The fractal dimension of the pore-volume inside soils

This paper deals with the applicability of the theory of fractal geometry on the pore-volume V. It describes the porous medium [soil] by characterizing the pore-size distribution over a range of pore sizes through a single number, the fractal dimension D. D is calculated from the relation Vp ± P^D-3, where P is the pressure of the water-retention curve and Vp is the cumulated pore volume. The fractal dimension D for the different pore-size distributions ranges from 2.1 to 2.8 and is an intrinsic measurement for the degree of space filling of the different pore systems.     

Investigation into the fractal scaling of the structure and strength of soil aggregates

The hierarchical nature of soil structure is examined by measuring the physical properties of a range of aggregate sizes obtained using repetitive fracture. Fractals are used to assess the change with aggregate size of the specific volume, the proportion of pre-existing cracks which link to form the aggregate failure surface, and the aggregate failure stress. The pore size distribution, evaluated using mercury porosimetry and the application of the box counting algorithm to thin sections and thick sections, is also used to obtain a fractal dimension, D. Our results show that D depends upon the measurement approach for mass fractal scaling. This finding may limit the application of fractals to predict the scaling behaviour of soil physical properties.     

The relation between silty soil structures and their mercury porosimetry curve counterparts: fractals and percolation

Mercury porosimetry data can be interpreted in terms of soil structure using ideas drawn from (i) network modelling and percolation theory and (ii) fractal geometry. We linked mercury intrusion to soil structure quantified by image analysis within a relevant common pore radius scale. We compared (i) three independent methods for computing fractal dimensions of the matrix and of the solid–pore interface, namely fitted square boxes method and pore chord distribution on scanning electron microscope images of soil thin sections, and mercury porosimetry, and (ii) two independent methods for characterizing pore connectivity (image analysis) and percolation process (pressure threshold from mercury porosimetry). The results from analyses of the pore size distribution by mercury porosimetry differed from those from the image analysis. Mercury intrusion is controlled by both the connectivity of the pore space network and locally by pore throats leading to larger pore bodies. By contrast, image analysis is unaffected by pore connectivity and measures pore bodies. On the other hand, the chord length method might not adequately capture the scaling properties of the solid–pore interface, whereas the mercury porosimetry data were also difficult to interpret in terms of fractal geometry because of the effects of pore connectivity. However, fractal dimension values of both the solid phase and the solid–pore interface increased as a function of clay content, whereas both percolation probability values and throat radius values at the mercury percolation threshold decreased. The results show the merit of applying both fractals and percolation theory for determining structural parameters relevant to mercury and water transport in soil.     

On the relation between number-size distributions and the fractal dimension of aggregates

Number-size distributions (i.e. particle- and aggregate-size distributions) have historically been used as indicators of soil structure, and recent work has aimed to quantify this link using fractals to model the soil fabric. This interpretation of number-size distributions is evaluated, and it is shown that a number-size relation described by a power law does not in itself imply fractal structure as suggested, and a counter example is presented. Where fractal structure is assumed, it is shown that the power-law exponent, φ, describing the number-size distribution cannot be interpreted as the mass-fractal dimension, D_M, of the aggregate. If the probability of fragmentation is independent of fragment diameter, then the exponent may be identified with the boundary dimension, D_B, of the original matrix. If, however, as is likely, this probability is scale-dependent, then φ may over- or under-estimate the boundary dimension depending on whether the fragmentation probability increases or decreases with fragment size. The significance of these conclusions is discussed in terms of the interpretation of number-size distributions, and alternative methods for quantifying and interpreting soil structure are evaluated.     

The relationship between structure and the hydraulic conductivity of soil

A random fractal matrix comprising a hierarchical aggregation of primary structural elements is used to capture the characteristics of a heterogeneous soil structure with a tortuous pore space. The influence of heterogeneity of both the solid matrix and the pore space, as well as the shape of the pore boundary, on the saturated and unsaturated hydraulic conductivity is studied. For such random structures, the fractal (Hausdorff) dimension alone is not enough to characterize the structure from the point of view of fluid flow and additional characterizations are introduced. The porosity, ρ_p, of the primary elements has a critical value, ρ_c. With probability 1, both the saturated and unsaturated conductivities are found to be dependent as a power law on the length scale, L, at which the measurement is made when ρ_p>ρ_c. When ρ_p<ρ_c, only the unsaturated conductivity is scaling in length scale, while the saturated conductivity becomes dominated, with probability close to 1, by the conductivity of the largest connecting pores in the structure, i.e. preferential pathways. The relationships between the parameters of the power laws and structure are derived and are found to depend on the fractal (Hausdorff) and spectral dimensions of the solid matrix, denoted d_m and respectively. A discussion of the importance of these results for the interpretation and extrapolation of measurements is presented, and the implications for variability and predictability of the hydraulic properties of soil is discussed.     

The fractal structure of soil aggregates: its measurement and interpretation

The theory of fractal geometry is presented with reference to soil structure. Recent work on relating fractal structure to pore structure in soils is reviewed. It is suggested that the connection made in previous work between the fractal dimension and soil moisture retention curves is based on simplified assumptions that complicate the interpretation of results. A simple method for estimating the fractal dimension, D, of natural aggregates which circumvents some of these assumptions is presented. Preliminary results of aggregates from soils under different management systems show that, for the soils examined, D ranged from 2.75 to 2.93. The use of D to quantify heterogeneity in soil is explored.     

From fractal analysis along a line to fractals on the plane

Self-similar fractals are useful models for soil solid and pore sets. The scaling properties of these fractals along a line and across an area can be described by the fractal dimensions. One method, for estimating the soil areal fractal dimension from the solid and pore set distributions along the lines, was proposed and tested with real macro- and micromorphological data on three soils of Mexico. The soil areal fractal dimension (D_a) was compared with the soil mass fractal dimension (D_m) estimated by two-dimensional binary image analysis, separately for solids and pores. Both methods are based on the box-counting technique and are suitable for determining the soil ‘box' or ‘capacity' fractal dimension, that seems to be apt to estimate the alternative filling of an area by a fractal set of solids and pores. This paper examines the relations between the fractal dimensions obtained along a line, across an area and directly from the image. Analysis of D_a and D_m data seems to suggest that both soil genesis and management practice can contribute to areal fractal dimension dynamics. It was shown that fractal dimensions are useful parameters able to monitor tillage influence on soil properties and to estimate the degree of soil compaction.     

A multiscale study of silty soil structure

Dependency of soil properties on scale is a crucial issue in soil physics. In this paper, fractal approaches are used in two case studies in France and Australia, respectively, to study how measured physical soil properties change with the sample spacing and the scale of observation. At a scale of 10–1000 m (10⁴ to 10⁶ mm), fractals were applied to sample data from a linear transect, while at the 10^?6 to 10² mm scale, fractals were applied in two dimensions to analyse both soil micro‐ and macrostructure, based on thin section samples. Porosity was characterized by short‐range spatial variations using sample spacings of 0.5 and 5 m (from the transect data), and a sample spacing of 1 cm (from the thin section analysis). The size of the representative elementary volume (REV) or representative elementary area (REA), required to represent statistically the elementary soil structure, was identified in three ways: (i) by the correlation length of a representative interconnected pore network, (ii) by the upper limit of the non‐linear increase with observation scale of mean porosity (upper limit of the solid mass fractal domain), and (iii) by the non‐linear decrease with observation scale of the coefficient of variation, CV, of mean porosity. Two embedded REAs were identified: the first (0.1–0.4 mm) related to the soil microstructure whereas a second (11–44 mm) related to the soil macrostructure. The solid mass fractal dimensions of the two embedded structural domains showed that hierarchical heterogeneity of soil structure was more pronounced for microstructures than for macrostructures. The mean area ratio of microstructural matrix/total surface and the CV of mean microporosity both scale similarly at observation scales smaller than the REA size. Their scaling exponents were both related to the fractal dimension of microstructural matrix. This preliminary study shows that the theory of fractals applied to soil structures at a specific scale range cannot be directly applied to predict soil physical properties at another scale range. This is because there are different interdependent structuring processes operating at different scales resulting in fractal dimensions being consistent only over particular domain limits.     

基于Menger海绵模型的煤矸石粉改良膨胀土微结构特征

为解决膨胀土对工程结构以及农业生态环境的危害，进行煤矸石粉改良膨胀土的试验研究。对煤矸石粉掺量为0、3%、6%、9%的膨胀土土样进行压汞试验，测得微观孔隙特征值；选取Menger海绵模型建立孔隙分分形模型，计算土体孔隙分形维数，探究土体孔隙分形维数与孔隙特征参数以及煤矸石粉掺量变化的关系。结果表明：随着煤矸石粉掺量增加，土中大孔隙所占的含量较素膨胀减少61.5%，孔隙类型从团粒间孔隙转化为颗粒间孔隙；煤矸石粉的掺入改变了土体的孔隙结构特征，煤矸石粉与膨胀土发生胶结反应，孔隙连通性降低，使得总孔隙体积、孔隙率、孔隙平均孔径、孔隙临界孔径等孔隙特征参数呈减小趋势；基于分形理论分析孔隙分形维数，分形维数随煤矸石粉掺量的增加而增加，且与孔隙特征参数呈显著相关性。孔隙分形维数反应了孔隙特征参数以及孔隙发育程度，为土的孔隙表征提供方法借鉴。     

Dual-porosity modelling of the pore structure and transport properties of a contaminated soil

We have developed a new method to characterize the pore structure of mineral soils. We combined data from the analysis of back-scattered scanning electron microscope (BSEM) images of resin-impregnated pore-casts, and mercury intrusion porosimetry (MIP) data, with analytical percolation models and inverse modeling algorithms. The pore space is regarded as a dual-pore network consisting of a primary Euclidean pore-and-throat network and a secondary, fractal, pore system that is accessed through primary pores. The digitized 2-D BSEM images of resin-impregnated soil samples are employed to determine the autocorrelation function. The Fourier transform of this function provides the small-angle neutron scattering (SANS) intensity function, which is extended by using the surface fractal dimension obtained from high-pressure MIP data. Inversion of the extended scattering intensity function produces the volume-based radius distribution function of spherical pore bodies (PBRD). The complete volume-based PBRD is fitted with a composite number-based PBRD composed of a lognormal primary PBRD and a power (fractal) secondary PBRD with upper and lower cut-offs. Based on the concepts of invasion percolation, an analytic mathematical model that describes Hg intrusion into dual pore networks is developed. The complete PBRD and pore-throat radius distribution (PTRD) functions of the primary network along with the drainage accessibility functions (DAFs) of the primary and secondary pore networks are estimated with inverse modelling of the Hg intrusion curve. Based on critical path analysis of percolation theory, approximate analytical relationships are developed to calculate explicitly the absolute permeability and electrical formation factor from the geometrical and topological parameters of the primary pore network. The method is demonstrated with application to four soil samples.     

Water retention characteristics of soils with double porosity

A soil with double porosity is modelled as a collection of aggregated particles, in which a single aggregate is made up of discrete particles bonded together. Separate fractal distributions for pore sizes around and within aggregates are defined. The particle size distribution of the double porosity soil is also modelled using a fractal distribution, which may have a fractal dimension very different to those defining the pore sizes. The surface areas of the particles and the pores within the aggregates are assumed to be equal, enabling an expression linking two fractal dimensions to be defined. It is necessary to introduce ratios between maximum and minimum particle and pore sizes into the expression. A theoretical soil‐water characteristic curve is then derived for a double porosity soil. The curve, and the underlying assumptions regarding the distributions of pore and particle sizes, showed good agreement with experimental data for a range of soils having double porosity. A discontinuity is observed in the soil‐water characteristic curve at a second air entry value related to the maximum pore size within the aggregates, a feature also observed in experimentally obtained soil‐water characteristic curves for double porosity soils.     

土壤和润湿研究中的分形现象

分形几何学的理论与方法已被用于土壤表面不规则性、旋耕土块轮廓及表面裂纹形状和分布、土壤值的空间分布及固体表面润湿性的研究中,利用分形维数定量探讨它们的规律性。本文综合评述了土壤表面、土块轮廓与裂纹、土壤值及润湿研究中的分形特性。     

土壤孔隙质量分数维D_m二元图像分析及其影响因素研究

本文简要介绍了利用土壤切片的二元图像分析了土壤孔隙结构的质量分数维Dm 的方法 ,并研究比较了四类土壤类型的Dm 及其影响因素。结果表明Dm 可以较好地定量描述土壤孔隙的空间分布特征 ,Dm 与土壤孔隙面积、孔隙孔径分布、土壤深度以及土壤质地之间均具有一定的联系     

A casting method for the three-dimensional analysis of the intraprism structural pores in vertisols

The structural voids in vertisols contain easily available water for plants and their volume can be calculated from the shrinkage curve. Access by plants to that water depends also on the geometric arrangement of the pores so that the water can flow through them. We have devised a method for studying the structural porosity by casting the pores in resin. The intraprism pore space of wet soil clods is impregnated with a UV fluorescent polyester resin under vacuum. When this has set we use the swelling properties of the clay to separate the clay matrix from the resin. A cast so obtained is the real three-dimensional solid reproduction of the structural porosity. This representation of the pore system is easier to study than results from computerized reconstitution of the three-dimensional space from two-dimensional images of soil in thin sections. Channels, packing pores and planar voids can be observed directly in three dimensions as the method saves the integrity and continuity of pores as small as 10 μm in diameter. The geometry of the cast shapes agrees with the interpretation of shrinkage and moisture characteristic curves. The method offers direct qualitative observation of pore organization and volume measurements of the intraprism structural porosity in vertisols.     

子午岭林区典型植被下土壤结构及稳定性指标分析

运用分形理论,研究了子午岭林区5种天然次生植被(以6 a天然恢复弃耕地为对照)下土壤结构特征,分析了土壤水稳性团聚体分维、孔隙分维、平均重量直径等3个指标在描述土壤结构稳定性方面的差异.研究表明,相对于弃耕地,各个植被群落均能明显改善土壤结构,降低土壤水稳性团聚体分形维数,提高孔隙分形维数,增强土壤结构的稳定性.土壤水稳性团聚体分维、孔隙分维q>0.25 mm水稳性团聚体含量、土壤有机碳、容重的相关系数均达到了极显著水平,均能作为评价土壤结构稳定性的指标;而团聚体平均重量直径与土壤有机碳含量、容重相关性不显著,只与>5 mm团聚体含量和>0.25 mm团聚体含量有极显著正相关关系,因此,仅可作为大团聚体含量的评价指标.     

Effect of rain on the macroporosity at the soil surface

Rain falling on soil causes slaking, mechanical disruption of aggregates and compaction. Too few data exist to predict the changes likely to occur in particular soil, landscape and management conditions. Experiments with simulated rain were set up to study and to model mathematically the changes of the pore system within the surface layer of a soil when rain was applied on a field cropped with maize. Macroporosity, pore‐size and pore‐shape distributions, and the pore volume were measured by image analysis of thin sections and the fractal dimensions of the pore surface roughness were estimated. The general trends of changes in porosity indicated the presence of two different sets of processes at the surface (0–3 cm) and in the layer immediately underneath (3–6 cm). In both layers most of the variation in macroporosity was due to a loss of elongated porosity. A theoretical approach recently developed to link rain and erosion to sealing properties was extended to describing the effect of rain on the elongated porosity and the pore volume fractal dimension in these two layers. The resulting set of equations describe in detail the evolution of soil porosity near the soil surface. Our approach could be useful when modelling the effects of sealing processes in soil erosion.     

鲁中南山地典型植被土壤颗粒与土壤水分特征曲线的分形学特征

土壤粒径分布与水分特征曲线是土壤的重要物理性质,对土壤侵蚀状况和土壤肥力有显著影响。以鲁中南山地典型植被下土壤为研究对象,运用分形学理论研究5种典型植被土壤颗粒与水分特征曲线的分形学特征。结果表明:1)不同植被类型的土壤颗粒单重分形维数、多重分形参数和土壤水分特征曲线分形维数具有显著差异,均表现为麻栎+刺槐混交林>黑松+黄连木混交林>黑松林>核桃林>荒草地;2)土壤分形维数有林地大于荒草地,混交林大于纯林;3)土壤分形维数与黏粒体积分数、粉粒体积分数呈显著正相关,与砂粒体积分数呈显著负相关;4)土壤颗粒单重分形维数与土壤水分特征曲线分形维数呈显著的正相关关系。说明土壤颗粒分布与土壤水分特征曲线的分形参数可以作为反映土壤结构性状变化的定量指标,可利用土壤颗粒分形与土壤水分特征曲线分形维数的相关关系来描述对应的土壤水分特征曲线。研究成果可为鲁中南山地退耕还林与生态造林工程建设及其生态效益评价提供理论参考。     

灰色关联及非线性规划法构建传递函数估算黑土水力参数

土壤水分特征曲线和饱和导水率是重要的水力参数,为了简便准确获取这些参数,以松嫩平原黑土区南部为研究区域,采集136个采样点土样用于测定不同土层土壤水分特征曲线、饱和导水率以及土壤理化性质,并运用灰色关联分析确定影响土壤水力参数的主要土壤理化性质,采用非线性规划构建土壤分形维数、有机质、干容重、土壤颗粒组成与土壤水分特征曲线、饱和导水率之间的土壤传递函数,并通过与现有土壤传递函数对比分析进行精度验证。结果表明:1)土壤分形维数是估算土壤水分特征曲线模型参数和饱和导水率的主要参数之一,同时,干容重和有机质含量也在不同土层土壤传递函数中起到重要的作用;2)通过验证分析,不同土层各参数平均绝对误差接近于0,均方根误差值也都较小,其中在不同土层土壤传递函数估算的土壤含水率均方根误差分别为0.022、0.017cm~3/cm~3;3)对比分析其他已存的土壤水分特征曲线和饱和导水率的土壤传递函数,该文构建的土壤传递函数均方根误差值均较小,决定系数值都在0.66以上,表明估算精度较高,均好于其他方法估算精度,具有良好的区域适应性。综上,所构建的土壤水分特征曲线和饱和导水率土壤传递函数可以用于松嫩平原黑土区土壤水力参数估算。     

Water retention models for fractal soil structures

A review of water retention functions based upon fractal soil structures is presented. We consider the modelling approach for a fractal fabric, a fractal pore boundary and a fractal pore space, identifying the latter case as one of particular complexity. In each case, the water retention function is derived from the pore volume distribution arising from the structural model in question. We examine published models and highlight problems, namely lack of generality and inconsistency with the assumed fractal structure. The models considered in this paper do not account for the effects of pore connectivity, and as such their validity as a necessary condition for the existence of fractal structure is questionable.     

土壤图像孔隙轮廓线分形特征及其应用

孔隙轮廓线分形维数是用来描述孔隙本身所具有的分形特征，它反映了土壤孔隙与固体颗粒接触界限的不规则性。该文采用数字图像处理技术研究了中国科学院封丘农业生态实验站内3种不同质地土壤样本图像的孔隙轮廓线分形特征。结果表明：土壤孔隙轮廓线分形特征与土壤质地之间存在着一定的相关关系，即土壤质地越细(黏粒含量越高)分形维数取值越大。同时，根据这3种土壤的孔隙轮廓线分形维数，结合不同分形方法分别预测了它们的水分特征曲线，并与实测的土壤水分特征曲线进行了比较。