Velocity Statistics of Flows in Porous Media

Abstract:

In later phases of transport in porous media, dispersion might be modelled by Fickian diffusion using an effective diffusion coefficient in the transport equation. Early phases of the transport process are however more difficult to deal with due to non-Fickian behavior. On the macro-scale, modelling of non-Fickian dispersion is complicated by scale dependence and therefore empirical correlations, experiments or numerical simulations on the micro-scale must be employed [1]. A fully resolved solution of the Navier-Stokes and transport equations which yields a detailed description of the flow properties, dispersion, interfaces of fluids, etc. however, is not practical for domains containing more than a few thousand grains, due to the huge computational effort that resolving such geometries would require. Through Probability Density Function (PDF) based methods, the velocity distribution in the pore space can facilitate the understanding and modelling of non-Fickian dispersion [2,3,4].
Our aim is to model the transition between non-Fickian and Fickian dispersion in a random sphere pack within the framework of a PDF based transport model proposed by Meyer and Tchelepi [5]. In addition to [5], we consider the effects of pore scale diffusion and formulate a different stochastic equation for the increments in velocity space from first principles. To assess the terms in this equation, we performed Direct Numerical Simulations (DNS) for solving the Navier-Stokes equation. We extracted the PDFs and statistical moments (up to the 4^th moment) of velocity and first and second order velocity derivatives both independent and conditioned on velocity. This data enables us to quantify the fluxes in the velocity space. We observe that the components of velocity fluxes derived through the combination of the Taylor expansion and the Langevin equation, point to a drift and diffusion behavior in the velocity space.