Ground Surface Moisture Flux Boundary Conditions - Unsaturated Soil Mechanics in Engineering Practice - page 315

8. The runoff R can be calculated using Eq. 6.7. The
variables AE and I should be known when this cal-
culation is performed.
9. Check the solution to see if the convergence has
been reached for hydraulic head h . The difference in
hydraulic head between the last two iterations must
be less than the designated tolerance.
10. If the solution has not converged, repeat steps
4-9 while checking for convergence on hydraulic
head.

• Soil freezing/thawing processes are the same as in the
previously mentioned coupled solutions.
• The latent heat due to phase change including evapora-
tion and freezing/thawing is considered for heat transfer
beneath the soil surface.
• The AE is based on an empirical mathematical equation
that fits the data shown in Fig. 6.44.

Partial Differential Equation Governing Moisture and
Heat Flow. The moisture mass and heat transfer governing
equations are defined by Eqs. 6.50 and 6.53.
Initial Water Content Conditions. The starting water
content profile must be initialized using one of the previ-
ously mentioned procedures.
Initial Temperature Conditions. The temperatures
throughout the soil profile can be initialized to the air
temperature in accordance with Eq. 6.62.
Boundary Condition for Moisture Flow. The ground
surface boundary condition for moisture flow can be defined
using Eq. 6.65.
Thermal Boundary Condition at Soil Surface. The
thermal boundary condition at the soil surface can be spec-
ified as a constant temperature or a temperature expression
(Dirchlet boundary condition, Eq. 6.63) or computed from
a thermal flux (Neumann boundary condition, Eq. 6.64).
Actual Evaporation. The rate of evaporation from the
ground surface can be estimated using the Wilson et al.,
(1997a) empirical experimental-based relationship. The ratio
AE/PE is approximated using an equation similar in form
to the thermodynamic equilibrium relationship between rel-
ative humidity and total suction (Edlefsen and Anderson,
1943). The AE can be written as a function of PE using the
following experimental-based equation:

6.3.19.5 Experimental-Based Relationship between AE
and PE (Coupled Solution)
Wilson (1997) also presented experimental results that
showed a unique relationship for all soils between total
suction at the soil surface and the ratio of actual evaporation
to potential evaporation , AE/PE. In 1997, Wilson et al.,
(1997a) presented an equation that provided a reasonable fit
through the experimental data. Wilson (1990) showed that it
is primarily the soil suction at the ground surface of any soil
that mainly controls the rate of evaporation. Consequently,
the soil type at ground surface is not a controlling factor
when assessing the actual rate of evaporation.
Figure 6.44 shows a plot of the experimental data along
with the proposed mathematical relationships for calculating
AE. The laboratory tests involved the evaporation of water
from thin layers of sand, silt, and clay soils. The soils were
initially deposited as slurry (i.e., high water contents). The
rates of evaporation from the soils were compared to the rate
of evaporation from a pan of water at the same temperature.
The comparison of the evaporation rates from the soil and
the water allowed the computation of the relative evapora-
tion or the ratio of AE to PE (i.e., AE / PE). The rate of
evaporation from the soils became increasingly slower with
time than evaporation from a pan of water.
The water content versus soil suction relationship was
independently measured for each of the soils. This allowed
the relative rates of evaporation to be compared to the suc-
tion in the soil. The results in Fig. 6.44 show that the actual
evaporation data for sand, silt, and clay essentially form a
single unique curve. This means that the rate of water evap-
oration from any soil is the same when soil suction is used
as the basis of comparison. In other words, if the soil suction
is known at the ground surface, then the rate of evaporation
from the ground surface can be estimated from the unique
relationship shown in Fig. 6.44.
The following assumptions are made when performing a
coupled analysis along with the experimentally based AE/PE
equation:
• The temperature of the soil at ground surface is equal
to the air temperature.
• The moisture flow and heat transfer beneath the ground
surface are the same as in the previously mentioned
coupled solutions.

exp

AE
PE =

− ψgω v

ζ 1 h a γ w R T soil + 273 . 15

(6.75)

−

where:

ζ = a dimensionless empirical parameter equal to about

0.7 based on experimental data (Wilson, 1990).

Potential Evaporation. The PE can be either measured
or calculated using one of several empirical methods [e.g.,
Penman (1948) equation 6.20, Thornthwaite (1948) equation
6.19, Priestley-Taylor (1972) equation 6.23, or Monteith
(1965) equation 6.22].
Calculation Procedure.

1. Initialize the water content profile.
2. Initialize the soil temperature profile.
3. Apply the moisture flux boundary condition according
to Eq. 6.65.
4. Apply the thermal boundary condition at the ground
surface using Eq. 6.63 or 6.64.