Analyzing the role of the permeability heterogeneous structure in developing vorticity in porous media

Abstract:

Modeling flow in porous media is relevant for many environmental, energy and industrial applications. From an environmental perspective, the relevance of porous media flow becomes evident in subsurface hydrology. In general, flow in natural porous media is at low Reynolds numbers, yet the large variability in the hydraulic conductivity values encountered in natural aquifers leads to highly heterogeneous flow fields. This natural variability in the conductivity field will affect both dilution rates of non reactive tracers and reactive mixing. A physical consequence of this heterogeneity is also the presence of localized kinematical features such as straining, shearing and vorticity in the fluid element. These kinematical features will influence the shape of solute clouds and its fate. While in porous media straining and shearing have been considered as important kinematical features controlling the shape of the plume and consequently mixing, much less attention has been devoted to vorticity, which can be defined as the anti-symmetric component of the velocity gradient tensor. In the present work we analyze how vorticity is related to large scale heterogeneity, as described by global parameters such as the log-conductivity variance and integral scale, and porous microstructure. Following a micro-mechanical approach, the medium is schematized as an ensemble of independent inclusions placed at random in a homogeneous matrix. Adopting this model permits to solve the flow problem by means of only few parameters, such as the conductivity contrast, the volume fraction and the shape and inclination of the inclusions, the latter two properties representing the microstructure. Through this conceptualization, it is possible to approximate the structure of natural heterogeneous porous media. We show that vorticity depends on both conductivity contrast, epitomized by the log-conductivity variance, and the microstructure. In addition, vorticity and lateral dispersion are related, as both increase with conductivity contrast. The latter is the critical parameter controlling the spreading rate. However this relationship is not univocal, since increasing anisotropy ratio leads to an opposite trend in vorticity and dispersion values.