Newly developed high-speed, synchrotron-based X-ray computed microtomography enabled us to directly image pore-scale displacement events in porous rock in real time. Common approaches to modeling macroscopic fluid behavior are phenomenological, have many shortcomings, and lack consistent links to elementary pore-scale displacement processes, such as Haines jumps and snap-off. Unlike the common singular pore jump paradigm based on observations of restricted artificial capillaries, we found that Haines jumps typically cascade through 10–20 geometrically defined pores per event, accounting for 64% of the energy dissipation. Real-time imaging provided a more detailed fundamental understanding of the elementary processes in porous media, such as hysteresis, snap-off, and nonwetting phase entrapment, and it opens the way for a rigorous process for upscaling based on thermodynamic models.

https://www.pnas.org/content/pnas/110/10/3755.full.pdf

Real-time 3D imaging of Haines jumps in porous media flow

The extended Darcy’s law is a commonly used equation for the description of immiscible two-phase flow in porous media. It dates back to the 1940s and is essentially an empirical relationship. According to the extended Darcy’s law, pressure gradient and gravity are the only driving forces for the flow of each fluid. Within the last two decades, more advanced and physically based descriptions for multiphase flow in porous media have been developed. In this work, the extended Darcy’s law is compared to a thermodynamically consistent approach which explicitly takes the important role of phase interfaces into account, both as entities and as parameters. In this theoretically derived approach, forces related to capillarity and interfaces appear as driving/resisting forces, in addition to the traditional terms. It turns out that the extended Darcy’s law and the thermodynamically based approach are compatible if either (i) relative permeabilities are a function of saturation only, but capillary pressure is a function of saturation and specific interfacial area or (ii) relative permeabilities are a function of saturation and saturation gradients. Theoretical considerations suggest that the former alternative is only valid in case of reversible displacement while in the general case (irreversible displacement), the latter alternative is relevant.

/pdf/comparison-of-two-phase-darcy-s-law-with-a-thermodynamically-1886u4vb23.pdf

Comparison of Two-Phase Darcy's Law with a Thermodynamically Consistent Approach

Modified Johnson–Bossler–Naumann method to incorporate capillary pressure boundary conditions in drainage relative permeability estimation


 Two-phase immiscible drainage experiments are performed on strongly water-wet sandstone sample to assess and illustrate capillary end effects at different rates on relative permeability. All the experiments were done under unsteady state conditions with oil displacing brine in fully brine saturated cores. First, relative permeability curves are calculated using the Johnson–Bossler–Naumann (JBN) method (Johnson et al. 1959). Then, modified JBN model is defined to account for capillary pressure. Lastly, numerical simulator is used to generate relative permeability curves based on history matching. Results of the experiments show that as the rate increases (higher capillary number), the water relative permeability increases while the irreducible water saturation decreases. With high rate, less water saturation is being accumulated at the outlet hence less irreducible water and higher relative permeability. Moreover, the oil relative permeability curves are getting higher with higher rates confirming CEE impact on the tests. The calculated irreducible water saturation based on the core weights re-affirms the JBN calculation and decreases with higher rates. This paper proves capillary end effects on relative permeability curves at low rates which have low capillary numbers on water-wet sandstone cores.

Effect of Injection Rate on Oil Water Relative Permeability Accounting for Capillary Pressure and Outlet end Effects

/pdf/comparison-of-two-phase-darcy-s-law-with-a-thermodynamically-3alxmj16vi.pdf

Comparison of two-phase Darcy's law with a thermodynamically consistent approach

k , ro k are compared to corresponding ones measured with the steady-state method on the same porous medium. Potential differences of the two-phase flow coefficients are interpreted in the light of the transient growth pattern, and the steady-state two-phase flow regimes identified at the pore network scale. The macroscopic two-phase flow equations of the transient immiscible displacement of two fluids in a 2-D porous medium are simulated with finite elements by using as input data the steady-state and transient rw k , ro k and ignoring or accounting for the capillary pressure term. For the same porous medium and comparable experimental conditions, the steady-state relative permeability functions may differ significantly from the corresponding transient ones, whereas both are sensitive to the capillary number and wettability.

/pdf/simulation-of-the-immiscible-displacement-in-porous-media-55gks0mp04.pdf

Simulation of the immiscible displacement in porous media using capillary pressure and relative permeability curves from transient and steady-state experiments

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Simulation of the immiscible displacement in porous media using capillary pressure and relative permeability curves from transient and steady-state experiments