In the subsurface, fractures and the surrounding porous rock can deform in interaction with fluid flow. Advanced mathematical models governing these coupled processes typically combine fluid flow, poroelasticity, and fracture contact mechanics, representing fractures as co-dimension one objects within the porous medium. The resulting system of equations is complex and highly nonlinear. As a result, convergence issues with nonlinear solvers are common, which hinders the practical implementation of such models. One particular source of difficulty for the nonlinear solvers comes from the fracture contact mechanics, due to its inherently nonsmooth character. In this paper, we investigate solvers based on the augmented Lagrangian formulation of the frictional contact problem. This includes two classical solvers, namely the generalized Newton method (using complementarity functions) and the return map method. In addition, we propose a new augmented Lagrangian solver that combines the two approaches. Numerical experiments in two and three dimensions are conducted to assess the performance of the solvers on problems of mixed-dimensional poromechanics with fracture contact mechanics. The experiments indicate that for simpler setups, the solvers behave quite predictably and in accordance with established heuristics from the contact mechanics literature. However, as the complexity of the problem increases, the solvers become more unpredictable, and break with these heuristics. In particular, the two classical solvers become highly sensitive to the augmentation parameter. The new solver converges across a larger range of this parameter in most cases, but nevertheless also displays unpredictable behavior.

翻译：在地下环境中，裂缝与周围多孔岩石的变形会与流体流动相互作用。描述这些耦合过程的先进数学模型通常结合了流体流动、多孔弹性力学和裂缝接触力学，将裂缝表示为多孔介质中的余维一对象。由此产生的方程组复杂且高度非线性。因此，非线性求解器常出现收敛性问题，阻碍了此类模型的实际应用。非线性求解器的一个特殊困难来源是裂缝接触力学，因其固有的非光滑特性。本文研究了基于摩擦接触问题增广拉格朗日公式的求解器。这包括两种经典求解器：广义牛顿法（使用互补函数）和返回映射法。此外，我们提出了一种结合这两种方法的新型增广拉格朗日求解器。我们进行了二维和三维数值实验，以评估这些求解器在含裂缝接触力学的混合维数多孔介质力学问题上的性能。实验表明，对于较简单的设置，求解器的行为相当可预测，且符合接触力学文献中既有的启发式准则。然而，随着问题复杂性的增加，求解器变得难以预测，并偏离这些启发式准则。特别是，两种经典求解器对增广参数变得高度敏感。新求解器在大多数情况下能在更大的参数范围内收敛，但同样表现出不可预测的行为。