Soil Science Society of America Journal 64:1947-1950 (2000)
© 2000 Soil Science Society of America
DIVISION S-1-SOIL PHYSICS
A Note on Calculating Hysteretic Behavior
A. Poulovassilis and
G. Kargas
Agricultural University of Athens, Iera odos 75, 11855 Athens, Greece
lhyd4kag{at}auadec.aua.gr
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ABSTRACT
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A computational procedure for approximating the hysteretic relationship between pore water content and pressure head in porous bodies is described. A distribution function 
is introduced that retains the features of the distribution function F of the independent domain model, but its values are obtained by partitioning the slopes of the wetting boundary curve of a reproducible hysteresis loop proportionately to the slopes of the drying boundary curve or vice versa. With the help of data provided by a primary drying scanning curve the procedure can be extended to approximate the hysteretic curves in the presence of nonindependent water elements. Calculated scanning curves were compared with experimental ones and satisfactory agreement between calculated and experimental scanning curves was observed.
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INTRODUCTION
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THE INDEPENDENT DOMAIN MODEL (Preisach, 1935; Neel, 1942; Everett and Whitton, 1952; Everett and Smith, 1954; Everett, 1954, 1955; Enderby, 1955, 1956) has been shown to be able to account fairly satisfactorily for the hysteretic relationship between pore water content and pressure head in some porous bodies (Poulovassilis, 1962, 1970; Talsma, 1970). Nevertheless, it has been found that the independent domain concept cannot account in general for the hysteretic behavior of pore water properties (Topp and Miller, 1966; Poulovassilis and Childs, 1971; Poulovassilis and Tzimas, 1974; Poulovassilis and El-Ghamry, 1978), and for this Poulovassilis and Childs (1971) and Poulovassilis and El-Ghamry (1978) extended the concept of domains by introducing the hypothesis that, in some porous bodies, their accumulation or removal may depend on the values of the variables at reversal.
Much of the work on the hysteretic pore water properties referred to above was aimed mainly at elucidating the interaction between the solid matrix of a porous body and the water contained in it. However, apart from the interest the hysteresis phenomenon presents as such, hysteresis is inextricably involved in crucial cases of soil water movement and soil water profile development and therefore the need arose to develop computational schemes for approximating the hysteretic behavior of porous materials. The late John Philip was first in recognizing this need; he presented (1964) an elegant computational scheme based on the similarity hypothesis, which according to Philip implied that the distribution of geometrical relationships between wetting and drying meniscus curvatures is independent of pore size. Later, he was followed by other researchers who presented simpler computational schemes (Mualem, 1973, 1974; Mualem and Dagan, 1975; Mualem, 1977; Mualem and Miller, 1979; Mualem, 1984; Parlange, 1976).
Here another computational approach is described. It utilizes the information provided by the boundary curves of a hysteretic loop for reproducing hysteretic behaviors conforming to that prescribed by the independent domain concept. It is based on an assumption that falls into line with the physical characteristics of the phenomenon, and it allows the distribution of the water elements (domains) to be obtained by a simple procedure. The proposed approach can be extended to predict hysteretic curves in the presence of nonindependent water elements.
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Theory
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Introduction of a Distribution Function
This function retains all the features of the distribution function F introduced by the independent domain concept (see Poulovassilis, 1962). Thus, the slope of the wetting boundary curve of a reproducible hysteresis loop at a suction value
(P being the soil water pressure head) is given by (see Fig. 1 and 2)
 | (1) |
and the slope of the drying boundary curve at the value Sn by
 | (2) |
as in the case of the independent domain model where, however, the distribution function F replaces . The subscripts e and f in the above equations denote the variables during the process of the drying of the porous body (emptying the pore space, withdrawal of the pore water) and of the wetting of the porous body (filling the pore space, reentering of the pore water), respectively. Further, f
'i denotes the slope of the wetting boundary curve at the suction value Si to which corresponds a pore water content fi and e'i the slope of the drying boundary curve at the same value Si to which now corresponds a pore water content ei where ei > fi, except at the two end points of the hysteresis loop where these two variables have the same value, that is, when
and
, the former being the suction value the two boundary curves meet at the saturation and the latter the maximum suction value applied (see Fig. 1).
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Fig. 1 Drying boundary curve (a) and wetting boundary curve (b) of a hypothetical reproducible hysreresis loop describing the water content–suction relationship of a porous body
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For obtaining the values of the distribution function F across the area OAB of the distribution diagram shown in Fig. 2 according to the independent domain concept a set of experimental primary wetting or drying scanning curves must be available apart from the boundary curves of a reproducible hysteresis loop (see Poulovassilis, 1962). However, the values of the distribution function can be obtained by partitioning the slopes of the wetting boundary curve of an available reproducible hysteresis loop proportionately to the slopes on the drying boundary curve of the same loop or vice versa, that is, by partitioning the slopes of the drying boundary curve proportionately to the slopes on the wetting boundary curve. Water elements that enter the pore space when the suction relaxes from Si - 
S to Si following the wetting boundary curve (see Fig. 1) have to leave the pore space in the suction range from Si to Smax, after a reversal of the trend of suction from wetting to drying at a Si, contributing thus to the slope of the drying boundary curve in the suction range from Si to Smax. It is supposed that the contribution of these elements to the slope of drying boundary curve is proportional to the magnitude of that slope in the suction range from Si to Smax. Similarly, water elements that leave the pore space when the suction increases from Si to Si + S following the drying boundary curve have to reenter the pore space in the suction range from Si + S to Smin, after a reversal of the trend of suction from drying to wetting at Si + S, contributing thus to the slope of the wetting boundary curve in the suction range from Si + S to Smin. It is again supposed that the contribution of these elements to the slope of the wetting boundary curve is proportional to the magnitude of that slope in the suction range from Si + S to Smin. The procedure for obtaining the values is as follows. The difference e - f for all suction values applied is recorded. This difference for the suction value Si is ei - fi (see Fig. 1) and it is defined that
 | (3) |
the quantity under the integral being the pore water standing over the area CDEBC of the distribution diagram shown in Fig. 2. Similarly for the suction value Sn
 | (4) |
this quantity being the pore water standing over the area HQRBH. The value of the distribution function at the point G (fSi,eSn) is now given by
 | (5) |
when the slope of the wetting boundary at Si is partitioned proportionately to the slopes of the drying boundary for eS 
eSi or by
 | (6) |
when the slope of the drying boundary at Sn is partitioned proportionately to the slopes of the wetting boundary for fS 
fSn. The two i,n values given by Eq. [5] and [6] are equal since, as it can be shown, the right hand side of both equations is equal to
 | (7) |
the denominator representing the pore water standing over the area HGEBH, which is common to the area CDEBC and HQRBH. Thus, the partitioning of the slopes in both cases results in giving the same distribution of that satisfies the slope equations (Eq. [1] and [2]) for all values of fS and eS.
The quantities under the integrals in Eq. [5] and [6] must be determined beforehand by computing the n values for fS > fSi in the case of Eq. [5] or the i values for eS < eSn in the case of Eq. [6]. These computations may start at the dry end of the hysteresis loop in the former case or at the wet end in the latter. It is supposed that the function has zero values along the line OA of the distribution diagram (see also Philip, 1964). In the case the area OAB is subdivided by a grid of square mesh the above condition is approached by rendering the size of the mesh small enough and assigning zero values to the little triangles formed along the line OA. (See Fig. 3
, where an example of computing is given).
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Results and Discussion
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In Fig. 4 and 5
scanning curves calculated by the procedure described are compared with those obtained experimentally. It is evident that when the wetting boundary curve presents a substantial slope in a suction range from Smin to a value Sa over which the slope of the drying boundary is practically zero the partitioning of the slope of the wetting boundary curve proportionately to the slope of the drying curve over the above suction interval does not have meaning. This is the case for porous bodies for which practically no water leaves the pore space upon applying the process of drying from saturation till the drying suction reaches a suction value Sa (air entry value). This is the case where the application of the independent domain concept failed to describe the hysteretic behavior of such porous bodies and for which the concept of the nonindependence of domains was introduced in order to describe their behavior (Poulovassilis and Childs, 1971). It was supposed that nonindependent elements are formed in the pore space during wetting when the suction relaxes to values smaller that Sa. Their presence becomes evident when, upon reversing the trend of suction from wetting to drying at a reversal suction Sr < Sa, the resulting primary drying scanning curve shows a substantial slope for S < Sa, in contrast with the zero slopes of the drying boundary curve for suctions smaller than Sa. It was further supposed that nonindependent elements gradually acquire greater drying suction values than those at which they were initially formed so that after saturation all elements independent and nonindependent leave the pore space only when the drying suction acquires values greater that Sa. The process of the translocation of the nonindependent domains was clarified experimentally by drawing scanning curves of higher order than the primary ones.
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Fig. 4 Experimental boundary curves (solid lines and solid circles) and experimental primary wetting scanning curves (solid circles) presented originally by Poulovassilis (1970) and primary wetting scanning curves computed by the computational scheme proposed (dashed lines). Solid circles represent experimental points
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Fig. 5 Experimental boundary curves (solid lines and solid circles) and experimental primary drying scanning curves (solid circles) presented originally by Poulovassilis (1970) and primary drying scanning curves computed by the computational scheme proposed (dashed lines). Solid circles represent experimental points
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However, it may be attempted to partition, as it has been described here, the slopes of the drying boundary curve proportionately to the slopes of the wetting boundary in an effort to calculate the set of the primary wetting scanning curves. In Fig. 6
, calculated primary wetting scanning curves are compared with experimental ones. For approximating the paths of the primary drying curves an experimental primary drying curve is needed, preferably that with the larger slopes for suction values smaller than Sa. In the partial hysteresis loop formed by this scanning curve and the wetting boundary (see Fig. 7)
, one may attempt to approximate the path of the primary drying curves inside this loop by partitioning again the slopes of the wetting boundary proportionately to the slopes of the experimental primary drying curve or vice versa. Calculated by the procedure described here, primary drying scanning curves enclosed in the partial hysteresis loop are compared with the corresponding experimental ones. For approximating the paths of the primary drying curves outside the partial hysteresis loop one may compare the distribution diagram of the partial loop with that obtained when calculating the primary wetting curves. This comparison reveals the sites to which the existing nonindependent elements have finally to be transported when saturation is reached. For intermediate rS values the translocation may be approximated by supposing that it is proportional to the size of the remaining rS range as was originally proposed (Poulovassilis and Childs, 1971). A calculated primary drying curve with
is compared with the corresponding experimental one in Fig. 7.
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Fig. 6 Experimental boundary curves (solid lines and solid circles) and experimental primary wetting curves (solid circles) presented originally by Poulovassilis and Childs (1971) and primary wetting scanning curves computed by the computational scheme proposed (dashed lines). Solid circles represent experimental points
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Fig. 7 Experimental boundary curves (solid lines and solid circles) and experimental primary drying curves (solid circles) presented originally by Poulovassilis and Childs (1971) and primary drying scanning curves computed by the computational scheme proposed (dashed lines). Experimental primary drying curve ABC (solid line and solid circles) forms the drying boundary of the partial hysteresis loop used for calculating the primary drying curves inside the partial loop. Solid circles represent experimental points
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It may be seen from Fig. 4, 5, 6, and 7 that a good agreement between experimental scanning curves and calculated ones by means of the approach proposed here is observed. The approach is based on an assumption that is rather simple but quite relevant to the hysteresis phenomenon. The distribution of the function obtained by its application is unique, and the calculations involved are easy to perform.
Received for publication November 4, 1999.
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REFERENCES
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TOP
ABSTRACT
INTRODUCTION
. The volume 