Environmental Engineering Reference
In-Depth Information
W w =
wind speed, km/h,
7. Calculate net percolation I or infiltration at the soil
surface. The net infiltration flux at the ground surface
is defined as k∂h/∂y .
8. Calculate runoff R using Eq. 6.7 using the previously
calculated AE and infiltration I .
9. Check the solution to see if the convergence has been
achieved with respect to hydraulic head h . The differ-
ence between subsequent iteration must be less than
the designated tolerance.
10. If the solution has not yet converged, repeat steps 5-10
until convergence has been achieved.
Atmospheric (quasi) coupling has been solved using a
closed-form solution for soil surface temperature as one of
the governing equations (i.e., Eq. 6.72).
net radiation, J/m 2 /day,
R n =
ground surface thermal flux, J/m 2 /day,
R g
=
volumetric latent heat of vaporization, J/m 3 , and
L v =
AE
=
actual evaporation, mm/day.
Equation 6.71 can be rearranged and solved for the tem-
perature at the surface of the soil, T soil assuming ground
surface flux is zero.
1000 R n
L v AE
C f ηf (u) L v
T soil =
T a +
(6.72)
Equation 6.72 does not take the ground thermal flux into
consideration (i.e., Q g =
0) and can be considered as an
empirical estimation for the calculation of the soil temper-
ature. It is not fully known whether there are limitations
associated with the use of Eq. 6.72 in geotechnical engi-
neering practice.
Initial Soil-Water Content Profile. The initial water
content profile needs to be set to an initial set of values.
The water content profile may correspond to the initial pore-
water pressures dictated by the SWCC or the initial water
content may be measured (or estimated) based on the loca-
tion of the water table.
Initial Soil Surface Temperature. The soil tempera-
tures 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.
Actual Evaporation. The Wilson-Penman equation
(Wilson et al., 1994) can be used for calculating AE (i.e.,
Eq. 6.66). The uncoupled solution must be solved satisfying
the governing moisture PDE (i.e., Eq. 6.50) and soil temper-
ature equation 6.72 using the initial moisture and thermal
conditions. The moisture flow boundary condition equation
6.65 must be satisfied along with the AE equation 6.66.
It is necessary to know the soil temperature at the ground
surface when calculating AE (Eq. 6.66). When using
Eq. 6.72 to calculate the soil temperature, the AE must
also be known. In other words, Eqs. 6.72 and 6.66 must be
solved simultaneously to get AE and soil temperature.
Calculation Procedure.
1. Initialize water content profile.
2. The soil surface temperatures are made equal to the air
temperature.
3. Apply the moisture flux boundary condition in accor-
dance with Eq. 6.65.
4. Ensure the model domain is initialized.
5. Calculate AE using Eq. 6.66 based on a soil surface
temperature T soil given by Eq. 6.72.
6. Solve the moisture flow PDE (i.e., Eq. 6.50) and soil
temperature equation 6.72 simultaneously based on the
initial conditions and moisture flux boundary condition
designated by Eq. 6.65.
6.3.19.3 Limiting Function for AE (Coupled Solution)
A limiting function relationship was proposed by Wilson
et al. (1997a) that provided a relationship between AE and
PE. The vapor pressures associated with the relative humid-
ity at ground surface and the relative humidity in the air
above ground surface are scaled based on the assumption
that the air and soil temperatures were the same.
The limiting function solution is based on the following
assumptions:
• The soil temperature at the ground surface is equal to
the air temperature.
• The moisture flow and heat transfer beneath the ground
surface are the same as in the coupled rigorous solution.
• Soil freezing/thawing processes are the same as in the
coupled rigorous solution.
• The latent heat due to phase change, including evapo-
ration and freezing/thawing, must be considered in the
heat transfer beneath the ground surface.
• AE is calculated using the limiting function equation
proposed by Wilson et al. (1997a).
Partial Differential Equations Governing Moisture
and Heat Flow. The moisture mass transfer and heat
transfer partial differential equations are described by Eqs.
6.50 and 6.53.
Initial Water Content Conditions. Initial water con-
tents throughout the soil profile must be initialized as pre-
viously described.
Initial Temperature Conditions. The soil temperatures
throughout the soil profile can be initialized to the air tem-
perature (Eq. 6.62).
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
(i.e., Dirchlet boundary condition) or computed from a ther-
mal flux (Neumann boundary condition), in accordance with
Eqs. 6.63 and 6.64, respectively.
Boundary Condition for Moisture Flow. The atmo-
spheric moisture flux at the ground surface needs to satisfy
Eq. 6.65.
 
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