Biometeorology, ESPM 129
Lecture 34 Soil Physics, Moisture Transfer in Soils, Part 2
November 24, 2010
Instructor: Dennis Baldocchi
Professor of Biometeorology
Ecosystem Science Division
Department of Environmental Science, Policy and Management
345 Hilgard Hall
University of California, Berkeley
Berkeley, CA 94720
Topics
1. Theory, Moisture Transfer
a. Moisture transfer, Darcy’s Law and the Richard’s Equation
Soil Release curve.
c. Force Restore models, Bucket models
d. Role of different boundary conditions
e. Evaporation models
f. CO2 diffusion models
2. Observations, Moisture profiles
a. Seasonal patterns
b. Influence of soil texture
3. Soil Evaporation
a. measurements
b. model calculations
34.1 INTRODUCTION
The soil is the reservoir for moisture that is available to roots. How much moisture it
contains and the rate that water is lost from the soil depends on its texture,
physical/hydraulic capacity and the activity of plant contained within it and the extent of
their root system. The presence or absence of moisture and active biology in soils affects
how it weathers (Jenny 1994).
To understand changes in soil moisture we must start with an understanding of the
sources and losses of water.
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Biometeorology, ESPM 129
Sources of water to the biosphere from the atmosphere include precipitation, snow (in the
later form of melt water), condensation/dew, fog interception and stemflow. Sources of
water from the soil include capillary rise from the water table, direct axis of the water
table by roots and hydraulic lift of moisture by roots.
Losses of water from the biosphere include losses to the atmosphere and to deep soil
layers. Losses to the atmosphere include direct evaporation of surface water, evaporation
from the soil matrix, sublimination of snow, plant transpiration. Losses through the soil
column involve transport via perculation and saturated flow and unsaturated flow. Flow
through the soil column is vertical and lateral, a distinctly three-dimensional process.
Fog
Interception/
StemFlow
Evaporation/
Transpiration
Evaporation
Precipitation/
Condensation
Evaporation/
Sublimation
Snow Melt
snow/ice
Surface Water
Perculation/
saturated flow
Hydraulic Lifting
by roots
Wetting Front
unsaturated
flow
Capillary Rise
Water Table
Figure 1 Flows of moisture in and out of soil column. Adapted from Tindall and Kunkel (1998)
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Biometeorology, ESPM 129
A thermodynamic quantity, soil water potential, is used to describe soil hydrodynamics.
The total soil water potential is the amount of work done, per unit quantity of pure water,
to transport an infinitesimal amount of water from a pool of pure water. The process is
isothermal and a reference pressure (Tyndall and Kunkel, 2000).
The components of soil water potential for shallow soils are:
Gravitational potential: The force of gravity exerted on a water column produces the
gravitation potential. The gravitation potential is related to the work done to transport
water from one pool to another, as when lifting a column of water up the xylem of a tree:
 g   l gz
Turgor or pressure potential:
Water potential exerted by the pressure or weight of water
p
P
w
e
Vapor potential:  v  Rv T ln( )
es
R is the universal gas constant
Matric potential: It is water potential due to attraction between water and soils.
Adhesive and cohesive forces bind water to soil particles (Campbell and Norman, 1999).
These interactions reduce the potential of water, giving it a negative sign.
 m ~ aw b
Osmotic potential: Osmotic potential arises from the dilution of solutes in water, eg
salts, sugars etc. For the osmotic potential to drive water flow, a semi-permeable
membrane must separate two bodies of water, such as cells, and pools of water.
 o  cvRT
c is the concentration,  is the osmotic coefficient and  is the number of ions per mole
(Campbell and Norman 1998).
The total water potential is thus:
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Biometeorology, ESPM 129
   p  o  g  m
Which components are significant or not depends on the system we are studying (e.g.
Baver et al, 1976).
Water potential is expressed as energy (kg m2 s-2) per unit volume (m-3), giving it units
of kg m-1 s-2, which is equivalent to pressure units, or force per unit area, kg m s-2 m-2).
E/V = F x/V = F/A=P
In other instances pressure potentials may be normalized by water density
P
 |mass 
-1 -2
3
-1
 w (kg m s ) * (m kg )
We know the mass of water is 1000 kg = m3 and water is incompressible, we can
substitute mass for density, effectively 1 J/kg = 1 kPa
Soil-Water-Plant Relations (apolplast)
   o  m
Osmotic and matric potential are important for plant-water relations and water movement
in the apoplast and through the xylem. Gravimetric potential is negligible as the suction
needed to raise water, typically 1 m is less than 0.1 bar.
Organisms, cells (symplast):
   p  o
Inside cells turgor potential (eg pressure potential) and osmotic potentials are most
important
Unsaturated Flow:
   g  m
Gravitational and matrix potential are dominant
Saturated Flow
   p  g
Pressure and gravity are the main components driving water flow in saturated soils. The
pressure term includes overburden effects. At a point a point beneath the water table, the
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pressure potential is equal and opposite to the gravitational potential. Osomotic potential
is important only if there are solutes in the water.
0   p  g;
 p   g
Flow in the field
   p  g  m
This terminology considers mixed flow in the saturated and unsaturated zones. The
pressure term is zero above the water table. The matrix potential is zero below the water
table.
Physical Properties
The relative humidity of the soil pores was evaluated from thermodynamic principles as:
g
  exp(
)
(7
RwT
g is the acceleration due to gravity,  is the capillary potential, Rw is the gas constant for
water vapor and T is absolute temperature.
Many popular pedo-transfer functions exist that relate soil water potential and
volumetric soil moisture content. Early relations were reported by Gardner et al and
Clapp and Hornberger (Clapp and Hornberger 1978) that fitted power functions:
  sat
 b
s
(8
for sand,  equals 0.1 m3 m-3, b is 4.05, and  is -0.121 m. .
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Biometeorology, ESPM 129
Soil Water Potential (m),
10000
1000
100
10
sand
loam
clay
1
0.1
0.0
0.2
0.4
0.6
0.8
1.0
Volumetric Soil Moisture Content
Figure 2 Soil moisture retention using equation of Gardner
Table 1 Important physical and hydraulic properties of soils include their textural fractions
(silt/clay/sand, the hydraulic conductivity (K) and the volumetric water contents at field capacity (0.33 bar) and wilting point (-15 bars) (Rawls et al., 1992; Campbell and Norman, 1998)
Texture
silt
clay
Sand
loamy sand
sandy loam
Loam
silt loam
sandy clay loam
clay loam
silty clay loam
sandy clay
silty clay
clay
0.05
0.12
0.25
0.4
0.65
0.13
0.34
0.58
0.07
0.45
0.20
0.03
0.07
0.10
0.18
0.15
0.27
0.34
0.33
0.40
0.45
0.60

J/kg
0.7
0.9
1.5
1.1
2.1
2.8
2.6
3.3
2.9
3.4
3.7
b
1.7
2.1
3.1
4.5
4.7
4
5.2
6.6
6
7.9
7.6
6
Ks
kg s m-3
0.0058
0.0017
0.00072
0.00037
0.00019
0.00012
0.000064
0.000042
0.000033
0.000025
0.000017

M3 m-3
0.09
0.13
0.21
0.27
0.33
0.26
0.32
0.37
0.34
0.39
0.4

m3 m-3
0.03
0.06
0.1
0.12
0.13
0.15
0.2
0.32
0.24
0.25
0.27
Biometeorology, ESPM 129
Next we see how different soils range between field capacity and wilting point. Note how
peat soils become stressed at vary high water contents!
Volumetric water content (m3 m-3)
1.0
0.8
0.6
field capacity
permanent wilting point
0.4
0.2
0.0
d sand loam loam oam loam loam clay loam clay
Sanm
L cl ay
y andy clay
silt sandlyty clay
silty
s andy
loa
si
s
7
clay
peat
Biometeorology, ESPM 129
Soil moisture release curve
160
Water potential (-bar)
140
Data from dew point
Data from DANR
120
100
80
60
40
20
0
0
10
20
30
40
Gravimetric soil water content (%)
Figure 3 Data from Tonzi Ranch, Liukang Xu and ddb
The theory of van Genuchten (Van Genuchten 1980; van Genuchten and Sudicky 1999)
is more general and falls in the form of:
   resid 
 sat   resid
[1  ( ) n ]m
It is able to better mimic the S shape that is observed with soil water retention curves.
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Biometeorology, ESPM 129
1000
silt loam
Van Genuchten
Head (cm)
100
10
1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Volumetric Water Content
Figure 4 Soil moisture retention curve. Computed using the relation of van Genuchten (1980)
b. Theory and Concepts
Flux of Water
Flux density (mass per unit area per unit time) is related to a density (M/V) times a
velocity, F   u , (kg m-2 s-1). We can also express the flux density using flux-gradient
theory where the flux is related to the product of a density (kg m-3) times a diffusivity, D
(m2 s-1), times a driving gradient (m-1), e.g in terms of volumetric water content (m3 m-3).
Fm   Dmass
 (mwater / V )
 (  w )
  D
z
z
The time rate of change of a density (mass or energy per unit volume) equals the flux
F

 
divergence of that density,
t
z
The time rate of change of mass per unit volume (kg m-3) (s-1) is kg m-3 s-1
F

w
  m ; kg m-3 s-1
t
z
The flux density can be described as:

Fm    w D ( )
; kg m-2 s-1.
z
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Re-substitution of the flux equation into the conservation equation produces:
w
F

   
     w  Dh
t
z
z  z 
If normalized by w we have a simple budget equation for volumetric water content:


t

F
w
z

   
Dh
z  z 
When studying water transfer, water potential tends to be the currency defining the
driving gradient.
Here, the flux density of Energy is a product of the energy density (J/V) times a velocity:
(J m-3) m s-1 =J m-2 s-1= (kg m2 s-2) m-2 s-1 = kg s-3 = Pa m s-1
Fw   D ( )

z
If I was to start from first principles I’d define a flux in terms of water potential
F   D
d
dz
And solve for water potential with the advection-diffusion equation, or conservation
budget
 )
2
F  ( D

z  D   D  



t
z
z
z z
z 2
A difficulty/complexity/confusion arises because:
1. the continuity equation solves for water potential, but soil moisture tends to be measured in terms of volumetric water content and inputs and outflows are measured in terms of velocity per unit area, eg mm d‐1. 2. The water transfer equations in the hydraulic literature are typically expressed in terms of a conductivity, K, instead of and diffusivity, D, which is counter‐intuitive based on equations in the atmospheric literature. 3. Different flux equations have different units of K (m s‐1 vs kg s m‐3). 10
Biometeorology, ESPM 129
4. Some flux equations express the driving potential is terms of head, h, rather than pressure and others express the driving potential in terms of Energy normalized by mass, which produce pressure units too. Darcy’s Law
One of the earliest and most applicable theories of soil moisture through soils is attributed
to Henry Darcy (1803-1858), a French scientist working for Napoleon (Philip 1995)
Darcy performed experiments on water transfer through a bed of sand and derived what
came to be known as Darcy’s Law. The volume of water passing through a bed of sand
per unit time is a function of the cross-sectional area, A, the thickness of the bed, L, the
depth of water on top of the bed, h, and the hydraulic conductivity of the sand, K:
QK
Ah
L
In terms of a flux density
q
h
Q V

K
(volume of water crossing a known area in a given time)
A At
L
The hydraulic conductivity is defined from experiment as:
K
VL
Ath
Civil and environmental engineers and soil scientists often measure head, h, instead of
water potential, so Darcy’s law is often presented in this form:
q   K ( )
 K ( )
q  K
( h  z )
z
h
 K ( h)
z
Ph
P
z
  K ( t  g )
s
s
s
The time rate of change of volumetric water content can be expressed in terms of Fick’s
second law.
 
h

K ( h)  K ( h)  S
t z
z
LM
N
OP
Q
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We add a term S, which represents sources and sinks, such as root uptake or hydraulic lift
by plants.
Boundary conditions for assessing water content in soils the top and bottom layers. At
the top soil content depends upon infiltration, transpiration and evaporation from the soil.
At the bottom, one needs to know the water table.
Darcy's Law is valid for low Reynolds numbers, where flow is laminar and viscous forces
dominate (Re <1). Darcy’s law may be violated in karst limestone and dolomite soils.
Darcy’s law also fails in dense clay soils, which have low permeability.
The conventional form of Darcy’s Law does not work in the Vadose Zone, the
unsaturated soil zone. Water transfer through unsaturated soils is very complex, as
soils contain both small and large size pores. Large pores will form from cracks,
earthworm or animal holes, and roots. Macro pores can be routes of preferential flow.
In unsaturated soils large pores drain first, increasing the potential difference. But a
paradox is observed as water flow rates diminish because small pores, which remain wet,
are pore conductors of water. The path of water transfer also becomes more tortuous as
the soil dries. One can only account for diminishing flow rates with increasing water
potential gradients by the co-occurrence of a diminishing hydraulic conductivity, K.
Let’s start with the fundamental equations describing the flux of water and re-derive the
budget equations and numerical schemes with a consistent form of K that can be
evaluated with standard pedotransfer functions and applied correctly to the conservation
equation.
Darcy’s Law says a flux of water is defined as the product of a hydraulic conductivity
and a water potential gradient.
Discharge, volume of water per unit time in terms of a pressure or head gradient is:
Q
 P
h
A
  KA
 z
z
h is head, m P
z h
w g
P is pressure, F/A, Pa = kg m‐1 s‐2 A is area, m2 K is conductivity, m s‐1  is permeability, m2  is dynamic viscosity, kg m‐1 s‐1 12
Biometeorology, ESPM 129
Q is the Discharge of water per unit area (m s‐1)  g h
h
Q
 P
 Kh
 w
q 
z
A
 z
 z
 g
K h  w ; (m2) (kg m-3) (m s-2)(kg-1 m s) = m s-1

And mass flow of water is
 g h
h
 P
  w Kh
  w w
F  wq   w
z
 z
 z
kg m-2 s-1 = (kg m-3) (m2) (kg-1 m s) (kg m-1 s-2) (m-1) Campbell defines a mass flux as follows:
d |mass
F   K
dz
(kg s m-3) (J/kg) (m-1) = J s m-4 = kg m2 s-2 = kg m-2 s-1
Consequently, the hydraulic conductivities used by Campbell, K , and Darcy, K h , are
not equal and have different units.
Let’s try and resolve and reconcile this difference. First, let’s examine all the forms of
the flux equation. They may have different drivers and rate factors, but they all must
produce a mass flux with units of kg m-2 s-1:
F  wq   w
 g h
h
 P
  |mass
 w Kh
  w w
  w w
z
 z
 z
 z
From these equations we can deduce several definitions for hydraulic conductivities. The
conductivity used by Campbell has units of kg m-3 s and must only be applied against a
water potential normalized by mass. This conductivity can be defined as:
K   w  w

; (kg m-3)(kg m-3) m2 (kg-1 m s) = kg m-3 s

The hydraulic conductivity used in the Richards Equation and has velocity units and must
only be applied against head. It can be deduced to equal:
Kh 
 w g
; (m2) (kg m-3) (m s-2) (kg-1 m s) = m s-1

So it is clear that K  K h .
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Biometeorology, ESPM 129
Next let’s relate them to one another:
K   w  w
K  K h
 g

 Kh  w


w
g
So the mass flux equation can be re-written in terms of Kh as:
F  wq   w Kh
 |mass
  |mass
h
  K
 Kh w
g
z
z
z
It is also critical to recognize that one cannot estimate the mass flux by multiplying Kh
times the pressure or water potential gradient:
F  wq   w Kh
P

 w Kh
z
z
But I want to express the fluxes in terms of  and used pedotransfer functions for Kh.
With substitution it should be
F  wq   w
 w g h
K 
K 
h
 P
  w Kh
  w
  w h
 h
 z
 z
 w g z
z
g z
So the form of the water flux density equation I want to work with is:
Fw  
K h 
; (m s-1)(m-1 s2)(kg m-1 s-2)(m-1) = kg m-2 s-1
g z
Water Flow in Soil, case 1: No Evaporation
The history of modeling soil moisture contains a mix of theories based on units of head,
pressure, specific energy etc. Soil physicists start with Darcy’s Law and Richard’s
Equation and define a hydraulic conductivity that has units of m s-1 and they measure
water in terms of head (m). For saturated flow one can convert head to pressure potential
by multiplying head by the density of water times the acceleration due to gravity,
 head  w h g (Pa).
For example it is standard practice to define the time rate of change of volumetric soil
moisture with the Richard’s equation for unsaturated flow in terms of head, h:
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Biometeorology, ESPM 129
   m  g ~  w g h   w g z

~ h z
w g
Here I am considering z as the depth relative to the soil surface. So z is negative when it
is below the ground and it is positive when above the ground.
In some applications you’ll see
   m  g ~  w g h   w g z
This is valid when z is represented as the distance from the soil surface.
Hence, we see two forms of the Richard’s equation. This form if z is the distance from
the surface:
 d h  
 (h  z )   
h
h
  


  K h ( h)
  K h (h)  K h (h)    K h (h)(  1) 

t dh t z 
z  z 
z
z
 z 

And this form if z is negative when below the surface.
 d h  
 (h  z )   
h
h
  


  K h ( h)
  K h (h)  K h (h)    K h (h)(  1) 

t dh t z 
z  z 
z
z
 z 

Here Kh(h) is the hydraulic conductivity and it has units of velocity (m s-1). The
equation is very complex and non-linear because the conductivity is a function of the
d
driving force, head. Typically, pedo-transfer functions are used to compute K and
dh
using models by such authors as Brooks and Corey (1964), Campbell (1980) and van
Genutchen et al (1980).
Because we are interested in assessing changes in soil moisture due to inputs from
rainfall (mm d-1) and loses from evaporation (mm d-1) there are advantages towards
evaluating changes in volumetric water content with time, but as a function of water
potential.
w
F

 m
t
z (kg m-3 s-1)
The mass flux can be computed as:
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Biometeorology, ESPM 129
Fm  
K h 
g z
And on substitution we have
w
  K h ( ) 
 [
]
t z
z
g
And normalizing by water density produces:
  K h ( ) 
 [
]
t z  w g z ; (m s-1)(kg m-1 s-2) (m-1) (m3 kg-1) (s2 m-1) (m-1)= s-1
We want to explicitly consider effects of matrix and gravitational potential. Replacing
 ~  m  g
The gravitational potential is computed as:
 g    w gz
But I need to be careful about sign convention because I am working in z units where 0 is
the soil surface and depths below the surface have a negative sign. Gravitational water
potential is positive if above a reference level and is negative if below a reference level.
 g   w gz
And rearranging terms produces
  K h ( ) d m K h ( ) d g
 [

]
t z  w g dz
 w g dz
 K h ( ) d m  w g K h ( )d z

[
]
z  w g dz
w g
dz
 K h ( ) d m
 K h ( ) ]
[
z  w g dz
s-1
Simplifying to:
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Biometeorology, ESPM 129
  K h ( ) d m
 [
 K h ( ) ]
t z  w g dz
The solution to the soil moisture problem is complicated because the conductivity is a
function of the soil moisture content and it is a very non-linear function of moisture.
Expanding terms leads to:

1 [ K h ( m )

t  w g
z
 m
z ]

K h ( m )

z
1 K h ( m ) d m K h ( )  2 m K h ( )


dz
w g
 w g z 2
z
z
or

1 K h ( m ) d m K h ( )  2 m K h ( )



t  w g
z
dz
 w g z 2
z
Finally, we want to solve the equations for water potential, so we apply the chain rule
operation:
 d 

t d t
d
d is computed from the slope of the water retention curve
 K ( ) d m
d 
 [ h
 K h ( ) ]
d t z  w g dz
d  m
1 K h ( m ) d m K h ( )  2 m K h ( )



d m t
w g
z
dz
 w g z 2
z
And regrouping terms
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Biometeorology, ESPM 129
d  m  K h ( )  2 m K h ( m ) 1 d m



[
 1]
d m t
t
 w g z 2
z
 w g dz
One can also apply the chain rule to the distance derivatives and produce a conservation
equation solely in terms of volumetric water content.
  

z
z 
1 

  m K h
[Kh
]

t  w g z
z 
z
Numerical Solution
 Define soil layer system in finite element with Resistance-Analog for flux of
water between layers

Flux is defined by the product of a conductance and the gradient of water
potential

The conductance will be expressed in terms of water potential
o
K ( m )  K s (
 e 23/ b
)
m

System of non-linear equations is defined and must be solved in time and space

Define a mass balance that equals zero, change in storage equals flux divergence

If we define the soil moisture equation in the Backward difference mode, we
solve for water potential at a future time step, which is a function of itself. This
non-linear equation needs to be solved via iteration.

Newton-Raphson iteration method may be needed to solve for root of equation for
the mass balance, which yields a water potential at node I and time.

System of simultaneous equations must be solved with Matrix Algebra and use of
Thomas Algorithm

Use Implicit solution, for time j+1, which is more stable numerically since the soil
moisture budget contains numerous non-linear terms.
Schematic for Finite Difference solution to water column problem.
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Biometeorology, ESPM 129
 (0)
U (0)
f (1)
K (1)
 (1)
U (1)
 ( I  1)
U ( I  1)
f ( I  1)
 (I )
U (I )
f (I )
K (I )
U, infiltration
f, flux
P, pressure
potential
K, conductivity
 ( I  1)
U ( I  1)
 (M )
U (I )
f (M )
U ( M  1)
K (M )
 (M  1)
Figure 5 Resistance scheme from Campbell
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Biometeorology, ESPM 129
Hydraulic Conductivity
Soil moisture quantities of interest include the hydraulic conductivity and the saturated
hydraulic conductivity.
Hydraulic Conductivity (K, cm/d)
Saturated Hydraulic Conductivity: the value when the soil is saturated with water.
K ( m )  K s (
K ( )  K s (
e 23/ b
)

 2 b 3
)
s
Campbell computes the saturated hydraulic conductivity as a function of the clay (c) and
silt (s) fractions:
Ks  0.004(
13
.
b
)1.3b exp( 6.9c  3.7 s)
20
Biometeorology, ESPM 129
Hydraulic Conductivity, kg s m-3
1000
100
sand
loam
clay
10
1
0.1
0.01
0.001
0.0001
0.00001
0.000001
0.0
0.1
0.2
0.3
0.4
0.5
0.6
volumetric water content
Figure 6 Hydraulic conductivity as a function of soil testure and volumetric water content. Computed
with the Equations of Campbell.
Field Measurements of Soil Moisture
Soil moisture can be sampled with an array of sensors and methods. Some are conducive
to hourly and daily measurements. Others require intensive sampling and may only be
done weekly, when investigators visit the field site.
Figure 6 shows the seasonal variation of soil moisture at a grassland field site. Soil
moisture is greatest during the winter rain season, and exhibits weekly cycles of wetting
and drying. After the cessation of the rains, the soil moisture content diminishes to about
5%.
21
Biometeorology, ESPM 129
0.5
Daily Sampling
Weekly Sampling
Soil moisture (cm3 cm-3)
0.4
0.3
0.2
0.1
0.0
0
50
100
150
200
250
300
350
day
Simulations
Variation in soil moisture after rainfall to a relatively dry soil column
22
Biometeorology, ESPM 129
0.01
0.1
-z (m)
T = 1 hour
T = 2 hour
T = 12 hour
T = 24 hours
T = 48 hours
T= 96 hours
T = 192 hours
T= 382 hours
1
10
0.0
0.1
0.2
0.3
0.4
0.5
volumetric water content
Change in soil moisture after rain from a soil that has a distributed root system and
is transpiring and evaporating.
ppt 25 mm/day on day 1; et 5 mm/day; Root Uptake
0.01
0.1
-z (m)
T = 1 hour
T = 2 hour
T = 12 hour
T = 24 hours
T = 48 hours
T= 96 hours
1
10
0.0
0.1
0.2
0.3
volumetric water content
23
0.4
0.5
Biometeorology, ESPM 129
Gaseous Diffusion Through Soils
Diffusion and production of trace gases (eg CO2) in the soil

c
2c
 D 2 
z
t
ε is the free air porosity of the soil, D is the molecular diffusivity and φ is the production
rate (Amundson, 1998).
At steady state there is a balance between the production rate and the diffusion flux
divergence:
D
2c
 
z 2
Using a no flux lower boundary condition (dc/dz =0 at L and C at 0 equals the
atmospheric value, then
C

Ds
( Lz 
z2
)  Catm
2
Such a calculation is useful for giving one some idea of the build up of CO2 in the soil
based on the production rate, depth and molecular diffusivity.
24
Biometeorology, ESPM 129
25
Biometeorology, ESPM 129
Figure 7after Suwa et al 2004 GCB
Soil Evaporation
Soil evaporation experiences two stages as soils dry. The initial stage is steady and at the
highest rates. It is determined by the atmospheric demand. The lower the demand, the
longer the first stage period lasts. The second stage is the drying stage. Here the supply
of water is limited by the soil.
Changing water content with time forms a linear log-log plot between water content and
time in days.
Ecumulative  k t  t0
One can measure daily soil evaporation with such simple tools as a weighing lysimeter
Figure 8 Weighing lysimeter in a ponderose pine forest, Kelliher and Baldocchi
The following figure shows the amount of moisture lost by the soil on successive days
26
Biometeorology, ESPM 129
mini-lysimeter
young PIPO
0.14
Soil Evaporation (mm/d)
0.12
0.10
0.08
0.06
0.04
0.02
0.00
190
192
194
196
198
200
202
204
206
208
Day
Figure 9 Time course of soil evaporation with a mini-lysimeters, data of Kelliher, Baldocchi and
Law.
Alternatively the eddy covariance method can be used to measure fluxes of water and
heat exchange above the soil under forest stands.
27
Biometeorology, ESPM 129
Energy Flux Density (W m-2)
125
Jack pine
Forest Floor
D144-259
100
Rn
LE
75
H
50
G
25
0
-25
0
4
8
12
16
20
24
Time (hours)
energy flux density (W m-2)
300
250
Ponderosa pine
forest floor
200
Rn
150
LE
100
H
50
Gsoil
0
-50
0
4
8
12
16
20
24
Time (Hour)
Figure 10 Energy Exchange at the floor of ponderosa pine and deciduous forest
With the eddy covariance method we have information with high temporal resolution on
surface fluxes. Since the method is non-disruptive, it provides information that cannot be
detected with lysimeters or chambers. For example we can investigate the relation
between evaporation and available energy.
28
Biometeorology, ESPM 129
70
jack pine forest
ponderosa pine forest
60
LE forest floor (W m-2)
temperate deciduous
forest
50
40
30
20
10
0
-10
-50
0
50
100
150
200
Rn -G: forest floor (W m-2)
Figure 11 Non-linear response of forest floor evaporation as a function of available energy
Note that it saturates! Interactions between the time scale of equilibrium evaporation and
the repeat time of coherent structures have been hypothesized to cause this effect
(Baldocchi and Meyers, 1991).
Modeling Soil Evaporation
Soil evaporation models are key components of meteorological SVAT models, as well as
climate, hydrology and biogeochemistry models.
Mahfouf and Noilhan (1991) surveyed the literature and present algorithms for bare soil
evaporation and have evaluated several algorithms. The most basic form is:
LE g 
L(hqs (T )  qa )
Ra  RSoil
h is the relative humidity at the site of evaporation, q is the mixing ratio, qs is the
saturated mixing ratio at T, Ra is the soil boundary layer resistance and Rsoil is the soil
resistance.
Note that the end point of the moisture potential is not as simples as with a leaf, which we
assume is saturated. We need to assess the relative humidity in soils, which is not one as
the soil dries.
29
Biometeorology, ESPM 129
Alpha and Beta schemes are used to evaluate h or Rs.
Alpha Model:
E

Ra
( qsat ( Ts )  qa )
  min( 1,
1.8 wg
wg  0.30
)
Beta models
Esoil 

Ra
 ( hqsat ( Ts )  qa )
case 1: (Sun)
Ra

;
Ra  Rsoil
h=1
w
Rsoil  3.5( sat )2 .3  33.5
wg
case 2: (Camillo and Gurney)
Rsoil  4104( ws  wg )  805
h  exp(
g
)
RwT
Mihailovic et al. (1995) tested several schemes and report that this simple one of
Deardorff(1978) was one of the best
h  min(1,  /  fc )
The soil resistance was computed using the algorithm of Sun (1972) as:
Rsoil  33.5   /  sat
2.38
Yet, in order to apply these algorithms, on must first simulate the microenvironment of
the overlaying canopy, that derives the environmental forcing variables sensed at the soil
surface and can drive the soil model.
30
Biometeorology, ESPM 129
A simple algorithm for the soil surface resistance is used by Kustas and Norman in a two
layer model. The equation is:
Ra 
1
a  bu
a is 0.004 m s-1 and b is 0.012. u is the wind speed near the soil.
We found it necessary to account for thermal stratification when computing the soil
surface aerodynamic resistance (Ra). We used the method reported by Daamond and
Simmonds (1996).
zd 2
(ln(
))
z0
Ra 
(1   )
k 2u
where
5 g( z  d )(Ts  Ta )

Ta u2
 is -0.75 when  is greater than zero and is -2 when  is less than zero.
Model Tests
Using the theory of Daamon and Simmonds we are able to match our eddy flux
measurements. Without this refinement we fail!
31
Biometeorology, ESPM 129
300
Rnet (W m-2)
250
Ponderosa Pine
Forest Floor
D187-205, 1996
200
150
measured
calculated
100
50
0
-50
LE (W m-2)
75
50
25
0
-25
200
-2
H (W m )
150
100
50
G (W m-2)
0
150
125
100
75
50
25
0
-25
-50
-75
0
4
8
12
16
20
24
Time (hours)
Figure 12 Measurements and calculations of energy exchange from the forest floor of a ponderosa
pine with CANVEG. Mean diurnal patterns
Sensitivity Tests
1. Litter
32
Biometeorology, ESPM 129
Litter depth, 0.01 m
litter depth, 0.02 m
Ponderos Pine
Forest Floor
-2
H (W m )
-2
LE (W m )
-2
Rn (W m )
litter depth, 0.05 m
300
250
200
150
100
50
0
-50
60
50
40
30
20
10
0
150
125
100
75
50
25
0
-2
G (W m )
150
100
50
0
-50
0
4
8
12
16
20
24
Time (hours)
Figure 13 Role of litter layer thickness on mass and energy exchange from the soil of a forest
2. Role of Stability on energy balance partitioning
Role of Leaf area and photosynthetic capacity on Partitioning soil and vegetation
evaporation
33
Biometeorology, ESPM 129
0.30
0.25
QE,soil/QE
0.20
0.15
0.10
0.05
0.00
0
20
40
60
80
100 120 140 160 180 200
LAI * Vcmax
Figure 14 Fraction of soil evaporation to total canopy evaporation as leaf area index and
photosynthetic capacity change. Soil evaporation can range from 5 to 20 % of total, assuming moist
soil surface.
34
Biometeorology, ESPM 129
Ra=f(stability)
Ra: neutral
300
250
200
150
100
50
0
-50
70
60
50
40
30
20
10
0
-2
H (W m )
-2
LE (W m )
-2
Rn (W m )
Ponderos Pine
Forest Floor
150
125
100
75
50
25
0
-2
G (W m )
200
150
100
50
0
-50
0
4
8
12
16
20
24
Time (hours)
Figure 15 Role of thermal stratification on heat and energy exchange at the floor of a forest. By
ignoring thermal convection, the surface gets too hot and radiates away more long wave energy,
thereby diminishing Rn.
Resources
Soil Water Characteristics Hydraulic Properties Calculator
http://www.bsyse.wsu.edu/saxton/soilwater/
35
Biometeorology, ESPM 129
Resources
Global Texture and Water Holding Capacities
http://www-eosdis.ornl.gov/SOILS/Webb.html
References:
Amundson et al. 1998. Geoderma. 82: 83-114.
Baldocchi, D.D. and T.P. Meyers. 1991. Trace gas exchange at the floor of a deciduous
forest: I. evaporation and CO2 efflux. Journal Geophysical Research,
Atmospheres. 96: 7271-7285
Baldocchi, D.D., B.E. Law and P. Anthoni. 2000. On measuring and modeling energy
fluxes above the floor of a homogeneous and heterogeneous conifer forest.
Agricultural and Forest Meteorology 102, 187-206.
Baver,L.D., Garnder. W.H. and Gardner, W.R. 1971. Soil Physics. Wiley and Sons.
Campbell, G.S. 1985.Soil Physics with Basic: transport models for soil-plant systems.
Elsevier.
Kabat, P., Hutjes, R. and Feddes, R.A. 1997. The scaling characteristics of soil
parameters: from plot scale heterogeneity to subgrid parameterization. Journal of
Hydrology.
Kabat, P. and Beekma, J. Water in the unsaturated zone. In. Drainage Principles and
Application. Wageningen, The Netherlands. Pp. 383-434
Monteith, J.L. and M.H. Unsworth. 1990. Principles of Environmental Physics. E.A.
Arnold.
Parlange, M., Hopmans JW (eds). 1999. Vadose Zone Hydrology
Tindall, J.A. and J.R. Kunkel. 1999. Unsaturated Zone Hydrology for Scientists and
Engineers. Prentice Hall.624 pp.
Tietje, O. and Tapkenhinrichs. 1993. Evaluation of Pedo-Transfer Functions. Soil
Science Soc. America. 57, 1088-1095.
van Geunuchten. M. and D.R. Nielsen. 1985. On describing and predicting hydraulic
properties of unsaturated soils. Annales Geophysicae. 3, 615-628.
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Campbell, G.S. and J.M. Norman 1998. An Introduction to Environmental Biophysics.
Springer Verlag, New York. 286 p.
Clapp, R.B. and G.M. Hornberger 1978. Empirical Equations for Some Soil HydraulicProperties. Water Resources Research. 14:601-604.
Jenny, H. 1994. Factors of Soil Formation: A System of Quantitative Pedology. Dover
Press.
Philip, J.R. 1995. Desperately Seeking Darcy in Dijon. Soil Sci Soc Am J. 59:319-324.
Van Genuchten, M.T. 1980. A Closed-Form Equation for Predicting the Hydraulic
Conductivity of Unsaturated Soils. Soil Science Society of America Journal.
44:892-898.
van Genuchten, M.T. and E.A. Sudicky 1999. Recent advances in Vadose zone flow and
transport modeling. In Vadose Zone Hydrology Eds. M. Parlange and J.W.
Hopmans. Oxford Press, New York, pp. 155-193.
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