arXiv subjects

Jakub Wiktor Both

Publications and source records attributed to Jakub Wiktor Both.

At least 19 recordsLinked to original sources

arXiv2026

Numerical analysis of the Biot equations coupled to frictional contact mechanics

We consider a mathematical model of a poro-visco-elastic medium subject to frictional contact with a rigid obstacle, and study its numerical approximation. This model couples the Biot equations and contact conditions in the form of normal compliance and Coulomb friction. The resulting variational problem consists of a linear partial differential equation coupled to a nonlinear variational inequality. We propose and analyze a fully discrete numerical scheme for this problem, using conformal finite elements in space and the implicit Euler method in time. Existence and uniqueness of the discrete solution is established, and stability and a priori error estimates are derived. A numerical experiment is performed in which numerical error estimates are computed and compared to the theoretical results.

math.NA↗

arXiv2026

Splitting-strategies for arbitrary-order fully mixed finite element discretizations of the Biot equations

We study the fully-mixed formulation of the Biot equations which is characterized by a symmetric coupling between flow and deformation while each subphysics has internally a saddle point structure. This enables the use of stable mixed finite elements for each subproblem without a strong compatibility condition across the two subphysics. To exploit this flexibility while preserving the conservation structure of both subproblems, we consider fully mixed finite element methods in which the symmetry of the elastic stress tensor is enforced weakly, often referred to as a five-field formulation. The resulting mixed formulation exhibits an overarching saddle-point structure whose stability is determined by suitable inf-sup conditions. Inf-sup stability is established for several families of discrete spaces of arbitrary order, leading to optimal a priori error estimates. Iterative splitting strategies following the classical fixed-stress split with additional tuning are specifically investigated for the fully mixed formulation with proof of convergence and rates depending on the coupling strength. Contrary to previous analyses on coupled problems with a symmetric structure, we theoretically prove the efficacy of negative stabilization, consistent with Schur-complement ideas. Numerical results based on analytical solutions and the classical 2D Mandel problem and a 3D footing problem are presented supporting the theory.

math.NA↗

arXiv2026

Efficient design of continuation methods for hyperbolic transport problems in porous media

Full-physics modeling of multiphase flow in porous media, e.g., for carbon storage and groundwater management, requires the nonlinear coupling of various physical processes. Industry standard nonlinear solvers, typically of Newton-type, are not unconditionally convergent and computationally expensive. Homotopy continuation solvers have recently been studied as a robust and versatile alternative. They tackle challenging nonlinear problems by first solving a simple auxiliary problem and then tracing a solution curve towards the more complex target problem. Robustness and efficiency of the method depends on the iterative numerical curve tracing algorithm as well as on careful design of the auxiliary problem. We assess the traceability of the solution curve for different choices of the auxiliary problem. For the Buckley-Leverett equation, modeling two-phase flow in one dimension, we exemplarily compare the previously introduced vanishing-diffusion and linear constitutive laws homotopy continuation, and a new approach based on the entropy solution of the problem. This provides insight toward systematically and robustly designing homotopy continuation methods for solving complex multiphase flow in porous media.

math.NA↗

arXiv2026

Partial Differential Equations in the Age of Machine Learning: A Critical Synthesis of Classical, Machine Learning, and Hybrid Methods

Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature but epistemologically distinct paradigms for PDE solution, classical numerical methods and machine learning approaches, through a unified evaluative framework organized around six fundamental computational challenges. Classical methods are assessed for their structure-preserving properties, rigorous convergence theory, and scalable solver design; their persistent limitations in high-dimensional and geometrically complex settings are characterized precisely. Machine learning approaches are introduced under a taxonomy organized by the degree to which physical knowledge is incorporated and subjected to the same critical evaluation applied to classical methods. Classical methods are deductive -- errors are bounded by quantities derivable from PDE structure and discretization parameters -- while machine learning methods are inductive -- accuracy depends on statistical proximity to the training distribution. This epistemological distinction is the primary criterion governing responsible method selection. We identify three genuine complementarities between the paradigms and develop principles for hybrid design, including a framework for the structure inheritance problem that addresses when classical guarantees propagate through hybrid couplings, and an error budget decomposition that separates discretization, neural approximation, and coupling contributions. We further assess emerging frontiers, including foundation models, differentiable programming, quantum algorithms, and exascale co-design, evaluating each against the structural constraints that determine whether current barriers are fundamental or contingent on engineering progress.

cs.LG↗

arXiv2026

Consistent initialization of mixed-dimensional multiphysics models for fractured reservoirs under geomechanical constraints and field measurements

Modeling coupled processes in fractured porous media -- flow, deformation, fracture mechanics, and thermal/chemical effects -- often relies on mixed dimensional multiphysics formulations. These systems are nonlinear and depend on physical states and state dependent material laws. While in-situ field measurements consistently describe the deformed equilibrium configuration, computational models typically start from an idealized reference configuration and require explicit initialization of the in-situ stress state. This mismatch complicates initialization and linearization of constitutive laws. As a consequence, due to the two scale nature of fractured media, this can induce large deviations in fracture aperture directly impacting flow predictions. To address this, a discrete fracture model is introduced whose constitutive laws are expressed with respect to the unknown equilibrium state. This is paired with a fixed point initialization strategy that consistently reconstructs the reference configuration, consistent with both geomechanical constraints and field measurements up to load-path dependence. This data-consistent strategy provides a foundation for extending models to more complex scenarios, including multiphase and multicomponent flow in fractured reservoirs.

physics.geo-ph↗

arXiv2026

A Machine-Learned Near-Well Model in OPM Flow

Recent advances in reservoir simulation increasingly utilize hybrid approaches that couple physics-based simulators with machine-learning (ML) components. ML components offer high fidelity to training data and fast inference, enabling efficient and accurate modeling of complex multi-scale or multi-physics phenomena. Modern reservoir simulators rely on automatic differentiation (AD) to support efficient and flexible strategies for nonlinear solvers, inverse problems, and optimization problems. Efficient hybrid modeling therefore requires tight integration of the ML components with the simulator's AD framework. We present the first integration of neural networks into the high-performance reservoir simulator OPM Flow. Networks are trained in TensorFlow and imported into OPM, where they are accessed as native AD functions. This presents an efficient framework for hybrid modeling and enables seamless integration in existing simulator workflows. As an application, we introduce a novel, data-driven near-well model. Near-well models are essential in reservoir simulation for accurately representing singular pressure gradients around wells. Commonly used are the Peaceman near-well model and its extensions, or local grid refinement around the wells. Peaceman-type models are limited to simplified flow regimes, whereas local grid refinement is computationally expensive. We address these limitations by training a neural network to infer a Peaceman-like well index from fine-scale ensemble simulations of the near-well region. It is then integrated into OPM Flow with the new framework. Tested on relevant examples for CO$_2$ storage, the method offers high fidelity to fine-scale results at low computational cost, demonstrating the potential of the OPM Flow-Neural Network framework for hybrid modeling.

math.NA↗

arXiv2025

Performance of an open-source image-based history matching framework for CO$_2$ storage

We present a history matching (HM) workflow applied to the International FluidFlower benchmark study dataset, which features high-resolution images of CO$_2$ storage in a meter-scale, geologically complex reservoir. The dataset provides dense spatial and temporal observations of fluid displacement, offering a rare opportunity to validate and enhance HM techniques for geological carbon storage (GCS). The combination of detailed experimental data and direct visual observation of flow behavior at this scale is novel and valuable. This study explores the potential and limitations of using experimental data to calibrate standard models for GCS simulation. By leveraging high-resolution images and resulting interpretations of fluid phase distributions, we adjust uncertain parameters and reduce the mismatch between simulation results and observed data. Simulations are performed using the open-source OPM Flow simulator, while the open-source Everest decision-making tool is employed to conduct the HM. After the HM process, the final simulation results show good agreement with the experimental CO$_2$ storage data. This suggests that the system can be effectively described using standard flow equations, conventional saturation functions, and typical PVT properties for CO$_2$-brine mixtures. Our results demonstrate that the Wasserstein distance is a particularly effective metric for matching multi-phase, multi-component flow data. The entire workflow is implemented in a Python package named pofff (Python OPM Flow FluidFlower), which organizes all functionality through a single input file. This design ensures reproducibility and facilitates future extensions of the study.

physics.flu-dyn↗

arXiv2025

Trends in porous media laboratory imaging and open science practices

Understanding processes in porous media is fundamental to a broad spectrum of environmental, energy, and geoscience applications. These processes include multiphase fluid transport, interfacial dynamics, reactive transformations, and interactions with solids or microbial components, all governed by wettability, capillarity, and reactive transport at fluid-fluid and fluid-solid interfaces. Laboratory-based multiscale imaging provides critical insights into these phenomena, enabling direct visualization and quantitative characterization from the nanometer to meter scale. It is essential for advancing predictive models and optimizing the design of subsurface and engineered porous systems. This review presents an integrated overview of imaging techniques relevant to porous media research, emphasizing the type of information each method can provide, their applicability to porous media systems, and their inherent limitations. We highlight how imaging data are combined with quantitative analyses and modeling to bridge pore-scale mechanisms with continuum-scale behavior, and we critically discuss current challenges such as limited spatio-temporal resolution, sample representativity, and restricted data accessibility. We conduct an in-depth analysis on open-science trends in experimental and computational porous media research and find that, while open-access publishing has become widespread, the availability of imaging data and analysis code remains limited, often restricted to 'upon request'. Finally, we underscore the importance of open sharing of imaging datasets to enable reproducibility, foster cross-disciplinary integration, and support the development of robust predictive frameworks for porous media systems.

physics.flu-dyn↗

arXiv2025

Augmented Lagrangian Solvers for Poroelasticity with Fracture Contact Mechanics

In the subsurface, fractures and the surrounding porous rock can deform in interaction with fluid flow. Advanced mathematical models governing these coupled processes typically combine fluid flow, poroelasticity, and fracture contact mechanics. The resulting system of equations is complex and highly nonlinear. As a result, convergence issues with nonlinear solvers are common, causing a bottleneck for the numerical solution of such models. One source of difficulty for the nonlinear solvers comes from the fracture contact mechanics, due to its inherently nonsmooth character. In addition, depending on the chosen constitutive model, the degree of nonlinearity is increased through coupling of flow and contact mechanics. In this paper, we investigate solvers based on the augmented Lagrangian formulation of the frictional contact problem. This includes two classical solvers, namely the generalized Newton method (using complementarity functions) and the return map method (equivalent to an Uzawa method). In addition, we propose a new solver that combines features of both approaches. Numerical experiments in two and three dimensions, designed to simulate hydraulic stimulation of geothermal reservoirs, are conducted to assess the performance of the solvers on problems of poromechanics with fracture contact mechanics. The return map method has more difficulty handling the nonlinear coupling between flow and contact mechanics than the other solvers, in many cases not converging or using an excessive number of iterations. Our new combined solver performs the most robustly across the experiments, its performance being less sensitive to the value of the augmentation parameter than the other solvers.

math.NA↗

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

High-fidelity experimental model verification for flow in fractured porous media

Mixed-dimensional mathematical models for flow in fractured media have been prevalent in the modeling community for almost two decades, utilizing the explicit representation of fractures by lower-dimensional manifolds embedded in the surrounding porous media. In this work, for the first time, direct qualitative and quantitative comparisons of mixed-dimensional models are drawn against laboratory experiments. Dedicated displacement experiments of steady-state laminar flow in fractured media are investigated using both high-resolution PET images as well as state-of-the-art numerical simulations.

physics.flu-dyn↗

arXiv2023

A global existence result for weakly coupled two-phase poromechanics

Multiphase poromechanics describes the evolution of multiphase flow in deformable porous media. Mathematical models for such multiphysics system are inheritely nonlinear, potentially degenerate and fully coupled systems of partial differential equations. In this work, we present a thermodynamically consistent multiphase poromechanics model falling into the category of Biot equations and obeying to a generalized gradient flow structure. It involves capillarity effects, degenerate relative permeabilities, and gravity effects. In addition to established models it introduces a Lagrange multiplier associated to a bound constraint on the effective porosity in particular ensuring its positivity. We establish existence of global weak solutions under the assumption of a weak coupling strength, implicitly utilizing the gradient flow structure, as well as regularization, a Faedo-Galerkin approach and compactness arguments. This comprises the first global existence result for multiphase poromechanics accounting for degeneracies that are consistent with the multiphase nature of the flow.

math.AP↗

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

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↗

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↗

arXiv2021

Free energy diminishing discretization of Darcy-Forchheimer flow in poroelastic media

In this paper, we develop a discretization for the non-linear coupled model of classical Darcy-Forchheimer flow in deformable porous media, an extension of the quasi-static Biot equations. The continuous model exhibits a generalized gradient flow structure, identifying the dissipative character of the physical system. The considered mixed finite element discretization is compatible with this structure, which gives access to a simple proof for the existence, uniqueness, and stability of discrete approximations. Moreover, still within the framework, the discretization allows for the development of finite volume type discretizations by lumping or numerical quadrature, reducing the computational cost of the numerical solution.

math.NA↗

arXiv2020

Efficient Solvers for Nonstandard Models for Flow and Transport in Unsaturated Porous Media

We study several iterative methods for fully coupled flow and reactive transport in porous media. The resulting mathematical model is a coupled, nonlinear evolution system. The flow model component builds on the Richards equation, modified to incorporate nonstandard effects like dynamic capillarity and hysteresis, and a reactive transport equation for the solute. The two model components are strongly coupled. On one hand, the flow affects the concentration of the solute; on the other hand, the surface tension is a function of the solute, which impacts the capillary pressure and, consequently, the flow. After applying an Euler implicit scheme, we consider a set of iterative linearization schemes to solve the resulting nonlinear equations, including both monolithic and two splitting strategies. The latter include a canonical nonlinear splitting and an alternate linearized splitting, which appears to be overall faster in terms of numbers of iterations, based on our numerical studies. The (time discrete) system being nonlinear, we investigate different linearization methods. We consider the linearly convergent L-scheme, which converges unconditionally, and the Newton method, converging quadratically but subject to restrictions on the initial guess. Whenever hysteresis effects are included, the Newton method fails to converge. The L-scheme converges; nevertheless, it may require many iterations. This aspect is improved by using the Anderson acceleration. A thorough comparison of the different solving strategies is presented in five numerical examples, implemented in MRST, a toolbox based on MATLAB.

math.NA↗