Augmented Lagrangian Solvers for Poroelasticity with Fracture Contact Mechanics

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.