Spelunkers look at a cave and wonder how to explore its deepest reaches. But physicists look at it and wonder how it got there in the first place.

A new mathematical analysis solves a longstanding cave-formation puzzle: how a trickle of water laced with carbonic acid manages to quickly dissolve rock to create massive conduits. The trick, it seems, is that fluid flow focuses rapidly in certain channels, which grow at the expense of others and allow the acid to penetrate deeply.

“Most of the models in cave formation today don’t have this mechanism at all,” says Piotr Szymczak, a physicist at the University of Warsaw. He and his colleague Anthony Ladd, a chemical engineer at the University of Florida in Gainesville, lay out their new equations in a paper to appear in Earth and Planetary Science Letters.

The work could improve understanding of the safety of dams, waste storage sites, or anywhere else fluid might be seeping through the ground.

For more than a century, researchers have known the basics of how limestone caves form: A tiny fracture opens in the rock, perhaps due to some internal stress, and water begins percolating through it. Most water contains some carbon dioxide, making it a weak acid that can eat away at the calcium carbonate in limestone. The question is how that dissolution can happen fast enough to produce deep penetration and allow long cave systems to form. The longest known system in the world is Mammoth Cave in Kentucky, with at least 580 kilometers of passageways.

Earlier work had suggested that the rate at which rock dissolved could slow down dramatically when the fluid is nearly saturated with carbon dioxide, allowing more solution to reach deep inside the fracture. But the new study can explain cave formation without invoking such a mechanism, says Szymczak.

The researchers showed how the equations describing fluid flow in the rock always contain a mathematical instability. The fact that this instability exists means that very soon after a fracture opens, fluid flow begins to focus along tiny ripples and build some bigger channels at the expense of others. “This mechanism of channeling speeds your dissolution time quite a lot,” says Szymczak. “That’s what allows it to penetrate so deep.”

The mathematical analysis is likely to bring new insight into ideas that have been circulating since the 1990s, when the notion of focused flow in limestone was first proposed, says Harihar Rajaram, a hydrologic engineer at the University of Colorado at Boulder. The new work, says Szymczak, builds on that foundation by showing that the instability always exists in the math, no matter what materials are involved.

The work, he adds, could help explain why caves sometimes form faster than expected beneath dams. The equations might also help improve modeling of how fluid seeps through rocks, a ke question raised about the once-planned nuclear waste repository at Yucca Mountain, Nevada.

The researchers next want to look into what happens when engineers inject liquid carbon dioxide from power plants deep underground, in an attempt to keep the carbon from entering the atmosphere.

Image: A simulation shows how fluid flow (in this case, from bottom to top) can lead to deep penetration in a rock and, eventually, a cave. At first (lower panel) channels begin developing almost equally, but over time (upper panel) certain channels gain in size at the expense of others. Credit: Piotr Szymczak and Tony Ladd

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