1. Introduction

When fluids move through porous media, the detailed flow field is complex owing to the intricate topology of the pore spaces between the solid matrix and the no slip condition on the matrix surface. This leads to a wide range of travel times of fluid particles as they follow different streamlines from one plane to a second parallel plane downstream. This process forms a progressively more convoluted interface when one fluid displaces a second miscible fluid through the porous layer. When the effects of molecular diffusion are included, the interaction with the mechanical dispersion leads to a gradually growing zone across which the average concentration of fluid varies from the upstream to the downstream value.

Experiments and numerical simulations suggest that the mixing depends on the Peclet number of the flow, $Pe = U \delta /D$ , defined in terms of the mean Darcy speed $U$ , the molecular diffusivity $D$ and the typical length scale of the flow $\delta$ , often taken to scale with the mean pore size. The Peclet number can be understood as the ratio of the molecular diffusion time to the advection time across the pore-spaces. High $Pe$ corresponds to relatively fast flow, and it has been argued that the effective dispersion coefficient scales linearly with $Pe$ , whereas for smaller $Pe$ the dispersion coefficient follows a more complex set of power law scalings (Saffman Reference Saffman1959; Koch & Brady Reference Koch and Brady1988).

In the paper by Liu et al. (Reference Liu, Xiao, Aquino, Dentz and Wang2025), an argument is made that both the fluctuations in the fluid speed within individual pore spaces and also the fluctuations in the flow speed between different pore spaces, of different size, may both play a role in determining the dispersive mixing, and the authors propose that a continuous time random walk can be used to model these processes (cf. Berkowitz, Scher & Silliman Reference Berkowitz, Scher and Silliman2000). In quantifying this dispersion, they build on the idealised model that porous media consists of a series of pore throats with a range of radii and lengths, which connect at junctions. The fluid particles arriving at a junction from one pore throat may separate into different pore throats as they move downstream.

Within a single pore throat, the low-Reynolds-number flow develops a Pouseille type of shear flow, with a characteristic mean travel time across the pore, as well as a travel time distribution for the fluid particles on different streamlines. With low flow rates, and hence low $Pe$ , the fluid concentration diffuses so as to be uniform at each point along the pore throat. The travel time for the upstream fluid parcels and any associated tracer to migrate along a pore throat therefore depends inversely on the average speed along the pore throat, unless the flow is so slow that diffusion along the pore throat dominates. At higher flow rates, and hence higher $Pe$ , the travel time distribution associated with the Pouseille flow controls the advance of the upstream flow along the pore throat. By combining the travel time distribution for the pore throats of a given size and hence $Pe$ , with the distribution of pore throat sizes, the rate of dispersion of the fluid–fluid front can be calculated. If there is a wide range of pore throat sizes, then both the advective and diffusive limits may apply in the larger and smaller pore throats, respectively, whereas with a smaller range of sizes, one or other $Pe$ number limit typically pertains. Liu et al. (Reference Liu, Xiao, Aquino, Dentz and Wang2025) have explored the predictions of this model with an idealised series of distributions for the pore throat sizes, and they obtain results for the dispersion in good accordance with published experimental and numerical results. As well as providing an elegant idealised model which is able to characterise these asymptotic dispersion regimes, and which suggests that at larger $Pe$ the dispersion scales as $D_m Pe \ln (Pe)$ , the model presented by the authors promises to uncover the transient adjustment to these regimes.

Figure 1. (a) Photograph of the Bridport Sandstone, a typical porous rock, with multiple layers. (b) Image of a line of tracer (brown) moving from left to right, from a layer of uniform permeability, through a cross-bedded layer, and back to a layer of uniform permeability, illustrating the distortion of the flow in passing between the different regions of rock, and the generation of shear (Bhamidipati & Woods Reference Bhamidipati and Woods2020). (c) False colour image of the growth of buoyancy-driven fingers as a dense fluid migrates downwards through a bead pack, displacing a less-dense fluid (image courtesy of N. Mingotti). (d) Image of a bead-pack experiment, including a lens of high permeability. A band of red dye migrates along the pack, then rapidly moves through the lens and back into the pack, leading to large distortion of the band of dye (Woods Reference Woods2015).