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\title{From viscous fingers to wormholes - interactions between emergent fingers in unstable growth}

\authors{
\underline{A. Budek}$^{1,2}$, K. Kwiatkowski $^2$ \& P. Szymczak$^2$
}

\affiliations{
$^1$Institute of Geophysics, Polish Academy of Sciences, Poland\\
$^2$Institute of Theoretical Physics, University of Warsaw, Poland\\
}

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Dissolution of porous and fractured rock can lead to
instabilities, where long finger-like channels or ``wormholes'' are
spontaneously formed, focusing the majority of the flow. Formation
of those structures leads to a significant increase in
permeability of the system, and is thus important in many engineering
applications, e.g. in acidization during oil and gas recovery
stimulation. In this communication, we analyse this process using two
different numerical models (a network model and a Darcy scale one). We
show that wormhole patterns depend strongly on the amount of
soluble material in the system, as quantified by the permeability
contrast $\kappa$ between the dissolved and undissolved medium. For
small and intermediate values of $\kappa$, a large number of
relatively thin and strongly interacting channels are formed.
The longer channels attract shorter ones, with loops being formed as a result.
However, for large values of $\kappa$
the pattern gets sparse with individual wormholes repelling each
other.

Interestingly, a similar succession of  patterns can be observed in
viscous fingering simulations, with fingers growing in a rectangular
network of channels. In such a system, anisotropy of the network
promotes the growth of long and thin fingers which behave similarly to
wormholes. The attraction rate between growing fingers depends
strongly on the viscosity ratio and the distance between the fingers.
To explain this behaviour, we have created a simple analytical model
of interacting fingers, allowing us to quantify their mutual
interaction as a function of finger lengths, distances between them
and -- most importantly -- relative permeabilities. The theoretical
predictions are in a good agreement with simulation data for both
dissolution and viscous fingering processes.

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