Mass fractal dimensions and some selected physical properties
of contrasting soils and sediments of Mexico
K. Oleschko
a,*, B. Figueroa, S.
b, M.E. Miranda
c, M.A. Vuelvas
d, E. Solleiro, R.
aaInstituto de GeologõÂa, Universidad Nacional AutoÂnoma de MeÂxico (UNAM), Aportado Postal 70-296, Ciudad Universitaria, C.P. 04510, Coyacan, MeÂxico, D.F., Mexico
bColegio de Postgraduados, km.35.5 Carretera MeÂxico-Texcoco, Montecillos, Estado de MeÂxico, C.P. 56230, Mexico
cPosgrado en Ciencias de la Tierra, Instituto de GeologõÂa, Universidad Nacional AutoÂnoma de MeÂxico (UNAM), Aportado Postal 70-296, Ciudad Universitaria, C.P. 04510, CoyacaÂn, MeÂxico, D.F., Mexico
dInstituto Nacional de las Investigaciones Forestales y Agropecuarias (INIFAP), km.6.5, Carretera Celaya ± San Miguel Allende, Celaya, Guanajuato 38001, Mexico
Received 2 February 1999; received in revised form 17 December 1999; accepted 28 January 2000
Abstract
The fractal approach to the study of soil structure, its dynamics, and physical processes appears to be a useful tool in reaching a better understanding of system performance. Nevertheless, the precise concurrent analysis of soil physical properties and fractal parameters seems to be important in order to evaluate the applicability of fractal concepts to soil science. The objective of this study was to relate the solid and pore mass fractal dimensions of some soils of Mexico to in situ measured bulk density, electrical permittivity and mechanical resistance. For this purpose, the fractal structure of some soils and sediments with contrasting physical properties is documented, using microscopic images, in the scale range from 0.009 to 0.2 cm. A single and statistically representative mass fractal dimension, independent of scale, was found for analysed solid and pore networks. Non-invasive techniques were used for in situ measurement of selected soil physical properties. The physical measurements were made simultaneously with sampling for the micromorphological analysis. Simple but not unique relations were established between the solid and pore mass fractal dimensions, soil bulk density, apparent dielectric constant (except for Eutric Vertisol), and mechanical resistance (except for Tepetates). A high correlation was found between mass fractal dimension and horizon depth. We conclude that the fractal analysis is a useful tool to distinguish between soils and sediments of different genesis, and that the solid and pore mass fractal dimensions may be useful indicators of the horizon compaction status.#2000 Elsevier Science B.V. All rights reserved.
Keywords:Mass fractal dimension; Compaction status; Porosity; Mechanical resistance
1. Introduction
Fractal geometry may be a powerful tool in describ-ing heterogeneity and understanddescrib-ing the relationships between the soil structure and physical, chemical and *Corresponding author. Tel.:52-5622-4267;
fax:52-5622-4317.
E-mail addresses: [email protected] (K. Oleschko), [email protected] (K. Oleschko),
[email protected] (B. Figueroa, S.), [email protected] (M.A. Vuelvas)
biological processes (Anderson et al., 1998). However, Baveye and Boast (1998) have emphasised that, in spite of the impressive volume of literature on the subject, the lack of a consensus on what it means for a soil to be or to behave ``like a fractal'', is apparent. In this paper the original de®nition of fractals, regarding them as geometrical constructions that are similar to their parts (Baveye and Boast, 1998) is accepted. Therefore, the fractal dimension of all analysed sets is strictly constrained to be between the topological dimension of the fractal and the Euclidean dimension of the space in which this is embedded. Notwithstand-ing, the solid and pore networks are physical objects, and therefore the Pareto distribution (or power law relationship) can be established for them with known precision only between lower and upper cut-offs (length scale limits, Anderson et al., 1998). For the mentioned scale range, the solid and pore geometrical properties are statistically similar to those of mathe-matical (or deterministic) fractals (Baveye and Boast, 1998), and the respective networks can be described as statistical fractals (Oleschko et al., 1998).
Rieu and Sposito (1991) have stated that precise concurrent analysis of soil physical properties and fractal parameters are required in order to evaluate the applicability of fractal concepts to soil science. There are several fractal dimensions needed for the complete characterisation of soil structure (Anderson et al., 1996). Experimental evidence shows fractal scaling of mass, pore space, pore surface and the size distribution of fragments, between upper and lower limits of scale, but typically there is no coincidence in the values of the fractal dimensions characterising different properties (GimeÂnez et al., 1997). The static structural properties, including heterogeneity and space-®lling ability of an object, are described by the mass fractal dimension Dm, and showed to be directly related to the material compactness (Bartoli et al., 1991; Anderson et al., 1998; Lipiec et al., 1998). Anderson et al. (1996) have speci®ed that objects regarded as mass fractals are those that are porous and do not have a uniform internal mass distribution. The more general ``pore±solid fractal'' model (PSF) was proposed for soil by Perrier et al. (1999). Neither the pore phase nor the solid phase of a general PSF exhibits mass fractal scaling. Two groups of empirical methods are useful to functionally characterise a porous medium using fractal approach (Crawford
et al., 1999). The ®rst one is based on the direct imaging and analysis of soil structural units. In the second group all indirect evaluations of structural parameters are undertaken, including water retention, water vapour adsorption data, mercury porosimetry and X-ray scattering (Gomendy et al., 1999; Rice et al., 1999; Sokolowska and Sokolowski, 1999). At present, the analysis of soil images using computer programs is potentially the most reliable method for measuring the geometry of fractals (Crawford et al., 1993; Oleschko et al., 1998). The objective of this study is to relate the solid and pore mass fractal dimensions of some soils and sediments of Mexico, obtained from microscopic images, to in situ measured bulk density, electrical permittivity and mechanical resistance. These physi-cal properties are more common measures of soil compaction status and can be measured in situ by modern, indirect and non-invasive techniques (Jury et al., 1991). The relation between the mass fractal dimensions and the horizon depth was also investi-gated.
2. Materials
Four contrasting geosystems, in steady- or station-ary-state moisture conditions (steady state ¯uxes), were selected for the present study. Their location is shown in Fig. 1. The complete pro®les of three representative soils and one set of lacustrine sedi-ments, containing the materials with a broad variety of physical and morphological properties, were studied.
2.1. Geosystems description
2.1.1. Melanic Andosol, Veracruz state
were distinguished by in situ bulk density measure-ment. The ®rst of these, at 0±18 cm, was more com-pacted and corresponded to the Andosol arable horizon. The last one corresponded to the fresh ash deposit.
2.1.2. Eutric Vertisol, Guanajuato state
This study area is located in Celaya, Guanajuato state at the National Institute of Forest and Agricul-tural Research (INIFAP) experimental ®eld (BajõÂo). Eutric Vertisols are the typical soils for this area, with a high shrink±swell capacity which is dependent on the water content (Oleschko et al., 1993, 1996;
Coulombe et al., 1996). The soils of this zone origi-nated from different types of alluvium derived from basalt and other igneous extrusive rocks.
The study area is representative of the Bajio agri-cultural zone, under continuous drip irrigation sys-tems. A 1.5 m deep pro®le was opened, and four different horizons were identi®ed. The arable layer was limited by the plough-pan at approximately 30 cm depth. This horizon is typical for Vertisols under intensive agricultural use (two crops per year during the last 50 years). The boundary with the next horizon is at 80 cm depth, and is characterised by a change in colour from black to deep brown. A change in soil
texture from clay to sandy loam marks the transition to the third horizon at 120 cm. The detailed physical characterisation of the Eutric Vertisol was accom-plished only in the upper 45 cm of the pro®le.
2.1.3. Texcoco lake sediments, Mexico state
This sampling area is located in Montecillo, Mexico state, and covers the eastern zone of the dry Texcoco lake bank. The lake is a ¯at geomorphological zone with total surface area of 1000 ha (Luna, 1980). The Texcoco lake stratigraphy is composed of a basaltic bed, and limestone and marine deposits of the Upper Cretaceous period (at 2000 m depth below the sur-face). The upper strata are dominated by lacustrine sediments (from 40 to 20 m) and by speci®c lacustrine clay accumulations (from 6 to 0 m). The amorphous clay, locally referred to as ``jaboncillo'', is charac-terised by speci®c physical and chemical properties: an extremely low bulk density (280±430 kg mÿ3
), a high speci®c surface of 225 m2gÿ1
and a very high water retention capacity (e.g. more than 3 g of water is retained by 1 g of clay).
This area represents the most interesting case for the micromorphological and physical studies. Six hori-zons with clear differences in morphology as well as physical and chemical properties and regular horizon-tal boundaries were identi®ed in the reference pro®le. The natural water table occurs at 1.8 m depth and is the lower boundary for the morphological description which follows.
The sediment pro®le is divided into two contrasting parts by an intermediate horizon of compacted basaltic volcanic ash deposit with a bulk density of 1420 kg mÿ3
and a particle size varying from silt to sand, at a depth of 39 and 69 cm (Fig. 2). This horizon represents the textural screen between Mollic surface layer (from 0 to 39 cm), and the clay accumulation (from 69 to 180 cm). The detailed seasonal studies have shown that these two parts of the pro®le are not hydrologically connected. Therefore, the moisture content in the upper part of the pro®le depends only on the precipitation regime, whereas the water content in the lower part is determined by the ground water table ¯uctuations. The ®eld survey was carried out in the dry season (January), after more than four months had elapsed since the last rain. Thus, the studied area was in quasi-equilibrium with respect to water content.
On one side of the reference pro®le, three ¯at surfaces 5 m5 m were opened for detailed physical measurements. The ®rst surface was at 15 cm depth, while the second coincided with the centre of the volcanic ash deposits (50 cm), and the third corre-sponded to the pure lacustrine clay deposit at 80 cm depth.
The dielectric constants of these materials ranged from 13.01 in the upper zone to 65.9 in the water saturated lower part of the pro®le. The bulk density varied from 980 to 280 kg mÿ3
and the gravimetric water content in steady-state conditions from 21.48 to 363.35% (estimated in grams of water retained by 100 g of solid material). These data emphasise the high porosity of clays in the lower part of the studied pro®le. However, microscopic observations of this zone disclosed a predominance of pores of sub-micro-scopic size as well as presence of a few ®ssures.
2.1.4. Tepetates, Mexico state
In Latin America, hardened soils of volcanic origin are often referred to by their vernacular names (Zeb-rowski, 1992). In Mexico, these formations are called ``Tepetates'', which means ``hard'' in the NaÂhuatl Indian language. Commonly, Tepetates are interstra-ti®ed with volcanic and sedimentary deposits. All pro®les include different paleosols characterised by clear and abrupt horizontal limits between the layers. In general, Tepetates in the Mexican high plateau are natural, massive, compact and hard formations, cemented by different chemical agents, including clays and silica. Hardness is the distinctive character-istic of these materials.
All Tepetates have a speci®c geomorphological position. Hardened horizons are inserted in piedmont and glacis soils, and associated with brown clay soils, de®ned as Cambisols (Zebrowski et al., 1991). This location coincided with sub-humid and sub-arid regimes (ustic), and suggests the climate and pedo-genic effects on Tepetates formation (Quantin, 1992). The opened pro®le presents several advantages for the fractal study: (1) All cemented horizons are sepa-rated by paleosols with vertic properties. (2) The limits between horizons are abrupt and horizontal, since Tepetates are related to ancient volcanic events. (3) The period of moisture quasi-equilibrium can be selected easily. (4) Tepetates are characterised by speci®c pore space morphology: main pores are iso-lated and occupied by cementing agents (Fig. 3d). The apparent dielectric constant was measured only in the upper 15 cm (Ka4.4), since the hardness and the presence of numerous ®ssures make the measurements in other layers statistically incoherent.
Nine horizons of different origin with contrasting properties were described in this pro®le, and three of them were identi®ed as cemented layers (Tepetates), separated by clayey layers.
3. Methods
3.1. Micromorphological analysis
Three undisturbed samples (8 cm4 cm) were col-lected with metal samplers from each horizon of the four opened pro®les. All samples were taken at ®eld moisture and carried in plastic bags without drying. In
the laboratory, samples were dried by the acetone replacement (in liquid phase) method and then impreg-nated with a 1:1 polyester resin (HU-543) and acetone mixture (Murphy, 1986). Soil cores were re-impreg-nated with the same resin under vacuum conditions. When the resin was suf®ciently hardened, samples were horizontally sectioned parallel to the soil surface.
Thin sections (2 cm4 cm, with average thickness of 30mm) were prepared by standard petrographic
procedures (Brewer, 1964) and analysed under the petrographic microscope (Olympus, BH-2). Four black and white photographs were taken at the same scale from each thin section. In total 12 microphotographs were used for the fractal analysis of each soil horizon. More contrasting examples are shown in Fig. 3.
3.2. Fractal analysis
Each photograph (15 cm10 cm) of the thin section was scanned, using a 600 dpi resolution commercial scanner. The obtained image was rendered on a grid of 10001000 pixels, and used for the automatic fractal analysis. Adobe(TM) software was used to change all grey scale ``tiff'' images into ``raw'' format. The distribution of grey tones, varied from 0 (totally black) to 255 (completely white) and was estimated using the image histogram. On the microphotograph the bound-ary between the solid and pore networks is fuzzy; therefore the limit between both sets was quanti®ed by a range of grey tones. The Adobe PhotoShop software was used to measure with pipette the grey values corresponding to the solid and pore networks. The obtained grey tone distributions were compared with histogram data. The fractal programme, designed by Parrot and Rico (1997) in Borland environment and coded using C
progressively with larger boxes. The box size was equal tom-pixels. The size ofmbegins at 1 and was increased to a maximum value of 1000. Sixteen different box sizes are used in the Parrot and Rico (1997) programme. The number ofm-pixels or boxes that ®t in the solid (or pore) part of the image, whenm is increased, was counted. This method was described and shown diagrammatically by Anderson et al. (1996), who have mentioned that, the value of Dms (andDmp) can be estimated from a plot of ln (number of m-pixels corresponding to the solid or pore net-work) versus ln(m). The theoretical line was adjusted by least-squares regression and Dms and Dmp were estimated from the negative of the slope.
3.3. Physical measurements
3.3.1. Time domain re¯ectometry
The time domain re¯ectometry technique (TDR), was used to measure the apparent dielectric constant (Ka) and the volumetric water content (yi) of the
studied soils and sediments. The Trase System Model 6050x1 is designed to measure dielectric constants over a frequency bands between 100 and 1000 MHz. A step pulse of electromagnetic radiation was sent along the two parallel waveguides placed in the soil. The permittivity of the material between the wave-guides causes the pulse velocity deviation from the known light velocity in vacuum (Jury et al., 1991). The permittivity is estimated from the pulse travel time, and the water content is calculated by a known empirical third-order polynomial regression, by relat-ing these two parameters. This model was proposed by Topp et al. (1980). The obtained calibration curve was used for the in situ volumetric content measurements. In total 16 measurements were accomplished by TDR on each horizon of interest.
Simultaneously to the TDR measurements, three samples were taken for gravimetric water content (Wi)
and soil bulk density (rb). In the arable horizon of Eutric Vertisol more detailed analysis was carried out, taking more than 50 samples forKa,Wiandrb(Fig. 4d).
Fig. 4. Relation between the material bulk density (rb) and apparent dielectric constant (Ka): (a) all materials compared together; (b) Melanic
The undisturbed samples (10 cm10 cm10 cm) were obtained by using a coring tube of known volume suitable to measurerb. The gravimetric water and bulk density data were used for the volumetric water cal-culation. The volumetric water content, measured by TDR and estimated fromrbandWidata, were
com-pared, and used: (1) to de®ne the precision of the TDR calibration curve in contrasting compactness condi-tions; (2) to establish the relation betweenyiandrb. Subsequently, the last relation was used for the indir-ect bulk density estimation from the volumetric water content data.
3.4. Mechanical resistance
A cone penetrometer (Rimik CP20) was used to measure the mechanical resistance of soils and sedi-ments. The cone penetrometer measures and records
cone index (CI) data down to a depth of 600 mm to give CI values up to 5000 kPa. The most important feature of this penetrometer is that it uses an ultrasonic method for measuring depth. CI data at all of the studied sites were recorded on each of 16 sampling points, every 15 mm down to 600 mm depth, and are presented in Fig. 7 in values divided by 100.
3.5. Statistical analysis
The data variation was estimated by classical sta-tistical procedures. For all mass fractal dimensions the mean value of 12 samples was used (Table 1). The number of physical data has varied, depending on the used technique, and not always has coincided with the number of analysed fractal parameters, in this case only mean values of variables were used to establish the relationships of interest, where it was possible all
Table 1
Solid (Dms) and pore (Dmp) fractal dimensions of studied materialsa
Soil Depth
(cm)
Dms Dmp Da
(kg mÿ3)b
Kac Morphological description
Melanic Andosol 0±18 1.9374 (0.007)d 1.8864 (0.009) 0.39 18.7 Melanic horizon
18±55 1.9304 (0.008) 1.8768 (0.012) 0.37 25.6 Sandy loam 55±95 1.9246 (0.008) 1.8969 (0.009) 0.32 36.0 Sandy loam 95±130 1.9278 (0.009) 1.9082 (0.010) 0.3 44.4 Sandy
Eutric Vertisol 0±15 1.9310 (0.009) 1.8845 (0.090) 1.07 29.55 Arable layer 15±30 1.9363 (0.011) 1.8352 (0.003) 1.23 Arable layer 30±45 1.9350 (0.008) 1.7987 (0.007) 1.35 Plough-pan
Texcoco Lake deposits 0±10 1.9385 (0.005) 1.8233 (0.01) 0.98 13.01 Mollic horizon 27±38 1.9479 (0.010) 1.8620 (0.020) 1.42 16.43 Loamy horizon 56±67 1.9351 (0.009) 1.8741 (0.019) 1.21 21.84 Basaltic volcanic ashes
69±80 1.9570 (0.012) 1.7651 (0.002) 0.43 44.85 Loamy with small clay subhorizons 90±100 1.9697 (0.009) 1.7174 (0.006) 0.37 56.09 Clay
110±120 1.9580 (0.008) 1.7808 (0.020) 0.28 65.9 Clay
Tepetate profile 0±25 1.9520 (0.001) 1.8360 (0.009) 1.57 4.4 Colluvium 69±87 1.9537 (0.004) 1.8033 (0.032) 1.63 Vertic properties 87±124 1.9600 (0.004) 1.8875 (0.006) 1.15 Vertic properties 124±150 1.9408 (0.005) 1.7768 (0.003) 1.25 Tepetate, t2a 200±220 1.9593 (0.010) 1.7815 (0.050) 1.68 Tepetate, t2b, hard 220±235 1.9773 (0.002) 1.8013 (0.008) 1.68 Tepetate, t2b, hard 235±260 1.9150 (0.004) 1.9079 (0.004) 1.19 Paleosoil 260±420 1.9645 (0.010) 1.7606 (0.016) 1.30 Paleosoil
420±460 1.9888 (0.007) 1.6219 (0.165) 1.66 Tepetate, t3, very hard
aSome data of this table were published before by Oleschko (1999). bBulk density.
cApparent dielectric constant.
measured values were graphed (Fig. 4d, Figs. 7a±d, 8a±d and 9a±c).
4. Results and discussion
4.1. Fractal analysis of the microscopic images
Pentland (1984) has shown that a fractal surface and its bi-dimensional photographic image are mathema-tically equivalent. In the present study, the fractal structure of the solid and pore networks of soils and sediments in four contrasting geosystems, was docu-mented in the scale range from 0.009 to 0.2 cm, using their microscopic images. All analysed networks were characterised by a single, near ``ideal'' fractal dimen-sion, independent of scale. The solid (Dms) and pore (Dmp) mass fractal dimensions have shown a clear dependence on the soil and sediment genesis (Table 1). A maximum fractal dimension of 1.9888 was obtained for the solid set of fragipan (t3), in the Tepetate pro®le, identi®ed as the hardest horizon. This means that the solids in this layer occupy considerably more space than the pores. The minimum value of the pore mass fractal dimension (Dmp1.6219) was found for the same horizon. The Melanic Andosol and the paleosol with vertic properties in the Tepetates pro®le have the lowest values of the solid set fractal dimension (1.9246 and 1.9150, respectively). These results coin-cide with our previous data (Oleschko et al., 1997, 1998), and showed that the fractal dimension, within the studied scale range, is a useful parameter for distinguishing between materials of different genesis. It means that this parameter may possibly be used to monitor the tillage in¯uence on soil physical proper-ties, including the soil compaction status. Neverthe-less, from time to time the differences among the fractal dimension of extremely contrasting materials, were small. For instance, the solid set fractal dimen-sions in the arable horizon of the Melanic Andosol and Eutric Vertisol soils with clear bulk density differences (390 compared to 1070 kg mÿ3
), were 1.9374 and 1.9310, respectively. The highest contrasting example is related toDmpof the amorphous clay layer (Texcoco Lake, Dmp1.7651), compared with the cemented horizons (Tepetates, t2a formation, Dmp1.7768), which are estimated in the same range of scales and are not statistically different. These results agree with
Anderson et al. (1998), who have concluded that the common box-counting technique takes into account only pores that are visible at the resolution of the analysed photograph. The visible porosity is similar in the compacted clay and cemented Tepetate horizon, when the sub-microscopic porosity is out of the method resolution. Therefore, it should be empha-sised, that it is essential to refer all fractal analysis conclusions to the exact scale range at which the data were acquired, and that more than one fractal para-meter is needed for soil structure description. The standard deviation varied between 0.001 and 0.011 for the solid mass fractal dimension, and between 0.003 and 0.165 for the pore mass fractal dimension (Table 1).
4.2. Relationship between the soil and sediment physical properties and mass fractal dimensions
For a better understanding of the origin of fractal dimension dependence on the material genesis, the relationships between some physical properties of the studied soils and sediments and their solid and pore fractal dimensions were analysed. A number of dif®-culties, including some incoherent data and results opposite to the theoretically expected tendencies, were encountered. This was due to either the speci®c prop-erties of all studied materials, which vary widely, or independence of fractal parameters from soil genesis. Nevertheless, it was considered that even if relations between solid and pore mass fractal dimensions and physical properties were not clearly obvious, it would be very helpful to know some important trends in the dynamics of these fractal parameters in soils and sediments with different compaction status. A repre-sentative analysis of both fractal dimensions and physical properties would ensure, whether such cor-relation of fractal dimensions and physical properties exists, or it is underlined by large measurement noise and random sample spatial variation.
4.2.1. Relation between bulk density, apparent dielectric constant and mass fractal dimensions
Dirksen and Dasberg (1993) tried to establish a single relation between the soil permittivity and volu-metric water content related directly to the material bulk density. They concluded that the bulk densities differed not only between soils, but also among arti®-cially packed columns of the same soil: bulk densities ®rst decreased and then increased with water content increase. Therefore, the relationship between rband Ka has two components with completely opposite tendencies. Dirksen and Dasberg (1993) strongly recommended comparing this pair of variables indi-vidually for each studied material.
In the present study, a nearly linear relation (R2=0.55) was obtained among rb and Ka when all horizons of the four pro®les were compared (Fig. 4a). The established trend has shown that the higher the value ofrb(the more compact a soil), the lower the dielectric constant. Therefore, the more porous mate-rial has a greater permittivity. All compared matemate-rials, except the Tepetates pro®le, had water contents near ®eld capacity at the time of sampling. Therefore, it seems reasonable that, lower values of bulk density correlated with higher values of the volumetric water content, and also, with larger apparent dielectric con-stants.
In the Melanic Andosol, this relation has shown the best ®t (R20.98). The layer with maximum bulk
density (390 kg mÿ3
) has the minimum dielectric constant (18.7, Fig. 4b). It can be noted that in this volcanic and highly porous soil, a change in bulk density from 390 to 300 kg mÿ3
coincides with a drastic change in the volumetric water content (from 25.3 to 58.3%) and a considerable increase in apparent dielectric constant (from 18.7 to 44.4).
In order to validate the established relation in more contrasting materials, detailed studies were conducted in the Texcoco Lake and in the Eutric Vertisol. In the latter area, the only arable layer was analysed in detail. A strong direct linear relation (R20.96) was obtained for the Texcoco lacustrine deposits (Fig. 4c). The near saturation clay horizons with bulk densities of 280 and 370 kg mÿ3
were characterised by the maximum values of apparent dielectric constant (65.9 and 56.09, respectively).
Nevertheless in contrast, an inverse tendency was observed in the arable layer of the Eutric Vertisol,
where more than 100 TDR measurements were made, with simultaneous sampling for bulk density (the coring tube method) and gravimetric water content. High data variability was noted, but the dielectric constant increase with bulk density was statistically con®rmed (R20.57, Fig. 4d). This atypical behaviour of Vertisol with high shrink±swell capacity is well documented in the literature (Oleschko et al., 1993, 1996; Coulombe et al., 1996), and is considered one of the possible reasons of the reported differences. The unsuitability of the commonly used tube method to obtain the precise Vertisol bulk density measurements, and the absence of some alternative techniques may be the other reason for the observed trend.
The mass fractal dimensions of soil solid (Dms) and pore (Dmp) networks were related to the bulk density (rb) and apparent dielectric constant (Ka) of the studied materials. A good ®t (R2=0.75) was obtained for the Melanic Andosol, where a direct linear relation between theDmsandKawas observed (Fig. 5a). The horizon with the maximum value of solid fractal dimension (1.9374) had the lowest apparent dielectric constant (18.7). From a theoretical point of view, the larger the solid fractal dimension the higher is the solid occupied space, and as a consequence the larger the bulk density and the lower the volumetric water content. A similar, but only slightly correlated statis-tical relationship (R2=0.46) was observed between the Andosol bulk density and the solid set mass fractal dimension: the more compacted horizon (Da390 kg mÿ3) was characterised by the maximum Dms (1.9374, Fig. 5b). This conclusion agrees with general assessments about the solid set mass fractal dimension dynamics. Previously, comparing the same soils under different tillage conditions, it was estab-lished that the solid mass fractal dimensions were higher and the pore mass fractal dimensions were lower for the more compacted soil (Bartoli et al., 1991; Oleschko et al., 1997, 1998). Therefore, it was shown thatDmswas inversely related to average soil porosity.
Relationships between the solid fractal dimension and the bulk density observed in the Texcoco Lake pro®le were unclear (Fig. 6a). The high data dispersion was typical for this experimental area. However, R20.59 suggests that one of the horizons with the highest clay content (90±100 cm) had the largestDms (1.9697), high apparent dielectric constant (56.09, Fig. 6b), and low bulk density (370 kg mÿ3
). This is in line with the ®ndings of Perfect and Kay (1995) and others who also found thatDmsincreased with increas-ing clay content.
The relationship between the pore mass fractal dimension and bulk density in this pro®le, was oppo-site to the theoretical trend. The horizons with the largestDmp(and therefore with the smallestDmsand highest porosity) were characterised by higher bulk density (R20.86, Fig. 6c). The speci®c nature of this
lacustrine amorphous clay with bulk density decreas-ing down to 280 kg mÿ3
, is that of a high invisible sub-microscopic porosity. This was considered responsible for the observed tendency, where the increment of
pore mass fractal dimension coincided with the increase in horizon compactness. This is consistent with the relation established in the same pro®le between the pore fractal dimension and the apparent dielectric constant: the larger theDmp, the smaller is the Ka(R20.60, Fig. 6d).
Rieu and Sposito (1991), using the Chepil data, came to a similar conclusion, establishing that increases inDm(only solid dimension was estimated in their study) were correlated to increases in clay content. Crawford et al. (1993) and Anderson et al. (1996) have concluded that, generally, a soil structure with large continuous pores had larger values ofDmp than a structure with small discrete pores and low lacunarity. However, Anderson et al. (1996) have underlined thatDmpdoes not depend wholly on poro-sity, and different values of Dmp are obtained for structures of equal porosity but with different spatial distribution of pore sets (distinct lacunarity). It was not possible to check the above trend for the Tepetates pro®le, since the data of the apparent dielectric
con-Fig. 5. Relation between Melanic Andosol: (a) solid set fractal dimension (Dms) and apparent dielectric constant (Ka); (b)Dmsand bulk
stant were obtained only for the upper, unconsolidated horizon. In the Vertisol, the high data dispersion, explained by the presence of ®ssures, was responsible for the statistically incoherent ®ts between the vari-ables of interest. However, if only mean values of variables were considered, the largest pore fractal dimension (1.8845) would coincide with the minimal bulk density (1070 kg mÿ3
andR2=0.83).
In the present study it was impossible to establish some general relation between Dms and Ka, for the strongly contrasting soils and sediments studied. This relation should be established individually for each pro®le.
It was concluded that, in general, for the studied soils and sediments with water contents near ®eld capacity, the apparent dielectric constant tends to increase with decrease in bulk density (except the Eutric Vertisol). However, the relation between the material bulk density, apparent dielectric constant and mass fractal dimensions strongly depends on the soil and sediment origin.
4.2.2. Relation between the mechanical resistance and mass fractal dimensions
An inverse intuitively valid linear relation was detected in the Texcoco Lake pro®le between the soil apparent dielectric constant and mechanical resistance (R20.717, Fig. 7a). The clay horizon has signi®-cantly lower CI than the compacted basaltic ash deposits. Nevertheless, in this pro®le, all obtained data were derived from materials with extremely contrasting properties, therefore the experimental points form two clouds concentrated at both extreme ends of Ka versus CI graph, and never form a real continuous function. The mass fractal dimension of the pore set also shows the inverse, contrary to the theoretically expected, relation: the layer with bigger pore dimension (1.8620) has the largest mechanical resistance (8.1 kPa cmÿ2
, Fig. 7c). The same unex-pected trend was found for the relation between solid mass fractal dimension and mechanical resistance of the Texcoco Lake sediments (Fig. 7b). The decrease in theDmsvalues (in the upper part of the pro®le with less
Fig. 6. Texcoco Lake pro®le, relation between: (a) solid set fractal dimension (Dms) and bulk density (rb); (b)Dmsand apparent dielectric
clay content) coincided with the increase in the mate-rial's mechanical resistance (R20.50). Similar rela-tions were observed for the solid and pore mass fractal dimensions versus mechanical resistance in the Mela-nic Andosol pro®le. The deeper horizons (from 55 to 130 cm) have larger CI (600 kPa cmÿ2
in comparison with 380 kPa cmÿ2
in the melanic horizon) and smal-ler solid set mass fractal dimensions (1.9278 versus 1.9374,R20.92). Therefore, the mass fractal dimen-sion is not the direct indicator of soil compaction status, depending more on the material genesis.
The results obtained for Eutric Vertisol con®rm this hypothesis. The solid mass fractal dimension in the ®rst 45 cm of Eutric Vertisol, sampled at 15 cm inter-vals, was almost constant at 1.93, however the pore dimension has diminished from 1.8845 (0±15 cm) to 1.7987 (30±45 cm), and the mechanical resistance increased from 797 to 1317 kPa cmÿ2
. Changes in the pore mass fractal dimension and mechanical resistance re¯ect the presence of a plough-pan, while Dmsis independent of this. An inverse linear relation was established between CI andDmp: higher mechan-ical resistance coincided with smaller pore mass frac-tal dimensions (R20.96, Fig. 7d). These results are in
agreement with bulk density data and the soil mor-phological description.
In the Tepetates pro®le, it was not possible to analyse the relation between the mechanical resistance and mass fractal dimensions of different layers, because the cone penetrometer was unsuitable for use in these consolidated materials.
4.2.3. Relation between horizon depth and mass fractal dimensions
Hatano and Booltink (1992) have estimated the mass fractal dimension (Dmp) from the stained ¯ow pattern, using two-dimensional images. They con-cluded that, in general, values of Dmp decrease with depth. Similar results were obtained in the present study for the pore fractal sets of the Texcoco Lake, Tepetates and Vertisol soils, where the pore set fractal dimension decreased with depth (R20.94, 0.63 and
0.82, respectively, Fig. 8b±d). These data indicate a lower porosity in the deeper layers, linked with a solid set fractal dimension increase in the same direction. However, the inverse trend was obtained for the Andosol pro®le (Fig. 8a) with the deeper and more
Fig. 7. Relation between material mechanical resistance (CI/100) and: (a) apparent dielectric constant (Ka); (b) solid set fractal dimension
(d) Eutric Vertisol.
porous horizons having higher pore fractal dimensions (R20.61).
A trend opposite to the pore networks, was obtained for the solid set fractal dimensions in relation to pro®le depth (except Vertisol, Fig. 9a±c). Different and some-times opposite tendencies, depending on the origin of soil and sediment, were detected. In the Melanic Andosol (R20.71) Dms decreased with depth (Fig. 9a). Then the more recent, more porous and unweathered ashes constitute the deeper layers. The opposite trend, and therefore the increase of the solid mass fractal dimension with depth, was detected in the Texcoco Lake and Tepetates pro®les (R20.81 and 0.70, respectively, Fig. 9b and c). These relationships are strongly linked with the deposition history of the materials. For instance, in the Texcoco Lake the clay horizon comprises a more recent undisturbed material, that was never in direct interaction with the surface. In the Tepetates pro®le, the deep Tepetate (t3) is the hardest layer with clear geological origin, which shows only a few features of perturbation. In the Eutric Vertisol, the pedoturbation is responsible for the continuous mixing of materials, and the homo-genisation of the pro®le up to one or more meters of depth. In this case, theDmsappears to be independent of depth (Table 1).
5. Conclusions
The fractal dimensions of solid and pore networks are useful parameters capable of distinguishing between materials with contrasting genesis. The cemented horizons (Tepetates) are characterised by the maximum solid and minimum pore fractal dimen-sions. The solid mass fractal dimension near 1.99, and the pore mass fractal dimension near 1.62, determined for these hardest and extremely compacted layers, may be proposed as the bounded values for the soil solid and pore networks. The former value is close to the upper topological limit of a two-dimensional image. However, the traditional box-counting techni-que applied to image of one scale is able to estimate only visible porosity, and therefore, on occasions, the soils and sediments with extremely contrasting phy-sical properties may be described by similar mass fractal dimensions, inside the studied scale range. The pore mass fractal dimension re¯ects the soil
compaction status. Nevertheless, the solid mass fractal dimension depends more on material genesis. On the same image, the pore mass fractal dimension is always smaller than the solid one. A complete description of this pattern requires the multiscale fractal analysis, and the use of some additional fractal parameters (i.e. lacunarity). It seems clear that for many soil properties affecting water movement and root growth, the pore-size distribution pattern has much greater importance than total porosity, bulk density and mass fractal dimensions. It is possible to establish simple and statistically coherent relations between fractal para-meters and some selected physical properties of mate-rials. These relations should be established individually for each soil and sediment of interest. Nevertheless, for the same soil or sediment, the pore mass fractal dimension is a useful indicator of soil compaction status and visible porosity.
Acknowledgements
This research was supported by DGAPA (PAPIT programme), UNAM (project IN-106697) and CON-ACYT (project 3617P-A), Mexico. We thank A.M. Rocha T. for technical support and M. Alcayde O. and Dr. G. Tolson for improving the English version of the manuscript.
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