Pore-solid Fractal Model Research Articles

An improved pore-solid fractal model for predicting coastal saline soil hydraulic properties based on changepoints determined by genetic algorithm-support vector regression

Coastal soil is an important reserve land resource. The accurate prediction of soil hydraulic properties plays an important role in understanding the improvement of soil properties after reclamation in coastal areas. With the development of modern mathematical theory, the pore-solid fractal (PSF) model has become an important basis for simulating soil hydraulic properties. The accuracy of PSF model depends on the accurate acquisition of changepoints and fractal dimensions. Previous studies have found that micro-CT scanning combined with image processing could accurately obtain fractal dimensions. However, we still need soil water retention data to determine changepoints, which greatly limits the application of fractal models for estimating soil hydraulic properties. In this case, we determined the relationship between changepoints and soil physical and chemical properties and established a genetic algorithm-support vector regression (GA-SVR) model to predict changepoints. Then an improved PSF model was adopted to predict soil water retention curve in coastal areas of Jiangsu Province based on image processing. The results showed that soil physical and chemical properties changed the soil water movement and changepoints by affecting soil pore distribution. In coastal saline soil, four parameters (BD, silt, EC, SOM) were selected to predict changepoints. The mass fractal dimension (Dm) was mainly influenced by the porosity and the heterogeneity of pore space. The porosity, pore diameter and specific surface area were the determining factors of DS value. Reclamation activities in coastal reclamation area changed Dm value but had no clear effect on DS value. By combining predicted changepoints, measured water content at the suction of changepoints and fractal dimensions, the accuracy of the PSF model had been greatly improved.

Permeability Prediction of Saturated Geomaterials with Revised Pore–Solid Fractal Model and Critical Path Analysis

The revised pore–solid fractal (PSF) model is presented by using the largest inscribed circle-based geometries of squares or cubes to replace the original pore or solid subregions as the new pore or solid phase in porous media. The revised PSF model changes the discrete lacunar pore and solid phases in the original PSF model to integrated. Permeability is an intrinsic property of geomaterials and has broad applications in exploring fluid flow and species transport. Based on the revised PSF model and critical path analysis, a fractal model for predicting the permeability of saturated geomaterials is proposed. The permeability prediction model is verified by comparison with the existing predicted model and the laboratory testing. The results show that the predicted permeabilities match the measured values very well. This work provides a theoretical framework for the revised PSF model and its application in predicting the permeability of geomaterials.

Modelling soil hydraulic properties with an improved pore‐solid fractal (PSF) model through image analysis

AbstractSoil hydraulic properties are important for studying Earth science. The pore‐solid fractal (PSF) model, combined with a critical path analysis from percolation theory, seems to be more promising in the modelling of soil hydraulic properties. The accuracy of the PSF model depends on the accurate acquisition of fractal dimensions, which requires the combination of micro‐CT scanning and image analysis technology. In addition, there is a changepoint in soil water movement due to the coexistence of soil micro‐ and macromorphology. Determining the changepoint and using different fractal dimensions to predict hydraulic properties on different sides of the changepoint can further improve the accuracy of the PSF model. Therefore, in this study, we tested the changepoint in soil water movement and adopted an improved PSF model to predict hydraulic parameters in saline soil based on image analysis. The results showed that the two‐sample t‐test could identify the changepoint accurately. There was only one changepoint in coastal saline soil when predicting hydraulic properties. Micro‐CT scanning and image analysis can obtain fractal dimensions more accurately and quickly. The coefficients of determination of all treatments were above 0.9. The improved PSF model was more accurate than the previous model in predicting soil hydraulic properties. A comparison of goodness‐of‐fit criteria showed that it is better to adopt the geometrical mean error ratio () and geometrical standard deviation error ratio () as the judgement standard. Due to the anisotropy of soil, the improved PSF model demonstrated a higher accuracy in predicting water content than hydraulic conductivity. The hydraulic conductivity prediction accuracy was negatively correlated with the degree of anisotropy () parameter, and the improved model was more suitable for soils with weak anisotropy. Our research can provide a simple and accurate method for parameter calculation of the PSF model to predict soil hydraulic properties more accurately.Highlights The two‐sample t‐test can find the changepoint accurately in the process of soil drying. Micro‐CT scanning and image analysis can calculate fractal dimension more accurately. The improved PSF model is more accurate when predicting soil hydraulic properties. Anisotropy is an important factor that restricts the prediction accuracy of PSF model.

A user-friendly modified pore-solid fractal model

The primary objective of this study was to evaluate a range of calculation points on water retention curves (WRC) instead of the singularity point at air-entry suction in the pore-solid fractal (PSF) model, which additionally considered the hysteresis effect based on the PSF theory. The modified pore-solid fractal (M-PSF) model was tested using 26 soil samples from Yangling on the Loess Plateau in China and 54 soil samples from the Unsaturated Soil Hydraulic Database. The derivation results showed that the M-PSF model is user-friendly and flexible for a wide range of calculation point options. This model theoretically describes the primary differences between the soil moisture desorption and the adsorption processes by the fractal dimensions. The M-PSF model demonstrated good performance particularly at the calculation points corresponding to the suctions from 100 cm to 1000 cm. Furthermore, the M-PSF model, used the fractal dimension of the particle size distribution, exhibited an accepted performance of WRC predictions for different textured soils when the suction values were ≥100 cm. To fully understand the function of hysteresis in the PSF theory, the role of allowable and accessible pores must be examined.

Estimation of soil water retention curve: An asymmetrical pore-solid fractal model

The soil water retention curve is an important hydraulic function for the study of flow transport processes in unsaturated soils. The objective of this study was to develop a soil water retention function using a generalized fractal approach. The model exhibits asymmetry between the solid phase and pore phase, which is in marked contrast to the symmetry between phases present in a conventional fractal model. The retention function includes 4 parameters: the saturated water content θs, the air entry value ha, the fractal dimension Df, and an empirical parameter β, characterizing the complicated soil pore structures. Sixty one data sets, covering a wide range of soil structure and textural properties, were used to evaluate the applicability of the proposed soil water retention function. The retention function is shown to be a general model, which incorporates several existing retention models. The values of β/πs and (πs−πr)/β were used as indexes to quantify the relationships between the proposed retention function and the existing retention models. The proposed function fits all the data very well, whereas other tested models only match about 16%–48% of the soil retention data.

Estimation of soil water retention function based on asymmetry between particle- and pore-size distributions

By examining the symmetry between the distributions of particle-size (PSD) and pore-size (POD) in a soil, as hypothesized by early pore-solid fractal (PSF) models, we found significant discrepancies in fractal dimensions between the PSD and the water retention curve (WRC) of a soil. Therefore, we developed an asymmetry-based PSF model to estimate better the WRC directly from the PSD data of a soil. To do so, we adopted the concept of a microscopic arrangement of different-sized particles to address such asymmetry, and evaluated the performance of the modified PSF model on five soil textural classes (coarse-, moderately coarse-, medium-, moderately fine- and fine-textured soils) using experimental PSD and WRC data from the UNSODA database (159 undisturbed soils for model calibration and 70 undisturbed soils for model validation). The fit of the symmetry-based PSF model to the calibration dataset showed that the fractal dimension of the WRC (Dp) was slightly larger than that of cumulative mass distribution of particles (Ds) for most soils. The asymmetry-based PSF model performed better than the symmetry-based PSF model. In addition, the asymmetry-based PSF model reduced the tendency to under estimate soil water content for a given matric head and the performance of the asymmetry-based model was consistent irrespective of soil texture, indicating that the adoption of asymmetry between the PSD and the POD was adequate in predicting the WRC of a porous, particulate system such as soil.

A review of fractal, prefractal and pore-solid-fractal models for parameterizing the soil water retention curve

Ghanbarian-Alavijeh, B., Millán, H. and Huang, G. 2011. A review of fractal, prefractal and pore-solid-fractal models for parameterizing the soil water retention curve. Can. J. Soil Sci. 91: 1–14. The soil water retention curve is an important hydraulic parameter for characterizing water flow and contaminant transport in porous media. Therefore, many empirical, semi physical, and physical models of the soil water retention curve have been proposed. Among them, fractal models appear to be a useful approach for modeling soil as a heterogeneous porous medium and its hydraulic characteristics. Fractal models are mathematically based, and their parameters have physical meanings. In this study, we review published fractal, prefractal and pore-solid-fractal models for soil water retention curves including Tyler and Wheatcraft, Rieu and Sposito, Perrier et al., Perfect, Bird et al., Millán and González-Posada, and Cihan et al. models. In the pore-solid fractal (PSF) approach the pore phase and matrix phase have a finite volume even for an infinite number of iterations. The results of fitting the PSF model to measured soil water retention data indicate that this model works well, particularly at lower water contents.

Scaling analysis of water retention curves for unsaturated sandy loam soils by using fractal geometry

Fractal geometry was deployed to analyse water retention curves (WRC). The three models used to estimate the curves were the general pore‐solid fractal (PSF) model and two specific cases of the PSF model: the Tyler & Wheatcraft (TW) and the Rieu & Sposito (RS) models. The study was conducted on 30 undisturbed, sandy loam soil samples taken from a field and subjected to laboratory analysis. The fractal dimension, a non‐variable scale factor characterizing each water retention model proposed, was estimated by direct scaling. The method for determining the fractal dimension proposed here entails limiting the analysis to the interval between an upper and lower pressure head cut‐off on a log‐log plot, and defining the dimension itself as the straight regression line that interpolates the points in the interval with the largest coefficient of determination, R2. The scale relative to the cut‐off interval used to determine the fractal behaviour in each model used is presented. Furthermore, a second range of pressure head values was analysed to approximate the fractal dimension of the pore surface. The PSF model exhibited greater spatial variation than the TW or RS models for the parameter values typical of a sandy loam soil. An indication of the variability of the fractal dimension across the entire area studied is also provided.

Fractal Features of Particle Size Redistributions of Deposited Soils on the Dam Farmlands

Fractal geometry concepts have been widely applied as a tool for describing complex natural phenomena, for instance, mechanics and physics of rocks and soils. However, they are seldom used to describe deposited soils. The objectives of this study were to classify soils collected from dam farmlands according to their particle size distributions (PSD) and to examine the relationship between scaling exponents and soil textures (clay and/or sand content). A total of 649 PSD of deposited soils were collected and determined using laser granulometry. The soils were grouped into four texture classes: silt loam, sandy loam, loamy sand, and sand. All the PSD fit the Pore-Solid fractal model well. The mean values of composite scaling constant (c) and fractal dimension (D) differed among soil texture classes. The significant relationships between the values of D and the contents of sand, silt, and clay fractions were found to follow a sigmoid trend. Our results show that the fractal dimension of PSD indicates the degree of soil detachment or soil desertification and the effect of check dams on soil conservation.

A comparison of fractal analytical methods on 2- and 3-dimensional computed tomographic scans of soil aggregates

Computed tomographic (CT) images of small (approximately 3 mm diameter) intact aggregates from an agricultural soil of two contrasting management histories were analyzed. Fractal analysis was employed to quantify the heterogeneity of mass and pore space within individual aggregates. Three fractal analytical methods were compared: 1) density scaling fitted to a solid mass fractal model, 2) density scaling fitted to a pore–solid fractal model, and 3) cumulative pore number scaling. Ten aggregates from a continuous corn treatment and another 10 aggregates from a five-year crop rotation treatment were each scanned to obtain a 2-D CT image. Two of these aggregates, one from each treatment, were scanned fully for 3-D reconstructions. The objectives were to 1) demonstrate the utility of industrial quality CT scanning and density scaling for deriving fractal parameters, including the ability of 2-D scans to represent 3-D structures, and 2) compare the three fractal analytical methods to identify which worked best for estimating fractal parameters of individual soil aggregates. The results showed that density scaling methods are compatible with CT scanning and that solid mass fractal models described the 20 aggregates better than pore–solid fractal models. Comparison of the 2-D and 3-D data suggested that some fractal parameters, including the fractal dimension, can be obtained from 2-D slices of a whole soil aggregate. This study demonstrated a non-disruptive approach to quantify internal heterogeneity of individual soil aggregates, which can be used to address scaling issues of soil structure and to predict soil behavior and soil quality.

The relationship between fractal properties of solid matrix and pore space in porous media

Measuring fractal dimensions has become a common practice for describing structural properties of porous media. Depending on the object of interest, different features of the structure can be measured: solid matrix, pores, and the interface between them. However, when measuring the fractal dimension of all these features, the question arises whether these dimensions are independent from each other or whether they can be related to an underlying property of the structure or image, respectively. For a variety of porous media we measured the fractal dimension of the matrix, the pore space, and the interface between them simultaneously using the box counting method. Analyzed images were obtained from soil thin sections and a void system in a clayey soil. Measured fractal dimensions were compared with fractal dimensions estimated by the pore-solid fractal (PSF) model, which derives the fractal properties of the matrix and the pore space completely as a function of the porosity, the size of the initiator and the fractal dimension of the interface. Measured results agree well with values obtained from the PSF model. A clear relationship between the fractal dimensions of the two phases (solid matrix and pore space) of a porous medium and their interface was observed. For all images the smallest fractal dimension was found for the interface between matrix and pores. Values for the fractal dimension of the two phases were between those for the interface and the Euclidian space with the phase with the lower mass fraction always having the smaller dimension. Porosity was found to act as weighing factor linking the dimension of the phases to those of the interface and Euclidian space. Model results also predict a dependency of the dimension of the phases on the spatial resolution of the analyzed image. For images having a high resolution (compared to the size of the initiator) phase dimensions are expected to be greater than for images having a low resolution.

Evaluation of soil water retention curve with the pore–solid fractal model

Empirical models have been developed to describe the soil water retention curve (SWRC). Applications of the fractal theory may provide a useful tool to fill the gap between the use of empirical models and physical interpretation of their parameters. Especially a more generalized model for the SWRC has been developed based on the pore–solid fractal (PSF) distribution. The PSF model covers several existing models as its special cases and theoretically provides a direct way to estimate the SWRC fractal dimension from particle size distributions (PSDs). The first objective of this study was to evaluate the SWRC with the PSF distribution using more than 400 data sets of PSD and SWRC. The second objective was to establish a relationship between the fractal dimension and soil texture. Analyses of the data sets showed that the fractal dimension values estimated using the PSD data were consistently smaller than those estimated using the SWRC data. Therefore, using the fractal dimension from PSD to predict the SWRC will result in underestimation. To resolve the problem, the soil data sets of PSD and SWRC were used to establish relationships between the fractal dimensions in the PSF model of SWRC and soil clay content as well as soil texture class. Independent SWRC data sets were used to test the methods. Predicted results using the PSF model of SWRC with the fractal dimension estimated using clay content or soil texture class were compared with the measured soil water retention data. Linear regressions of the predicted and measured SWRC showed good agreement for relatively fine texture soils with the coefficients of determination ( r 2) of 0.940.

FRAGMENTATION OF SOIL INITIATORS: APPLICATION OF THE PORE-SOLID FRACTAL MODEL

The quantification of fragmentation of natural, polidisperse, porous media using fractal models is well documented in the literature. However, in many cases, fractal exponents (fractal fragmentation dimension) and coefficients (fractal lacunarity) arising from a power law behavior do not make clear differences between different media. In the present work, the pore-solid fractal (PSF) model was used as a new fractal approach for deriving four scaling parameters (fractal dimension of the particle-size distribution, fractal fragmentation dimension of the fragment-size distribution, probability of fragmentation and scaling factor) from soil initiators subjected to different energy density input. The fractal fragmentation dimension for all soil samples was Df=2.42±0.16 without correlating with the energy expended in the fragmentation process. By contrast, probability of fragmentation and scaling factor correlated significantly with the energetic term. The PSF model is useful for estimating a group of scaling parameters more appropriate for the quantification of complex patterns associated to fragment-size distributions.

The pore–solid fractal model of soil density scaling

SummaryWe have developed the fractal approach to modelling variations in soil bulk density and porosity with scale of measurement or sample size. A new expression is derived for each quantity based on the pore–solid fractal (PSF) model of soil structure. This new general expression covers a range of fractal media and accommodates existing fractal models as special cases. Model outputs cover a range of scaling behaviour expressed in terms of monotonic functions, from increasing density and decreasing porosity, through constant porosity and density to decreasing density and increasing porosity with increasing scale of measurement. We demonstrate the link between this new model for the scaling of porosity and bulk density and the water retention model for the PSF. The model for scaling bulk density is fitted to data on aggregate bulk density and shown to yield good fits describing bulk density decreasing with increasing aggregate size. Porosity scaling is also inferred from the fitting of water retention data. Inferred porosities from different fittings are shown to follow decreasing, scale‐invariant and increasing values with decreasing size of structural unit, and these theoretical results emphasize the need for further experimental investigation on the basic issue of density scaling in soil science.

Modelling soil fragmentation: the pore solid fractal approach

Models have been developed to explain independently either the powerlaw aggregate size or the powerlaw particle-size distributions observed in many soils. We present a fragmentation model coupled with a recent model of soil structure called the pore solid fractal (PSF). The approach represents a generalisation of existing fragmentation models based on homogeneous and mass fractal soil structures. Scale invariant and incomplete fragmentation is seen to result in the creation of both distributions of aggregates and primary particles of comparable size. These two distributions are powerlaws and combine to yield a single powerlaw expression for the fragment number-size distribution. The powerlaw exponent determines a fragmentation dimension D f which integrates information about both the structure of the parent material and the intensity of the fragmentation process. This can be used to model experimental data which usually comprises a mixture of particles and aggregates. For a PSF model exhibiting scale invariant density, we obtain a powerlaw form for the cumulative distribution of fragment mass versus size that can be fitted easily to available experimental data as shown in an example. We provide a consistent theoretical background for a physically based interpretation of previous powerlaw models.