We present a complete numerical analysis for a general discretization of a coupled flow-mechanics model in fractured porous media, considering single-phase flows and including frictionless contact at matrix-fracture interfaces, as well as nonlinear poromechanical coupling. Fractures are described as planar surfaces, yielding the so-called mixed- or hybrid-dimensional models. Small displacements and a linear elastic behavior are considered for the matrix. The model accounts for discontinuous fluid pressures at matrix-fracture interfaces in order to cover a wide range of normal fracture conductivities. The numerical analysis is carried out in the Gradient Discretization framework, encompassing a large family of conforming and nonconforming discretizations. The convergence result also yields, as a by-product, the existence of a weak solution to the continuous model. A numerical experiment in 2D is presented to support the obtained result, employing a Hybrid Finite Volume scheme for the flow and second-order finite elements ($\mathbb P_2$) for the mechanical displacement coupled with face-wise constant ($\mathbb P_0$) Lagrange multipliers on fractures, representing normal stresses, to discretize the contact conditions.

翻译：本文针对裂隙多孔介质中的耦合流动-力学模型，提出了一种通用离散化的完整数值分析。该模型考虑单相流动，包含基质-裂缝界面的无摩擦接触以及非线性孔隙力学耦合。裂缝被描述为平面曲面，从而得到所谓的混合维或杂交维模型。基质考虑小位移和线弹性行为。该模型允许基质-裂缝界面处的流体压力不连续，以涵盖广泛的法向裂缝导流能力。数值分析在梯度离散化框架下进行，涵盖了大量协调与非协调离散格式。收敛性结果还附带证明了连续模型弱解的存在性。文中通过二维数值实验验证所得结果，该实验采用混合有限体积格式处理流动问题，二阶有限元（$\mathbb P_2$）处理力学位移，并结合裂缝上分段常数（$\mathbb P_0$）拉格朗日乘子（表示法向应力）来离散接触条件。