Mixed Formulation of a Linearized Lubrication Fracture Model in a Poro-elastic Medium
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.
KeywordsPoro-elasticity Biot Lubrication Mixed formulation Finite-elements Fixed stress split algorithm
- 1.Alboin C, Jaffré J, Roberts JE, Serres C (2001) Modeling fractures as interfaces for flow and transport in porous media. In: Chen Z, Ewing RE (eds) Fluid flow and transport in porous media: mathematical and numerical treatment. South Hadley, MA. (Vol 295 of Contemporary Mathematics). American Mathematical Society, Providence, RI, pp 13–24 (2002)Google Scholar
- 2.Babus̆ka I (1972/73) The finite element method with Lagrangian multipliers. Numer Math 20:179–192Google Scholar
- 9.Ciarlet PG (1991) Basic error estimates for elliptic problems. In: Handbook of numerical analysis, Vol II. North-Holland, Amsterdam, pp 17–351Google Scholar
- 16.Girault V, Raviart P-A (1986) Finite element methods for Navier-Stokes equations: theory and algorithms, vol 5. Springer Series in Computational Mathematics. Springer, BerlinGoogle Scholar
- 17.Girault V, Wheeler MF, Ganis B, Mear ME (2013) A lubrication fracture model in a poro-elastic medium. ICES Report 13-32, Institute for Computational Engineering and Sciences, University of Texas at AustinGoogle Scholar