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 effects even when individual gas fluxes are primarily diffusive; and (7) the measurement of pressure differentials necessary for a definitive check on the validity of these models in most natural systems is at (or beyond) the limits of present technology.