# About the Porous Media Group

The porous media group is located at the Department of Mathematics and collaborates with scientists from physics, geophysics, geology, chemistry, and medicine. Several long-term research topics are actively pursued within this collaboration, broadly based around either processes in geophysical porous materials (as applied to geothermal energy, CO2 and energy storage, and oil and gas recovery) and image processing (as applied to seismic data and medical imaging).

## Hovedinnhold

For more details on the individual projects connected to the applications described below, see the Projects page.

## Subsurface Energy Storage

Intermittent renewable energy (IRE) such as wind and solar are among the fastest growing energy sources and are critical for the transition to a low carbon emission society. A characteristic feature of these IRE sources is their great variability on both short (sub-daily) and long (seasonal) time-scales. In contrast, the majority of industrial demand, as well as seasonal household consumption such as heating and cooling, remain fixed or seasonal in nature.

A portfolio of scalable solutions must be deployed to mitigate the production-consumption gap. Today, conventional energy resources such as hydroelectric and fossil fuels are used to buffer the comparatively small output from IRE sources, however, this strategy becomes less viable as the market share of IRE increases. Furthermore, transmission bottlenecks in the power grid and inertia in the implementation of smart grids and local-scale batteries also call for additional storage solutions.

Illustration of subsurface energy storage systems in the context of intermittent renewable energy

We are interested in subsurface energy storage in the context of energy storage in geological permeable layers such as saline aquifers or depleted oil and gas reservoirs. These geological features are found globally, and as such provide an attractive venue for energy storage. Our particular interest lies in thermo-mechanical storage: hot fluids at temperatures above 50 °C are injected at elevated pressure, and both thermal and mechanical energy can be recovered during extraction.

To date, large-scale thermo-mechanical subsurface energy storage has not been applied in the context of IRE. Some experience has been gained over the last decades with high-temperature aquifer thermal energy storage (HT-ATES), and research needs related to fracturing of the confining layers and dissolution and precipitation due to water chemistry and temperatures have been identified. Our research is initiated around the following topics:

- Cyclical thermo-mechanical weakening of the rock leading to continuous development of fracture networks during operation.
- High fluid pressure fluctuations and high thermal fluctuations leading to coupled flow-mechanical-chemical systems.
- Extreme flow-rates needed to accommodate rapid oscillations in the production-demand gap leading to non-equilibrium flow conditions.

Contact person: Jan Martin Nordbotten

## Geothermal Energy

Unlocking the world’s vast geothermal energy resources depends on our ability to engineer profitable geothermal reservoirs, characterized by sufficient permeability corresponding to the distributed contact area between mobile fluids and hot reservoir rock.

Simulated temperature distribution during production from highly fractured rock

The reservoir performance of many geothermal systems depends on the presence of open interconnected fracture networks, or the ability to create such networks. In geothermal reservoirs, fractures increase not only permeability but also the contact surface between the rock and the fluid, which facilitates heat transport. Distributed and connected networks of fractures enable production from large reservoir volumes, but fractures may also result in significant water losses and a decrease in sweep efficiency due to water short-circuiting between injectors and producers.

The PMG at the University of Bergen has had a sustained effort on developing mathematical models and numerical tools for the stimulation and production of geothermal reservoirs. Research activities involve mathematical modeling, upscaling, and development of simulation tools for processes in fractured geothermal reservoirs, including coupled THCM processes. Several of the group’s ongoing research projects supporting this activity include strong inter-disciplinary and inter-sectoral collaboration.

Contact person: Inga Berre

## Enhanced Oil Recovery (EOR)

Efficient recovery is essential for optimal exploitation of existing and new oilfields. In spite of technological advances, between 25 and 50 % of the oil is left in reservoirs after conventional recovery. EOR refers to the techniques used to recover the rest of the oil in the reservoir. The success and optimization of an EOR technique is crucially dependent on mathematical models and numerical simulations.

Our group is currently involved in four interdisciplinary projects involving EOR. The projects focus on microbial EOR (MEOR), nano-particle EOR (NANO-EOR), CO_{2}-EOR and polymer EOR. The challenges behind all these projects are in dealing with the multi-phase, multi-scale, multi-physics character of the problems and to design reliable and efficient simulators. To exemplify, we present below more details of the MEOR project.

MEOR technologies are environmentally sound and cost-efficient tertiary recovery methods for water-injected oil reservoirs. Norway has a leading position in MEOR, with Statoil taking a first use position through the offshore development at Norne and later at the Statfjord field.

Two-phase flow through smooth and fluvial layers from the SPE 10 benchmark

MEOR is a complex process, where flow of multiple phases and their interaction with the rock surface are affected by bacterial growth and activity, metabolic components and structural changes. The project aims at studying MEOR mechanisms involved in adaptive bio-plugging of highly permeable structures in heterogeneous reservoir formations. The processes involved in MEOR are encountered at various scales, but is initiated at pore scale. Our approach is to develop a pore scale and core model where the mathematical modelling is based on experimental data from both scales.

The objective of the MEOR project is to develop accurate mathematical models and numerical approaches for MEOR, capable of describing the relevant biological, physical and chemical processes occurring at various scales. Given the complexity of MEOR, this can only be gained by combining the practical insights from the laboratory experiments with a consistent treatment of the pore scale and transition to a heterogeneous porous system at Darcy scale, from both analytical and numerical point of view. This challenging task requires a tight, interdisciplinary collaboration.

Contact person: Kundan Kumar

## Mixed-dimensional modeling

Two trends are evident in modern science: A pursuit of ever more complex models, and a desire to use computer simulation to give these models predictive capabilities. At the same time, CPU speeds have stagnated, and while high-performance computing resources are available, the majority of computations are still performed on desktops with very limited parallelization. Common engineering computations are therefore conducted on 10^{5}-10^{8} degrees of freedom, which in three dimensions implies that only 2-3 orders of magnitude separate the scale of the computational grid from the full computational domain. Despite significant advances in grid adaptivity, numerical multiscale methods, and linear solvers, resolving high-aspect features in computational domains thus remains a significant

Many high-aspect features can be adequately modeled on a manifold embedded in an ambient space of higher dimensionality. Classical examples are rods (1D), shells and membranes (2D) from mechanics; pumping wells (1D) and groundwater aquifers (2D) from hydrology; and the vascular network of the human body (1D network with 0D links). When physical models of different topological dimensions are coupled, we refer to the resulting system as mixed-dimensional (abbreviated mD herein: other researchers have used hybrid-dimensional or multiscale as nomenclature). Mixed-dimensional systems are inherently challenging from a mathematical perspective since from the perspective of the higher-dimensional domain, the inclusion of a physical model on a lower-dimensional manifold will often lead to singularities in the solution.

Mixed-dimensional (2d-3d) geometry

While the examples above show that physical models based on dimension reduction have a long and rich history, mixed-dimensional models have only recently started to receive serious attention as a field in itself. Thus much of the mathematical foundation which is taken granted for fixed-dimensional equations, such as generalizations of vector and tensors, differential operators on these, and the resulting Hilbert and Sobolev spaces, is not available in a systematic and unified way for mixed-dimensional equations.

Consequently, our ongoing research has the overarching main objective to establish modeling, analysis, and numerical methods for mixed-dimensional partial differential equations.

Contact person: Jan Martin Nordbotten

## Discretisation Techniques

Typical mathematical models for the applications above are fully coupled systems of partial differential equations on complex domains. A special focus of our group is on mass conservative and structure-preserving discretization methods (FV, MPFA, MFEM), and on higher-order space-time elements (continuous or discontinuous Galerkin in time).

The group has developed several discretization schemes including a finite volume method (MPSA) for elasticity which allows for the application of a pure finite volume discretization (MPSA-MPFA) for thermo-poroelasticity. Recently, a special focus has been on the development of discretization schemes and a posteriori analysis for flow in (and deformation of) fractured media, ultimately implemented in the in-house simulator PorePy.

Flow as predicted by a new hierarchical mixed finite element discretization for fractured media

Another core area of our research is the development of multiscale simulators by homogenisation or numerical multiscale methods. We apply homogenisation to develop new models, which adequately describe the evolving geometry at the micro-scale (due to e.g. precipitation/dissolution processes or biofilm formation). These models are capable of describing features like clogging or structure damage in a rational manner. Multigrid methods can be obtained by defining projectors and restriction operators based on homogenisation.

Contact person: Eirik Keilegavlen

## Solvers

Typical mathematical models for the applications above are systems of non-linear, fully coupled partial differential equations, with coefficients ranging over many order of magnitudes, which are very challenging to be solved. A special focus of our group is on iterative linear (iterative coupling) and non-linear solvers (Newton method, L-scheme or combination of them) for coupled and non-linear problems. We particularly emphasise our recent works on multi-phase flow, poromechanics and upscaling.

Regarding multi-phase flow: we have developed a very robust iterative scheme for solving two-phase flow in porous media or for the Richards equation. The scheme, called L-scheme, is only first order convergent but does not employ the computation of any derivatives and, moreover, the linear systems to be solved within each iteration are typically much better conditioned than the corresponding systems in the Newton method. One can also combine the robust L-scheme with the quadratic convergent Newton method by performing a few iterations using the L-scheme and then switching to Newton (based on a posteriori indicators).

Our group has developed a range of iterative schemes for (non-)linear poromechanics with emphasis on nonlinear constitutive laws, unsaturated poromechanics, and thermo-poromechanics with convection. The methods are extensions of the widely used fixed-stress split. Furthermore, Anderson acceleration can be applied to accelerate the schemes, similar to preconditioning for linear problems. A special focus is also on developing new solvers for coupled multi-phase flow and mechanics. These challenges must always be seen in context with the need to handle the presence of fractures in the material.

Contact person: Florin Radu

Our research will formulate and investigate an alternative interpretation strategy where the tracer concentration for an individual voxel will be considered in the context of a global flow problem that connects all voxels in the image domain. By modelling the flow between voxels from first principles and calibrate the models to observations via systematic assimilation techniques that include rigorous error estimates, our goal is to advance understanding and clinical utility of dynamic imaging interpretation.

Contact person: Erik Hanson