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

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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.

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Abbreviations

A :

Section of the medium, m2

Ca κ :

κ-Region capillary number

d :

Grain size, m

f α :

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

K α :

α-Phase effective permeability tensor, m2

k κ :

Intrinsic permeability in the κ-region, m2

k r α :

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

M α :

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

N :

Number of grid blocks

n e :

Unit vector normal to the outlet face

n i :

Unit vector normal to the inlet face

n l :

Unit vector normal to the lateral surfaces

n ω η :

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

p :

Fluid pressure for one-phase flow, Pa

p atm :

Atmospheric pressure, Pa

p α :

α-Phase pressure, Pa

p 0 :

Initial oil-phase pressure, Pa

p c :

Capillary pressure, Pa

p c0 :

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

r e :

Position vector relative to the outlet face, m

Re :

Reynolds number

Re α :

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

Re cl :

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

S α :

α-Phase saturation

S wi :

Irreducible water saturation

S or :

Residual oil saturation

S 0 :

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

u α :

α-Phase seepage velocity, m s−1

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

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

u t :

Total velocity (=u o  + u w ), m s−1

u tx , u ty , u tz :

Components of the total velocity, m s−1

W :

Front velocity, m s−1

x :

Position variable, m

β α :

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

β :

Intrinsic inertial resistance factor, m−1

β r α :

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

β κ :

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

Γ ω η :

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

μ α :

α-Phase dynamic viscosity, Pas

ρ α :

α-Phase density, kg/m3

σ :

Interfacial tension, N m−1

ξ, γ, θ:

Constant exponents

τ :

Tortuosity

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Authors and Affiliations

  1. TREFLE, UMR CNRS 8508, Arts et Métiers ParisTech, University Bordeaux 1, 33405, Talence Cedex, France

    Azita Ahmadi, Ali Akbar Abbasian Arani & Didier Lasseux

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  1. Azita Ahmadi
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  2. Ali Akbar Abbasian Arani
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  3. Didier Lasseux
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Correspondence to Didier Lasseux.

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Ahmadi, A., Abbasian Arani, A.A. & Lasseux, D. Numerical Simulation of Two-Phase Inertial Flow in Heterogeneous Porous Media. Transp Porous Med 84, 177–200 (2010). https://doi.org/10.1007/s11242-009-9491-1

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