The Impact of Pore‐Scale Flow Regimes on Upscaling of Immiscible Two‐Phase Flow in Porous Media

Picchi, Davide; Battiato, Ilenia

doi:10.1029/2018wr023172

Cited by 52 publications

(73 citation statements)

References 99 publications

(163 reference statements)

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“…In the proposed model, the flow regime parameter does not appear in due to the choice of the CAF as the viscous limiting case. A possibility to account for k w dependence on α is the choice of the plug flow closures (section 4.4 in Picchi & Battiato, ) as the viscous limit. However, this choice would require the determination of an additional parameter, i.e., the film thickness, and, to the best of our knowledge, a model which correlates the film thickness and the capillary number for ganglia flow in a porous matrix has not been proposed yet.…”

Section: Discussionmentioning

confidence: 99%

“…In this section we propose a semiempirical generalization of the relative permeabilities developed in Picchi and Battiato () to account for pore‐scale flow regimes in complex porous media. For completeness and clarity, we first briefly review some of the results derived by Picchi and Battiato (, section ) and, then, we present its generalization to model complex porous media (sections and ).…”

Section: Theoretical Considerationsmentioning

confidence: 99%

“…The general model reduces to Darcy's law when (i) the contribution of inertia is negligible, i.e., at low Reynolds numbers; (ii) the driving force is the same for both phases; (iii) Eotvos number is small at the pore scale: this condition is often satisfied in real rocks where the characteristic length scale of the pores is sufficiently small to consider microgravity conditions. Since most of the data available in the literature and used in this study are collected at low Reynolds and Eotvos numbers, here we report results in the Darcy's limit only (see Picchi & Battiato, , for a complete set of equations), i.e.,

(⟨⟩, {\overset{u}{true}}_{w}) = - \frac{k_{w}}{32} \nabla p_{w},

(⟨⟩, {\overset{u}{true}}_{n w}) = - \frac{k_{n w}}{32 M} \nabla p_{n w},

where

⟨ {\overset{u}{true}}_{w} ⟩

⟨ {\overset{u}{true}}_{n w} ⟩

, k w , k nw , p w , and p nw are the average velocity, the relative permeabilities, and the macroscopic pressures of the wetting and the nonwetting phases, respectively, and ⟨ · ⟩ is the average over a representative volume, while

\overset{\cdot}{true}

is the temporal average over a representative time interval. The viscosity ratio M is defined as

M = \frac{μ_{n w}}{μ_{w}},

where μ w and μ nw are the dynamic viscosities of the wetting phase and the nonwetting phase, respectively.…”

Section: Theoretical Considerationsmentioning

confidence: 99%

“…The dependence on pore‐scale flow regimes is embedded in the relative permeabilities since they are directly affected by the spatial distribution and connectivity of the flowing phases. For the capillary tube analogy, they can be analytically expressed through one‐dimensional closures relationships (Picchi et al, , ; Ullmann & Brauner, ) for the quasi‐static CP and the CAF regimes (Picchi & Battiato, ). Specifically, the CP regime can be conceptualized as a bundle of capillary tubes, see Figure a (right), and leads to the following relative permeability‐saturation relationship

k_{w, CP} = S_{w}^{2}, k_{n w, CP} = {false(1 - S_{w} false)}^{2} .

…”

Section: Theoretical Considerationsmentioning

confidence: 99%

“…The macroscale equations presented in Picchi and Battiato () include a conservation law for the saturation that, in the Darcy's limit, can be cast in terms of fractional flow, i.e.,

\frac{\partial S_{w}}{\partial t} + ⟨ \overset{u}{true} ⟩ \cdot \nabla f_{w} = 0,

where

⟨ \overset{u}{true} ⟩ = (⟨⟩, {\overset{u}{true}}_{w}) + (⟨⟩, {\overset{u}{true}}_{n w})

is the total Darcy velocity and f w is the fractional flow of the wetting phase defined as

f_{w} = \frac{q_{w}}{q_{w} + q_{n w}} = \frac{1}{1 + \frac{q_{n w}}{q_{w}}},

with

q_{w} = A (⟨⟩, {\overset{u}{true}}_{w})

and

q_{n w} = A (⟨⟩, {\overset{u}{true}}_{n w})

the volumetric flow rates of the wetting and nonwetting phases, respectively, and A the cross‐sectional area. Under the hypothesis that two‐phase Darcy's law is valid, equations (1) and can be combined to obtain

f_{w} = \frac{1}{1 + \frac{1}{M} \frac{k_{n w}}{k_{w}}} .

…”

Section: Theoretical Considerationsmentioning

confidence: 99%

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Relative Permeability Scaling From Pore‐Scale Flow Regimes

Picchi

Battiato

2019

Water Resources Research

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Relative permeability measurements are commonly fitted with the Corey and Brooks‐Corey correlations. Despite such correlations fit experimental data generally well, they are mostly of an empirical nature. Here, we propose a semiempirical model to determine relative permeabilities of the wetting and the nonwetting phases in real 3‐D porous media that accounts for pore‐scale flow regimes. The starting point is the homogenization framework proposed by Picchi and Battiato (2018, https://doi.org/10.1029/2018WR023172), where the upscaling is conducted for different spatial distributions of the flowing phases in the capillary tube setting. First, we extend the approach to realistic media by allowing pore‐scale flow regimes to coexist in a complex geometry while accounting for capillary and viscous limits in the dynamics. Then, we discuss the scaling behavior of normalized relative permeabilities in terms of the phases viscosity ratio and identify three classes which govern their scaling. We also derive an analytical expression for the fractional flow. Finally, we provide a detailed validation of the proposed model for both relative permeabilities and fractional flow against data from numerical simulations and experiments available in the literature. The data set used for validation covers a wide range of systems, ranging from brine‐CO2 to oil‐water flows. The equations derived capture well the trend of both numerical and experimental data.

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Section: Discussionmentioning

confidence: 99%

Section: Theoretical Considerationsmentioning

confidence: 99%

(⟨⟩, {\overset{u}{true}}_{w}) = - \frac{k_{w}}{32} \nabla p_{w},

(⟨⟩, {\overset{u}{true}}_{n w}) = - \frac{k_{n w}}{32 M} \nabla p_{n w},

where

⟨ {\overset{u}{true}}_{w} ⟩

⟨ {\overset{u}{true}}_{n w} ⟩

\overset{\cdot}{true}

is the temporal average over a representative time interval. The viscosity ratio M is defined as

M = \frac{μ_{n w}}{μ_{w}},

where μ w and μ nw are the dynamic viscosities of the wetting phase and the nonwetting phase, respectively.…”

Section: Theoretical Considerationsmentioning

confidence: 99%

k_{w, CP} = S_{w}^{2}, k_{n w, CP} = {false(1 - S_{w} false)}^{2} .

…”

Section: Theoretical Considerationsmentioning

confidence: 99%

“…The macroscale equations presented in Picchi and Battiato () include a conservation law for the saturation that, in the Darcy's limit, can be cast in terms of fractional flow, i.e.,

\frac{\partial S_{w}}{\partial t} + ⟨ \overset{u}{true} ⟩ \cdot \nabla f_{w} = 0,

where

⟨ \overset{u}{true} ⟩ = (⟨⟩, {\overset{u}{true}}_{w}) + (⟨⟩, {\overset{u}{true}}_{n w})

is the total Darcy velocity and f w is the fractional flow of the wetting phase defined as

f_{w} = \frac{q_{w}}{q_{w} + q_{n w}} = \frac{1}{1 + \frac{q_{n w}}{q_{w}}},

with

q_{w} = A (⟨⟩, {\overset{u}{true}}_{w})

and

q_{n w} = A (⟨⟩, {\overset{u}{true}}_{n w})

f_{w} = \frac{1}{1 + \frac{1}{M} \frac{k_{n w}}{k_{w}}} .

…”

Section: Theoretical Considerationsmentioning

confidence: 99%

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Relative Permeability Scaling From Pore‐Scale Flow Regimes

Picchi

Battiato

2019

Water Resources Research

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Real-time imaging reveals distinct pore scale dynamics during transient and equilibrium subsurface multiphase flow

Spurin

Bultreys

Rücker

et al. 2020

Preprint

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Many subsurface fluid flows, including the storage of CO 2 underground or the production of oil, are transient processes incorporating multiple fluid phases. The fluids are not in equilibrium meaning macroscopic properties such as fluid saturation and pressure vary in space and time. However, these flows are traditionally modeled with equilibrium (or steady-state) flow properties, under the assumption that the pore-scale fluid dynamics are equivalent. In this work, we used fast synchrotron X-ray tomography with 1 s time resolution to image the pore-scale fluid dynamics as the macroscopic flow transitioned to steady state. For nitrogen or decane, and brine injected simultaneously into a porous rock, we observed distinct pore-scale fluid dynamics during transient flow. Transient flow was found to be characterized by intermittent fluid occupancy, whereby flow pathways through the pore space were constantly rearranging. The intermittent fluid occupancy was largest and most frequent when a fluid initially invaded the rock. But as the fluids established an equilibrium the dynamics decreased to either static interfaces between the fluids or small-scale intermittent flow pathways, depending on the capillary number and viscosity ratio. If the fluids were perturbed after an equilibrium was established, by changing the flow rate, the transition to a new equilibrium was quicker than the initial transition. Our observations suggest that transient flows require separate modeling parameters. The time scales required to achieve equilibrium suggest that several meters of an invading plume front will have flow properties controlled by transient pore-scale fluid dynamics.

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