The FEST Model

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The FEST Model for Testing the Importance of
Hysteresis in Hydrology
J. Philip OKane Department of Civil
Environmental Engineering, Environmental Research
Institute UCC
Int. Workshop on HYSTERESIS MULTI-SCALE
ASYMPTOTICS, University College Cork, Ireland,
March 17-21, 2004
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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
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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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Some soil 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
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Ratios describe the multi-phase material
Total porosity ff (Va Vw)/(Va Vw Vs
) Void ratio e (Va Vw)/Vs m3
m-3 Particle density rs Ms/Vs Dry bulk
density rb Ms/(Va Vw Vs ) Water density
rw Mw/Vw Mg m-3
Air Ma, Va
Water Mw, Vw
Soil-solids Ms, Vs
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Moisture content
Volumetric wetness q Vw /(Va Vw Vs ) In
clay soils the 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 lt q lt qs lt 1, at
saturation Va 0 Mass wetness w Mw /Ms
q w rb /rw in rigid soils
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Potential energy of soil water
A mass m of soil water of volume V and 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
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Potential energy per unit mass, volume and weight
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Total potential is a sum of partial
potentials y yg ym yo yp ya yW yg
gravitational potential ym matric potential yo
osmotic potential yp hydrostatic potential ya
atmospheric yW overburden potential
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Partial potentials with common reference state -
free water at z0
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Soil-moisture characteristic - matric potential,
soil suction or drying
?m ?m(?), ?m ? ?e lt 0, 0 lt ? ? ? s, ?e
air-entry potential, ? ? s ? ? (?m)
inverse function Specific water capacity
C(?) d?/d?m Drying and wetting are different
- hysteresis - usually ignored !
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Ym(z) partitions q(z) into liquid and vapour
fractions
h(z) relative humidity of soil-air Mw is 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).
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The BASE model - bare soil with evaporation and
drainage
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P
E
T
Water flow in a column of soil
Vertical coordinate from the ground surface pos
itive downwards to the watertable (no air)
0
Soil 1
I
Soil 2
10 m
z
Soil 3
1 m
1 m
C
D
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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 z
i.e. downwards,
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Generalised Darcys Law
Philip, 1955 Buckingham, 1907
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Philip-Richards equation ? form
Solutions sought in the space of continuous
functions ym(z,t) Discontinuities allowed in
q(z,t) to match discontinuous soil horizons
Philip 1955, Richards, 1931
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Boundary conditions forcing
Flux Boundary conditions Precipitation Evaporati
on Overland flow - ignore initially Interflow
- ignore in one dimension Potential Boundary
condition Ponded infiltration Fixed water
table Mixed Boundary condition Evaporation Drain
age to a moving water table
Forcing function Transpiration
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Standard hydrological questions
Infiltration surface runoff Evaporation Transpir
ation Redistribution Capillary rise Drainage
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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
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The raining and drying cycle
td
Potential evaporation
Actual evaporation
Soil drying begins
EaEp
EaltEp
tE
tQ
Soil wetting begins
q0ltqR
q0qR
Actual infiltration
Potential infiltration
tp
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Alternating control
td
Potential evaporation
Actual evaporation
Atmosphere control! EaEp
Soil control ? EaltEp
Soil drying begins
tE
tQ
Soil wetting begins
Soil control ? q0ltqR
Atmosphere control! q0qR
Actual infiltration
Potential infiltration
tp
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Richards equation ? form
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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)
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Infiltration - soil control
D \ K constant K linear K non-linear
K delta function D Green Ampt
(1911) constant D Carslaw Philip Philip Ja
eger (1969) (1974) (1946) Fujita
D Fujita not solved not solved (1952)
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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
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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
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The FEST model -fully vegetated soil slab with
transpiration
Goal from plausible biophysics an ode - for
testing hysteresis operators
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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

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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
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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
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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)
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Feedback loops for transpiration
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Transpiration loops with soil physics parameters
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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.
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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
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Infiltration feedback loops
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Infiltration with parameters
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Feedback loops for drainage/capillary rise with
soil physics parameters
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Feedback loops for drainage, capillary rise
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Cut the feedback loops
Multiple equilibria Bifurcation
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Bifurcation to desert
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E
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E1
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A
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B
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C
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D
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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)
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Insertion of Preisach operator
One insertion makes everything hysteretic Extensi
on in space horizontally with a scalar wave
equation bifurcation in space vertically with
Philip-Richards equation