Solute Mass Flux Research Articles

A Laboratory Apparatus to Measure Chemico-Osmotic Efficiency Coefficients for Clay Soils

Abstract A laboratory apparatus for measuring the chemico-osmotic efficiency coefficient, ω, for clay soils in the presence of electrolyte solutions is described. A chemico-osmotic experiment is conducted by establishing and maintaining a constant difference in electrolyte concentration across a soil specimen while preventing the flow of solution through the specimen. The chemico-osmotic efficiency coefficient is derived from a measured pressure difference induced across the specimen in response to the applied concentration difference. The effective diffusion coefficient (D*) and retardation factor (Rd) of the electrolytes (solutes) also can be determined simultaneously by measuring the diffusive solute mass flux through the specimen until steady-state diffusion is achieved. Experimental results using specimens of a geosynthetic clay liner subjected to potassium chloride solutions indicate that the measurement of ω may be affected by soil-solution interactions, as well as by changes in the induced chemico-osmotic pressure difference due to solute diffusion. As a result, ω should be evaluated using the induced pressure difference at steady state. The time required to achieve a steady-state response in induced pressure difference is related to the time required to achieve steady-state diffusion of all solutes, and may be affected by the circulation rate at the specimen boundaries. The circulation rate should be sufficiently rapid to minimize changes in the boundary concentrations due to diffusion, but sufficiently slow to allow measurement of solute mass flux at the lower concentration boundary for evaluating D* and Rd.

Effect of buoyancy ratio on the development of double‐diffusive finger convection in a Hele‐Shaw Cell

We consider the evolution of double‐diffusive finger convection for a two‐ solute (salt‐sucrose) system in a Hele‐Shaw cell. A high‐resolution, full‐field, light transmission technique was used to study the development of the instability that resulted from layering a lighter sucrose solution over a denser salt solution. The buoyancy ratio (Rρ), which is a ratio of fluid density contributions by the two solutes and defines the degree of system disequilibrium, was varied systematically from conditions that were nearly stable (Rρ = 2.8) to those that were moderately unstable (Rρ = 1.4). In all experiments, fingers are found to form continuously throughout time from a finger “generation” zone that straddles the location of the initial interface between solutions. At low Rρ, fingers develop rapidly, merge with adjacent fingers, and grow far beyond the finger generation zone through a series of finger “conduits.” In the higher Rρ experiments, fingers are slower to evolve, do not interact as dynamically, and do not grow far beyond the generation zone. Solute mass fluxes at low Rρ quickly reach a constant many times greater than that of a purely diffusive system; at high Rρ, mass fluxes decay as t−½ and behave diffusively but with effective diffusion coefficients much greater than those for molecular diffusion.

Numerical study of boundary conditions for solute transport through a porous medium

AbstractA transition region may be defined as a region of rapid change in medium properties about the interface between two porous media or at the interface between a porous medium and a reservoir. Modelling the transition region between different porous media can assist in the selection of the most appropriate boundary conditions for the standard advection–dispersion equation (ADE). An advantage of modelling the transition region is that it removes the need for explicitly defining boundary conditions, though boundary conditions may be recovered as limiting cases. As the width of a transition region is reduced, the solution of the transition region model (TR model) becomes equivalent to the solution of the standard ADE model with correct boundary conditions. In this paper numerical simulations using the TR model are employed to select the most appropriate boundary conditions for the standard ADE under a variety of configurations and conditions. It is shown that at the inlet boundary between a reservoir and porous medium, continuity of solute mass flux should be used as the boundary condition. At the boundary interface between two porous media both continuity of solute concentration and solute mass flux should be used. Finally, in a finite porous medium where the solute is allowed to advect freely from the exit point, both continuity of solute concentration and solute mass flux should be used as the outlet boundary condition. The findings made here are discussed with reference to a detailed review of previous relevant theoretical and experimental observations. Copyright © 2001 John Wiley & Sons, Ltd.

Prediction of plume migration in heterogeneous media using artificial neural networks

Because of the many uncertainties associated with most flow and transport parameters, studies often implement the numerical simulations within a Monte Carlo frame of work. The large numbers of realizations needed to achieve convergence for the statistics of concern make the Monte Carlo approach computationally demanding. In this study we attempt to develop an empirical‐numerical approach to generate Lagrangian particle trajectories in two‐dimensional domain given a certain input heterogeneity model without repeatedly solving the flow equation for each realization, with the purpose of evaluating ensemble plume statistics. Artificial neural networks are used to map the relationship between the particle trajectories and the physical properties of the formation. This is achieved by training the neural network through a set of sufficient examples derived from a few realizations using an exponential log‐K covariance. The trained network is then used to predict particle trajectories in heterogeneity characterized by an exponential, a Gaussian, a hole‐type, or a fractal covariance model. The network that is trained using the exponential model successfully predicts transport results for other models with high accuracy and low computational effort. Accuracy of prediction of the trajectory (percent of explained variance) reaches 94% in the direction of mean flow and 91% normal to it. For the cases studied here, the speed of calculation is ∼7.8 times faster than the traditional approach for 1000 realizations. As the number of realizations increases, the speed factor approaches 12.8. Solute mass flux (mean and standard deviation) and mean‐plume concentrations predicted by the trained network agree very closely with the target result obtained via traditional Monte Carlo simulations.

Water flow and solute mass flux in heterogeneous porous formations with spatially random porosity

The impact of the spatial variability of porosity on the solution of the flow equation is investigated using standard first-order perturbation analysis. We use existing theories of flow in heterogeneous media and incorporate spatially varying porosity that may or may not be correlated to the hydraulic conductivity. The first-order analysis shows that longitudinal velocity covariance and its cross covariance with the log-conductivity and head fields are significantly affected by porosity variability with increase or decrease in the magnitude of the covariance if there is a correlation between porosity and conductivity. In the absence of any correlation, only longitudinal velocity covariance is affected, but to a much lesser extent and with an increase in the velocity variance. In some cases where porosity variability is only one order of magnitude lower than that of the conductivity, the impact is so strong that it reverses the sign of the longitudinal velocity-head cross covariance. Transverse velocity covariance and its cross covariance with other flow quantities are not affected by porosity variability. The analytical results are confirmed by Monte Carlo simulations in terms of general patterns and porosity effects. The consequences of these results for transport processes are explored using the solute flux approach. The solute discharge and mass flux across an arbitrary control plane are strongly affected by porosity variability only when it is correlated to conductivity. The mean solute discharge is equally affected in both absolute and relative dispersion frameworks. The solute discharge variance computed within the absolute dispersion framework is more influenced by porosity variability than that computed within the relative dispersion framework. Accounting for the two-particle correlation of travel time and transverse displacement renders the effect of porosity variability less significant than when this correlation is dropped from the formulation of the point solute flux variance.

Stochastic analysis of transport in unsaturated heterogeneous soils uder transient flow regimes

Approximate analytic solutions are developed to predict the transport of inert solutes in variably saturated, randomly heterogeneous porous media subject to a random boundary flux condition. The mean and variance of solute mass flux past a series of vadose zone control planes are determined using a Lagrangian approach. The method requires the specification of an advective travel time probability density function (pdf) between the soil surface and the control planes and an impulse response function which quantifies the specific flow and transport processes that affect the solute along an individual streamline in the soil. In this study an advective‐dispersive impulse response function for inert solute is assumed, and an advective travel time pdf is derived from the statistics of the boundary flux and soil properties using a one‐dimensional transient vadose zone flow model [Foussereau et al., this issue]. Monte Carlo simulations are conducted using random boundary flux sequences and spatially correlated random hydraulic conductivity fields in a multidimensional variably saturated flow and transport code to test the analytic model assumptions. The analytic model accurately predicts the ensemble mean solute flux breakthrough curves for a variety of soil‐climate combinations. The general behavior of the coefficient of variation of solute flux is also accurately reproduced; however, the minimum coefficient of variation at the mean center of mass was slightly underpredicted in most cases. Results show that for shallow soils in humid climates, boundary flux variability dominates soil variability in determining the uncertainty of solute transport predictions in the vadose zone.

Reconstruction of complex passageways for simulations of transport phenomena: development of a graphical user interface for biological applications

Flow of fluids, such as blood, lymph and air, plays a major role in the normal physiology of all living organisms. Within individual organ systems, flow fields may significantly influence the transport of solutes, including nutrients and chemical toxicants, to and from the confining vessel walls (epithelia and endothelia). Computational fluid dynamics (CFD) provides a potentially useful tool for biologists and toxicologists investigating solute disposition in these flow fields in both normal and disease states. Application of CFD is dependent upon generation of accurate representations of the geometry of the system of interest in the form of a computational reconstruction. The present investigations, which were based on studies of the toxicology of inhaled reactive gases in the respiratory tract of rodents, provide computer programs for the generation of finite element meshes from serial tissue cross-sections. These programs, which interface with a commercial finite element fluid dynamics simulation package (FIDAP 7.05, Fluid Dynamics International, Evanston, IL), permit simulation of fluid flow in the complex geometries and local solute mass flux to the vessel walls of biological systems. The use of these programs and their application to studies of respiratory tract toxicology are described.

A solute flux approach to transport in heterogeneous formations: 1. The general framework

It is common to represent solute tranport in heterogeneous formations in terms of the resident concentration C(x, t), regarded as a random space function. The present study investigates the alternative representation by q, the solute mass flux at a point of a control plane normal to the mean flow. This representation is appropriate for many field applications in which the variable of interest is the mass of solute discharged through a control surface. A general framework to compute the statistical moments of q and of the associated total solute discharge Q and mass M is established. With x the direction of the mean flow, a solute particle is crossing the control plane at y = η, z = ζ and at the travel (arrival) time τ. The associated expected solute flux value is proportional to the joint probability density function (pdf) g1 of η, ζ and τ, whereas the variance of q is shown to depend on the joint pdf g2 of the same variables for two particles. In turn, the statistical moments of η, ζ and τ depend on those of the velocity components through a system of stochastic ordinary differential equations. For a steady velocity field and neglecting the effect of pore‐scale dispersion, a major simplification of the problem results in the independence of the random variables η, ζ and τ. As a consequence, the pdf of η and ζ can be derived independently of τ. A few approximate approaches to derive the statistical moments of η, ζ and τ are outlined. These methods will be explored in paper 2 in order to effectively derive the variances of the total solute discharge and mass, while paper 3 will deal with the nonlinear effect of the velocity variance upon the moments of η, ζ and τ

Regional water flow and pesticide leaching using simulations with spatially distributed data

Simulation modeling of water and chemical movement and techniques of Geographical Information Systems (GIS) were used to integrate, condense and summarize the large-scale (order of kilometers) behavior of spatially variable soils to provide management guidance on issues related to pesticide movement in soil. A one-dimensional, convection-dispersion-based solute transport model was used to simulate the movement of four classes of chemicals through layered soils for a 7 km X 10 km site near Albany, New York. The site was divided into a 163 X 112 grid (0.4 ha per cell), with multiple model executions used as a means of incorporating the influence of spatial variability of soil hydraulic properties into the study. Pedotransfer functions based on regression equations were used to relate soil physical properties to the mean and variance of hydraulic properties within each cell. Each model execution used hydraulic properties selected from frequency distributions derived from these parameters. Land use, soil, and slope maps were overlain with a Geographical Information System (GIS) to stratify the grid cells appropriate for modeling. Time series plots for mean and variance of water flux, solute mass, and solute flux were produced based on 25 1-year simulations for each hydraulic group, using local climatic data. Two climatic conditions were analyzed, a very ‘wet’ year (90th percentile of the population of total annual precipitation) and a mean precipitation year. The calculated solute fluxes are discussed in terms of the implications for pesticide management and for estimating potential leaching hazards at the study site.

The Cell Analytical‐Numerical Method for Solution of the Advection‐Dispersion Equation: Two‐Dimensional Problems

In this paper we present the development and evaluation of a new numerical scheme for efficient solution of groundwater solute transport problems. The scheme, which we have named the cell analytical‐numerical (CAN) method, is an extension of the so‐called nodal methods which have been developed in the nuclear engineering area for solving neutron diffusion problems. The CAN method is based upon decomposition of the solution domain into a number of rectangular cells (volume subdomains); each cell is homogeneous so that a local analytic solution to the solute transport equation can be obtained. A first‐order accurate finite difference approximation of the time derivative followed by a transverse averaging procedure is used to transform the governing partial differential equation into a set of coupled one‐dimensional ordinary differential equations for the transverse moments of the concentration. The local analytical solution is therefore found in terms of these moments. Solute mass flux continuity across cell surfaces, along with the local analytical solution, is used to construct an algebraic relationship between concentration moment values at adjacent cell surfaces. Assembling all the cells together results in a set of coupled tridiagonal matrix equations, one set for each spatial direction, which can be solved very efficiently. The CAN method is applied to several simple test problems involving uniform flow in homogeneous aquifers. While the method lacks geometrical flexibility, it offers a spatial approximation that is demonstrated to have high accuracy and minimal grid orientation error, even when applied to coarse meshes.

A derivation of the macroscopic solute transport equation for homogeneous, saturated, porous media

The macroscopic transport equation for a conservative solute in a homogeneous, water‐saturated porous medium is derived on the basis of a rigorous cumulant expansion applied to the equation of mass balance. The essential physical concept underlying the derivation is that of a local volume‐averaged solute velocity which fluctuates on a time scale that is orders of magnitude smaller than its autocorrelation time scale, which, in turn, is much smaller than the time scale of interest in a typical solute transport experiment. This clear separation of time scales is illustrated with representative data on solute transport in homogeneous, water‐saturated soils and is employed to justify the truncation of an exact cumulant expansion of the divergence of the volume‐averaged solute mass flux density. With the cumulant expansion terminated at first order in the ratio of the solute velocity autocorrelation time to the macroscopic solute transport time interval, an expression for the macroscopic solute mass flux density is produced which is the same as Fick's Law extended to porous media.

Improved representation of velocity correlations in aqueous electrolyte solutions

A general definition of the salt molecule for any asymmetric polyvalent electrolyte in solution is given which fulfils the requirement that the solute mass flux in the presence of an electric field strength vanishes when referred to the proper representative point of the molecule. Effectively this is the definition of the mass flux in the microscopic representation of an electrolyte solution. The electric current density arising from a gradient in the chemical potential also vanishes when referred to the same representative point. Time integrals over the velocity cross-correlation functions have been calculated for aqueous NaCl, LiCl, CaCl2 and BaCl2 solutions using conductivity data, transport numbers and mutual diffusion coefficients fulfilling the above requirements. A first approximation is also given for the calculation of velocity correlation coefficients in the case where transport numbers are not available.