**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:**

The flow of immiscible fluids in porous media is a complex problem at any length scale, being the pore scale measured in microns or the field scale measured in kilometers. The way one describes such flow at the pore scale must be very different from how one describes it at the field scale. Still, it is the same problem. The situation today is that one has a pretty good handle on how to describe the behavior of immiscible fluids at the pore scale. However, at the continuum scale has seen little progress over the last decades. The problem is that the complexity explodes as one tries to coarse grain the pore level description. One may perhaps say that the problem has been solved in principle, but no practical applications are possible with the current state of affairs. So, one is stuck with the eighty year old phenomenological and highly approximative relative permeability theory on the continuum scale.

It is worthwhile to look to a very different field where in fact the scale-up problem has been solved, namely statistical mechanics which binds together the description of the motion of atoms and molecules at the nano-scale with thermodynamics at the macro-scale. Let us be even more specific. Let us consider a magnet. The flipping spins of the atomic nuclei cause the magnet to be a magnet on the macro-scale. It sounds like a horribly complex problem to get from the microscopic dynamics governed by the interactions between the atoms in the magnet at the atomistic scale to the thermodynamics of magnets on the macro-scale. But, it turns out that the details at the atomic level are not that important. One may in fact use rather crude models, e.g., the Ising model at this scale. Such models may be seen as effective descriptions of the problem at a somewhat coarse grained scale. Statistical mechanics then takes one from this coarse grained scale to the macro-scale. How to do this in practice? The renormlization group techniques developed by Wilson and others in the seventies essentially solved the problem.

We now return to the immiscible flow in porous media problem. Comparing this field to that of statistical mechanics, one is still struggling

to define proper coarse grained models that can be used as input for an upscaling to the continuum scale, but progress is being made. Up to now, the question of what happens after such models have been constructed has not been addressed. Rather, one has assumed that this one-step coarse graining in fact constitutes the final coarse graining. It does not. This is in fact where an equivalent of statistical mechanics is necessary to bring the description to the continuum scale.

The aim of this Research Theme is to construct an equivalent to statistical mechanics for immisible and incompressible two-phase flow in porous media. And from this, construct an equivalent to thermodynamics where the variables are fluid currents and forces. Are we succeeding? Look at the papers listed below.

**Deliverables**

A comprehensive statistical mechanics description of immiscible two-phase flow in porous media linking pore scale description of the flow with Darcy scale differential equations controlling the flow patterns.

New variables describing the flow at the Darcy scale and relations between these variables resembling those found in thermodynamics.

**Investigators**

**Principal Investigator Research Theme 1**: Professor Alex Hansen.

**Partners**: José Soares Andrade Jr., Saman Aryas, Carl Fredrik Berg, Sarah Codd, Eirik Grude Flekkøy, Marcel Moura, Knut Jørgen Måløy, Alberto Rosso, Joseph Seymour, Laurent Talon

**Postdoctoral Fellows and Researchers:** Quirine Krol, Santanu Sinha

**PhD students:** Hursanay Fyhn, Federico Lanza, Håkon Pedersen.

**Selected articles:**

- A. Hansen, E. G. Flekkøy, S. Sinha and P. A. Slotte, “A statistical mechanics for immiscible and incompressible two-phase flow in porous media,” arXiv:2205.13791
- S. Roy, H. Pedersen, S. Sinha and A. Hansen, “The co-moving velocity in immiscible two-phase flow in porous media,” Transp. in Porous Media, 143, 69 (2022).
- S. Roy, S. Sinha and A. Hansen, “Flow-Area Relations in Immiscible Two-Phase Flow in Porous Media,” Front. Phys. 8, 4 (2020).
- A. Hansen, S. Sinha, D. Bedeaux, S. Kjelstrup, M. Aa. Gjennestad and M. Vassvik, “Relations Between Seepage Velocities in Two-Phase Flow in Homogeneous Porous Media,” Transp. Porous Med. 125, 565 (2018).