Miscible transfer of solute in different types of rough fractures: from random to multiscale fracture walls heights

Miscible tracer dispersion measurements in transparent model fractures with different types of wall roughness are reported. The nature (Fickian or not) of dispersion is determined by studying variations of the mixing front as a function of the traveled distance but also as a function of the lateral scale over which the tracer concentration is averaged. The dominant convective dispersion mechanisms (velocity profile in the gap, velocity variations in the fracture plane) are established by comparing measurements using Newtonian and shear thinning fluids. For small monodisperse rugosities, front spreading is diffusive with a dominant geometrical dispersion (dispersion coefficient $D \propto Pe$) at low P'eclet numbers $Pe$; at higher $Pe$ values one has either $D \propto Pe^2$ ({\it i.e.} Taylor dispersion) for obstacles of height smaller than the gap or $D \propto Pe^{1.35}$ for obstacles bridging the gap. For a self affine multiscale roughness like in actual rocks and a relative shear displacement $\vec{\delta}$ of complementary walls, the aperture field is channelized in the direction perpendicular to $\delta$. For a mean velocity $\vec{U}$ parallel to the channels, the global front geometry reflects the velocity contrast between them and is predicted from the aperture field. For $\vec{U}$ perpendicular to the channels, global front spreading is much reduced. Local spreading of the front thickness remains mostly controlled by Taylor dispersion except in the case of a very strong channelization parallel to $\vec U$.

💡 Research Summary

This paper presents a systematic experimental investigation of miscible tracer dispersion in transparent model fractures whose walls exhibit a range of roughness characteristics, from simple monodisperse protrusions to multiscale self‑affine topographies, and that may be laterally displaced relative to each other. The authors use both Newtonian (water) and shear‑thinning polymer solutions to explore how fluid rheology interacts with geometric heterogeneity. Tracer concentration fields are recorded with high‑speed imaging, and the evolution of the mixing front is quantified both in terms of its mean position and its thickness (standard deviation) as functions of travel distance, Péclet number (Pe), and the lateral averaging scale L. By analysing the scaling of the front thickness with Pe and L, the study distinguishes between Fickian (asymptotically diffusive) and non‑Fickian dispersion regimes and identifies the dominant physical mechanisms responsible for each regime.

Key findings can be grouped into three thematic areas.

1. Geometric versus Taylor‑type dispersion – For fractures populated with small, monodisperse obstacles, the dispersion coefficient D scales linearly with Pe (D ∝ Pe) at low Pe, indicating a “geometric” dispersion regime in which velocity fluctuations induced by the roughness dominate. As Pe increases, two distinct behaviors emerge depending on obstacle height relative to the fracture aperture. When obstacles are shorter than the aperture, the classic Taylor‑dispersion law (D ∝ Pe²) is recovered, reflecting the role of the parabolic velocity profile across the gap. When obstacles bridge the gap, the authors observe an anomalous scaling D ∝ Pe¹·³⁵, which they attribute to a complex interplay of channeling, stagnation zones, and shear‑induced mixing that cannot be captured by simple Taylor or geometric models.

2. Effect of fluid rheology – Shear‑thinning fluids display a muted transition between the geometric and Taylor regimes. Because the effective viscosity drops in high‑shear regions, velocity gradients across the gap are reduced, leading to smaller dispersion coefficients at a given Pe compared with Newtonian water. Consequently, the D‑Pe curve for the polymer solution is smoother, and the anomalous D ∝ Pe¹·³⁵ regime is less pronounced. This demonstrates that fluid rheology must be accounted for when extrapolating laboratory dispersion data to field conditions where non‑Newtonian fluids (e.g., polymer‑enhanced water floods) are common.

3. Channelization induced by lateral shear displacement – When the two complementary rough walls are laterally shifted by a vector δ, the aperture field becomes strongly anisotropic: channels form perpendicular to δ, producing a highly heterogeneous velocity field. The authors examine two flow orientations. With the mean flow U parallel to the channels, the global front geometry mirrors the distribution of channel apertures; the front width grows proportionally to the standard deviation of the channel‑averaged velocities, and this relationship can be predicted directly from the measured aperture map. When U is orthogonal to the channels, the front remains comparatively flat, and overall spreading is dramatically reduced, although the local front thickness continues to be governed by Taylor dispersion except in cases of extreme channelization.

A further important observation is the dependence of measured dispersion on the lateral averaging scale L. At small L (on the order of a few millimetres), the front exhibits non‑Fickian behavior, reflecting unresolved velocity fluctuations. As L increases, the dispersion converges toward a Fickian limit, confirming that the apparent non‑Fickian signatures are largely a scale‑effect rather than a fundamental breakdown of diffusive transport.

The authors conclude that realistic fracture transport models must incorporate (i) the multiscale nature of wall roughness, (ii) possible shear‑induced channelization, and (iii) the rheological properties of the flowing fluid. Simple one‑dimensional advection‑dispersion equations with constant D are insufficient for predicting solute migration in fractured rock, especially when the flow direction aligns with channelized apertures. The experimental scaling laws and the aperture‑based front‑prediction methodology presented here provide a practical framework for upscaling laboratory observations to field‑scale predictions in hydrogeology, enhanced oil recovery, and contaminant transport applications.