Subsurface flows are influenced by the presence of faults and large fractures which act
as preferential paths or barriers for the flow. In literature models were proposed to
handle fractures in a porous medium as objects of codimension 1. In this work we consider
the case of a network of intersecting fractures, with the aim of deriving physically
consistent and effective interface conditions to impose at the intersection between
fractures. This new model accounts for the angle between fractures at the intersections
and allows for jumps of pressure across intersections. This fact permits to describe the
flow when fractures are characterized by different properties more accurately with respect
to other models that impose pressure continuity. The main mathematical properties of the
model, derived in the two-dimensional setting, are analyzed. As concerns the numerical
discretization we allow the grids of the fractures to be independent, thus in general
non-matching at the intersection, by means of the extended finite element method
(XFEM). This increases the flexibility of the method in the case of complex
geometries characterized by a high number of fractures.