10 water in soil-rev 1

Water in the Soil and Subsoil
Soils, Groundwater, Water, Ice

Jay Stratton Noller, Nekia at Springbrook, 2009

Riccardo Rigon

Friday, September 10, 2010

“The medium is the message”
Marshall McLuham



Objectives:

•To define what soils are
•To introduce aquifers and groundwater
•To define the dynamics of flows in soils, to introduce Darcy’s Law and
the elements that appear in the it, and to verify the validity of the
continuum hypothesis

•To verify the presence of multiple scales in the soil and subsoil hydrology

3

Riccardo Rigon



What are soils?

The term soil indicates the surface portion of the ground which is composed
of inorganic and organic matter in proportions that vary from place to
place. It is characterised by its own chemical and mineralogical composition,
its own atmosphere, its own particular hydrology, and specific flora and
fauna.

This meaning is different from the more common usage indicating the
surface of the ground upon which we walk.

4

Riccardo Rigon



Soil

• it is a natural and living body, resulting from

http://www.directseed.org/soil_quality.htm
long evolutionary processes dictated by a
series of environmental factors (climate,
parent material, morphology, vegetation,
living organisms)
• it is an essential element of terrestrial
ecosystems
• it is in dynamic equilibrium: it interacts
• it is a non-renewable natural resource

5

Giacomo Sartori



Microflora and
Fauna of the Soil
Cunningham/Saigo, Environmental Science, 1999

6

Giacomo Sartori



An Overview
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Riccardo Rigon



Pedogenesis
The process by which the parent rock forms soils is called pedogenesis. This
process is a series of physical, chemical, and biological actions that contribute to
structuring soils in horizons.

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8

Riccardo Rigon



Pedogenesis
Substratum/Regolith/Soil

Regolith :=
parent
material

Substratum := Rock

http://gis.ess.washington.edu/grg/courses05_06/ess230/lectures/257,1,Soils 9

Giacomo Sartori



Definitions
parent material (or regolith): the unconsolidated material (non-coherent,
slightly coherent, or pseudo-coherent) from which soils result

substratum: the consolidated rock formation, from which the soil
originated, or which indirectly affected the soil formation, or which did not
affect the soil formation at all, as in the case of a limestone substratum
covered with a thin layer of allochthonous material (glacial...) from which
the soil resulted

soil: surface layer of the earth’s surface that shows signs of alteration and
is affected by living organism

10

Giacomo Sartori



Definitions

profile: vertical section of the soil that evidences the sequence
of soil horizons

horizons: strata of varying thickness within a soil profile,
normally with a disposition that is nearly parallel to the soil
surface, that have homogeneous characteristics with regards to
colour, texture, structure, pH, carbonates etc.

11

Giacomo Sartori



Pedogenesis
Substratum/Regolith/Soil

thin layer of soil on parent
material composed of glacial
very thin layer of soil on material
basaltic rock (= substratum) 12

Giacomo Sartori



Horizons

O horizon

O horizon
A horizon

real soil
A horizon
B horizon layer
real soil
layer
B horizon
C horizon

C horizon

unconsolidated rock

rocky substratum rocky substratum

13

Riccardo Rigon



Parent materials

Classification of Minerals
that compose rocks
Base elements Minerals
Silicon + Oxygen Silicates
Silicon + Aluminium + Hydrogen + Oxygen Aluminosilicates
Aluminium + Oxygen + Hydroxyl Metal Oxides and Hydroxides
Iron + Oxygen + Hydroxyl Metal Oxides and Hydroxides
Manganese + Oxygen + Hydroxyl Metal Oxides and Hydroxides
Cation + Carbon + Oxygen Carbonates
Cation + Sulphur + Oxygen Sulphates

from: prof. Dazzi, University of Palermo

THE MOST IMPORTANT ARE:
- silicates and aluminosilicates
- carbonates
14

Giacomo Sartori



The Pedogenesis Timescale
The formation of soils generally requires a very long time.

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15

Riccardo Rigon



Pedogenesis

An example of soil evolution
Mineral fragments and Organic
Organic matter organic matter matter
Humus

A Horizon A Horizon

B Horizon

Altered rock Parent material Parent material Parent material
C Horizon C Horizon

Unaltered rock Unaltered rock Unaltered rock Unaltered rock

Organic matter The evolved soil
The rock begins to Horizons
facilitates the sustains dense
disintegrate form
disintegration process vegetation

As time passes the profile gets deeper and more differentiated
(= more distinct horizons) 16

Giacomo Sartori



What are soils?
From the Hydrologist’s point of view we can extend the concept to include
everything that derives from the alteration/demolition of the bedrock (regolith)
and also the products of repeated phases of erosion/accumulation/alteration
etc. even in the absence of well-defined horizons.
Photo by Onorevoli, 2009

Valentini Refuge
Sella Pass,
17

Riccardo Rigon



The Colour of the Soil
The colour of the soil is indicative of some very important characteristics.

• Characteristics that can be deduced from the colour:
- dark colours: a lot of organic material
- light colours: little organic material
- brown colours: clay-humus complexes (originating from worms)
- reddish colours: iron oxides in anhydrous form (warm climates)
- yellowish colours: iron oxides in hydrated form (wet climates)
- green or blue colours: permanent hydromorphic condition (no O2)

- dappled colours: temporary hydromorphic condition (water-table
oscillations)

18

Giacomo Sartori



The Colour of the Soil

NB: certain strongly coloured rocks (e.g. Gardena Sandstone, Scaglia Rossa
limestones, both red), pass on their colouring to the soil; the coloration of
the soil in these cases is therefore hereditary rather than due to alteration
processes.

19

Giacomo Sartori



Soil Classification
There are numerous attempts of soil classification. These are not, of
course, based solely on hydrological characteristics, but on a series of
general characteristics. The classification criteria are based on the analysis
of:
After Erika Micheli, 2004

- the soil formation factors

- the processes involved

- the horizons, properties e materials present

There results a soil taxonomy.

e.g. http://eusoils.jrc.it/
20

Riccardo Rigon



Soil Classification

For example, the European Union has set about classifying soils; that is
to say, any material present within the first 2 m of the land surface with
the exclusion of:

- living creatures,

- continuous glacier areas not covered by other material,

- bodies of water deeper than 2 m

http://eusoils.jrc.it/
21

Riccardo Rigon



Soil Classification

The definition, therefore, includes:

- exposed (naked) rock

- paved urban soils

And it must contain, when available, information on the spatial structure of
the soils.

http://eusoils.jrc.it/
22

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Soil Classification

For more information see Micheli (2004)

23

Riccardo Rigon



Soil Classification
Folic horizon (from Latin folium, leaf)
consists of well-aerated organic material

Defined SOM % content, and thickness
24

Riccardo Rigon



Soil Classification
Albic horizon (from Latin albus, white)
is a light-coloured subsurface horizon

Defined colour and thickness
25

Riccardo Rigon



Soil Classification

Spodic horizon (from Greek spodos, wood ash)
is a subsurface horizon that contains illuvial amorphous substances composed
of organic matter and Al, or of illuvial Fe.

Defined pH, color or chemical
requirements, and thickness 26

Riccardo Rigon



Soil Classification
Reducing conditions (defined by low rH or presence of Fe++,
iron sulphide or methane), that appear in staging colour patterns

Abrupt textural change
(defined by clay content and increase) 27

Riccardo Rigon



Soil Classification
Step 1
Diagnostics

Folic horizon

Albic Horizon

Spodic horizon

Reducing conditions
Staging colour patterns

Abrupt textural change

28

Riccardo Rigon



Soil Classification

Step 2
The key

Soils….!

!
!
Other soils having a spodic horizon
starting within 200 cm of the mineral
soil surface
" PODZOLS

29

Riccardo Rigon



Soil Classification
1. Soils with thick organic layers: HISTOSOLS

30

Riccardo Rigon



Soil Classification
2. Soils with strong human influence
Soils with long and intensive agricultural use: ANTHROSOLS
Soils containing many artefacts: TECHNOSOLS

31

Riccardo Rigon



Soil Classification

3. Soils with limited rooting due to shallow permafrost or stoniness
Ice-affected soils: CRYOSOLS
Shallow or extremely gravelly soils: LEPTOSOLS

32

Riccardo Rigon



Soil Classification
4. Soils influenced by water
Alternating wet-dry conditions, rich in clays: VERTISOLS
Floodplains, tidal marshes: FLUVISOLS
Alkaline soils: SOLONETZ
Salt enrichment upon evaporation: SOLONCHAKS

Groundwater affected soils: GLEYSOLS

33

Riccardo Rigon



Soil Classification
5. Soils set with Fe/Al chemistry
Allophanes or Al-humus complexes: ANDOSOLS
Cheluviation and chilluviation: PODZOLS
Accumulation of Fe under hydromorphic conditions: PLINTHOSOLS

Low-activity clay, strongly structured: NITISOLS
Dominance of kaolinite and sesquioxides: FERRALSOLS

34

Riccardo Rigon



In 3D

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Riccardo Rigon



landscape
Soils and Landscape

soil

As a result of the actions of the different factors of soil
evolution, different parts of the landscape have different soils
characterised by different soil profiles. 36

Giacomo Sartori



Soil Map

37

Giacomo Sartori



38

Riccardo Rigon



Soil + Water
After Warric, 2003

39

Riccardo Rigon



What is there below the soil?

40

Riccardo Rigon



Below the soil: aquifers

http://ga.water.usgs.gov/edu/earthgwaquifer.html
41

Riccardo Rigon



Below the soil: unconfined aquifers
After de Marsily, 1986

42

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43

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44

Riccardo Rigon




45

Riccardo Rigon



Below the soil: confined aquifers

46

Riccardo Rigon




47

Riccardo Rigon




48

Riccardo Rigon



Flusso nei suoli

Water moves through the pores of unconsolidated sedimentary
formations and through the cracks and fissures of rock

49

Alberto Bellin



Basic Notation
The column of soil with water
Mass Volume

Aria
Air
Mag Vag

Ghiaccio
Ice

Mi Vi

Ms Vs
Acqua (Liquida)
Water (liquid)

Mlw Vlw

Soil
Suolo
Msp Vsp

50

Riccardo Rigon



Basic Notation

Ms = Mag + Mw + Mi + Msp
Mtw = Mv + Mw + Mi

51

Riccardo Rigon



Basic Notation

Mtw = Mv + Mw + Mi

Mass of water

51

Riccardo Rigon



Basic Notation

Mtw = Mv + Mw + Mi

Mass of water
Mass of vapour

51

Riccardo Rigon



Basic Notation

Mtw = Mv + Mw + Mi

Mass of water
Mass of vapour

Mass of liquid
water 51

Riccardo Rigon



Basic Notation

Mtw = Mv + Mw + Mi

Mass of water
Mass of ice
Mass of vapour

Mass of liquid
water 51

Riccardo Rigon



Basic Notation

Mass of soil

Mtw = Mv + Mw + Mi

Mass of water
Mass of ice
Mass of vapour

Mass of liquid
water 51

Riccardo Rigon



Basic Notation

Mass of soil
Mass of air

Mtw = Mv + Mw + Mi

Mass of water
Mass of ice
Mass of vapour

Mass of liquid
water 51

Riccardo Rigon



Basic Notation
Mass of soil
Mass of soil particles

Mass of air

Mtw = Mv + Mw + Mi

Mass of water
Mass of ice
Mass of vapour

Mass of liquid
water 51

Riccardo Rigon



Basic Notation
The volumes are indicated with the same indices as the masses

Vs = Vag + Vw + Vi + Vsp
Vtw = Vv + Vw + Vi

52

Riccardo Rigon



Basic Notation
Soil particle density

Msp
ρsp :=
Vsp
Soil bulk density

Msp Msp
ρb := =
Vs Vag + Vw + Vi + Vsp

53

Riccardo Rigon



Basic Notation
Volume fraction of condensed water in soil pores (liquid water +ice)

Vw + Vi
θcw :=
Vag + Vw + Vi + Vsp

Volume fraction of liquid water in soil pores

Vw
θw :=
Vag + Vw + Vi + Vsp

54

Riccardo Rigon



Basic Notation

Volume fraction of frozen water (ice) in soil pores

Vi
θi :=
Vag + Vw + Vi + Vsp

55

Riccardo Rigon



Basic Notation
Soil porosity

Vag + Vw + Vi
φs :=
Vag + Vw + Vi + Vsp

Effective soil porosity

Vag + Vw
φse := = φs − θi
Vag + Vw + Vi + Vsp

56

Riccardo Rigon



Basic Notation
Relative saturation of soil

θw
Ss =
φse
Effective saturation of soil

Ma la ‘r’ a pedice e`
θw − θr giusta?
Se =
φse − θr

57

Riccardo Rigon



Soil Texture
Particle Diameter (logarithmic scale) [mm]

Very Very
Clay fine Fine Medium Coarse coarse
Silt Gravel
Sand

Fine Coarse
Clay Silt Gravel
Sand

Fine Coarse
Clay Silt Gravel
Sand

Fine Medium Coarse
Gravel
Clay
Silt Sand

Fine Medium Coarse
Clay Silt Gravel
Sand

58

Riccardo Rigon



Texture
Dimensional relationships
sand-silt-clay
Clay:
<0,002 mm

Sand:
2- 0,050 mm Silt:
0,050- 0,002 mm

59

Giacomo Sartori



Texture

Sand
• sand: 2 mm - 0.05 mm (50 – 2000
μm)

• visible without a microscope
• either rounded or angular in form
• the grains of quartz are white,
other minerals have different
colours
• however, dark colours, reds and
yellows, can be caused by Fe, Al,
and Mn coatings that cover the
grains

60

Giacomo Sartori



Texture

Silt

silt: 0.050 - 0.002 mm (2-50 μm)

microscopic image
(non visible to the naked eye)

61
from: prof. Vittori, Università di Bologna

Giacomo Sartori



Texture

Clay

• argilla: <0.002 mm (<2μm)

• large surface area

• negatively charged

62

Giacomo Sartori



Texture

Clay

• being colloids, they can be found in the soil
either dispersed or flocculated (Ca2+ is a
flocculating agent, Na+ is deflocculating): a
very important role in soil aggregation

• some clays have the capacity to absorb
water between platelets, which can bring
about large changes in volume during the
wetting-drying cycle: expanding clays (e.g.
montmorillonite, typical of vertisols)

63

Giacomo Sartori



Soil Texture

Pe
rce
clay

nta
y
cla

ge
ht

we
eig

igh
ew

ts
tag

ilt
en

silty
rc

clay
Pe

sandy
clay
silty
clay loam clay
loam

loam
sandy loam silt loam

loamy silt
sand sand

Percentage weight sand

64

Riccardo Rigon



Soil Texture

Clay
% particles < d

Loam
Unevenly distributed
sand
Evenly distributed
sand

Particle diameter in mm (d)
65

Riccardo Rigon



Soil Structure

Individual clay platelet
interaction (rare)

Individual silt or sand
particle interaction

Clay platelet face-face
group interaction

Partly discernible
Clothed silt or sand particle interaction

66

Riccardo Rigon



Soil Structure

Individual clay platelet
interaction (rare)

Intra-elemental pores

Individual silt or sand

Clay platelet face-face
group interaction

Partly discernible
Clothed silt or sand particle interaction

66

Riccardo Rigon



Soil Structure

Connectors

Connectors
Regular
aggregations

Irregular
aggregations

Interweaving
bunches

Clay
matrix Granular
matrix

67

Riccardo Rigon



Soil Structure

Intra-assemblage
pores
Connectors

Connectors
Regular
aggregations

Irregular
aggregations

Interweaving
bunches

Clay
matrix Granular
matrix

67

Riccardo Rigon



Soil Structure

Intra-assemblage
pores
Connectors

Connectors
Regular
aggregations

Irregular
aggregations

Interweaving Inter-assemblage pores
bunches

Clay
matrix Granular
matrix

67

Riccardo Rigon



Representative Elementary Volume (REV)
porosity-structure-texture of soil

68

Riccardo Rigon



And therefore?

What are the consequences of this complexity on hydrology?

What experiments can be carried out in order to characterise the behaviour
of soils?

Which laws of motion does water in the soil and in aquifers obey?

With what instruments can we characterise these equations?

How can we resolve these equations?

And, by all means, which are the relevant problems that we need to solve
with the equations that we will find?

69

Riccardo Rigon


Darcy, Buckingham, Richards

Jay Stratton Noller, Great Basin Soil #2, 2009

Riccardo Rigon



Objectives:

•To define the flow dynamics of groundwater and introduce Darcy’s Law,
the elements that appear in the law, and verify the continuum
hypothesis.

•To verify the presence of multiple scales in soil and subsoil hydrology.

•To introduce the Richards equation, Buckingham’s law, water retention
curves

71

Riccardo Rigon



Darcy’s experiment

72

Riccardo Rigon



Q ∝ (A/l)(h2 − h1 )

Q (h2 − h1 )
Jv = =K
A l

(h2 − h1 ) dh
=
l dz
dh
Jv = K
dz

73

Riccardo Rigon



K is the hydraulic conductivity

dh
Jv = K
dz

Furthermore, the pressure
at the base of the column
is:

p = ρw g(h − z)

Therefore:
p
h=z+
ρw g
74

Riccardo Rigon



p
h=z+
ρw g

It should be observed
that h is the hydraulic
load (the energy per unit
volume) of a volume of
water set at height z and
subjected to a relative
pressure p

75

Riccardo Rigon



Hydraulic conductivity

Studies subsequent to Darcy’s experiment have shown that the hydraulic
conductivity is, in non-homogeneous soils, a vector with components along
three preferential directions

¯
K = (Kx , Ky , Kz )

And it is therefore a tensor in the direction of an arbitrary system of
coordinated axes (x,y,z)

76

Riccardo Rigon



Hydraulic Conductivity
upper limit
(matrix deformation)

lower limit
of validity
77

Riccardo Rigon



Laminar flow in a capillary tube: Poiseuille’s Law

q

q γ (2Rh )2
A A
q=v ω= ∇h a
8µ

R=2RH
section A_A

a

78

Riccardo Rigon




The hydraulic conductivity is, generally, a tensor. However, for the sake
of simplicity, we shall consider it a scalar. This factor is a lumped
parameter that pulls together all the physical factors that interact with
the motion of a fluid in a porous medium:

- the mechanical properties of the fluid

- and the geometric characteristics of the medium

79

Riccardo Rigon




The mechanical properties of the fluid:

- kinematic viscosity
µ [L T
2 −1
]
- fluid density
ρ [ML −3
]
- (or their combination, the dynamic viscosity) ν [M(LT) −1
]

The geometric characteristics of the medium

- the scale of the particles (the structure of d [L]
the pores)

- the geometric form of the pore factor
N
80

Riccardo Rigon



Given that K has units with dimensions of velocity, it follows that hydraulic
conductivity can be expressed with a monomial that combines the quantities
seen in the previous slide raised to appropriate powers:

[N d ν ] = [T L
a b −1
]

From where, equalising the exponents, there results:

K = N d2 ν −1 ≡ k ν −1

k is called the permeability. It depends solely on the geometry of the
medium
81

Riccardo Rigon



Darcy scale

82

Alberto Bellin



and saturation

83

Alberto Bellin



and saturation
The hydraulic conductivity varies greatly in
space. Organised connections can be
observed between areas with high
conductivities that create preferential
paths.

84

Alberto Bellin



Heterogeneity at intermediate scales

85

Alberto Bellin



Heterogeneity at intermediate scales

86

Alberto Bellin



Heterogeneity at regional scale
Regional scale

87

Alberto Bellin



At different scales

different measuring instruments are used 88

Alberto Bellin



Due to heterogeneities

effective quantities are necessary 89

Alberto Bellin



Conservation of mass
A conservation law of a quantity is expressed as follows:

The variation of the quantity in the control volume is equal to the sum of
all the quantity that enters less all the quantity that leave from the
surface of the control volume summed algebraically with the quantity
that is transformed to other things.

∂Jv
Jv ∆y ∆z (Jv + ∆x)∆y ∆z
∂x

90

Riccardo Rigon




When speaking of conservation of mass the conservation law becomes:

The variation in the mass of water in a volume is equal to the amount of
incoming water reduced by the amount of water that leaves from the surface of
the volume, less the water that is transformed (e.g. to ice or vapour)

∂Jv
Jv ∆y ∆z (Jv + ∆x)∆y ∆z
∂x

91

Riccardo Rigon



If, momentarily, we omit phase changes, then the variation in water mass
per unit time can be written:

dMw d(ρw Vw )
=
dt dt

Assuming the density of water to be constant:

dMw d(Vw )
= ρw
dt dt

and, in general, rather than considering the variations in mass flows we
consider the volumetric variations

92

Riccardo Rigon




The volumetric variation is then usually expressed in terms of the
dimensionless water content:

d(Vw ) Vs d(Vw ) dθw
= = Vs
dt Vs dt dt

where it is assumed that the soil volume Vs is constant in time

93

Riccardo Rigon



The continuity equation
The flow of water through the surfaces of an elementary volume of size

∆x ∆y ∆z

∂Jv
Jv ∆y ∆z (Jv + ∆x)∆y ∆z
∂x

is the sum of three contributions, one for each pair of faces
94

Riccardo Rigon



For example, for the faces parallel to the yz plane, as can be deduced
from the figure, we have:
∂Jv
(Jv + ∆x)∆y ∆z − (Jv )∆y ∆z
∂x

∂Jv
Jv ∆y ∆z (Jv + ∆x)∆y ∆z
∂x

95

Riccardo Rigon



Repeating the operation for the other two pairs of faces, and having
carried out the appropriate subtractions, there results:
∂Jv ∂Jv ∂Jv
∆x∆y ∆z + ∆x∆y ∆z + ∆x∆y ∆z
∂x ∂y ∂z

∂Jv
Jv ∆y ∆z (Jv + ∆x)∆y ∆z
∂x

that is to say, if the volume is infinitesimal, the divergence theorem.
96

Riccardo Rigon




∂Jv
Jv ∆y ∆z (Jv + ∆x)∆y ∆z
∂x
divergence theorem

∂θw
= ∇ · Jv (ψ)
∂t

97

Riccardo Rigon




∂θw
= ∇ · Jv (ψ)
Richards, 1931

∂t

98

Riccardo Rigon




∂θw
= ∇ · Jv (ψ)
Richards, 1931

∂t

Variation in water
content of the soil
per unit time

98

Riccardo Rigon




∂θw
= ∇ · Jv (ψ)
Richards, 1931

∂t

Variation in water
content of the soil
per unit time

Divergence of the
volumetric flow
through the surface of
the infinitesimal
volume 98

Riccardo Rigon



Darcy-Buckingham Law
Buckingham, 1907, Richards, 1931

Jv = K(θw )∇ h

99

Riccardo Rigon




Jv = K(θw )∇ h

Volumetric flow
through the surface
of the infinitesimal
volume 99

Riccardo Rigon




Jv = K(θw )∇ h

Hydraulic conductivity X
gradient of the load

Volumetric flow
through the surface
of the infinitesimal
volume 99

Riccardo Rigon



The hydraulic load is an energy per unit volumeand it is measured in units
of length

h=z+ψ
Richards, 1931

100

Riccardo Rigon



of length

h=z+ψ
Richards, 1931

Hydraulic load

100

Riccardo Rigon



of length

h=z+ψ
Richards, 1931

Hydraulic load

Gravitational field

100

Riccardo Rigon



of length

h=z+ψ
Richards, 1931

Hydraulic load

Gravitational field

Capillary forces - pressure

100

Riccardo Rigon



A flash-back to non-saturated soil

Liquid phase

Biphasic fluid
Humid air
(gas)

Solid matrix

101

Alessandro Tarantino



Capillarity

cos θ
pw = −2γ
r
h

pa=0

pw=0
p =0
uw0

102




Capillarity

cos θ
pw = −2γ
r
h

pa=0

pw=0
p =0
uw0

If the contact angle is θ90°, the liquid enters the capillary tube and is said to
wet the surface. It rises within the tube to height that is inversely
proportional to the radius of the tube 102




Capillary effects in soils

particle
interstitial water

103





pw 0 The contact angle is less than 90°

The meniscus is concave in the
direction of the air and the pressure
particle
interstitial water is negative

103





pw 0 The contact angle is less than 90°

The meniscus is concave in the
direction of the air and the pressure
particle
interstitial water is negative

T The particles are held together by
surface tension and the negative
pressure
-pw

103




The soil is like a complex system of capillary tubes

104




A flash-back to non-saturated soils
suction

Unsaturated soil

105




suction

Unsaturated soil

A non-saturated soil is capable of absorbing water
in the liquid and gaseous phase. This property is
called suction
105




saturation

S=1
pw 0

Suction is generated solely by the curvature of the menisci at
the surface. The soil is saturated. The air is dissolved in the
water.
106




in proximity of saturation

0.85-0.90 S 1
pw 0

Suction is generated by the curvature of the menisci
at the surface and the air cavities between the pores.
The liquid phase is continuous, the gaseous phase is
discontinuous.
107




still less saturated

0-0.1 S 0.85-0.90
uw 0

Suction is generated by the curvature of the menisci in the
pores. There are parts of the volume that are saturated and
parts where menisci form. Both phases are continuous.

108




residual saturation

S 0-0.1
pw 0

Suction is generated by menisci in the pores, and the
menisci form in contact with the particles. The
gaseous phase is continuos, the liquid phase is
discontinuous.

109




The Relation between Saturation
(water content) and Suction
It is the Soil Water Retention Curve (SWRC) and it illustrates the
various states of water in soil.
Chahal and Yong, 1965

Soil-water retention curve for Soil-water retention curve for
initially saturated coarse silt. initially saturated coarse silt 110

Riccardo Rigon




Saturated soil

S Nearly saturated soil
1

Partially saturated soil

Residual saturation

ln (s)
111





S
1
sb = value of air intake
sr = residual suction
Sr = degree of residual saturation

Sr

sb sr ln (s)
112




The SWRC is not a curve
Hydraulic Hysteresis

The hypothesis is made that the
solid matrix is rigid

S
1
Drainage curve

“Scanning curves”

Infiltration curve

ln (s)
113




But usually we ignore this
and think of the SWRC as a function

∂θ(ψ) ∂θ(ψ) ∂ψ ∂ψ
= ≡ C(ψ)
∂t ∂ψ ∂t ∂t

Hydraulic capacity of
the soil

114

Riccardo Rigon



The hydraulic capacity of soil is proportional
to the pore-size distribution E’ giusto pore-size
distribution?

O forse e’ pore
distribution?

SWRC

Derivative

Water content
115

Riccardo Rigon



to the pore-size distribution

r
θw = φs f (r) dr
0

116

Riccardo Rigon




r
θw = φs f (r) dr
0

Porosity

116

Riccardo Rigon




r
θw = φs f (r) dr
0

Porosity

Pore-size distribution, i.e.
how much of Vs is
occupied by pores of a
certain size 116

Riccardo Rigon




r
θw = φs f (r) dr
0

2γ 2γ
ψ=− =⇒ r = −
r ψ

117

Riccardo Rigon




r
θw = φs f (r) dr
0

2γ 2γ
ψ=− =⇒ r = −
r ψ

suction potential 117

Riccardo Rigon




r
θw = φs f (r) dr
0

2γ 2γ
ψ=− =⇒ r = −
r ψ

energy per unit
surface area


Riccardo Rigon




r
θw = φs f (r) dr
0

2γ 2γ
ψ=− =⇒ r = −
r ψ

energy per unit
surface area pore radius


Riccardo Rigon




r
θw = φs f (r) dr
0

2γ 2γ
ψ=− =⇒ r = −
r ψ
− 2σ
ψ f (r(ψ)
θw = φs 2
dψ
0 ψ
118

Riccardo Rigon



− 2γ
ψ f (r(ψ)
θw = φs dψ
0 ψ2

119

Riccardo Rigon



− 2γ
ψ f (r(ψ)
θw = φs dψ
0 ψ2

=⇒

dθw
= φf (r(ψ))
dψ

119

Riccardo Rigon




Where the following identity was used:

b(x)
d db(x) da(x)
s(y) dy = s(b(x)) − s(a(x))
dx a(x) dx dx

120

Riccardo Rigon



dθw α m n(α ψ)n−1
= −φs (θr + φs )
dψ [1 + (α ψ)n ]m+1

SWRC

Derivative

Water content
121

Riccardo Rigon



Parametric forms of the SWRC

Equation Author Validity

Infiltration

Drainage

Redistribution

Drainage

122

Riccardo Rigon


10 water in soil-rev 1

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