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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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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DOI: https://doi.org/10.1007/s11242-009-9491-1