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A History-Dependent Dynamic Biot Model

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A History-Dependent Dynamic Biot Model

European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2023

作者： Stokke, Jakob Seierstad Jakobsen, Morten Kumar, Kundan Radu, Florin Adrian Center for Modeling of Coupled Subsurface Dynamics University of Bergen Bergen Norway

ISBN: (纸本)9783031861680

In this work, we consider a fully dynamic Biot model that includes memory effects due to evolving permeability. Time integrals are used to account for the change in structure. We propose an iterative splitting scheme for this model, extending the fixed-stress split for the quasi-static Biot. We use finite elements in space and a backward Euler discretization in time. The performance of the method is demonstrated through a numerical experiment. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.

关键词： Bioinformatics

A History-dependent Dynamic Biot Model

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arXiv 2024年

作者： Stokke, Jakob S. Jakobsen, Morten Kumar, Kundan Radu, Florin A. Center for Modeling of Coupled Subsurface Dynamics University of Bergen Bergen Norway

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A Line Search Algorithm for Multiphysics Problems with Fracture Deformation

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arXiv 2024年

作者： Stefansson, Ivar Center for Modeling of Coupled Subsurface Dynamics Department of Mathematics University of Bergen Norway

Models for multiphysics problems often contain strong nonlinearities. Including fracture contact mechanics introduces discontinuities at the transition between open and closed or sliding and sticking fractures. The resulting system of equations is highly challenging to solve. The naïve choice of Newton’s method frequently fails to converge, calling for more refined solution techniques such as line search methods. When dealing with strong nonlinearities and discontinuities, a global line search based on the magnitude of the residual of all equations is at best costly to evaluate and at worst fails to converge. We therefore suggest a cheap and reliable approach tailored to the discontinuities. Utilising adaptive variable scaling, the algorithm uses a line search to identify the transition between contact states. Then, a solution update weight is chosen to ensure that no fracture cells move too far beyond the transition. We demonstrate the algorithm on a series of test cases for poromechanics and thermoporomechanics in fractured porous media. We consider both single- and multifracture cases and study the importance of proper scaling of variables and equations. © 2024, CC BY.

关键词： Fracture

Fixed Stress Splitting Approach for Contact Problems in a Porous Medium

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Fixed Stress Splitting Approach for Contact Problems in a...

European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2023

作者： Almani, Tameem Kumar, Kundan Dhahran Saudi Arabia Center for Modeling of Coupled Subsurface Dynamics University of Bergen Bergen Norway

ISBN: (纸本)9783031861680

We consider a poromechanics model including frictionless contact mechanics. The resulting model is the Biot equation with contact boundary conditions leading to a variational inequality modelling mechanical deformations coupled to a linear parabolic flow equation. We propose a fully discrete iterative scheme for solving this model, extending the fixed-stress splitting scheme. We use finite elements in space and a backward Euler discretization in time. We show that the fixed stress split scheme is a contraction. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.

关键词： Friction

Fixed stress splitting approach for contact problems in a porous medium

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arXiv 2024年

作者： Almani, Tameem Kumar, Kundan Saudi Arabia Center for Modeling of Coupled Subsurface Dynamics University of Bergen Bergen Norway

关键词： Porous materials

Mixed finite element and TPSA finite volume methods for linearized elasticity and Cosserat materials

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arXiv 2024年

作者： Nordbotten, J.M. Boon, W.M. Duran, O. Keilegavlen, E. Center for Modeling of Coupled Subsurface Dynamics Department of Mathematics University of Bergen Norway Norway Bergen Norway

Cosserat theory of elasticity is a generalization of classical elasticity that allows for asymmetry in the stress tensor by taking into account micropolar rotations in the medium. The equations involve a rotation field and associated "couple stress" as variables, in addition to the conventional displacement and Cauchy stress fields. In recent work, we derived a mixed finite element method (MFEM) for the linear Cosserat equations that converges optimally in these four variables. The drawback of this method is that it retains the stresses as unknowns, and therefore leads to relatively large saddle point system that are computationally demanding to solve. As an alternative, we developed a finite volume method in which the stress variables are approximated using a minimal, two-point stencil (TPSA). The system consists of the displacement and rotation variables, with an additional "solid pressure" unknown. Both the MFEM and TPSA methods are robust in the incompressible limit and in the Cauchy limit, for which the Cosserat equations degenerate to classical linearized elasticity. We report on the construction of the methods, their a priori properties, and compare their numerical performance against an MPSA finite volume method. © 2024, CC BY.

关键词： Finite element method

Two-point stress approximation: A simple and robust finite volume method for linearized (poro-)mechanics and Stokes flow

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arXiv 2024年

作者： Nordbotten, Jan Martin Keilegavlen, Eirik Center for Modeling of Coupled Subsurface Dynamics Department of Mathematics University of Bergen Norway Bergen Norway

In this paper, we construct a simple and robust two-point finite volume discretization applicable to isotropic linearized elasticity, valid in also in the incompressible Stokes’ limit. The discretization is based only on co-located, cell-centered variables, and has a minimal discretization stencil, using only the two neighboring cells to a face to calculate numerical stresses and fluxes. The discretization naturally couples to finite volume discretizations of flow, providing a stable discretization of poroelasticity. We show well-posedness of a weak statement of the continuous formulation in appropriate Hilbert spaces, and identify the appropriate weighted norms for the problem. For the discrete approximations, we prove stability and convergence, both of which are robust in terms of the material parameters. Numerical experiments in 3D support the theoretical results, and provide additional insight into the practical performance of the discretization. © 2024, CC BY.

关键词： Hilbert spaces

A simulation study of the impact of fracture networks on the co-production of geothermal energy and lithium

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arXiv 2025年

作者： Banshoya, Shin Irgens Berre, Inga Keilegavlen, Eirik Center for modeling of Coupled Subsurface Dynamics Department of Mathematics University of Bergen Allegaten 41 Bergen5020 Norway

Co-production of geothermal energy and lithium is an emerging opportunity with the potential to enhance the economic potential of geothermal operations. The economic reward of extracting lithium from geothermal brine is determined by how the lithium concentration evolves during brine production. In the initial stage, production will target lithium contained in the brine resident close to the production well. While lithium recharge, in the form of rock dissolution and inflow from other parts of the reservoir, is possible, the efficiency of such recharge depends on the geology of the reservoir. In this work, we study how structural heterogeneities in the form of fractures impact the flow of lithium-carrying brine. Using a numerical simulation tool that gives high resolution of flow and transport in fractures and the host rock, we study how the presence of fractures influences energy and lithium production. Our simulations show that, due to heat conduction and the lack of mineral recharge from the rock, differences in fracture network geometries have a much larger impact on lithium production than energy production. The simulations thus confirm that in addition to the geochemical characterisation of lithium in geothermal brines, understanding fracture characterisation and its impact on production is highly important for lithium production. © 2025, CC BY.

关键词： Geothermal fields

WEAK SOLVABILITY OF ELLIPTIC VARIATIONAL INEQUALITIES coupled WITH A NONLINEAR DIFFERENTIAL EQUATION

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arXiv 2024年

作者： Taki, Nadia Skoglund Center for Modeling of Coupled Subsurface Dynamics Department of Mathematics University of Bergen Postbox 7800 Bergen5020 Norway

In this paper we establish existence, uniqueness, and boundedness results for an elliptic variational inequality coupled with a nonlinear ordinary differential equation. Under the general framework, we present a new application modelling the antiplane shear deformation of a static frictional adhesive contact problem. The adhesion process has been extensively studied, but it is usual to assume a priori that the intensity of adhesion is bounded by introducing truncation operators. The aim of this article is to remove this restriction. The proof is based on an iterative approximation scheme showing that the problem has a unique solution. A key ingredient is finding uniform a priori bounds for each iterate. These are obtained by adapting versions of the Moser iteration to our system of equations. © 2024, CC BY.

关键词： Adhesion