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Critical point network for drainage between rough surfaces

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Abstract

In this paper, we present a network method for computing two-phase flows between two rough surfaces with significant contact areas. Low-capillary number drainage is investigated here since one-phase flows have been previously investigated in other contributions. An invasion percolation algorithm is presented for modeling slow displacement of a wetting fluid by a non wetting one between two rough surfaces. Short-correlated Gaussian process is used to model random rough surfaces.The algorithm is based on a network description of the fracture aperture field. The network is constructed from the identification of critical points (saddles and maxima) of the aperture field. The invasion potential is determined from examining drainage process in a flat mini-channel. A direct comparison between numerical prediction and experimental visualizations on an identical geometry has been performed for one realization of an artificial fracture with a moderate fractional contact area of about 0.3. A good agreement is found between predictions and observations.

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Abbreviations

g :

Gravity acceleration

h(x,y):

Local aperture field between surfaces

h 0 :

Local aperture of critical points

h :

Local aperture associated with the interface whose position along coordinate y is y

H = ∂2 ij h :

Is the local Hessian matrix of the aperture field

H xx & H yy :

Are the Hessian principal eigenvalues

ℓ:

Local width of the saddle-point throats

cap :

Capillary length

c :

Spacial correlation length of aperture field

L :

Total width of the micro-channel

ΔP :

Difference of pressure between gas and liquid

Q :

Volumetric flow rate per unit width

R i , i = 1,2:

Are the principal radii of curvature of the interface

R 0 :

Is the in-plane radius of curvature of the interface at the origin O

y(x):

Is the in-plane (i.e., (x,y) plane) position of the interface in direction y at coordinate x

y :

Is the in-plane (i.e., (x,y) plane) asymptotic position of the interface in direction y

β,β′ :

Local angle associated with the slope of the upper and lower surfaces

ε:

Small parameter associated with the aspect ratio of the critical points: local aperture divided by width

ε′:

Small parameter associated with the surface slopes

κ :

Mean curvature of the interface

μ :

Dynamic viscosity of the fluid

ρ :

Fluid density

σ i :

Is the root mean square roughness of surface i = 1,2

\(\sigma=\sqrt{\sigma_1^2+\sigma_2^2}\) :

Is the composite roughness

θ,θ′ :

Wettability angles

References

  • Adler P.M. and Thovert J.F. (1999). Fractures and fracture networks. Kluwer Academic Publishers, Amsterdam

    Google Scholar 

  • Amundsen H., Wagner G., Oxaal U., Meakin P., Feder J. and Jossang T. (1999). Slow two-phase flow in artificial fractures: experiments and simulations. Wat. Res. Res. 35: 2619–2626

    Article  Google Scholar 

  • Amyot O. and Plouraboué F. (2007). Capillary pinching in a pinched micro-channel. Phys. Fluids 19: 033101

    Article  Google Scholar 

  • Berkowitz B. (2002). Characterizing flow and transport in fractured geological media : a review. Adv. Wat. Res. 25: 861–884

    Article  Google Scholar 

  • Blunt M.J. (2001). Flow in porous media: pore networks models and multiphase flows. Curr. Opin. Colloid Interface Sci. 6: 187–207

    Article  Google Scholar 

  • Blunt M.J., Jackson M.D., Piri M. and Valvatne P.H. (2002). Detailed physics, predictive capabilities and macroscopic consequences for pore-network models of multiphase flow. Adv. Wat. Res. 25: 1069–1089

    Article  Google Scholar 

  • Bories, S., Prat, M.: Isothermal Nucleation and Bubble Growth in Porous Media at Low Supersaturations-Transport Phenomena in Porous Media, pp. 276–315. D.B. Ingham et K. Lambert Ed., UK (2002)

  • Brown S.R. and Scholz C.H. (1985). Broad bandwidth study of the topography of natural rock surfaces. J. Geoph. Res. 90: 12–575–12–582

    Google Scholar 

  • Bryant S.L., Mellor D.W. and Cade C.A. (1993). Physically representative network models of transport in porous media.. AIChE J. 39(3): 387–396

    Article  Google Scholar 

  • Chrysikopoulos C.V. and Abdel-Salam A. (1997). Modeling colloid transport and deposition in saturated fractures. Colloids Surfaces A: Physicochem. Eng. Aspects 121: 189–202

    Article  Google Scholar 

  • Chrysikopoulos C.V. and James S.C. (2003). Transport of neutrally buoyant and dense variably sized colloids in a two-dimensional fracture with anisotropic aperture. Trans. Porous Med. 51: 191–210

    Article  Google Scholar 

  • Evans, D.D., Nicholson, T.J., Rasmussen, T.E.C.: Flow and Transport Through Unsaturated Fractured Rock. AGU Geophysical Monograph Series, vol 42, New York (2001)

  • Faybishenko, B., Witherspoon, P.A., Benson, S.M.: Dynamics of Fluid I Fractured Rock. AGU Geophysical Monograph Series, vol 122, New York (2000)

  • Flukiger, F., Plouraboué, F., Prat, M.: Non-universal conductivity exponents in continuum percolating Gaussian fracture. in revision for Phys. Rev. E (2006)

  • Fourar M., Bories S., Lenormand R. and Persoff R.F. (1993). Two-phase flow in smooth and rough fractures: measurement and correlation by porous media and pipe-flow models. Wat. Res. Res. 29(11): 3699–3708

    Article  Google Scholar 

  • De Gennes P.G. (1983). Theory of slow biphasic flow in porous media. Phys. Chem. Hydrol. 4: 175–185

    Google Scholar 

  • Geoffroy S. and Prat M. (2004). On the leak through a spiral-groove metallic static ring gasket. ASME J. Fluids Eng. 126(1): 48–54

    Article  Google Scholar 

  • Geoffroy S., Plouraboué F., Prat M. and Amyot O. (2006). Quasi-static liquid-air drainage in narrow channels with variations in the gap. J. Colloids Interface Sci. 294: 165–175

    Article  Google Scholar 

  • Glass R.J., Nicholl M.J. and Yarrigton L. (1998). A modified invasion percolation model for low-capillary number immiscible displacements in horizontal rough-walled fractures: influence of local in-plane curvature. Wat. Res. Res. 34(12): 3215–3234

    Article  Google Scholar 

  • Glass R.J., Rajaram H., Nicholl M.J. and Detwilder R.L. (2001). The interaction of two fluid phases in fractured media. Curr. Opin. Colloid Interface Sci. 6: 223–235

    Article  Google Scholar 

  • Lenormand R., Touboul E. and Zarcone C. (1988). Numerical models and experiments on immiscible displacements in porous media. J. Fluid Mech. 189: 165–187

    Article  Google Scholar 

  • Letalleur N., Plouraboué F. and Prat M. (2002). Average flow model of rough surface lubrication: flow factors for sinusoidal surfaces. ASME J. Tribol. 124: 539

    Article  Google Scholar 

  • Loggia D., Gouze P., Greswell R. and Parker D.J. (1995). Investigation of the geometrical dispersion regime in a single fracture using positron emission projection imaging journal transport in porous media. Trans. Por. Med. 55(1): 1–20

    Article  Google Scholar 

  • Makse H.A., Havlin S., Schwartz M. and Stanley H.E. (1996). Method for generating long-range correlation for large systems. Phys. Rev. E 53(5): 5445

    Article  Google Scholar 

  • Mendoza, C.A., Sudicki, E.A.: Hierarchical scaling of constitutive relationships controlling multi-phase flow in fractured geologic media, in reservoir chracterization. In: 3rd International Technical Conference: Papers, Pennwell, Tulsa, Okla., B.Linville (1991)

  • Mourzenko V.V., Thovert J.F. and Adler P.M. (1996). Geometry of simulated fractures. Phys. Rev. E 53(6): 5606–5626

    Article  Google Scholar 

  • National Research council: Rock Fractures and Fluid Flow: Contemporary Understanding and Applications. Technical Report, Washington D.C.: National Academic Press., Avril (1996)

  • Pereira G.G., Pinczewski W.V., Chan D.Y.C., Paterson L. and Oren P.E. (1996). Pore-scale network model for drainage-dominated three-phase flow in porous media. Trans. Por. Med. 24(2): 167–201

    Article  Google Scholar 

  • Plouraboué F., Flukiger F., Prat M. and Crispel P. (2006). Geodesic network method for flows between two rough surfaces in contact. Phys. Rev. E 73: 036305

    Article  Google Scholar 

  • Plouraboué F., Geoffroy S. and Prat M. (2004). Conductances between confied rough walls. Phys. Fluids 16(3): 615–624

    Article  Google Scholar 

  • Plouraboué F., Kurowski P., Hulin J.P., Roux S. and Schmittbuhl J. (1995). Aperture of rough cracks. Phys. Rev. E 51(3): 1675–1685

    Article  Google Scholar 

  • Plouraboué F., Prat M. and Letalleur N. (2001). Sliding lubricated anisotropic rough surfaces. Phys. Rev. E 64(1): 011202

    Article  Google Scholar 

  • Polycarpou A. and Etsion I. (2000). A Model for satic sealing performance of compliant metallic gas seals including surface roughness and rarefaction effects. Tribol. Transac. 43(2): 237–244

    Article  Google Scholar 

  • Prat M. (2002). Recent advances in pore-scale models for drying of porous media. Chem. Eng. J. 86(1–2): 153–164

    Article  Google Scholar 

  • Prat M., Letalleur N. and Plouraboué F. (2002). Averaged Reynolds equation for flow between rough surfaces in sliding motion. Trans. Por. Med. 48: 291–313

    Article  Google Scholar 

  • Sahimi M. (1995). Flow and Transport in Porous Media and Fractured Rock. VCH Wienheim, New York

    Google Scholar 

  • Satik C. and Yortsos Y.C. (1996). A pore network study of bubble growth in porous media driven by heat transfer. J. Heat Trans. T. ASME 118: 455–462

    Google Scholar 

  • Satik C., Li X. and Yortsos Y.C. (1995). Scaling of single bubble growth in a porous medium. Phys. Rev. E 51: 3286

    Article  Google Scholar 

  • Sok R.M., Knackstedt M.A., Sheppard A.P., Pinczewski W.V., Lindquist W.B., Venkatarangan A. and Paterson L. (2002). Direct and stochastic generation of network models from tomographic images; effect of topology on residual saturations. Trans. Por. Med. 46: 2–3

    Google Scholar 

  • Tarjan R. (1983). Data Structures and Network Algorithms. Society for Industrial and Applied mathematics, New-York, USA

    Google Scholar 

  • Vandersteen K., Carmeliet J. and Feyen J. (2003). A network modeling approach to derive unsaturated hydraulic properties of a rough -walled fracture. Trans. Por. Med. 50: 197–221

    Article  Google Scholar 

  • Wagner G., Meakin P., Feder J. and Jossang T. (1997). Invasion percolation on self-affine topographies. Phys. Rev. E 55(2): 1698–1703

    Article  Google Scholar 

  • Weinrib A. (1982). Percolation threshold of two-dimensional continuum system. Phys. Rev. B 26(3): 1352–1361

    Article  Google Scholar 

  • Wilkinson D. (1984). Percolation model of immiscible displacement in the presence of buoyancy forces. Phys. Rev. A 30: 520–531

    Article  Google Scholar 

  • Wilkinson D. and Willemsen J.F. (1983). Invasion percolation: a new form of percolation theory. J. Phys. A-Math. Gen. 16: 3365–3376

    Article  Google Scholar 

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Amyot, O., Flukiger, F., Geoffroy, S. et al. Critical point network for drainage between rough surfaces. Transp Porous Med 70, 257–277 (2007). https://doi.org/10.1007/s11242-007-9098-3

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