10 water in soil-rev 1

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

Riccardo Rigon

Jay

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Friday, September 10, 2010

“The medium is the message” Marshall McLuham


Water in the Soil and Subsoil

Riccardo Rigon

Objectives:

3

•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



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.

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• it is a natural and living body, resulting from

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

Soil


http://www.directseed.org/soil_quality.htm

http://www.directseed.org/soil_quality.htm


Giacomo Sartori

6

Microflora and Fauna of the Soil

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Riccardo Rigon

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An Overview

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Riccardo Rigon

Pedogenesis

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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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Giacomo Sartori

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PedogenesisSubstratum/Regolith/Soil

Substratum := Rock

Regolith := parent material

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





Giacomo Sartori

10

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



Giacomo Sartori

11

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.

Definitions



Giacomo Sartori

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thin layer of soil on parent material composed of glacial materialvery thin layer of soil on

basaltic rock (= substratum)

PedogenesisSubstratum/Regolith/Soil



Riccardo Rigon

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Horizons

O horizon

rocky substratum

unconsolidated rock

real soil

layer

O horizon

rocky substratum

real soil

layer

A horizon

B horizon

C horizon

A horizon

B horizon

C horizon



Giacomo Sartori

14

Parent materials

Classification of Mineralsthat compose rocks

from: prof. Dazzi, University of Palermo

THE MOST IMPORTANT ARE:

- silicates and aluminosilicates

- carbonates

Base elements MineralsSilicon + Oxygen SilicatesSilicon + Aluminium + Hydrogen + Oxygen AluminosilicatesAluminium + Oxygen + Hydroxyl Metal Oxides and HydroxidesIron + Oxygen + Hydroxyl Metal Oxides and HydroxidesManganese + Oxygen + Hydroxyl Metal Oxides and HydroxidesCation + Carbon + Oxygen CarbonatesCation + Sulphur + Oxygen Sulphates



Riccardo Rigon

The Pedogenesis Timescale

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The formation of soils generally requires a very long time.

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Giacomo Sartori

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Pedogenesis

An example of soil evolution

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

The rock begins to disintegrate

Organic matter facilitates the

disintegration process

Horizons form

The evolved soil sustains dense

vegetation

Unaltered rock

Altered rock Parent material

Unaltered rock Unaltered rock Unaltered rock

Parent material Parent material

C Horizon C Horizon

Organic matterOrganic matter

Humus

Mineral fragments and organic matter

A Horizon A Horizon

B Horizon



Riccardo Rigon

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.

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What are soils?



Giacomo Sartori

18

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)



Giacomo Sartori

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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.

The Colour of the Soil



Riccardo Rigon

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:

- 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

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http://eusoils.jrc.it



Riccardo Rigon

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

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Soil Classification





Riccardo Rigon

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





Riccardo Rigon

For more information see Micheli (2004)

23

Soil Classification



Riccardo Rigon

Folic horizon (from Latin folium, leaf)

consists of well-aerated organic material

Defined SOM % content, and thickness24

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Soil Classification



Riccardo Rigon

Defined colour and thickness

Albic horizon (from Latin albus, white)

is a light-coloured subsurface horizon

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Soil Classification



Riccardo Rigon

Defined pH, color or chemical

requirements, and thickness

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.

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Soil Classification



Riccardo Rigon

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

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Soil Classification



Riccardo Rigon

Folic horizon

Albic Horizon

Spodic horizon

Reducing conditions

Staging colour patterns

Abrupt textural change

Step 1

Diagnostics

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Soil Classification



Riccardo Rigon

Step 2

The key

Soils….!

!

!

Other soils having a spodic horizon

starting within 200 cm of the mineral

soil surface

" PODZOLS

29

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Soil Classification



Riccardo Rigon

1. Soils with thick organic layers: HISTOSOLS

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Soil Classification



Riccardo Rigon

2. Soils with strong human influence

Soils with long and intensive agricultural use: ANTHROSOLS

Soils containing many artefacts: TECHNOSOLS

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Soil Classification



Riccardo Rigon

3. Soils with limited rooting due to shallow permafrost or stoniness

Ice-affected soils: CRYOSOLS

Shallow or extremely gravelly soils: LEPTOSOLS

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Soil Classification



Riccardo Rigon

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

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Soil Classification



Riccardo Rigon

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

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Soil Classification



Riccardo Rigon

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In 3D



Giacomo Sartori

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soil

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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.

landscape



Giacomo Sartori

37

Soil Map



Riccardo Rigon

38

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Riccardo Rigon

Soil + Water

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Riccardo Rigon

What is there below the soil?

40



Riccardo Rigon

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

Below the soil: aquifers

41





Riccardo Rigon

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Riccardo Rigon

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Riccardo Rigon

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Alberto Bellin

49

Flusso nei suoli

Water moves through the pores of unconsolidated sedimentary

formations and through the cracks and fissures of rock



Riccardo Rigon

Acqua (Liquida)

Suolo

Ghiaccio

Aria

Mass Volume

Ms Vs

Vag

Vi

Vlw

VspMsp

Mlw

Mi

Mag

Basic NotationThe column of soil with water

50

Air

Ice

Water (liquid)

Soil



Riccardo Rigon

Ms = Mag + Mw + Mi + Msp

Mtw = Mv + Mw + Mi

Basic Notation

51



Riccardo Rigon

Mass of water


Mtw = Mv + Mw + Mi

Basic Notation

51



Riccardo Rigon

Mass of water


Mtw = Mv + Mw + Mi

Mass of vapour

Basic Notation

51



Riccardo Rigon

Mass of water


Mtw = Mv + Mw + Mi

Mass of vapour

Mass of liquid water

Basic Notation

51



Riccardo Rigon

Mass of water


Mtw = Mv + Mw + Mi

Mass of vapour


Mass of ice

Basic Notation

51



Riccardo Rigon

Mass of water


Mtw = Mv + Mw + Mi

Mass of vapour


Mass of ice

Mass of soil

Basic Notation

51



Riccardo Rigon

Mass of water


Mtw = Mv + Mw + Mi

Mass of vapour


Mass of ice

Mass of soil

Mass of air

Basic Notation

51



Riccardo Rigon

Mass of water


Mtw = Mv + Mw + Mi

Mass of vapour


Mass of ice

Mass of soil

Mass of air

Mass of soil particles

Basic Notation

51



Riccardo Rigon

Vs = Vag + Vw + Vi + Vsp

Vtw = Vv + Vw + Vi

The volumes are indicated with the same indices as the masses

52

Basic Notation



Riccardo Rigon

Soil particle density

53

Soil bulk density

ρsp :=Msp

Vsp

ρb :=Msp

Vs=

Msp

Vag + Vw + Vi + Vsp

Basic Notation



Riccardo Rigon

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

54

Volume fraction of liquid water in soil pores

θw :=Vw

Vag + Vw + Vi + Vsp

θcw :=Vw + Vi

Vag + Vw + Vi + Vsp

Basic Notation



Riccardo Rigon

55

Volume fraction of frozen water (ice) in soil pores

θi :=Vi

Vag + Vw + Vi + Vsp

Basic Notation



Riccardo Rigon

56

Soil porosity

Effective soil porosity

φs :=Vag + Vw + Vi

Vag + Vw + Vi + Vsp

φse :=Vag + Vw

Vag + Vw + Vi + Vsp= φs − θi

Basic Notation



Riccardo Rigon

Relative saturation of soil

57

Effective saturation of soil

Ss =θw

φse

Se =θw − θr

φse − θr

Basic Notation

Ma la ‘r’ a pedice e` giusta?



Riccardo Rigon

Soil Texture

58

Clay

Clay

Clay

Clay

Clay

Silt

Silt

Silt

Silt

SiltSand

Sand

Sand

Sand

Sand

Gravel

Gravel

Gravel

Gravel

Gravel

Particle Diameter (logarithmic scale) [mm]

Very fine

Very coarseFine Medium Coarse

Fine Coarse

CoarseFine

Fine

Fine

Coarse

Coarse

Medium

Medium



Giacomo Sartori

59

Sand:2- 0,050 mm Silt:

0,050- 0,002 mm

Clay:<0,002 mm

Texture

Dimensional relationships sand-silt-clay



Giacomo Sartori

60

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



Giacomo Sartori

61

silt: 0.050 - 0.002 mm (2-50 μm)

microscopic image(non visible to the naked eye)

from: prof. Vittori, Università di Bologna

Texture

Silt



Giacomo Sartori

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• argilla: <0.002 mm (<2μm)

• large surface area

• negatively charged

Texture

Clay



Giacomo Sartori

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• 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)

Texture

Clay



Riccardo Rigon

Soil Texture

64Percentage weight sand

Percentage weight silt

Perc

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eigh

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siltyclay

sandyclay

sandsiltloamy

sand

sandy loam loam

silt loam

clay loamsilty clay loam



Riccardo Rigon

65

Soil Texture

Particle diameter in mm (d)

% p

arti

cles

< d

Clay

Loam

Evenly distributed sand

Unevenly distributed sand



Riccardo Rigon

Individual clay platelet interaction (rare)

Individual silt or sand particle interaction

Clay platelet face-face group interaction

Clothed silt or sand particle interaction

Partly discernible particle interaction

66

Soil Structure



Riccardo Rigon

Individual clay platelet interaction (rare)

Individual silt or sand particle interaction

Clay platelet face-face group interaction

Clothed silt or sand particle interaction

Partly discernible particle interaction

Intra-elemental pores

66

Soil Structure



Riccardo Rigon

Regular

aggregations

Connectors

Connectors

Irregularaggregations

Interweaving bunches

Granular matrix

Claymatrix

67

Soil Structure



Riccardo Rigon

Regular

aggregations

Connectors

Connectors



Granular matrix

Claymatrix

Intra-assemblage pores

67

Soil Structure



Riccardo Rigon

Regular

aggregations

Connectors

Connectors



Granular matrix

Claymatrix

Intra-assemblage pores

Inter-assemblage pores

67

Soil Structure



Riccardo Rigon

Representative Elementary Volume (REV)

68

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soil



Riccardo Rigon

69

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?


Water in the Soil and SubsoilDarcy, Buckingham, Richards

Riccardo Rigon

Jay

Stra

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Noll

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Bas

in S

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, 20

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Riccardo Rigon

Objectives:

71

•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



Riccardo Rigon

Darcy’s experiment

72



Riccardo Rigon

Q ∝ (A/l)(h2 − h1)

Jv =Q

A= K

(h2 − h1)l

(h2 − h1)l

=dh

dz

Jv = Kdh

dz

73



Riccardo Rigon

Jv = Kdh

dz

K is the hydraulic conductivity

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

p = ρwg(h− z)

Therefore:

h = z +p

ρwg

74



Riccardo Rigon

h = z +p

ρwg

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

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)

Hydraulic conductivity

76



Riccardo Rigon

Hydraulic Conductivity

77

lower limit

of validity

upper limit

(matrix deformation)



Riccardo Rigon

q

R=2RH

A A

section A_A

a

q

Laminar flow in a capillary tube: Poiseuille’s Law

q = v ω =γ (2Rh)2

8µ∇h 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

- fluid density

- (or their combination, the dynamic viscosity)

The geometric characteristics of the medium

- the scale of the particles (the structure of the pores)

- the geometric form of the pore factor

d [L]N

µ [L2T−1]

ρ [ML−3]

ν [M(LT)−1]

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:

[Ndaνb] = [TL−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




Alberto Bellin

Darcy scale

82



Alberto Bellin

83

Hydraulic conductivity and saturation



Alberto Bellin

84

Hydraulic conductivity and saturation

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



Alberto Bellin

85

Heterogeneity at intermediate scales



Alberto Bellin

86

Heterogeneity at intermediate scales



Alberto Bellin

87

Heterogeneity at regional scale

Regional scale



Alberto Bellin

88

At different scales

different measuring instruments are used



Alberto Bellin

89

Due to heterogeneities

effective quantities are necessary



Riccardo Rigon

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∆y ∆z (Jv +∂Jv

∂x∆x)∆y ∆z

90

Conservation of mass



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)


∂x∆x)∆y ∆z

91




Riccardo Rigon

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

dMw

dt=

d(ρwVw)dt

Assuming the density of water to be constant:

dMw

dt= ρw

d(Vw)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:

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

93

d(Vw)dt

=Vs

Vs

d(Vw)dt

= Vsdθw

dt




Riccardo Rigon

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

∆x ∆y ∆z

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


∂x∆x)∆y ∆z

94

The continuity equation



Riccardo Rigon

For example, for the faces parallel to the yz plane, as can be deduced from the figure, we have:

(Jv +∂Jv

∂x∆x)∆y ∆z − (Jv)∆y ∆z


∂x∆x)∆y ∆z

95




Riccardo Rigon

Repeating the operation for the other two pairs of faces, and having

carried out the appropriate subtractions, there results:


∂x∆x)∆y ∆z

∂Jv

∂x∆x∆y ∆z +

∂Jv

∂y∆x∆y ∆z +

∂Jv

∂z∆x∆y ∆z

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

96




Riccardo Rigon


∂x∆x)∆y ∆z

97

∂θw

∂t= ∇ · �Jv(ψ)

divergence theorem




Riccardo Rigon

Ric

har

ds,

19

31

98

∂θw

∂t= ∇ · �Jv(ψ)




Riccardo Rigon

Variation in water content of the soil per unit time

Ric

har

ds,

19

31

98

∂θw

∂t= ∇ · �Jv(ψ)




Riccardo Rigon

Variation in water content of the soil per unit time

Divergence o f the v o l u m e t r i c f l o w through the surface of t h e i n f i n i t e s i m a l volume

Ric

har

ds,

19

31

98

∂θw

∂t= ∇ · �Jv(ψ)




Riccardo Rigon

Darcy-Buckingham Law

Bu

ckin

gh

am, 1

90

7, R

ich

ard

s, 1

93

1

99

�Jv = K(θw)�∇ h



Riccardo Rigon


Volumetric flow through the surface of the infinitesimal volume

Bu

ckin

gh

am, 1

90

7, R

ich

ard

s, 1

93

1

99




Riccardo Rigon


Volumetric flow through the surface of the infinitesimal volume

Hydraulic conductivity X gradient of the loadB

uck

ingh

am, 1

90

7, R

ich

ard

s, 1

93

1

99




Riccardo Rigon

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

h = z + ψ

Ric

har

ds,

19

31

100




Riccardo Rigon


h = z + ψ

Hydraulic load

Ric

har

ds,

19

31

100




Riccardo Rigon


h = z + ψ

Hydraulic load

Gravitational field

Ric

har

ds,

19

31

100




Riccardo Rigon


h = z + ψ

Hydraulic load

Gravitational field

Capillary forces - pressure

Ric

har

ds,

19

31

100




Alessandro Tarantino

A flash-back to non-saturated soil

Liquid phase

Humid air (gas)

Solid matrix

Biphasic fluid

101




102

Capillarity

uw<0

pw=0

pw=0

h

pa=0

pw = −2γcos θ

r




If the contact angle is θ<90°, 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

Capillarity

uw<0

pw=0

pw=0

h

pa=0

pw = −2γcos θ

r




particleinterstitial water

Capillary effects in soils

103






pw < 0 The contact angle is less than 90°

The meniscus is concave in the direction of the air and the pressure

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

is negative

The particles are held together by surface tension and the negative

pressure-pw

T


103





The soil is like a complex system of capillary tubes

104





Unsaturated soil

105

A flash-back to non-saturated soilssuction




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

Unsaturated soil

105

A flash-back to non-saturated soilssuction




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

A flash-back to non-saturated soilssaturation




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

A flash-back to non-saturated soilsin proximity of saturation




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

A flash-back to non-saturated soilsstill less saturated




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

A flash-back to non-saturated soilsresidual saturation



Riccardo Rigon

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.

Soil-water retention curve for initially saturated coarse silt

Soil-water retention curve for initially saturated coarse silt.

Ch

ahal

an

d Y

on

g, 1

96

5

110




Saturated soil

S

ln (s)

1Nearly saturated soil

Partially saturated soil

Residual saturation

111






S

ln (s)

1

Sr

sb sr

sb = value of air intake

sr = residual suction

Sr = degree of residual saturation

112






S

ln (s)

1

The hypothesis is made that the solid matrix is rigid

Drainage curve

Infiltration curve

“Scanning curves”

The SWRC is not a curve

Hydraulic Hysteresis

113



Riccardo Rigon

But usually we ignore thisand think of the SWRC as a function

∂θ(ψ)∂t

=∂θ(ψ)∂ψ

∂ψ

∂t≡ C(ψ)

∂ψ

∂t

Hydraulic capacity of the soil

114



Riccardo Rigon

The hydraulic capacity of soil is proportional to the pore-size distribution

115

SWRC

Derivative

Water content

E’ giusto pore-size distribution?

O forse e’ pore distribution?



Riccardo Rigon

116

θw = φs

� r

0f(r) dr




Riccardo Rigon

Porosity

116

θw = φs

� r

0f(r) dr




Riccardo Rigon

Pore-size distribution, i.e. how much of Vs is

occupied by pores of a certain size

Porosity

116

θw = φs

� r

0f(r) dr




Riccardo Rigon

117

θw = φs

� r

0f(r) dr

ψ = −2 γ

r=⇒ r = −2 γ

ψ




Riccardo Rigon

117

θw = φs

� r

0f(r) dr

ψ = −2 γ

r=⇒ r = −2 γ

ψ

suction potential




Riccardo Rigon

117

θw = φs

� r

0f(r) dr

ψ = −2 γ

r=⇒ r = −2 γ

ψ

suction potential

energy per unit surface area




Riccardo Rigon

117

θw = φs

� r

0f(r) dr

ψ = −2 γ

r=⇒ r = −2 γ

ψ

suction potential

energy per unit surface area pore radius




Riccardo Rigon

118

θw = φs

� r

0f(r) dr

ψ = −2 γ

r=⇒ r = −2 γ

ψ

θw = φs

� − 2σψ

0

f(r(ψ)ψ2

dψ




Riccardo Rigon

119

θw = φs

� − 2γψ

0

f(r(ψ)ψ2

dψ




Riccardo Rigon

119

θw = φs

� − 2γψ

0

f(r(ψ)ψ2

dψ

=⇒

dθw

dψ= φf(r(ψ))




Riccardo Rigon

d

dx

� b(x)

a(x)s(y) dy = s(b(x))

db(x)dx

− s(a(x))da(x)dx

Where the following identity was used:

120




Riccardo Rigon

121

dθw

dψ= −φs

α m n(α ψ)n−1

[1 + (α ψ)n]m+1(θr + φs)


SWRC

Derivative

Water content



Riccardo Rigon

122

Parametric forms of the SWRC

Equation ValidityAuthor

Infiltration

Drainage

Redistribution

Drainage



Riccardo Rigon

Which has five parameters:

123

Se ≡θw − θr

φs − θr=

� 11 + (αψ)n

�m

θr

φs

αnm

Parametric forms of the SWRCThe most used is Van Genuchten’s



Riccardo Rigon

(a)

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

0 0.2 0.4 0.6 0.8 1! (-)

po

ten

zia

le c

ap

illa

re, "

(h

Pa

)

n=1.0

n=1.4

n=2

n=3

n=4

n=6

n=10

n=15

m=0.1

(b)

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

0 0.2 0.4 0.6 0.8 1! (-)

po

ten

zia

le c

ap

illa

re, "

(h

Pa

)

n=0.4

n=0.6

n=1

n=2

n=6

m=1

Figura 2: Curve di ritenzione idrica per diversi valori del parametro n ed avendoposto m = 0.1 (a) e m = 1.0 (b).

8

Par

amet

ric

form

s of

the

SWR

C

124



Riccardo Rigon

125




Riccardo Rigon

126




Riccardo Rigon

Aft

er M

ual

em, 1

97

6

127

Even the hydraulic capacity varies with the water content of the soil !



Riccardo Rigon

K(Se) = KsSve

�f(Se)f(1)

�2

f(Se) =� Se

0

1ψ(x)

dx

Where v is an exponent expressing the connectivity between pores, evaluated by Mualem

for various soil types.

Aft

er M

ual

em, 1

97

6

PARAMETRIC FORMS OF THE HYDRAULIC CONDUCTIVITY

128



Riccardo Rigon

K = Ks Kr

Having defined the relative hydraulic conductivity:

ψ =1α

�S−1/m

e − 1�1/n

And expressed the suction in terms of van Genuchten’s expression::

The integral can be calculated:

Aft

er M

ual

em, 1

97

6

129




Riccardo Rigon

The integral can be calculated:

f(Se) =� Se

0

1ψ(x)

dx

resulting:

f(Se) = α

� Se

0

1(x−1/m − 1)1/n

dx

Aft

er M

ual

em, 1

97

6

130




Riccardo Rigon

which results in, having changed the variable,

f(Se) = α

� Se

0

1(x−1/m − 1)1/n

dx

f(Se) = α

� S1/me

0

mym−1

(y−1 − 1)1/ndy

x = ym

f(Se) = α m

� S1/me

0ym−1+1/n(1− y)−1/ndy

Aft

er M

ual

em, 1

97

6

131




Riccardo Rigon

f(Se) = α m

� S1/me

0ym−1+1/n(1− y)−1/ndy

The integral:

can be calculated numerically, as a function of the

Hypergeometric2F1 function (i.e. as in Mathematica).

If m=1-1/n (van Genuchten-Mualem model), then:

f(Se) = −α�1− S1/m

e

�m+ α (m = 1− 1/n)

132




Riccardo Rigon

Lastly, by substituting:

f(Se) = −α�1− S1/m

e

�m+ α (m = 1− 1/n)

in :

K(Se) = KsSve

�f(Se)f(1)

�2

133




Riccardo Rigon

there results:

K(Se) = KsSve

�1−

�1− S1/m

e

�m�2

(m = 1− 1/n)

or, by expressing everything as a function of the suction potential:

K(ψ) =Ks

�1− (αψ)mn [1 + (αψ)n]−m

�2

[1 + (αψ)n]mv (m = 1− 1/n)

134




Riccardo Rigon

so obtaining:

135


Water content

Curves of the hydraulic conductivity, K, as a function of the water content (dimensionless), for various values of the parameter m. The curves are calculated with the application of the van Genuchten equation and the Mualem model, having used v=0.5 and m-1-1/n.



Riccardo Rigon

HY

DR

AU

LIC

CO

ND

UC

TIV

ITY

136



Riccardo Rigon

Other parametric forms can be derived, based on different hypotheses than those used by Mualem. The best known of these is Burdine’s, that starts with different form of f(Se).

137


Equation Author



Riccardo Rigon

138



Riccardo Rigon

139

HY

DR

AU

LIC

CO

ND

UC

TIV

ITY


Water in the Soil and SubsoilThe Richards Equation

Riccardo Rigon

Jay

Stra

tton

Noll

er, G

lob

al S

oil

Sca

pes

, Des

ert

Det

ail, 2

00

7



Riccardo Rigon

The Richards Equation!

141

Se = [1 + (−αψ)m)]−n

Se :=θw − θr

φs − θr

C(ψ)∂ψ

∂t= ∇ ·

�K(θw) �∇ (z + ψ)

�

K(θw) = Ks

�Se

��1− (1− Se)1/m

�m�2

C(ψ) :=∂θw()∂ψ



Riccardo Rigon


141

Se = [1 + (−αψ)m)]−n

Se :=θw − θr

φs − θr

C(ψ)∂ψ

∂t= ∇ ·

�K(θw) �∇ (z + ψ)

�

K(θw) = Ks

�Se

��1− (1− Se)1/m

�m�2

Mass balance

C(ψ) :=∂θw()∂ψ



Riccardo Rigon


141

Se = [1 + (−αψ)m)]−n

Se :=θw − θr

φs − θr

C(ψ)∂ψ

∂t= ∇ ·

�K(θw) �∇ (z + ψ)

�

K(θw) = Ks

�Se

��1− (1− Se)1/m

�m�2

Mass balance

ParametricMualem

C(ψ) :=∂θw()∂ψ



Riccardo Rigon


141

Se = [1 + (−αψ)m)]−n

Se :=θw − θr

φs − θr

C(ψ)∂ψ

∂t= ∇ ·

�K(θw) �∇ (z + ψ)

�

K(θw) = Ks

�Se

��1− (1− Se)1/m

�m�2

Mass balance

ParametricMualem

Parametricvan Genuchten

C(ψ) :=∂θw()∂ψ



Riccardo Rigon

142

C(ψ)∂ψ

∂t= �∇K(θw) · �∇ (z + ψ) + K(θw)

�∇2 (z + ψ)

�

∂ψ

∂t=

1C(ψ)

�∇K(θw) · �∇z +1

C(ψ)�∇K(θw) · �∇ψ +

K(θw)C(ψ)

�∇2 (z + ψ)

�




Riccardo Rigon

Gravitational term related to the downward gradient of the hydraulic

conductivity

142

C(ψ)∂ψ

∂t= �∇K(θw) · �∇ (z + ψ) + K(θw)

�∇2 (z + ψ)

�

∂ψ

∂t=

1C(ψ)

�∇K(θw) · �∇z +1

C(ψ)�∇K(θw) · �∇ψ +

K(θw)C(ψ)

�∇2 (z + ψ)

�




Riccardo Rigon


conductivity

Advection term with transfer of psi in the direction of the hydraulic

conductivity gradient

142

C(ψ)∂ψ

∂t= �∇K(θw) · �∇ (z + ψ) + K(θw)

�∇2 (z + ψ)

�

∂ψ

∂t=

1C(ψ)

�∇K(θw) · �∇z +1

C(ψ)�∇K(θw) · �∇ψ +

K(θw)C(ψ)

�∇2 (z + ψ)

�




Riccardo Rigon

143

C(ψ)∂ψ

∂t= �∇K(θw) · �∇ (z + ψ) + K(θw)

�∇2 (z + ψ)

�

∂ψ

∂t+ �u(ψ) · ∇ψ =

1C(ψ)

�∇K(θw) · �∇z +K(θw)C(ψ)

∇2ψ

�u(ψ) := −�∇ K(ψ)C(ψ)




Riccardo Rigon

velocity of pressure advection

143

C(ψ)∂ψ

∂t= �∇K(θw) · �∇ (z + ψ) + K(θw)

�∇2 (z + ψ)

�

∂ψ

∂t+ �u(ψ) · ∇ψ =

1C(ψ)

�∇K(θw) · �∇z +K(θw)C(ψ)

∇2ψ

�u(ψ) := −�∇ K(ψ)C(ψ)




Riccardo Rigon

144

C(ψ)∂ψ

∂t= �∇K(θw) · �∇ (z + ψ) + K(θw)

�∇2 (z + ψ)

�

Dψ

Dt=

1C(ψ)

�∇K(θw) · �∇z +K(θw)C(ψ)

∇2ψ

Dψ

Dt:=

∂ψ

∂t+ u(ψ) · ∇ψ




Riccardo Rigon

Total Derivative

144

C(ψ)∂ψ

∂t= �∇K(θw) · �∇ (z + ψ) + K(θw)

�∇2 (z + ψ)

�

Dψ

Dt=

1C(ψ)

�∇K(θw) · �∇z +K(θw)C(ψ)

∇2ψ

Dψ

Dt:=

∂ψ

∂t+ u(ψ) · ∇ψ




Riccardo Rigon

145

C(ψ)∂ψ

∂t= �∇K(θw) · �∇ (z + ψ) + K(θw)

�∇2 (z + ψ)

�

Dψ

Dt=

1C(ψ)

�∇K(θw) · �∇z +K(θw)C(ψ)

∇2ψ

D(ψ) :=K(ψ)C(ψ)




Riccardo Rigon

Diffusive term

145

C(ψ)∂ψ

∂t= �∇K(θw) · �∇ (z + ψ) + K(θw)

�∇2 (z + ψ)

�

Dψ

Dt=

1C(ψ)

�∇K(θw) · �∇z +K(θw)C(ψ)

∇2ψ

D(ψ) :=K(ψ)C(ψ)




Riccardo Rigon

Diffusive term

Hydraulic diffusivity

145

C(ψ)∂ψ

∂t= �∇K(θw) · �∇ (z + ψ) + K(θw)

�∇2 (z + ψ)

�

Dψ

Dt=

1C(ψ)

�∇K(θw) · �∇z +K(θw)C(ψ)

∇2ψ

D(ψ) :=K(ψ)C(ψ)




Riccardo Rigon

146

C(ψ)∂ψ

∂t= �∇K(θw) · �∇ (z + ψ) + K(θw)

�∇2 (z + ψ)

�

∂ψ

∂t=

1C(ψ)

�∇K(θw) · �∇z +1

C(ψ)�∇K(θw) · �∇ψ +

K(θw)C(ψ)

�∇2 (z + ψ)

�



conductivity

Advection term with transfer of psi in the direction of the hydraulic

conductivity gradient

Diffusive term



Riccardo Rigon

147

How is the equation resolved?(partial differential equation)

Does an analytic solution exist?

Initial conditions are determined

Boundary conditions are determined



Riccardo Rigon

148

Numeric resolution

Does an analytic solution exist ?

no

yes

Results published

How is the equation resolved?(partial differential equation)



Riccardo Rigon

149

Numerical Resolution

A numerical method is selected

The equations are discretised

A programme to resolve the equations

is written

Code that will resolve the equations is

“bought”



Riccardo Rigon

150

Execution

The parameters are determined

Initial conditions

Boundary conditions

Execution of the numerical code

Results published



Riccardo Rigon

151

Parameters!

Se = [1 + (−αψ)m)]−n

C(ψ)∂ψ

∂t= ∇ ·

�K(θw) �∇ (z + ψ)

�

K(θw) = Ks

�Se

��1− (1− Se)1/m

�m�2

Se :=θw − θr

φs − θrC(ψ) :=

∂θw()∂ψ



Riccardo Rigon

152

How are the parameters determined?The fundamental idea is

that the hydraulic

properties of soils, at the

Darcy scale, are functions

of:

soil texture

organic matter

soil structure

The first attempts to

express these relations were

represented in tables like

this one to the left



Riccardo Rigon

m =n− 1

n

153

How are the parameters determined?



Riccardo Rigon

154




Riccardo Rigon

155




Riccardo Rigon

156

The procedure used to develop with the preceding tables, though it varies

from author to author, essentially consists in:

•estimating the SWRC parameters by means of samples collected in the field

and analysed in the laboratory

•simultaneously measuring the texture of the same samples

Both measurements are carried out with the appropriate geotechnical

analyses.




Riccardo Rigon

Pedotransfer Functions

Bouma (1989) introduced the term pedotransfer function (PTF), which might be

described as a pedofuntion or a pedo-hydrological function, in order to define

the approaches used to estimate the hydrological parameters in the expressions

of van Genuchten, and Brooks and Corey, starting from data that was fast and

economical to obtain, as opposed to data obtained from field and laboratory

analyses that were costly and onerous (Romano and Santini, 1997).

157

The pedotransfer functions represent a generalisation of the preceding tables, in a

statistical sense.

These PTF ara multivariate relations.



Riccardo Rigon

Depending on the level of available information, it is possible to define 5 classes

of PTF according to the classification of Ungaro and Calzolari (2001)

1) Level 1: granulometric fractions (at least three), texture classes;

2) Level 2: granulometric fractions (at least three), and apparent density OR organic matter;

3) Level 3: granulometric fractions, apparent density AND organic matter;

4) Level 4: granulometric fractions, apparent density, organic matter, and water content at -33 and -1500 kPa;

5) Level 5: granulometric fractions, apparent density, organic matter, and hydraulic conductivity at saturation Ks.

158




Riccardo Rigon

159

Ref. Lev. Input Output Model Soil Loc. ObservationsCosby sand%et al., 1 Ks

1984 clay%3 continuous equations for

Saxton sand% SWRC Brooks and capillary potential values inet al., 1 clay% and Corey; United the range: 1500-10; 10-ψe;

ψe-01986 θr = 0 conductivity Campbell States valid texture range:

sand 5-30% and clay 8-58%sand 30-95% and clay 5-60%discrete function of the

Rawls sand% values of θ water content for 10et al., 2 silt% for 12 values United capillary potential values:1982 clay% of potential States 10, 20, 33, 60, 100, 200, 400,

Sorg% 700, 1000, 1500 kPaBrakensiek sand%et al., 2 clay% Ks

1984 porosity(φ)Rawls and sand% θr

Brakensiek 2 clay% λ Brooks and United valid texture range:1989 porosity(φ) ψb Corey States sand 5-70% and clay 5-60%

sand%Gupta e silt% values of θ discrete function of the waterLarson 3 clay% for 12 values van Belgium content for 12 capillary1979 ρb [g cm−3] of potential Genuchten potential values

Sorg%sand% the van Genuchten model is

Vereecken silt% θr,θs van applied with m = 1et al., 3 clay% α, n Genuchten Belgium Conductivity refers to1989, 1990 ρb [g cm−3] Ks,Kψ the Gardner model (1958)

Corg%clay%

Scheinost [kg kg−1] θr, θs vanet al., 3 ρb [g cm−3] α, n Genuchten Germany1997a Corg %

silt%clay%

Wˆsten ρb [g cm−3] α, n van PTF obtained from a largeet al., 3+ Sorg % θs Genuchten Europe European database (HYPRES,1999 topsoil or n= 4020)

subsoilθr = 0



Riccardo Rigon

The association between the elements listed in the previous slides and hydraulic

properties is achieved with multivariate statistical regression on multiple soil

samples, or else with forecasting techniques using cellular automaton or other

tools. For example, Rawls, 1982, proposes:

160


All the parameters are functions of the suction, as shown in the following table.

parameter = a + b (% sand) + c (% silt) + d (% clay)+ e (% organic matter) + f (apparent density)



Riccardo Rigon

Raw

ls, 1

98

2

161


parameter = a + b (% sand) + c (% silt) + d (% clay)+ e (% organic matter) + f (apparent density)

Suction a b c d e f R2

potential intercept % sand % silt % clay %organic apparent correlation(kPa) matter density coefficient

[g cm−3]4 0.7899 -0.0037 0.0100 -0.1315 0 0.587 0.7135 -0.0030 0.0017 -0.1693 0.7410 0.4118 -0.0030 0.0023 0.0317 0.8120 0.3121 -0.0024 0.0032 0.0314 0.8633 0.2576 -0.0020 0.0036 0.0299 0.8760 0.2065 -0.0016 0.0040 0.0275 0.87100 0.0349 0.0014 0.0055 0.0251 0.87200 0.0281 0.0011 0.0054 0.0220 0.86400 0.0238 0.0008 0.0052 0.0190 0.84700 0.0216 0.0006 0.0050 0.0167 0.811000 0.0205 0.0005 0.0049 0.0154 0.811500 0.0260 0.0050 0.0158 0.80



Riccardo Rigon

162


Nemes (2006) proposes an association between the texture classes

(identified in the following slides) and the hydraulic properties of the soil.

texture class %sand %claysand 92 5loamy-sand 82 6sandy-loam 65 10sandy-clay-loam 60 28loam 40 18silty-loam 20 15silty-clay 8 45silty-clay-loam 10 35clay-loam 35 35clay 20 60



Riccardo Rigon

Nem

es, 2

00

6

163




Riccardo Rigon

Nem

es, 2

00

6

164




Riccardo Rigon

165


Nem

es (

20

06

)



Riccardo Rigon

SOILPAR (http://www.sipeaa.it/ASP/ASP2/SOILPAR.asp) – By Acutis and Donatelli

ROSETTA (http://www.ars.usda.gov/Servi[3] the USDA, uses artificial neural networks

RETC - van Genuchten, M. Th., F. J. Leij, and S. R. Yates. 1991.The RETC Code for Quantifying the Hydraulic Functions of Unsaturated Soils, Version 1.0. EPA Report 600/2-91/065, U.S. Salinity Laboratory, USDA, ARS, Riverside, California.

Software:

166


http://www.sipeaa.it/ASP/ASP2/SOILPAR.asp


http://www.ars.usda.gov/Servi%5B3%5Dthe

http://www.ars.usda.gov/Servi%5B3%5Dthe


Riccardo Rigon

Geometry of the Integration Domain

In order to resolve the Richards equation, the first thing to do is assign

the geometry of the integration domain. It can be assigned, for example,

on the basis of GIS terrain analysis.

Mod

ified

from

Abb

ot e

t al.,

198

6

167



Riccardo Rigon

Unsaturated Layer

Surface Layer

Saturated Layer:

Surface boundary condition

Bottom Boundary condition

Boundary Conditions

Mod

ified

from

Abb

ot e

t al.,

198

6

168



Riccardo Rigon

The boundary condition at the surface of the soil is:

−Kz∂ψ

∂z+ Kz = J(t)

where J(t) is the rain, valid if the first soil layer is not saturated or else

the water is forced to run off on the surface

169

Boundary Conditions



Riccardo Rigon

The boundary condition at the bottom of the domain can either be a

gravitational flow condition:

or an impermeable bottom condition:

∂ψ

∂z= 0

∂ψ

∂z= 1

or intermediary conditions (N.B. the boundary conditions are of the

second type, or of Neumann, that is to say they set the derivative of the

unknown quantity)170

Boundary Conditions



Riccardo Rigon

In order to resolve the differential equation it is also necessary to assign

initial conditions that correspond to the suction distribution at the time

t=0.

In general, assigning the initial conditions is not a trivial problem because

it involves either guessing the values of an entire domain or extrapolating

some point measurements over the domain; the same domain of which the

evolution is then to be evaluated.

For example, the condition in the figure is a condition of “hydrostatic distribution along the vertical” ...

171

Initial Conditions



Riccardo Rigon

Now, ideally, we have assigned:

- the geometry of the domain- the initial conditions- the boundary conditions

In general, analytic solutions of the Richards equation do NOT exist, except for some very particular cases in which the parameters are “linearised”. To obtain a solution, therefore, we need to:

- simplify the equation

or

- resolve it numerically

172

How to



Riccardo Rigon

C(ψ)∂ψ∂t = ∂

∂z

�Kz

�∂ψ)∂z − cosθ

��+ ∂

∂y

�Ky

∂ψ∂y

�+ ∂

∂x

�Kx

�∂ψ)∂x − sinθ

��

The Richards equation on a plane hillslope

Iver

son

, 20

00

; Cord

ano a

nd

Rig

on

, 20

08

173



Riccardo Rigon

C(ψ)∂ψ∂t = ∂

∂z

�Kz

�∂ψ)∂z − cosθ

��+ ∂

∂y

�Ky

∂ψ∂y

�+ ∂

∂x

�Kx

�∂ψ)∂x − sinθ

��

ψ ≈ (z − d cos θ)(q/Kz) + ψs

Bearing in mind the previous positions, the Richards equation, at hillslope

scale, can be separated into two components. One, boxed in red, relative

to vertical infiltration. The other, boxed in green, relative to lateral flows.

174


Iver

son

, 20

00

; Cord

ano a

nd

Rig

on

, 20

08



Riccardo Rigon

175




Riccardo Rigon

C(ψ)∂ψ

∂t=

∂

∂z

�Kz

�∂ψ

∂z− cos θ

��+ Sr

Vertical infiltration: acts in a

relatively fast time scale because

it propagates a signal over a

thickness of only a few metres

175




Riccardo Rigon

Sr =∂

∂y

�Ky

∂ψ

∂y

�+

∂

∂x

�Kx

�∂ψ

∂x− sin θ

��

176




Riccardo Rigon

Sr =∂

∂y

�Ky

∂ψ

∂y

�+

∂

∂x

�Kx

�∂ψ

∂x− sin θ

��

Properly treated, this is reduced to

groundwater lateral flow, specifically to the

Boussinesq equation, which, in turn, have

been integrated from SHALSTAB equations

176




Riccardo Rigon

C(ψ)∂ψ

∂t=

∂

∂z

�Kz

�∂ψ

∂z− cos θ

��+ Sr

In literature related to the determination of slope stability this equation

assumes a very important role because fieldwork, as well as theory, teaches

that the most intense variations in pressure are caused by vertical infiltrations.

This subject has been studied by, among others, Iverson, 2000, and D’Odorico

et al., 2003, who linearised the equations.

177




Riccardo Rigon

ψ ≈ (z − d cos θ)(q/Kz) + ψs

Iver

son

, 20

00

; D’O

dori

co e

t al

., 2

00

3,

Cord

ano a

nd

Rig

on

, 20

08

178

s




Riccardo Rigon

Assuming K ~ constant and neglecting the source terms

179

The Richards Equation 1-D

C(ψ)∂ψ

∂t= Kz 0

∂2ψ

∂z2

D0 :=Kz 0

C(ψ)



Riccardo Rigon

Assuming K ~ constant and neglecting the source terms

∂ψ

∂t= D0 cos2 θ

∂2ψ

∂t2

179


C(ψ)∂ψ

∂t= Kz 0

∂2ψ

∂z2

D0 :=Kz 0

C(ψ)



Riccardo Rigon

∂ψ

∂t= D0 cos2 θ

∂2ψ

∂t2

180




Riccardo Rigon

The equation becomes LINEAR and, having found a solution

with an instantaneous unit impulse at the boundary, the

solution for a variable precipitation depends on the

convolution of this solution and the precipitation.

∂ψ

∂t= D0 cos2 θ

∂2ψ

∂t2

180




Riccardo Rigon

181




Riccardo Rigon

For a precipitation impulse of constant intensity, the solution can be

written:

ψ0 = (z − d) cos2 θ

D’O

dori

co e

t al

., 2

00

3

182

ψ = ψ0 + ψs

ψs =

qKz

[R(t/TD)] 0 ≤ t ≤ T

qKz

[R(t/TD)−R(t/TD − T/TD)] t > T




Riccardo Rigon

In this case the equation admits an analytical solution

D’O

dori

co e

t al

., 2

00

3

183

R(t/TD) :=�

t/(π TD)e−TD/t − erfc��

TD/t�

ψs =

qKz

[R(t/TD)] 0 ≤ t ≤ T

qKz

[R(t/TD)−R(t/TD − T/TD)] t > T

TD :=z2

D0




Riccardo Rigon

D’O

dori

co e

t al

., 2

00

3

184

TD

TD

TD

TD

Th

e R

ich

ard

s Eq

uat

ion

1

-D



Riccardo Rigon

The analytical solution methods for the advection-dispersion equation

(even non-linear), that results from the Richards equation, can be found

in literature relating to heat diffusion (the linearised equation is the

same), for example Carslaw and Jager, 1959, pg 357.

Usually, the solution strategies are 4 and they are based on:

- variable separation methods

- use of the Fourier transform

- use of the Laplace transform

- geometric methods based on the symmetry of the equation (e.g.

Kevorkian, 1993)

All methods aim to reduce the partial differential equation to a system

of ordinary differential equations185

Th

e R

ich

ard

s Eq

uat

ion

1

-D



Riccardo Rigon

186

Th

e R

ich

ard

s Eq

uat

ion

1

-D



Riccardo Rigon

Sim

on

i, 2

00

7

187

Th

e R

ich

ard

s Eq

uat

ion

1

-D



Riccardo Rigon

188

Sim

on

i, 2

00

7

Th

e R

ich

ard

s Eq

uat

ion

1

-D



Riccardo Rigon

Software:

- Hydrus-1D - by Simunenk et al.

- Lavagna - by Cordano, 2008

189

Th

e R

ich

ard

s Eq

uat

ion

1

-D



Riccardo Rigon

190

Bibliography, further reading, web

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•Abbott M., J. Bathurst, J. Cunge, and P. O'Connell. An introduction to the european hydrological system

"Systeme Hydrologique Europeén" SHE, 2: Structure of a physically based, distributed modelling system. J.

Hydrol., vol. 87, p. 6177, 1986b.

•ASCE, Hydrology Handbook. ASCE Manuals and Reports of Engineering Practice, n.28, 1996

•Baver, Gardner e Gardner, 1972

•Beven K.J. & M.J. Kirkby, A physically based, variable contributing area model of basin hydrology, Hydrological

Science Bulletin, vol. 24, p. 43-69, 1979

•



Riccardo Rigon

191

• Black and Dunne, 1978 Dunne T., Field studies of hillslope processes. In Hillslope Hydrology, Kirkby MJ (ed.).

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runoff production in permeable soils. Water Resources Research, vol. 6, p. 478-499, 1970)

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•Brooks, R.H., and A.T. Corey. 1964. Hydraulic properties of porous media. Civil Engineering Dept., Colorado

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•Brooks R.H. & Corey A.T., Properties of Porous Media Affecting Fluid Flow, J. Irrig. Drain. Div., vol. 6, n.61, 1966

•Buckingham E., Studies on the movement of soil moisture. Bull 38 USDA, Bureau of Soils, Washington DC, 1907

•Calzolari C. & Ungaro F., I modelli MACRO e SOILN: l'esperienza del progetto SINA-Carta pedologica in aree a

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192

Carsel R.F., & Parrish R. S., Developing joint probability distributions of soil water retention

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vol. 14, n. 4, 601-604, 1978.



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193

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200

Web

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ROSETTA: http://www.ars.usda.gov/Services/docs.htm?docid=8953, uses artificial neural networks, last

accessed, 2009- 11-01

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Riccardo Rigon

201

Authors

Riccardo Rigon - University of Trento, Department of Civill and Environmental Engineering/

CUDAM, riccardo.rigon <at> ing.unitn.it

Alberto Bellin - University of Trento, Department of Civill and Environmental Engineering/ CUDAM,

alberto.bellin <at> ing.unitn.it

Alessandro Tarantino - University of Trento, Department of Structural Mechanics,

alessandro.tarantino <at> ing.unitn.it

Giacomo Sartori - University of Trento, adjunct professor, giacomo.sartori <at> ing.unitn.it



Riccardo Rigon

Notationmass

Symbol Name nickname UnitMs mass of soil ms [M]Mag mass of air gas in soil mags [M]Mv mass of water vapor in soil mwvs [M]Mw mass of liquid water in soil mlws [M]Mi mass of ice in soil mlws [M]Msp mass of soil particle msp [M]Mtw mass of water mtw [M]Mcw mass of condensed water mcw [M]



Riccardo Rigon

Symbol Name nickname UnitVs volume of soil vs [L3]Vag volume of air gas in soil vags [L3]Vv volume of water vapor in soil vwvs [L3]Vw volume of liquid water in soil vlws [L3]Vi volume of ice in soil vlws [L3]Vsp volume of soil particle vsp [L3]Vtw volume of water vw [L3]Vcw volume of condensed water vcw [L3]

Notationvolume



Riccardo Rigon

204

Symbol Name nickname Unitρb bulk density of soil bds [M L−3]ρag density of air gas dag [M L−3]ρv density of water vapor in soil dwvs [M L−3]ρw density of liquid water dlw [M L−3]ρi density of ice di [M L−3]ρis density of ice in soil dis [M L−3]ρsp density of soil particle dsp [M L−3]ρtw mean density of water dw [M L−3]

Notationdensity



Riccardo Rigon

205

Symbol Name nickname Unitφs soil porosity sp 1φse effective soil porosity esp 1θcw volume fraction of condensed water in soil pores vfcwsp 1θw volume fraction of liquid water in soil pores vflwsp 1θi volume fraction of frozen water in soil pores vffwsp 1θr residual volume fraction of water in soil pores rvfwsp 1Ss relative saturation saturation of soil rss 1Se effective saturation of soil ess

Notationvarious



Riccardo Rigon

206

Thank you for your attention!

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10 water in soil-rev 1

Education