fundamentals of soil water - atmosphere interaction · 2016-11-06 · fundamentals of soil water -...

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