fractal fragmentation, soil porosity, and soil water properties: i. theory

Fractal Fragmentation, Soil Porosity, and Soil Water Properties: I. TheoryMichel Rieu* and Garrison Sposito

ABSTRACTRecent efforts to characterize soil water properties in terms of

porosity and particle-size distribution have turned to the possibilitythat a fractal representation of soil structure may be especially apt.In this paper, we develop a fully self-consistent fractal model ofaggregate and pore-space properties for structured soils. The conceptunderlying the model is the representation of a soil as a fragmentedfractal porous medium. This concept involves four essential com-ponents: the mathematical partitioning of a bulk soil volume intoself-similar pore- and aggregate-size classes, each of which is iden-tified with a successive fragmentation step; the definition of a uniformprobability for incomplete fragmentation in each size class; the def-inition of fractal dimensions for both completely and incompletelyfragmented porous media; and the definition of a domain of lengthscales across which fractal behavior occurs. Model results includea number of equations that can be tested experimentally: (i) a fractaldimension <3; (ii) a decrease in aggregate bulk density (or an in-crease in porosity) with increasing aggregate size; (iii) a power-lawaggregate-size-distribution function; (iv) a water potential that scalesas an integer power of a similarity ratio; (v) a power-law expressionfor the water-retention curve; and (vi) an expression for hydraulicconductivity in terms of the conductivities of single-size arrange-ments of fractures embedded in a regular fractal network. Futureresearch should provide experimental data with which to evaluatethese predictions in detail.

THE STRUCTURE OF SOIL and its close relationshipto aeration and water movement have been stud-

ied intensively by soil physicists throughout the pres-ent century (Hillel, 1980, Ch. 6). Of perhaps mostimportance in these studies is the characterization ofthe soil fabric and its associated pore space (Kemperand Rosenau, 1986; Danielspn and Sutherland, 1986).

This classic problem in soil physics recently has tak-en on new dimensions with the recognition that, be-cause soil is both a fragmented material and a porousmedium, a fractal representation may be especially ap-propriate (Mandelbrot, 1983; Turcotte, 1986; Feder,1988). Fractal representations of soils as fragmentedmaterials produced by weathering processes have beendeveloped in a variety of ways during the past 20 yr,usually based on a power-law distribution function forparticle size (Hartmann, 1969; Turcotte, 1986, 1989).Similar fractal representations of colloidal media havebeen derived on the basis of assumptions about thephysics of aggregate growth (Jullien and Botet, 1987;Schaefer, 1989). Fractal representations of sedimentsas porous media also are abundant in the literature(see, e.g., the review by Thompson et al. [1987]) and,very recently, Tyler and Wheatcraft (1989, 1990) havedescribed fractal models of soil that relate particle-sizeMichel Rieu, Centre ORSTOM Bondy, JO-74 Route d'Aulnay,93143 Bondy Cedex, France; Garrison Sposito, Dep. of Soil Science,Univ. of California, Berkeley, CA 94720. Received 12 Feb. 1990."Corresponding author.

Published in Soil Sci. Soc. Am. J. 55:1231-1238 (1991).

distribution to pore-size distribution, and thus to soilwater properties. As noted by Arya and Paris (1981),Saxton et al. (1986), and Haverkamp and Parlange(1986), laboratory or field measurements of the rela-tion between water potential and water content, andthe relation between hydraulic conductivity and eitherwater potential or water content, are time consumingand expensive. For this reason, studies often have beencarried out using an alternative procedure, which con-sists of predicting soil water properties from simpler,routine laboratory measurements of soil bulk densityor particle-size distribution (Cosby et al., 1984; Saxtonet al., 1986). Arya and Paris (1981) have presented anexhaustive review of different models that have beendeveloped for use in this approach, and they have de-scribed a physical model of soil porosity based on theparticle-size distribution. In their model, the relationbetween pore radius and particle radius involves anexponent that has been interpreted recently through afractal concept by Tyler and Wheatcraft (1989). In asimilar vein, Haverkamp and Parlange (1986) consid-ered a constant packing parameter that relates poreand particle radii, suggesting that pores of differentsizes were nonetheless similar in shape. Tyler andWheatcraft (1990) have suggested that this similarity,interpreted with fractal concepts, is basic to the com-monly observed power-law relationships among soilwater properties.

We have developed a general theoretical frameworkfor a self-consistent fractal representation of soil asboth a fragmented natural material and a porous me-dium. We began with a description of virtual pore-size fractions in a porous medium, motivated by anapproach of Childs and Collis-George (1950), that per-mits a facile introduction of fractal concepts of thesolid matrix and pore space. These concepts led toequations for the porosity and bulk density of boththe size fractions and the porous medium in terms ofa characteristic fractal dimension, D. The equationsderived apply to materials that have been fragmentedinto a collection of unconnected aggregates, such asoccurs in particle-size analysis. The aggregate-size dis-tribution for this collection follows a power-lawexpression involving D. The fractal representation wasthen extended to describe a structured field soil whoseaggregates are interconnected by solid material to forma stable pore space. Using a concept of incompletefragmentation of a fractal porous medium (Turcotte,1986, 1989), we derived equations for the porosity andbulk density that involve a "bulk" fractal dimension,DT. These equations can be tested with suitable ex-perimental measurements on the structure and poros-ity of natural soils. Once a fractal, incompletelyfragmented structure is described for a soil, its porespace can be represented by conducting network ofsimilar, size-scaled fractures and it is possible to derivetestable relationships among soil water content, waterpotential, and hydraulic conductivity.

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1232 SOIL SCI. SOC. AM. J., VOL. 55, SEPTEMBER-OCTOBER 1991

FRACTAL STRUCTURE AND POROSITYVirtual Pore-Size Fractions

Consider a porous medium whose porosity resultsfrom a broad range of pore sizes, decreasing in mean(or median) diameter from p0 to pm_i (m> 1). A bulkelement of the porous medium, of volume V0 largeenough to include all sizes of pore, has porosity 0 andthe dry bulk density a0. Following Childs and Collis-George (1950), we shall divide the pore- volume dis-tribution of V0 mathematically into m virtual pore-size fractions, with the rth virtual size fraction dennedby

PI - Vt ~ [1]where Pt is the volume of pores solely of size pi con-tained in Vt, the /th partial volume of the porous me-dium, which itself contains all pores of size < pt. Thepartial volume VM thus is contained in Vt and thepartial volume Vm_l contains the smallest pore-sizefraction, />„_,, along with the residual solid volume,denoted Vm. In general, the solid material whose vol-ume is Vm will not be chemically or mineralogicallyhomogeneous. Its mass density, denoted am, is thus anaverage "primary particle" density. Given Eq. [1], thebulk volume of the porous medium can be representedmathematically as the sum of m increments of thevirtual pore-size fraction P0 to Pm.l; plus a residualsolid volume V-

i-O[2]

Successive increments of pore volume P, are ratioedto the corresponding partial volumes F, to define thepore coefficient:

r, - P/F, (r, < i) [3]or, using Eq. [1],

VM = (1 - r,)F, (/ = 0, ..., m - 1) [4]The pore coefficient F, is interpreted as the partial po-rosity contributed to F, by pores of size pt. It followsfrom Eq. [4] that the residual solid volume FOT can beexpressed

*-iO - IV,)m-l

n-j n o - r,.) (/=!,.., [5]

where the second step comes from repeated applica-tion of Eq. [4]. The porosity of the medium then canbe expressed as a product involving the pore coeffi-cients:

(PC- Vm)/V0m-l

1 - no -r,)[6]

[7]

where Eq. [5] has been inserted withy set equal to m.Equation [7] is a representation of the porosity as thediiference between 1 and the product of successivepartial-volume fractions, F(>1/F, = 1 — r,.

Fractal Porous MediaA fractal porous medium is one in which there is

self-similarity of both the pore space and the solidmatrix (Mandelbrot, 1983, Ch. 14; Jullien and Botet,1987, Ch. 3). In the context of Eq. [1], this means thatboth the increments of pore volume, Ph and the partialvolumes, F;>, are similar in shape and obey scaling re-lations in size. To introduce the scaling property, wedefine a linear similarity ratio, denoted r, which relatessuccessive pore sizes pt or successive partial volumesrepresented by their mean (or median) diameters d{.

PM = rp,dM = rdt

(r < 1)(r < 1)

[8a][8b]

Corresponding to Eq. [8], the pore-volume incrementsand the partial volumes v.hen scale as r 3:

PM =VM =

[9a][9b]

which impliesPJVt = constant = T (/ = 0, . . ., m - 1) [10]In a scaling porous medium the pore coefficients areuniform across all virtual; pore-size fractions.

The self-similarity property of a fractal porous me-dium means that, for each virtual pore-size fraction,a constant number N of smaller volumes VM (or PM)can be associated with any volume Vt (or P,) (Man-delbrot, 1983, Ch. 6). Simply put, every volume Vtcontains N smaller volumes VM and one associatedpore volume Pf:

+ P, (i = 0,...,m- 1) [11]In turn, each volume VM contains N volumes Vi+2 andone associated pore volume PM, and so on. Therefore,N pore volumes PM can also be associated with eachpore volume />• and Eq. [2] takes the form

m-l

i=0[12]

where now Nm Vm represents the residual solid volume.Equation [4] becomes

VMIVt = (1 - DIN (i = 0, . . ., m - 1)[13]because of Eq. [10] and [1 1]. Equation [5] correspond-ingly is transformed to the expression

(j=\,...,m) [14]

[15][16]

The porosity of the medium is then0 -(F0-7V»Fm)/F0

= 1 - (1 - r)m

after the insertion of Eq. [14] with j set equal to m.Similarly, the partial porosity 0, of a partial volumeF, is denned by

0, =(JV»F, - AT-KJ/TV-'F, [17]= i - (i - ry*-1 (/= o, . . ., m - i) [is]

where Eq. [14] has been used with; = m — i to obtainEq. [18]. Equation [18] leads to the important result

RIEU & SPOSITO: FRACTAL FRAGMENTATION, SOIL POROSITY, AND SOIL WATER PROPERTIES: I. 1233

that, in a fractal porous medium, the porosities of suc-cessive partial volumes decrease with the pore size, asthe index / approaches m. The fractal dimension of aporous medium is defined by (Mandelbrot, 1983, p.37)

D = logAT/log(l/r) (Z» 0) [19]where N appears in Eq. [11] and r is defined in Eq.[8]. The fractal dimension is related closely to the porecoefficient, r. This can be seen by combining Eq. [9b]and [13] to obtain the expression

r = 1 - Nr3 [20]which, with Eq. [19], leads to the result

T = 1 - r3-0 (r < 1, r < 1) [21]Equation [21] shows that, in a fractal porous mediumwhere pore sizes are scaled by the similarity ratio r <1, the magnitude of the pore coefficient decreases asthe fractal dimension increases. The relation betweenthe porosity and the fractal dimension now followsfrom Eq. [16] and [21]:

</> = 1 - [22]For a given value of the exponent m, the porosity of afractal porous medium decreases as the fractal dimen-sion increases. Equation [22] shows further that the frac-tal dimension of a porous medium must be <3.

Natural porous media typically exhibit only a lim-ited domain (or domains) of pore length scale acrosswhich fractal behavior is observed (Thompson et al.,1987; Turcotte, 1989), such that the index m alwayshas a finite value. The upper and lower limits of thisdomain can be identified with the corresponding limitson the partial volume diameters d0 and dm respec-tively. It follows from Eq. [8b] that

djd0 = rm

and from Eq. [22] that4, = 1 - (dm/d0?-D

[23]

[24]The partial porosities 0, also can be connected withpartial volume diameters by combining Eq. [8b], [18],[21], and [23]:

0, = 1 - (dm/d,y-D [25]As D ] 3, the 0, become smaller, corresponding to amore finely partitioned porosity, whereas a small valueof D results in a more coarsely partitioned porositywith larger 0, for a given set of df. Thus, the fractaldimension can be interpreted to represent the parti-tioning of the porosity into size classes, with a largerfractal dimension representing a smaller, more finelypartitioned porosity. If 3 — D is very small, Eq. [18]can be expanded in a MacLaurin series in r to showthat 0,- «s (m — i)T for the partial porosities in thelimit of small pore coefficients.

FRAGMENTED FRACTAL POROUS MEDIAComplete Fragmentation

The concepts of virtual pore-size fractions, pore coef-ficients, and porosity were developed in Eq. [1]

(b)

(c) (d)Fig. 1. Conceptual models of porous media: (a) a porous medium that

is not necessarily fractal; (b) the Menger sponge (similarity ratio (r)= 1/3, N = 20, fractal dimension (D) = 2.7268), after Mandelbrot(1983, p. 145); (c) a completely fragmented fractal porous medium(r = 0.476, number of self-similar elements [N] = 8, D = 2.8); (d)cross-section of a random, incompletely fragmented fractal porousmedium (r = 0.485, bulk fractal dimension (Dr) = 2.92, D = 2.87)showing active networks of fractures that are not crossed by largerfractures.

through [7] without consideration of scaling or self-similarity. They apply to any porous medium with anarbitrary distribution of pore sizes (Fig. la). Equations[10] through [25] specialized these concepts (plus thatof partial porosity) to a porous medium that exhibitsscaling relations and self-similarity as defined by Eq.[8], [9], and [11]. A well-known example of this kindof fractal porous medium is the Menger sponge (Man-delbrot, 1983, p. 134), shown in Fig. Ib for the case r= 1/3, N = 20, D = 2.7268. The solid matrix andthe pore space each form connected sets of similarelements. The volume Pt comprises a single rectan-gular pore in each partial volume V( and the pore spaceitself comprises size-scaled, similar pores that can bedescribed by Eq. [10] through [25]. This example il-lustrates the point that a fractal porous medium is notrestricted as to the shape of its solid matrix elementsor its pores.

Figure Ic depicts an arrangement of pore space ina fractal medium that is quite different from the Men-ger sponge. The volume P, now comprises pore ele-ments that are distributed in a regular manner aroundthe partial volumes VM contained in the partial vol-ume Vj. This arrangement represents a fragmentedfractal porous medium: clusters of size-scaled, similarpartial volumes separated from one another by a net-work of size-scaled, similar fractures. In this case, thesolid matrix is a disconnected set, while the pore spaceremains a connected set. The latter is partitioned intovirtual pore-size fractions in the usual way in order tobring out its self-similarity. The example shown in Fig.


Ic is a three-dimensional analog of the well knownCantor dust (Mandelbrot, 1983, Ch. 8) for the case r= 0.476, N = 8,D = 2.S. The alternative choices, r= 0.4665, N = 8, or r = 0.2986, N = 27, would haveresulted in fragmented, fractal porous media whosefractal dimension is the same as that of the Mengersponge in Fig. Ib. Equations [8] through [25] apply tothese fractal media as well as to the Menger sponge:the connectedness of the solid matrix does not lead toquantitative characteristics differentiating fractal po-rous media.

The fragmented fractal porous medium is concep-tually more similar to a natural soil, wherein aggre-gates of differing size can be associated with the partialvolumes Vt. The pore space of soil consists of fracturesor cracks that form ped boundaries and diminish thecollusiveness of the soil. The bulk element, of volumeF0, is any element that includes all sizes of fracture,such that no system of larger fractures separates bulkelements from one another. This element has the po-rosity 4> and corresponding dry bulk density <TO. Thelatter property is related to the solid mass M0 of theelement by:

M0 = a0V0 [26]An alternative expression to Eq. [26] is found by not-ing that the same mass of solid is contained in theaggregates of volume Vt at any step of fragmentation,i.e.,

M0 = Nay, (i = 0,...,m) [27]where <r, is the bulk density of the partial volume Vt.As i increases from 0 to m, the partial volumes includeless and less of the pore volume (cf. Fig. Ic), such thatthe density <r, must increase to its limiting value am inagreement with the implication of Eq. [ 1 8]. Indeed,Eq. [13], [26], and [27] can be combined to derive therelation

aja0 = (1 - T)-' (i = 0, . . ., m) [28]Given Eq. [8b] and [21], one can transform this equa-tion into

<r,/<r0 = (4W-3 (i = 0, . . ., m) [29]Equation [29] is a well-known expression for the dis-tribution of mass in a fractal object (Jullien and Botet,1987, p. 31; Feder, 1988, p. 34). The density of a fractalporous medium decreases, as larger and larger com-ponent aggregates are considered, because larger andlarger pores are thereby enclosed. If aggregates of dif-fering size had the same bulk density a0, the fractaldimension of the medium would necessarily be 3 andits porosity would be zero, as follows from Eq. [24]and [29].

Incomplete FragmentationThe fragmented, fractal porous medium is, in fact,

merely a hypothetical construct, since, if every aggre-gate were to be surrounded by void space, the porousmedium would collapse and the bulk volume wouldreduce to a collection of primary particles of size dm.To ensure the stability of the pore space and structure,one must take into account the interaggregate bridgesholding aggregates together. But these contacts will in-

terrupt the fractures separating the aggregates. In otherwords, the fragmentation is not complete. In a fractalporous medium, this lack of complete fragmentationcan be considered a scale-invariant property, ex-pressed by the partitioning

V0 = FV0 + (1 - F)V0 (0<F< 1) [30]where F is a clustering factor. Following Turcotte(1986, 1989), we interpret F as the (uniform) proba-bility that an element of volume will not fragment ata given fragmentation step. Thus the product FV0 isthe portion of the volume V0 that remains unfrag-mented at the first step, whereas the portion (1 — F) V0is fragmented into aggregates of volume F,. Followingthis approach, we can express the first step of incom-plete fragmentation by

0 = [FV0 - F)NVl] - F)P0 [31]which reduces to Eq. [11] with / = 0 when F vanishes.The first increment of pore volume, given by the sec-ond term on the right side of Eq. [31], can be rewrittenin terms of V0 after combining Eq. [10] with the def-inition of the reduced pore coefficient:

r, - (1 - F)T (Tr < 1)The result is

(i - F)P0 = rrF0so that Eq. [31] becomes

FV0 + (1 - F)NV, = (1 rr)F0

[32]

[33]

[34]The next step of fragmentation will produce an incre-ment of pore space rr(l — rr)F0 and a volume ofaggregates (1 — rr)2F0, and so on. The porosity of anincompletely fragmented fractal porous medium isthus

= 1 - (1 - rr)» [35]and the partial porosity and corresponding bulk den-sity of an aggregate are6- = 1 - (1 - rr)""- (i = 0, ..., m - 1) [36a]

erAo = (1 - Tr)-> (i = 0, . .., m) [36b]These equations reduce to Eq. [ 16], [ 18], and [28] whenF vanishes.

The physical significance of the clustering factor Fcan be understood as follows. We can express the vol-ume of aggregates resulting from the i— 1 step of in-complete fragmentation of a bulk soil element ofvolume V0 by

(1 - rr)'F0 = (F + (1 - F)(l - r)]<F0 [37]The expansion of the rij^ht side of Eq. [37] involvesthe term FVm which represents the portion of the bulksoil element remaining unaffected by fragmentation.The other terms, which involve coefficients like (1 —F)k(\ — T)', can be interpreted as volumes F, to F, ofaggregates of sizes d\ to dt. These aggregates are sep-arated by fractures of size p0 to pM. The zth step offragmentation partitions, in proportion to (1 — F), thevolume denoted FV0 by opening fractures of size /?,only. There will remain a volume P+1F0, which willbe, in turn, partially fragmented by fractures of open-ing pi+l only, and so on. Thus, incomplete fragmen-


tation of a bulk soil element produces a complexfracture network, where m — 1 particular networkscomprising single-size arrangements of fractures areembedded in a regular fractal network of fractures thatdoes not extend throughout the bulk soil element. Thesingle-size particular networks will be termed activenetworks. The volume of each active network is rVFF0,and the corresponding partial porosity is IVF.

If some mechanical process completed the fragmen-tation, the result would be a collection of loose, self-similar aggregates scaled by the ratio r. In this totallyfragmented porous medium, a pore coefficient can bedefined as in Eq. [21]. But, considering Eq. [19] and[32], we hypothesize that the number of smaller ag-gregates contained in a larger one (by virtue of self-similarity) must be greater in an incompletely frag-mented medium than in the completely fragmentedmedium. Consequentially, we shall write

r r= 1 - Ntr3 [38]where Nr > N in Eq. [20]. The difference, Nr - N,characterizes the bridges bonding the aggregates in theactual porous-medium structure. Equation [38] leadsto the definition of a bulk fractal dimension:

DT«logAfr/log(l/r) [39]where DT > D in Eq. [19]. The fractal dimension ofthe completely fragmented porous medium representsboth the fracture porosity of the actual medium anda complement of porosity obtained by the completionof fragmentation. The bulk fractal dimension, on theother hand, reflects only the porosity of the actual me-dium. From Eq. [38] and [39], the reduced pore coef-ficient may be expressed

Tr = 1 - r3-^ [40]and Eq. [35] and [36] take the form

(A = 1 - (djdj3-* [41]<A, = 1 - (djd,)3-0* (i = 0, ..., m - 1) [42a]

(7,/<70 = (djd^3 (i = 0, ..., m) [42b]which are analogous to Eq. [24], [25], and [29]. Finally,it follows from Eq. [21] and [40] that

fD-D, _ 1 p

PD-3 _ 1[43]

where DT is the bulk fractal dimension of the actualporous medium and D is the fractal dimension of itsaggregate distribution. As can be deduced from Eq.[43], the less completely developed is the fragmenta-tion of the medium, the greater is the bulk fractal di-mension Dr. Indeed, if a medium were not fragmented,the clustering factor would take on its maximum val-ue: the condition F = 1 is equivalent to DT = 3.

Aggregate-Size DistributionMeasurements of the aggregate-size distribution

(Kemper and Rosenau, 1986) have long been used tocharacterize soil structure. In these measurements, asperformed conventionally, for each size class repre-sented by a mean (or median) diameter dh a certainmass M(dt) is measured that defines the contribution

of the size class to the total sample distribution. Gard-ner (1956) has shown, in a study of more than 200aggregate-size distributions for soils, that the distri-bution ofM(dl) can be represented well by a lognormalfunction (Crow and Shimizu, 1988) whose indepen-dent variable is the ratio of the aggregate diameter toits median value.

The quantity M(d^/djah where er, is the average massdensity of the aggregates with diameter dh is propor-tional to the number of aggregates in the size class /.(A geometric factor, dependent on aggregate shape,that ensures equality between d]at and the averagemass per aggregate has been omitted under the as-sumption that it is the same for all size classes.) Giventhe results of Gardner (1956) and Eq. [8b] and [29],the number of aggregates of size dt in a fragmentedfractal porous medium can be expressed in the form

[44]r(p-i)D

where Eq. [29] has been applied to the size classes /and p to derive the general equation

djffj = d^pG^dJd^f [45a]and Eq. [8b] has been used recursively to yield thescaling relationship

dt = [45b]for anyp = 0, 1,..., m — 1. Equation [44] representsthe number distribution of aggregate diameters d inthe size class i, as referenced to a particular size classp, via the scaling relationships that apply to a fractalporous medium (Eq. [8b] and [45]). In the simplestcase, M(d/d,) is a rectangular "spike," centered on dfand extending across the range of d values that is rep-resented by the mean or median dt (Gardner, 1956).In Eq. [44], M(d/dj) can be any mathematical functionof d/dj that encloses a finite area.

If the choice p = m — 1, j = p — Us made in Eq.[44], then the sum of terms over j = 0, 1, ..., k hasthe form (dropping the constant factor

.) [46]The function F^d/dm^\ is proportional to the numberof aggregates with diameters less than dk+i (= r^k+l)dm.i),i.e., the conventional cumulative aggregate-size-distri-bution function. Another number-distribution func-tion, related to Ffo is defined by the choice p = 0 inEq. [44] (again dropping ,N^d/d0) = M(d/d0) +

The function Nk(d/d0) is proportional to the numberof aggregates with diameters larger than dk+i (= rk+ld0),i.e., the complement of the cumulative aggregate-size-distribution function. In Eq. [46], the first term on theright side is proportional to the number of smallest-size aggregates (class m — 1, see Eq. [1] and [8b]); thesecond term is proportional to the number of next-


smallest-size aggregates (diameter dm_\jr\ with theweighting factor r° arising from the relationships inEq. [45]; and so on. Each term after the first describesa scaling up of the smallest aggregate size to a new sizevia self-similarity, as expressed in Eq. [45]. In Eq. [47],the first term on the right side is proportional to thenumber of largest-size aggregates (Class 0); the secondterm is proportional to the number of next-largest sizeaggregates (diameter rd0), with the weighting factor r~D;and so on. Each term after the first in Eq. [47], as in Eq.[2] and [12], describes a scaling down of the largest ag-gregate size via self-similarity. In both Eq. [46] and [47],the number-distribution function defined reflects a frag-mentation process in which aggregate size is amplifiedsequentially (cf. Montroll, 1987).

The dependence ofNk(d/d0) on dean be found afterderiving a recursion relation for N^rd/d0):Nk(rd/d0) = M(rd/d0) + r

= T-D[rDM(rd/d0) + M(d/d0)-DM(r-ld/d0)

[48]The second through the last terms in the square brack-ets in Eq. [48] are included in Nk(d/d0) (Eq. [47]), buta term in M(r-kd/d^ is missing. After adding and sub-tracting this term in Eq. [48], the recursion relation isobtained:

= r-D[Nk(d/d0) + rDM(rd/d0)[49]

Since d0 is the largest aggregate size, M(rd/d0), corre-sponding to the diameter djr, must be zero. The termin M(r-kd/d0) does not vanish identically, but its do-main is restricted to the range of d values that arerepresented by the smallest aggregate size dk (= rkd0).The general behavior of Nk(rd/d0) thus will be con-trolled by the first term on the right side of Eq. [49]:

Nk(rd/d0) = r-DNk(d/d0) [50]This equation is satisfied by a power-law relationship(cf. Montroll, 1987):

Nk(d/d0) = Bk(r)(d/d0YD [51]as can be demonstrated by direct substitution into Eq.[50], where the function B^r) is independent of d butmay depend on r. Equation [51] is an aggregate-sizenumber-distribution function for a fragmented fractalporous medium. Its dependence on the choice of ref-erence diameter d0 is not essential, and the alternateform

where A^r) = Bk(r)d%, expresses the singular behaviorof Nk in a way that can be tested experimentally. Itfollows from Eq. [52] that the graph of log N(dk) vs.log dk should be approximately a straight line withslope equal to — D for a fragmented fractal porousmedium.

If Eq. [52] is applied to the aggregates in a soil thathas been modeled as an incompletely fragmented frac-tal porous medium, a linear plot of log N(dk) vs. log

dk will lead to an estimate of the fractal dimension Dinstead of the bulk fractal dimension DT. This is be-cause D represents a completely fragmented fractal po-rous medium, and a soil whose aggregate-sizedistribution is being measured is merely a collectionof aggregates resulting from the destruction of thestructured soil. Indeed, if a nonporous solid (e.g., apiece of quartz) were fragmented by crushing it intoa collection of scaled, self-similar particles, the parti-cle-size distribution would obey Eq. [52] (approxi-mately). Likewise, if an incompletely fragmentedporous medium were crusihed into fractal aggregates,their size distribution also would follow Eq. [52], eventhough the original porous medium would have thefractal dimension Dr.

WATER IN A FRAGMENTED FRACTALPOROUS MEDIUM

To simplify the mathematical description of waterin a porous medium having a fragmented fractal struc-ture, it will be assumed that the fracture walls are ap-proximately parallel and that the fractures exhibit thesame cross-section porosity in all directions, such thatthe hydraulic conductivity is a scalar quantity.

Water ContentConsider a fragmented fractal porous medium

wherein the fractures of size <p,_i are filled with waterby a capillary-flow process. The volume of water con-tained in the bulk volume: V0 is equal to the sum ofthe successive pore-volume increments Pm^ to Pf. Asimplied in the discussion following Eq. [34], each ofthese pore-volume increments has the form (1 —rryrrK0 and the resulting volumetric water content is

m-le< = Srr(i - rr)>

f-i

= (r3-*)! - (T3-D-)m [53]where the second step is derived after substitution ofEq. [40].

Water PotentialThe pressure of water in a fracture (^) must satisfy

the Laplace equation:

where 7 is the liquid-vapor surface tension of water,RI is the radius of the meniscus curvature in a planeperpendicular to the fracture wall, and R2 is the radiusof the meniscus curvature in the plane of the fracture.If the lateral extension of the fracture is »/?,-, R2 canbe assumed infinite and Eq. [54] becomes

&i = 2y cosct/pi (i = 0, . .., m - 1) [55]where a is the contact angle between liquid and solid.Assuming a = 0 and expressing the pressure in metersof water, one derives an equation for the water poten-tial in a fracture:

^ = (2y/Pwg)(l/Pi) (/ = 0, ..., m - 1) [56]where pw is the mass density of water and g is thegravitational acceleration. At 20 °C, 7 = 0.0727 kg s-2


and pw = 0.99823 Mg nr3. (Eq. [56] is identical to thewell-known expression for the water pressure in a cap-illary tube of diameter 2p,.)

The general expression in Eq. [56] is specialized toa fragmented fractal porous medium by imposing ascaling relation on the fracture opening pt. With re-peated application of Eq. [8a], the water-potentialexpression becomes

h, = h0r-> (i = 0, . . ., m - 1) [57]where h0 is given by Eq. [56] for the case / = 0. Equa-tion [57] expresses the fact that, in a fragmented fractalporous medium, the values of the water potential arescaled by r~', where i is a positive integer. In conse-quence, a plot of log (hi/h0) vs. an arbitrary integerscale will be a straight line of slope minus log r. Yetanother form of Eq. [57] can be derived by introducingEq. [35], [40], and [53]:

h, = h0[(l - 0) + fl,.]1/'*-3' [58]A log-log plot of this equation is a straight line withslope l/(Z>r — 3). Equations [57] and [58] can be testedwith data on the moisture characteristic for an undis-turbed soil, and linear plots of the transformed datapermit the estimation of both the similarity ratio andthe bulk fractal dimension.

Hydraulic ConductivityThe steady flow of water through fractures of a given

opening PJ can be modeled by an integrated form ofthe Navier-Stokes equation (de Marsily, 1986):

<?z, = (djfjXrf/UpMPi ~ PJ/L] ~ pwg} [59]where dj)j represents the cross-sectional areas of ver-tically oriented (z axis) fractures of opening PJ andlateral extension d0, n is the kinematic viscosity ofwater (1.007 X 1Q-6 m2 s-' at 20 °C), L is the lengthof the fractures, and a pressure gradient is denned byPI at the top of the fracture and P2 at the bottom, withPI > P2. If unit area across a network of fractures ofopening PJ contains n fracture cross-sections djtj, itscross-section porosity is (j>sj = ndjjj, and the total flowacross this unit area is:Qzi - nqzj

[60]If the cross-sectional porosity is identical in all direc-tions of space, Eq. [60] may be generalized to anydirection of space, f.

Qjy = -(l/12M)(0s,;>,2)[grad P + Pw£grad z]r[61]where grad z is a vector with coordinates (0,0,1) (thez axis is vertical and oriented upwards) and [grad P+ Pv,g grad z]yis the component of the total hydraulicgradient along the direction/ Assuming that the wateris incompressible, we can introduce the hydraulichead, H:

H = (PPwg) - zand Eq. [61] becomes:

Qfj= -[(

[62]

[63]

The general expression in Eq. [63] is specialized toa fragmented, fractal porous medium by modeling thecross-sectional porosity 0SJ. In a regular fractal networkof fractures, every fracture of opening p, < p0 intersectsa larger fracture. This fracture network, therefore, haszero hydraulic conductivity for any water content lessthan that at saturation. On the other hand, followingthe discussion of Eq. [37], we expect that, in an in-completely fragmented porous medium, there are m— 1 active networks of single-size fractures that arenot crossed by larger fractures but are connected tothe regular fractal network. This situation is illustratedin Fig. Id for a random, incompletely fragmented frac-tal porous medium with similarity ratio r = 0.485 andDT = 2.92, D = 2.87. If it is assumed that, at the watercontent 0, (Eq. [53]), fractures of opening <p,_i arefilled with water, these active networks will have anonzero hydraulic conductivity. Their cross-sectionalporosity is the two-dimensional analog of the partialporosity FfP (see the discussion following Eq. [37]):

0S, = ft(? [64]where ft is the two-dimensional analog of Tr and G isthe analog of F.

Explicit expressions for ft and G can be derivedfollowing the approach used to obtain Eq. [20] and[38], but with F2/3, N2'3, and N2^ replacing V, N, andNr, respectively (area-to-volume ratio; cf. Turcotte[1989, p. 186]). The two-dimensional analog of T is

[65]= i - ( i - r)2/3

= 1 - (r3-*)2/3

where the second step comes from Eq. [13] and thethird from Eq. [21]. The corresponding two-dimen-sional analog of Tr is:

ft = 1 - (1 - rr)2/3 = 1 - (r3-*)2/3 [66]The two-dimensional analog of F then follows:

ft =r2/3(£>-3) _ [67]

If a uniform hydraulic gradient Jf is applied alonga direction f, water will flow through a pore area ofm-l

„• for each unit cross-sectional area perpendicularto the direction / The flow rate may be expressed as

m-l

Qf- iQfj [68]

where Qf} is the flow per unit cross-sectional area pro-duced by the hydraulic gradient ̂ through the "active"network of fracture opening PJ, as given in Eq. [63].Thus:

Qr= - [69]

and the hydraulic conductivity of the fracture systemfor the water content 0, is


K, = (pvg/UMp^tfGJ) [70]

Equation [70] shows that, in an incompletely frag-mented, fractal porous medium, the hydraulic con-ductivity is compounded of the partial hydraulicconductivities of the active networks of fractures thatare filled with water.

CONCLUDING REMARKSThe concepts developed here lead to a self-consist-

ent description of a fractal porous medium that canbe realized in nature. Scaling and self-similarity prop-erties are imposed on both the pore space and the solidmatrix, with the result that the porosity, bulk density,and aggregate-size distribution can be related to themean (or median) aggregate diameters in successivepartial volumes (Eq. [1]) of the porous medium. Theserelationships were derived first for a fractal porousmedium in which the pore space is a connected set,while the solid matrix may be either connected (Fig.Ib) or not (Fig. le), the latter case being termed a(completely) fragmented fractal porous medium. Themathematical description of this kind of fractal porousmedium is epitomized in Eq. [24], [25], [29], and [52],which relate the total porosity, the porosity and bulkdensity of a partial volume, and the aggregate-size dis-tribution to aggregate diameter in terms of the fractaldimension of the medium, D. The equation derivedfor the total porosity shows that D < 3.

The completely fragmented fractal porous medium,however, cannot represent a soil in nature becauseevery aggregate in it is surrounded by pore space, asituation that exists only for soil in the laboratorywhen collected for aggregate-size analysis. What ismissing are the interaggregate bridges that cement ag-gregates together and endow natural soils with theirfield structures. These bridges occur because the frag-mentation of the soil is not complete. Thus, it is nec-essary to define a probability of incompletefragmentation (the clustering factor, F) and to incor-porate it into the concept of a fractal porous medium.This can be done directly, if F has a uniform valuefor all partial volumes. The resulting expressions forthe porosity and bulk density in terms of aggregatediameter (Eq. [41] and [42]) now involve the bulk frac-tal dimension of the medium, DT. This latter parameterobeys the inequality D < Dr < 3. It represents thematrix and pore space of a medium that is not com-pletely fragmented; Le., Dr is the fractal dimension ofthe structured porous medium in nature and D is thatof a collection of its aggregates.

Incomplete fragmentation produces a network ofpores in which some are single-size fractures that areconnected to the regular fractal network that does notextend throughout the porous medium (Fig. Id). Thesesingle-size fractures are comprised in "active net-works" that will exhibit a nonzero hydraulic conduc-tivity when they are filled with water. At a given watercontent, defined by Eq. [53] for an incompletely frag-mented fractal porous medium, the corresponding hy-draulic conductivity can be expressed in terms of thesimilarity ratio and the fractal dimensions, D and Z)r

(Eq. [65], [66], and [70]). The moisture characteristicof the medium also can be expressed in terms of theseparameters (Eq. [57] and |[58J). Thus, the fractal con-cepts lead to a testable model of soil water properties.Very few data sets, however, are available with whichto examine the applicability of Eq. [41], [42], [52], [57],[58], and [70] to natural soils. It is our hope that futureresearch will provide the complete data on soil aggre-gate and soil water properties for undisturbed soils thatwill permit a conclusive test of the fractal model de-veloped here.

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. League and L.J. Waldron arethanked for their technical assistance and helpful advice dur-ing the preparation of this article, and Ms. Susan Durhamand Ms. Joan Van Horn are thanked for their excellent typ-ing of the manuscript.

fractal fragmentation, soil porosity, and soil water properties: i. theory

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