arXiv subjects

Jan Martin Nordbotten

Publications and source records attributed to Jan Martin Nordbotten.

At least 19 recordsLinked to original sources

arXiv2026

Elastic wave propagation in fractured media with spring-type and frictional contact deformation laws

Elastic wave propagation in fractured media is relevant to applications such as analysis of seismic waves and non-destructive characterization of materials. Understanding attenuation and scattering behavior arising from wave-fracture interaction is important for interpreting observations at both field and laboratory scales. This work presents a computational framework for elastic wave propagation in fractured media based on a mixed-dimensional discrete fracture-matrix representation. Fracture deformation is governed by four models of increasing complexity, ranging from widely used spring-based formulations to fracture contact mechanics models with friction, all incorporated within a unified computational framework. Many previous studies are often restricted to simplified wave fields, single fractures or subsets of the relevant fracture deformation mechanisms. In contrast, the proposed framework enables fully coupled simulation of elastic wave propagation with fracture deformation models that account for elastic normal deformation, frictional contact and fracture opening and closure. The elastic wave equation is discretized in space using the cell-centered finite volume method Multi-Point Stress Approximation with weak symmetry and in time using the Newmark method. The spatial discretization is locally conservative and applicable to general polyhedral grids, making it well suited for media containing fractures, material heterogeneities and anisotropy. The proposed framework is verified through numerical convergence analyses and is subsequently applied to wave propagation and fracture deformation in two- and three-dimensional media containing multiple intersecting fractures.

math.NA↗

arXiv2026

Approximation properties of double complexes

We consider the simplicial de Rham complex and the Čech-de Rham complex, two bigraded Hilbert complexes whose Hodge-Laplace problems govern spatially coupled problems in mixed dimension and homogeneous dimension, respectively. The former complex can be realized as a subcomplex of the latter. In this paper, we quantify how close these complexes are to each other by constructing bounded cochain complexes between them, and thus we quantify how close a mixed-dimensional formulation of a problem is to an equidimensionally coupled formulation of the same problem. From this construction, we derive a priori- and a posteriori error estimates between the associated Hodge-Laplace problems on the two complexes. These estimates represent the error which is introduced by treating a spatially coupled problem as mixed-dimensional, rather than an equidimensional problem with thin overlaps.

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

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

A Finite Volume Method for Elastic Waves in Heterogeneous, Anisotropic and Fractured Media

Numerical modeling of elastic wave propagation in the subsurface requires applicability to heterogeneous, anisotropic and discontinuous media, as well as support of free surface boundary conditions. Here we study the cell-centered finite volume method Multi-Point Stress Approximation with weak symmetry (MPSA-W) for solving the elastic wave equation. Finite volume methods are geometrically flexible, locally conserving and they are suitable for handling material discontinuities and anisotropies. For discretization in time we have utilized the Newmark method, thereby developing an MPSA-Newmark discretization for the elastic wave equation. An important aspect of this work is the integration of absorbing boundary conditions into the MPSA-Newmark method to limit possible boundary reflections. We verify the MPSA-Newmark discretization numerically for model problems. Convergence analysis of MPSA-Newmark is performed using a known solution in a medium with homogeneous Dirichlet boundary conditions. The analysis demonstrates the expected convergence rates of second order for primary variables (displacements) and between first and second order for secondary variables (tractions). Further verification is conducted through convergence analysis with the inclusion of absorbing boundary conditions. The stability of the scheme is shown through numerical energy decay analyses for waves travelling with various incidence angles onto the absorbing boundaries. Lastly, we present simulation examples of wave propagation in fractured, heterogeneous and transversely isotropic media to demonstrate the versatility of the MPSA-Newmark discretization.

math.NA↗

arXiv2025

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

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.

math.NA↗

arXiv2024

An abstract approximation tool for mixed-dimensional and equidimensional modeling

Many coupled problems in engineering and science can be described by elliptic partial differential equations on adjacent domains, where the coupling can be considered either as a thin equidimensional overlap between the model domains, or as a lower-dimensional interface. Thereby we distinguish equidimensional and mixed-dimensional models of the same system, and the relationship between these modeling approaches is of natural interest. In this paper, we construct an overlapping open cover for a class of simplicial geometries and construct a bounded cochain map from the simplicial de Rham complex to the Čech-de Rham complex associated with the overlapping cover. Thus, we establish an isomorphism between simplicial de Rham complexes (i.e. functions and forms on mixed-dimensional partitions and their differentials) and subcomplexes of Čech-de Rham complexes (i.e. functions and forms on equidimensional partitions and their differentials), which serves as an abstract approximation tool for comparing mixed-dimensional problems to the equidimensional version of the same problem.

math.AT↗

arXiv2024

Mixed finite element methods for linear Cosserat equations

We consider the equilibrium equations for a linearized Cosserat material and provide two perspectives concerning well-posedness. First, the system can be viewed as the Hodge Laplace problem on a differential complex. On the other hand, we show how the Cosserat materials can be analyzed by inheriting results from linearized elasticity. Both perspectives give rise to mixed finite element methods, which we refer to as strongly and weakly coupled, respectively. We prove convergence of both classes of methods, with particular attention to improved convergence rate estimates, and stability in the limit of vanishing characteristic length of the micropolar structure. The theoretical results are fully reflected in the actual performance of the methods, as shown by the numerical verifications.

math.NA↗

arXiv2024

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

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.

math.NA↗

arXiv2024

Network Inpainting via Optimal Transport

In this work, we present a novel tool for reconstructing networks from corrupted images. The reconstructed network is the result of a minimization problem that has a misfit term with respect to the observed data, and a physics-based regularizing term coming from the theory of optimal transport. Through a range of numerical tests, we demonstrate that our suggested approach can effectively rebuild the primary features of damaged networks, even when artifacts are present.

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

Numerical simulations of viscous fingering in fractured porous media

The effect of heterogeneities induced by highly permeable fracture networks on viscous miscible fingering in porous media is examined using high-resolution numerical simulations. We consider the planar injection of a less viscous fluid into a two-dimensional fractured porous medium which is saturated with a more viscous fluid. This problem contains two sets of fundamentally different preferential flow regimes; the first is caused by the viscous fingering and the second is due to the permeability contrasts between the fractures and the rock matrix. We study the transition from the regime where the flow is dominated by the viscous instabilities, to the regime where the heterogeneities induced by the fractures define the flow paths. Our findings reveal that even minor permeability differences between the rock matrix and fractures significantly influence the behavior of viscous fingering. The interplay between the viscosity contrast and permeability contrast leads to the preferential channeling of the less viscous fluid through the fractures. Consequently, this channeling process stabilizes the displacement front within the rock matrix, ultimately suppressing the occurrence of viscous fingering, particularly for higher permeability contrasts. We explore three fracture geometries; two structured and one random configuration, and identify a complex interaction between these geometries and the development of unstable flow. While we find that the most important factor determining the effect of the fracture network is the ratio of fluid volume flowing through the fractures and the rock matrix, the exact point for the cross-over regime is dependent on the geometry of the fracture network.

physics.flu-dyn↗

arXiv2023

The Hodge-Laplacian on the Čech-de Rham complex governs coupled problems

By endowing the Čech-de Rham complex with a Hilbert space structure, we obtain a Hilbert complex with sufficient properties to allow for well-posed Hodge-Laplace problems. We observe that these Hodge-Laplace equations govern a class of coupled problems arising from physical systems including elastically attached strings, multiple-porosity flow systems and 3D-1D coupled flow models.

math.AP↗

arXiv2023

Scaling up FluidFlower results for carbon dioxide storage in geological media

The partial differential equations describing immiscible, but soluble, carbon dioxide (CO2) displacement of brine are developed including local mass-transfer effects. Scaling relationships for characteristic time among laboratory and representative storage formation conditions are found upon assumption that free-phase CO2 transport during injection is dominated by convection. The implication is that an hour in the FluidFlower (large-scale visual model) scales to hundreds of years of elapsed time in the storage formation. The scaling criteria permit extrapolation of the effects of changes in parameters and operating conditions. Interphase mass transfer allows CO2 to saturate the brine phase and such mass transfer is a significant nonequilibrium phenomenon. Significant mixing of CO2 dissolved into formation brine with original brine is found experimentally and is also predicted. The magnitude of onset time for downward migrating fingers containing CO2 is typically only a fraction of the duration of CO2 injection and in general agreement with theoretical analysis in the literature. Predictions for onset time of convective mixing at representative storage formation conditions likewise teach that the onset time for viscous fingering is significantly less than the duration of CO2 injection in some cases. The implications of this observation include that mixing of CO2 with brine and the subsequent settling due to gravity are relatively rapid and coincide with the period of active CO2 injection.

physics.geo-ph↗

arXiv2023

Engineering meter-scale porous media flow experiments for quantitative studies of geological carbon sequestration

This technical note describes the FluidFlower concept, a new laboratory infrastructure for geological carbon storage research. The highly controlled and adjustable system produces a strikingly visual physical ground truth of studied processes for model validation, comparison, and forecasting, including detailed physical studies of the behavior and storage mechanisms of carbon dioxide and its derivative forms in relevant geological settings for subsurface carbon storage. The design, instrumentation, structural aspects and methodology are described. Furthermore, we share engineering insights on construction, operation, fluid considerations, and fluid resetting in the porous media. The new infrastructure enables researchers to study variability between repeated CO2 injections, making the FluidFlower concept a suitable tool for sensitivity studies on a range of determining carbon storage parameters in varying geological formations.

physics.geo-ph↗

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↗