Gas transport in unsaturated zones: Multicomponent systems and the adequacy of Fick's laws

Abstract

A complete understanding of multicomponent gas transport in porous media (unsaturated zones) requires a knowledge of Knudsen transport, the molecular and nonequimolar components of diffusive flux, and viscous (pressure‐driven) flux. The constitutive equations relating these flux components are available from the “dusty gas” model of Mason et al. (1967). This paper presents a brief discussion of the principles underlying each of the above flux mechanisms, illustrated with binary systems, and then casts the constitutive equations in forms thought to be most useful for the study of natural unsaturated zones. A derivation is presented showing that the constitutive equations maintain the same form when expressed in terms of the potentiometric head of a gas column in a gravitational field, a conclusion of considerable practical importance for the study of natural systems. Very small pressure gradients (1 Pa/m or less) can produce viscous fluxes greater than or equal to diffusive fluxes; conversely, pressure gradients of this magnitude can be generated by diffusive processes. The viscous and diffusive fluxes are coupled in the constitutive equations through the Knudsen diffusivities; a knowledge of Knudsen diffusivities is necessary to calculate the viscous component of flux and pressure gradients. Equations are derived allowing calculation of Knudsen diffusivities from measurements of the Klinkenberg effect. The accuracy of Fick's first law (and by inference, Fick's second law) is shown to depend primarily on the relative magnitudes of the viscous and diffusive flux components. Methods are presented for approximating these flux components, in order to determine whether the multicomponent equations are needed for a given problem. These estimates depend on a knowledge of concentration profiles of stagnant (zero flux) gases. To a first approximation, Ar and N2 are considered to be stagnant gases in subsoil environments. Fick's law(s) are, essentially by definition, inadequate to deal with stagnant gases. In the examples presented, the error associated with estimating total fluxes of nonstagnant gases by Fick's law ranges from a few percent to orders of magnitude. Other conclusions are as follows: (1) major gas concentration gradients can be produced solely by transport phenomena; (2) any study of natural systems that requires a knowledge of N2 (or Ar) fluxes, as opposed to assuming them to be stagnant, must be based on a multicomponent analysis; (3) any study of systems in which a nonatmospheric gas constitutes a significant fraction of the gases present must be based on multicomponent analysis; (4) concentration profiles of stagnant gases can be used to determine the direction, and semiquantitatively the magnitude, of the net gas flux into or out of unsaturated zones; (5) with permeability equal to 10−12 m2, for pressure gradients greater than 10 N m−3 viscous fluxes predominate, the Stefan‐Maxwell equations become inapplicable, and the general equations must be used; (6) the net gas flux may be dominated by viscous effe...

Buoyant Advection of Gases in Unsaturated Soil

“…When the contaminant is not dilute in the gas mixture, Fick's law requires modification to account for bulk gas motion using the Stefan-Maxwell equations (Bird et al, page 570, 1960). Other diffusive processes have some importance in porous media, but for atmospheric pressures and gas-phase permeabilities above 0.1 darcy, inclusion of Knudsen diffusion and the dusty gas model are not necessary (Massmann 1989;Thorstenson and Pollock 1989;Baehr and Bruell 1990).…”

Pore-gas dynamics in overburden and reclamation soil covers

“…Moreover, diffusion is assumed to be at steady-state and doesn't consider temporal variations in gas production or consumption (Martin and Schack-Kirchner 2015). There are also limitations to the applicability of Fick's Law to for multi-component gas mixtures (Jaynes and Rogowski 1983;Leffelaar 1987;Thorstenson and Pollock 1989;Voudrias and Chiayang 1992). Fick's laws don't consider Knudsen diffusion, which occurs in finer-grained soils when molecules collide more with pore walls than with other molecules (Klinkenberg 1941).…”

“…Barometric pressure influences rates and direction of soil-atmosphere gas exchange (Buckingham 1905;Fukuda 1955;Jaynes and Rogowski 1983, Thorstenson and Pollock 1989, Nilson et al 1991Cepiel 1996;Poulsen 2003;Massman 2006;Redecker 2015). This phenomenon is denoted in the literature as "barometric pumping"; falling pressures induce gas flow from soils to atmosphere, whereas rising pressures suppress gas flow from atmosphere to soils (Young 1990;Massman 2006;).…”

Development of a Method to Estimate the Dispersion and Knudsen Diffusion Coefficients in Soil by Using the Dusty Gas Model

“…，Thorstenson and Pollock 11) )．ただし， Massmann and Farrier 12) によると，Knudsen拡散係数が物質 移動に大きく影響する透過度の範囲は10 …”