Large scale flow and dispersion in heterogeneous karst aquifers under laminar and turbulent flow conditions

Non-linear flow and dispersion in fractured and karstic media are key issues in different fields of science and engineering, ranging from the assessment of groundwater vulnerability and flood risks to geothermal energy and speleogenesis. Spatial variability in the physical medium properties lead to scale effects in the flow and dispersion processes that manifest in non-Fickian transport and non-Darcian flow behaviors. We study the mechanisms of flow and dispersion in two- and three-dimensional heterogeneous networks. Flow is modeled by the Darcy-Weisbach equation, which for low Reynolds numbers describes laminar and for high Reynolds numbers turbulent flow conditions. Due to spatial heterogeneity, the Reynolds number and thus the flow conditions may strongly vary in  space. That is, laminar flow regions alternate with regions of dominantly turbulent flow. The aim is to understand and predict large scale flow and dispersion in such media by understanding their relation to medium geometry and heterogeneity. To this end, the flow fields are characterized statistically in terms of the distribution of Eulerian and Lagrangian flow velocities and their correlation properties with emphasis on the relation between network heterogeneity and flow statistics. We find that large scale flow can be characterized by a Darcy-Weisbach law in terms of a  large scale friction factor that depends on the medium heterogeneity. Solute dispersion is measured in terms of particle breakthrough curves and displacement statistics. We observe broad distributions of particle arrival times and non-linear evolution of the displacement variance, which are manifestations of memory processes that occur due to broadly distributed flow velocities and mass transfer rates. These behaviors are linked to the medium structure and Eulerian flow statistics. Based on this analysis, we propose a stochastic time domain random walk approach to quantify the impact of the network heterogeneity on large-scale flow and dispersion.