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

Abstract

A consistent linearisation has been carried out for a monolithic solution procedure of a poroelastic medium with fluid-transporting fractures, including a comprehensive assessment of the convergence behaviour. The fracture has been modelled using a sub-grid scale model with a continuous pressure across the fracture. The contributions to the tangential stiffness matrix of the fracture have been investigated to assess their impact on convergence. Simulations have been carried out for different interpolation orders and for Non-Uniform Rational B-Splines as interpolants vs Lagrangian polynomials. To increase the generality of the results, Newtonian as well as non-Newtonian (power-law) fluids have been considered. Unsurprisingly, a consistent linearisation invariably yields a quadratic convergence, but comes at the expense of a loss of symmetry and recalculation of the contribution of the interface to the stiffness matrix at each iteration. When using a linear line search however, the inclusion of only those terms of the interface stiffness which result in a symmetric and constant tangential stiffness matrix is sufficient to obtain a stable and convergent iterative process.

Extended isogeometric analysis of a progressively fracturing fluid‐saturated porous medium

“…[45][46][47] For poroelasticity IGA was first adopted for the intact porous materials, 48 and developed subsequently for fractured and/or fracturing porous media using hydromechanical interface elements. 20,21,[49][50][51] Later, the extension was made to the simulation of non-Newtonian fluids and multi-phase flows. [52][53][54][55] Moreover, other isogeometric applications, namely collocation methods, 56 are studied for poromechanics problems.…”

“…A more promising alternative to provide higher‐order continuity is isogeometric analysis (IGA), originally proposed to connect the design and the analysis tools in order to obtain the highest possible precision in geometric parametrization and to reduce the computational cost through bypassing the mesh generation stage 45‐47 . For poroelasticity IGA was first adopted for the intact porous materials, 48 and developed subsequently for fractured and/or fracturing porous media using hydromechanical interface elements 20,21,49‐51 . Later, the extension was made to the simulation of non‐Newtonian fluids and multi‐phase flows 52‐55 .…”

A refined two-scale model for Newtonian and non-Newtonian fluids in fractured poroelastic media

“…Due to the nonlinearity of the fracture flow an iterative Newton-Raphson method is used to obtain converged pressures and displacements at the new time. To obtain a well-converging scheme, it is important to use correct tangential matrices to perform the iterations [58]. These matrices are straightforward for the interior of the porous material.…”

“…It gives the fluid flux as a function of the current fracture opening height h and the pressure gradient ∂p d /∂x d . For this subgrid model to be fully integrated in the macro-scale formulation, and to preserve quadratic converging [58], the derivative of q x with regard to the pressure gradient and the fracture opening height are also needed.…”