Analytical Solutions for Water Flow and Solute Transport in the Unsaturated Zone

Analytical solutions for the water flow and solute transport equations in the unsaturated zone are presented. We use the Broadbridge and White nonlinear model to solve the Richards’ equation for vertical flow under a constant infiltration rate. Then we extend the water flow solution and develop an exact parametric solution for the advection-dispersion equation. The method of characteristics is adopted to determine the location of a solute front in the unsaturated zone. The dispersion component is incorporated into the final solution using a singular perturbation method. The formulation of the analytical solutions is simple, and a complete solution is generated without resorting to computationally demanding numerical schemes. Indeed, the simple analytical solutions can be used as tools to verify the accuracy of numerical models of water flow and solute transport. Comparison with a finite-element numerical solution indicates that a good match for the predicted water content is achieved when the mesh grid is one-fourth the capillary length scale of the porous medium. However, when numerically solving the solute transport equation at this level of discretization, numerical dispersion and spatial oscillations were significant.     

Lattice Boltzmann Models for Flow and Transport in Saturated Karst

Flow and transport simulation in karst aquifers remains a significant challenge for the ground water modeling community. Darcy's law–based models cannot simulate the inertial flows characteristic of many karst aquifers. Eddies in these flows can strongly affect solute transport. The simple two-region conduit/matrix paradigm is inadequate for many purposes because it considers only a capacitance rather than a physical domain. Relatively new lattice Boltzmann methods (LBMs) are capable of solving inertial flows and associated solute transport in geometrically complex domains involving karst conduits and heterogeneous matrix rock. LBMs for flow and transport in heterogeneous porous media, which are needed to make the models applicable to large-scale problems, are still under development. Here we explore aspects of these future LBMs, present simple examples illustrating some of the processes that can be simulated, and compare the results with available analytical solutions. Simulations are contrived to mimic simple capacitance-based two-region models involving conduit (mobile) and matrix (immobile) regions and are compared against the analytical solution. There is a high correlation between LBM simulations and the analytical solution for two different mobile region fractions. In more realistic conduit/matrix simulation, the breakthrough curve showed classic features and the two-region model fit slightly better than the advection-dispersion equation (ADE). An LBM-based anisotropic dispersion solver is applied to simulate breakthrough curves from a heterogeneous porous medium, which fit the ADE solution. Finally, breakthrough from a karst-like system consisting of a conduit with inertial regime flow in a heterogeneous aquifer is compared with the advection-dispersion and two-region analytical solutions.     

Upstream Dispersion in Solute Transport Models: A Simple Evaluation and Reduction Methodology

This article outlines analytical solutions to quantify the length scale associated with “upstream dispersion,” the artificial movement of solutes in the opposite direction to groundwater flow, in solute transport models. Upstream dispersion is an unwanted artifact in common applications of the advection-dispersion equation (ADE) in problems involving groundwater flow in the direction of increasing solute concentrations. Simple formulae for estimating the one-dimensional distance of upstream dispersion are provided. These show that under idealized conditions (i.e., steady-state flow and transport, and a homogeneous aquifer), upstream dispersion may be a function of only longitudinal dispersivity. The scale of upstream dispersion in a selection of previously presented situations is approximated to highlight the utility of the presented formulae and the relevance of this ADE anomaly in common transport problems. Additionally, the analytical solution is applied in a hypothetical scenario to guide the modification of dispersion parameters to minimize upstream dispersion.     

Methods and Results of Numerical Modeling of Reactive Solute Transport in River Flow

The equation of advection–diffusion transport of a reactive solute in a stream is solved by different methods. Procedures for evaluating the parameters of a four-node finite-difference approximation used to numerically integrate this equation are proposed. Numerical experiments based on the developed numerical procedures, software, and models of solute transport allowed, in particular, the evaluation of the sensitivity of pollutant concentration estimates to variations in the parameters of interaction processes and hydrodynamic dispersion for a lowland river.     

Eolian Transport of Geogenic Hexavalent Chromium to Ground Water

A conceptual model of eolian transport is proposed to address the widely distributed, high concentrations of hexavalent chromium (Cr⁺⁶) observed in ground water in the Emirate of Abu Dhabi, United Arab Emirates. Concentrations (30 to more than 1000 μg/L Cr⁺⁶) extend over thousands of square kilometers of ground water systems. It is hypothesized that the Cr is derived from weathering of chromium-rich pyroxenes and olivines present in ophiolite sequence of the adjacent Oman (Hajar) Mountains. Cr⁺³ in the minerals is oxidized to Cr⁺⁶ by reduction of manganese and is subsequently sorbed on iron and manganese oxide coatings of particles. When the surfaces of these particles are abraded in this arid environment, they release fine, micrometer-sized, coated particles that are easily transported over large distances by wind and subsequently deposited on the surface. During ground water recharge events, the readily soluble Cr⁺⁶ is mobilized by rain water and transported by advective flow into the underlying aquifer. Chromium analyses of ground water, rain, dust, and surface (soil) deposits are consistent with this model, as are electron probe analyses of clasts derived from the eroding Oman ophiolite sequence. Ground water recharge flux is proposed to exercise some control over Cr⁺⁶ concentration in the aquifer.