Algebraically stabilized Lagrange multiplier method for frictional contact mechanics with hydraulically active fractures

Andrea Franceschini, Nicola Castelletto, Joshua A. White, and Hamdi A. Tchelepi

Computer Methods in Applied Mechanics and Engineering


Accurate numerical simulation of coupled fracture/fault deformation and fluid flow is crucial to the performance and safety assessment of many subsurface systems. In this work, we consider the discretization and enforcement of contact conditions at such surfaces. The bulk rock deformation is simulated using low-order continuous finite elements, while frictional contact conditions are imposed by means of a Lagrange multiplier method. We employ a cell-centered finite-volume scheme to solve the fracture fluid mass balance equation. From a modeling perspective, a convenient choice is to use a single grid for both mechanical and flow processes, with piecewise-constant interpolation of Lagrange multipliers, i.e., contact tractions and fluid pressure. Unfortunately, this combination of displacement and multiplier variables is not uniformly inf–sup stable, and therefore requires a stabilization technique. Starting from a macroelement analysis, we develop two algebraic stabilization approaches and compare them in terms of robustness and convergence rate. The proposed approaches are validated against challenging analytical two- and three-dimensional benchmarks to demonstrate accuracy and robustness. These benchmarks include both pure contact mechanics problems and well as problems with tightly-coupled fracture flow.

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A simple example of frictional sliding between two surfaces. Using a piecewise constant Lagrange multiplier interpolation, the resulting traction field would show severe oscillations. With stabilization, a smooth and accurate solution is recovered.