  The present work deals with the numerical resolution of coupled 3D-2D
problems arising from the simulation of fluid flow in fractured porous media
modeled via the Discrete Fracture and Matrix (DFM) model. According to the DFM
model, fractures are represented as planar interfaces immersed in a 3D porous
matrix and can behave as preferential flow paths, in the case of conductive
fractures, or can actually be a barrier for the flow, when, instead, the
permeability in the normal-to-fracture direction is small compared to the
permeability of the matrix. Consequently, the pressure solution in a DFM can be
discontinuous across a barrier, as a result of the geometrical dimensional
reduction operated on the fracture.
  The present work is aimed at developing a numerical scheme suitable for the
simulation of the flow in a DFM with fractures and barriers, using a mesh for
the 3D matrix non conforming to the fractures and that is ready for domain
decomposition. This is achieved starting from a PDE-constrained optimization
method, currently available in literature only for conductive fractures in a
DFM. First, a novel formulation of the optimization problem is defined to
account for non permeable fractures. These are described by a filtration-like
coupling at the interface with the surrounding porous matrix. Also the extended
finite element method with discontinuous enrichment functions is used to
reproduce the pressure solution in the matrix around a barrier.
  The method is presented here in its simplest form, for clarity of exposition,
i.e. considering the case of a single fracture in a 3D domain, also providing a
proof of the well posedness of the resulting discrete problem. Four validation
examples are proposed to show the viability and the effectiveness of the
method.
