# Fractal Fragmentation, Soil Porosity, and Soil Water Properties: II. Applications

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Fractal Fragmentation, Soil Porosity, and Soil Water Properties: II. ApplicationsMichel Rieu* and Garrison Sposito

ABSTRACTThe fractal model of Rieu and Sposito contains seven predictive

equations that can be tested experimentally with data on aggregatecharacteristics and soil water properties for structured soils. How-ever, data with which to test the model are extremely limited atpresent because of the need to have precise, concurrent measure-ments of aggregate physical properties (bulk density and size dis-tribution) along with soil water properties for an undisturbed soil.For the five sets of suitable physical soil aggregate data currentlyavailable, good agreement was found with the fractal model bulkdensity-aggregate size and size-distribution relationships. For thesingle set of aggregate/soil water properties data available, goodagreement also was found with the fractal model water-potentialscaling relationship, moisture characteristic, and hydraulic conduc-tivity-water content relationship. Model simulations of the last twosoil water relationships for hypothetical sandy and clayey soils alsowere qualitatively accurate and showed the sensitivity of the modelto the value of the fractal dimension. These encouraging results sug-gest that the model should have success in further experimental testswith natural soils.

THE CONCEPT of a fragmented fractal structure in aporous medium was developed by Rieu and Spos-ito (1991) to derive equations that relate porosity, bulkdensity, and aggregate size-distribution-characteristicsof soil structureas well as water content, water poten-tial, and hydraulic conductivityto fractal parameterssuch as the similarity ratio and the fractal dimension.The equations derived (see Appendix) are testable withsuitable experimental data. We performed a limitedtesting of these equations with available data on aggre-gate porosity, bulk density, and size distribution, andwith data on soil water properties.

MATERIALS AND METHODSAggregate Properties

Numerous studies of aggregate-size distribution have beencarried out, as reported by Gardner (1956), but almost noneincludes other physical properties of aggregates, such as bulkdensity. An exception is the work of Chepil (1950), whocompared three different methods of measurement of thebulk density of aggregates in three soils of differing texture.The results considered best by Chepil (1950) are reported inTable 1. Wittmus and Mazurak (1958) studied the aggregatesin the Sharpsburg soil (fine, montmorillonitic, mesic TypicArgiudoll) and determined both their size distribution andbulk density. Their data are reported in Table 2. No otherprecise published data of this type were found in our searchof the literature. For each soil, the mean diameter, 'dh of eachsize class i was computed (second column in Tables 1 and 2)along with the quantities djd0 and ajaa, where

1240 SOIL SCI. SOC. AM. J., VOL. 55, SEPTEMBER-OCTOBER 1991

Table 2. Mass distribution and bulk density of aggregates of Sharps-burg soil (Wittmuss and Mazurak, 1958).

Size class Mean size Oversize mass Aggregate densitymm

9.250-4.764.760-2.382.380-1. >91.190-0.590.590-0.2970.297-0. J 490.149-0.0740.074-0.0370.037-0.0185

mm

7.0053.5701.7850.8850.4460.2240.1110.05550.02775

kg IV0.01770.00630.01080.01880.02160.01380.00210.0060 :0.0025

Ignr3

.320f

.373

.410

.480

.510

.540

.6502.1002.360

t Extrapolated.

Table 3. Physical properties of the Ariana soil.Aggregate-size distribution

Class number

012345678-

Size classmm

2.00-1.601.60-1.251.25-1.001.00-0.800.80-0.630.63-0500.50-0.315

0.315-0.200.20-0.10

RIEU & SPOSITO: FRACTAL FRAGMENTATION, SOIL POROSITY AND, SOIL WATER PROPERTIES: II. 1241

Table 4. Parameters resulting from linear regression of log (

1242 SOIL SCI. SOC. AM. J., VOL. 55, SEPTEMBER-OCTOBER 1991

Table 6. Fractal parameters for the Ariana and hypothetical sandyand clayey soils.

Property Symbol Ariana Sandy soil Clayey soilBulk density (Mg nr3)Particle density (Mg nr3)Similarity ratioClustering factorBulk fractal dimensionFractal dimensionFracture opening (mm)Fragmentation numberReduced pore coefficient

"offmrFD,DP,mr.

1.4092.610.820.4182.902.830.068

310.01965

1.412.610.820.4182.882.790.081

260.02353

1.4162.610.820.4182.952.910.034

620.0099

jeo

o>o

ARIANASOIL

I I I I-1.0 -0.75 -0.50 -0.25 0 0.25 0.50

log rdi.(mm) jFig. 5. Test of Eq. [A3] with the aggregate-size distribution for the

Ariana soil.

is expected to be smaller than DT determined from Eq.[A2]. The difference, Dr D, expresses the decreasein fractal dimension resulting from the completion offragmentation.

Figure 5 shows a log-log plot ofN(dk) vs. dk for theAriana soil, based on data in Table 3 and Eq. [1] and[2]. (The values of

RIEU & SPOSITO: FRACTAL FRAGMENTATION, SOIL POROSITY AND, SOIL WATER PROPERTIES: II. 1243

0.1 0.2 0.3 0.4 0.5Water Content {m3rrf3)

Fig. 8. Test of Eq. [A5] and [A7] with water-retention data for theAriana soil and hypothetical moisture characteristics for fractalsandy and clayey soils in Table 6.

1), the soil water content was calculated with Eq. [A7],the water potential with Eq. [A5], and the hydraulicconductivity with Eq. [A6]. The results are comparedwith experimental values taken from Table 3 in Fig.8 and 9. The excellent agreement between the modelequations and experiment in Fig. 8 and the good agree-ment in Fig. 9 suggest that the concept of an incom-pletely fragmented fractal porous medium isconsistent with the structure of the Ariana soil. It ispossible that the somewhat poorer agreement betweenthe model and data in Fig. 9 results both from lesserprecision in the conductivity measurements than inthe matric-potential measurements and from a likelygreater sensitivity of the conductivity to the nonfractalstructure in a porous medium.

The sensitivity of the model to the fractal param-eters was examined by repeating the calculation aboveusing different fractal dimensions under the assump-tion of constant similarity ratio, r, F, and $. Values ofDt = 2.88 and Dr = 2.95 were determined for sandyand clayey soils, respectively, in Table 1, whereas theAriana soil (silty clay loam) has an intermediate valueof 2.90. Fractal fragmented models of hypotheticalsandy and clayey soil structures thus can be developedwith the Ariana soil as a reference (Table 6). The valueof D was calculated with Eq. [43] in Rieu and Sposito(1991) and the exponent m was calculated with Eq.[Al] following the method used to compute this pa-rameter for the Ariana soil. Since the parameter rr canbe interpreted physically as the partial porosity con-tributed by pores of a given size (Rieu and Sposito,1991, Eq. [35]), the fracture opening p0 should scalewith r

PO'/PO = rr'/rr [10]Equation [10] was used to calculate p0 values for thesandy and clayey soils with Tr and p0 (Eq. [4] and [8])for the Ariana soil taken as a reference. The resultingmodel relations between water content, water poten-tial, and hydraulic conductivity are presented in Fig.

oo

T3CoO

0.1 0.2 0.3 0.4 0.5Water Content (m3 m"3)

Fig. 9. Test of Eq. [A6] and [A7] with hydraulic-conductivity datafor the Ariana soil (dashed line) and hypothetical hydraulic-con-ductivity curves for fractal sandy and clayey soils based on theparameters in Table 6. The solid line associated with the Arianasoil is a plot of the laboratory-based relation for hydraulic con-ductivity (Table 3).

8 and 9. The models of water potential and hydraulicconductivity are seen to be rather sensitive to the valueof the fractal dimension. It should be noted also thatthe shapes of the model curves are consistent withconventional experimental results for sandy and clay-ey soils (Hillel, 1980, p. 150), indicating the ability ofEq. [A5] to [A7] to describe fundamental soil waterproperties.

CONCLUSIONSThe fractal model of a soil developed by Rieu and

Sposito (1991) is accessible to experiment through theseven equations in the Appendix. Equations [Al] to[A3] express three physical properties that characterizea fragmented fractal porous medium: decreasing ag-gregate bulk density with increasing aggregate size, apower-law aggregate-size distribution, and incompletefractal fragmentation, which is reflected in the differ-ence between D and DT. The corresponding soil waterproperties are expressed by Eq. [A4] to [A7]: a waterpotential that scales in inverse powers of the similarityratio and whose dependence on water content is ex-pressed by a power-law relationship with an exponentequal to the inverse of the difference between the bulkfractal dimension and the Euclidian dimension; for agiven water content, a hydraulic conductivity that isthe sum of partial hydraulic conductivities contributedby the active single-size arrangements of fractures thatare water filled. Very few experimental data are avail-able with which to test these equations.

The limited comparisons of Eq. [Al] through [A7]with experimental measurements, illustrated in Fig. 1through 9, suggest that aggregates in soils may be frac-

1244 SOIL SCI. SOC. AM. J., VOL. 55, SEPTEMBER-OCTOBER 1991

tal objects and that the pore space in undisturbed soilsmay exhibit structure characteristic of an incompletelyfragmented fractal medium. Precise data on soil ag-gregate physical properties and soil water parameters,taken concurrently, will be required in order to eval-uate the applicability of fractal concepts to soils.

ACKNOWLEDGMENTSGratitude is expressed by the senior author for the hos-

pitality of the Department of Soil Science, University ofCalifornia at Berkeley, during the tenure of a sabbatical leavefrom ORSTOM. Professors K. Loague and L J. Waldron arethanked for their technical assistance and helpful advice dur-ing the preparation of this article, Ms. Susan Durham andMs. Joan Van Horn are thanked for their excellent typingof the manuscript, and Mr. Frank Murillo is thanked forpreparation of the figures.

APPENDIXEquations describing a fragmented fractal porous medium,

derived by Rieu and Sposito (1991), where i is an integerindex running from O to m.Porosity: *, = 1 - (djd? ~ D- [Al]Bulk density:

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