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Fundamentals of soil water - atmosphere
interaction
Alessandro Tarantino
Department of Civil and Environmental Engnieering
University of Strathclyde
Scotland
Soils above the water table are unsaturated
and have negative pore-water pressures
Water table
Slope Foundation
Embankment Dam
Unsaturated zone
Unsaturated zoneUnsaturated zone
Water table
Unsaturated zone
Soils above the water table are exposed
to the atmosphere
Water content (and suction) of soils above the water
tables changes over time
Water
content
z
Saturated
zone
Unsaturated
zone
Water
content
z
Saturated
zone
Unsaturated
zone
Modelling effects of rainwater infiltration
Saturated zone
Unsaturated zone𝝏
𝝏𝒛−𝑲 𝒖𝒘
𝝏
𝝏𝒛
𝒖𝒘𝜸𝒘+ 𝒛 = −
𝝏𝜽
𝝏𝒖𝒘
𝝏𝒖𝒘𝝏𝒕
𝒒 ≡ 𝒓𝒂𝒊𝒏𝒇𝒂𝒍𝒍 𝒊𝒏𝒕𝒆𝒏𝒔𝒊𝒕𝒚
Water flow equation
Modelling effects of evaporation
Saturated zone
Unsaturated zone𝝏
𝝏𝒛−𝑲 𝒖𝒘
𝝏
𝝏𝒛
𝒖𝒘𝜸𝒘+ 𝒛 = −
𝝏𝜽
𝝏𝒖𝒘
𝝏𝒖𝒘𝝏𝒕
𝒒 ?
Water flow equation
Physics of evaporation from free water
Liquid-vapour equilibrium
(pure water - flat interface)
water, uw
vapour, p°v
The pressure of the vapour in equilibrium with the liquid is referred
to as saturated vapour pressure, p°v (denoted with superscript 0)
Temperature
Pre
ssu
re
Water phase diagram
Liquid
VapourSolid
20°
p°v =2.3 kPa
Driver of evaporation from free water
Temperature
Pre
ssu
re
Vapour pressure
differential
T
p°v
p°v
𝑹𝑯 =𝒑𝒗
𝒑𝟎𝒗
pv
pv
Relative humidity
Vapour flow is driven by the vapour pressure differential p°v - pv
Evaporating
surface
Surrounding atmosphere
Dynamics of evaporation from free water
p°v
pv
Evaporation from free water never stops as the vapour pressure
differential p°v – pv remains constant over time
p°v
pv
Evaporation from free water
Water
Balance
Time
Mass
Water
Atmosphere
pv
T
Evaporation drivers (1)
p°v
pv
Evaporation rate increases with the energy supply via
solar radiation
Evaporating
surface
p°v
Energy needs to be supplied to transform liquid into
vapour (latent heat of evaporation)
Example from our day-to-day life
Energy supply increases the evaporation rate
Evaporation drivers (2)
p°v
pv
Evaporation rate increases with the velocity of the air flow
(which ‘sweeps’ the vapour at the evaporating surface)
Evaporating
surface
Vapour sublayer
p°v
pv p°v
Example from our day-to-day life
• The higher the wind speed
• The higher the evaporation rate
• The higher the energy extracted from our body to supply the latent
heat of evaporation
• The lower the body temperature
Potential evaporation (from free water):
Penman equation
Vapour pressure differential
triggers evaporation
𝑃𝐸𝑇 =1
𝜆
Δ ∙ 𝑅𝑛 + 𝜌𝑎 ∙ 𝑐𝑝∙ 𝑝𝑣0 𝑧 − 𝑝𝑣 𝑧 ∙ 𝑟𝑎
Δ + 𝛾
pv0
pv = RHpv0
Solar radiation supplies energy for
transforming liquid into vapour
Evaporating surface
Vapour sublayerAerodynamic resistance
(wind function)
Evaporating surface
Derivation of Penman equation (1)
Evaporating interface (T=Ts)
Convective heat - vapour
hv(Ts)(wE*) Atmosphere (T=Ta)
Liquid (T=Ts)
Radiation
Rn
Turbulent diffusion
sensible heat
H
q
Conductive heathv(Ts) qmv
Convective heat - vapour
hl(Ts) qml
Convective heat - liquid
wE* (vapour water)
qml (liquid water)
Mass Balance
Energy Balance
Evaporating interface
Atmosphere
Liquidqmv (vapour water)
Derivation of Penman equation (2)In the dynamic sublayer (up to 10 m from ground surface):
• Equilibrium statically neutral (Coriolis forces and buoyancy forces can be neglected)
• Molecular diffusion of mass and energy negligible with respect to turbulent advective
transfer of mass and energy
• Viscous shear stresses negligible with respect to turbulent shear stress
• Air flow in proximity of horizontal surface fully turbulent and under steady-state
Logarithmic profile of wind mean velocity
Smooth surface
Viscous sub-layer
xv
z
Rough surface
z0
xv
z
z0m
Velocity profileExtrapolated
logarithmic profile
Derivation of Penman equation (3)
From dimensional analysis and experimental evidence :
zpp
z
dz
zvk
TREE vvs
m
x
d
l
2
0
0
2*
ln
622.0 Vapour flux profile
zTT
z
dz
zvckH s
m
xpa
2
0
0
2
ln
Sensible heat flux profile
Penman equation
(Not magic box but is derived from mass and energy balance equations
and assumptions on air flow at the interface)
Derivation of Penman equation (4)
Brutsaert, W. 1982. Evaporation into the atmosphere.
Kluwer Academic Publisher, Dordrecht
.
Wish to understand more?
Physics of evaporation from soil
Soil as a system of capillary tubes
Effect of curvature of the liquid-gas interface
q
R
water
air
r
T
R
T
r
Tuu aw
2cos2
q
ua = air pressure [F/L2]
uw = water pressure [F/L2]
q = contact angle
T = surface tension [F/L]
r = radius of capillary tube [L]
R = radius of curvature of spherical cup [L]
q cos222 rTruru wa
If q < 90°, water pressure is negaitve
(lower than air pressure)
The smaller is the radius of curvature,
the more negative is the water pressure
Mechanical equilibrium
Rise in capillary tube
uw<0
uw=0
uw=0
uw=0
r
Thu ww
q
cos2
h
If q<90°, the liquid enter the cavities in the solid surface
the liquid is said to wet the surface
ua=0
Hysteresis of the contact angle
q
qr qa
qr = receding angle
qa = advancing angle
qr=qmin
qmax=qa
In a capilary tube, the contact
angle ranges from qa to qr
Evaporation from a capillary tube
q=qrq>qr
q=qr
1 2 3 4
uw=0 uw<0 uw= -2T cosqr /r
r
uw= -2T cosqr /r
Evaporation from a system of capillary
tubes
a
c
b
ra
rc
rb
c
c
b
b
a
a
rrr
qqq coscoscos
wcwbwa uuu
La = Lb = Lc
ra = 2 rb = 4 rc
Meechanical equilibrium
Geometry
Water retention of a sytem of capillary
tubes
1 2 3 4 5
Vw / V
- uw= suction
1 2
34
5
S
Liquid-vapour equilibrium (free water - curved interface)
water, uw
Evaporation of water molecules is hampered by the
tensile state of stress in water and
Vapour pressure pv is lower than the ‘saturated’
vapour pressure p0v associated with the flat surface
pv < p°v
vapour, pv
air, pa
Owing to the meniscus
uw < pa = 0
Vapour pressure for decreasing (more
negative) water pressures
Temperature
Pre
ssu
re
Liquid
Vapour
Solidp°v
p°v
pv, soil
pv, soil
soillv
soilv
lwa RHv
RT
p
p
v
RTuus lnln
0
Psychrometric Law
pv, soil
pv, soil
Zero liquid pressure
Negative liquid
pressure
0,4
0,5
0,6
0,7
0,8
0,9
1
10 100 1000 10000 100000
Rela
tive H
um
idit
y, R
H
Suction, s (kPa)
Psychrometric Law
p°v
pv, soil
pv, soil
Zero liquid pressure 𝑹𝑯 =𝒑𝒗
𝒑𝟎𝒗
Evaporation from an initially saturated soil
Evaporation from saturated soil with zero
water pressure
Temperature
Pre
ssu
re
Liquid
Vapour pressure
differential
T
p°vpv, soil= p°v
𝑹𝑯 =𝒑𝒗
𝒑𝟎𝒗
pv
pv
Relative humidity
Vapour pressure in the soil is the same as vapour pressure in free water
Surrounding atmosphere
Driver of evaporation from soil
Temperature
Pre
ssu
re
Liquid
T
p°v, soil
𝑹𝑯 =𝒑𝒗
𝒑𝟎𝒗
pv
pv
Relative humidity
Vapour pressure in the soil decreases as evaporation proceeds eventually
matching the value in the surrounding atmosphere
Surrounding atmosphere
pv, soil < p°v
Vapour pressure
differential
0,4
0,5
0,6
0,7
0,8
0,9
1
10 100 1000 10000 100000
Rela
tive H
um
idit
y, R
H
Suction, s (kPa)
Psychrometric Law
p°v
pv, soil
pv, soil
Zero liquid pressure
RH atmosphere
Vapour pressure in the soil decreases as evaporation proceeds eventually
matching the value in the surrounding atmosphere
Evaporation from soil
Sample
Balance
Time
Mass Saturated / quasi-saturated
To residual state
Soil
Atmosphere
SoilAtmosphereSoil
pv
T
pv
TFree water
Boundary condition associated with
evaporation
Saturated zone
Unsaturated zone
𝒒 =1
𝜆
Δ ∙ 𝑅𝑛 + 𝜌𝑎 ∙ 𝑐𝑝∙ 𝑝𝑣0 𝑧 − 𝑝𝑣 𝑧 ∙ 𝑟𝑎
Δ + 𝛾= 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
Free water (Penman equation)
𝒒 =1
𝜆
Δ ∙ 𝑅𝑛 + 𝜌𝑎 ∙ 𝑐𝑝∙ 𝑝𝑣0 𝑧 − 𝑝𝑣 𝑧 ∙ 𝑟𝑎
Δ + 𝛾∙ 𝑭 𝒔𝒖𝒄𝒕𝒊𝒐𝒏
Soil water (Penman equation modified)
Reduction
function
Vapour pressure differential should
be made dependent on suction
Reduction function
Wilson et al. 1997
• The reduced evaporation rate is referred to as ‘actual evaporation’ by
Wilson et al. 1997
• This is not correct, they still consider the potential evaporation but take
into account the reduction of potential evaporation as suction increases
Warning !
𝑔𝑣 = −𝛿𝑑𝑝𝑣𝑑𝑥
• Evaporation is a non-isothermal process with phase transition
• The problem should be therefore modelled considering
𝑞 = −𝜆𝑑𝑇
𝑑𝑥
𝑣 = −𝑘𝑑ℎ
𝑑𝑥
Vapour flow (Fick’s law)
Heat flow (Fourier’s law)
Liquid flow (Darcy’s law)
• By considering only liquid flow, we squeeze a complex coupled multi-
physics process into a line (the boundary condition)
𝜕
𝜕𝑧−𝐾 𝑢𝑤
𝜕
𝜕𝑧
𝑢𝑤𝛾𝑤+ 𝑧 = −
𝜕𝜃
𝜕𝑢𝑤
𝜕𝑢𝑤𝜕𝑡
• This is generally acceptable as long as liquid flow dominates (high and
medium degrees of saturation) and temperature gradients are not significant
Modelling evaporation process
Unsaturated zone
𝒒 =1
𝜆
Δ ∙ 𝑅𝑛 + 𝜌𝑎 ∙ 𝑐𝑝∙ 𝑝𝑣0 𝑧 − 𝑝𝑣 𝑧 ∙ 𝑟𝑎
Δ + 𝛾∙ 𝑭 𝒔𝒖𝒄𝒕𝒊𝒐𝒏
Soil water (Penman equation modified)
uw
z
Why suction at the ground surface tends to go to infinite and the
numerical solution does not converge anymore?
200
kPa
1000000
kPa
1000
kPa
Potential and actual evaporation
The concept of potential and actual evapotranspiration
Evaporative demand of the atmosphere
(Potential ‘energy-limited’ evapotranspiration, PET)
Water that soil ‘hydraulic system’ can supply
(‘water-limited’ evapotranspiration ETlim)
Energy-limited regime (potential evapotranspiration)
PET < ETlim
The soil hydraulic system CAN
accommodate the evaporative
demand of the atmosphere
Actual ET = Potential ET
Controlled by the atmosphere, i.e. solar
radiation, wind speed, air humidity
The Penman equation
Evaporative demand of the atmosphere
(Potential ‘energy-limited’ evapotranspiration, PET)
Saturated zone
Unsaturated zone
𝒒 =1
𝜆
Δ ∙ 𝑅𝑛 + 𝜌𝑎 ∙ 𝑐𝑝∙ 𝑝𝑣0 𝑧 − 𝑝𝑣 𝑧 ∙ 𝑟𝑎
Δ + 𝛾= 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
Free water (Penman equation)
𝒒 =1
𝜆
Δ ∙ 𝑅𝑛 + 𝜌𝑎 ∙ 𝑐𝑝∙ 𝑝𝑣0 𝑧 − 𝑝𝑣 𝑧 ∙ 𝑟𝑎
Δ + 𝛾∙ 𝑭 𝒔𝒖𝒄𝒕𝒊𝒐𝒏
Soil water (Penman equation modified)
Water-limited regime
PET > ETlim
The soil hydraulic system CANNOT
accommodate the evaporative
demand of the atmosphere
Actual ET = ETlim
Controlled by the soil hydraulic system
regardless of solar radiation, wind speed,
air humidity, etc.
Water limited evapotranspiration:
steady-state case
Pore-water pressure
z
uw0
Hydrostatic
Water flux, q0
Pore-water pressure
ETlim
𝑞0 = −𝑘 𝑢𝑤𝜕
𝜕𝑧
𝑢𝑤𝛾+ 1
0
When uw -
q0
Water limited versus potential evapotranspiration
z
Water flux, q0
uw
ETlim
PET
uw
z
uw
uw
PET
ETlim
𝑞0 = −𝑘 𝑢𝑤𝜕
𝜕𝑧
𝑢𝑤𝛾+ 1
The instability of the numerical solution
PET > ETlim
𝑞0 = −𝑘 𝑢𝑤𝜕
𝜕𝑧
𝑢𝑤𝛾+ 1
10 mm/day
2 mm/day
• To accomdate the flux imposed, uw is
decrased at the surface to increase the
gradient
• However, the hydraulic conductivity
decreases and the resulting flux does not
match the one imposed
• The numerical model ignores the limiting
evaporation, and uw is decreased again in
the vain attempt to match the imposed flux
ETlim for bare soil (steady-state)
z
Water flux
uw
ETlim
uw
q0
−𝜕
𝜕𝑧−𝑘 𝑢𝑤
𝜕
𝜕𝑧
𝑢𝑤𝛾+ 𝑧 = 0
𝑘 𝑢𝑤 = 𝑘𝑠𝑒𝛼𝑢𝑤𝛾𝑤
𝐸𝑇𝑙𝑖𝑚 = lim𝑢𝑤 𝐿 →−∞
𝑞0 =𝐾𝑠𝑒𝛼𝐿 − 1
𝑢𝑤𝛾𝑤=1
𝛼𝑙𝑛 𝑒−𝛼𝑧 +
𝑞0𝐾𝑠𝑒−𝛼𝑧 − 1
-
ks – saturated hydraulic conductivity
– unsaturated hydraulic conductivity
L – water table depth
Yuan and Lu 2005
Actual evapotranspiration in water-limited regime
PET > ETlim
Actual ET = ETlim
ETlim
water table depth
root depth
hydraulic properties
Actual ETwater table depth
root depth
soil hydraulic properties
Methods to estimate actual ET based on water pressure only (as in several commercial codes) are conceptually incorrect
ETlim for bare soil (transient-state)
𝜕𝜃
𝜕𝑡= −𝜕
𝜕𝑧−𝑘 𝑢𝑤
𝜕
𝜕𝑧
𝑢𝑤𝛾+ 𝑧 𝐾 𝑢𝑤 = 𝐾𝑠𝑒
𝛼𝑢𝑤𝛾𝑤 ; 𝜃 𝑢𝑤 = 𝜃𝑠𝑒
𝛼𝑢𝑤𝛾𝑤
𝑢𝑤𝛾𝑤=1
𝛼𝑙𝑛 𝑒−𝛼𝑧 − 8𝑞1
𝛼
𝐾𝑠𝑒𝛼 𝐿−𝑧2
𝑛=1
∞𝑠𝑖𝑛 𝜆𝑛𝐿 𝑠𝑖𝑛 𝜆𝑛𝑧
2𝛼 + 𝛼2𝐿 + 4𝐿𝜆𝑛2 1 − 𝑒
−𝐷 𝜆𝑛2+𝛼2
4𝑡
Yuan and Lu 2005
TIME
uw
zq1
Water flux
• There is a critical time where uw - ∞
• Desperate attempt to accommodate imposed water flux
• This critical time marks the transition to water-limited ET
Water limited evapotranspiration in transient regime
(bare soil)
𝐸𝑇𝑙𝑖𝑚 = lim𝑢𝑤 𝐿 →−∞
𝑞0 =
𝐾𝑠𝛼 𝑒𝑥𝑝 −𝛼𝐿
8 𝑛=1∞ 𝑠𝑖𝑛2 𝜆𝑛𝐿2𝛼 + 𝛼2𝐿 + 4𝐿𝜆𝑛
2 1 − 𝑒𝑥𝑝 −𝐷 𝜆𝑛2 +𝛼2
4 𝑡
PET
TIME
Transpiration
The misconception about plants extracting water
• Transpiration is the upward movement of water to replace what is lost by evaporation
• Transpiration in itself is neither an essential physiological function, nor a direct result
of the living process
Spongy Mesophyll
Microfibrils
Menisci
Stoma
uw<0
Guard cell
Water vapourC02 (Photosynthesis)
LEAF
The driving mechanism of transpiration
\
Guard cell
pv,leaf
Spongy Mesophyll
Microfibrils
Menisci
Stomauw<0
Menisci
• Water is extracted by the atmosphere THROUGH the plant (and not by the plant)
• The driving mechanism of water extraction is the same for bare and vegetated soil
Xy
lem
Roots
uw<0
pv, atmosphere
pv,soil
pv, atmosphere
Potential evapotranspiration:
Penman-Monteith equation
Vapour pressure differential
triggers evaporation
𝑃𝐸𝑇 =1
𝜆
Δ ∙ 𝑅𝑛 + 𝜌𝑎 ∙ 𝑐𝑝∙ 𝑝𝑣0 𝑧 − 𝑝𝑣 𝑧 ∙ 𝑟𝑎
Δ + 𝛾 ∙ 𝑟𝑎 + 𝑟𝑐 𝑟𝑎
pv0
pv = RHpv0
Solar radiation supplies energy for
transforming liquid into vapour
Evaporating surface
Vapour sublayerAerodynamic resistance
(wind function)
Stomatal (canopy)
resistance
ETlim for vegetated soil (steady-state)
z
Water
flux
uw
ETlim
S
−𝜕
𝜕𝑧−𝑘 𝑢𝑤
𝜕
𝜕𝑧
𝑢𝑤𝛾+ 𝑧 − S 𝑧 = 0
Sink term to simulate
root water uptake
𝐸𝑇𝑙𝑖𝑚 = lim𝑢𝑤 𝐿 →−∞
𝑞0 = 𝛼𝐾𝑠𝛿
𝑒𝛼𝐿 − 𝛼𝛿 − 𝑒𝛼 𝐿−𝛿
d-
d – depth of root zone
4 6 8 10Water table depth, L (m)
4
8
12
ET
lim (
mm
/da
y)d=1.2m
d=0.4m
bare
Water limited regime under steady-state
PET=8 mm/day
Actual ET
Water table depth (m)
ET
lim
(mm
/day)
Water limited regime under steady-state:
Pore-water pressure and FoS profiles
Under water limited regime, vegetation has beneficial effects on factor of safety (lower pore-water pressures)
This is associated with the different mode of extraction, concentrated at the ground surface for the bare soil, distributed over depth for vegetated soil
-800 -600 -400 -200 0Pore-water pressure (kPa)
8
6
4
2
0
De
pth
(m
)
0 1 2 3 4FoSvegetated-FoSbare
8
6
4
2
0
De
pth
(m
)
d=1.2m
d=0.4m
bare
d=1.2m
d=0.4m
Energy limited regime under steady-state
4 6 8 10Water table depth, L (m)
4
8
12
ET
lim (
mm
/da
y)
d=1.2m
d=0.4m
barePET=8 mm/day = Actual ET
Water table depth (m)
ET
lim
(mm
/day)
Energy limited regime under steady-state:
Pore-water pressure and FoS profiles
Under ENERGY limited regime, vegetation may not have beneficial effects on factor of safety (higher pore-water pressures)
At the same overall water flux, lower pressures are generated by the bare soils because higher gradients need to be generated
-160 -120 -80 -40 0Pore-water pressure (kPa)
5
4
3
2
1
0
De
pth
(m
)
-1.6 -1.2 -0.8 -0.4 0FoSvegetated-FoSbare
4.5
3.5
2.5
1.5
0.5
De
pth
(m
)
d=1.2m
d=0.4m
bared=1.2m
d=0.4m
Andrew Simon’s data (Tuesday)100 CM
-20
0
20
40
60
80
100
12/2
9/99
1/29
/00
2/29
/00
3/29
/00
4/29
/00
5/29
/00
6/29
/00
7/29
/00
8/29
/00
9/29
/00
10/2
9/00
11/2
9/00
12/2
9/00
1/29
/01
2/28
/01
MA
TR
IC S
UC
TIO
N,
IN K
PA
0
20
40
60
80
100
RA
INF
AL
L,
IN M
M
100 cm
2 4 6 8 10Water table depth, L (m)
4
8
12
ET
lim (
mm
/da
y)
d=1.2m
d=0.4m
bare
30 CM
-20
0
20
40
60
80
100
12/2
9/99
1/29
/00
2/29
/00
3/29
/00
4/29
/00
5/29
/00
6/29
/00
7/29
/00
8/29
/00
9/29
/00
10/2
9/00
11/2
9/00
12/2
9/00
1/29
/01
2/28
/01
MA
TR
IC S
UC
TIO
N,
IN K
PA
0
20
40
60
80
100
RA
INF
AL
L,
IN M
M
Rainfall
Bare
Gamma Grass
Mixed Trees
PET Spring-Summer
(water-limited evapotranspiration)
PET Autumn-Winter
(energy-limited evapotranspiration)
Conclusions (1)
• Evaporation from soils s a complex multi-physics process involving heat, liquid, and
vapour flux
• We tend to simplify the process by modelling liquid water flow only (hence squeezing
multi-physical processes into the boundary condition
• Potential evaporation from free water remains constant during evaporation
• Potential evaporation from soil decreases as degree of saturation decreases and
suction increases
• The soil hydraulic system has limited capacity to transfer water to the atmosphere
• The transition to potential (energy-limited) regime to actual (water-limited) regime
occurs when the soil hydraulic system cannot accommodate anymore the evaporative
demand of the atmosphere
Conclusions (2)
• The driving mechanisms of water extraction from vegetated and bare soil are
essentially the same
• Water-limited evapotranspiration may differ significantly due to the different mode of
extraction (under steady-state and most of all under transient state)
• Differences are significant and in favour of vegetation in the water- limited regime
• Differences are less significant but in favour of bare soil in the energy-limited regime