Principles of early development of karst conduits under natural and man-made conditions revealed by mathematical analysis of numerical models

Numerical models of the enlargement of primary fissures in limestone by calcite aggressive water show a complex behavior. If the lengths of the fractures are large and hydraulic heads are low, as is the case in nature, dissolution rates at the exit of the channel determine its development by causing a slow increase of water flow, which after a long gestation time by positive feedback accelerates dramatically within a short time span. Mathematical analysis of simplified approximations yields an analytical expression for the breakthrough time, when this happens, in excellent agreement with the results of a numerical model. This expression quantifies the geometrical, hydraulic, and chemical parameters determining such karat processes. If the lengths of the enlarging channels are small, but hydraulic heads are high, as is the case for artificial hydraulic structures such as darns, it is the widening at the entrance of the flow path which determines the enlargement of the conduit. Within the lifetime of the dam this can cause serious water losses, This can also be explained by mathematical analysis of simplified approximations which yield an analytical threshold condition from which the safety of a dam can be judged. Thus in both cases the dynamic processes of karstification are revealed to gain a deeper understanding of the early development of karst systems. As a further important result, one finds that minimum conditions, below which karstification cannot develop, do not exist.