Microsoft Word - 15-47894


1961 
Bioscience Journal  Original Article 

Biosci. J., Uberlândia, v. 36, n. 6, p. 1961-1974, Nov./Dec. 2020 
http://dx.doi.org/10.14393/BJ-v36n6a2020-47894 

FRACTAL FEATURES OF SOIL TEXTURE AND PHYSICAL ATTRIBUTES 
IN INDIAN DARK EARTH UNDER DIFFERENT USES IN WESTERN 

AMAZON 
 

CARACTERÍSTICAS FRACTAIS DA TEXTURA E ATRIBUTOS FÍSICOS DO SOLO 
EM AREAS COM TERRA PRETA DE ÍNDIO SOB DIFERENTES USOS NA 

AMAZÔNIA OCIDENTAL 
 

Half Weinberg Corrêa JORDÃO1; Milton César Costa CAMPOS2; 
 José Maurício da CUNHA3; Ivanildo Amorim de OLIVEIRA4; Laércio Santos SILVA5; Elilson 

Gomes de BRITO FILHO3; Bruno Campos MANTOVANELLI6;  
Ludmila de FREITAS4; Romário Pimenta GOMES5 

1. Universidade Estadual Paulista - UNESP, Departamento de Ciências Agrárias, Botucatu, SP, Brasil; 2. Universidade Federal da 
Paraíba – UFPB, Departamento de Solos e Engenharia Rural, Areia, PB, Brasil. mcesarsolos@gmail.com; 3. Universidade Federal do 
Amazonas - UFAM, Instituto de Educação, Agricultura e Ambiente, Humaitá, AM, Brasil; 4. Instituto Federal de Rondônia - IFRO, 

Departamento de Agronomia, Ariquemes, RO, Brasil; 5. Universidade Estadual Paulista - UNESP, Departamento de Solos, Jaboticabal, 
SP, Brasil; 6. Universidade Federal de Santa Maria – UFSM, Departamento de Solos, Santa Maria, RS, Brasil. 

 
ABSTRACT: Studying particle size distribution is important to understand soil structure and formation 

processes. This research aimed to assess the fractal dimension of soil texture in Indian Dark Earth (IDE) areas 
in southern Amazonas state under different land uses, as follows: two areas in the municipality of Apuí, one 
growing cocoa and the other coffee; a grassland area in the municipality of Manicoré; and a forest area in the 
municipality of Novo Aripuanã. A sampling grid containing 88 collection points (intersecting points on the 
grid) was established in each area, measuring 80 x 42 m for the cocoa and coffee-growing sites, and 80 x 56 m 
and 60 x 42 m for the grassland and forest areas, respectively. Soil samples were collected in soil core and as 
clumps at a depth of 0.0-0.20m to determine the structural physical properties and texture of the soil. The 
following physical attributes were assessed: texture (PSD), bulk density (BD), macroporosity (Macro), 
microporosity (Micro), total porosity (TP) and aggregate stability (GMD and WMD). The fractal dimension (D) 
of the soil texture was determined, followed by analysis of variance and comparison of the means using 
Tukey’s test (p≤0.05). Pearson’s correlation was applied to assess the correlation between variables. There was 
a significant difference between the IDEs studied, with a higher D value in the cocoa-growing area in relation 
to the other sites. Additionally, the larger the clay fraction, the higher the D value. Fractal dimension (D) 
showed a positive correlation with sand, clay, BD, Macro, GMD and WMD, and a negative correlation with 
silt, micro, TP. Based on the D values obtained, the ADE cultivated with cocoa showed superior quality in 
relation to the other areas studied. 

 
KEYWORDS: Fractal dimension. Soil physics. Soil use. 

 
INTRODUCTION 

 
Applications of fractal geometry in soil 

science have shown that soil exhibits fractal 
characteristics, being a porous medium having 
different particle compositions, with irregular shape 
and self-similar structure (TYLER; 
WHEATCRAFT, 1989; KRAVCHENKO; 
ZHANG, 1998). Fractal geometry, proposed and 
established by Mandelbrot (1982), is a method for 
describing systems with non-characteristic scales 
and self-similarity. In recent years, this theory has 
been used to quantitatively describe the particle size 
distribution of soil, attracting the interest of 
pedologists worldwide (DENG et al., 2017). Particle 
size distribution is one of the most important 

physical characteristics of soil because of its 
significant influence on water flow and soil erosion 
(XU; LI; LI, 2013). 

In this respect, broad and precise knowledge 
of particle size distribution is vital to understanding 
soil structures and formation, since it is closely 
related to soil erosion, organic matter content and 
moisture content (DU et al., 2017). Deng et al. 
(2017) studied the fractal features of soil particle 
size distribution and found an association between 
fractal dimensions and the physical and chemical 
properties of the soil analyzed, indicating that the 
lower the fractal dimension, the worse the soil 
physical and chemical properties. Recently, the 
fractal method was applied to estimate soil structure 
and proved to be an efficient tool in analyzing soil 

Received: 01/04/2019 
Accepted: 30/01/2020 



1962 
Fractal features…  JORDÃO, H. W. C. et al. 

Biosci. J., Uberlândia, v. 36, n. 6, p. 1961-1974, Nov./Dec. 2020 
http://dx.doi.org/10.14393/BJ-v36n6a2020-47894 

properties and their relationship with other 
environmental factors (XIA et al., 2015; DENG et 
al., 2017). 

Textural analysis is commonly used to 
characterize particle size distribution (PSD) in soil, 
but the size definitions of the three main fractions 
(sand, silt and clay) are quite random and therefore 
cannot provide comprehensive information (DENG 
et al., 2017). In this case, according to the results of 
a study conducted by Xia et al. (2015), fractal 
dimension can be used to identify soil particle size 
characteristics. Additionally, Filgueira et al. (2006) 
reported that fractal theory and analysis can 
effectively describe soil texture. 

According to Cunha et al. (2016), 
exhaustive soil use and management can irreparably 
damage its physical quality, thereby reducing its 
yield potential. In this context, a quantitative 
description of particle size distribution is important 
in soil structure research, and fractal dimension is a 
useful approach in quantitatively evaluating 
different land-use patterns (DENG et al., 2017). 

In this research, verified the PSD and 
fractural dimensions of soils under different crops 
and natural environment, and identified the 
relationships between soil physical properties. The 
objective of this research was to evaluate the 
possibility that the fractal dimension of PSD can be 
used as an integrative index to quantify crop effects 
associated with structural quality in Indian Dark 
Earth in the Western Amazon. 

MATERIAL AND METHODS 
 
Experiment location 

The soils were located in the region of the 
municipality of Apuí-AM, Brazil and Novo 
Aripuanã-AM, Brazil (Figure 1). The climate in the 
region was classified as humid tropical according to 
Köppen’s classification, with a short dry period 
(between May and September), a mean annual 
temperature of 25 to 26 °C, rainfall from 2,200 to 
2,700 mm and a relative air humidity of 85% to 
90% (BRASIL, 1978; CAMPOS et al., 2012). The 
characteristic vegetation of the region is tropical 
dense forest, consisting of dense multilayered trees 
between 15 and 20 meters high (ZEE, 2008). 

The study was conducted at the four Indian 
Dark Earth (IDE) under different land uses. Two 
were located in the municipality of Apuí (7º 12’ 05” 
S and 59º 39’ 35” W), one under cultivation for 14 
years, growing rice, maize, beans and watermelon 
for six years and cocoa (Theobroma cacao) 
thereafter, and the other used a grassland for two 
years, followed by coffee (Coffea canephora) 
cultivation for four years. No machinery was used in 
either area to plant or maintain the crops. The soil in 
both IDEs areas was classified as Typic Hapludults, 
according to Soil Taxonomy (SOIL SURVEY 
STAFF, 2014) and “Argissolo Amarelo Eutrófico” 
to criteria established by the Brazilian Soil 
Classification System (SANTOS et al., 2018). 

 

Amazonas

0        250      500                1000  km

Manicoré N ovo 
A ripuanã

Humaitá

Sampling Scheme

A B

C
 

Figure 1. Location and representation of the sampling grid in the IDEs studied, with the respective spacing 
between collection points.  

  A – Cocoa and Coffee; B – Grassland; C – Forest. 
 
The third area is located in the municipality 

of Manicoré (Figure 1) (7º 59’ 22” S and 61º 39’ 
51.2” W, mean altitude of 83 m), used for extensive 
grazing (Brachiaria brizantha) for approximately 7 



1963 
Fractal features…  JORDÃO, H. W. C. et al. 

Biosci. J., Uberlândia, v. 36, n. 6, p. 1961-1974, Nov./Dec. 2020 
http://dx.doi.org/10.14393/BJ-v36n6a2020-47894 

years and capable of supporting livestock at around 
one unit/animal per hectare. The soil was classified 
as Typic Rhodudults (SOIL SURVEY STAFF, 
2014) or “Argissolo Vermelho Amarelo Eutrófico” 
(SANTOS et al., 2018) and primary vegetation in 
the region is characterized as dense tropical forest. 
The final area (forest) was a forest fragment in the 
municipality of Novo Aripuanã (Figure 1) (07º 51' 
30" S and 61º 18' 01" W), preserved for more than 
25 years and containing 15 to 20-meter-high 
secondary trees, the soil types in this area classified 
as Xanthic Eutrudox (SOIL SURVEY STAFF, 
2014) or “Latossolo Vermelho Amarelo latossólico” 
(SANTOS et al, 2018). 

 
Soil-sampling and evaluation of soil attributes 

A sampling grid was established in each 
area, with 88 collection points per grid. Grid 
dimensions were 80 x 42 m in the cocoa and coffee 
areas, 80 x 56 m in the grassland area and 60 x 42 m 
in the forest area, with 6 x 8 m spacing between 
points for the coffee and cocoa areas and 8 x 8 m 
and 6 x 6 m for the grassland and forest areas 
respectively (Figure 1). 

In order to determine the structural and 
textural properties of the soil, deformed (clumps) 
and non-deformed samples were collected between 
0.0-0.20 m depth, using a 4.0 cm high soil-core with 
an internal diameter of 5.1 cm. 

The following physical properties were 
determined to correlate fractal dimension with 
particle size: texture (sand, silt and clay), bulk 
density (BD), macroporosity (Macro), microporosity 
(Micro), total porosity (TP) and aggregate stability 
(GMD and WMD).  

The particle size distribution (PSD) was 
determined with 1.0 mol L-1 NaOH solution was 
used as chemical dispersant, with a resting time of 
16 hours. Next, the suspension was transferred to 
steel cups containing water and coupled to an 
electric stirrer at 12,000 RPM for 15 minutes 
(TEIXEIRA et al., 2017). The clay fraction (<0.002 
mm) was separated using the pipette method, sand 
by sieving and silt (0.05-0.002 mm) was calculated 
based on the difference. The sand fraction was 
divided into coarse (0.5-2.0 mm), medium (0.25-0.5 
mm) and fine (0.105-0.24 mm) according to Arraes, 
Bueno and Pissarra (2010), in order to calculate the 
fractal dimension. 

Next, the fractions obtained were sieved to 
determine the size of the solid particles analyzed, 
using a SOLOTEST sieve shaker with digital time 
and frequency adjustment to separate the particles 
through vibrations that accelerate sieving. Each 
sample was agitated for 3 minutes, using sieves with 

mesh sizes of 2mm; 1mm; 0.5mm; 0.250mm; 
0.125mm and 0.053mm. 

The samples were prepared in the laboratory 
by removing the excess of soil from the soil core 
edges; they were then saturated by a gradual 
increase in water depth until reaching approximately 
2/3 of the soil core height. Total porosity was 
determined by the saturation method. Macroporosity 
was obtained from the balance of the set soil core-
soil after applying --6 kPa in a tension table. In its 
turn, microporosity was obtained by subtracting the 
weight of the soil core-soil set equilibrated at -6 kPa 
and its respective oven-dried weight at 105 °C. 

Bulk density was determined by the soil 
core method as described in Grossman and Reinsch 
(2002). In this case, the soil in the soil core was 
oven dried at 105 °C until constant weight. 

Aggregate stability was assessed according 
to Kemper and Chepil (1965), with modifications in 
the following size classes:4.76-2.0 mm; 2.0-1.0 mm; 
1.0-0.50 mm; 0.50-0.25 mm; 0.25-0.125 mm; 0.125-
0.063 mm. The aggregates were placed in contact 
with water in a 2.0 mm mesh sieve and submitted to 
vertical agitation in a Yoder sieve shaker 
(SOLOTEST) for 15 min, at 32 oscillations per 
minute. 

The material from each class retained in the 
sieve was dried in an oven at 105ºC and the 
respective masses were measured on a digital 
balance. The weighted mean diameter (WMD) was 
calculated based on the formula proposed by Castro 
Filho, Muzilli and Podanoschi (1998) and the 
geometric mean diameter (GMD) in line with 
Schaller and Stockinger (1953) as cited by 
Alvarenga et al. (1986), using the following 
equations: 

 

  and  

 
where: 
ni = is the percentage of aggregates retained 

in the sieve, Di = the mean diameter of the sieve and 
N = the number of sieve classes.  

Soil fractal model theory 
The definition of a fractal can be given 

based on the relationship between number and size 
in a statistically self-similar system and defined by 
the following Equation 1 (MANDELBORT, 1982; 
TURCOTTE, 1986): 

 
                                   (1) 

 
In an effort to compensate for the lack of N 

values, Tyler and Wheatcraft (1989) estimated the 



1964 
Fractal features…  JORDÃO, H. W. C. et al. 

Biosci. J., Uberlândia, v. 36, n. 6, p. 1961-1974, Nov./Dec. 2020 
http://dx.doi.org/10.14393/BJ-v36n6a2020-47894 

fractal dimension of soil particle-size distribution, 
Dm, based on the following Equation 2. Fractal 
dimension (D) was estimated using the particle size 
distribution (PSD) method proposed by Taylor and 
Wheatcraft (1992), based on the size distribution 
values for particles of coarse, medium and fine sand, 
silt and clay, according to the following equation: 

 

 (2) 

 
were: 
M = is the accumulated mass of the r-sized 

soil fractions (coarse, medium and fine sand, silt and 
clay) smaller than; R = determined by the diameter 
of the sieves; MT = the total mass; RL = the 
parameter that estimates the largest particle size; D 
= the fractal dimension of the particles.  

The equation is limited by the variation of 
D, with 0<D<3, and is applied to describe the 
particle size distribution of dry soil. 
 
Statistical analysis 

The fractal dimension results obtained for 
each area were submitted to analysis of variance 
and, when significance was observed according to 
the F-test, the means were compared using Tukey’s 
test at 5% probability. The correlation between 
fractal dimension and the physical attributes of the 
respective IDEs was analyzed using Pearson’s 
correlation coefficient. All analyses were performed 
using Assistat software, version 7.7 (SILVA; 
AZEVEDO, 2016). D values were calculated using 
an electronic spreadsheet and R statistical software, 
version 3.4.1 (R CORE TEAM, 2013). 
 
RESULTS AND DISCUSSION 
 
Soil particle-size distribution, fractal features 
and physical attributes 

As shown in Table 1, there are considerable 
differences in PSD among the four areas the IDEs in 
this study. The predominant soil particle size is 
sand, followed by silt. The average of the sand 
fraction range from 375.0 g/kg-1 at 707.0 g/kg-1, the 
clay content is relatively lower, range from 17.0 
g/kg-1 to 248.0 g/kg-1 and the silt content between 
199.0 g/kg-1 at 607.0 g/kg-1 (Table 1). Dominance of 
coarse particles in these areas of IDEs (forest and 
grassland) can be explained by the contribution of 

lithic and ceramic fragments, thus presenting higher 
levels for the sand fraction. Dominance of the sand 
and silt fraction was also reported by Campos et al. 
(2012); Mota Júnior et al. (2017) in anthropic soils 
in the city of Apuí-AM, associating this 
granulometric characteristic with the formation of 
these soils and constituent materials. In addition, it 
is important to note that the use of fire retardants in 
the presence of sand is more likely to occur in the 
sand (2000). 

Considering IDEs under grassland and 
forest areas, the relationship of PSD has an 
influence on processes related to movement and 
retention of water, solute transport, heat and air in 
the soil. In the grassland area, due to the forces 
acting on the soil, the processes of degradation, 
decreased water retention capacity, soil nutrient loss 
and soil structure decrease are indicative of the 
selective removal of fine fractions and can be 
attributed to the anthropic processes and the source 
material acting in the formation of these soils. Xu, 
Li and Li (2013) and Deng et al. (2017), observed 
similar behavior in non-anthropogenic soils as the 
granulometric fractions of sand. Cocoa and coffee 
areas presented domain of the silt fraction, in this 
case, the silt fraction together with clay, acted in the 
processes related to greater structuring and 
aggregation in these areas. 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 



1965 
Fractal features…  JORDÃO, H. W. C. et al. 

Biosci. J., Uberlândia, v. 36, n. 6, p. 1961-1974, Nov./Dec. 2020 
http://dx.doi.org/10.14393/BJ-v36n6a2020-47894 

Table 1. Descriptive statistics of the fractal dimension and physical attributes in Indian Dark Earth under different uses in the Western Amazon. 

Cocoa area Coffee area 

Variable Mean StDev CoefVar Minimum Maximum R² Variable Mean StDev CoefVar Minimum Maximum R² 

D 2.80 0.02 0.79 2.76 2.85 0.851 D 2.43 0.02 0.79 2.39 2.54 0.822 
Silt (g/kg-¹) 549.89 21.00 3.82 491.13 590.56  Silt (g/kg-¹) 607.96 26.64 4.38 531.20 666.06  
Clay (g/kg-¹) 248.03 39.27 15.83 174.09 341.12  Clay (g/kg-¹) 17.29 2.94 16.99 13.09 38.74  
Sand (g/kg-¹) 201.41 33.38 16.57 121.10 274.11  Sand (g/kg-¹) 375.55 26.74 7.12 317.57 454.13  
BD (Mg m³) 0.93 0.09 9.22 0.59 1.15  BD (Mg m³) 1.14 0.14 12.63 0.72 1.43  
Macro (cm³ cm-3) 0.20 4.01 19.49 11.74 31.03  Macro (cm³ cm-3) 0.19 3.94 20.49 9.54 27.52  
Micro (cm³ cm-3) 0.46 3.26 7.03 35.64 53.49  Micro (cm³ cm-3) 0.38 1.99 5.21 34.25 45.09  
TP (cm³ cm-3) 0.66 3.14 4.69 58.85 73.25  TP (cm³ cm-3) 0.57 3.54 6.16 50.27 66.57  
GMD (mm) 2.60 0.28 10.74 1.80 3.34  GMD (mm) 2.55 0.28 10.86 1.73 3.19  
WMD (mm) 2.87 0.21 7.24 2.19 3.35  WMD (mm) 3.01 0.14 4.75 2.38 3.29  

Grassland area Forest area 

D 2.61 0.03 1.32 2.53 2.69 0.985 D 2.68 0.02 0.64 2.64 2.73 0.954 
Silt (g/kg-¹) 226.43 28.79 12.71 160.14 326.39  Silt (g/kg-¹) 199.41 29.81 14.95 139.30 278.56  
Clay (g/kg-¹) 65.80 16.12 24.50 34.84 112.85  Clay (g/kg-¹) 94.51 10.79 11.42 70.98 123.17  
Sand (g/kg-¹) 707.76 28.63 4.05 606.14 759.59  Sand (g/kg-¹) 705.69 30.88 4.38 627.90 767.96  
BD (Mg m³) 1.27 0.09 7.26 1.04 1.47  BD (Mg m³) 1.05 0.06 5.82 0.93 1.23  
Macro (cm³ cm-3) 0.17 2.22 12.71 13.08 24.62  Macro (cm³ cm-3) 0.18 2.00 10.74 13.02 23.46  
Micro (cm³ cm-3) 0.38 2.31 6.06 27.30 45.09  Micro (cm³ cm-3) 0.22 1.69 7.41 19.50 26.40  
TP (cm³ cm-3) 0.55 2.56 6.13 36.14 47.51  TP (cm³ cm-3) 0.40 2.60 6.28 32.90 46.54  
GMD (mm) 2.67 0.22 8.31 2.06 3.03  GMD (mm) 2.03 0.31 15.09 1.42 2.73  
WMD (mm) 3.10 0.10 3.19 2.82 3.25  WMD (mm) 2.56 0.27 10.52 1.96 3.08  

D = Fractal dimension; BD = Bulk density; Macro = Macroporosity; Micro = Microporosity; GMD = Geometric mean diameter; WMD = Weighted mean diameter; StDev = standard deviation; CoefVar = 
coefficient of variation. 
 



1966 
Fractal features…  JORDÃO, H. W. C. et al. 

Biosci. J., Uberlândia, v. 36, n. 6, p. 1961-1974, Nov./Dec. 2020 
http://dx.doi.org/10.14393/BJ-v36n6a2020-47894 

The fractal dimension (D) values presented 
in Table 1 show differences between the IDEs under 
management and forest, in which they are 
characterized by a decrease in the fractal dimension, 
indicating that soils in the cultivated lands with 
cocoa are better, and coffee and grassland as highly 
influenced by management. This is because the sites 
with more expressive antropic spot were used for 
cultivation of species that provide greater input of 
vegetal residues. It is possible to verify that the 
greater the value of the clay fraction and the lower 
the BD, the greater the D, this corroborates with 
researches done by Liu et al. (2009) and Deng et al. 
(2017), in which they verified that the fractal 
dimension of the soil particle was larger when the 
texture was finer, corresponding to the particle size 
distribution of the soil. Overall, D values in the 
forest and cocoa area varied respectively from 2.68 
to 2.80. The variation of D values in all areas was 
relatively high, so the use of D based on PSDs could 
provide more information than just soil texture.  

The coefficients of determination (R2) 
showed, in general, values were satisfactory, that is, 
they can satisfactorily explain the results obtained 
for D, since they ranged from 0.822 to 0.985, 
indicating that on average 90% of the results 
obtained are explained by model. Huang et al. 
(2017) also found high R2 values, ranging from 
0.961 to 0.992, indicating that it is reasonable to use 
the fractal model to evaluate the particle size 
distribution of soils. 

From the physical attributes of soil physical 
quality (macroporosity, microporosity, TP and BD), 
the degree of similarity between the different areas 
is evident, decreasing only to the forest area and 
grassland in attributes such as TP and BD. In the 
condition of physical quality of soils, management 
areas and natural area, values higher than those 
established by the literature, mainly macroporosity, 
always higher than 15%, as already highlighted in 
Tormena et al. (2002). In the forest area, physical 
variables of macro, micro and total porosity 
presented, respectively, 0.18, 0.22 and 0.4 cm³ cm-3 
(Table 1), this behavior observed in forest area, 
leaves strong evidence of the low magnitude of the 
anthropic processes of formation of the IDEs at this 
site, thus considering low variation with areas where 
the anthropic formation process was more 
accentuated. Cocoa area presented significant 
results, in which the TP reached 0.66 cm³ cm-3, 
influencing a low BD of 0.93 Mg m³ (Table 1). 
Contrary to what was observed in the forest area, 
anthropic action was more expressive in this 
environment, even more associated with the great 

contribution of vegetal cover, which incorporates 
residues and lead to a greater formation of biopores 
by agents of the micro and meso fauna, favoring the 
diffusion of oxygen in the soil to the roots.  

Effects of the influence of management on 
physical properties are more significant in the 
grassland area, mainly on the BD, with values of 
1.27 Mg m³, making it evident that, anthropic soils 
are highly subject to degradation due to their 
management, and that the effects of the excess of 
animal grazzing can lead to the strong compaction 
of these soils, as it was evident the comparison with 
areas where the management is not very intense 
(cocoa and coffee) and mainly in natural area 
(forest).   

Compared to the GMD values of the 
different systems of use in the IDEs (Table 1), there 
is reduced GMD in forest area with a value of 2.03 
mm, this behavior of aggregate diameter, disagrees 
with the characteristics of occurrence in soils of 
native conditions, making evident the influence of 
the sandy texture in the aggregate stabilization 
capacity. On the other hand, ADE under grassland 
presented higher GMD and WMD results, 
respectively 2.67 mm and 3.10 mm. However, this 
behavior as highlighted by Mantovanelli et al. 
(2015) does not show in this specific case, better 
structural conditions, since these values of 
aggregates are due to the fact that the soil is at the 
level of compaction exerted by the animal grazzing 
and thus present greater resistance to rupture. 

 
Pearson correlation 

From Table 2 we present the Pearson 
correlation with emphasis on the correlation 
between D and other soil physical attributes. It is 
observed that the correlation coefficient was 
significant at 1% between D and the clay fraction, 
and significant at 5 % probability between D and 
BD. D showed positive correlation with sand, clay, 
BD, macro, GMD and WMD and negative 
correlation with silt, microporosity and TP. 

 
 
 
 
 
 
 
 
 
 
 

 



1967 
Fractal features…  JORDÃO, H. W. C. et al. 

Biosci. J., Uberlândia, v. 36, n. 6, p. 1961-1974, Nov./Dec. 2020 
http://dx.doi.org/10.14393/BJ-v36n6a2020-47894 

Table 2. Pearson correlation of fractal dimension and physical attributes in Indian Dark Earth under different 
uses in the Western Amazon. 

 
D Sand Silt Clay BD Macro Micro TP GMD WMD 

D 1 0,09 -0,08 0,29** 0,21* 0,06 -0,13 -0,11 0,20 0,13 
Sand 

 
1 -0,04 0,27* 0,23* -0,01 -0,17 -0,25* -0,06 -0,07 

Silt   1 0,02 -0,03 -0,08 0,13 0,05 0,17 0,21* 
Clay    1 0,21* -0,11 -0,08 -0,10 -0,02 -0,06 
BD     1 -0,52** 0,08 -0,65** -0,08 -0,14 
Macro      1 -0,46** 0,71** 0,12 0,18 
Micro       1 0,14 0,01 0,002 
TP        1 0,14 0,23* 
GMD         1 0,72** 
WMD          1 
** Significant at 1% probability level; * Significant at the 5% probability level; D = fractal dimension; BD = Bulk density; Macro = 
Macroporosity; Micro = Microporosity; TP = total porosity; GMD = Geometric mean diameter; WMD = Weighted mean diameter. 

 
From the soil texture, high correlation of the 

clay fraction with D was observed, corroborating 
works by Xu, Li and Li (2013); Xia et al. (2015) and 
Deng et al. (2017), who found a positive and strong 
correlation of the clay fraction with D. However, in 
their respective studies, Xu, Li and Li (2013) and 
Huang et al. (2017) found a strong positive 
correlation between D with silt fraction and negative 
correlation with sand fraction, differently from what 
was observed in this work, where silt presented 
negative correlation (-0.08) and sand positive 
correlation (0.09), thus not corroborating with the 
authors mentioned above. Huang et al. (2017) 
evaluated changes in texture, structure stability and 
nutrient availability of artificial soils on the slopes 
due to restoration time in Southeast China and found 
significant correlation coefficient between D and 
BD, but this correlation was negative in order of -
0.805. These results, different from the present 
study, however should be considered that the 
authors worked in areas that underwent intensive 
soil structure management, which certainly 
influence BD values. 

In a study by Liao et al. (2017), verified that 
the soil moisture showed a weak spatial dependence 
according to the fractal dimension criteria. Thus, it 
is stated that, in general, the fractal dimension is 
better than the ratio of nugget/range effect in 
describing the spatial dependence of soil properties. 

In this context, the D of the soil particle size 
distribution, is an important tool to describe mainly 
attributes that characterize soil types, in addition, 
Deng et al. (2017) make an association of values of 
D with the quality of the physical-chemical 
properties of the soil, so the lower the value of D the 
lower the quality of the soil. 
 
Relationship between fractal dimension and soil 
particle-size distribution 

Linear regression analyzes were performed 
to determine the relationships between D and the 
sand, silt and clay contents (Figures 2 and 3) in the 
different areas of IDEs. The statistical results 
obtained from the 88 sample points show that the 
fractal dimension of the PSD has a strong positive 
correlation with the clay content in all IDEs (Figure 
2B, n = 88, R2 = 0.99, P <0.05, Figure 2E, n = 88, 
R2 = 0.94, P <0.05, Figure 3A, n = 88, R2 = 0.88, P 
<0.05, Figure 3E, n = 88, R2 = 0.97, P <0.05). 
Results indicate a weak relationship between D and 
sand content in the different systems of IDEs use, 
except in the cocoa are (R2 = 0.71, Figure 2C), the 
negative correlation is evident in the other areas, 
with R2 ranging from 0.08 to 0.13. Thus, regression 
analysis indicates that IDEs with higher clay 
contents and smaller sand fractions have higher D 
values. Other studies have shown similar results in 
very different landscapes and under contrasting 
climatic conditions (LIU et al., 2009). 

From the linear regressions in the areas 
cultivated with cocoa (Figure 2 A-B-C) and forest 
area (Figure 3 A-B-C), it was verified that there was 
a significant decrease in the coefficient of 
determination between the different textural 
fractions, thus having a difference between the D of 
a natural and cultivated environment. A similar 
result was found by Xu, Li and Li (2013), where 
they verified that the differences between cultivated 
areas and forest environments are characterized by a 
decrease in the fractal dimension, indicating that 
soils in agricultural lands are better. However, it is 
worth mentioning that the relationships between the 
cultivated areas with cocoa (Figure 2 A-B-C) and 
coffee (Figure 2 D-E-F) were too deferential, with 
the cultivated area with coffee, lower values in 
relation, thus evidencing that not only the 
environment, but also the type of crop cultivated can 
influence the D. Despite the fact that these two 



1968 
Fractal features…  JORDÃO, H. W. C. et al. 

Biosci. J., Uberlândia, v. 36, n. 6, p. 1961-1974, Nov./Dec. 2020 
http://dx.doi.org/10.14393/BJ-v36n6a2020-47894 

cultivated areas are close to each other and present 
the same soil class over the years, their management 
was different, since the cocoa area was previously 
cultivated by other crops (rice, corn, beans and 
watermelon), while the coffee area, besides having a 
few years of cultivation, in the first two years was 
used by grassland. 

The characteristics of D verified from the 
linear regressions and the relationships with the 
PSD (Figures 2 and 3), thus indicate, accelerated the 
physical soil structuring process in the evolution of 

the IDE formation process under cocoa cultivation. 
The effects associated to higher ratio of clay fraction 
in this crop, associated with the high organic matter 
input, lead to a maintenance of organic carbon and 
nutrients, mainly attributed to fine particles that 
enrich with stable organic C and N, as well as 
increase of formation of particulate organic matter. 
As the proportion of coarse fractions increases, the 
negative correlation obtained is evident, thus 
evidencing the tendency of decrease in the levels of 
organic carbon. 

 
 

Soil fractal dimension

2.70 2.75 2.80 2.85 2.90

S
il

t 
(g

/k
g

-¹
)

480

500

520

540

560

580

600

620

y = -497.32+1944.56*x
R² = 0.27

Soil fractal dimension

2.70 2.75 2.80 2.85 2.90

C
la

y 
(g

/k
g
-¹

)

150

180

210

240

270

300

330

360

y = 1755.87-4676.09*x
R² = 0.99

 

Soil fractal dimension

2.70 2.75 2.80 2.85 2.90

S
an

dy
 (

g
/k

g
-¹

)

90

120

150

180

210

240

270

300

y = -1269.21-3760.75*x
R² = 0.71

Soil fractal dimension

2.30 2.35 2.40 2.45 2.50 2.55 2.60

S
il

t 
(g

/k
g

-¹
)

510

540

570

600

630

660

690

y = 274.98-60.66*x
R² = 0.03

 Soil fractal dimension Soil fractal dimension

Soil fractal dimension

2.30 2.35 2.40 2.45 2.50 2.55 2.60

C
la

y 
(g

/k
g

-¹
)

10

15

20

25

30

35

40

45

y = 149.96-347.34*x
R² = 0.94

Soil dimension fractal

2.30 2.35 2.40 2.45 2.50 2.55 2.60

S
an

dy
 (

g/
kg

-¹
)

300

330

360

390

420

450

480

y = -426.82+1413.37*x
R² = 0.09

 
Figure 2. Relationship between fractal dimension and particle size distribution on Indian Dark Earth under 

different uses in the Western Amazon.  
A-B-C: Cocoa area; D-E-F: Cooffe area. 
 



1969 
Fractal features…  JORDÃO, H. W. C. et al. 

Biosci. J., Uberlândia, v. 36, n. 6, p. 1961-1974, Nov./Dec. 2020 
http://dx.doi.org/10.14393/BJ-v36n6a2020-47894 

 

Soil dimension fractal

2.60 2.65 2.70 2.75

S
il

t 
(g

/k
g
-¹

)

120

150

180

210

240

270

300
y = 65.81+22.80*x
R² = 0.001

Soil dimension fractal

2.60 2.65 2.70 2.75

C
la

y 
(g

/k
g-

¹)

60

80

100

120

140

y = 585.77-1477.18*x
R² = 0.88

 

Soil dimension fractal

2.60 2.65 2.70 2.75

S
an

dy
 (

g/
k

g-
¹)

600

630

660

690

720

750

780

y = -650.04+2449.82*x
R² = 0.13

Soil dimension fractal

Soil dimension fractal

2.50 2.55 2.60 2.65 2.70

S
il

t 
(g

/k
g-

¹)

150

180

210

240

270

300

330

360

y = -215.20+787.69*x
R² = 0.06

 Soil dimension fractal Soil dimension fractal

Soil dimension fractal

2.50 2.55 2.60 2.65 2.70

C
la

y 
(g

/k
g-

¹)

0

30

60

90

120

y = 461.29-1137.28*x
R² = 0.97

Soil dimension fractal

2.50 2.55 2.60 2.65 2.70

S
an

d
 (

g/
kg

-¹
)

570

600

630

660

690

720

750

780

y = -246.09+1349.59*x
R² = 0.08

 
Figure 3. Relationship between fractal dimension and particle size distribution on Indian Dark Earth under 

different uses in the Western Amazon.  
    A-B-C: Forest area; D-E-F: Grassland area. 

 
Relationship between fractal dimension, GMD 
and WMD 

Linear regressions were also performed to 
determine the relationships between D and GMD 
and WMD aggregation variables for the different 
crops (Figure 4). In spite of the other physical 
attributes studied, the relationships with GMD and 
WMD were those that presented coherent relations 
with the values of D, however these same values, 

are very low as to the determination index R2, in 
which area under cocoa cultivation presented (R2 = 
0.03 and 0.06), with a negative linear relationship 
(Figure 4 A-B). ADE under grassland presents 
constant relation, tending to a positive linear, 
however again the values of R2 are considered 
extremely low varying between 0.006 for GMD and 
0.008 for WMD (Figure 4 C-D), however, as 
previously mentioned, this specific condition does 



1970 
Fractal features…  JORDÃO, H. W. C. et al. 

Biosci. J., Uberlândia, v. 36, n. 6, p. 1961-1974, Nov./Dec. 2020 
http://dx.doi.org/10.14393/BJ-v36n6a2020-47894 

not demonstrate expressive results of adequate 
aggregation conditions for this area, only possible 
compaction effects induce this condition. 

IDEs areas, present high proportions of 
organic carbon, being verified this condition in 
studies of Soares et al. (2018) in grassland in 
southern Amazonas and also by Silva et al. (2017) 
that found high levels of native forest in the South 
of Amazonas. In particular, the increase in total 
organic carbon content generally results in an 
increase in the size and stability of the aggregates 

(KAY; ANGERS, 2000). The organic carbon 
content in the soil may reflect the number of 
polysaccharides that must increase cementation 
between mineral particles and influence the 
arrangement of these particles between soil failure 
zones (CHENU; GUÉRIF, 1991). Therefore, the 
data of this work show that the fractal dimension 
was not expressively correlated with the stability of 
the aggregates, so there is some process in particular 
of the relation of formation of the aggregates with 
the PSD or organic matter of the soil. 

 

2.70 2.75 2.80 2.85 2.90

G
eo

m
et

ri
c 

m
ea

n
 d

ia
m

et
er

 (
m

m
)

1.5

1.8

2.1

2.4

2.7

3.0

3.3

3.6

y = -2.28+8.99*x
R² = 0.03

2.70 2.75 2.80 2.85 2.90

W
ei

gh
te

d 
av

er
ag

e 
d

ia
m

et
er

 (
m

m
)

2.1

2.4

2.7

3.0

3.3

3.6

y = -2.45+9.74*x
R² = 0.06

Soil dimension fractal

2.30 2.35 2.40 2.45 2.50 2.55

W
ei

gh
te

d 
av

er
ag

e 
d

ia
m

et
er

 (
m

m
)

2.1

2.4

2.7

3.0

3.3

3.6
y = -0.21+3.53*x
R² = 0.008

Soil dimension fractal

2.30 2.35 2.40 2.45 2.50 2.55

G
eo

m
et

ri
c 

m
ea

n 
di

am
et

er
 (

m
m

)

1.5

1.8

2.1

2.4

2.7

3.0

3.3 y = -0.39+3.51*x
R² = 0.008

 

Soil dimension fractal

2.50 2.55 2.60 2.65 2.70

G
eo

m
et

ri
c 

m
ea

n 
di

am
et

er
 (

m
m

)

1.8

2.1

2.4

2.7

3.0

3.3

y = 0.51+1.31*x
R² = 0.006

Soil dimension fractal

2.50 2.55 2.60 2.65 2.70

W
ei

gh
te

d
 a

v
er

ag
e 

d
ia

m
et

er
 (

m
m

)

2.8

2.9

3.0

3.1

3.2

3.3

y = 0.26+2.39*x
R² = 0.008

G
eo

m
et

ri
c 

m
ea

n
 d

ia
m

et
er

 (
m

m
)

 

2.70

Soil dimension fractal

2.60 2.65 2.70 2.75 2.80

G
eo

m
et

ri
c 

m
ea

n 
di

am
et

er
 (

m
m

)

1.2

1.5

1.8

2.1

2.4

2.7

3.0

y = -0.53+3.45*x
R² = 0.009

Soil dimension fractal

2.60 2.65 2.70 2.75 2.80

W
ei

gh
te

d
 a

ve
ra

ge
 d

ia
m

et
er

 (
m

m
)

1.8

2.1

2.4

2.7

3.0

3.3
y = 0.04+2.43*x
R² = 0.003

 
Figure 4. Relationship between fractal dimension, GMD and WMD in Indian Dark Earth under different uses 

in Western Amazonia.  
A-B: Cocoa area; C-D: Cooffe area; E-F: Grassland area; G-H: Forest area. 



1971 
Fractal features…  JORDÃO, H. W. C. et al. 

Biosci. J., Uberlândia, v. 36, n. 6, p. 1961-1974, Nov./Dec. 2020 
http://dx.doi.org/10.14393/BJ-v36n6a2020-47894 

There are recognized difficulties in relating 
the soil structure to the specific functions exerted by 
them in the processes of pedogenesis (GUBER; 
PACHEPSKY; LEVKOVSKY, 2005). Therefore, 
these difficulties are caused by the multiplicity of 
factors that affect the soil structure and multiplicity 
of effects that the structure has on the processes in 
the soil. Therefore, in spite of the complexity of the 
anthropic process of IDE formation, the aggregation 
scale requires parameterization that can better relate 
the effects with fractal dimension studies, since the 
textural variation associated with conditions of 
different diameters of aggregates presented low 
relations. 
 
CONCLUSIONS 

 
The mean fractal dimension values in the 

Indian Dark Earth studied here indicate better soil 
structure in the cocoa-growing area when compared 
to the other sites, possibly due to the land use and 
management of this environment. The clay fraction 
directly influenced fractal dimension values in the 
Indian Dark Earth areas. 

There were considerable linear relationships 
between the clay content in the different areas. It has 
been suggested that the fractal dimension of PSD 
may have significant implications for soil 
degradation or land-use structuring. 

The search for informative parameters of the 
soil structure associating parameters of aggregation 
with fractal dimension, is the hope that the research 
in the aggregate scale can generate useful 
complements for the parameterization of the soil 
structure in Amazonian anthropic soils, evident the 
low relation found, thus showing the peculiar 
characteristic of these soils of the Amazon region. 
 
ACKNOWLEDGEMENTS 

 
This work was carried out with the support 

of the “Coordenação de Aperfeiçoamento de Nível 
Superior (CAPES)”, “Conselho Nacional de 
Desenvolvimento Científico e Tecnológico, 
(CNPQ)”, “Fundação de Amparo à Pesquisa do 
Estado do Amazonas (FAPEAM)” and 
“Universidade Federal do Amazonas (UFAM)”. 

 
 
 
RESUMO: Estudar a distribuição do tamanho das partículas é importante para entender a estrutura do 

solo e os processos de formação. Esta pesquisa teve como objetivo avaliar a dimensão fractal da textura do solo 
em áreas de Terra Preta de Índio (TPI) no sul do Estado do Amazonas sob diferentes usos da terra: duas áreas 
no município de Apuí, uma com cultivo de cacau e outra de café; uma área de pastagem no município de 
Manicoré; e uma área florestal no município de Novo Aripuanã. Uma malha de amostragem contendo 88 
pontos de coleta (pontos de interseção na grade) foi estabelecida em cada área, medindo 80 x 42 m para as 
áreas de cacau e café, e 80 x 56 m e 60 x 42 m para as áreas de pastagem e floresta, respectivamente. Amostras 
de solo foram coletadas em torrões a uma profundidade de 0,0-0,20 m para determinar as propriedades físicas 
estruturais e a textura do solo. Os seguintes atributos físicos foram avaliados: textura, densidade do solo (DS), 
macroporosidade (Macro), microporosidade (Micro), porosidade total (PT) e estabilidade de agregados (DMG e 
DMP). Determinou-se a dimensão fractal da textura do solo (D), seguida da análise de variância e comparação 
das médias pelo teste de Tukey (p≤0,05). A correlação de Pearson foi aplicada para avaliar a correlação entre as 
variáveis. Houve uma diferença significativa entre as TPIs estudadas, com um maior valor D na área de cultivo 
de cacau em relação aos outros locais. Além disso, quanto maior a fração argila, maior o valor de D. A 
dimensão fractal (D) apresentou correlação positiva com areia, argila, DS, Macro, DMG e DMP, e correlação 
negativa com silte, micro, PT. Com base nos valores de D obtidos, as TPIs cultivadas com cacau apresentaram 
qualidade superior em relação às demais áreas estudadas. 

 
PALAVRAS-CHAVES: Dimensão Fractal. Física do solo. Uso do solo. 

 
 
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