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The Independent Domain Model for

HysteresisA practical approach to understand and quantify

hysteretic effects

A practical approach to understand and quantify

hysteretic effects

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The basic assumptions(1) Assume that media can be characterized as an aggregation of independent pores, each with a characteristic filling pressure (hf) dictated by the body radius and emptying pressure (he) controlled by the neck radius.

(1) Assume that media can be characterized as an aggregation of independent pores, each with a characteristic filling pressure (hf) dictated by the body radius and emptying pressure (he) controlled by the neck radius.

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Second Assumption(2) Each pore is hydraulically connected to the bulk media so that if a pressure is established at one of the media’s boundaries, all pores will experience that pressure. Each pore responds independently (the system is fully funicular).filling and draining of each pore is determined strictly by

that pore’s geometry, regardless of the connection of that pore to surrounding pores (thus the term “independent”).

Pore necks are, by definition, smaller than pore bodies, so the absolute value of he is necessarily larger than hf for a given pore

(2) Each pore is hydraulically connected to the bulk media so that if a pressure is established at one of the media’s boundaries, all pores will experience that pressure. Each pore responds independently (the system is fully funicular).filling and draining of each pore is determined strictly by

that pore’s geometry, regardless of the connection of that pore to surrounding pores (thus the term “independent”).

Pore necks are, by definition, smaller than pore bodies, so the absolute value of he is necessarily larger than hf for a given pore

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Pros and ConsFine for the wet end of the characteristic curve,

when water fills most pores

In dry media pores become isolated by empty pores: the independent domain assumption breaks down, although vapor phase re-connects, but at slow pace.

Note that the distribution functions for he and hf are not independent, since pores with very small necks are more likely to have similarly small bodies

Fine for the wet end of the characteristic curve, when water fills most pores

In dry media pores become isolated by empty pores: the independent domain assumption breaks down, although vapor phase re-connects, but at slow pace.

Note that the distribution functions for he and hf are not independent, since pores with very small necks are more likely to have similarly small bodies

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Quantification of pore distribution Pores will have a range of volumes for he and hf described by a joint probability density function

Pores will have a range of volumes for he and hf described by a joint probability density function

h = 2rfmax bmin

h = 2rfmin bmax

h = 2remax nmin

h = 2remin nmax

h f

he

EMPTYING

f

On 45 line r = r and h = h .

nb

The value of f gives the fraction of pores which fill at h and empty at hf e

o

f e

This line identifies all pores which empty at pressure

This line identifies all pores which fill at pressure

h + hf f

h + he e

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Observations about f(he,hf)Distribution functions for he and hf are not

independent, since pores with very small necks are more likely to have similarly small bodies

The total volume (probability) under the curve is 1, corresponding to the fact that all pores will have some combination of the two characteristic radii

Distribution functions for he and hf are not independent, since pores with very small necks are more likely to have similarly small bodies

The total volume (probability) under the curve is 1, corresponding to the fact that all pores will have some combination of the two characteristic radii

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Features of the joint density functionEnd-points of the pressure scales are defined by the

largest and smallest bodies and necks.

Horizontal line with a value h1 going from the right boundary (hemax) to the 45° line, this delineates all of the pores which fill at pressure h1.

Vertical line at pressure defines all the pores which empty at pressure

End-points of the pressure scales are defined by the largest and smallest bodies and necks.

Horizontal line with a value h1 going from the right boundary (hemax) to the 45° line, this delineates all of the pores which fill at pressure h1.

Vertical line at pressure defines all the pores which empty at pressure

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How to use this representationIt is straightforward to obtain the various characteristic

curves once you have determined the joint density function. boils down to figuring out the range over which to integrate the

density function.

Example:Consider the series of characteristic curves shown in Figure

2.10Starting from point 1 where the media is dry, all the pores

are empty. So now we will go from h = - to h = 0 adding up all the pores with emptying pressures between h to -, integrating the density function as we go.

It is straightforward to obtain the various characteristic curves once you have determined the joint density function. boils down to figuring out the range over which to integrate the

density function.

Example:Consider the series of characteristic curves shown in Figure

2.10Starting from point 1 where the media is dry, all the pores

are empty. So now we will go from h = - to h = 0 adding up all the pores with emptying pressures between h to -, integrating the density function as we go.

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Procedure cont.To obtain the main draining curve we follow the same

procedure. Pressure starts at 0 and becomes more negative. Integrate along

vertical lines all the pores which empty at a given pressure regardless of the pressure at which they filled.

Now re-fill the media. Pores which are already filled with water cannot be refilled! Same as before, but only add the pores which fill between pressures h1

and hf

KEY: determining which area of the domain to integrate over to determine the moisture content for any sequence of pressure changes. This procedure is quite amenable to numerical implementation.

To obtain the main draining curve we follow the same procedure. Pressure starts at 0 and becomes more negative. Integrate along

vertical lines all the pores which empty at a given pressure regardless of the pressure at which they filled.

Now re-fill the media. Pores which are already filled with water cannot be refilled! Same as before, but only add the pores which fill between pressures h1

and hf

KEY: determining which area of the domain to integrate over to determine the moisture content for any sequence of pressure changes. This procedure is quite amenable to numerical implementation.

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Comparing the two representations

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1962 Poulovasilis Data-model

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Did it graphically, now mathematicallyStated mathematically as integrating over the domain of filled pores.

Main wetting:

Main Draining:

Defining the turning point as:

Primary wetting:

Stated mathematically as integrating over the domain of filled pores.

Main wetting:

Main Draining:

Defining the turning point as:

Primary wetting:

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Notation for hysteretic processNeed to keep track of turning points

Subscripts denote the order of pressures, and relative position indicates whether the transition was wetting or drying. In the case shown, the media wetted from h0 to h1, dried to h2, and then re-wetted to the present pressure h.

Need to keep track of turning points

Subscripts denote the order of pressures, and relative position indicates whether the transition was wetting or drying. In the case shown, the media wetted from h0 to h1, dried to h2, and then re-wetted to the present pressure h.

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‘dis-functionalCurly brackets {}: is not a function

of h but is a functional of h. There is not a one-to-one mapping between and h without consideration of the antecedent conditions.

Can relate and h from a known initial state and through a known sequence of either or h as stated in equation [2.66]

Curly brackets {}: is not a function of h but is a functional of h. There is not a one-to-one mapping between and h without consideration of the antecedent conditions.

Can relate and h from a known initial state and through a known sequence of either or h as stated in equation [2.66]

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How to obtain joint density function?

Carry out a terrific number of experiments where you map out the entire domain of possible filling and draining pressures to obtain f(he,hf) by brute force.

With computer control this is feasible using an automated pressure cell.

Carry out a terrific number of experiments where you map out the entire domain of possible filling and draining pressures to obtain f(he,hf) by brute force.

With computer control this is feasible using an automated pressure cell.

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Similarity Theories1973 Mualem introduced a simplification of this model: noted that

the joint density function f(he,hf) could be well approximated by the product of two univariate density functions

f(he,hf) g(he)l(hf) [2.67]

g() and l() are probability density functions that depend only on he and hf, respectively.

The filling pressure distributions are the same, up to a constant multiplier, along draining pressure lines

Using this, only need the main filling and emptying curves to obtain g() and l().

1973 Mualem introduced a simplification of this model: noted that the joint density function f(he,hf) could be well approximated by the product of two univariate density functions

f(he,hf) g(he)l(hf) [2.67]

g() and l() are probability density functions that depend only on he and hf, respectively.

The filling pressure distributions are the same, up to a constant multiplier, along draining pressure lines

Using this, only need the main filling and emptying curves to obtain g() and l().

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Graphical Representation

Similarity assumption of Mualem (1973)Similarity assumption of Mualem (1973)

00

Emptying Pressure he

f

45of:

Fra

ctio

n of

Moi

stur

e C

onte

nt

h3 h4

h1

h2

y

xz

w

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Mualem’s similarity data-model

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And to make life even easier...

Parlange (1976) similarity model based on data from the main draining curve alone is sufficient to reproduce the full family of scanning curves.

Parlange (1976) similarity model based on data from the main draining curve alone is sufficient to reproduce the full family of scanning curves.

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Parlange’s model-data

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Summary of hysteresis•One Approach to handling (semi-physical) shown – lots of others out there.•In the real world all soils are hysteretic.•Shown to be very influential in movement of NAPLs, fingered flow, and desert recharge.•Typically ignored due to lack of data and models.•With contemporary models, little reason to leave out this factor.

•One Approach to handling (semi-physical) shown – lots of others out there.•In the real world all soils are hysteretic.•Shown to be very influential in movement of NAPLs, fingered flow, and desert recharge.•Typically ignored due to lack of data and models.•With contemporary models, little reason to leave out this factor.