An investigation of localised soil heterogeneities on solute transport using a multisegment percolation system

A multisegment percolation system (MSPS) consisting of 25 individual collection wells was constructed to study the effects of localised soil heterogeneities on the transport of solutes in the vadose zone. In particular, this paper discusses the transport of water and nutrients (NO3-, Cl-, PO43-) through structurally stable, free-draining agricultural soil from Victoria, Australia. A solution of nutrients was irrigated onto the surface of a large undisturbed soil core over a 12-h period. This was followed by a continuous irrigation of distilled water at a fate which did not cause pending for a further 18 days. During this time, the volume of leachate and the concentration of nutrients in the leachate of each well were measured. Very significant variation in drainage patterns across a small spatial scale was observed. Leaching of nitrate-nitrogen and chloride from the core occurred two days after initial application. However, less than 1% of the total applied phosphate-phosphorus leached from the soil during the 18-day experiment, indicating strong adsorption. Our experiments indicate considerable heterogeneity in water flow patterns and solute leaching on a small spatial scale. These results have significant ramifications for modelling solute transport and predicting nutrient loadings on a larger scale.

A mathematical model for estimating the extent of solute- and water-flux heterogeneity in multiple sample percolation experiments

Multiple sampling is widely used in vadose zone percolation experiments to investigate the extent in which soil structure heterogeneities influence the spatial and temporal distributions of water and solutes. In this note, a simple, robust, mathematical model, based on the beta-statistical distribution, is proposed as a method of quantifying the magnitude of heterogeneity in such experiments. The model relies on fitting two parameters, alpha and zeta to the cumulative elution curves generated in multiple-sample percolation experiments. The model does not require knowledge of the soil structure. A homogeneous or uniform distribution of a solute and/or soil-water is indicated by alpha = zeta = 1, Using these parameters, a heterogeneity index (HI) is defined as root 3 times the ratio of the standard deviation and mean. Uniform or homogeneous flow of water or solutes is indicated by HI = 1 and heterogeneity is indicated by HI > 1. A large value for this index may indicate preferential flow. The heterogeneity index relies only on knowledge of the elution curves generated from multiple sample percolation experiments and is, therefore, easily calculated. The index may also be used to describe and compare the differences in solute and soil-water percolation from different experiments. The use of this index is discussed for several different leaching experiments. (C) 1999 Elsevier Science B.V. All rights reserved.

Unsaturated liquid percolation flow through nonwetted packed beds

The unsaturated flow of liquid through packed beds of large particles was studied using six different liquids, all with contact angles greater than 90degrees on the bed packing (wax spheres of 9, 15 and 19.4 mm diameter). The liquid flow was discrete in nature, as drops for low flow rates and rivulets for high flow rates. For unsaturated liquid flows, the actual percolation velocity, not superficial velocity, should be used to characterize the flow. The percolation velocity did not vary with packed-bed depth, but was a strong function of liquid flow rate, liquid and particle properties. Effects of liquid and particle properties (but not flow rate) are well captured by a simple correlation between the liquid-particle friction factor and Reynolds number based on actual percolation velocities. Liquid dispersion, characterized by the maximum dispersion angle, varies significantly with liquid and particle properties. The tentative correlation suggested here needs further validation for a wider range of conditions.

Percolation Phenomena in Micropore Adsorption: Influence on Single and Multicomponent Adsorption Equilibria

A novel and simple method for determination of micropore network connectivity of activated carbon using liquid phase adsorption is presented in this paper. The method is applied to three different commercial carbons with eight different liquid phase adsorptives as probes. The effect of the pore network connectivity on the prediction of multicomponent adsorption equilibria was also studied. For this purpose, the Ideal Adsorbed Solution Theory (IAST) was used in conjuction with the modified DR single component isotherm. The results of comparison with experimental data show that incorporation of the connectivity, and consideration of percolation processes associated with the different molecular sizes of the adsorptives in the mixture, can improve the performance of the IAST in predicting multicomponent adsorption equilibria.

Equilibrium self-assembly of colloids with distinct interaction sites: Thermodynamics, percolation, and cluster distribution functions

We calculate the equilibrium thermodynamic properties, percolation threshold, and cluster distribution functions for a model of associating colloids, which consists of hard spherical particles having on their surfaces three short-ranged attractive sites (sticky spots) of two different types, A and B. The thermodynamic properties are calculated using Wertheim's perturbation theory of associating fluids. This also allows us to find the onset of self-assembly, which can be quantified by the maxima of the specific heat at constant volume. The percolation threshold is derived, under the no-loop assumption, for the correlated bond model: In all cases it is two percolated phases that become identical at a critical point, when one exists. Finally, the cluster size distributions are calculated by mapping the model onto an effective model, characterized by a-state-dependent-functionality (f) over bar and unique bonding probability (p) over bar. The mapping is based on the asymptotic limit of the cluster distributions functions of the generic model and the effective parameters are defined through the requirement that the equilibrium cluster distributions of the true and effective models have the same number-averaged and weight-averaged sizes at all densities and temperatures. We also study the model numerically in the case where BB interactions are missing. In this limit, AB bonds either provide branching between A-chains (Y-junctions) if epsilon(AB)/epsilon(AA) is small, or drive the formation of a hyperbranched polymer if epsilon(AB)/epsilon(AA) is large. We find that the theoretical predictions describe quite accurately the numerical data, especially in the region where Y-junctions are present. There is fairly good agreement between theoretical and numerical results both for the thermodynamic (number of bonds and phase coexistence) and the connectivity properties of the model (cluster size distributions and percolation locus).

Percolation of colloids with distinct interaction sites

We generalize the Flory-Stockmayer theory of percolation to a model of associating (patchy) colloids, which consists of hard spherical particles, having on their surfaces f short-ranged-attractive sites of m different types. These sites can form bonds between particles and thus promote self-assembly. It is shown that the percolation threshold is given in terms of the eigenvalues of a m x m matrix, which describes the recursive relations for the number of bonded particles on the ith level of a cluster with no loops; percolation occurs when the largest of these eigenvalues equals unity. Expressions for the probability that a particle is not bonded to the giant cluster, for the average cluster size and the average size of a cluster to which a randomly chosen particle belongs, are also derived. Explicit results for these quantities are computed for the case f = 3 and m = 2. We show how these structural properties are related to the thermodynamics of the associating system by regarding bond formation as a (equilibrium) chemical reaction. This solution of the percolation problem, combined with Wertheim's thermodynamic first-order perturbation theory, allows the investigation of the interplay between phase behavior and cluster formation for general models of patchy colloids.

On Lundh's percolation diffusion

A collection of spherical obstacles in the unit ball in Euclidean space is said to be avoidable for Brownian motion if there is a positive probability that Brownian motion diffusing from some point in the ball will avoid all the obstacles and reach the boundary of the ball. The centres of the spherical obstacles are generated according to a Poisson point process while the radius of an obstacle is a deterministic function. If avoidable configurations are generated with positive probability Lundh calls this percolation diffusion. An integral condition for percolation diffusion is derived in terms of the intensity of the point process and the function that determines the radii of the obstacles.

On Lundh's percolation diffusion

A collection of spherical obstacles in the unit ball in Euclidean space is said to be avoidable for Brownian motion if there is a positive probability that Brownian motion diffusing from some point in the ball will avoid all the obstacles and reach the boundary of the ball. The centres of the spherical obstacles are generated according to a Poisson point process while the radius of an obstacle is a deterministic function. If avoidable con gurations are generated with positive probability Lundh calls this percolation di usion. An integral condition for percolation di ffusion is derived in terms of the intensity of the point process and the function that determines the radii of the obstacles.

Percolation thresholds in chemical disordered excitable media

The behavior of chemical waves advancing through a disordered excitable medium is investigated in terms of percolation theory and autowave properties in the framework of the light-sensitive Belousov-Zhabotinsky reaction. By controlling the number of sites with a given illumination, different percolation thresholds for propagation are observed, which depend on the relative wave transmittances of the two-state medium considered.

Water percolation estimated with time domain reflectometry (TDR) in drainage lysimeters

Due to the difficulty of estimating water percolation in unsaturated soils, the purpose of this study was to estimate water percolation based on time-domain reflectometry (TDR). In two drainage lysimeters with different soil textures TDR probes were installed, forming a water monitoring system consisting of different numbers of probes. The soils were saturated and covered with plastic to prevent evaporation. Tests of internal drainage were carried out using a TDR 100 unit with constant dielectric readings (every 15 min). To test the consistency of TDR-estimated percolation levels in comparison with the observed leachate levels in the drainage lysimeters, the combined null hypothesis was tested at 5 % probability. A higher number of probes in the water monitoring system resulted in an approximation of the percolation levels estimated from TDR - based moisture data to the levels measured by lysimeters. The definition of the number of probes required for water monitoring to estimate water percolation by TDR depends on the soil physical properties. For sandy clay soils, three batteries with four probes installed at depths of 0.20, 0.40, 0.60, and 0.80 m, at a distance of 0.20, 0.40 and 0.6 m from the center of lysimeters were sufficient to estimate percolation levels equivalent to the observed. In the sandy loam soils, the observed and predicted percolation levels were not equivalent even when using four batteries with four probes each, at depths of 0.20, 0.40, 0.60, and 0.80 m.

Estimation of water percolation by different methods using TDR

Detailed knowledge on water percolation into the soil in irrigated areas is fundamental for solving problems of drainage, pollution and the recharge of underground aquifers. The aim of this study was to evaluate the percolation estimated by time-domain-reflectometry (TDR) in a drainage lysimeter. We used Darcy's law with K(θ) functions determined by field and laboratory methods and by the change in water storage in the soil profile at 16 points of moisture measurement at different time intervals. A sandy clay soil was saturated and covered with plastic sheet to prevent evaporation and an internal drainage trial in a drainage lysimeter was installed. The relationship between the observed and estimated percolation values was evaluated by linear regression analysis. The results suggest that percolation in the field or laboratory can be estimated based on continuous monitoring with TDR, and at short time intervals, of the variations in soil water storage. The precision and accuracy of this approach are similar to those of the lysimeter and it has advantages over the other evaluated methods, of which the most relevant are the possibility of estimating percolation in short time intervals and exemption from the predetermination of soil hydraulic properties such as water retention and hydraulic conductivity. The estimates obtained by the Darcy-Buckingham equation for percolation levels using function K(θ) predicted by the method of Hillel et al. (1972) provided compatible water percolation estimates with those obtained in the lysimeter at time intervals greater than 1 h. The methods of Libardi et al. (1980), Sisson et al. (1980) and van Genuchten (1980) underestimated water percolation.

Percolation and epidemic thresholds in clustered networks

We develop a theoretical approach to percolation in random clustered networks. We find that, although clustering in scale-free networks can strongly affect some percolation properties, such as the size and the resilience of the giant connected component, it cannot restore a finite percolation threshold. In turn, this implies the absence of an epidemic threshold in this class of networks, thus extending this result to a wide variety of real scale-free networks which shows a high level of transitivity. Our findings are in good agreement with numerical simulations.

Potts fully frustrated model: Thermodynamics, percolation and dynamics in two dimensions

We consider a Potts model diluted by fully frustrated Ising spins. The model corresponds to a fully frustrated Potts model with variables having an integer absolute value and a sign. This model presents precursor phenomena of a glass transition in the high-temperature region. We show that the onset of these phenomena can be related to a thermodynamic transition. Furthermore, this transition can be mapped onto a percolation transition. We numerically study the phase diagram in two dimensions (2D) for this model with frustration and without disorder and we compare it to the phase diagram of (i) the model with frustration and disorder and (ii) the ferromagnetic model. Introducing a parameter that connects the three models, we generalize the exact expression of the ferromagnetic Potts transition temperature in 2D to the other cases. Finally, we estimate the dynamic critical exponents related to the Potts order parameter and to the energy.