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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
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
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
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

slide7
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
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
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

slide10
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
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
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
Ym(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
T

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
Conservation of water mass in one dimension

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
Generalised Darcy’s Law

Philip, 1955

Buckingham, 1907

philip richards equation form
Philip-Richards equation – ψ 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
Boundary conditions & forcing

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
Standard hydrological questions

Infiltration & surface runoff

Evaporation

Transpiration

Redistribution

Capillary rise

Drainage

two pairs of switched boundary conditions atmosphere or soil control of fluxes
Two pairs of switched boundary conditions - 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
The raining and drying cycle

td

Potential

evaporation

Actual

evaporation

Soil

drying

begins

Ea

Ea=Ep

tE

tQ

Soil

wetting

begins

q0

q0=qR

Actual

infiltration

Potential

infiltration

tp

alternating control
Alternating control

td

Potential

evaporation

Actual

evaporation

Atmosphere

control!

Ea=Ep

Soil control ?

Ea

Soil

drying

begins

tE

tQ

Soil

wetting

begins

Soil control ?

q0

Atmosphere

control!

q0=qR

Actual

infiltration

Potential

infiltration

tp

infiltration atmosphere control
Infiltration - atmosphere control

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
Infiltration - soil control

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
Evaporation - atmosphere control

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
Evaporation - soil control

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
The FEST model -fully vegetated soil slab with transpiration

Goal: from plausible biophysics

an ode - for testing hysteresis operators

fest ordinary differential equation
FEST ordinary differential equation
  • 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
Transpiration
  • 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
Potential transpiration - given

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
Actual transpiration
  • 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
Infiltration

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
Potential infiltration 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

cut the feedback loops
Cut the feedback loops

Multiple equilibria

Bifurcation

titles
Titles

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
Insertion of Preisach operator

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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