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Fractures are normally present in the underground and are, for some physicalprocesses, of paramount importance. Their accurate description is fundamentalto obtain reliable numerical outcomes useful, e.g., for energy management.Depending on the physical and geometrical properties of the fractures, fluidflow can behave differently, going from a slow Darcian regime to morecomplicated Brinkman or even Forchheimer regimes for high velocity. The mainproblem is to determine where in the fractures one regime is more adequate thanothers. In order to determine these low-speed and high-speed regions, this workproposes an adaptive strategy which is based on selecting the appropriateconstitutive law linking velocity and pressure according to a thresholdcriterion on the magnitude of the fluid velocity itself. Both theoretical andnumerical aspects are considered and investigated, showing the potentiality ofthe proposed approach. From the analytical viewpoint, we show existence of weaksolutions to such model under reasonable hypotheses on the constitutive laws.To this end, we use a variational approach identifying solutions withminimizers of an underlying energy functional. From the numerical viewpoint, wepropose a one-dimensional algorithm which tracks the interface between the low-and high-speed regions. By running numerical experiments using this algorithm,we illustrate some interesting behaviors of our adaptive model on a singlefracture and small networks of intersecting fractures.

Model adaptation in a discrete fracture network: existence of solutions and numerical strategies