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The FEST Model for Testing the Importance of Hysteresis in Hydrology J. Philip O’Kane Department of Civil & Environmental Engineering, Environmental Research Institute UCC Int. Workshop on HYSTERESIS & MULTI-SCALE ASYMPTOTICS, University College Cork, Ireland, March 17-21, 2004 Content

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Slide1 l.jpg

The FEST Model

for Testing the Importance of

Hysteresis in Hydrology

J. Philip O’Kane

Department of Civil & Environmental Engineering,

Environmental Research Institute UCC

Int. Workshop on HYSTERESIS & MULTI-SCALE ASYMPTOTICS,

University College Cork, Ireland, March 17-21, 2004


Content l.jpg
Content

1. Introduction

soil physics

2. The BASE model

bare soil with evaporation and drainage

3. The FEST model

fully vegetated soil slab with transpiration

4. The structure of FEST

feedback structure

bifurcation


1 introduction l.jpg
1. Introduction

1. Hysteresis in hydrology, climatology, ecohydrology

Is it significant? For what questions?

2. Hysteresis in open channel flow

Rate dependent

3. Hysteresis in soil physics

Rate independent

4. Method

Build “test rigs” to answer the questions

BASE model - pde - soil physics

FEST model - ode - plausible soil bio-physics



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Soil: a multi-phase material

Each phase has

mass M and volume V

The REV –

representative elementary volume

1 cm

Air Ma, Va

Water Mw, Vw

Soil-solids Ms, Vs

1 cm

1 cm


Slide8 l.jpg

Ratios describe the multi-phase material

Total porosity ff = (Va + Vw)/(Va + Vw + Vs )

Void ratio e = (Va + Vw)/Vs[m3m-3]

Particle densityrs = Ms/Vs

Dry bulk densityrb = Ms/(Va + Vw + Vs )

Water densityrw = Mw/Vw [Mgm-3]

Air Ma, Va

Water Mw, Vw

Soil-solids Ms, Vs


Slide9 l.jpg

Moisture content

Volumetric wetnessq = Vw /(Va +Vw +Vs )

In clay soilsthe soil matrix swells, Vs = f(Vw),

q has no well-defined maximum value

In gravel, sand and silt, the soil matrix is “rigid”

q has a maximum at saturation

0 < q < qs < 1, at saturation Va= 0

Mass wetnessw = Mw /Ms

q = wrb/rw in rigid soils


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Potential energy of soil water

  • A mass m of soil water of volume Vand density w = m/V

  • is moved on an arbitrary path through a vertical distance z

  • by a force

  • mg = wVg

  • The dissipationless work done against the force of gravity is

  • mgz = (wVg)z

  • There are three alternative ways of representing the

  • potential energy of this water as dissipationless work

  • (a) per unit mass, (b) per unit volume, and (c) per unit weight



Slide12 l.jpg

Total potential

is a sum of partial potentials

y = yg + ym + yo + yp + ya + yW

yggravitational potential

ymmatric potential

yo osmotic potential

yp hydrostatic potential

ya atmospheric

yW overburden potential



Soil moisture characteristic matric potential soil suction or drying l.jpg
Soil-moisture characteristic - matric potential, soil suction or drying

m = m(), m e < 0, 0 <  s,

e air-entry potential,  =  s

=  (m) inverse function

Specific water capacity

C() = d/dm

Drying and wetting are different - hysteresis -

usually ignored !


Y m z partitions q z into liquid and vapour fractions l.jpg
Y suction or dryingm(z) partitions q(z)into liquid and vapour fractions

  • h(z) relative humidity of soil-air

  • Mwis the molar mass of water (0.018 kg/mol),

  • R the molar gas constant (8.314 J/mol K)

  • T the constant temperature in degrees Kelvin (293 K at 200C).



Slide17 l.jpg

T suction or drying

E

P

Water flow in a

column of soil

Vertical

coordinate

from the

ground

surface

positive

downwards

to the

watertable (no air)

0

Soil 1

I

Soil 2

10 m

z

Soil 3

1 m

1 m

C

D


Slide18 l.jpg

Conservation of water mass in one dimension suction or drying

fl is the flux density of liquid water (kg m-2s-1)

fv is the flux density of water vapour (kg m-2s-1),

in the direction of positive zi.e. downwards,


Slide19 l.jpg

Generalised Darcy’s Law suction or drying

Philip, 1955

Buckingham, 1907


Philip richards equation form l.jpg
Philip-Richards equation – suction or dryingψ form

Solutions sought in the space of continuous functions

ym(z,t)

Discontinuities allowed inq(z,t)

to match discontinuous soil horizons

Philip 1955,

Richards, 1931


Boundary conditions forcing l.jpg
Boundary conditions & forcing suction or drying

Flux Boundary conditions

Precipitation

Evaporation

Overland flow - ignore initially

Interflow - ignore in one dimension

Potential Boundary condition

Ponded infiltration

Fixed water table

Mixed Boundary condition

Evaporation

Drainage to a moving water table

Forcing function

Transpiration


Standard hydrological questions l.jpg
Standard hydrological questions suction or drying

Infiltration & surface runoff

Evaporation

Transpiration

Redistribution

Capillary rise

Drainage


Two pairs of switched boundary conditions atmosphere or soil control of fluxes l.jpg
Two pairs of switched boundary conditions suction or drying - atmosphere or soil control of fluxes?

Outer pair - fluxes at potential rates

Raining or drying

atmosphere control

Inner pair - fluxes at smaller actual rates

Surface ponding or phase 2 drying

soil control


The raining and drying cycle l.jpg
The raining and drying cycle suction or drying

td

Potential

evaporation

Actual

evaporation

Soil

drying

begins

Ea<Ep

Ea=Ep

tE

tQ

Soil

wetting

begins

q0<qR

q0=qR

Actual

infiltration

Potential

infiltration

tp


Alternating control l.jpg
Alternating control suction or drying

td

Potential

evaporation

Actual

evaporation

Atmosphere

control!

Ea=Ep

Soil control ?

Ea<Ep

Soil

drying

begins

tE

tQ

Soil

wetting

begins

Soil control ?

q0<qR

Atmosphere

control!

q0=qR

Actual

infiltration

Potential

infiltration

tp


Richards equation form l.jpg
Richards equation – suction or dryingθform


Infiltration atmosphere control l.jpg
Infiltration - atmosphere control suction or drying

D \ K constant K linear K non-linear K

delta function D Mein & Larson (1973)

constant D Breaster Breaster Clothier et al

(1973) (1973) (1981)

‘Fujita D’ Knight & Rogers et al. Sander et al. Philip (1983) (1988)

(1974)


Infiltration soil control l.jpg
Infiltration - soil control suction or drying

D \ K constant K linear K non-linear K

delta function D Green & Ampt (1911)

constant D Carslaw & Philip Philip

Jaeger (1969) (1974)

(1946)

‘Fujita D’ Fujita not solved not solved

(1952)


Evaporation atmosphere control l.jpg
Evaporation - atmosphere control suction or drying

D \ K constant K linear K non-linear K

delta function D not applicable

constant D Breaster* Breaster* Kühnel

(1973) (1973) (1989 [C])

‘Fujita D’ Knight & Sander & Sander & Philip* Kühnel Kühnel

(1974) (19**) (19**)

*complementary to infiltration solution


Evaporation soil control l.jpg
Evaporation - soil control suction or drying

D \ K constant K linear K non-linear K

delta function D not applicable

constant D Carslaw & Kühnel & Kühnel

Jaeger* Sander (1989 [C]) (1946) (19**)

‘Fujita D’ Fujita* not solved not solved

(1952)

*complementary to infiltration solution


The fest model fully vegetated soil slab with transpiration l.jpg
The FEST model suction or drying-fully vegetated soil slab with transpiration

Goal: from plausible biophysics

an ode - for testing hysteresis operators


Fest ordinary differential equation l.jpg
FEST ordinary differential equation suction or drying

  • Uniform moisture in the root zone

  • Gradients in potential become differences

  • Brooks-Corey-Campbell parametric expressions for the matric potential and hydraulic conductivities of soils

  • Square wave atmospheric forcing


Slide33 l.jpg

  • Transpiration suction or drying

  • Roots completely penetrate the uniform root zone

  • A 3-D wick sucks water from the uniform roots to a uniform canopy

  • Leaf potential is matric potential of soil water plus change in gravitational potential between the roots and canopy

  • Potential transpiration (given) drives actual transpiration


Potential transpiration given l.jpg
Potential transpiration - given suction or drying

The Philip boundary condition

Leaf evaporation is proportional to the difference in humidity between

(a) the atmosphere, and

(b) the stomatal air

in “thermodynamic” equilibrium

with its plant water in the canopy


Slide35 l.jpg

  • Actual transpiration suction or drying

  • drops below

  • the potential rate

  • when stomates close

  • at leaf potentials between

  • some higher value (e.g. -5,000cm)

  • and the wilting potential (e.g. -10,000cm)




Infiltration l.jpg
Infiltration suction or drying

Actual infiltration is the minimum of the

rainfall rate and the potential infiltration rate

Infiltration is assumed

to occur throughout the soil slab

through preferential paths

due to worm holes, animal burrows and dead roots

presenting the infiltrating water

uniformly to the soil matrix.


Slide39 l.jpg

Potential infiltration suction or drying rate

is equal to the hydraulic conductivity

at the soil water potential

times

the difference

between that potential

and the air entry potential of the rain divided by an arbitrary pore spacing


Infiltration feedback loops l.jpg
Infiltration feedback loops suction or drying


Infiltration with parameters l.jpg
Infiltration with parameters suction or drying




Cut the feedback loops l.jpg
Cut the feedback loops parameters

Multiple equilibria

Bifurcation



Slide46 l.jpg
E parameters


Slide47 l.jpg
E1 parameters


Slide48 l.jpg
A parameters


Slide49 l.jpg
B parameters


Slide50 l.jpg
C parameters


Slide51 l.jpg
D parameters


Titles l.jpg
Titles parameters

Bifurcation on e over p = 1.2, 1, 0.8; period 10

Bifurcation on e over p = 1.2, 1, 0.8; period 20

Bifurcation on e over p = 1.2, 1, 0.8; period 40

Bifurcation on theta(0)


Insertion of preisach operator l.jpg
Insertion of Preisach operator parameters

One insertion makes everything hysteretic

Extension in space

horizontally with a scalar wave equation

bifurcation in space

vertically with Philip-Richards equation


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