Robust fixed stress splitting for Biot’s equations in heterogeneous media

Abstract

This thesis concerns iterative solvers for poromechanics problems. The problems in the studies have involved linear poromechanics, non-linear poromechanics, and poromechanics under large deformation. We included high order discretizations, applied linearization techniques and splitting methods to develop new solvers. We studied the robustness and convergence of these solvers. By studying the fixed stress method as an iterative solver for poromechanics, we developed an optimized version of it. Furthermore, by extending the convergence analysis in the time domain, we developed a new version of the fixed stress method that is partially parallelized. This splitting method was combined with linearization techniques to develop solvers for non-linear poromechanics. By studying the convergence of the linearisation schemes, we developed new solvers and extended the applicability to more complex phenomena, for instance poromechanics with large deformation.

On the optimization of the fixed‐stress splitting for Biot's equations

“…The authors show that the scheme is a contraction for any stabilization parameter . This analysis was confirmed in the work of Both et al for heterogeneous media using a simpler technique, and the same result was obtained for both continuous and discontinuous Galerkin higher‐order space‐time finite elements in the works of Bause et al and Bause, implying that the value of the stabilization parameter does not depend on the order of the spatial discretization. The question of which stabilization parameter is the optimal one (in the sense that it requires the least number of iterations to converge) arises, and the aim of this paper is to answer this open question.…”

“…They determined numerically the optimal stabilization parameter for each considered case. This study, together with the previous results presented in the works of Mikelić et al and Both et al, suggests that the optimal parameter actually is a value in the interval , depending on the data. In particular, the optimal parameter depends on the problem's boundary conditions and flow parameters, and not only on its mechanical properties and coupling coefficient.…”

“…In this paper, we propose for the first time that the optimal stabilization parameter for the fixed‐stress splitting scheme lies in the interval and depends also on the fluid flow properties and stability properties of the spatial discretization. This is achieved through refining the proof techniques in the work of Both et al to obtain an improved linear rate of convergence; minimizing this rate with respect to the stabilization parameter gives the “theoretical” optimal choice. Although the trends for the practical and the proposed theoretically optimal stabilization parameter are sound for varying material parameters, the theoretically calculated one does not show great practical promise in terms of being optimal (see the work of Storvik et al for a supplementary numerical study).…”

On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics

“…In addition, stability and convergence of the fixed-stress split method have been rigorously established in [8]. Recently, in [9] the authors have proven the convergence of the fixed-stress split method in energy norm for heterogeneous problems. Estimates for the 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 case of the multirate iterative coupling scheme are obtained in [10], where multiple finer time steps for flow are taken within one coarse mechanics time step, exploiting the different time scales for the mechanics and flow problems.…”

Monte Carlo Assessment of the Impact of Oscillatory and Pulsating Boundary Conditions on the Flow Through Porous Media

“…Furthermore, the fully coupled scheme was employed which involves solving the coupled governing equations of flow and geomechanics simultaneously at every time step. Another approach that is widely used in coupling the flow and the mechanics in porous media is the fixed stress split method (Both et al 2017;Kim et al 2009;Mikelić and Wheeler 2013). In this manuscript, we consider a nonlinear relation between the permeability and the dilatation.…”