Augmented Lagrangian Solvers for Poroelasticity with Fracture Contact Mechanics

TL;DR

This paper develops and compares augmented Lagrangian-based solvers for complex coupled poroelasticity and fracture contact problems, introducing a new combined method that improves robustness and convergence in numerical simulations.

Contribution

It introduces a novel combined solver that merges classical approaches, enhancing robustness and efficiency for nonlinear poromechanics with fracture contact.

Findings

The return map method struggles with nonlinear coupling, often failing to converge.

The new combined solver demonstrates superior robustness across various tests.

Performance of the combined solver is less sensitive to augmentation parameters.

Abstract

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. The resulting system of equations is complex and highly nonlinear. As a result, convergence issues with nonlinear solvers are common, causing a bottleneck for the numerical solution of such models. One source of difficulty for the nonlinear solvers comes from the fracture contact mechanics, due to its inherently nonsmooth character. In addition, depending on the chosen constitutive model, the degree of nonlinearity is increased through coupling of flow and contact mechanics. 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 (equivalent to an Uzawa method). In addition, we propose a new solver that combines features of both approaches. Numerical experiments in two and three dimensions, designed to simulate hydraulic stimulation of geothermal reservoirs, are conducted to assess the performance of the solvers on problems of poromechanics with fracture contact mechanics. The return map method has more difficulty handling the nonlinear coupling between flow and contact mechanics than the other solvers, in many cases not converging or using an excessive number of iterations. Our new combined solver performs the most robustly across the experiments, its performance being less sensitive to the value of the augmentation parameter than the other solvers.