Abstracts
Fractal scaling has been applied to soils, both for void and solid phases, as an approach to characterize the porous arrangement, attempting to relate particlesize distribution to soil water retention and soil water dynamic properties. One important point of such an analysis is the assumption that the void space geometry of soils reflects its solid phase geometry, taking into account that soil pores are lined by the full range of particles, and that their fractal dimension, which expresses their tortuosity, could be evaluated by the fractal scaling of particlesize distribution. Other authors already concluded that although fractal scaling plays an important role in soil water retention and porosity, particlesize distribution alone is not sufficient to evaluate the fractal structure of porosity. It is also recommended to examine the relationship between fractal properties of solids and of voids, and in some special cases, look for an equivalence of both fractal dimensions. In the present paper data of 42 soil samples were analyzed in order to compare fractal dimensions of poresize distribution, evaluated by soil water retention curves (SWRC) of soils, with fractal dimensions of soil particlesize distributions (PSD), taking the hydraulic conductivity as a standard variable for the comparison, due to its relation to tortuosity. A new procedure is proposed to evaluate the fractal dimension of poresize distribution. Results indicate a better correlation between fractal dimensions of poresize distribution and the hydraulic conductivity for this set of soils, showing that for most of the soils analyzed there is no equivalence of both fractal dimensions. For most of these soils the fractal dimension of particlesize distribution does not indicate properly the pore trace tortuosity. A better equivalence of both fractal dimensions was found for sandy soils.
Fractal; particle size distribution; pore size distribution; soil water retention; soil hydraulic conductivity
O escalonamento fractal tem sido aplicado a solos, levandose em conta tanto a sua distribuição de partículas como a distribuição de poros, na tentativa de se caracterizar o espaço poroso no que diz respeito às suas propriedades de retenção e condução de água. Um ponto importante nessas análises é a consideração de que a geometria do espaço poroso dos solos é o reflexo da geometria do espaço ocupado pelas partículas sólidas e que, portanto, a dimensão fractal da distribuição de poros do solo pode ser avaliada através da dimensão fractal da sua distribuição de partículas. Alguns autores, no entanto, reconhecem que a distribuição de partículas por si só não é suficiente para uma perfeita caracterização da geometria fractal do espaço poroso de um solo e que tal possibilidade é válida apenas para alguns casos especiais nos quais há uma correspondência entre ambas as dimensões fractais. No presente trabalho foram analisadas 42 amostras de solos de composições granulométricas distintas com o objetivo de avaliar a equivalência das dimensões fractais de suas distribuições de partículas às de suas distribuições de poros, tomandose a condutividade hidráulica dos solos como uma propriedade indicadora da dimensão fractal da distribuição de poros, uma vez que tal propriedade, assim como a dimensão fractal, deve refletir diretamente a tortuosidade dos poros do solo. Os resultados indicam uma melhor correlação entre as dimensões fractais das distribuições de poros com as condutividades hidráulicas do que as dimensões fractais avaliadas pelas distribuições de partículas e que, para a maioria dos solos analisados, a distribuição de partículas não é um bom indicador para
NOTA
FRACTAL SCALING OF PARTICLE AND PORE SIZE DISTRIBUTIONS AND ITS RELATION TO SOIL HYDRAULIC CONDUCTIVITY
O.O.S. BACCHI ^{1,3 }; K. REICHARDT ^{1,2,3}; VILLA NOVA, N.A.^{2,3}
^{1}Centro de Energia Nuclear na Agricultura, USP, C.P. 96, CEP:13400970  Piracicaba, SP.
^{2}Departamento de Física e Meteorologia, ESALQ/USP, C.P. 9, CEP: 13418900  Piracicaba,SP.
^{3}Bolsista do CNPq.
ABSTRACT: Fractal scaling has been applied to soils, both for void and solid phases, as an approach to characterize the porous arrangement, attempting to relate particlesize distribution to soil water retention and soil water dynamic properties. One important point of such an analysis is the assumption that the void space geometry of soils reflects its solid phase geometry, taking into account that soil pores are lined by the full range of particles, and that their fractal dimension, which expresses their tortuosity, could be evaluated by the fractal scaling of particlesize distribution. Other authors already concluded that although fractal scaling plays an important role in soil water retention and porosity, particlesize distribution alone is not sufficient to evaluate the fractal structure of porosity. It is also recommended to examine the relationship between fractal properties of solids and of voids, and in some special cases, look for an equivalence of both fractal dimensions. In the present paper data of 42 soil samples were analyzed in order to compare fractal dimensions of poresize distribution, evaluated by soil water retention curves (SWRC) of soils, with fractal dimensions of soil particlesize distributions (PSD), taking the hydraulic conductivity as a standard variable for the comparison, due to its relation to tortuosity. A new procedure is proposed to evaluate the fractal dimension of poresize distribution. Results indicate a better correlation between fractal dimensions of poresize distribution and the hydraulic conductivity for this set of soils, showing that for most of the soils analyzed there is no equivalence of both fractal dimensions. For most of these soils the fractal dimension of particlesize distribution does not indicate properly the pore trace tortuosity. A better equivalence of both fractal dimensions was found for sandy soils.
Key Words: Fractal; particle size distribution; pore size distribution; soil water retention, soil hydraulic conductivity
ESCALONAMENTO FRACTAL DE DISTRIBUIÇÕES DE PARTICULAS E DE POROS E SUA RELAÇÃO COM A CONDUTIVIDADE HIDRAULICA DO SOLO
RESUMO: O escalonamento fractal tem sido aplicado a solos, levandose em conta tanto a sua distribuição de partículas como a distribuição de poros, na tentativa de se caracterizar o espaço poroso no que diz respeito às suas propriedades de retenção e condução de água. Um ponto importante nessas análises é a consideração de que a geometria do espaço poroso dos solos é o reflexo da geometria do espaço ocupado pelas partículas sólidas e que, portanto, a dimensão fractal da distribuição de poros do solo pode ser avaliada através da dimensão fractal da sua distribuição de partículas. Alguns autores, no entanto, reconhecem que a distribuição de partículas por si só não é suficiente para uma perfeita caracterização da geometria fractal do espaço poroso de um solo e que tal possibilidade é válida apenas para alguns casos especiais nos quais há uma correspondência entre ambas as dimensões fractais. No presente trabalho foram analisadas 42 amostras de solos de composições granulométricas distintas com o objetivo de avaliar a equivalência das dimensões fractais de suas distribuições de partículas às de suas distribuições de poros, tomandose a condutividade hidráulica dos solos como uma propriedade indicadora da dimensão fractal da distribuição de poros, uma vez que tal propriedade, assim como a dimensão fractal, deve refletir diretamente a tortuosidade dos poros do solo. Os resultados indicam uma melhor correlação entre as dimensões fractais das distribuições de poros com as condutividades hidráulicas do que as dimensões fractais avaliadas pelas distribuições de partículas e que, para a maioria dos solos analisados, a distribuição de partículas não é um bom indicador para a distribuição de poros. As melhores equivalências entre ambas as dimensões fractais foram observadas para as amostras arenosas.
Descritores: Fractal, distribuição de partículas, distribuição de poros, retenção de água, condutividade hidráulica.
INTRODUCTION
Soil hydraulic characterization both in the laboratory and the field is tedious, time consuming and expensive, if not prohibitive, when considering the spatial variability of soil properties. Thus, the lack of information in the available soil surveys and the importance of such data for many studies has motivated researchers to develop predictive approaches from routinely measured soil properties. The attempt to relate soil particlesize distributions (PSD) to soil water retention curves (SWRC) is an old and well known challenge among soil physicists. Empirical relations have been proposed in frequency, inspite of their recognized limitations, including, more recently, some new approaches that make use of fractal concepts. This paper focuses basically the papers of Turcotte (1986); Tyler and Wheatcraft (1989); Tyler & Wheatcraft (1990); Tyler & Wheatcraft (1992) which comprise a very interesting sequence of new ideas started with the work done by Arya & Paris (1981) in the attempt of bridging the two above cited soil characteristics in a physically acceptable way. In the last mentioned paper of Tyler & Wheatcraft, the authors present a theoretical discussion of their earlier papers and recognize some limitations of their previous ideas. One important aspect discussed is the equivalence of both the fractal dimension of the solid and the void spaces, and the limitation in obtaining information on the fractal structure of the porosity solely from PSD data.
In this paper fractal dimensions of PSD and SWRC are analyzed for 42 soil samples and their correlation with soil hydraulic conductivity is established in order to test, for real soils, the above mentioned equivalence of both fractal dimensions. Basic soil data on PSD, SWRC and hydraulic conductivity were taken from Pukett et al. (1985) which present this information for each horizon of 7 different soils of Alabama, comprising the total of 42 different soil samples.
ESTIMATION OF THE FRACTAL DIMENSIONS
a) Fractal dimensions of PSD: Fractal dimensions of PSD for the soil samples were estimated by the procedure proposed by Tyler & Wheatcraft (1992) using the mass distribution of particles in substitution of the cumulative number of particles, according to the relation :
(1)
where M (r<r_{i}) represents the percentage of mass of soil grains of radius r smaller than a characteristic radius r_{i }; M_{t }is the total soil mass (100%); r_{l }is the upper size limit of the course particle fraction, and D_{1} is the fractal dimension of the particle distribution. Since PSD represents a threedimensional collection of soil particles, the fractal dimension D_{1 }corresponds to the volumetric fractal dimension. This value can be used to estimate the fractal dimension of a surface or a transect inside the matrix, as it was done by Tyler & Wheatcraft (1989) to evaluate the fractal dimension of the pore trace lined by soil particles. In this paper we will represent the fractal dimension of PSD based on mass distribution in threedimensions, by D_{MPSD}.
b) Fractal dimensions of SWRC: Tyler & Wheatcraft (1990) analyzed fractal characteristics in soil water retention processes, evaluating water retention properties of Sierpinski carpets and related these properties to the Brooks and Corey empirical water retention model. For this analysis they derived a relation to describe the pore size distribution:
(2)
where N_{p} represents the number of pores with radius R larger than a characteristic radius R_{i} and D_{2} is the fractal dimension of the pore size distribution. A similar equation was also proposed by these authors in their earlier work (Tyler & Wheatcraft, 1989) to describe soil particle size distributions, taking in this case Np as the number of soil particles.
There is a practical difficulty in applying equation (2) when one tries to estimate the number of pores of radius R>R_{i }from the water retention functions q(Y_{i}) or q(R_{i}). The values of volumetric soil water content q corresponding to values of soil water potential Y_{i} (or radius R_{i}) represent the cumulative volume of pores of radius R< R_{i} ,which could easily be converted into the corresponding volume of pores of radius R>R_{i }by taking the difference qq_{s} ; where q_{s} is the saturated soil water content. The main problem lies in the definition of R values which characterize the pore size of each range of cumulative values of q or qq_{s} , in order to estimate the volume of one characteristic pore of each range, and the number of such pores. The same difficulty is found when working with PSD using the number of particles.
The use of equation (3) is here proposed, which, by analogy to equation (1), represents the distribution of void volumes (equivalent to q ) according to the size of pores in the total range of the soil moisture retention curve:
(3)
where R_{l }is the upper size limit for the pore distribution, D_{2} is the fractal dimension estimated from the soil water retention curve (SWRC), represented by D_{swrc}.
For the application of equation (3) the first step involves the evaluation of the corresponding values of matric potential for different values of pore radia according to Laplace's equation, which was simplified into:
Y _{mi} = 0,149/ R_{i} (4)
where Y_{mi} is the soil water matric potential (cm) and R_{i }is the pore radius (cm).
In the next step q (cm^{3}/cm^{3}) values are evaluated from the corresponding matric potential values using Van Genuchten's equation. These q values, corresponding to R_{i} values, are the cumulative volumes of pores with radius R ranging from near zero to R_{i} which hold water at a matric potential Y _{mi}.
RELATIONSHIP BETWEEN FRACTAL DIMENSIONS AND SOIL HYDRAULIC CONDUCTIVITY
Since both fractal dimensions D_{MPSD }andD_{SWRC} reflect the pore trace tortuosity , it would be expected to find an inverse correlation of these values with the soil hydraulic conductivity. Simple regression was used to test this inverse relation and the equivalence of the two sets of fractal dimensions.
RESULTS AND DISCUSSION
TABLE 1 shows basic data on soil texture, Van Genuchten's equation parameters, saturated soil hydraulic conductivity for the 42 soil samples (From Puckett et al.,1985) and the values of D_{MPSD }and D_{SWRC }calculated according the described procedures, using equations (1) and (3).
The correlations between both fractal dimensions (D_{MPSD }and D_{SWRC} ) and the saturated soil hydraulic conductivity is shown in Figure 1 for the 42 soils. As expected, both fractal dimensions present an inverse relation with hydraulic conductivity, however, the fractal dimension of SWRC presented a better correlation (higher R^{2}), indicating that it reflects closer the tortuosity of the soil pore trace than the fractal dimension of PSD.
Figure1  Correlation between fractal dimensions and saturated soil hydraulic conductivity, for all 42 soil samples.
Dividing the set of 42 soils into two subsets according to their clay contents, both fractal dimensions were compared for each subset. Figure 2 shows the correlation between D_{MPSD} and D_{SWRC} for the sandy subset (clay content ranging from 1.4 to 20.9%) and Figure 3 shows the same correlation for the clayey subset (clay content ranging from 22.2 to 42.1%). As it can be seen, there is an evident better correlation between both fractal dimensions for sandy soils.
 Correlation between fractal dimension of PSD and SWRC for the clayey soil sample subset.
An interesting logarithmic relation was found between both fractal dimensions and soil clay content, with a very nice adjustment for fractal dimensions of PSD (R^{2}=0.9993), as shown in Figure 4. A fairly good fitting was also found for fractal dimensions of SWRC and clay content (R^{2}=0.8479). From these results it seems quite reasonable to possibly estimating fractal dimensions of PSD and SWRC from soil clay contents.
Recebido para publicação em 16.08.96
Aceito para publicação em 20.10.96
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 PUCKETT, W.E.; DANE, J.H.; HAJEK, B.F. Physical and mineralogical data to determine soil hydraulic properties. Soil Science Society of America Journal, v.49, n.4, p.831836, 1985.
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 TYLER, W.S.; WHEATCRAFT, S.W. Application of fractal mathematics to soil water retention estimation. Soil Science Society of America Journal, v.3, p.987996, 1989.
 TYLER, W.S.; WHEATCRAFT, S.W. Fractal Processes in soil water retention. Water Resources Research, v.26, n.5, p.10471054, 1990.
 TYLER, W.S.; WHEATCRAFT, S.W.. Fractal Scaling of Soil ParticleSize Distributions: Analysis and Limitations, Soil Science Society of America Journal, v.56, p.362369, 1992.
Publication Dates

Publication in this collection
10 Feb 1999 
Date of issue
May 1996
History

Received
16 Aug 1996 
Accepted
20 Oct 1996