  In many applications the accurate representation of the computational domain
is a key factor to obtain reliable and effective numerical solutions. Curved
interfaces, which might be internal, related to physical data, or portions of
the physical boundary, are often met in real applications. However, they are
often approximated leading to a geometrical error that might become dominant
and deteriorate the quality of the results. Underground problems often involve
the motion of fluids where the fundamental governing equation is the Darcy law.
High quality velocity fields are of paramount importance for the successful
subsequent coupling with other physical phenomena such as transport. The
virtual element method, as solution scheme, is known to be applicable in
problems whose discretizations requires cells of general shape, and the mixed
formulation is here preferred to obtain accurate velocity fields. To overcome
the issues associated to the complex geometries and, at the same time,
retaining the quality of the solutions, we present here the virtual element
method to solve the Darcy problem, in mixed form, in presence of curved
interfaces in two and three dimensions. The numerical scheme is presented in
detail explaining the discrete setting with a focus on the treatment of curved
interfaces. Examples, inspired from industrial applications, are presented
showing the validity of the proposed approach.
