Objective:
To provide a set of equations and relations for immiscible two-phase flow in porous media connecting the physics at the pore scale with the macroscopic scale where the porous medium may be described as a continuum.
Description:
Porous media may be defined as shapes where the surface area scales as the volume. This makes the mathematical description of the physics associated with porous media extraordinarily difficult as surface effects can never be ignored on any scale.
Porous media typically span orders of magnitude in scales. The molecular scale is in the nanometer range. Next comes the pore scale, typically on the micrometer scale. Next step towards increasing scales is the Darcy scale. This is the scale at which it makes sense to treat the porous medium as a continuum and still be seen as homogeneous. The last distinct scale is the reservoir scale, where inhomogeneities become important.
We may scale up from one scale to the next. Scaling up from the molecular scale to the pore scale is essentially the approach which leads from molecules to the Navier-Stokes equation and thermodynamics. Scaling up from the pore scale to the Darcy scale, however, is an essentially unsolved problem in the context of immiscible two-phase flow in porous media. It is furthermore the main theme of this Research Theme. The result of such an upscaling should be a set of differential equations reflecting the fluid dynamics at the pore scale. The scaling-up from the Darcy scale to the reservoir scale is also a challenge. The typical situation is that there are heterogeneities across all scales from the Darcy scale and up. How can one resolve these heterogeneities when constructing grid blocks?
So, the problem is to find the right equations describing immiscible two-phase flow in porous media at the Darcy scale. All practical applications today rely on the relative permeability framework. This framework, dating back to the mid-thirties, is purely phenomenological, and has clear weaknesses.
There exists in physics an upscaling procedure that works, namely equilibrium statistical mechanics. When used to describe a gas or a liquid, it takes us from a mechanical description on the molecular scale to a continuum scale description, which in this case is thermodynamics.
By its nature, equilibrium statistical mechanics demands that the system in question is in equilibrium; that is one cannot detect which way time proceeds. This is clearly not the case for flow in porous media. Filming the flow, it is easy to see whether the film runs forwards or backwards. This necessitates a complete rethinking of how statistical mechanics may be implemented for this problem. Our hypothesis is that it is possible to construct an equilibrium statistical mechanics based on Shannon’s information entropy for immiscible two-phase flow in porous media. The key is to shift focus from thermodynamic entropy, which is produced through viscous dissipation, to Shannon’s information entropy which measures the structure of the hydrodynamic flow field. This entropy is constant under steady-state flow and forms the basis for a statistical mechanics.
This approach is different from and unrelated to “standard” non-equilibrium statistical mechanics and thermodynamics, which focuses on the production of thermodynamic entropy together with conservation laws.
A successful outcome for Research Theme 1 has interest beyond the field of porous media. It is the first time a statistical mechanics framework for non-thermal, non-equilibrium systems with internal dynamics has been proposed. There are two examples of non-thermal systems where statistical mechanics is used: percolation theory and the Edwards statistical mechanics for granular packings. Neither of these describe dynamics.
This research theme is closely related to the ERC Advanced Grant AGIPORE, held by Alex Hansen.
Deliverables
To develop a set of equations for immiscible two-phase flow in porous media at the Darcy scale derived from the pore scale physics, which are simple enough to be useful in practical applications.
Investigators
Principal Investigator Research Theme 1: Professor Alex Hansen.
Collaborators: Dr. Santanu Sinha, Adjunct Professors Saman Aryana, Steffen Berg, and Sauro Succi.
PhD students: Eivind Bering, Zhengwei Chen, Hursanay Fyhn, Magnus Aashammer Gjennestad, Jonas Tøgersen Kjellstadli, Federico Lanza, Bjarte Matre, Anders Daltveit Melve, Håkon Pedersen, Isha Savani
Postdocs: Martin Hendrick, Quirine Krol, Subhadeep Roy, Pooja Singh, Mathias Winkler
Selected articles:
- Hansen, S. Sinha, D. Bedeaux, S. Kjelstrup, M. Aa. Gjennestad and m. Vassvik, Relations between seepage velocities in immiscible, incompressible two-phase flow in porous media, Transport in Porous Media, 125, 565 (2018); https://doi.org/10.1007/s11242-018-1139-6.
- Roy, S. Sinha and A. Hansen, Flow-area relations in immiscible two-phase flow in porous media, Front. Phys. 8, 4 (2020); https://doi.org/10.3389/fphy.2020.00004.
- Roy, H. Pedersen, S. Sinha and A. Hansen, The co-moving velocity in immiscible two-phase flow in porous media, Transport in Porous Media, 143, 69 (2022); https://doi.org/10.1007/s11242-022-01783-7.
- Feder, E. G. Flekkøy and A. Hansen, Physics of Flow in Porous Media (Cambridge U. Press, Cambridge, 2022).
- Hansen, E. G. Flekkøy, S. Sinha and P. A. Slotte, A statistical mechanics framework for immiscible and incompressible two-phase flow in porous media, Adv. Water Res. 171, 104336 (2023); https://doi.org/10.1016/j.advwatres.2022.104336.
- Pedersen and A. Hansen, Parameterizations of immiscible two-phase flow in porous media, Front. Phys. 11, 1127345 (2023); https://doi.org/10.3389/fphy.2023.1127345.
- Fyhn, S. Sinha and A. Hansen, Local statistics of immiscible and incompressible two-phase flow in porous media, Physica A, 616, 128626 (2023); https://doi.org/10.1016/j.physa.2023.128626.
- Alzubaidi, J. E. McClure, H. Pedersen, A. Hansen, C. F. Berg, P. Mostaghimi and R. T. Armstrong, The Impact of wettability on the co-moving velocity of two-fluid flow in porous media, Transport in Porous Media, 151, 1967 (2024); https://doi.org/10.1007/s11242-024-02102-y.
- Hansen, Linearity of the co-moving velocity, Transport in Porous Media, 151, 2477 (2024); https://doi.org/10.1007/s11242-024-02121-9.
- Hansen and S. Sinha, Thermodynamics-like formalism for immiscible and incompressible two-phase flow in porous media, Entropy, 27, 121 (2025); http://doi.org/10.3390/e27020121.
- P. Olsen, B. Hafskjold, A. Lervik and A. Hansen, A new thermodynamic function for binary mixtures: The co-molar volume, J. Chem. Phys. 163, 184504 (2025); https://doi.org/10.1063/5.0302106.
- Berg, R. T. Armstrong, M. Rücker, A. Hansen, S. Kjelstrup and D. Bedeaux, From interface dynamics to Darcy scale description of multiphase flow in porous media, arXiv:2510.19582; https://doi.org/10.48550/arXiv.2510.19582.