Saturation-Dependence of Dispersion in Porous Media

In this study, we develop a saturation-dependent treatment of dispersion in porous media using concepts from critical path analysis, cluster statistics of percolation, and fractal scaling of percolation clusters. We calculate spatial solute distributions as a function of time and calculate arrival time distributions as a function of system size. Our previous results correctly predict the range of observed dispersivity values over ten orders of magnitude in experimental length scale, but that theory contains no explicit dependence on porosity or relative saturation. This omission complicates comparisons with experimental results for dispersion, which are often conducted at saturation less than 1. We now make specific comparisons of our predictions for the arrival time distribution with experiments on a single column over a range of saturations. This comparison suggests that the most important predictor of such distributions as a function of saturation is not the value of the saturation per se, but the applicability of either random or invasion percolation models, depending on experimental conditions

Saturation-Dependence of Dispersion in Porous Media

In this study, we develop a saturation-dependent treatment of dispersion in porous media using concepts from critical path analysis, cluster statistics of percolation, and fractal scaling of percolation clusters. We calculate spatial solute distributions as a function of time and calculate arrival time distributions as a function of system size. Our previous results correctly predict the range of observed dispersivity values over ten orders of magnitude in experimental length scale, but that theory contains no explicit dependence on porosity or relative saturation. This omission complicates comparisons with experimental results for dispersion, which are often conducted at saturation less than 1. We now make specific comparisons of our predictions for the arrival time distribution with experiments on a single column over a range of saturations. This comparison suggests that the most important predictor of such distributions as a function of saturation is not the value of the saturation per se, but the applicability of either random or invasion percolation models, depending on experimental conditions

Temporal complexity of daily precipitation records from different atmospheric environments: Chaotic and Lévy stable parameters

Rainfall events are very erratic at short and large temporal and spatial scales. The main objectives of the present study were (i) to describe different time series of daily precipitation records using both chaos theory and stable distribution methods and (ii) to search for potential relationships between chaotic and Lévy-stable parameters. We studied eight time series of daily rainfall from different latitudes around the world. Each rainfall signal spanned nine years (1997-2005). We used methods derived from chaos theory (embedding delays, spectrum of Lyapunov exponents, determinism tests and others) and parameters computed after fitting a stable distribution model to each differenced time series of rainfall data. All the rainfall signals showed chaotic structures with two positive Lyapunov exponents. The stability index was α < 2 which accounts for the scale-free behavior of rainfall dynamics. There were found latent statistical relationships between chaotic and Lévy stable parameters. That represents a potential connection between chaotic behavior, sub-Gaussian statistics and rainfall dynamics. Future research should deal with the connection between chaotic invariants, stable parameters and rainfall phenomenology. © 2011 Elsevier B.V

Scaling of Geochemical Reaction Rates via Advective Solute Transport

Transport in porous media is quite complex, and still yields occasional surprises. In geological porous media, the rate at which chemical reactions (e.g., weathering and dissolution) occur is found to diminish by orders of magnitude with increasing time or distance. The temporal rates of laboratory experiments and field observations differ, and extrapolating from laboratory experiments (in months) to field rates (in millions of years) can lead to order-of-magnitude errors. The reactions are transport-limited, but characterizing them using standard solute transport expressions can yield results in agreement with experiment only if spurious assumptions and parameters are introduced. We previously developed a theory of non-reactive solute transport based on applying critical path analysis to the cluster statistics of percolation. The fractal structure of the clusters can be used to generate solute distributions in both time and space. Solute velocities calculated from the temporal evolution of that distribution have the same time dependence as reaction-rate scaling in a wide range of field studies and laboratory experiments, covering some 10 decades in time. The present theory thus both explains a wide range of experiments, and also predicts changes in the scaling behavior in individual systems with increasing time and/or length scales. No other theory captures these variations in scaling by invoking a single physical mechanism. Because the successfully predicted chemical reactions include known results for silicate weathering rates, our theory provides a framework for understanding changes in the global carbon cycle, including its effects on extinctions, climate change, soil production, and denudation rates. It further provides a basis for understanding the fundamental time scales of hydrology and shallow geochemistry, as well as the basis of industrial agriculture. VC 2015 AIP Publishing LLC

Minasny and McBratney, 2007), and the use of tables * 1 Supported by the National Natural Science Foundation of China

ABSTRACT The van Genuchten (vG) function is often used to describe the soil water retention curve (SWRC) of unsaturated soils and fractured rock. The objective of this study was to develop a method to determine the vG model parameter m from the fractal dimension. We compared two approaches previously proposed by van Genuchten and Lenhard et al. for estimating m from the pore size distribution index of the Brooks and Corey (BC) model. In both approaches we used a relationship between the pore size distribution index of the BC model and the fractal dimension of the SWRC. A dataset containing 75 samples from the UNSODA unsaturated soil hydraulic database was used to evaluate the two approaches. The statistical parameters showed that the approach by Lenhard et al. provided better estimates of the parameter m. Another dataset containing 72 samples from the literature was used to validate Lenhard&apos;s approach in which the SWRC fractal dimension was estimated from the clay content. The estimated SWRC of the second dataset was compared with those obtained with the Rosetta model using sand, silt, and clay contents. Root mean square error values of the proposed fractal approach and Rosetta were 0.081 and 0.136, respectively, indicating that the proposed fractal approach performed better than the Rosetta model. Many soil and water management and environmental protection practices require knowledge of the evolution of water and solutes in the subsurface. During the past several decades, a large number of computer models have been developed to simulate water flow and contaminant transport in saturated and unsaturated soils and fractured rock. Their application is often restricted by a lack of hydraulic property information involving the soil water retention curve (SWRC) and the unsaturated hydraulic conductivity. Furthermore, due to inherent temporal and spatial variability of the hydraulic properties in the field, large numbers of samples are generally required to properly characterize the spatial distribution of the hydraulic properties. Accurate characterization and estimation of the SWRC has been a major focus of research for more than 60 years. Many empirical model

Dispersion of solutes in porous media

A recently introduced theory of solute transport in porous media is tested by comparison with experiment. The solute transport is predicted using an adaptation of the cluster statistics of percolation theory to critical path analysis together with knowledge of how the structure of such percolation clusters affects the time of transport across them. Only the effects of a single scale of medium heterogeneity are incorporated, and a minimal amount of information regarding the structure of the medium is required. This framework is used to find effectively the distributions of solute velocities and travel distances and thus generate arrival time distributions. The comparison with experiment focuses on the dispersivity (the ratio of the second to the first moment of the spatial solute distribution). The predictions of the theory in the absence of diffusion are verified by comparing with over 2200 experiments over length scales from a few microns to 100 km. At larger length scales (centimeters on up) about 95% of the data lie within our predicted bounds. At smaller length scales approximately 99.8% of the data lie where we predict. These comparisons are not trivial as the typical values of the dispersivity increase by ten orders of magnitude over ten orders of magnitude of length scale. Noteworthy is that the classical advection-dispersion (ADE) equation predicts that the dispersivity should be independent of length scale! This agreement with experiment requires rethinking of the relevance of diffusion and multi-scale heterogeneity and would also appear to signal the complete inappropriateness of using the classical ADE or any of its derivatives to model solute transport

Dispersion of solutes in porous media

A recently introduced theory of solute transport in porous media is tested by comparison with experiment. The solute transport is predicted using an adaptation of the cluster statistics of percolation theory to critical path analysis together with knowledge of how the structure of such percolation clusters affects the time of transport across them. Only the effects of a single scale of medium heterogeneity are incorporated, and a minimal amount of information regarding the structure of the medium is required. This framework is used to find effectively the distributions of solute velocities and travel distances and thus generate arrival time distributions. The comparison with experiment focuses on the dispersivity (the ratio of the second to the first moment of the spatial solute distribution). The predictions of the theory in the absence of diffusion are verified by comparing with over 2200 experiments over length scales from a few microns to 100 km. At larger length scales (centimeters on up) about 95% of the data lie within our predicted bounds. At smaller length scales approximately 99.8% of the data lie where we predict. These comparisons are not trivial as the typical values of the dispersivity increase by ten orders of magnitude over ten orders of magnitude of length scale. Noteworthy is that the classical advection-dispersion (ADE) equation predicts that the dispersivity should be independent of length scale! This agreement with experiment requires rethinking of the relevance of diffusion and multi-scale heterogeneity and would also appear to signal the complete inappropriateness of using the classical ADE or any of its derivatives to model solute transport. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2011