PhD postion at IFPEN : Up-scaling and averaging of flows in karstic networks

This project is part of an ERC funded Synergy, one of the laureates of an ERC synergy

grant. The overall goal consists of developing a predictive flow model in an entire karst net- work. It will be necessary to simulate water flows possibly marked with tracer in networks that may be described with millions of nodes. The flows will not necessarily be saturated, and nonlinear flow/Dp relationships between the inlet and outlet of the ducts lead to the resolution of a large system of nonlinear equations. We recall that 30 % of drinkable water flows through karstic aquifers that are very sensitive to global climate change. Ultimately, we will have to focus on large systems of equations of discretized Laplacian type, with a hollow character, and destructured in the majority of cases, since karst networks can be made of large conduits intersecting a large number of other poorly con-nected conduits. The weights related to the edges of the graph correspond to the hydraulic conductivity of the ducts, and are themselves random. It is therefore necessary to solve large linear systems of the Laplacian type, on weighted graphs with complex topology. The PhD student will be interested in the question of up-scaling on a discrete network, allowing to manage in particular the intrinsic hazard of this modeling chain, due to uncertainties on the values of conductivities and volumes of ducts. We will work with fixed network topology by focusing on the averaging on conductivities. The difficulty is to give meaning to homogenization when working in a discrete context where the underlying Euclidean metric is lost, making the notion of change of scale delicate. One idea will be to work on the spectra of Laplacian matrices generalizing the notion of Fourier transform to graphs, a technique very close to convolutional neural networks.

Supervision by a joint team of several physicists and geoscientists belonging to the ERC funded the karst team and work periods in the partners institutions will be organized (Barcelona, Ljubljana, Neûchatel)

Keywords: Karsts, climate change, floodings, drought, quantitative geosciences, statistical physics, percolation, coupling, applied mathematics, programming, numerical simulation