A new multifractal-based grain size distribution model

Highlights

• Grey-scale soil image is an indicator of a soil density field.

• The grain size distribution (GSD) is predicted by up-scaling the grey-scale image.

• It is done by treating only densities above the fixed threshold while up-scaling.

• The representation of grains at different resolutions follows a multifractal law.

• This multifractal GSD model shows rather good agreement with dry sieving data.

Abstract

Previous works related to the application of the multifractal theory for analyzing the grain size distribution (GSD), showed the potential of this approach to deal with this complex issue. However, absence of the practical application of this kind of statistical analysis raised some doubts among the soil scientists. Compared to the experimental dry sieving method, which is based on mass representations of different grain sizes, the approach presented in this work relies on the analysis of grain densities (density indicators) scanned by means of X-ray CT (Computed Tomography). By reducing the resolution of the scanned soil image(s), the cumulative representation of solid particles equal to or larger than the actual discretization element can be determined, and described analytically by means of the universal multifractals (UM).

For validation of the new UM approach, the X-ray CT results of three different soils were used: the volcanic substrate covering Green Wave (a green roof of Champs-sur-Marne in France), and two horizons of the soil collected from the low land mountain area of Sierra de Guadarrama in Spain. Comparison between the proposed UM model and the experimental data of these three materials confirms that the GSD can be reasonably well predicted from the scanned images of soils covering wide range of grain sizes. The UM model, unlike the fractal-based models, accounts for fractal dimension that depends on grain size, and hence, based on the preliminary results presented in this work, it could be rather useful in case of multi-modal soils whose GSD curves are described with multiple fractal dimensions.

Introduction

The grain size distribution (GSD) is one of the fundamental properties of granular soils that, besides the influence on mechanical characteristics, also affects the packing arrangement of grains (Nolan and Kavanagh, 1993, He et al., 1999 among the others), and hence the distribution of pores that further impacts the hydraulic properties of the porous medium (Segal et al., 2009). Most often the GSD curve is experimentally determined based on the mass fractions of different grain sizes extracted either by using sieves of different void sizes, for grains larger than 80 µm (dry sieving method - AFNOR, 1996), or by means of sedimentation test (AFNOR, 1992, Beuselinck et al., 1998) for finer particles. The alternative approach proposed to measure GSD is a laser diffraction method (Miller and Schaetzl, 2012).

Detailed overview of different approaches used for describing the complexity of GSD curves can be found in Ghanbarian and Hunt (2017). One of them is the self-similarity principle which is included in fractal-based models and which assumes occurrence of the same pattern of the soil structure at all scales. According to Ghanbarian-Alavijeh et al. (2011), the three-phase PSF (pore-solid-fractal) approach (Perrier et al., 1999, Bird et al., 2000) is the most consistent and with the strongest physical-basis among the fractal-based approaches. Besides pores and grains, it assumes one additional “fictive” type of soil elements – fractals - that are successively broken at smaller scales in a self-similar way, leading finally to the structure consisting of fractal-distributed pore and grain sizes. Thus, the GSD can be represented by means of a power (fractal) law, where the fractal coefficient is included in the exponent. However, unlike assumed in the PSF model, grain densities are non-homogeneous, which also contributes to the complexity of distribution of different mass fractions that often cannot be described with a single fractal dimension (Bittelli et al., 1999).

Multifractal formalism, that takes into account different fractal coefficients for different threshold values, was also used for analyzing the complexity of GSD. Grout et al., 1998, Posadas et al., 2001 used Renyi dimensions, one of the multifractal parameters, to characterize the heterogeneous distribution of different mass fractions. Besides this type of multifractal analysis, the singularity spectra analysis is also applied for analyzing the dry soil volume-size distribution obtained by using a laser distraction method (Martín and Montero, 2002). Recently, Torre et al. (2016) used a X-ray CT, a non-destructive technique for obtaining a three-dimensional grey-scale image of a porous material (Hseih, 2003, Banhart, 2008) in order to compare the three-dimensional structural complexity of spatial arrangement of grains and pores, with that of differently oriented two-dimensional planes. The multifractal analysis has also proved to be convenient in this case. Even though the multifractal theory brings great potential for understanding the complexity of GSD (Ghanbarian and Hunt, 2017), up to date this kind of analysis has not found practical application.

This work is focused on development of a new physically-based GSD model founded on the Universal Multifractal (UM) framework (Schertzer and Lovejoy, 1987, Schertzer and Lovejoy, 1997). Based on a grey-scale soil image scanned by means of X-ray CT, it is possible to recognize solid particles of different sizes by progressively decreasing the resolution of the image while keeping the fixed value of the threshold. Change of the representation of solid particles with the resolution of the image can be directly linked with the grain size distribution, and described analytically in a mathematically-elegant way by means of the UM framework. Compared to work of Lai and Chen (2019), where a sophisticated machine learning tool was used for particle recognition, this approach is much simpler and more convenient for practical application.

The UM framework in combination with X-ray CT imaging was firstly validated for some artificial volcanic substrate (Stanić et al., 2020b, Stanić et al., 2020a) used for covering green roof named Green Wave (Versini et al., 2018, Versini et al., 2020). Results of the model, whose parameters are directly determined from scanned images, were first compared with the experimental data obtained by means of the standard dry sieving method (AFNOR, 1996) and sedimentation test (AFNOR, 1992). Furthermore, the UM model was tested on scanned images of two horizons of an intact soil sample collected from the low land mountain area of Sierra de Guadarrama (Schmid et al., 2016) called La Herreria. In this case, results of the model were compared with measured percentages of sand, silt and clay particles, since detailed GSD curves are lacking. Finally, for published experimental GSD data of the GW substrate and Walla Walla soil (Bittelli et al., 1999), the UM model was compared with the fractal-based PSF model.