Abstract
Heterogeneous porous media are found in many engineering and natural processes, either by design to improve the efficiency of the system, or as a consequence of the process itself. Here, we focus on dispersive transport in porous media displaying spatially evolving heterogeneities, characterized by continuous spatial variations of their properties. Using upscaling, we derive the macroscopic transport equations for momentum and species dispersion and identify additional terms implying spatial derivatives of the porosity. While these terms do not influence the definition of the effective diffusion-dispersion tensor, we find that they remain in the closure problems for momentum transport and lead to the definition of two new effective permeability tensors. Length-scale considerations show that the closure problems for momentum transport can be simplified to facilitate their solving. To assess the validity of the derived model, we solve the macroscopic transport equations in a stationary dispersive mixing process: a Y -junction mixing chamber filled with porous media with spatially evolving heterogeneities. The consequences of distribution and strength of the porosity gradients on fluid velocity and concentration field are systematically compared to direct numerical simulations for various Péclet numbers, showing excellent agreement even in disordered porous media. Notably, we show that Péclet-dependent non-symmetric mixing layers can be produced using porous media with controlled porosity gradients. Our results highlight the potential for the development of novel industrial processes utilizing porous media with spatially evolving heterogeneities such as continuous flow chemistry.