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Transport in Porous Media

, Volume 84, Issue 1, pp 177–200 | Cite as

Numerical Simulation of Two-Phase Inertial Flow in Heterogeneous Porous Media

  • Azita Ahmadi
  • Ali Akbar Abbasian Arani
  • Didier LasseuxEmail author
Article

Abstract

In this study, non-Darcy inertial two-phase incompressible and non-stationary flow in heterogeneous porous media is analyzed using numerical simulations. For the purpose, a 3D numerical tool was fully developed using a finite volume formulation, although for clarity, results are presented in 1D and 2D configurations only. Since a formalized theoretical model confirmed by experimental data is still lacking, our study is based on the widely used generalized Darcy–Forchheimer model. First, a validation is performed by comparing numerical results of the saturation front kinetics with a semi-analytical solution inspired from the Buckley–Leverett model extended to take into account inertia. Second, we highlight the importance of inertial terms on the evolution of saturation fronts as a function of a suitable Reynolds number. Saturation fields are shown to have a structure markedly different from the classical case without inertia, especially for heterogeneous media, thereby, emphasizing the necessity of a more complete model than the classical generalized Darcy’s one when inertial effects are not negligible.

Keywords

Inertial two-phase flow Heterogeneous porous media Numerical simulations Generalized Darcy–Forchheimer model 

List of Symbols

A

Section of the medium, m2

Caκ

κ-Region capillary number

d

Grain size, m

fα

Fractional flow for the α-phase

g

Gravitational acceleration, m s−2

I

Unit tensor

K

Intrinsic permeability tensor (= k I for an isotropic case), m2

Kα

α-Phase effective permeability tensor, m2

kκ

Intrinsic permeability in the κ-region, m2

krα

Relative permeability tensor for the α-phase (= k r α I for an isotropic case)

l

Characteristic scale of the problem, m

L

Length of the medium, m

M

Total mobility tensor (= M o  + M w ), m3 kg−1 s

Mα

α-Phase mobility tensor (= M α I for an isotropic case), m3 kg−1 s

N

Number of grid blocks

ne

Unit vector normal to the outlet face

ni

Unit vector normal to the inlet face

nl

Unit vector normal to the lateral surfaces

nωη

Unit vector normal to the ωη interface pointing from the ω-region toward the η-region

p

Fluid pressure for one-phase flow, Pa

patm

Atmospheric pressure, Pa

pα

α-Phase pressure, Pa

p0

Initial oil-phase pressure, Pa

pc

Capillary pressure, Pa

pc0

Maximum capillary pressure at S w  = S wi, Pa

\({p_{c}^{\kappa}}\)

Capillary pressure in the κ-region, Pa

q

Flow rate of water injected at the inlet of the medium, m3 s−1

r

Position vector, m

re

Position vector relative to the outlet face, m

Re

Reynolds number

Reα

Reynolds number associated to the α-phase, \({\left(=\frac{\rho_{\alpha}\left\Vert {\bf u}_{\alpha}\right\Vert l}{\mu_{\alpha}}\right)}\)

Recl

Classical Reynolds number associated to the α-phase, \({(=\underset{\alpha}{\max}(\rho_{\alpha}/\mu_{\alpha})\left\Vert {\bf u}_{t}\right\Vert d)}\)

Sα

α-Phase saturation

Swi

Irreducible water saturation

Sor

Residual oil saturation

S0

Initial water-phase saturation

S*

Reduced saturation \({\left(=\frac{S_{w}-S_{\rm wi}}{1-S_{\rm wi}-S_{\rm or}}\right)}\)

t

Time, s

u

Seepage velocity for one-phase flow, m s−1

uα

α-Phase seepage velocity, m s−1

\({{\bf u}_{\alpha}^{\kappa}}\)

α-Phase seepage velocity in the κ-region, m s−1

ut

Total velocity (=u o  + u w ), m s−1

utx, uty, utz

Components of the total velocity, m s−1

W

Front velocity, m s−1

x

Position variable, m

Greek Letters

βα

α-Phase effective inertial resistance tensor (= β α I for the isotropic case), m−1

β

Intrinsic inertial resistance factor, m−1

βrα

α-Phase relative inertial resistance tensor (= β r α I for the isotropic case)

βκ

Intrinsic inertial resistance factor for the κ-region, m−1

Γωη

Interface between the ω-region and the η-region, m2

Δt

Time step, s

Δx, Δy, Δz

Grid sizes in the x, y and z directions, m

ε

Porosity

μα

α-Phase dynamic viscosity, Pas

ρα

α-Phase density, kg/m3

σ

Interfacial tension, N m−1

ξ, γ, θ

Constant exponents

τ

Tortuosity

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Azita Ahmadi
    • 1
  • Ali Akbar Abbasian Arani
    • 1
  • Didier Lasseux
    • 1
    Email author
  1. 1.TREFLE, UMR CNRS 8508, Arts et Métiers ParisTechTalence CedexFrance

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