Wormholing in anisotropic media: Pore orientation effect on large-scale patterns

The dissolution of fractured or porous media by reactive flow is often occurring preferentially, forming highly conductive channels, so-called “wormholes”. Wormhole formation prevails in subsurface karst where it can form extensive speleological systems, and is also significant for a large range of applications, e.g. well acidizing or CO₂ geo-sequestration. The underlying mechanism involves positive feedback between reaction and transport— the flow pathways that focus the reactive flow dissolve preferentially and increase their conductivity, and in turn their flow. An increased pressure ahead of the longer wormholes screens off the shorter ones, which ultimately cease to grow. Over time, the characteristic spacing between active (growing) wormholes increases, while their number declines, which results in a hierarchical scale-invariant distribution of wormhole lengths. Interestingly, a variety of other pattern-forming processes in nature show a similar competitive dynamics and emergent of hierarchical structures, with examples ranging from viscous fingering to crack propagation in brittle solids and side-branches growth in crystallization [1].

The importance of wormholing and its intriguing dynamics motivated intensive research, including on the emergence of reactive-infiltration instabilities [2], as well as on the effects of medium heterogeneity on the wormhole growth. Here, we study wormholing in anisotropic media using a network model of regular geometry— longitudinal channels (aligned along the main flow direction) and transverse ones, of a different average cross-section. Our simulations show that anisotropy substantially affects wormholing, controlling the characteristic spacing between the wormholes and the overall permeability evolution. In the case of wider transverse channels, wormhole interaction via the pressure field is enhanced, resulting in stronger wormhole competition and hence larger spacing. Conversely, in the extreme case of very narrow transverse channels, spacing becomes minimal and neighboring wormholes tend to merge. Simulations further reveal that narrow transverse channels promote the emergence of thinner and more conical wormholes with several side-branches.

Additionally, we discuss the relation between the wormhole development in an anisotropic medium and viscous fingering phenomena in a network of microfluidic channels [3]. Despite many similarities between these systems we also find important differences— while the spacing between viscous fingers increases linearly with anisotropy, the corresponding relation for wormholes turns out to be nonlinear. This nonlinearity could be attributed to the effect of anisotropy on wormhole shape and advancement velocity and is of interest for future investigation. Our findings contribute to the understanding of wormholing in geological systems and demonstrate how the small-scale features can fundamentally affect the resulting large-scale morphologies.

[1] Krug, J., Adv. Phys., 46, 139, 1997

[2] Ortoleva, P. et al, Amer. J. Sci., 287, 1008, 1987

[3] Budek, A. et al, Phys. Fluids, 27, 112109, 2015