  Fractures are normally present in the underground and are, for some physical
processes, of paramount importance. Their accurate description is fundamental
to obtain reliable numerical outcomes useful, e.g., for energy management.
Depending on the physical and geometrical properties of the fractures, fluid
flow can behave differently, going from a slow Darcian regime to more
complicated Brinkman or even Forchheimer regimes for high velocity. The main
problem is to determine where in the fractures one regime is more adequate than
others. In order to determine these low-speed and high-speed regions, this work
proposes an adaptive strategy which is based on selecting the appropriate
constitutive law linking velocity and pressure according to a threshold
criterion on the magnitude of the fluid velocity itself. Both theoretical and
numerical aspects are considered and investigated, showing the potentiality of
the proposed approach. From the analytical viewpoint, we show existence of weak
solutions to such model under reasonable hypotheses on the constitutive laws.
To this end, we use a variational approach identifying solutions with
minimizers of an underlying energy functional. From the numerical viewpoint, we
propose 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 single
fracture and small networks of intersecting fractures.
