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耦合流-力学模型的一般离散化方法开展完整数值分析，考虑单相流、基质-裂缝界面无摩擦接触条件及非线性孔隙力学耦合效应。裂缝被描述为平面表面，形成所谓的混合或混合维度模型。假定基质发生小变形并呈现线弹性行为。为覆盖广泛的法向裂缝导流能力，模型在基质-裂缝界面处考虑非连续流体压力。数值分析在梯度离散化框架下进行，涵盖大量相容与非相容离散格式。收敛结果同时间接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论，采用混合有限体积格式处理流动问题，力学位移由二阶有限元（$\mathbb P_2$）离散，裂缝面上采用面元常数（$\mathbb P_0$）的拉格朗日乘子离散接触条件，以表征法向应力。