1 The Independent Domain Model for Hysteresis A practical approach to understand and quantify hysteretic effects

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

3 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 h e is necessarily larger than h f 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 h e is necessarily larger than h f for a given pore

4 Pros and Cons  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 h e and h f 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 h e and h f are not independent, since pores with very small necks are more likely to have similarly small bodies

5 Quantification of pore distribution Pores will have a range of volumes for h e and h f described by a joint probability density function

7 Observations about f(h e,h f )  Distribution functions for h e and h f 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 h e and h f 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

8 Features of the joint density function  End-points of the pressure scales are defined by the largest and smallest bodies and necks.  Horizontal line with a value h 1 going from the right boundary (h emax ) to the 45° line, this delineates all of the pores which fill at pressure h 1.  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 h 1 going from the right boundary (h emax ) to the 45° line, this delineates all of the pores which fill at pressure h 1.  Vertical line at pressure defines all the pores which empty at pressure

9 How to use this representation  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 2.10  Starting 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 2.10  Starting 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.

10 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 h 1 and h f 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 h 1 and h f 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.

12 Comparing the two representations

Poulovasilis Data-model

14 Did it graphically, now mathematically Stated 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:

15 Notation for hysteretic process 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 h 0 to h 1, dried to h 2, 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 h 0 to h 1, dried to h 2, and then re-wetted to the present pressure h.

16 ‘dis-functional  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]  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]

17 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(h e,h f ) 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(h e,h f ) by brute force.  With computer control this is feasible using an automated pressure cell.

18 Similarity Theories  1973 Mualem introduced a simplification of this model: noted that the joint density function f(h e,h f ) could be well approximated by the product of two univariate density functions f(h e,h f ) ­ g(h e )l(h f )[2.67]  g() and l() are probability density functions that depend only on h e and h f, 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(h e,h f ) could be well approximated by the product of two univariate density functions f(h e,h f ) ­ g(h e )l(h f )[2.67]  g() and l() are probability density functions that depend only on h e and h f, 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().

19 Graphical Representation Similarity assumption of Mualem (1973)

20 Mualem’s similarity data-model

21 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.

22 Parlange’s model-data

23 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.