Edmund Perfect

Applications of fractals in soil and tillage research: a review

Abstract Fractals are spatial and temporal model systems generated using iterative algorithms with simple scaling rules. This paper reviews the literature on spatial fractals as it applies to soil and tillage research. Applications of fractals in this area can be grouped into three broad categories: (i) description of soil physical properties; (ii) modeling soil physical processes; (iii) quantification of soil spatial variability. In terms of physical properties, fractals have been used to describe bulk density, pore-size distribution, pore surface area, particle-size distribution, aggregate-size distribution, ped shape and soil microtopography. In terms of physical processes, fractals have been used to model adsorption, diffusion, transport of water and solutes, brittle fracture and fragmentation. In terms of spatial variability, fractals have been applied to quantify distributions of soil properties and processes using semivariograms, power spectra and multifractal spectra. Further research is needed to investigate the specificity of different fractal models, to collect data for testing these models, and to move from the current descriptive paradigm towards a more predictive one. Fractal theory offers the possibility of quantifying and integrating information on soil biological, chemical and physical phenomena measured at different spatial scales.

Generalized Modeling of Spontaneous Imbibition Based on Hagen–Poiseuille Flow in Tortuous Capillaries with Variably Shaped Apertures

Spontaneous imbibition of wetting liquids in porous media is a ubiquitous natural phenomenon which has received much attention in a wide variety of fields over several decades. Many traditional and recently presented capillary-driven flow models are derived based on Hagen-Poiseuille (H-P) flow in cylindrical capillaries. However, some limitations of these models have motivated modifications by taking into account different geometrical factors. In this work, a more generalized spontaneous imbibition model is developed by considering the different sizes and shapes of pores, the tortuosity of imbibition streamlines in random porous media, and the initial wetting-phase saturation. The interrelationships of accumulated imbibition weight, imbibition rate and gas recovery and the properties of the porous media, wetting liquids, and their interactions are derived analytically. A theoretical analysis and comparison denote that the presented equations can generalize several traditional and newly developed models from the literature. The proposed model was evaluated using previously published data for spontaneous imbibition measured in various natural and engineered materials including different rock types, fibrous materials, and silica glass. The test results show that the generalized model can be used to characterize the spontaneous imbibition behavior of many different porous media and that pore shape cannot always be assumed to be cylindrical.

Disruptive methods for assessing soil structure

The description and quantification of soil structure is very important because of the many agronomic and environmental processes related to the arrangement of secondary soil units (aggregates, peds or clods) and their stability. The purpose of this review is to present and discuss methods and indices used to characterize soil structure based on the size distribution and stability of fragments produced by breaking apart the soil matrix. The size of fragments is inversely related to the mechanical stress applied. Thus, the selection of an appropriate fragmentation procedure is critical if information on soil structure is to be recovered, and often depends upon the soil process of interest. Soil fragmentation starts at sampling in the field and continues during laboratory separation of soil units by sieving. It is useful to characterize the fragment mass-size distribution with parameters from a model, such as the log-normal distribution function. Fractal theory provides a physically based link between the size distribution and stability of fragments. Structural stability is based on the ratio of fragment mass-sizes measured before and after low and high mechanical stresses, respectively. Thus, an adequate description of the applied stress conditions is essential for the parameterization of structural stability as well as the fragment mass-size distribution.

Management versus inherent soil properties effects on bulk density and relative compaction

Abstract Bulk density ( D b ) is a soil physical parameter used extensively to quantify soil compactness. The D b varies with management as well as with inherent soil properties. Because of its dependence on inherent soil properties, measurements of D b are of limited value as a measure of the effect of management on soil compaction when soils with different inherent characteristics are compared. Researchers have used the concept of relative compaction or relative bulk density, ( D b-rel ), the ratio of the D b of a soil to the D b under some standard compaction treatment ( D b-ref ), in order to compare the response of different soils to stresses arising from management. This research was conducted to determine D b , D b-ref and D b-rel across a range of soils under different tillage practices and to assess the relative importance of texture, organic matter content and management on these parameters. Thirty-six paired sampling sites were located along two parallel transects in a side by side comparison of no-tillage and conventional tillage treatments. The transects crossed three soil types: Aquic Hapludalfs, Psammentic Hapludalfs, and Typic Hapludalfs. Clay content (CLAY) varied from 5.8 to 42.3%, and organic matter (OM) varied from 1.4 to 11.6%. Multiple regression analyses showed that D b was related ( R 2 = 0.83) with CLAY, OM, tillage, position, and the interactions OM∗CLAY and tillage∗position, whereas the D b-ref was related ( R 2 = 0.86) with OM, CLAY, and tillage. Normalizing D b with respect to D b-ref , effectively eliminated the influence of CLAY, OM and their interaction on D b-rel . The analyses indicated that it is possible to quantify the separate effects of inherent soil properties and management on D b , when soils derived from similar parent materials and under similar climatic conditions are considered, either by using multiple regression analyses to describe D b , or by normalizing D b with respect to D b-ref .

An Improved Fractal Equation for The Soil Water Retention Curve

The water retention curve, θ(ψ), is important for predicting soil physical properties and processes. Until recently, equations for the θ(ψ) were empirical. Advances in fractal geometry have led to the derivation of physical models for the θ(ψ). However, both existing fractal equations have only two parameters and thus are relatively inflexible. We derived a new three-parameter fractal model for the θ(ψ). This equation was fitted to 36 θ(ψ)s for a silt loam soil with a wide range of structural conditions. The new equation fitted these data much better than the existing equations. The parameters of the new equation, ψa, ψd and D are physical entities, corresponding to the air-entry value, tension draining the smallest pores, and fractal dimension, respectively. Estimates of the ψa, ψd and D were physically reasonable, with median values of 2.9 × 10−1 kPa, 1.6 × 104 kPa, and 2.87, respectively. In contrast, the existing equations yielded anomalous estimates of either ψa or D. The new equation was able to fit θ(ψ) for a variety of porous media, including sandstone, glass beads, sands, sieved soil, and undisturbed soils ranging from very fine sandy loam to heavy clay. The ψa and ψd were more sensitive to structural and textural variation than D. The new equation represents an improvement over existing models in terms of both goodness of fit and the physical significance of its parameters.

Estimating soil mass fractal dimensions from water retention curves

The drying branch of the water retention curve is widely used for modeling hydrologic processes and contaminant transport in porous media. A prefractal model is presented for this function based on the capillary equation and a randomized Menger sponge algorithm with upper and lower scaling limits. The upper limit is the air entry value (Ψ0) and the lower limit is the tension at dryness (Ψj). Between these two limits the theoretical curve is concave when plotted as relative saturation (S) vs. the log of tension (Ψ). The mass fractal dimension (D) controls the degree of curvature, with decreasing concavity as D→3. The theoretical equation was fitted to water retention data for six soils from Campbell and Shiozawa [Campbell, G.S., Shiozawa, S., 1992. Prediction of hydraulic properties of soils using particle size distribution and bulk density data. International Workshop on Indirect Methods for Estimating the Hydraulic properties of Unsaturated Soils. University of California Press, Berkeley, CA, pp. 317–328]. These data consisted of between 31 and 39 paired measurements of S and Ψ for each soil, with Ψ ranging from 3.1×100 to 3.3×105 kPa. All of the fits were excellent with adjusted R2 values ≥0.96. The resulting estimates of D were all significantly less than three at P<0.05. The lowest value of D was 2.60 for a sandy loam soil, and the highest was 2.90 for a silty clay soil. Refitting the same data, but over a restricted subset of Ψs≤1.5×103 kPa, produced errors in the estimation of D. Two of the estimates of D were significantly greater than three at P<0.05. To estimate D accurately, water retention data covering the entire tension range from saturation to sero water content are required. In the absence of such data, it is possible to obtain physically reasonable estimates of D by setting Ψj=106 kPa, the approximate tension at oven dryness, and fitting the proposed equation as a two parameter model.

Simulation and testing of self-similar structures for soil particle-size distributions using iterated function systems

Abstract Particle-size distribution (PSD) is a fundamental soil physical property. The PSD is commonly reported in terms of the mass percentages of sand, silt and clay present. A method of generating the entire PSD from this limited description would be extremely useful for modeling purposes. We simulated soil PSDs using an iterated function system (IFS) following Martin and Taguas [ Martin, M.A., Taguas, F.J., 1998 . Fractal modeling, characterization and simulation of particle-size distribution in soil. Proc. R. Soc. Lond. A 454, 1457–1468]. By means of similarities and probabilities, an IFS determines how a fractal (self-similar) distribution reproduces its structure at different length scales. The IFS allows one to simulate intermediate distributional values for soil textural data. A total of 171 soils from SCS [ Soil Conservation Service, 1975 . Soil taxonomy: a basic system of soil classification for making and interpreting soil surveys. Agricultural Handbook no. 436. USDA-SCS, USA, pp. 486–742] were used to test the ability of different IFSs to reconstruct complete PSDs. For each soil, textural data consisting of the masses in eight different size fractions were used, and different PSDs were predicted using different combinations of three similarities. The five remaining data points were then compared with the simulated ones in terms of mean error. Those similarities that gave the, lowest mean error were identified as the best ones for each soil. Fifty-three soils had an error less than 10%, and 120 had an error less than 20%. The similarities corresponding to the sand, silt and clay fractions, i.e., IFS {0.002, 0.05}, did not, in general, produce good results. However, for soils classified as sand, silt loam, silt, clay loam, silty clay loam and silty clay, the same similarities always produced the lowest mean error, indicating the existence of a self-similar structure. This structure was not the same for all classes, although loams and clays were both best simulated by the IFS {0.002, 0.02}. It is concluded that IFSs are a powerful tool for identifying self-similarity in soil PSDs, and for reconstructing PSDs using data from a limited number of textural classes.

Fractal characterization of fracture networks: An improved box‐counting technique

given by N / rD , where N is the number of boxes containing one or more fractures and r is the box size, then the network is considered to be fractal. However, researchers are divided in their opinion about which is the best box-counting algorithm to use, or whether fracture networks are indeed fractals. A synthetic fractal fracture network with a known D value was used to develop a new algorithm for the box-counting method that returns improved estimates of D. The method is based on identifying the lower limit of fractal behavior (rcutoff) using the condition ds/dr ! 0, where s is the standard deviation from a linear regression equation fitted to log(N) versus log(r) with data for r < rcutoff sequentially excluded. A set of 7 nested fracture maps from the Hornelen Basin, Norway was used to test the improved method and demonstrate its accuracy for natural patterns. We also reanalyzed a suite of 17 fracture trace maps that had previously been evaluated for their fractal nature. The improved estimates of D for these maps ranged from 1.56 ± 0.02 to 1.79 ± 0.02, and were much greater than the original estimates. These higher D values imply a greater degree of fracture connectivity and thus increased propensity for fracture flow and the transport of miscible or immiscible chemicals.

Temporal Variation and Persistence of Bacteria in Streams

Better understanding of bacterial fate and transport in watersheds is necessary for improved regulatory management of impaired streams. Novel statistical time series analyses of coliform data can be a useful tool for evaluating the dynamics of temporal variation and persistence of bacteria within a watershed. For this study, daily total coliform data for the Little River in East Tennessee from 1 Oct. 2000 to 31 Dec. 2005 were evaluated using novel time series techniques. The objective of this study was to analyze the total coliform concentration data to: (i) evaluate the temporal variation of the total coliform, and (ii) determine whether the total coliform concentration data demonstrated any long-term or short-term persistence. For robust analysis and comparison, both time domain and frequency domain approaches were used for the analysis. In the time domain, an autoregressive moving average approach was used; whereas in the frequency domain, spectral analysis was applied. As expected, the analyses showed that total coliform concentrations were higher in summer months and lower in winter months. However, the more interesting results showed that the total coliform concentration exhibited short-term as well as long-term persistence ranging from about 4 wk to approximately 1 yr, respectively. Comparison of the total coliform data to hydrologic data indicated both runoff and baseflow are responsible for the persistence.

Percolation Thresholds in 2-Dimensional Prefractal Models of Porous Media*

Considerable effort has been directed towards the application of percolation theory and fractal modeling to porous media. We combine these areas of research to investigate percolation in prefractal porous media. We estimated percolation thresholds in the pore space of homogeneous random 2-dimensional prefractals as a function of the fractal scale invariance ratio b and iteration level i. The percolation thresholds for these simulations were found to increase beyond the 0.5927l... porosity expected in Bernoulli (uncorrelated) percolation networks. Percolation in prefractals occurs through large pores connected by small pores. The thresholds increase with both b (a finite size effect) and i. The results allow the prediction of the onset of percolation in models of prefractal porous media and can be used to bound modeling efforts. More fundamental applications are also possible. Only a limited range of parameters has been explored empirically but extrapolations allow the critical fractal dimension to be estimated for a large combination of b and i values. Extrapolation to infinite iterations suggests that there may be a critical fractal dimension of the solid at which the pore space percolates. The extrapolated value is close to 1.89 – the well-known fractal dimension of percolation clusters in 2-dimensional Bernoulli networks.