Uncertainty quantification in Discrete Fracture Network models: Stochastic fracture transmissivity

We consider flows in fractured media, described by Discrete Fracture Network (DFN) models. We perform an Uncertainty Quantification analysis, assuming the fractures’ transmissivity coefficients to be random variables. Two probability distributions (log-uniform and log-normal) are used within different laws that express the coefficients in terms of a family of independent stochastic variables; truncated Karhunen-Loève expansions provide instances of such laws. The approximate computation of quantities of interest, such as mean value and variance for outgoing fluxes, is based on a stochastic collocation approach that uses suitable sparse grids in the range of the stochastic variables (whose number defines the stochastic dimension of the problem). This produces a non-intrusive computational method, in which the DFN flow solver is applied as a black-box. A very fast error decay, related to the analytical dependence of the observed quantities upon the stochastic variables, is obtained in the low dimensional cases using isotropic sparse grids; comparisons with Monte Carlo results show a clear gain in efficiency for the proposed method. For increasing dimensions attained via successive truncations of KarhunenLoève expansions, results are still good although the rates of convergence are progressively reduced. Resorting to suitably tuned anisotropic grids is an effective way to contrast such curse of dimensionality: in the explored range of dimensions, the resulting convergence histories are nearly independent of the dimension.