application of fractal mathematics to soil water retention estimation

SOIL SCIENCE SOCIETY OF AMERICAJOURNAL

VOL. 53 JULY-AUGUST 1989 No. 4

DIVISION S-l-SOIL PHYSICS

Application of Fractal Mathematics to Soil Water Retention EstimationScott W. Tyler* and Stephen W. Wheatcraft

ABSTRACTIn this paper, we present an analysis correlating the fitting pa-

rameter a in the Arya and Paris (1981) soil water retention modelto physical properties of the soil. Fractal mathematics are used toshow that a is equal to the fractal dimension of the pore trace andexpresses a measure of the tortuosity of the pore trace. The fractaldimension of the particle-size distribution can be easily measuredand related to the a parameter of the Arya and Paris model. Bysuggesting a physical significance of the coefficient, the universalityof the model is greatly improved. Soil water retention data, estimatedstrictly from particle-size distributions, are proven to match mea-sured data quite well. The fractal dimension of pore traces rangefrom 1.011 to 1.485 for all but one soil tested.

NUMEROUS ATTEMPTS have been made to relateparticle-size distribution to soil water retention

data (Hall et al., 1977; Clapp and Hornberger, 1978;Gupta and Larson, 1979). The relative ease withwhich this data may be attained, as well as the simi-larity in shape of the retention and cumulative distri-bution curves for soils, suggested such efforts were jus-tified. In recent years, two approaches, Arya and Paris(1981) and Haverkamp and Parlange (1986), havebeen presented that show significant promise in relat-ing the particle-size distribution data to the retentioncurve in nonswelling soils. These simplified tech-niques provide a valuable method for bridging thedata availability gap. Traditional soil surveys providea wealth of spatially variable particle-size distributiondata, but are often lacking in retention data. Studiesof flow and transport in field soils, however, requireinput data on the variability of hydraulic properties(retention and conductivity data). Techniques thatcan bridge this data gap are therefore critical in esti-mating the impacts of agriculture and industry on soiland groundwater quality.Desert Res. Inst., Univ. of Nevada System, 7010 Dandini Blvd.,Reno, NV 89506. Received 10 June 1988. *Corresponding author.

Published in Soil Sci. Soc. Am. J. 53:987-996 (1989).

The Arya and Paris model (reviewed below) has re-ceived some criticism (Haverkamp and Parlange,1982, 1986; Arya and Paris, 1982) for its empiricism.In this paper, we significantly reduce this empiricism.We propose that the technique is physically sound andprovides a valuable alternative when it is necessary toestimate soil water retention when such data are lack-ing. This paper focuses on the technique of Arya andParis (1981) to apply physical significance and developestimative techniques for their curve-fitting param-eter, a. The physical significance is based upon theconcepts of fractal mathematics and scaled similari-ties.

MODEL DESCRIPTIONArya and Paris (1981) present a "physicoempirical" ap-

proach to water retention data. The underlying assumptionof the model is that the soil particle size is related to a cor-responding pore diameter. This forms the "physical" basisfor the model. The model treats the soil as a bundle of cap-illary tubes. Each capillary tube corresponds to a user-de-fined particle-size class. The capillary tube volume is takento be a function of the particle size, the weight fraction ofthe particle size and an empirical fitting coefficient, a. Thefitting parameter is derived from least squares regression ofthe predicted water content to the measured water content.This term forms the "empirical" side of the model. As cor-rectly pointed out by Haverkamp and Parlange (1982), thistechnique is somewhat sensitive to the user-defined particle-size classes. In this paper, we recognize this limitation, how-ever, we show that the model is conceptually based on phys-ical principles.

Upon division of the particle-size distribution data intoM size fractions, the solid mass in the Mi particle class isequated to the mass of N, spherical particles of radius Rt.Their volume, Vfi is given by

Vpi = 4/3 Nt TT [1]The volume of the voids, KV1, is represented by a single cap-illary tube of radius r,.

Vvi = TT r}h, [2]where ht is the capillary tube length. The length (h,) of a

987

988 SOIL SCI. SOC. AM. J., VOL. 53, JULY-AUGUST 1989

straight capillary tube in a cubic close-packed arrangement,as measured in units of Rh is simply the product of thenumber of particles of radius Rf, multiplied by their diam-eter (2R,). Arya and Paris (1981) assume that the particlesare not spherical and represent the true capillary pore lengthof the RJ class as

hi = 2 Ri N" [3]where a is an empirical constant between 1 and 2. The useof Eq. [3] is justified by Arya and Paris to account for thenonspherical nature of the particles.

By combining Eq. [1], [2], and [3], the authors arrive atan expression for the capillary tube radius, r,

r, = R,{2/3eNfl - «>} [4]

where e is the sample void ratio. The void ratio for eachpore class is assumed constant and equal to the bulk samplevoid ratio. The tube radius is then related to the capillarypressure, ^,, via the equation of rise in a capillary tube. Thewater content corresponding to the capillary pressure \l/if isevaluated by summing the available pore space containedin all particle classes from the smallest class to the rth class.

The technique does not account for hysterisis nor en-trapped air. The latter condition may be overcome by as-

h = Nd constant

= N,

Fig. l.(a) Straight capillary tube comprised of uniform spheres, (b)Fractal capillary tube in which length is a function of measurementscale

suming that any entrapped air is equally divided among allpore classes (Haverkamp and Parlange, 1986). By the ap-plication of the equation of capillary rise (and assuming azero contact angle), the technique most closely models themain drying curve for a nonswelling soil.

EVALUATION OF THE a TERMThe power law relationship given by Eq. [3] indicates that

as Rt decreases (assuming Nt increases proportionally) h, willgrow exponentially. This apparent length-increase behaviorhas received considerable attention through the subject offractal mathematics (Mandelbrot, 1983).

The development of fractal mathematics has shown thathighly irregular features such as coastlines, rivers, and frac-tures (and capillary tubes) can be categorized and quantifiedin a fundamentally new and different way (Mandelbrot,1983). One of the most important features of a fractal objectis that its "degree of irregularity" is independent of scale.

In normal Euclidean geometry, we can easily measure thelength of a straight line of length L. If our measuring unitis of length e (« < L), then

L(e) = Nfl = constant [5]where N is the number of measuring units needed to coverthe straight line and the exponent of unity is consistent withthe topologic dimension of a line. This analogy may be ex-panded to two or three dimensions (a plane or cube, re-spectively) where the exponent reflects exactly the topolog-ical dimension (plane = 2.0 and cube = 3.0).

If the line we are measuring is irregular, Eq. [5] remainstrue except that the length, L(e), for any e is not constant. Ithas been found that the following relationship holds true forirregular lines, such as a coastline

F = N(D = constant. [6]By analogy with Eq. [5], F is taken to be a measure of theline length that is independent of e, and D is taken to be thedimension that yields the constant length (F) of the line.

By combining Eq. [5] and [6], we obtainL(e) = F€I'D [7]

that can be thought of as a transformation relationship be-tween the topological dimension of one and the fractal di-mension of D for a fractal line. The reader is referred toMandelbrot (1983) and Feder (1988) for a more completereview of fractal concepts.

Instead of the straight capillary tube approach of Arya andParis (1981), fractal measures can be used to evaluate thepore length as a function of measuring scale. In this ap-proach, smaller and larger particles line the pore wall. Figurela represents a Euclidean (nonfractal) pore wall whoselength is independent of the scale at which it is measured.The volume of the pore space is easily measured and con-stant. Figure Ib, however, represents a more realistic soilpore. Both large and small grains line the pore. If we mag-nified any portion of the pore, we would see yet more smallgrains lining the pore wall. The volume of such a pore istherefore a function of the scale at which it is measured. Itslength is also a function of scale. The pore length and radiusare "measured" in a retention experiment by varying thetension applied to the soil water. An equivalent radius maybe estimated from Laplace's equation, while pore lengthmay be estimated from the change in water content. Theleft-hand trace in Fig. Ib is shown measured with a rulerlength of di, and the total pore length, hi, is simply N, dt.The right-hand side of Fig. Ib shows the same pore as mea-sured with a smaller ruler size, d2. As we decrease the rulersize from dl to d2, it is clear that the estimated pore tracelength, h2 will be larger than h, since we can more closely

TYLER & WHEATCRAFT: APPLICATION OF FRACTAL MATHEMATICS TO SOIL WATER RETENTION ESTIMATION 989

follow the irregularities of the pore wall. In contrast, as d ismade smaller in Fig. la, the estimate of the line length (orpore trace) does not change. For the fractal pore trace in Fig.Ib, however, the length continues to increase as d is madesmaller due to irregularities that exist at all scales. The poretrace is therefore denned as nonrectifiable. These conceptsof fractal measures are physically related to a measure oftortuosity (Wheatcraft and Tyler, 1988).

The true length of the pore channel, /z,*, as a function ofRi is given by Eq. [7] as

ht* = [8]where we have replaced«with 2R,, the measuring unit. Theconstant F may be evaluated from Eq. [6] by setting ourmeasuring length, 2Rh equal to the straight line length of/z,*. In this case; N> = 1 and

F = h,D. [9]Substitution of Eq. [9] into Eq. [8] yields an expression

for the true fractal pore length at the rth scale of measure-ment in terms of the particle size and straight line length

h* = h?(2R,)l-D. [10]But the estimated length hh as given by Arya and Paris is

simply equal to 2/?,Ar,; (Fig. la). Arya and Paris invoke apower law scaling relationship with the justification that thiswill account for nonspherical particles suggesting that A, islonger than 2RiNi. This same scaling relation also defines afractal scaling or self-similar pore channel. The term self-similarity implies that the pore trace contains quantifiableirregularities at all scales of observation. By substituting therelationship that h, = 2/?,-7V/ into Eq. [10], we obtain equiv-alent results to Arya and Paris based upon fractal concepts

h', = 2RiN?. [H]The exponent D in Eq. [11] is the fractal dimension de-

scribing the tortuosity of the pore channels and is equivalentto Arya and Paris' a (Eq. [3]). A low fractal dimension in-dicates a fairly straight path while a D of 1.5 yields a verytortuous path. Numerical experiments (Wheatcraft and Ty-ler, 1988) indicate that a fractal dimension > 1.5 appearsto yield physically unrealistic pore channel representations.

The volume of the fractal pore channel, Vvi can now beexpressed as

TT rf N?2Ri. [12]We now replace Eq. [2] with Eq. [12] in the Arya and

Paris model, which results in a similar equation for the ef-fective pore radius

ri = Rl{2/3eNil-D]1'2. [13]

The difference, however, is that the exponent D is a truemeasure of scale-dependent tortuosity of the pore trace.

The capillary pressure, ^,, in terms of equivalent heightof water for the corresponding pore radius, rh is obtainedusing the equation of rise in a capillary tube and is givenbelow

r 1-1/22/3ety'-D [14]

where 7 is the surface tension of water, ft is the contact angle,pw is the density of water, and g is the accelerations, due togravity.

In order to apply Eq. [14], it is necessary to assume thatthe bulk sample void ratio is equivalent to the void ratio ofeach particle class. This assumption, in light of the powerlaw scaling used to estimate the pore length, has interestingramifications. Most pore-size/grain-size relations (for ex-ample, see Haverkamp and Parlange, 1986) assume a linear

relationship of the form: rt = CRh where C is a constantindicative of the grain packing. Such linear behavior is notpreserved in porous media exhibiting fractal pores.

Assuming a constant void ratio for all pore classes, Eq.[1] and [12] may be used to show the relationship of poresize to grain size. The void ratio is given as

e = TT /f A/f 2Rt

4/3 TTRearranging terms yields

r,

[15]

[16]where C = (2/3e)1/2.

The relationship between pore and grain size is now afunction of the distribution and number of soil grains. Equa-tion [16] shows several interesting features. For soils exhib-iting nonfractal pores (D = 1), Eq. [16] reduces to the tra-ditional linear proportionality of grain size to pore size. Thesame is also true if the soil exhibits a constant number ofgrains in each particle-class size. Such a soil would be dom-inated (on a weight basis), however, by a few large grains.

Most soils, however, show an increasing number of grainswith decreasing grain size. For such soils, Eq. [16] indicatesa more rapid decrease in effective pore radius with decreas-ing grain size than a purely linear decrease. Such behavioris consistent with fractal pore traces. Such traces will showincreased tortuosity as the scale of observation is decreased.This increased tortuosity effectively lengthens the pore, re-sulting in a smaller cross-sectional area (pore radius) nec-essary to maintain the assumption of a constant void ratio.

ESTIMATION OF THE FRACTAL DIMENSIONArya and Paris (1981), in their study of soils and soil mois-

ture, based their estimates of a (our D) on a mean squareddifference between measured and predicted capillary pres-sures. Water content at each capillary pressure was esti-mated based upon the particle mass at the rth particle sizeand the sample void ratio. Based upon this, Arya and Parisreport good agreement between measured and predictedwater retention data on most of the 15 soils reported. Theirfitting coefficient, a, was found to vary between 0.9 and 1.5.

If fractal concepts are to be of use in estimating a, it isnecessary that an estimation technique be available for thefractal dimension. Turcotte (1986) has shown that particlesizes of geologic material exhibit fractal behavior of the form

NR? = constant [17]where TV is the total number of particles of radius greater

than RJ and D is the fractal dimension of the particle-sizedistribution. The fractal dimension defines the distributionof particles by size. For D = 0, the distribution is composedsolely by particles of equal diameter. When the fractal di-mension is equal to 3.0, the number of particles greater thana given radius doubles for each corresponding decrease inparticle mass by one-half [or particle radius decrease of(1/2)1/3]. A fractal dimension between 0 and 3.0 thereforereflects a greater number of larger grains, while a dimension> 3.0 reflects a distribution dominated by smaller particles.Turcotte (1986) presents data on the fractal dimension of 21particle-size distributions. Of the distributions presented,those representing soils had fractal dimensions approaching3.0.

It is possible to estimate the fractal dimension of the par-ticle-size distribution by plotting the cumulative number ofparticles larger than a given sieve size. Equation [17] maybe rearranged and plotted on a log-log scale. The fractal di-mension D will be equivalent to the negative of the slope.Since mechanical sieving yields a distribution of particlesizes between two successive sieves and it is impractical to


count the number of particles directly, it is necessary tochoose a "representative" particle radius for a given sievesize. For this analysis, this radius was chosen as the arith-metic mean between two successive sieve sizes. The numberof particles assigned to each sieve was calculated by dividingthe retained weight by the weight of a particle of mean radiusbetween the two successive sieve sizes. The particle densitywas assumed to be 2.65 g/cm3 for all analyses in this paper.

The choice of an arithmetic mean particle size to estimatethe fractal dimension is consistent with the Arya and Parismethod of representative radii of successive pore classes.Other averages (harmonic, geometric, etc.) could be used,however, it would also be necessary to estimate /?, (Eq. [14])using an equivalent averaging process. The use of the meanparticle radius will have some affect on the estimated fractaldimensions. We are currently investigating various othertechniques to more uniquely define the dimension from par-ticle-size distributions.

In our case, it is necessary to estimate the fractal dimen-sion of the one-dimensional trace of the pore channel. Thescale of the measurement unit (2R,) will determine the porechannel length. Mandelbrot et al. (1984) has suggested thatthe difference between an object's fractal dimension and itstraditional topologic dimension (DT) be denoted as the frac-tal increment, £>,. It has also been suggested (Mandelbrot etal., 1984) that this fractal increment can be used to estimatethe fractal dimension of lower topological-dimensioned ob-jects taken from the original fractal process. For example,the fractal dimension of a profile taken across a fractal sur-face will have the same fractal increment as the surface. Ifthe fractal dimension of the surface is 2.5 (a highly irregularsurface), the fractal increment is then 0.5 and the profile'sfractal dimension would be 1.5 (1 +0.5). This "slit island"technique has been used extensively in the literature to es-timate the fractal dimension of surfaces and volumes fromtransects, cross sections, and contours. In this study, wemake the assumption that the fractal increment (D — DT)obtained from the grain-size distribution may be used toestimate the fractal dimension of the pore trace, where D isthe fractal dimension of the particle-size distribution. Sincea particle-size distribution represents a three-dimensionalcollection of soil grains (DT = 3), the fractal increment isgiven by D — 3. The fractal dimension of the pore trace(viewed as a one-dimensional transect through a three-di-mensional soil matrix) is then 1 + Dt. The fractal dimensionof the pore trace can range from 1 (a pore trace whose lengthis independent of measurement scale) to 2 (a trace that com-pletely fills the plane).

MODELS AND MATERIALSArya and Paris (1981) present complete cumulative dis-

tribution data and water retention data for five soils (soilsB-F). Both particle-size and water retention data were dig-itized from their original paper for calculation and compar-ison. In addition, five soils cataloged in Mualem (1976) wereanalyzed. The soils ranged in texture from sand (Oakleysand) to silty clay loam (Arya and Paris' soil B). Cumulativenumber of particles (N) were calculated based upon 1-g sam-ples and an assumed particle density of 2.65 g/cm3. Particle-size classes (R,) were chosen as the mean radius betweensuccessive sieve sizes.

The fractal dimensions (D) of the particle-size distribu-tions were calculated from the slope of the log particle sizevs. log number of particles. A least squares regression wasused to estimate D from the log-log plot.

The capillary pressure for the r'th particle-size class wascalculated using Eq. [14] assuming a zero contact angle andfluid properties at 25 °C. The soil water content 0,, for each<l/i was calculated as in Arya and Paris (1981) by summingthe available pore space from the smallest size class up to

the fth class. The void ratios were estimated from bulk den-sity or saturated water content data depending upon the datasource. The resulting pairs of capillary pressure and watercontent (&,, 6,) were fitted using van Genuchten's retentionmodel (1980)

(1 + [18]

where a, n, and m, and 6r were taken to be fitting parameters.The fitted equation was used to estimate root mean

squared (RMS) error between measured and predicted waterretention data using the fractal approach to the Arya andParis model. The RMS error was calculated as

RMS =1- 4 ''measured 'predicted.

1/2

[19]

where Nf is the number of measured pairs of water contentand pressure head, with 4 degrees of freedom removed dueto the fitting parameters used in Eq. [18]. For two soils,saturation data were used to improve the fit of the data. Inthese cases, RMS error was calculated in terms of satura-tion^) divided by the arithmetic mean of reported saturatedwater content and bulk density-derived porosity.

RESULTS AND DISCUSSIONFractal particle-size relations are shown in Fig. 2a-

d, 3a-d, and 4a, b for the 10 soils analyzed. With theexception of the Sable de Riviere sand (Fig. 4b), thesoils showed clear fractal behavior according to Eq.[17]. The fractal dimension of the distributions rangedfrom 2.7 to 3.485, while fractal dimensions of the es-timated pore traces ranged from 0.7 to 1.485. The frac-tal dimensions (D) of all but the Sable de Riviere sandwere >3.0. It is interesting to note that the powerfunction fit is rather poor for this soil. Table 1 showsthe distribution of fractal dimensions for the soils ana-lyzed. The fractal dimensions of the particle-size dis-tributions were generally larger than those reported byTurcotte (1986). It is not clear if these differences arerelated to the choice of the mean particle radius orthe more coarse texture of soils investigated by Tur-cotte (1986).

Most of the fractal dimensions of pore traces cal-culated from the particle-size distributions fall in therange of previous work. Jacquin and Adler (1985) re-port the fractal dimension of a one-dimensional, fluid-gas interface in a packed column to be approximately1.3. Jacquin and Adler (1988) observed fractal incre-ments of 0.23 and 0.34 in pore structure of dolomiticlimestone. Pines-Rojanski et al. (1987), in a series ofadsorption experiments, estimated fractal incrementsof silica gels to be between 0.05 and 0.82. Katz andThompson (1985), in a study of the fractal nature ofsandstone porosity, clearly show fractal scaling of thepores over a wide range of scales. They report porespace fractal dimension increments between 0.57 and0.87. The fractal increments from the cemented sand-stones used by Katz and Thompson were higher thanfor the soils investigated in this study, but are con-sistent with the type of materials investigated. It isexpected that sandstone pores will be significantly"rougher," i.e., higher dimensioned, due to cementa-tion and diagenetic alteration. In this case, the pore


structure may be less dependent upon the initial grain-size distribution than in unconsolidated soils.

Calculated water retention data for each of the 10soils are shown along with reported retention data inFig. 5 through 14. Of the 10 soils analyzed, 7 showed

good to excellent agreement with the measured reten-tion data. The three soils showing poor agreementwere the Sable de Riviere, Oakley sand, and Arya andParis soil F. These are the coarsest textured soils in-vestigated. These soils generally also had the lowest

OL

A|I/)z

o 10N

UJm2

10'-

10'

SIEVE DATAN = 1.63R-3'4M

io~3 io~2 i<rMEAN RADIUS, R (mm)

a:o

Um2:Dz

10'°-

107-

10'-

10'

SIEVE DATAN = 1.73R-"19

10"3 10~2 10~1

MEAN RADIUS, R (mm)

SIEVE DATAN = 3.11 FT3264

10'MEAN RADIUS, R (mm)

o:A|i/iz

10'°-

g 107-|

UJm=>Z

10"-

SIEVE DATAN=6.2R-3163

ID'3 1C'2 10'1

MEAN RADIUS, R (mm)

Fig. 2. Fractal particle distribution for (a) Arya and Paris soil B; (b) Arya and Paris soil C; (c) Arya and Paris soil D; and (d) Arya and Parissoil E.

I/)z

101'

107.

104

101

« SIEVE DATA—— N = 9.4R-3011

1<T3 1C'2 10-MEAN RADIUS, R (mm)

« 1°8

£ 106-|

* io'-|u.O

u 102-]m

10°

• SIEVE DATA—— N = 2.7R-3485

10-1

MEAN RADIUS, R (mm)

A| 1(Jio

Z

o 107

oo: 10".uim

101

. SIEVE DATA—— N = 14.92R-3M°

1Q-3 ID'2 1C'1MEAN RADIUS, R (mm)

10"

o 107

Cf.UJm

104

10-

« SIEVE DATA—— N = 19.95R'3071

10- 10-C M E A N RADIUS, R (mm)

Fig. 3. Fractal particle distribution for (a) Arya and Paris soil F; (b) Columbia silt; (c) Gilat sandy loam; and (d) Yolo light clay.


fractal dimensions. Since the pore trace dimension ofthe Sable de Riviere sand was <1.0, implying non-

Table 1. Fractal dimensions from particle-size data.

Soil typeSand

Sable de RiviereOakley sand

Particle fractaldimension

2.703.138

Pore fractaldimension

0.701.138

Sandy loamArya and Paris soil F 3.011Gilat sandy loam 3.160

Clay loamYolo light clay 3.071

Silty clay loamArya and Paris soil B 3.404

LoamArya and Paris soil D 3.264Arya and Paris soil E 3.163

1.0111.160

1.071

1.404

1.2641.163

Silty loamArya and Paris soil C

SiltColumbia silt

3.419

3.485

1.419

1.485

fractal behavior, water retention data was calculatedusing both D = 0.7 and D = 1.0 (Fig. 14). It is ap-parent, however, that a poor agreement exists usingeither D = 0.7 or D = 1.0. The reasons for this areunclear, however, the low fractal dimension of theparticle-size distribution (2.7) indicates that the soil ispredominantly composed of larger particles, with asignificantly decreasing number of smaller size parti-cles. This is not surprising since the soil is a sand andtherefore it may not be appropriate to apply the scale-dependent tortuosity concepts inherent in the Aryaand Paris model. This soil was also poorly fit by Eq.[17] and therefore the assumption of fractal scalingfor this soil is suspect. This soil exhibits a very sharpretention curve implying a very narrow range of poresizes that may not be distinguishable with a relativelycoarse sieve analysis. These results are consistent withSchuh et al. (1988), who reported fitted values of a tobe highly variable near saturation for coarse texturedsoils. Schuh et al. (1988) also suggests that a (in ourstudy D) was not constant over all pore classes andgenerally increased with decreasing pore size. Al-

o;Al

in5o;Ou.oo;um3

101

107

104

101

SIEVE DATAN = 3.5R'3138

10- 10-2

MEAN RADIUS, R

OL

A| 106

(ft

g 104

o

g 1°2m

~?o°A

10°10"

SIEVE DATAN = 15.7FT27

10-1

MEAN RADIUS, R(mm) A MEAN R A D I U S , R (mm)Fig. 4. Fractal panicle distribution for (a) Oakley sand and (b) Sable de Riviere sand.

10°B

106-,

EV 10'LJIT3COCOUlDCDL

ft

E 102 -o

10° I0.1

TENSIOMETERPRESSURE PLATEPREDICTED

I0.2

I0.3

I0.40.0 0.1 0.2 0.3 0.4 0.5

VOLUMETRIC WATER CONTENTFig. 5. Predicted and measured retention data for Arya and Paris

soil B.

105-,

10"-

103 -COCOUJccOL

io2-Q.

O

10'

10°

TENSIOMETERPRESSURE PLATEPREDICTED

0.0T

0.1I

0.2I

0.3T

0.4

.00

I0.5

VOLUMETRIC WATER CONTENT

Fig. 6. Predicted and measured retention data for Arya and Parissoil C.


though the fractal approach presented in this paperpermits the fractal dimension to vary, the estimationfrom particle-size distribution may not be straightfor-ward. It is important to note, however, that in therange of —10 to —1000 cm of tension, Schuh et al.(1988) found that a was fairly constant with valuessimilar to those found in this study. Such behavior

104

UJ£E3COtoUJ(E0.

1C

0.

o

102-

10°

suggests that a constant value of a as determined fromparticle-size distribution may be appropriate over therange of tensions typically found in many soil waterproblems. The finer textured soils that showed a widerdistribution in particle sizes (and hence, a larger va-riety of pore classes) showed characteristically higherfractal dimensions, while the coarse textured soilsshowed smaller fractal dimensions.

* TENSIOMETERo PRESSSURE PLATE

PREDICTED

TENSIOMETER

PRESSURE PLATEPREDICTED

0.0 0.1I

0.2I

0.3I

0.4I

0.5VOLUMETRIC WATER CONTENT

10°0.0

Fig. 7. Predicted and measured retention data for Arya and Parissoil D.


105-,

104-

UJccDCOCOUlDCD.

£C

OLo

102

10'

10°

TENSIOMETERPRESSURE PLATE

PREDICTED

Fig. 9. Predicted and measured retention data for Arya and Parissoil F.

MEASUREDPREDICTED

0.0I

0.1I

0.2I

0.3 0.4I

0.5VOLUMETRIC WATER CONTENT

Fig. 8. Predicted and measured retention data for Arya and Parissoil E.

10

0.0 0.2 0.4 0.6 0.8 1.0

SATURATION

Fig. 10. Predicted and measured retention data for Columbia silt.


Based upon these results, it appears that the appl-icability of the fractal model can be determined byinspection of the plot of particle number vs. particlesize. Soils exhibiting fractal increments close to or lessthan zero are likely to produce poor approximationswhile increments in the range of 0.1 to 0.5 should

10s _,

104-

(E

COto111OC0.

C

0.

o

103-

102 -J

10°

MEASUREDPREDICTED

0.0I

0.1 0.2 0.3 0.4 0.5


Fig. 11. Predicted and measured retention data for Gilat sandy loam.

104-i

103H

9,uioc3COCO

102-

0.

o101-

10° •

MEASUREDPREDICTED

0.0 0.1 0.2 0.3 0.4 0.5


Fig. 12. Predicted and measured retention data for Yolo light clay.

produce accurate retention data. Particular interest iscalled to the Yolo light clay soil (Fig. 3d and 12). Thefractal dimension as well as the water retention datawere calculated using only three particle-size classes:sand, silt, and clay. Even with such minimal data, the

104

io3-

COCO

102-

c

a.O

101 -

10°

o o o o o oo

o MEASURED——— PREDICTED

I0.2

T0.4

I0.6

I0.80.0 0.2 0.4 0.6 0.8 1.0

SATURATION

Fig. 13. Predicted and measured retention data for Oakley sand.

103-!

o MEASUREDPREDICTED (D = 0.7)

--- PREDICTED (D = 1.O)

O o

10°0.0 0.1


Fig. 14. Predicted and measured retention data for Sable de Rivieresand.


calculated retention data closely fits the measureddata. It is important to note that the Oakley sand andColumbia silt are plotted as saturation. Based uponthe bulk densities reported for these soils, the reten-tion experiments did not completely saturate the soils.Table 2 shows the RMS error for each of the soilsanalyzed.

The apparent residual water contents predicted withthe fractal model were generally less than the reportedwater contents at high tensions. This is not unreason-able since the Arya and Paris model assumes completedesorption of all pores of a given class size at the crit-ical pressure. At low tensions, this assumption appearsreasonable, however, at high tensions, a significantpercentage of water may be held as films and in dead-end or poorly connected pores. As a result, the modelwill tend to underpredict the water content in the hightension regions. Fortunately, the range of interest formany flow and transport problems is at low to inter-

Table 2. Calculated root mean squared (RMS) error (in water con-tent).

Soil RMS, cm3/cm3

Arya and Paris soil BArya and Paris soil CArya and Paris soil DArya and Paris soil EArya and Paris soil FColumbia siltGilat sandy loamYolo light clayOakley sandSable de Riviere sand

0.0140.04030.0370.0770.1410.023f0.0640.0380.137f0.249(0 = 0.7)

t Calculated from saturation data and the average of calculated and reportedsaturated water content.

104_

moc3ininuis.Q.

102-

Q.<o

10° \

t- TENSIOMETERo PRESSURE PLATE

———— D = 1.4O4———— D = 1.00— - - - D = 1.20———— D = 1.60

I I I0.2 0.3 0.4 0.5

Fig. 15.sion.

0.0 0.1VOLUMETRIC WATER CONTENT

Sensitivity of predicted water content to the fractal dimen-

mediate tensions where the model appears to be mostaccurate.

The sensitivity of the analysis to the fractal dimen-sion is shown in Fig. 15. Water retention data werecalculated for Arya and Paris' soil B for three differentvalues ofD (1.0, 1.2, and 1.6) and are shown in com-parison to the retention data generated with the fractaldimension estimated from the particle-size distribu-tion (D = 1.404). Figure 15 shows the extreme sen-sitivity to the fractal dimension and clearly shows thatthe fractal dimension obtained independently fromthe particle-size distribution is appropriate to estimatethe retention data.

CONCLUSIONSThe empirical constant (a) used in the Arya and

Paris model is shown to be equivalent to the fractaldimension of a tortuous fractal pore. This analysis sig-nificantly reduces the empiricism of their water reten-tion model. A relationship is developed to estimatethe pore trace fractal dimension and hence, the porechannel length that is strictly scale-dependent basedupon the mean particle radius. The concepts of fractalpore spaces also imply a nonlinear relationship be-tween grain size and equivalent pore radius in conflictwith traditional analysis.

A simple method is presented to relate the fractaldimension of the particle-size distribution to the di-mension of the fractal pore trace. The relationship be-tween particle-size distribution and pore geometry re-duces the model input variables to bulk density andcumulative particle-size distribution data.

Ten soils in which retention and particle-size datawere available were analyzed to obtain both the fractaldimension and, subsequently, the water retention datausing the Arya and Paris model. The soil texturesranged from sand to silty clay loam to silt. The fractaldimension of the particle-size distributions generallyincreased with decreasing texture. Of the soils inves-tigated, all but one soil showed clear fractal or powerlaw scaling behavior. Pore fractal dimensions rangedfrom < 1.0 to 1.4, in agreement with other reporteddata. Using the fractal dimension from the particle-size distribution, the estimated water retention dataclosely matched the observed data for all but the threecoarsest soils. For most of the soils analyzed, the watercontents at high tensions were slightly underesti-mated. The results are consistent with those reportedby Schuh et al. (1988), suggesting that a constant valueof«(equivalent to the fractal dimension in our model)may not be appropriate at high soil water tensions.

These results indicate that water retention data maybe estimated with reasonable accuracy for soils inwhich the particle-size data shows power law scalingwith a fractal dimension of >3.0. Such soils are thosewith a wide range of particle sizes. The results alsoindicate that a nonlinear relationship between poreand grain size is appropriate for such soils and shouldbe incorporated into retention and conductivitymodels. Such work is currently underway.

With the physical significance of the empirical termexplained, the Arya and Paris model is shown to be


physically based and should prove useful for efficientestimation of water retention data where field or lab-oratory measurements are not available.

ACKNOWLEDGMENTSThis work was funded by the Nevada Agency for Nuclear

Projects/Nuclear Waste Projects Office under Dep. of En-ergy Grant no. DE-FG08-85-NV10461. The opinions ex-pressed in this paper do not necessarily represent those ofthe State of Nevada or the U.S. Dep. of Energy.

application of fractal mathematics to soil water retention estimation

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