Estimation of uncertainties in the density driven flow in fractured porous media using MLMC

Abstract We use the Multi Level Monte Carlo method to estimate uncertainties in a Henry-like salt water intrusion problem with a fracture. The flow is induced by the variation of the density of the fluid phase, which depends on the mass fraction of salt. While the fracture’s location is fixed, its aperture is uncertain. In our setting, porosity and permeability vary spatially and recharge is time-dependent. So we introduce three random variables, one controlling both the porosity and permeability fields, one for the fracture width and one for the intensity of recharge. For each realization of these uncertain parameters, the evolution of mass fraction and pressure fields is modeled using a system of non-linear, time-dependent PDEs with a solution discontinuity at the fracture. These uncertainties propagate, affecting the distribution of salt concentration, a key factor in water resource quality. We show that the MLMC method can be successfully applied to this problem. It significantly reduces the computational cost compared to classical Monte Carlo methods by effectively balancing discretisation and statistical errors, and by evaluating multiple scenarios over different spatial and temporal mesh levels. The deterministic PDE solver, using the ug4 library, runs in parallel to compute all stochastic scenarios.