Analytical Solutions for Reactive Transport of Multiple Volatile Contaminants in the Vadose Zone

Groundwater contamination usually originates from surface contamination. Contaminants then move downward through the vadose zone and finally reach the groundwater table. To date, however, analytical solutions of multi-species reactive transport are limited to transport only in the saturated zone. The motivation of this work is to utilize analytical solutions, which were previously derived for single-phase transport, to describe the reactive transport of multiple volatile contaminants in the unsaturated zone. A mathematical model is derived for describing transport with phase partitioning of sequentially reactive species in the vadose zone with constant flow velocity. Linear reaction kinetics and linear equilibrium partitioning between vapor, liquid, and solid phases are assumed in this model.     

Implementing random scan Gibbs samplers

Summary  The Gibbs sampler, being a popular routine amongst Markov chain Monte Carlo sampling methodologies, has revolutionized the application of Monte Carlo methods in statistical computing practice. The performance of the Gibbs sampler relies heavily on the choice of sweep strategy, that is, the means by which the components or blocks of the random vector X of interest are visited and updated. We develop an automated, adaptive algorithm for implementing the optimal sweep strategy as the Gibbs sampler traverses the sample space. The decision rules through which this strategy is chosen are based on convergence properties of the induced chain and precision of statistical inferences drawn from the generated Monte Carlo samples. As part of the development, we analytically derive closed form expressions for the decision criteria of interest and present computationally feasible implementations of the adaptive random scan Gibbs sampler via a Gaussian approximation to the target distribution. We illustrate the results and algorithms presented by using the adaptive random scan Gibbs sampler developed to sample multivariate Gaussian target distributions, and screening test and image data. Research by RL and ZY supported in part by a US National Science Foundation FRG grant 0139948 and a grant from Lawrence Livermore National Laboratory, Livermore, California, USA.     

On equilibrium and primary variables in transport in porous media

Thermodynamic equilibrium, which involves mechanical, thermal, and chemical equilibria, in a multiphase porous medium, is defined and discussed, both at the microscopic level, and at the macroscopic one. Conditions are given for equilibrium in the presence of forces between the surface of the solid matrix and the fluid phases. The concept ofapproximate thermodynamic equilibrium is introduced and discussed, employing the definition of athermodynamic potential. This discussion serves as the basis for the methodology of determining the number of degrees of freedom in models of phenomena of transport (of mass, energy, and momentum) in porous media. Equilibrium and nonequilibrium cases are considered. The proposed expressions for the number of degrees of freedom in macroscopic transport models, represent the equivalent ofGibbs phase rule in thermodynamics. Based on balance considerations and thermodynamic relationships, it is shown that the number of degrees of freedom, NF, in a problem of transport in a deformable porous medium, involving NP fluid phases and NC components, under nonisothermal conditions, with equilibrium among all phases and components, is $${\text{NF = NC + NP + 4}}{\text{.}}$$ Under nonequilibrium conditions among the phases, the rule takes the form $${\text{NF = NC }} \times {\text{ NP + 2NP + NC + 4}}{\text{.}}$$ In both cases, when fluid phase velocities are determined by Darcy's law, NF is reduced by NP. When the solid matrix is nondeformable, NF is reduced by 3. The number of degrees of freedom is also determined for conditions of approximate chemical and thermal equilibria, and for conditions of equilibrium that prevail only among some of the phases present in the system. Examples of particular cases are presented to illustrate the proposed methodology.