arXiv subjects

Erlend Storvik

Publications and source records attributed to Erlend Storvik.

14 recordsLinked to original sources

arXiv2026

On the convergence and efficiency of splitting schemes for the Cahn-Hilliard-Biot model

In this paper, we present a novel solution strategy for the Cahn-Hilliard-Biot model, a three-way coupled system that features the interplay of solid phase separation, fluid dynamics, and elastic deformations in porous media. It is a phase-field model that combines the Cahn-Hilliard regularized interface equation and Biot's equations of poroelasticity. Solving the system poses significant challenges due to its coupled, nonlinear, and non-convex nature. The main goal of this work is to provide a consistent and efficient solution strategy. With this in mind, we introduce a semi-implicit time discretization such that the resulting discrete system is equivalent to a convex minimization problem. Then, using abstract theory for convex problems, we prove the convergence of an alternating minimization method to the time-discrete system. The solution strategy is relatively flexible in terms of spatial discretization, although we require standard inverse inequalities for the guaranteed convergence of the alternating minimization method. Finally, we perform some numerical experiments that show the promise of the proposed solution strategy, both in terms of efficiency and robustness.

math.NA↗

arXiv2024

Sharp-Interface Limit of the Cahn-Hilliard-Biot Equations

In this letter, we derive the sharp-interface limit of the Cahn-Hilliard-Biot equations using formal matched asymptotic expansions. We find that in each sub-domain, the quasi-static Biot equations are obtained with domain-specific material parameters. Moreover, across the interface, material displacement and pore pressure are continuous, while volumetric fluid content and normal stress are balanced. By utilizing the energy of the system, the phase-field potential is shown to be influenced by the curvature, along with contributions from both flow and elasticity at the interface. The normal velocity of the interface is proportional to the jump in normal derivative of the phase-field potential across the interface. Finally, we present a numerical experiment that demonstrates how the location of each phase evolves consistently as the diffuse-interface width parameter becomes smaller; only the width of the diffuse interface changes.

math.AP↗

arXiv2024

Sequential solution strategies for the Cahn-Hilliard-Biot model

This paper presents a study of solution strategies for the Cahn-Hilliard-Biot equations, a complex mathematical model for understanding flow in deformable porous media with changing solid phases. Solving the Cahn-Hilliard-Biot system poses significant challenges due to its coupled, nonlinear and non-convex nature. We explore various solution algorithms, comparing monolithic and splitting strategies, focusing on both their computational efficiency and robustness.

math.NA↗

arXiv2023

Well-posedness analysis of the Cahn-Hilliard-Biot model

We investigate the well-posedness of the recently proposed Cahn-Hilliard-Biot model. The model is a three-way coupled PDE of elliptic-parabolic nature, with several nonlinearities and the fourth order term known to the Cahn-Hilliard system. We show existence of weak solutions to the variational form of the equations and uniqueness under certain conditions of the material parameters and secondary consolidation, adding regularizing effects. Existence is shown by discretizing in space and applying ODE-theory (the Peano-Cauchy theorem) to prove existence of the discrete system, followed by compactness arguments to retain solutions of the continuous system. In addition, the continuous dependence of solutions on the data is established, in particular implying uniqueness. Both results build strongly on the inherent gradient flow structure of the model.

math.AP↗

arXiv2023

Physical variability in meter-scale laboratory CO$_2$ injections in faulted geometries

Carbon, capture, and storage (CCS) is an important bridging technology to combat climate change in the transition towards net-zero. The FluidFlower concept has been developed to visualize and study CO$_2$ flow and storage mechanisms in sedimentary systems in a laboratory setting. Meter-scale multiphase flow in two geological geometries, including normal faults with and without smearing, is studied. The experimental protocols developed to provide key input parameters for numerical simulations are detailed, including an evaluation of operational parameters for the FluidFlower benchmark study. Variability in CO$_2$ migration patterns for two different geometries is quantified, both between 16 repeated laboratory runs and between history-matched models and a CO$_2$ injection experiment. The predicative capability of a history-matched model is then evaluated in a different geological setting.

physics.geo-ph↗

arXiv2023

Room-scale CO2 injections in a physical reservoir model with faults

We perform a series of repeated CO2 injections in a room-scale physical model of a faulted geological cross-section. Relevant parameters for subsurface carbon sequestration, including multiphase flows, capillary CO2 trapping, dissolution, and convective mixing, are studied and quantified. As part of a forecasting benchmark study, we address and quantify six predefined metrics for storage capacity and security in typical CO2 storage operations. Using the same geometry, we investigate the degree of reproducibility of five repeated experimental runs. Our analysis focuses on physical variations of the spatial distribution of mobile and dissolved CO2, multiphase flow patterns, development in mass of the aqueous and gaseous phases, gravitational fingers, and leakage dynamics. We observe very good reproducibility in homogenous regions with up to 97 % overlap between repeated runs, and that fault-related heterogeneity tends to decrease reproducibility. Notably, we observe an oscillating anticline CO2 leakage behavior from an open anticline with a spill point in the immediate footwall of a normal fault, and discuss the underlying causes for the observed phenomenon within the constraints of the studied system.

physics.geo-ph↗

arXiv2023

DarSIA: An open-source Python toolbox for two-scale image processing of dynamics in porous media

Understanding porous media flow is inherently a multi-scale challenge, where at the core lies the aggregation of pore-level processes to a continuum, or Darcy-scale, description. This challenge is directly mirrored in image processing, where grains and interfaces may be clearly visible, yet continuous parameters are desirable to measure. Classical image processing is poorly adapted to this setting, as most techniques do not explicitly utilize the fact that the image contains explicit physical processes. Here, we adapt classical image processing concepts to what we define as physical images of porous materials and processes within them. This is realized through the development of a new open-source image analysis toolbox specifically adapted to time-series of images of porous materials.

cs.MS↗

arXiv2023

An adaptive solution strategy for Richards' equation

Flow in variably saturated porous media is typically modelled by the Richards equation, a nonlinear elliptic-parabolic equation which is notoriously challenging to solve numerically. In this paper, we propose a robust and fast iterative solver for Richards' equation. The solver relies on an adaptive switching algorithm, based on rigorously derived a posteriori indicators, between two linearization methods: L-scheme and Newton. Although a combined L-scheme/Newton strategy was introduced previously in [List & Radu (2016)], here, for the first time we propose a reliable and robust criteria for switching between these schemes. The performance of the solver, which can be in principle applied to any spatial discretization and linearization methods, is illustrated through several numerical examples.

math.NA↗

arXiv2022

Splitting Schemes for Coupled Differential Equations: Block Schur-Based Approaches and Partial Jacobi Approximation

Coupled multi-physics problems are encountered in countless applications and pose significant numerical challenges. Although monolithic approaches offer possibly the best solution strategy, they often require ad-hoc preconditioners and numerical implementations. Sequential (also known as splitted, partitioned or segregated) approaches are iterative methods for solving coupled problems where each equation is solved independently and the coupling is achieved through iterations. These methods offer the possibility to flexibly add or remove equations from a model and to rely on existing black-box solvers for every specific equation. Furthermore, when problems are non-linear, inner iterations need to be performed even in monolithic solvers, therefore making a sequential iterative approach a viable alternative. The cost of running inner iterations to achieve the coupling, however, could easily becomes prohibitive, or, in some cases the iterations might not converge. In this work we present a general formulation of splitting schemes for continuous operators, with arbitrary implicit/explicit splitting, like in standard iterative methods for linear systems. By introducing a generic relaxation operator we find the conditions for the convergence of the iterative schemes. We show how the relaxation operator can be thought as a preconditioner and constructed based on an approximate Schur-complement. We propose a Schur-based Partial Jacobi relaxation operator to stabilise the coupling and show its effectiveness. Although we mainly focus on scalar-scalar linear problems, most results are easily extended to non-linear and higher-dimensional problems. Numerical tests (1D and 2D) for two PDE systems, namely the Dual-Porosity model and a Quad-Laplacian operator, are carried out to confirm the theoretical results.

math.NA↗

arXiv2022

A robust solution strategy for the Cahn-Larché equations

In this paper we propose a solution strategy for the Cahn-Larché equations, which is a model for linearized elasticity in a medium with two elastic phases that evolve subject to a Ginzburg-Landau type energy functional. The system can be seen as a combination of the Cahn-Hilliard regularized interface equation and linearized elasticity, and is non-linearly coupled, has a fourth order term that comes from the Cahn-Hilliard subsystem, and is non-convex and nonlinear in both the phase-field and displacement variables. We propose a novel semi-implicit discretization in time that uses a standard convex-concave splitting method of the nonlinear double-well potential, as well as special treatment to the elastic energy. We show that the resulting discrete system is equivalent to a convex minimization problem, and propose and prove the convergence of alternating minimization applied to it. Finally, we present numerical experiments that show the robustness and effectiveness of both alternating minimization and the monolithic Newton method applied to the newly proposed discrete system of equations. We compare it to a system of equations that has been discretized with a standard convex-concave splitting of the double-well potential, and implicit evaluations of the elasticity contributions and show that the newly proposed discrete system is better conditioned for linearization techniques.

math.NA↗

arXiv2021

A Cahn-Hilliard-Biot system and its generalized gradient flow structure

In this work, we propose a new model for flow through deformable porous media, where the solid material has two phases with distinct material properties. The two phases of the porous material follow a Cahn-Hilliard type evolution, with additional impact from both elastic and fluid effects, and the coupling between flow and deformation is governed by Biot's theory. This results in a three-way coupled system which can be seen as an extension of the Cahn-Larché equations with the inclusion of a fluid flowing through the medium. The model covers essential coupling terms for several relevant applications, including solid tumor growth, biogrout, and wood growth simulation. Moreover, we show that this coupled set of equations follow a generalized gradient flow framework. This opens a toolbox of analysis and solvers which can be used for further study of the model. Additionally, we provide a numerical example showing the impact of the flow on the solid phase evolution in comparison to the Cahn-Larché system.

math-ph↗

arXiv2020

An accelerated staggered scheme for phase-field modeling of brittle fracture

There is currently an increasing interest in developing efficient solvers for phase-field modeling of brittle fracture. The governing equations for this problem originate from a constrained minimization of a non-convex energy functional, and the most commonly used solver is a staggered solution scheme. This is known to be robust compared to the monolithic Newton method, however, the staggered scheme often requires many iterations to converge when cracks are evolving. The focus of our work is to accelerate the solver through a scheme that sequentially applies Anderson acceleration and over-relaxation, switching back and forth depending on the residual evolution, and thereby ensuring a decreasing tendency. The resulting scheme takes advantage of the complementary strengths of Anderson acceleration and over-relaxation to make a robust and accelerating method for this problem. The new method is applied as a post-processing technique to the increments of the solver, hence, the implementation can be done with minor modifications to already available software. Moreover, the cost of combining the two acceleration schemes is negligible. The robustness and efficiency of the method are demonstrated through numerical examples.

math.NA↗

arXiv2019

The fixed-stress splitting scheme for Biot's equations as a modified Richardson iteration: Implications for optimal convergence

The fixed-stress splitting scheme is a popular method for iteratively solving the Biot equations. The method successively solves the flow and mechanic subproblems while adding a stabilizing term to the flow equation, which includes a parameter that can be chosen freely. However, the convergence properties of the scheme depend significantly on this parameter and choosing it carelessly might lead to a very slow, or even diverging, method. In this paper, we present a way to exploit the matrix structure arizing from discretizing the equations in the regime of impermeable porous media in order to obtain a priori knowledge of the optimal choice of this tuning/stabilization parameter.

math.NA↗

arXiv2018

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

In this work we are interested in effectively solving the quasi-static, linear Biot model for poromechanics. We consider the fixed-stress splitting scheme, which is a popular method for iteratively solving Biot's equations. It is well-known that the convergence of the method is strongly dependent on the applied stabilization/tuning parameter. In this work, we propose a new approach to optimize this parameter. We show theoretically that it depends also on the fluid flow properties and not only on the mechanics properties and the coupling coefficient. The type of analysis presented in this paper is not restricted to a particular spatial discretization. We only require it to be inf-sup stable. The convergence proof applies also to low-compressible or incompressible fluids and low-permeable porous media. Illustrative numerical examples, including random initial data, random boundary conditions or random source terms and a well-known benchmark problem, i.e. Mandel's problem are performed. The results are in good agreement with the theoretical findings. Furthermore, we show numerically that there is a connection between the inf-sup stability of discretizations and the performance of the fixed-stress splitting scheme.

math.NA↗