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 structure—as well as water content, water poten-tial, and hydraulic conductivity—to 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 <r, is the bulkdensity of the zth size class and <r0 is the bulk density of thelargest aggregate. Straight lines were fitted to an experimentalplot of log (<Tj/ff0) vs. Iog(4/d0) (see Eq. [A2]) for the four soilsM. Rieu, Centre ORSTOM Bondy, 70-74 Route d'Aulnay, 93143Bondy Cedex, France; Garrison Sposito, Dep. of Soil Science, Univ.of California, Berkeley, CA 94720. Received 12 Feb. 1990. "Cor-responding author.

Published in Soil Sci. Soc. Am. J. 55:1239-1244 (1991).

to estimate the bulk fractal dimension, Dr. A quantity pro-portional to the number of aggregates of each size class of theSharpsburg soil, N(d,), was calculated with the equation:

N(d.) = Mwwfa) a = o, i , . . . . ) [i]where M(d,) is the mass of class i. The quantity log[N(dk)],where

[2]1=0

was then plotted vs. log dk (see Eq. [A3]) to estimate thefractal dimension, D, for the Sharpsburg soil.

So/7 Water PropertiesThe fractal model of soil water properties was applied

illustratively to physical data obtained by Bousnina (1984)for the surface horizon (0-0.2 m) of Ariana silty clay loam,collected from an experimental plot at the Institut NationalAgronomique de Tunis (Tunisia). These data are presentedin Table 3. The soil bulk density was determined on undis-turbed core samples of 10~4 m3 volume and a pycnometerwas used for the particle-density measurement. Aggregateseparation was carried out by submerging undisturbed, large,dry clods in methanol, then drying and sieving (Braudeau,1982). The elutriation methods, with Na hexametaphos-phate as a dispersant, was used for mechanical analysis. Thesoil water retention curve (moisture characteristic) and hy-draulic conductivity were determined at the field plot, foreach layer of 0.2-m depth, by the method of zero-flow limitdeveloped by Vachaud et al. (1981). A comparative deter-mination of hydraulic conductivity (Table 3) was carried outfrom measurements of water potential and water contentduring the drainage of small, undisturbed cores using thelaboratory method developed by Rieu (1978). No other com-plete set of physical data like these for a single soil was foundin our review of the literature.

Values of the water potential in Table 3 were listed inascending order and assigned to a set of increasing integerindices. The water potential (H) value of 0.22 m H2O, meas-ured at a maximum water content (0max) of 0.46 m3 m"3, wasassumed to be the smallest value of the water potential inthe Ariana soil; i.e.,

hn = 0.22 m [3]The corresponding value of the fracture opening (p0) wasassumed to be the largest in the fractal length scale:

Table 1. Aggregate bulk densities in three soils of different texture(Chepil, 1950)._____________________________

Aggregate bulk density

Size class

mm6.40-2.002.00-1.191.19-0.840.84-0.590.59-0.420.42-0.250.25-0.150.15-0.10

Fine sandyMean size loam

mm4.2001.5951.0250.7150.505

.49

.58

.75

.82

.940.335 2.170.200 2.110.125 2.15

Silt loam—— Mg nr> ——

1.421.581.681.611.721.751.822.10

Clay

1.491.681.701.731.751.801,751.80

1239

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

mm7.0053.5701.7850.8850.4460.2240.1110.05550.02775

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

<0.10

Geometricmean size

mm1.791.4141.1180.890.710.5610.4000.2510.141-

Mass

kg0.013950.021000.012510.010400.009400.006370.009000.007200.007670.00312

Soil water properties!

Water content Water potential Hydraulic conductivitym3 m"3

0.4600.4550.4510.4400.4310.4000.3840.3730.3620.3530.3390.3330.3300.3250.3330.2870.2580.2580.2430.2380.2340.2210.2150.2090.2050.1860.180

mH2O0.220.230.260.280.320.400.500.600.700.700.900.950.981.160.951.662.002.302.342.402.502.803.103.244.205.806.20

Mechanical analysis

ms'1

4.44 X 10-52.22 X 10-'1.11 X ID'52.8 X 10-«1.1 X 10-«1.5 X 10-'5.8 X 10-"2.3 X 10-"

. 2.5 X 10-"—

3.3 X 10-"—

2.5 X 10-"6.4 X 10-9

————————___—-

Size class Massmm2.00-0.500.50-0.050.05-0.020.02-0.002<0.02Bulk density:Particle density:Hydraulic conductivity:^

kg0.001400.011300.044300.026000.01500

1.409 Mg m-3

2.610 Mg m'3

K(9) •= 5.169 X 1Q-5 flM14> m s'1

P0 = 0.068 mm [4]A plot of log (ht/ha) vs. the associated integer scale was thenfitted to a straight line in order to estimate the similarityratio (see Eq. [A4]). The porosity ($) of the Ariana soil wascalculated with the conventional formula (Danielson andSutherland, 1986):

0 1- [5]where aa is the bulk density and am is the particle densitygiven in Table 3. The aggregate-size distribution was cal-culated as described in Eq. [1] and [2], then plotted aslog[N(dk)] vs. log dk to test Kq. [A3] and estimate D. Meas-ured values of the total porosity, water content, and waterpotential were plotted as log h vs. log (1 — <t> + 6) to testEq. [A5] and estimate DT.

RESULTS AND DISCUSSIONBulk Density and Porosity

Figure 1 shows a log-log plot of <r,/(r0 vs. djd0 forthe three sets of soil data in Table 1. The data showconsiderable scatter, but they are consistent in a sta-tistical sense with Eq. [A2] as summarized in Table 4.The value of Dr, calculated by adding 3.00 to the valueof the slope of each line determined by linear regres-sion, fell in the range 2.88 to 2.95.

The plot of log(<r,/o-0) vs. log(d,/fi?0) for the Sharpsburgsoil (Fig. 2) showed a clear break around the aggregate-size value df = 0.111 mm, which corresponds to thesize class 0.149 to 0.074 mm in Table 2. As pointedout by Wittmus and Mazurak (1958), the aggregatesof <0.074-mm size contained more sand and silt thandid the larger aggregates. The density of their primaryparticles is thus greater and their porosity is coarser.These physical differences support the existence of twoseparate fragmented fractal structures through therange of aggregate sizes. (Indeed, the aggregates of <0.074-mm size may well have a different shape, sincethey comprise a sandy skeleton on which the floccu-lation of clay develops larger aggregates.) In order toexamine this possibility in more detail, Eq. [A2] wasapplied to two separate ranges of size class: betweenthe mean limits dol = 7.005 mm, dml = 0.111 mm,and between rfml = 0.111 rnm and dm2 = 0.02775 mm.Two straight lines were fitted, resulting in fractal di-mensions of 2.95 and 2.74, respectively (Table 5). Thesmaller value of the second fractal dimension can beinterpreted as a reflection of a more fragmented struc-ture, consistent with the physical properties of the sec-ond size-class range.

0.20

o fine sandy loom. silt loamx clay

-2

t Field measurement.$ Laboratory measurement.

-1.5 -1.0 -0.5Iog(dj/d0)j 0

Fig. 1. Tests of E<j. [A2] with the data of Chepil (1950).

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

Table 4. Parameters resulting from linear regression of log (<rf/<r0)vs. Iog(rf//rf0),where a, is the bulk density of the ith aggregate sizeclass, <70 is the bulk density of the largest aggregate, d, is the meandiameter of the ith size class, and </„ is the diameter of the largestaggregate, based on the data of Chepil (1950).

Level of FractalSoil Slope Intercept r2 Significance dimensionFine sandyloamSilt loamClay

-0.1199-0.0950-0.0469

-0.0037-0.001

0.021

0.920.900.78

P < 0.001P < 0.001P < 0.01

2.882.912.95

The existence of two fragmented fractal structuresin the Sharpsburg soil can be tested further by mod-eling the porosity of its aggregates. The experimentalvalues of this parameter were calculated by applyingEq. [5], with a0 replaced by an aggregate bulk density(Table 2) and am set equal to the largest bulk densityof the aggregates (i.e., am = am2 = 2.36 Mg m-3). Amodel of the aggregate porosity was developed by re-writing Eq. [5] in the form

/dm2^-^ [6]where the second step comes from substituting Eq.[A2]. Equation [A2] can be extended to describe theporosity of an arbitrary aggregate by writing it in theform:

#</,<?) = 1 - (d/dmtf-^(d'/dmtf-^ [7]where d = dml while dm2 < d' < dm{ and d' = dmiwhile dml < d < d0. The values of dml, dm2, Dri, andDr2 are available from Table 5. A plot of the resulting4>(d,d') vs. aggregate size is shown in Fig. 3 along withthe experimental aggregate porosity values. The agree-ment between Eq. [7] and the data is very good, show-ing that the hypothesis of two embedded fractals isconsistent with the observed porosity of the Sharps-burg soil.

Aggregate-Size DistributionFigure 4 shows a log-log plot ofN(dk) vs. dk for the

Sharpsburg soil (data from Table 2 were used in Eq.

0.25

0.20

"~p^ 0.15^

5 0.10

0.05

SHARPSBURGSOIL

-3.0 -2.5 -0.5-2.0 -1.5 -1.0Iog(dj/d0)

Fig. 2. Test of Eq. [A2] with the data of Wittmus and Mazurak (1958),indicating two embedded fractal structures (Lines 1 and 2).

Table 5. Parameters resulting from statistical fitting of the Sharps-burg soil data (Table 2) to Eq. [A2] or [A3].

Sloper2

Level of significanceFractal dimension

7.005-0.111 mmEq. [A2] Eq. [A3]

-0.04998 -2.8420.98 0.98

P<0.001 /><0.0012.95 2.84

0.1 11-0.02775 mmEq. [A2]-0.25748

0.96P<0.02

2.74

Eq. [A3]

-2.5810.98

/><0.012.58

[1] and [2]). Again, confirming the "embedded fractal"hypothesis, an apparent break was noted near the di-ameter of 0.111 mm. Two lines thus were fitted to theexperimental points corresponding to the size rangesgreater and less than 0.111 mm, respectively. The ob-served aggregate-size distribution satisfies Eq. [A3]fairly well when applied piecewise. The correspondingfractal dimensions were slightly smaller than those de-termined from Eq. [A2]: 2.84 instead of 2.95 for thesize range 7.005 to 0.111 mm, and 2.58 instead of 2.74for the smaller size range (Table 5). These differencesare not unexpected; as suggested by Rieu and Sposito(1991), D determined from Eq. [A3] reflects the me-chanical destruction of the incompletely fragmentedstructure of a soil. The value of this latter dimension

0.4 -

>> 0.3<noo 0.2Q.

0.1

SHARPSBURGSOIL

0.02 0.05 0.1 0.5 1.0d (mm)

5.0 10.0

Fig. 3. Test of Eq. [7] with aggregate porosities computed using thebulk-density data for the Sharpsburg soil (Table 2).

-1.5 -1.0 -0.5 0.5 1.0log [dk(mm)]

Fig. 4. Test of Eq. [A3] with an aggregate-size distribution calcu-lated with the data for the Sharpsburg soil (Table 2).

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

je•o

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 <r, needed in Eq. [1] were calculatedwith Eq. [A2] and the value of Dr determined from ananalysis of the moisture-characteristic data, as de-scribed below.) The slope of the line through the datapoints, fit by linear regression (r2 = 0.97*** [signifi-cant at P = 0.001]), led to D = 2.83, according to Eq.[A3].

Soil Water PropertiesFigure 6 shows a graph of log (/z,//z0) vs. size class i

for the Ariana soil using the water-potential data inTable 3 and h0 given by Eq. [3]. The conformity ofthe data to a straight line was excellent (r2 = 0.996***)and the slope value led to a similarity ratio r = 0.8209according to Eq. [A4]. With the value of <t> computedfrom Eq. [5], the measured values of the water poten-tial and water content were plotted according to Eq.[A5], as shown in Fig. 7. Linear regression of the data(r2 = 0.991***) yielded a slope equal to -9.5591 and

5 19integer i

Fig. 6. Test of Eq. [A4] with water-potential data for the Arianasoil (Table 3).

-0.15 -0.1 -0.05

log Q-<|>+e[|Fig. 7. Test of Eq. [A5] with water-retention data for the Ariana

soil (Table 3).

a corresponding Dr = 2.90. This latter value is largerthan D (= 2.83), indicating the lesser extent of frag-mentation of the undisturbed field soil. Given Eq. [Al]for the case / = 0 and the relation djd0 = rm (Rieuand Sposito, 1991, Eq. [23]), the value of the exponentm can be calculated, with the result m = 31. Thefractal parameters for the Ariana soil are summarizedin Table 6.

Given the physical parameters in Table 3 and thefractal parameters summarized in the third column ofTable 6, the values of the parameters r, rr, and clus-tering factor, F, can be calculated with formulas de-rived by Rieu and Sposito (1991, Eq. [21], [40], and[43]):

0.03375 rr = 0.01965 F = 0.418 [8]and the corresponding cross-section parameters re-quired to model the hydraulic conductivity can becomputed (Rieu and Sposito, 1991, Eq. [65]-[67]):

ft = 0.0226 ft = 0.01313 G = 0.419 [9]The parameters in Eq. [9] can be used in Eq. [A6]. Forsuccessive values of the increment i, (0 < / < m —

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.

o•o

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: <,Ja0 = (dJd0)D< ~ 3 [A2]Aggregate size distribution: Nk(d) = A£f)d-D [A3]Scaled water potential: ht = /z0H [A4]Water-retention curve: h, = ha[(\ -</>) + 0,]1/(D' -3) [A5]

Hydraulic conductivity: Kt = (pwg/12/i)(/3r ^ PjGJ)j -1 [A6]

Water content: 0, = (r3 - Dy - (r*-D.)m [AT]