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.

翻译：本文研究了由离散裂缝与基质（DFM）模型模拟的裂隙多孔介质流体流动所引发的3D-2D耦合问题的数值求解。根据DFM模型，裂缝被表示为浸入3D多孔基质中的平面界面，在导水裂缝情况下可作为优先流动通道，而当垂直于裂缝方向的渗透率远小于基质渗透率时，则可能成为流动屏障。因此，由于对裂缝进行的几何降维处理，DFM中的压力解在跨越屏障时可能不连续。本文旨在开发一种适用于含裂缝与屏障的DFM流动模拟的数值方案，该方案采用非一致于裂缝的3D基质网格，并可直接用于区域分解。通过从目前文献中仅适用于DFM导水裂缝的偏微分方程约束优化方法出发实现这一点。首先，定义了一种新型优化问题公式以考虑非渗透裂缝。这类裂缝通过类似过滤的耦合机制与周围多孔基质在界面上进行描述。同时采用具有非连续增强函数的扩展有限元方法，以再现屏障周围基质中的压力解。为清晰阐述，本文以最简单的形式呈现该方法（即考虑单个裂缝位于三维域中的情形），并给出离散问题适定性证明。通过四个验证算例展示了该方法的可行性与有效性。