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To model the development of karst aquifers from primary fissures in limestone rock, a numerical model of solutional widening of such fractures by calcite agressive water is suggested. The geological setting determines relevant geometrical parameters, i. e. , length of the fracture, its initial width and the hydraulic gradient driving water from the input to the output. To simulate the solutional widening as it proceeds in time, the solution rates must be known as a function of concentration c of dissolved calcite. They are given as a first-order kinetic rate law for c k c₄ₐ as F^ (1) = ₀ (c - c₄ₐ. For c > k c₄ₐ close to equilibrium, a fourth-order rate law becomes dominant: F^ (4) = (c - c₄ₐ) ^4. The parameters ₀, , and k depend on the chemistry of the carbonate system and rock chemistry. The saturation concentration, c₄ₐ, with respect to calcite depends on the initial CO₂ concentration of the inflowing solution, thus reflecting the influence of climate. The value of k for natural limestone ranges between 0. 5 and 0. 9. The results of the model show that the concerted action of both fast first-order kinetics and slow fourth-order kinetics is necessary to create early karst channels of several cm width in geologically reasonable times. At the beginning of the process the solutional widening occurs by slow fourth-order kinetics along the entire length of the fracture, because water reaches sufficiently high concentrations less than 1 m from the input. This widens the fissure slowly. As a consequence, increasing water flow rate increases the distance where first-order kinetics are operative. This region propagates through the length of the fracture with accelerating speed until it reaches the outlet. At the moment of breakthrough the flow rate increases dramatically by several orders of magnitude. From then on, further solutional widening is essentially constant along the entire length of the fracture, on the order of about 10^-2cm/yr. The breakthrough time is a significant measure in studies of karstification and has been calculated for a wide range of geometrical parameters. Expressions for its dependence on length, initial widths, and hydraulic gradient are derived, which allow ready calculation of breakthrough times for most realistic cases. Dependence of breakthrough times on chemical parameters reflects influences of lithology and climate. The lithologic parameters ₀, , , and k show only a small influence, which explains why karst features exist in so many different types of limestone. The dependence on c₄ₐ shows that karstification proceeds only at sufficiently high levels of CO₂ in the water entering the system. The model is applied to estimate evolution times of real karst aquifers in Germany, yielding results in agreement with geological observations. Depending on the geometrical parameters, especially length and hydraulic gradient, breakthrough times range from 10^4 to several 10^6 yr. The model is also applied to small-scale karst features, such as the evolution of solution dolines. In these cases fracture lengths are in the order of several tens of meters, and hydraulic gradients are high. This is also the case in land use projects where the groundwater table is lowered or artificial canals or dams are constructed. Karstification times of several tens to several hundred years are derived for these cases.
Wolfgang Dreybrodt (Sat,) studied this question.