In the following, we describe the work performed and main achievements during the first 18 months. The work performed evolves overall according to the WP time distribution indicated in Annex 1.
WP1: Physical laws at the conduit scale
Individual conduit geometry: The KARST team at UNINE has acquired a mobile LIDAR (Laser Imaging Detection and Ranging) instrument for the 3D scanning of karst conduits (caves). So far, a total of 17 conduits have been scanned in Slovenia, Switzerland, and France. The obtained point clouds are used to generate geographic meshes and to define geometrical descriptors in interaction with the teams at UL and IFPEN. The team at UNINE has generated synthetic rough conduits using a representation of the wall geometry by multi-Gaussian random fields.
Flow and transport experiments: The laboratory setup at IFPEN is ready and has been tested for flow in a smooth cylindrical conduit. Also, the particle imaging velocimetry (PIV) has been tested and is ready. The INRIA team has set up a laboratory and designed a workflow for the fabrication of scaled and transparent conduits (for PIV) and has manufactured inlet and test sections for the laboratory setup. The team has selected a set of representative conduits for 3D printing.
Flow and transport simulations: The CSIC team has set up three flow simulators for CFD simulations. The simulators are based on OpenFoam and NEK5000, for which an immersed boundary method was implemented to represent the complex conduit wall geometries. The simulators were validated for smooth conduits. Also, a particle tracking algorithm was implemented and validated.
Flow and transport laws at conduit scale: Using CFD simulations, the CSIC team has determined large scale descriptors for flow upscaling in terms of the Darcy friction factor for complex geometries and wavy conduits. The UNINE and CSIC teams have derived a methodology for the upscaling of laminar and turbulent flow in complex conduit geometries.
WP2: Karst network structure
Data collection and network geometry: The UNINE team has collected so far 206 data sets on cave network geometries from around the world from speleology clubs and public data bases. The team has established a workflow for data processing and extraction of network attributes. A data base structure and data base was established in the git repository of KARST, which is one of the most coherent data sets on cave geometries worldwide currently available.
Karst network characterization and analysis: For the characterization and analysis of graphs underlying karst networks, the UL team has explored possible mechanisms of speleogenesis. Also in this context, the UL and SFU teams have studied graphs on surfaces and flows in relation to spectral properties of discrete Laplacian operators, geometric representations and geometric optimization problems, nowhere-zero flows of graphs on surfaces, and nowhere-zero flows of graphs on surfaces.
Simulating cave networks: The UNINE team has been developing a graph simulator using neural networks that allows to generate an ensemble of graphs starting from a real mapped karst system preserving basic graph statistics. A workflow was developed that consists in extracting the data set from real world data, training and testing the neural network using the graph adjacency matrix, and subsequently training and testing a denoising diffusion probabilistic model (DDPM). The generators are available to the KARST team through the git repository.
WP3: Physical laws at the network scale
Network scale numerical flow and transport simulator: The CSIC and IFPEN teams have implemented network scale simulators (openkarst) for flow under pressurized and open-channel flow conditions based on the Darcy-Weisbach and Saint-Venant equations. The codes operate on arbitrary network topologies and heterogeneity in hydraulic and geometric properties. Transport is implemented based on a time-domain random walk method. The openkarst code is available to the KARST team through the git repository.
Flow upscaling: Based on the network simulators for pressurized flow, the IFPEN and CSIC teams have studied flow upscaling on two and three-dimensional spatial random networks characterized by a complex geometry (close to the percolation threshold) or by broad statistical distributions of conduit apertures on maze-like networks. The network effective conductivity (laminar flow) and effective friction factor (turbulent flow) was studied as well as the relation between heterogeneity and flow distribution.
Transport upscaling: Based on the detailed network scale flow and transport simulations, the CSIC team has derived a stochastic model for the prediction of network scale solute transport. The model is parameterized in terms of the network flow tortuosity and statistical distribution of flow rates.
