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

Abstract

Summary A 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.

Considerations on a phase‐field model for adhesive fracture

“…Different from earlier work on cohesive phase‐field models, we use NURBS for the spatial discretization . NURBS basis functions can be constructed for an arbitrarily high degree of continuity, and although the isotropic phase‐field model permits the use of classical ‐continuous elements, the use of a smooth basis is beneficial, as the stresses are represented more accurately . Bézier extraction constructs the minimal set of Bézier elements defining a NURBS .…”

A convergence study of monolithic simulations of flow and deformation in fractured poroelastic media

“…coordinate system is used for the fracture with an aperture h ous pressure model for fluid flow in fractured porous media. 8,25 In the remainder of this section, we will provide a succinct overview of this model for Newtonian 7,11,13 and for non-Newtonian (power-law) 26 fluids.…”

“…The expression closely resembles Reynolds' equation for the flow of a thin layer of fluid between two walls and corresponds to the cubic law often used to model fluid flow inside fractures. [1][2][3][4][5][6] It can be extended to non-Newtonian fluids 26 and then takes the form…”

“…It has been shown that this increased continuity is required in order to obtain accurate results based on the continuous pressure model used for the fracture. 26 In order to use the Lagrange and NURBS shape functions in a similar manner, Bézier extraction has been used to cast the NURBS shape functions into a standard finite element data structure. 29 This allows for the evaluation of the internal forces on a per-element basis, while retaining the higher order interelement continuity.…”

“…To show the effect of including the tangential stiffness terms, a typical boundary-value problem is simulated. 13,26 The problem consists of a rectangular domain of × 10 m, with a discontinuity through the centre at a 30 • angle, as shown in Figure 2 GPa have been used as values for the stiffnesses in the non-fractured interface elements. The domain has been discretised using 40 × 40 (Bézier extracted) elements, and a time step size Δt = 1 s has been used.…”