APPLICATION OF FRACTAL THEORY IN QUANTIFYING SOIL AGGREGATE STABILITY AS INFLUENCED BY VARYING TILLAGE PRACTICES AND COVER CROPS IN NORTHERN GUINEA SAVANNA, NIGERIA

Background. Soil aggregate stability varies with management practices like tillage and soil organic matter management. Objective. The objectives of this study were to (i) Explore a fractal theory to investigate the extent of soil fragmentation in soil of Samaru, Northern Nigeria, subjected to different tillage practices and covers crops for the purpose of improving soil aggregate stability (ii) Establish a relationship between fractal dimension (D) and mean weight diameter (MWD). Methodology. A three years field trials was conducted with three tillage practices (no-till, reduced till and conventional till) as main treatments and four cover crops ( Centrosema pascuorum, Macrotyloma uniflorum, Cucurbita maxima and Glyine max ), and a bare/control (no cover crop) as sub treatments. The trial was laid out in randomized complete block design, split plot arrangement and replicated three times. Soil samples were collected at 0-15 cm prior to trial establishment for routine soil physical and chemical properties and at depths 0-5, 5-10, 10-15 and 15-20 cm at the end of each cropping season each year for soil aggregate stability test. Results. No-till soils had 12.58% better soil aggregate stability than soils under reduced and conventional tillage systems. Also, no-till soils had 2.40 % less fragmented soil aggregates than Reduced and Conventional till soil as indicated by the fractal dimension of soil aggregates. Soils under Macrotyloma uniflorum cover crop, were on the average 2.33% less fragmented than soils under Centrosema pascuorum, Glycine max and Cucurbita maxima but 4.56% less fragmented than soils with no cover crops. Implication, No-till and reduced till systems; and cover cropping better enhanced soil aggregate stability due to better accumulation of organic matter emanating from crop residues return to soil in these practices. Conclusion. Conservation tillage and the use of cover crops lowered soil aggregate fragmentation relative to conventionally tilled system and bare soil with no cover crop. The strong linear relationship established between MWD and fractal dimension showed over 80% dependency, suggesting that fractal dimension is another useful index for evaluating soil aggregate stability.


INTRODUCTION
The ability of cohesive forces between soil particles to withstand applied disruptive forces (external stress) simulating the phenomena that occurs on field is known as aggregate stability. Good soil aggregate stability induces good soil structure, soil structure influences seed germination, plant root growth, and water and contaminants transport. However, soil tillage and management practice greatly modify soil structure and consequently soil aggregate stability (Pirmoradian et al., 2005). Blanco- Canqui and Lal (2004) indicated that soil aggregation is an important factor controlling plant growth and carbon sequestration, as well as nutrient flow (Tripathi et al., 2008). It is therefore necessary to implore a scale to quantify strength of stability of soil aggregate.
Aggregate size distribution had been used exhaustively to quantify soil aggregate stability; however, it is necessary to use a single parameter to characterize soil aggregate size distribution. As a result, several empirical indices have been proposed for describing the entire distribution with a single value. Van Bavel (1949) used mean-weight diameter (MWD) to integrate aggregate size distribution obtained by mechanical sieving. Mazurak (1950); suggested that the geometric mean diameter (GMD) may be more appropriate. However, Baldock and Kay (1987) used the following power function to describe the cumulative percentage of aggregates by weight less than a characteristic linear dimension x (e.g., equivalent diameter or height);

W<x = A(x) B
Where: W is the cumulative percentage weight of aggregate; x is the characteristic linear dimension; and A and B are regression constants. Since the coefficient B exhibited maximum variation, it was used as the index of aggregate size distribution.
Previous indices to quantify soil structure, often, were empirical. Recent advances in fractal theory introduced scaling parameters, as fractal dimension that may be suitable for characterizing aggregate-size distribution in soil. Several researchers explored this possibility (Perfect and Kay, 1991;Young and Crawford, 1991;Rieu and Sposito, 1991a, b;Perfect et al., 1992Perfect et al., , 1994Rasiah et al., 1992 andAnderson et al., 1997).
According to Mandelbrot (1982), fractals are characterized by a power-law relation between the number and size of objects. The value of fractal dimension (D) is equal to the absolute value of the exponent in the relation N >x = k(x) -D Where N >x is the cumulative number of objects greater than x, and k is a constant corresponding to the number of fragments of unit length. The value of D depends on the shape of individual objects within the distribution, and the overall extent of aggregate fragmentation. The larger the value of D, the greater the aggregate fragmentation; this means that the shape of aggregate may be similar in various ranges of aggregate size. However, it may be assumed that the value of D is scale invariant in shape.
A parameter which has been primarily estimated from the wet-sieving data is the fractal dimension (D). It is found from the regression of logarithms of mass (mass-based approach) or cumulative number (number-based approach) of stable aggregates to the logarithms of characteristic linear dimensions (such as radius or length). The fractal dimension is derived as the absolute value of the exponent in the relationship (Perfect and Kay, 1991). A review by Anderson et al. (1998) stated that the power-law distribution observed in the fragmentation of natural materials as a result of scale invariance of the fragmentation mechanism implies that the zones of weakness that are predisposed to failure exist at all scales.
According to the fractal fragmentation model of Turcotte (1986), the fractal dimension in soil is expected to be less than 3 since the inequality D ≥3 would require that the probability of grain fragmentation be ≥1 which is not valid. Perfect and Kay (1991), however, indicated that the value of D determined from aggregate-size distribution is a measure of soil fragmentation and showed that it can be as high as 3.5.
Fractal dimension has been used to characterize the influence of soil properties and cropping systems on the size distribution of aggregate, subsequent to fragmentation , these investigators have reported values of D>3. McBratney (1993) has also questioned the merit of values D>3 and their physical significance. Perfect et al. (1993) showed, however, that values of D>3 are theoretically possible if the fragmentation process exhibits multi fractal behavior. Physically, values of D>3 mean that fragments are retained at each level in the hierarchy than is possible for fractal fragmentation. Rasiah and Biederbeck (1995) have shown that values of D obtained using the non-linear fitting procedure in general were smaller and more accurate than those obtained using the linear procedure. Perfect et al., (1994) reported a range of 1.179 to 2.803 for values of D estimated for fragmented aggregates. Several authors (Young and Crawford, 1991;Tyler and Wheatcraft, 1992;Matsui, 1996 andKozak et al., 1996) have observed that the approach for estimating D values influences the results obtained. Rasiah et al. (1992) and Perfect et al., (1994) reported that the estimates of D from aggregate size distribution data varied with cropping, wetting and tillage treatments. Perfect and Kay (1991) noted that the value of D was a measure of fragmentation and irregularity, which increases with increasing input and time under corn production. Furthermore, Anderson et al. (1998) observed that the main factor that limits the estimation of fragmentation fractal dimension is that estimates are made from a distribution of aggregates or particles, which bear no resemblance to the original soil matrix. Perfect and Blevins (1997) have shown that fractal parameters are sensitive to tillage treatment. Mouldboard ploughing increases soil aggregate fragmentation in comparison with no-till. Sepaskhah et al. (2000) compared indirect number-size fractal dimension (Dn), mass-size fractal dimension (Dm) and mean-weight diameter (MWD) as measures of soil aggregate stability. The fractal dimensions Dn and Dm decreased with increasing amount of mulch application indicating an increase in aggregate stability as a result of the addition of the mulch. Salako et al. (1999) evaluated soil macro aggregate stability under different fallow management systems and cropping intensities in southwestern Nigeria; they reported higher D values (more fragmentation) in cultivated soils relative to fallowed soil where aggregates stability was enhanced. In addition, Salako et al. (1999) reported that pueraria system favoured better soil aggregate stability than the bush fallow and leucaena systems under continuous cropping. However, there is a need to implore fractal dimension, to test macro aggregate stability of low organic matter soil of Northern Nigeria, under different management practices. The objectives of this study are to: (i) use fractal theory to evaluate the extent of soil fragmentation in soil of Samaru, Northern Nigeria, subjected to different tillage practices and covers crops for the purpose of improving soil aggregate stability.
(ii) Establish a relationship between fractal dimension and mean weight diameter.

Description of Study Area
The trials were conducted at the Institute for Agricultural Research (IAR) farm (latitude 11.17358°N, longitude 7.63020°E and altitude of 691 m above sea level) Samaru, Zaria, Northern Guinea Savanna ecological zone of Nigeria. The study area has a long-term mean annual rainfall of 1101±16.1mm (Oluwasemire and Alabi, 2004), with a uni-modal rainfall pattern annually, beginning in April and ending in October. The minimum and maximum mean annual temperature being 18°C and 31.5°C respectively. The soil type is Typic haplustults derived from pre-Cambrian crystalline basement complex rocks with some quaternary aeolian deposits.

Experimental Layout and Soil Sampling
The experimental field was laid out in a randomized complete block design, split plot arrangement and replicated three times. The main treatments were three tillage practices namely: conventional tillage, CT, (ploughing, harrowing and ridging; with crop residue removed at the end of each cropping season as practiced by the local farmers in Northern Nigeria), reduced tillage, RT, (harrow once and crop residue incorporated) and no-till, NT, (no soil disturbance except for seed sowing, and crop residue were left on soil surface), while a control i.e. bare, except for sole maize (no cover crops) and four cover crops namely: Glycine max, Centrosema pascuorum, Macrotyloma uniflorum and Cucurbita maxima, were the sub treatments. The experimental field was grown to Maize (Zea mays) as the test crop for three rainy seasons (2011)(2012)(2013).
Prior to trial establishment, disturbed and undisturbed soil samples were taken from the experimental field at 0-15 cm depth, for routine soil physical and chemical analyses. After trial establishment, in each year, disturbed auger soil samples were collected in each treatment plot at depths 0-5, 5-10, 10-15 and 15-20 cm. At each sampling depth, in each replication, soil samples were taken at five different spots per plot then bulked as one.

Soil physical and chemical analysis
Disturbed soil samples taken at 0-15 cm depth, were air-dried and sieved through 2 mm mesh, for determination of particle size distibution (Gee and Or, 2002), organic carbon (Nelson and Sommers, 1982), pH (Mcleans, 1982), Total nitrogen (Bremner and Mulvaney, 1982) Available P by Bray No. 1 acid fluoride method and exchangeable bases (Rhodes, 1982). The undisturbed soil sample was used for determination of bulk density by core method (Grossman and Reinsch, 2002).

Macroaggregate stability and distribution
Macroaggregate stability was determined by wet sieving as described by Kemper and Rosenau (1986) and Angers and Mehuys (1993) with slight modification. Twenty grams of 5-10 mm sieved soil sample was placed on 5 mm mesh openings and immersed in water for 5 minutes. After which, it was sieved for 5 minutes using 5 mm, 2 mm, 1 mm, 0.25 mm and 0.125mm mesh openings. The fractionated aggregates were corrected for sand and stones; and MWD was determined as described below.
The proportional weight of sand free aggregates is given as:

Fractal approach to soil aggregate stability
The cumulative number approach was used to analyze cohesiveness and stability of soil aggregates (Perfect and Kay, 1991;Rasiah et al., 1992;Salako et al., 1999). The wet sieving data obtained during the macro aggregate stability was used to estimate fractal dimension. Briefly the mass of ten pieces of the air dried 5-10 mm aggregates sizes were taken. Then the volume was measured by the displacement method, where the aggregates were coated with paraffin wax and then immersed in water. The density of aggregates is thus given as the proportion of the mass of soil aggregates to its volume. The volume of each fragmented aggregate was calculated as described by Perfect and Kay (1991) and Salako et al. (1999) using the mean size obtained on successive sieves with the assumption that each soil aggregate was a cube. The aggregate density was multiplied by the cubical volume to give the mass of a single aggregate in the total mass of fragments retained on the sieve. The number of aggregates retained on each sieve was then obtained by dividing the total mass of aggregates on each successive sieve by the mass of a single aggregate.
The fractal dimension (D) of soil aggregates was obtained by the relationship: N>x = kx -D Where: N>x = Cumulative number of aggregates, X = Mean sieve size (obtained by finding the average value of successive sieve sizes), D = Fractal dimension, k = intercept obtained from the log-log regression analysis

Microaggregates within stable macro aggregates
Microaggregates occluded in stable macroaggregate (>0.25 mm) was determined as described by Six et al. (2000) but without using the micro aggregates isolator. A sub sample of 15g oven dried macro aggregates obtained during the wet sieving was slaked in deionized water for 20 minutes to break down large macroaggregates, by placing it on a 0.25 mm sieve and shake it with some glass beads on a reciprocal shaker at low speed (150 rpm) for 5 minutes. The content that passed through the 0.25 mm sieve was passed through a 0.053 mm sieve at a rate of 50 strokes in 2 minutes to ensure that the isolated micro aggregates were water stable (Elliot, 1986). After breaking up the macroaggregates, sand and coarse particulate organic matter were retained on the 0.25 mm screen. All fractions were washed in aluminum pans and oven dried at 60 o C to a constant weight. The difference between the initial weight of macroaggregate and weight of sand and coarse particulate organic matter was adopted as the weight of microaggregates occluded in stable macroaggregate. Hence, proportional weight of microaggregates occluded in stable macroaggregate is given as:

Weight of occluded microaggregate fraction
Weight macroaggregate

Data Analysis
Data collected for the three years of study were subjected to statistical analysis of variance for randomized complete block design, using the generalized linear model (GLM) procedure of statistical analytical software, SAS package (SAS, 2008) Significant difference among treatment means were separated using the Duncan multiple range test.The averages for the three years of study are presented for each parameter evaluated.

Characterization of Soil of the Study Area
The physical and chemical properties of soil of the study area prior to trial establishment is presented in Table 1. The soil is generally loam (L) in texture with 43% sand, 43% silt and 14% clay and moderately acidic in soil reaction, with moderate organic carbon (10.17 g kg -1 ) and bulk density (1.4 Mg m -3 ); but poor in total nitrogen (0.72 g kg -1 ). The soil has very low available phosphorus (2.56 mg kg -1 ), exchangeable calcium and cation exchange capacity. While exchangeable magnesium, potassium and Sodium are generally low, in line with published findings of savanna soils in Nigeria (Jones and Wild, 1975). Table 2 reveals the effect of tillage, cover crop and soil sampling depth on distribution of water stable soil aggregates at Samaru. Variations due to tillage practices showed that no-till significantly enhanced macroaggregates fractions (10-5, 5-2 and 2-1 mm) relative to RT and CT practices except in 10-5mm soil fraction were NT and RT had statistically similar soil proportional weight. Whereas, the CT practice had significantly higher microaggregates (≤0.25mm) fraction relative to NT soil.

Tillage, cover crop and sampling depth effects on distribution of water stable soil aggregates
Effect due to cover crops revealed that 10-5mm macroaggregate fraction was not significantly influenced by cover crop. However, Cucurbita maxima alongside all other cover crops significantly enhanced soil macroaggregate fractions 5-2, 2-1 and 1-0.25 mm relative to soil with no cover crop. The least microaggregate fraction 0.25-0.125 was observed in soil under Cucurbita maxima while soil aggregate fraction <0.125 mm was not significantly influenced by cover crops. The stability of macro aggregate fractions 10-5 and 5-2 mm decreased significantly with increasing sampling depth. While, macro aggregate fractions 2-1 and 1-0.25 mm were significantly least stable at depth 15-20 cm relative to other sampling depths. Conversely, 0.25-0125 and <0.125 mm microaggregate fractions were significantly more stable in soils sampled from depth 15-20 cm relative to all other soil sampling depths.

Effect of tillage, cover crops and sampling depth on soil macro aggregate Stability
The effect of tillage and cover crops on soil macro aggregate stability; as characterized by mean weight diameter (MWD) and fractal dimension (D) is presented in Table 3. Generally, soils under no-till practices had significantly higher soil aggregate stability relative to soils harrowed (reduced tillage) and those harrowed, ploughed and ridged (conventional tillage). No-till soils were 12.58% better than soils under the RT and CT systems with respect to mean weight diameter and they were 2.40 % less fragmented than RT and CT soil as indicated by the fractal dimension of soil aggregates (D). Conversely, the intercept (log k) was significantly higher at the CT soils compared to the RT and no-till.
Variation among soil cover crops did not significantly influence MWD and log k. However, the effect of cover crops on fractal dimension of soil aggregates showed that soils grown to Macrotyloma uniflorum as cover crops were the least fragmented; next to it were soils grown to Centrosema pascuorum, Glycine max and Cucurbita maxima while soils with no cover crops were the most significantly fragmented. Soils under Macrotyloma uniflorum were on the average 2.33% less fragmented than Centrosema pascuorum, Glycine max and Cucurbita maxima and 4.56% less fragmented than soils with no cover crops Mean weight diameter, log k and fractal dimension of soil aggregates were significantly influenced by soil sampling depth (Table 3) both MWD and log k decreased with increase in soil depth while D which indicates soil fragmentation increased with increase in soil sampling depth. All the coefficient of determination (r 2 ) values obtained in the regression graphs of log x (mean sieve sizes) versus log N (number of aggregates retained on each sieve) presented dependencies of 94 to 96 % (Table 3).
Variation due to tillage showed that microaggregate occluded in stable macroaggregate (MiAOSMA) were significantly higher in conservation tillage (NT and RT) practices relative to the conventional tillage practices. Effect due to cover crops showed soil covered with Glycine max. had significantly higher microaggregate occluded in stable macro aggregate than all other cover crop plots and the control except for soils under Cucurbita maxima that had statistically similar MiAOSMA as soils under Glycine max. The subsurface soils (0-15 and 15-20 cm) were significantly higher in MiAOSMA than the surface soils at depth 0-5 and 5-10 cm. However, the surface soil had significantly, the least MiAOSMA Figure 1 shows the regression graph of MWD VS fractal dimension, this relationship was described by a linear equation, with an r squared value of 0.832. This relationship suggests over 80% dependency of fractal dimension on mean weight diameter. The data in the plot area showed that as mean weight diameter increases fractal dimension decreases, reason being that mean weight diameter measures the degree of aggregation or cohesiveness of soil while fractal dimension on the other hand measures the extent of fragmentation of soil aggregate.

DISCUSSION
The higher stability of macroaggregate in the no-till soils could be explained in the context of field operation. The no-till plots which had remained undisturbed (untilled) throughout the experimental years possess aggregates which are less susceptible to disruption; due to the fact that plant roots remain are left intact in the NT soil after each cropping season unlike in the CT and RT system where tillage operation disrupt aggregate formation. Furthermore, soil tillage restricted to the planting rows in NT, warrant conditions for the preservation of fungal hyphae and favours the formation and distribution of soil macroaggregates; and consequently, increase nutrient availability and soil water retention (Wright et al., 2007). In addition, crop residues are left on soil surface in NT system, thereby improving organic matter content and organic matter derived binding agent which would enhance stability of soil aggregate. The removal of crop residues in CT system result to loss of soil organic matter which consequently, facilitates the aggregate breakdown processes thus explains the poor aggregate stability in these soils.
Higher proportion of all macroaggregates fractions >0.25 mm in cover crop soils may be attributed to higher root densities and better root system distribution in the cover crop soil relative to the bare soil with no cover crop. Consequently, favoring the binding of soil mineral particles (clay, silt, and sand) and the formation of stable aggregates, which indicates the importance of cover crops to improve soil aggregation. Furthermore, the presence of cover crop at the surface reduces slaking and disintegration of aggregates when wetted, thereby creating room for organic matter buildup and soil aggregate stability. Higher macroaggregates fractions in the surface soil could be attributed to higher organic matter content in the surface soil relative to sub-surface soil. Since surface soil is the immediate recipient of plant residues, therefore; organic matter generated from these plant residues is naturally high at the soil surface especially where crop residues are not incorporated or ploughed into soil as in the conventional tillage system. Presence of plant residues in soil encourages the activities of soil microbes especially bacteria in decomposing these residues and producing polysaccharides and other viscous microbial substances which resist dissolution by water thus binds microaggregates into macroaggregates (Weil and Brady, 2017). Furthermore, organic products of decay such as complex polymer chemically interact with particles of silicate clays and iron and aluminum oxides to form compounds which orient clays into pockets, which form bridges between individual soil particles, thereby binding them in water stable aggregates.
Higher fractal dimension (D) values in the RT and CT systems implies that these tillage practices had more fragmented soil aggregates than the NT system (Table  3). Higher D values are obtained with increasing fragmentation of soil (Tyler and Wheatcraft, 1989;Perfect and Kay, 1991) similarly Anderson et al. (1998) reiterated that large D values represent a highly fragmented soil with dominance of small aggregates. Furthermore, this is an indication that soil tillage renders soil aggregates less cohesive and unstable because pedoturbation by disc plough and shattering of soil peds by the disc harrow resulted in higher soil aggregates fragmentation and larger D values.
This finding is supported by that of Perfect et al. (2004), who investigated the effects of tillage treatments on mass fractal dimension for the soil moisture equation. They observed a significant difference in the values of D for tillage treatments with smaller values for No-till compared with those in the ploughed-disc treatment. This result corroborates those   Perfect and Blevins (1997); Pirmoradian et al. (2005) and Tripathi et al. (2012) they showed that fractal parameter can be used to characterize both soil aggregation and fragmentation, and this parameter is sensitive to tillage treatment. According to their results, Mould board plus disc ploughing increased soil fragmentation in comparison with no-till. This effect was partially reversed by secondary cultivation, indicating that discing broke up large clods and/or coalesced small fragments produced by mould board ploughing.
Expectedly, the lower values of D in soils with cover crops could be attributed to the abundance of roots in these soils, which could have contributed to the clustering of soil aggregates (less fragmentation) relative to the bare plots with no cover crops which were most fragmented. This inference finds evidence in the work of Salako et al. (1999) that indicated live mulching with Pueraria was responsible for numerical less fragmentation of soil aggregate despite continuous cropping. Furthermore, abundance of crop residue in cover crop soils could create room for increased microbial activity in producing microbial derived binding agents for soil aggregates thus higher macroaggregate proportion and improved soil aggregate stability.
Furthermore, the abundance of roots in soils under cover crops probably promoted the release of exudates, known as polysaccharides, cementing substances, which are responsible for stabilizing soil particles, these cementing substances are also interwoven with fungal hyphae and roots of cover crops and contribute to increase in the activity of microorganisms in the soil, especially Arbuscular Mycorrhiza Fungi (Rillig and Mummey, 2006), This soil biota activity influences the soil structure and, consequently, increases the aggregate stability.
The higher organic matter content at the top soil, which often times, recede with soil depth, could be responsible for better soil aggregation (MWD) and lower values of D (less fragmentation of soil aggregates) at the top soil (0-5 cm). The variation of fractal parameters with the tillage and cover crop treatments and at the different soil sampling depth, suggests that fractal dimension is another useful index for evaluating soil structural stability. This had earlier been documented by some authors (Young and Crawford, 1991;Burrough, 1993;Crawford and Matsui, 1996;Kozak et al., 1996;Anderson et al., 1997;Salako et al, 1999).
The intercepts (log k) of the regression graph between of log x (mean sieve sizes) versus log N (number of aggregates retained on each sieve) can be used as an index of the abundance of fragmented soil aggregates in a given mass, Eghball et al. (1993) used them in a fractal analysis to indicate the abundance of roots. Generally, data obtained in this study showed that the intercepts (log k) were influenced more by the number of aggregates retained on sieves >0.10 mm diameter, thereby suggesting that invariably, the number of aggregates making up the macro aggregates was particularly reflected by the intercepts. While the high r 2 value values (between 0.94 and 0.97) in the regression plots suggest that the data were adequately described by fractal analysis.
The higher proportion of MiAOSMA in NT and RT soil is an indication that non and minimal soil disturbance improve soil macroaggregate structure and stability, since MiAOSMA contain physically protected organic matter from microbial decomposition; MiAOSMA is therefore key to improve C sequestration. Similarly, enhanced C sequestration through C stabilization within the microaggregates occluded in macroaggregates has been confirmed in afforested (Six et al. 2002;Del Galdo et al. 2003) and forested soils (Six et al. 2002) compared to agricultural soils where tillage operation was carried out.
The broader leaves, thus higher leaf area in both Glycine max and Cucurbita maxima relative to other cover crops may be useful in sequestering C thereby offering better protection for MiAOSMA and stability of macroaggregate in the soils of these two cover crops. In addition, the biodegradability of Glycine max and Cucurbita maxima crop residues and microbial activity generated after their addition, could generate stabilizing substances such as polysaccharides and hydrophobic compounds or motivate the development of efficient stabilizing microorganisms like filamentous fungi thereby increase the proportion of MiAOSMA and stability of macroaggregate. High clay content in lower soil depths due to illuviation may be responsible for significantly higher MiAOSMA at these soil depths.

CONCLUSION
Conservation till (NT and RT) systems and cover cropping better enhanced soil aggregate stability due to better accumulation of organic matter emanating from crop residues return to soil in these practices. The strong linear relationship established between MWD and fractal dimension and the significant fragmentation (D) revealed amongst tillage treatments, indicates that fractal theory of soil aggregates using the cumulative number approach showed significant differences in stability of soil aggregates. Therefore, suggesting that fractal dimension is another useful index for evaluating soil aggregate stability.