Karst Conduit Breakthrough

Conduit Parameters

Initial fracture aperture 0.40 mm

Restarts the run — a typical joint in fresh limestone.

Soil CO₂ (water aggressiveness) 2.0%

Hydraulic gradient 1.5%

Time acceleration 500 yr/s

Live Readouts

Aperture width

0.40 mm

Elapsed time

0 yr

Flow regime

Laminar

Outlet saturation

— %

How it works

Groundwater charged with soil CO₂ dissolves limestone along a narrow fracture: CaCO₃ + CO₂ + H₂O → Ca²⁺ + 2HCO₃⁻. Flow through the fracture follows the parallel-plate (cubic) law:

v(w) = ρ·g·i·w² / (12·μ)
Re    = ρ·v·2w / μ   (turbulent above ≈2000)

Reaction kinetics follow the Plummer–Wigley–Parkhurst law with a "kinetic trigger": dissolution is roughly linear in the undersaturation (1 − c) far from equilibrium, but collapses to a much steeper 4th-order law once the water passes 90% saturation (c = C/C_eq):

dc/dx = (κ(w)/v)·f(c)
κ(w)  = 2·k₁ / (w·C_eq)
f(c)  = (1 − c)                 for c < 0.9
f(c)  = 0.1·((1 − c)/0.1)⁴      for c ≥ 0.9

The width itself grows from the CaCO₃ mass the outflow actually carries away:

dw/dt = v·w·C_eq·c_out / (ρ_rock·L)

The feedback: in a narrow fracture, flow is slow and the huge wall-area-to-volume ratio saturates the water within millimetres — dissolution nearly stalls (transport-limited). As w creeps up, v grows as w² while the reaction has less wall area per unit volume to work with, so the outlet water arrives ever less saturated. That extra undersaturation dissolves rock faster, which widens the fracture further — a runaway loop. The result is the real "breakthrough" behavior of speleogenesis (Palmer 1991; Dreybrodt 1996): a slow, near-imperceptible incubation lasting most of a conduit's history, then a comparatively sudden widening into a true cave passage once flow crosses the turbulence threshold.

Initial aperture — smaller starting cracks take far longer (or never, within reach of the sliders) to break through.
CO₂ aggressiveness — raises the equilibrium Ca²⁺ ceiling (C_eq), so the water can carry away more rock before saturating.
Hydraulic gradient — the head drop per metre driving flow through the fracture.

Constants are tuned so the incubation-then-breakthrough shape plays out on an interactive timescale; the governing equations and the qualitative behavior they produce are the real ones used in karst hydrogeology.

Transport-limited

Undersaturated water Near-saturated water

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