arXiv subjects

Eirik Keilegavlen

Publications and source records attributed to Eirik Keilegavlen.

At least 19 recordsLinked to original sources

arXiv2026

Isochoric thermodynamic preconditioning for resolving mechanically induced phase change in fractured porous media

Rapid pore-volume changes can trigger phase change on timescales shorter than characteristic transport times and may therefore be skipped by conventional nonlinear solves and adaptive time stepping. We present a persistent-variable framework for transport in fractured porous media, introducing specific volume as an independent variable. Starting from a fully coupled system, we derive volume-based models and recover classical pressure-based formulations by eliminating local thermodynamic variables. To resolve abrupt fracture opening, we introduce a nonlinear preconditioner that assumes instantaneous free expansion and resolves the fluid state through an isochoric equilibrium calculation before advancing the transport problem. The preconditioner applies to both volume- and pressure-based models. In the studied fracture-opening cases, unpreconditioned simulations miss transient vaporization, whereas the preconditioned models do not. Across the investigated aperture range, larger openings produce monotonic increases in gas content, expansion-induced cooling, and durations of transients. Thermal effects alter the phase evolution but have otherwise minor influence on the overall transient duration. Within the proof-of-concept setting, fracture opening generates substantial transient pressure reductions, indicating that geomechanical feedback may become important in fully coupled applications. Pressure-enthalpy and volume-temperature formulations recover identical physical solutions but exhibit different nonlinear robustness, with the pressure-enthalpy formulation proving more robust in some recompression-dominated cases. These results show that equilibrium specifications control the numerical properties of the nonlinear problem rather than recovered physical responses, while isochoric preconditioning connects the nonlinear initialization directly to the underlying thermodynamics.

physics.comp-ph↗

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

Mathematical Modeling of Salt Precipitation and Multi-Phase Flow in High Enthalpy Fractured Geothermal Systems

Simulating high-enthalpy fractured geothermal reservoirs is challenging due to the complex coupled processes of non-isothermal, multiphase, multicomponent flow, strongly nonlinear thermodynamics, and the dominant role of fractures. These complexities are amplified by mineral scaling, such as halite precipitation, which can impair reservoir permeability and well productivity. To address this, we present a new compositional flow model based on a persistent set of primary variables (pressure, enthalpy, and overall salt mass fraction). The formulation naturally handles phase transitions without manual switching, enhancing numerical stability. The model integrates a discrete fracture-matrix approach and employs an efficient, robust correlation-based phase-behaviour linearisation of saltwater thermodynamics, replacing expensive on-the-fly phase separation calculations. It incorporates the Kozeny-Carman relation to dynamically model porosity and permeability reduction from halite precipitation. Implemented in the open-source PorePy framework, the model is verified through a 1D salt dissolution benchmark against the established closed-source simulator CSMP++, showing strong agreement across geothermal conditions involving transitions between single- and multi-phase regions. Application to a 2D halite-saturated fractured reservoir with injection and production demonstrates the model's capability to predict halite precipitation patterns and their impact on permeability damage and energy recovery. Numerical results further show the model's value in predicting operational challenges such as wellbore blockage and the role of fracture connectivity. The model thus provides an open-source numerical tool for analysing complex heat and mass transport with mineral scaling in high-enthalpy fractured geothermal systems.

math.NA↗

arXiv2026

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↗

arXiv2025

A posteriori error estimates for mixed-dimensional Darcy flow using non-matching grids

In this article, we extend the a posteriori error estimates for hierarchical mixed-dimensional elliptic equations developed in [Varela et al., J. Numer. Math., 48 (2023), pp. 247-280] to the setting of non-matching mixed-dimensional grids. The extension is achieved by introducing transfer grids between the planar subdomain and interface grids, together with stable discrete projection operators for primal (potential) and dual (flux) variables. The proposed non-matching estimators remain fully guaranteed and computable. Numerical experiments, including three-dimensional problems based on community benchmarks for incompressible Darcy flow in fractured porous media, demonstrate reliable performance of the estimators for the non-matching grids and effectivity that is comparable to the estimators for matching grids.

math.NA↗

arXiv2025

Persistent-variable thermal compositional simulation of multiphase flow with phase separation in porous media

Thermal compositional multiphase flow in porous media with phase transitions involves complex nonlinear interactions among flow, transport, and phase equilibrium. This paper presents a persistent-variable formulation for thermal compositional flow using enthalpy to formulate the energy balance and the local equilibrium problem. Equilibrium conditions are derived from a thermodynamically consistent minimization problem using a persistent set of variables, allowing for seamless integration of equilibrium calculations into a fully coupled flow and transport model. This formulation does not require phase stability tests and provides a continuous and full mathematical description of the multiphysics system, suitable for challenging non-isothermal scenarios. To tackle the nonlinearities arising from phase transitions, we embed a local solver for the thermodynamic subproblem within a global Newton solver for the fully implicit system. The local solver exploits the locality of the subproblem for parallelization and leverages the modularity of the persistent-variable formulation for both isothermal and isenthalpic equilibrium conditions locally. We demonstrate the capability of our approach to simulate complex high-enthalpy systems, including narrow-boiling phenomena. The impact of the embedded local solver is analyzed through numerical experiments, demonstrating a reduction in global nonlinear iterations of up to 23 \% with increased use of the local solver. The number of local iterations is controlled with a local solver tolerance and no significant impact on the global iteration number was observed for local residual tolerances as high as $1e-3$. The persistent-variable approach using enthalpy and the modularity of the embedded local solver advance the usage of equilibrium calculations in multiphase flow simulations and are suitable for high-enthalpy applications.

physics.comp-ph↗

arXiv2025

Data-driven linear solver selection and performance tuning for multiphysics simulations in porous media

Modeling multiphysics processes in porous media requires preconditioned iterative linear solvers to enable efficient simulations at industry-relevant scales. These solvers are typically composed of sub-algorithms that target individual physical processes. Various options are available for each algorithm, with the corresponding ranges of numerical parameters. The choices of sub-algorithms and their parameters significantly affects simulation performance and robustness. Optimizing these choices for each simulation is challenging due to the vast number of possible combinations. Moreover, optimization relies on performance data from past simulations, which becomes less representative as the simulation setup changes. This paper addresses the problem of automated selection and tuning of preconditioned linear solvers for multiphysics simulations. The proposed solver selection algorithm collects performance data during the run of the target simulation and continuously updates a machine learning model responsible for solver selection, resulting in an adaptively refined selection policy. The algorithm is evaluated on two time-dependent nonlinear model problems: (i) coupled fluid flow and heat transfer in porous media and (ii) thermo-poromechanics in porous media with fractures, governed by frictional contact mechanics. These experiments demonstrate that the algorithm selects efficient and robust solvers with negligible overhead and performs comparably to a reference selection policy that has full access to the performance data of prior simulations. Our results indicate that the proposed approach effectively addresses the challenge of solver selection and tuning, providing particular value to simulation engineers and researchers, especially when expert knowledge on linear solver tuning is not readily available.

math.NA↗

arXiv2025

A block preconditioner for thermo-poromechanics with frictional deformation of fractures

The numerical modeling of fracture contact thermo-poromechanics is crucial for advancing subsurface engineering applications, including CO2 sequestration, production of geo-energy resources, energy storage and wastewater disposal operations. Accurately modeling this problem presents substantial challenges due to the complex physics involved in strongly coupled thermo-poromechanical processes and the frictional contact mechanics of fractures. To resolve process couplings in the resulting mathematical model, it is common to apply fully implicit time stepping. This necessitates the use of an iterative linear solver to run the model. The solver's efficiency primarily depends on a robust preconditioner, which is particularly challenging to develop because it must handle the mutual couplings between linearized contact mechanics and energy, momentum, and mass balance. In this work, we introduce a preconditioner for the problem based on the nested approximations of Schur complements. To decouple the momentum balance, we utilize the fixed-stress approximation, extended to account for both the porous media and fracture subdomains. The singularity of the contact mechanics submatrix is resolved by a linear transformation. Two variations of the algorithm are proposed to address the coupled mass and energy balance submatrix: either the Constrained Pressure Residual or the System-AMG approach. The preconditioner is evaluated through numerical experiments of fluid injection into fractured porous media, which causes thermal contraction and subsequent sliding and opening of fractures. The experiments show that the preconditioner performs robustly for a wide range of simulation regimes governed by various fracture states, friction coefficients and Peclet number. The grid refinement experiments demonstrate that the preconditioner scales well in terms of GMRES iterations, in both two and three dimensions.

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↗

arXiv2025

An efficient preconditioner for mixed-dimensional contact poromechanics based on the fixed stress splitting scheme

Numerical simulation of fracture contact poromechanics is essential for various applications, including CO2 sequestration, geothermal energy production and underground gas storage. Modeling this problem accurately presents significant challenges due to the complex physics involved in strongly coupled poromechanics and frictional contact mechanics of fractures. The robustness and efficiency of the simulation heavily depends on a preconditioner for the linear solver, which addresses the Jacobian matrices arising from Newton's method in fully implicit time-stepping schemes. Developing an effective preconditioner is difficult because it must decouple three interdependent subproblems: momentum balance, fluid mass balance, and contact mechanics. The challenge is further compounded by the saddle-point structure of the contact mechanics problem, a result of the Augmented Lagrange formulation, which hinders the direct application of the well-established fixed stress approximation to decouple the poromechanics subproblem. In this work, we propose a preconditioner hat combines nested Schur complement approximations with a linear transformation, which addresses the singular nature of the contact mechanics subproblem. This approach extends the fixed stress scheme to both the matrix and fracture subdomains. We investigate analytically how the contact mechanics subproblem affects the convergence of the proposed fixed stress-based iterative scheme and demonstrate how it can be translated into a practical preconditioner. The scalability and robustness of the method are validated through a series of numerical experiments.

math.NA↗

arXiv2025

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

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.

physics.geo-ph↗

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

A hybrid upwind scheme for two-phase flow in fractured porous media

Simulating the flow of two fluid phases in porous media is a challenging task, especially when fractures are included in the simulation. Fractures may have highly heterogeneous properties compared to the surrounding rock matrix, significantly affecting fluid flow, and at the same time hydraulic aperture that are much smaller than any other characteristic sizes in the domain. Generally, flow simulators face difficulties with counter-current flow, generated by gravity and pressure gradients, which hinders the convergence of non-linear solvers (Newton). In this work, we model the fracture geometry with a mixed-dimensional discrete fracture network, thus lightening the computational burden associated to an equi-dimensional representation. We address the issue of counter-current flows with appropriate spatial discretization of the advective fluid fluxes, with the aim of improving the convergence speed of the non-linear solver. In particular, the extension of the hybrid upwinding to the mixed-dimensional framework, with the use of a phase potential upstreaming at the interfaces of subdomains. We test the method across several cases with different flow regimes and fracture network geometry. Results show robustness of the chosen discretization and a consistent improvements, in terms of Newton iterations, compared to use the phase potential upstreaming everywhere.

math.NA↗

arXiv2024

Automated solver selection for simulation of multiphysics processes in porous media

Porous media processes involve various physical phenomena such as mechanical deformation, transport, and fluid flow. Accurate simulations must capture the strong couplings between these phenomena. Choosing an efficient solver for the multiphysics problem usually entails the decoupling into subproblems related to separate physical phenomena. Then, the suitable solvers for each subproblem and the iteration scheme must be chosen. The wide range of options for the solver components makes finding the optimum difficult and time-consuming; moreover, solvers come with numerical parameters that need to be optimized. As a further complication, the solver performance may depend on the physical regime of the simulation model, which may vary with time. Switching a solver with respect to the dominant process can be beneficial, but the threshold of when to switch solver is unclear and complicated to analyze. We address this challenge by developing a machine learning framework that automatically searches for the optimal solver for a given multiphysics simulation setup, based on statistical data from previously solved problems. For a series of problems, exemplified by successive time steps in a time-dependent simulation, the framework updates and improves its decision model online during the simulation. We show how it outperforms preselected state-of-the-art solvers for test problem setups. The examples are based on simulations of poromechanics and simulations of flow and transport. For the quasi-static linear Biot model, we demonstrate automated tuning of numerical solver parameters by showing how the L-parameter of the so-called Fixed-Stress preconditioner can be optimized. Motivated by a test example where the main heat transfer mechanism changes between convection and diffusion, we discuss how the solver selector can dynamically switch solvers when the dominant physical phenomenon changes with time.

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

Copula modeling and uncertainty propagation in field-scale simulation of CO$_2$ fault leakage

Subsurface storage of CO$_2$ is an important means to mitigate climate change, and to investigate the fate of CO$_2$ over several decades in vast reservoirs, numerical simulation based on realistic models is essential. Faults and other complex geological structures introduce modeling challenges as their effects on storage operations are uncertain due to limited data. In this work, we present a computational framework for forward propagation of uncertainty, including stochastic upscaling and copula representation of flow functions for a CO$_2$ storage site using the Vette fault zone in the Smeaheia formation in the North Sea as a test case. The upscaling method leads to a reduction of the number of stochastic dimensions and the cost of evaluating the reservoir model. A viable model that represents the upscaled data needs to capture dependencies between variables, and allow sampling. Copulas provide representation of dependent multidimensional random variables and a good fit to data, allow fast sampling, and coupling to the forward propagation method via independent uniform random variables. The non-stationary correlation within some of the upscaled flow function are accurately captured by a data-driven transformation model. The uncertainty in upscaled flow functions and other parameters are propagated to uncertain leakage estimates using numerical reservoir simulation of a two-phase system. The expectations of leakage are estimated by an adaptive stratified sampling technique, where samples are sequentially concentrated to regions of the parameter space to greedily maximize variance reduction. We demonstrate cost reduction compared to standard Monte Carlo of one or two orders of magnitude for simpler test cases with only fault and reservoir layer permeabilities assumed uncertain, and factors 2--8 cost reduction for stochastic multi-phase flow properties and more complex stochastic models.

math.NA↗

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

Flexible and rigorous numerical modelling of multiphysics processes in fractured porous media using PorePy

Multiphysics processes in fractured porous media is a research field of importance for several subsurface applications and has received considerable attention over the last decade. The dynamics are characterised by strong couplings between processes as well as interaction between the processes and the structure of the fractured medium itself. The rich range of behavior calls for explorative mathematical modelling, such as experimentation with constitutive laws and novel coupling concepts between physical processes. Moreover, efficient simulations of the strong couplings between multiphysics processes and geological structures require the development of tailored numerical methods. We present a modelling framework and its implementation in the open-source simulation toolbox PorePy, which is designed for rapid prototyping of multiphysics processes in fractured porous media. PorePy uses a mixed-dimensional representation of the fracture geometry and generally applies fully implicit couplings between processes. The code design follows the paradigms of modularity and differentiable programming, which together allow for extreme flexibility in experimentation with governing equations with minimal changes to the code base. The code integrity is supported by a multilevel testing framework ensuring the reliability of the code. We present our modelling framework within a context of thermo-poroelasticity in deformable fractured porous media, illustrating the close relation between the governing equations and the source code. We furthermore discuss the design of the testing framework and present simulations showcasing the extendibility of PorePy, as well as the type of results that can be produced by mixed-dimensional simulation tools.

math.NA↗