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Modeling Non-Darcian Single- and Two-Phase Flow in Transparent Replicas of Rough-Walled Rock Fractures

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Abstract

While it is generally assumed that in the viscous flow regime, the two-phase flow relative permeabilities in fractured and porous media depend uniquely on the phase saturations, several studies have shown that for non-Darcian flows (i.e., where the inertial forces are not negligible compared with the viscous forces), the relative permeabilities not only depend on phase saturations but also on the flow regime. Experimental results on inertial single- and two-phase flows in two transparent replicas of real rough fractures are presented and modeled combining a generalization of the single-phase flow Darcy’s law with the apparent permeability concept. The experimental setup was designed to measure injected fluid flow rates, pressure drop within the fracture, and fluid saturation by image processing. For both fractures, single-phase flow experiments were modeled by means of the full cubic inertial law which allowed the determination of the intrinsic hydrodynamic parameters. Using these parameters, the apparent permeability of each fracture was calculated as a function of the Reynolds number, leading to an elegant means to compare the two fractures in terms of hydraulic behavior versus flow regime. Also, a method for determining the experimental transition flow rate between the weak inertia and the strong inertia flow regimes is proposed. Two-phase flow experiments consisted in measuring the pressure drop and the fluid saturation within the fractures, for various constant values of the liquid flow rate and for increasing values of the gas flow rate. Regardless of the explored flow regime, two-phase flow relative permeabilities were calculated as the ratio of the single phase flow pressure drop per unit length divided by the two-phase flow pressure drop per unit length, and were plotted versus the measured fluid saturation. Results confirm the dependence of the relative permeabilities on the flow regime. Also the proposed generalization of Darcy’s law shows that the relative permeabilities versus fluid saturation follow physical meaningful trends for different liquid and gas flow rates. The presented model fits correctly the liquid and gas experimental relative permeabilities as well as the fluid saturation.

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Abbreviations

\(A\) :

Cross-sectional area of the fracture or porous media (\(\text{ m }^{2}\))

\(a\) :

Corey saturation power law exponent

\(b\) :

Corey saturation power law exponent

\(C\) :

Concentration (g/l)

\(Ca\) :

Capillary number

\(F\) :

\(F\)-function

\(H\) :

Local aperture (m)

\(h\) :

Hydraulic aperture of the fracture (m)

\(I\) :

Intensity

\(K\) :

Intrinsic permeability (\(\text{ m }^{2}\))

\(K_{\mathrm{a}}\) :

Apparent permeability (\(\text{ m }^{2}\))

\(k_{\mathrm{r}}\) :

Relative permeability

\(Q\) :

Volumetric flow rate (\(\text{ m }^{3}/\text{ s }\))

\(Re\) :

Reynolds number (\(Re = K\beta \rho Q/\mu A\))

\(Re_{\mathrm{c}}\) :

Weak inertia cubic law Reynolds number \(Re_{\mathrm{c}} =\sqrt{K\gamma }{\rho Q}/{\mu A}\)

\(S\) :

Fluid saturation

\(w\) :

Width of the fracture (m)

\(\Delta P/L\) :

Pressure drop per unit length (Pa/m)

\(\beta \) :

Inertial coefficient (\(\text{ m }^{-1}\))

\(\beta _{\mathrm{r}}\) :

Relative inertial coefficient

\(\gamma \) :

Inertial coefficient

\(\varepsilon \) :

Solute absorptivity (\(\text{ m }^{2}/\text{ kg }\))

\(\mu \) :

Dynamic viscosity (Pa s)

\(\rho \) :

Density (\(\text{ kg/m }^{3}\))

g:

Gas

l:

Liquid

s:

Single-phase flow

t:

Transition

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Authors and Affiliations

  1. LEMTA, Nancy-University, CNRS, 2 Avenue de la Forêt de Haye, BP 160, 54504 , Vandœuvre Cedex, France

    Giovanni Radilla, Ali Nowamooz & Mostafa Fourar

  2. Arts et Métiers ParisTech, rue St Dominique, BP 508, 51006 , Châlons-en-Champagne, France

    Giovanni Radilla & Ali Nowamooz

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  1. Giovanni Radilla
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  2. Ali Nowamooz
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Radilla, G., Nowamooz, A. & Fourar, M. Modeling Non-Darcian Single- and Two-Phase Flow in Transparent Replicas of Rough-Walled Rock Fractures. Transp Porous Med 98, 401–426 (2013). https://doi.org/10.1007/s11242-013-0150-1

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