# Mixed Formulation of a Linearized Lubrication Fracture Model in a Poro-elastic Medium

• Vivette Girault
• Mary F. Wheeler
• Kundan Kumar
• Gurpreet Singh
Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 47)

## Abstract

We analyse and discretize a mixed formulation for a linearized lubrication fracture model in a poro-elastic medium. The displacement of the medium is expressed in primary variables while the flows in the medium and fracture are written in mixed form, with an additional unknown for the pressure in the fracture. The fracture is treated as a non-planar surface or curve according to the dimension, and the lubrication equation for the flow in the fracture is linearized. The resulting equations are discretized by finite elements adapted to primal variables for the displacement and mixed variables for the flow. Stability and a priori error estimates are derived. A fixed-stress algorithm is proposed for decoupling the computation of the displacement and flow and a numerical experiment is included.

## Keywords

Poro-elasticity Biot Lubrication Mixed formulation Finite-elements Fixed stress split algorithm

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© Springer International Publishing AG, part of Springer Nature 2019

## Authors and Affiliations

• Vivette Girault
• 1
Email author
• Mary F. Wheeler
• 2
• Kundan Kumar
• 3
• Gurpreet Singh
• 2
1. 1.Sorbonne Universités, UPMC Univ. Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis LionsParisFrance
2. 2.Center for Subsurface Modeling, Institute for Computational Engineering and SciencesThe University of Texas at AustinAustinUSA
3. 3.Porous Media Group, Mathematics InstituteUniversity of BergenBergenNorway