Jan M. Nordbotten’s research while affiliated with University of Bergen and other places

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.

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. Verification of the MPSA-Newmark discretization is achieved through numerical convergence analyses in 3D relative to a known solution, demonstrating the expected convergence rates of order two in time and up to order two in space. With the inclusion of absorbing boundary conditions, the resulting discretization is verified by considering convergence in a quasi-1d setting, as well as through energy decay analyses for waves with various wave incidence angles. Lastly, the versatility of the MPSA-Newmark discretization is demonstrated through simulation examples of wave propagation in fractured, heterogeneous and transversely isotropic 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.

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.

Mitochondrial and chloroplast DNA (mtDNA and cpDNA) encode essential cellular apparatus. This organelle DNA (oDNA) exists at high copy number (ploidy) in eukaryotic cells, which must both mitigate mutational damage and allow adaptation to changing demands. Across eukaryotes, a range of inheritance strategies are used for oDNA, including uniparental and doubly uniparental inheritance (DUI), paternal leakage, recombination-mediated repair and gene conversion, and an effective "genetic bottleneck" imposed between generations. Here, we use modelling and simulation to investigate how these different strategies support the robustness and evolvability of oDNA populations under different challenges of mutation and changes in selection imposed by the environment. We find a general tradeoff between maintaining heterozygosity for flexible adaptation and supporting purifying selection against dysfunctional mutants. Different combinations of leakage and bottleneck size provide optimal resolutions to this tradeoff under different sets of challenges. The model explains many observed behaviours, including the appearance of non-minimal bottleneck sizes, a tradeoff between high ploidy for heterozygosity and repair and tight bottlenecks for segregation, and environmental dependence of the benefits of leakage and DUI. We connect different strategies observed across eukaryotes with the ecology of the organisms involved to explore support for the predictions of this theory.

High‐resolution modeling of the flow dynamics in fractured soils is highly complex and computationally demanding as it requires precise geometrical description of the fractures in addition to resolving a multiphase free‐flow problem inside the fractures. In this paper, we present an idealized model for saturated–unsaturated flow in fractured soils that preserves the core aspects of fractured flow dynamics using an explicit representation of the fractures. The model is based on Richards’ equation in the matrix and hydrostatic equilibrium in the fractures. While the first modeling choice is standard, the latter is motivated by the difference in flow regimes between matrix and fractures, that is, the water velocity inside the fractures is considerably larger than in the soil even under saturated conditions. On matrix/fracture interfaces, the model permits water exchange between matrix and fractures only when the capillary barrier offered by the presence of air inside the fractures is overcome. Thus, depending on the wetting conditions, fractures can either act as impervious barriers or as paths for rapid water flow. Since in numerical simulations each fracture face in the computational grid is a potential seepage face, solving the resulting system of nonlinear equations is a nontrivial task. Here, we propose a general framework based on a discrete‐fracture matrix approach, a finite volume discretization of the equations, and a practical iterative technique to solve the conditional flow at the interfaces. Numerical examples support the mathematical validity and the physical applicability of the model.

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.

We construct a simple and robust finite volume discretization for linearized mechanics, Stokes and poromechanics, based only on co-located, cell-centered variables. The discretization has a minimal stencil, using only the two neighboring cells to a face to calculate numerical stresses and fluxes. We fully justify the method theoretically in terms of stability and convergence, both of which are robust in terms of the material parameters. Numerical experiments support the theoretical results, and shed light on grid families not explicitly treated by the theoretical results.

We have developed a physical room-scale porous media flow rig for operating, measuring, and visualizing reservoir flows in real time – the FluidFlower. The flow rig scale is large enough to achieve true multiphase flow effects (including phase mixture, gravity segregation and geological heterogeneities), while small enough to work on weekly time-scales, and allow for repeatable experiments. Mirroring the FluidFlower, we have constructed a prototype of a digital twin for porous media flow – the PoroTwin. Essentially, we demonstrate that it is possible to achieve real-time transmissions of laboratory data from the FluidFlower to a cloud-based simulation- and machine learning environment, and complete the loop with applying optimal control algorithms to steer the experiment. As part of the proof of concept, we also demonstrate that the machine learning environment can identify, and learn to correct for, incomplete physical descriptions within a reservoir simulator. The PoroTwin thus shows the potential of a fully integrated experimental and automated learning environment.