6
Hydraulic Properties
The soil hydraulic properties are of utmost importance for a large variety of soil processes and
in particular for soil water flow. The two most common hydraulic properties are the soil water
characteristic function (SWC) (also called soil water retention curve) and the hydraulic
conductivity function k(ψ) or k(θ). The first describes the relationship between the soil water
potential (ψ) and the corresponding value of water content (θ), while the second describes the
relationship between the hydraulic conductivity (k) and either the water potential (k(ψ)) or the
water content (k(θ)).
6.1
Darcy’s Law
Darcy’s law describes water flux determined by a water potential gradient:
qw = −k
dψ
dx
(6.1)
where qw is the flux density [kg m−2 s−1 ], dψ/dx is the water potential gradient and k is the
hydraulic conductivity [kg s/m3 ], ψ is the water potential [J/kg] and x is the space dimension [m].
The soil water potential is defined as a energy per unit mass [J/kg] or energy per unit volume
[J/m3 ]. Since a Joule is equal to a Newton per meter [J = N ·m], a Joule per unit volume [J/m3 ]
is equal to a Newton per meter squared [N ·m/m3 ] = [N/m2 ], which is equivalent to a pressure or
Pascal[P a]. For this reason often the water potential is expressed in pressure units, and since the
density of water is equal to 1000 kg/m3 , one Joule is equal to 1 kiloPascal. It is recommended to
use units of [J/kg] for the water potential, first because they are consistent with the SI units and
second because the unit of energy per mass J/kg do not change with temperature or pressure,
while the units of energy per volume J/m3 do change with temperature and pressure. Units of
2
J
1
= Nkgm = kgm
]. Substituting the proper
energy per mass are expressed in the SI units as: [ kg
s2 kg
units into Darcy’s law:
kg
kgs kgm2 1 1
(6.2)
= 3
m2 s
m
s2 kg m
with hydraulic conductivity in units of [ kgs
]. To obtain units which can be more easily “vim3
cm
sualized”, hydraulic conductivity is often expressed in [ m
s ] or [ day ]. To convert units,the water
24
J
J
potential is expressed as water head [m] and the flux in [ m
s ]. To convert [ kg ] into [m = N ],
J
] by [ g1 ] where [g] is the gravitational constant, which results into water potential
multiply [ kg
2
units of meters: [ kgm
s2
s2
1
kg 9.81m
= m]. To convert the flux density into [ m
s ], divide by the density
kg
of water ρl = 1000 m
3 . Into Darcy’s law this conversion is performed by dividing the left hand
side by the density of water and the right hand side by the gravitational constant:
kg
m2 s
1
ρl
kgs
= 3
m
kgm2 1 1
s2 kg m
1
g
(6.3)
with units of hydraulic conductivity in [ m
s ] becomes:
kg
m2 s
1
1000
m3
kg
m
=
s
kgm2 1 1
s2 kg m
1
9.81
2
s
m
(6.4)
m
where the units of flux (Jw ) are in [ m
s ], hydraulic conductivity(k) in [ s ], water potential (ψ)
in[m] and space dimension (x) in [m]. Rearranging eq. 6.3 and eq. 6.4 and solving for the
kgs
hydraulic conductivity shows that to convert units of hydraulic conductivity from [ m
s ] in [ m3 ]
kg
it is necessary to multiply by the density of water (1000 [ m
3 ]) and divide by the gravitational
kg
m
◦ C, the
constant (9.81 s2 ). Assuming that the density of water is equal to 1000 [ m
3 ] at 4
conversion factor is 101.936. Correction of the conversion factor is necessary to account for the
] in
variation of the density of water with temperature. On the other hand to convert from [ kgs
m3
]
it
is
necessary
to
divide
by
the
density
of
water
and
multiply
by
the
gravitational
constant,
[m
s
with conversion factor of 0.00981.
Following is a table showing saturated hydraulic conductivity classes in the most common units.
Table 6.1. Saturated hydraulic conductivity classes in equivalent units
kgs
m3
1.02 * 10−2
1.02 * 10−3
1.02 * 10−4
m
s
10−4
10−5
10−6
cm
day
cm
hr
864
86.4
8.64
36
3.6
0.36
Usually hydraulic conductivity is reported in one of the units shown in Table 6.1. For
consistency with the International System of Units, hydraulic conductivity is expressed in [ kgs
].
m3
Conversions are easily performed by using the conversion factors described above. The variation
],
of the density of water with temperature is more easily accounted for by using units of [ kgs
m3
J
and by expressing the water potential in units of energy per mass [ kg ].
6.2
Water Potential
The soil water potential is characterized by different components, depending on different physical
and chemical phenomena in soils. The total poteantial is given by the following components:
ψ = ψm + ψo + ψp + ψh + ψω + ψg
25
(6.5)
where ψm refers to the matric, ψo to the osmotic, ψh to the hydrostatic, ψp to the pneumatic,
ψω to the overburden and ψg to the gravitational component.
6.3
Water Content
The soil water content is usually expressed on a gravimetric base (θg ) [kg/kg] or on a volumetric
base (θv ) m3 /m3 . To convert from volumetric to gravimetric or viceversa, the following formula
is used:
ρw θ v = ρb θ g
(6.6)
where ρw is the density of water, and ρb is the bulk density [kg/m3 ].
6.4
Soil water characteristic
The two components of the total soil water potential that are affected by water content are the
matric and the osmotic components. As previously described the soil water characteristic
describes the relationship between the soil matric water potential (ψ) and the water content
(θ). Campbell (1985) describes the soil moisture characteristic curve by a power-law relation:
⎧
⎨
θs
θ=
⎩ θ
s
ψm −1/b
ψe
if (ψm < ψe )
if (ψm ≥ ψe )
(6.7)
where ψm (J kg−1 ) is the water potential , ψe (J kg−1 ) is the air entry potential, θ (m3 m−3 )
is the volumetric water content, θs (m3 m−3 ) is the saturated volumetric water content, and b is
a shape parameter related to the particle size distribution of the porous media. This equation
is discontinuous at the air entry potential, and it is analytically integrable. An alternative
equation commonly used to describe the SWR is the van Genuchten (1980) equation, which has
the following form:
θ − θr
1
Se (ψ) =
=
(6.8)
θs − θr
[1 + (αψ)n ]m
which solving for θ can be written as:
1
θ = θr + (θs − θr )
[1 + (αψ)n ]m
(6.9)
where Se is the degree of saturation (0, 1), α, n, m, θs θr are fitting parameters. Fig. 6.1 shows a
SWR for Salkum soil, and the fitted Campbell (1985)(a) and van Genuchten (1980)(c) equations
with independent parameters. A non-linear least squares fitting algorithm was implemented
(Marquardt, 1963; Press et al., 1992) to obtain the data showed in fig. 6.1. Different restriction
can be imposed on the parameters n and m van Genuchten et al. (1991) depending on the shape
of the SWR curve. In particular, when only a limited range of water retention values are available
(usually in the “wet” range of the curve) it might be necessary to restrict the parameters n and
m. More stable results are generally obtained when the restriction m = 1−(1/n) is implemented
for incomplete data sets (van Genuchten et al., 1991).
26
Campbell
Conductivity log(10) [kg s / m3]
Water Content [m3 / m3]
0.7
Air Entry Potential
0.6
Experimental
Fitted
0.5
0.4
0.3
0.2
(a)
0.1
0
0
1
2
3
4
5
-2
-4
-6
-8
-10
-12
-14
(b)
-16
-18
-6 -5 -4 -3 -2 -1
0
1
2
3
4
5
6
5
6
Water Potential log(10) [J / kg]
Water Potential log(10) [J / kg]
van Genuchten/Mualem
Conductivity log(10) [kg s / m3]
Water Content [m3 / m3]
0.7
0.6
Experimental
Fitted
0.5
0.4
0.3
0.2
(c)
0.1
0
0
1
2
3
4
-2
-4
-6
-8
-10
-12
-14
-16
(d)
-18
-6 -5 -4 -3 -2 -1
5
0
1
2
3
4
Water Potential log(10) [J / kg]
Water Potential log(10) [J / kg]
Fig. 6.1. Fitted Campbell’s (a) and van Genuchten’s (c) SWR curves for Salkum soil. Estimated
unsaturated hydraulic conductivity using eq. 6.9(b) and eq. 6.12(d). The sample has saturated
hydraulic conductivity, ks =10−3 [kg s m−3 ].
6.5
Hydraulic Condictivity function
Derivation of hydraulic conductivity from the soil water retention is obtained for the Campbell
(1985) formulation which describes the unsaturated hydraulic conductivity curve based on the
combined probability of finding continuous pores within a cross section of the porous media. The
calculation of the combined probability requires to integrate twice(for each of the two sections)
over the pore space to obtain the hydraulic conductivity (Campbell, 1974). Since the SWR is
analytically integrable, the unsaturated hydraulic conductivity function is given by:
⎧
⎨
Ks
K=
⎩ K
s
ψe (2+3/b)
ψm
if (ψm < ψe )
if(ψm ≥ ψe )
(6.10)
where K (kg s m−3 ) is the unsaturated conductivity and Ks (kg s m−3 ) is the saturated conductivity. Figure 6.1(b) shows the unsaturated hydraulic conductivity function obtained by using
27
eq. 6.9. Because of the discontinuous nature of the Campbell (1985) equation, the unsaturated
hydraulic conductivity function is also discontinuous at the air entry potential point.
The derivation of the hydraulic conductivity function for the van Genuchten (1980) equation
is given by the Maulem (1976) model which is written in the form:
K(Se ) =
Ks Sel
f (Se
f (1)
2
(6.11)
where
f (Se ) =
1
ψ(θ)
(6.12)
where Se is the degree of saturation, ks is the saturated hydraulic conductivity and l is a pore
space connectivity parameter assumed to be equal to 0.5 as average for many soils. Note that
the integration is first normalized over the spore space and then it is squared, which is equivalent
to integrate twice as in Campbell (1985). Nevertheless this integration (when the parameters
n and m are independent) requires the use of “special” functions such as the Incomplete Beta
function (Press et al., 1992). Equation 6.11 is then rewritten as:
K(Se ) = Ks Sel [Iζ (p, q)]2
(6.13)
where p = m + (1/n), q = 1 − (1/n), assuming independent n and m parameters. The incomplete Beta function is more difficult to evaluate and in some cases convergence is not assured.
For this reason for scattered and incomplete SWR data sets, the restriction of m = 1 − (1/n)
allows integration of equation 6.11 without use of the Incomplete Beta function, which results:
K(h) =
Ks 1 − (αψ)mn [1 + (αψ)n ]−m
[1 + (αψ)n ]ml
2
(6.14)
Note that the integration was always performed in terms of the water potential. Figure 6.1(d)
shows the unsaturated hydraulic conductivity function obtained by following Maulem (1976)
and implementation of the Incomplete Beta function (Press et al., 1992).
6.6
Exercises
1. Using the soil water characteristics listed in the WorkSheet Retention Curves in the files
Ex4a and Ex4b, fit both the Campbell (1985) and the van Genuchten (1980) equations.
Produce a table showing the fitting parameters for the two equations, for each soil sample.
Discuss the results based on the different physical properties of the tested samples.
2. Using the algorithms in Ex4a and Ex4b, calculate the unsaturated hydraulic conductivity
for the soil sample Salkum using both the Campbell (1985) and the van Genuchten (1980)
equations. Plot the results and discuss the differences.
28