)
t
1
/
2
it can readily be shown that the flux at the surface, that is the rate of
evaporation can be written as
θ
1
2
De
0
t
−
1
/
2
E
=
(9.108)
where
De
0
can be referred to as the capillary desorptivity, and is defined by
θ
i
De
0
=
φ
d
θ
(9.109)
θ
s
which is a constant for a given soil and given values of
θ
i
and
θ
s
.
