Flow of non‐Newtonian fluids in fractured porous media: Isogeometric vs standard finite element discretisation

Abstract

SummaryA formulation has been derived for the flow of non‐Newtonian (power‐law) fluids in deformable, fractured porous media. The formulation is enhanced with a subgrid scale model to accurately represent the flow of the power‐law fluids inside the cracks. The resulting equations have been discretised using standard (Lagrangian) finite element shape functions and with non‐uniform rational B‐splines (NURBS), which have been cast into a standard finite element datastructure using Bézier extraction. The effect of the power‐law index on the velocity inside the fracture and on the total fluid flow through the porous medium has been analysed for a typical boundary‐value problem. It is shown that large differences between non‐Newtonian and linearised Newtonian fluids can occur for the fluid velocity inside the fracture. This can significantly influence the total fluid transport through the domain. A mesh sensitivity study has been carried out as well and shows that markedly smaller element sizes are required in order to obtain accurate results for the local flow inside the fracture, compared with the element sizes necessary for obtaining accurate results inside the porous medium away from the fracture. Moreover, a comparison has been made between the results obtained using standard Lagrange polynomials and those obtained using NURBS. It is shown that while both discretisation methods are able to accurately simulate the deformations and pressures in the porous medium, the higher interelement continuity of NURBS is mandatory for obtaining correct values of the fluid velocities inside the fracture, especially near the tips.