The use of Fick's law for modeling trace gas diffusion in porous media

Two models for combined gas-phase diffusion and advection in porous media, the advective-diffusive model (ADM) and the dusty-gas model (DGM), are commonly used. The ADM is based on a simple linear addition of advection calculated by Darcy's law and ordinary diffusion using Fick's law with a porosity-tortuosity-gas saturation multiplier to account for the porous medium. The DGM applies the kinetic theory of gases to the gaseous components and the porous media (or 'dust') to develop an approach for combined transport due to diffusion and advection that includes porous medium effect. The ADM and Fick's law are considered to be generally inferior for gas diffusion in porous media, and the more mechanistic DGM is preferred. Under trace gas diffusion conditions, Fick's law overpredicts the gas diffusion flux compared to the DGM. The difference between the two models increases as the permeability decreases. In addition, the difference decreases as the pressure increases. At atmospheric pressure, the differences are minor (<10%) for permeabilities down to about 10(-13) m(2). However, for lower permeabilities, the differences are significant and can approach two orders of magnitude at a permeability of 10(-18) m(2). In contrast, at a pressure of 100 atm, the maximum difference for a permeability of 10(-8) m(2) is only about a factor of 2. A molecule-wall tortuosity coefficient based on the DGM is proposed for trace gas diffusion using Fick's law. Comparison of the Knudsen diffusion fluxes has also been conducted. For trace gases heavier than the bulk gas, the ADM mass flux is higher than the DGM. Conversely, for trace gases lighter than the bulk gas, the ADM mass flux is lower than the DGM. Similar to the ordinary diffusion variation, the differences increase as the permeability decreases, and get smaller as the pressure increases. At atmospheric pressure, the differences are small for higher permeabilities (>10(-13) m(2)) but may increase to about 2.7 for He at lower permeabilities of about 10(-18) m(2). A modified Klinkenberg factor is suggested to account for differences in the models.