Multiphase flow modeling in multiscale porous media: An open-source micro-continuum approach

Carrillo, Francisco J.; Bourg, Ian C.; Soulaine, Cyprien

doi:10.1016/j.jcpx.2020.100073

Cited by 49 publications

(99 citation statements)

References 90 publications

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“…As shown in Carrillo et al. (2020),

{bold-italicF}_{c, 1} = ({, \begin{matrix} left \\ left - \frac{γ}{ϕ_{f}} \nabla \cdot (n_{w, n}) \nabla α_{w} & left in solid ‐ free regions \\ left - p_{c} \nabla α_{w} & left in porous regions \end{matrix})

{bold-italicF}_{c, 2} = ({, \begin{matrix} left \\ left 0 & left in solid ‐ free regions \\ left M^{- 1} (M_{w} α_{n} - M_{n} α_{w}) (\nabla p_{c} + (ρ_{w} - ρ_{n}) g) & left in porous regions \end{matrix})

where p c is the average capillary pressure within a given averaging volume, γ is the fluid‐fluid interfacial tension, M i = k 0 k i , r / μ i is the mobility of each fluid (a function of absolute permeability k 0 and relative permeability k i , r ), and M = M w + M n is the single‐field fluid mobility. Lastly, n w , n is the unit normal direction of the fluid‐fluid interface as calculated by the Continuum Surface Force (CSF) formulation (Brackbill et al., 1992).…”

Section: Model Derivationmentioning

confidence: 62%

“…As described in Carrillo and Bourg (2019) and Carrillo et al. (2020), these terms can be recast into the following expression through asymptotic matching to the standard multiphase Darcy equations:

\begin{matrix} right \frac{\partial ρ_{f} U_{f}}{\partial t} + \nabla \cdot (\frac{ρ_{f}}{ϕ_{f}} {U_{f} U}_{f}) = & left - ϕ_{f} \nabla p + ϕ_{f} ρ_{f} g + \nabla \cdot \overset{bold-italicS}{true} \\ right & left - ϕ_{f} μ k^{- 1} (U_{f} - {\bar{U}}_{s}) + ϕ_{f} F_{c, 1} + ϕ_{f} F_{c, 2} \end{matrix}

where μk −1 is the drag coefficient (a function of the fluid viscosities and permeability k ),

{\overset{bold-italicU}{true}}_{s}

is the averaged solid velocity,

ϕ_{f} μ k^{- 1} ({bold-italicU}_{f} - \bar{U}_{s})

is a solid‐fluid momentum exchange term that accounts for a moving porous medium in an Eulerian frame of reference, and F c , i represents the forces emanating from fluid‐fluid and fluid‐solid capillary interactions. As shown in Carrillo et al.…”

Section: Model Derivationmentioning

confidence: 94%

“…The equations presented above tend toward the standard Navier‐Stokes Volume‐of‐Fluid approach in solid‐free regions (where the drag term becomes negligible) and toward the multiphase Darcy equations in porous regions. The latter can be explained by the fact that the viscous stress tensor

\nabla \cdot \bar{S}

becomes negligible under the scale‐separation assumption, inertial terms become negligible under the assumption of low Reynold's number flow in the porous medium, and the F c , terms are set to fit standard multiphase Darcy's law (Carrillo et al., 2020; Whitaker, 1986):

Eqn. 14 \approx ({, \begin{matrix} left \frac{\partial ρ_{f} U_{f}}{\partial t} + \nabla \cdot (ρ_{f} {U_{f} U}_{f}) = - \nabla p + \nabla \cdot \overset{bold-italicS}{true} + ρ_{f} g + F_{c, 1} & left in solid ‐ free regions \\ left (U_{f} - {\bar{U}}_{s}) = - \frac{k}{μ} (\nabla p - ρ_{f} g - F_{c, 1} - F_{c, 2}) & left in porous regions \end{matrix})

…”

Section: Model Derivationmentioning

confidence: 99%

“…Given that F c ,1 = − p c ∇ α w in the porous domains (Carrillo et al., 2020), the previous equation can be rearranged to obtain

{bold-italicB}_{c a p} = ϕ_{s} {bold-italicF}_{c, 1} - ϕ_{f} {bold-italicF}_{c, 2}

…”

Section: Model Derivationmentioning

confidence: 99%

“…(2019) and Carrillo et al. (2020) extensively benchmarked and released an open‐source extension of the microcontinuum framework for multiphase flow in static multiscale porous media. This allowed accurate modeling of complex systems such as multiphase flow in a fractured porous medium, methane extraction from tight porous media, and wave absorption in coastal barriers.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Modeling Multiphase Flow Within and Around Deformable Porous Materials: A Darcy‐Brinkman‐Biot Approach

Carrillo

Bourg

2021

Water Resources Research

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We present a new computational fluid dynamics approach for simulating two‐phase flow in hybrid systems containing solid‐free regions and deformable porous matrices. Our approach is based on the derivation of a unique set of volume‐averaged partial differential equations that asymptotically approach the Navier‐Stokes Volume‐of‐Fluid equations in solid‐free regions and multiphase Biot Theory in porous regions. The resulting equations extend our recently developed Darcy‐Brinkman‐Biot framework to multiphase flow. Through careful consideration of interfacial dynamics (relative permeability and capillary effects) and extensive benchmarking, we show that the resulting model accurately captures the strong two‐way coupling that is often exhibited between multiple fluids and deformable porous media. Thus, it can be used to represent flow‐induced material deformation (swelling, compression) and failure (cracking, fracturing). The model's open‐source numerical implementation, hybridBiotInterFoam, effectively marks the extension of computational fluid mechanics into modeling multiscale multiphase flow in deformable porous systems. The versatility of the solver is illustrated through applications related to material failure in poroelastic coastal barriers and surface deformation due to fluid injection in poro‐visco‐plastic systems.

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“…As shown in Carrillo et al. (2020),

{bold-italicF}_{c, 1} = ({, \begin{matrix} left \\ left - \frac{γ}{ϕ_{f}} \nabla \cdot (n_{w, n}) \nabla α_{w} & left in solid ‐ free regions \\ left - p_{c} \nabla α_{w} & left in porous regions \end{matrix})

{bold-italicF}_{c, 2} = ({, \begin{matrix} left \\ left 0 & left in solid ‐ free regions \\ left M^{- 1} (M_{w} α_{n} - M_{n} α_{w}) (\nabla p_{c} + (ρ_{w} - ρ_{n}) g) & left in porous regions \end{matrix})

Section: Model Derivationmentioning

confidence: 62%

\begin{matrix} right \frac{\partial ρ_{f} U_{f}}{\partial t} + \nabla \cdot (\frac{ρ_{f}}{ϕ_{f}} {U_{f} U}_{f}) = & left - ϕ_{f} \nabla p + ϕ_{f} ρ_{f} g + \nabla \cdot \overset{bold-italicS}{true} \\ right & left - ϕ_{f} μ k^{- 1} (U_{f} - {\bar{U}}_{s}) + ϕ_{f} F_{c, 1} + ϕ_{f} F_{c, 2} \end{matrix}

where μk −1 is the drag coefficient (a function of the fluid viscosities and permeability k ),

{\overset{bold-italicU}{true}}_{s}

is the averaged solid velocity,

ϕ_{f} μ k^{- 1} ({bold-italicU}_{f} - \bar{U}_{s})

Section: Model Derivationmentioning

confidence: 94%

\nabla \cdot \bar{S}

Eqn. 14 \approx ({, \begin{matrix} left \frac{\partial ρ_{f} U_{f}}{\partial t} + \nabla \cdot (ρ_{f} {U_{f} U}_{f}) = - \nabla p + \nabla \cdot \overset{bold-italicS}{true} + ρ_{f} g + F_{c, 1} & left in solid ‐ free regions \\ left (U_{f} - {\bar{U}}_{s}) = - \frac{k}{μ} (\nabla p - ρ_{f} g - F_{c, 1} - F_{c, 2}) & left in porous regions \end{matrix})

…”

Section: Model Derivationmentioning

confidence: 99%

“…Given that F c ,1 = − p c ∇ α w in the porous domains (Carrillo et al., 2020), the previous equation can be rearranged to obtain

{bold-italicB}_{c a p} = ϕ_{s} {bold-italicF}_{c, 1} - ϕ_{f} {bold-italicF}_{c, 2}

…”

Section: Model Derivationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Modeling Multiphase Flow Within and Around Deformable Porous Materials: A Darcy‐Brinkman‐Biot Approach

Carrillo

Bourg

2021

Water Resources Research

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Modeling Multiphase Flow Within and Around Deformable Porous Materials: A Darcy-Brinkman-Biot Approach

Carrillo

Bourg

2020

Preprint

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A multiscale LBM–TPM–PFM approach for modeling of multiphase fluid flow in fractured porous media

Chaaban

Heider

Markert

2022

Num Anal Meth Geomechanics

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In this paper, we present a reliable micro-to-macroscale framework to model multiphase fluid flow through fractured porous media. This is based on utilizing the capabilities of the lattice Boltzmann method (LBM) within the phase-field modeling (PFM) of fractures in multiphase porous media. In this, we propose new physically motivated phase-field-dependent relationships for the residual saturation, the intrinsic as well as relative permeabilities. In addition, an anisotropic, phase-field-dependent intrinsic permeability tensor for the fractured porous domains is formulated, which relies on the single-and multiphasic LBM flow simulations. Based on these results, new relationships for the variation of the macroscopic theory of porous media (TPM)-PFM model parameters in the transition zone are proposed. Whereby, a multiscale concept for the coupling between the multiphasic flow through the crack on one hand and the porous ambient, on the other hand, is achieved. The hybrid model is numerically applied on a real microgeometry of fractured porous media, extracted via X-ray microcomputed tomography data of fractured Berea Sandstone. Moreover, the model is utilized for the calculation of the fluid leak-off from the crack to the intact zones. Additionally, the effects of the depth of the transition zone and the orientation of the crack channels on the amount of leakage flow rates are studied. The outcomes of the numerical model proved the reliability of the multiscale model to simulate multiphasic fluid flow through fractured porous media.

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Multiphase flow modeling in multiscale porous media: An open-source micro-continuum approach

Cited by 49 publications

References 90 publications

Modeling Multiphase Flow Within and Around Deformable Porous Materials: A Darcy‐Brinkman‐Biot Approach

Modeling Multiphase Flow Within and Around Deformable Porous Materials: A Darcy‐Brinkman‐Biot Approach

Modeling Multiphase Flow Within and Around Deformable Porous Materials: A Darcy-Brinkman-Biot Approach

A multiscale LBM–TPM–PFM approach for modeling of multiphase fluid flow in fractured porous media

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