A Three-dimensional Statistical Model of Karst Flow Conduits

Abstract

It already exists several three-dimensional models dealing with groundwater circulation in karst systems. However, few of them are able either to give a large scale prediction of the repartition of the flow conduits or to make a comparison with real field data. Therefore, our objective is to develop a three-dimensional model about the early formation of karst flow conduits and to compare it with actual field data. This geometric and statistical model is based on percolation and random walks. It is computational and can be run on a personal computer. We examine the influence of fissures (joints and bedding planes) of variable permeability and orientations on the development or early flow conduits. The results presented here correspond to computations up to 2015. Because of long runtimes, we focused on some particular stereotypical situations, corresponding to some particular values of the parameters. Regarding the conduit patterns, the opening and directions of fissures have the same qualitative influence in the model than in actual systems. Two other predictions in good accordance with real karst are that flow conduits can either develop close to the water table or deeper, depending on the distribution of permeable fissures; and that, when viewed in the horizontal plane, conduits don't always develop close to the straight line between inlet and outlet. From a quantitative point of view, in the case of weak dips, our model predicts a realistic relationship between the stratal dip, the length of the system and the averaged depth of the conduits. Eventually, we show that the repartition of conduits depends not only on obvious geometrical parameters such as directions and sizes, but also also on other quantities difficult to measure such as the probability of finding open fissures. The lack of such data doesn't enable, at the present time, a whole comparison between model and reality.