Unsaturated flow cases
‹
‹
⇒
‹
Infiltration
Hydrostatic conditions
– q=0
Steady state flow
– q = constant
– no variations in time
– variations with depth
‹
the process of water entry into the soil
‹
early stage: capillary forces dominate
‹
later stage: gravitational forces dominate
– infiltration rate → Ks
‹
important factors:
– initial suction
Non-steady flow
– variations with time and depth
» infiltration
» the general case
– rainfall rate
– soil properties
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Let us consider a simple system
‹
homogeneous soil
‹
1-D vertical system
‹
uniform initial conditions
‹
constant rainfall rate R
2
R < Ks
θi
3
θa θs
4
1
Ks < R < ic
R > ic
θi
θs
θi
θs
Various formula have been developed for this case, e.g. by Philip (1956)
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Theoretical observations
‹
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Water content profile during infiltration
Theoretical analyses have shown that
– infiltration rate approaches Ks
– wetting front assumes a constant shape when the
infiltration rate approaches a constant value
– velocity of wetting front approaches a constant value
given by
Vf =
K (θ s ) − K (θ i ) K (θ s )
≈
θs − θi
θs − θi
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2
Green and Ampt infiltration model
Green and Ampt infiltration model
h0
h0
constant properties θs, Ks
Flow equation
hf − h0
L
( ψ f − L) − 0
= −K s
L
ψf
= K s (1 − )
L
i = −K s
hf
hf
Mass conservation
I = ( θ s − θ i ) L = ∆θ L
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Green and Ampt infiltration model
Green and Ampt infiltration model
Combining the two equations and integrating
The traditional formulation can be expanded to also predict the
time of ponding
∆θψ f
) ⇒
I
dI
I − ∆θψ f
= Ks (
) ⇒
dt
I
I
I + ∆θψ f ln(1 −
) = Kst
∆θψ f
i = K s (1 −
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Redistribution
Characteristic flow velocities
Redistribution refers to the water movement after infiltration
‹
Darcy flux q
‹
pore water velocity
‹
infiltration front velocity
θi
θs
L
v=
q
θ
t=0
Z
t>0
Vf =
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Solute transport
K (θs ) − K (θ) K (θs )
≈
θs − θi
θs
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Dispersion
Advection
v=
q
θ
Pore scale
pore water velocity
Dispersion
Larger scale
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Dispersion
General formulation of the ADE equation
Transport due to dispersion usually described by Fick’s law
Flux of solute
J s = qc r − θD
Mass conservation
∂M
∂J
=− s
∂t
∂z
ADE equation
∂c r
∂ 2c
∂c
= D 2r − v r
∂t
∂z
∂z
J s = −θD
∂c r
∂z
dispersion coefficient
D = αv
∂c r
∂z
(θ, q and D assumed constant in space and time)
dispersivity
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Effect of dispersion
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Dimensionless variables
1. Dimensionless time defined as the number of porevolumes
qt vt
T=
=
θL L
2. Dimensionless distance
z
Z=
L
3. Dimensionless concentration
c
C= r
c0
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4. Column Peclet number
vL
P=
D
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Dimensionless equation
Boundary conditions
Upper boundary
∂C 1 ∂ 2 C ∂C
=
−
∂t P ∂Z 2 ∂Z
∂c r
) z = 0 = qc 0
∂z
D ∂c r
(c r −
) z =0 = c 0
v ∂z
(qc r − θD
This equation is the basis for CXTFIT software
⇒
discontinuity in concentration
across the boundary
Lower boundary
∂c r
(L, t ) = 0
∂z
∂c r
(∞, t ) = 0
∂z
for no backmixing at the outlet, the boundary
condition for the semi-infinite case is a good
approximation, mathematically more convenient
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Flux concentration ≠ Resident concentration
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Flux concentration
cf =
cr: resident concentration (volume average)
Js
q
J s = qc r − θD
∂c r
∂z
cf = cr −
D ∂c r
v ∂z
q = θv
cf: flux concentration (flux averaged)
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ADE in flux concentration mode
Flux and resident concentrations
∂c f
∂ 2c
∂c
= D 2f − v f
∂t
∂z
∂z
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Effect of various processes on solute transport
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