Diffusion limited mixing in heterogeneous porous media
Authors: Hamada Mayumi , de Anna Pietro 
Published: March 2021

&lt;p&gt;&lt;span&gt;&lt;span&gt;A pore-scale description of the transport and mixing processes is particularly relevant when looking at biological and chemical reactions. For instance, a microbial population growth is controlled by local concentrations of nutrients and oxygen, and chemical reaction are driven by molecular-scale concentration gradients. The heterogeneous flow field typically found in porous media results from the contrast of velocities that deforms and elongates the mixing fronts between solutes that often evolves through a lamella-like topology. For continuous Darcy type flow field a novel framework that describes the statistical distribution of concentration being transported was recently developed (Le Borgne et al., JFM 2015). In this model, concentrations in each lamella are distributed as a Gaussian-like profile which experiences diffusion in the transverse direction while the lamella is elongated by advection along the local flow direction. The evolving concentration field is described as the superposition of each lamella. We hypothesize that this novel view, while perfectly predicting the distribution of concentration for Darcy scale mixing processes, will breakdown when the processes description is at the pore scale. Indeed the presence of solid and impermeable boundaries prevents lamella concentration to diffuse freely according to the a Gaussian shape, and therefore changes the mixing front profile, the lamella superposition and elongation rules. P&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;revious work (Hamada et al, PRF, 2020) demonstrated that &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;the presence of solid boundaries leads to an enhanced diffusion and thus fast homogenization of concentrations. &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;In a purely diffusive process the local mixing time is reduced by a factor of ten with respect to the &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;continuous case and concentration gradient are dissipated exponentially fast while a &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;power law decrease &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;is &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;observed in continuous medium.&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; To investigate the impact of these mechanisms on mixing we developed a&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;n experimental set-up to visualize and quantify the displacement of a conservative tracer in a synthetic porous medium. The designed apparatus allows to obtain high resolution concentration measurement&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;s&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; at the pore scale. We show that the resulting mixing measures, computed in terms of concentration probability density function and dilution index values, diverge &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;qualitatively and quantitatively from what happens in a continuous domain. These observations suggest &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;that description of pore-scale diffusion-limited mixing requires model that takes into account the confined nature of porous medium, &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;otherwise we will tend to overestimate concentration value and neglect the fast diffusion dynamic taking place at microscopic level.&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;