Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-27T01:40:00.393Z Has data issue: false hasContentIssue false
  • English
  • Français

Gas flow in ultra-tight shale strata

Published online by Cambridge University Press:  27 September 2012

Hamed Darabi ,
A. Ettehad ,
F. Javadpour  and
K. Sepehrnoori
Hamed Darabi
Affiliation:
Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, 1 University Station, Austin, TX 78712, USA
A. Ettehad
Affiliation:
Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, 1 University Station, Austin, TX 78712, USA
F. Javadpour*
Affiliation:
Bureau of Economic Geology, Jackson School of Geosciences, The University of Texas at Austin, University Station, Box X, Austin, TX 78713, USA
K. Sepehrnoori
Affiliation:
Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, 1 University Station, Austin, TX 78712, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the gas flow processes in ultra-tight porous media in which the matrix pore network is composed of nanometre- to micrometre-size pores. We formulate a pressure-dependent permeability function, referred to as the apparent permeability function (APF), assuming that Knudsen diffusion and slip flow (the Klinkenberg effect) are the main contributors to the overall flow in porous media. The APF predicts that in nanometre-size pores, gas permeability values are as much as 10 times greater than results obtained by continuum hydrodynamics predictions, and with increasing pore size (i.e. of the order of the micrometre), gas permeability converges to continuum hydrodynamics values. In addition, the APF predicts that an increase in the fractal dimension of the pore surface leads to a decrease in Knudsen diffusion. Using the homogenization method, a rigorous analysis is performed to examine whether the APF is preserved throughout the process of upscaling from local scale to large scale. We use the well-known pulse-decay experiment to estimate the main parameter of the APF, which is Darcy permeability. Our newly derived late-transient analytical solution and the late-transient numerical solution consistently match the pressure decay data and yield approximately the same estimated value for Darcy permeability at the typical core-sample initial pressure range and pressure difference. Other parameters of the APF may be determined from independent laboratory experiments; however, a pulse-decay experiment can be used to estimate the unknown parameters of the APF if multiple tests are performed and/or the parameters are strictly constrained by upper and lower bounds.

JFM classification

Low-Reynolds-number flows: Porous media Rarefied Gas Flow: Rarefied Gas Flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 710 , 10 November 2012 , pp. 641 - 658
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Agrawal, A. & Prabhu, S. V. 2008 Survey on measurement of tangential momentum accommodation coefficient. J. Vac. Sci. Technol. A. 26 (4), 634–645.CrossRefGoogle Scholar
2. Auriault, J.-L. 1983 Effective macroscopic description of heat conduction in periodic composites. Intl J. Heat Mass Transfer 26, 861–869.CrossRefGoogle Scholar
3. Auriault, J.-L. 1991 Is an equivalent macroscopic description possible? Intl J. Engng Sci. 29 (4), 785–795.CrossRefGoogle Scholar
4. Auriault, J.-L., Strzelecki, T., Bauer, J. & He, S. 1990 Porous deformable media saturated by a very compressible fluid: quasi-statics. Eur. J. Mech. A 9 (24), 373–392.Google Scholar
5. Beskok, A. & Karniadakis, G. E. 1999 A model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Therm. Engng 3 (1), 43–77.Google Scholar
6. Borwein, J. M. & Lewis, A. S. 2000 Convex Analysis and Nonlinear Optimization, Theory and Examples. Springer.Google Scholar
7. Bourbie, T. & Walls, J. 1982 Pulse decay permeability: analytical solution and experimental test. SPE J. 22, 719–722.Google Scholar
8. Brace, W. F., Walsh, J. B. & Frangos, W. T. 1968 Permeability of granite under high pressure. J. Geophys. Res. 73 (6), 2225–2236.CrossRefGoogle Scholar
9. Brown, G. P., Dinardo, A., Cheng, G. K. & Sherwood, T. K. 1946 The flow of gases in pipes at low pressures. J. Appl. Phys. 17, 802–813.CrossRefGoogle Scholar
10. Burgdorfer, A. 1959 The influence of the molecular mean free path on the performance of hydrodynamic gas-lubricated bearings. Trans. ASME: J. Basic Engng 81, 94–100.CrossRefGoogle Scholar
11. Chastanet, J., Royer, P. & Auriault, J.-L. 2004 Does Klinkenberg’s law survive upscaling? Trans. Porous Med. 56, 171–198.CrossRefGoogle Scholar
12. Chen, Y. & Durlofsky, L. J. 2008 Ensemble-level upscaling for efficient estimation of fine-scale production statistics. SPE J. 13 (4), 400–411.CrossRefGoogle Scholar
13. Civan, F. 2010 Effective correlation of apparent gas permeability in tight porous media. Trans. Porous Med. 82, 375–384.CrossRefGoogle Scholar
14. Civan, F., Chandra, S. R. & Sondergeld, C. H. 2011 Shale gas permeability and diffusivity inferred by improved formulation of relevant retention and transport mechanisms. Trans. Porous Med. 86, 925–944.CrossRefGoogle Scholar
15. Coppens, M.-O. 1999 The effect of fractal surface roughness on diffusion and reaction in porous catalysts from fundamentals to practical applications. Catal. Today 53 (2), 225–243.CrossRefGoogle Scholar
16. Coppens, M.-O. & Dammers, A. J. 2006 Effects of heterogeneity on diffusion in nanopores from inorganic materials to protein crystals and ion channels. Fluid Phase Equilib. 246 (1–2), 308–316.CrossRefGoogle Scholar
17. Cui, X., Bustin, A. M. & Bustin, R. 2009 Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. J. Geofluids 9, 208–223.CrossRefGoogle Scholar
18. Errol, B. A., Kenneth, S. B. & Martin, A. S. 2001 Mass flow and tangential momentum accommodation in silicon micromachined channels. J. Fluid Mech. 437, 29.Google Scholar
19. Evazi, M. & Mahani, H. 2010 Unstructured-coarse-grid generation using background-grid approach. SPE J. 15 (2), 326–340.CrossRefGoogle Scholar
20. Farmer, C. L. 2002 Upscaling: a review. Intl J. Numer. Meth. Fluids 40, 63–78.CrossRefGoogle Scholar
21. Finsterle, S. & Najita, J. 1998 Robust estimation of hydrogeologic model parameters. Water Resour. Res. 34 (11), 2939–2947.CrossRefGoogle Scholar
22. Gad-el-Hak, M. 1999 The fluid mechanics of microdevices: the Freeman scholar lecture. J. Fluids. Engng 121, 5–33.CrossRefGoogle Scholar
23. Ho, C. M. & Tai, Y. C. 1998 Micro-electro-mechanical-systems (MEMS) and fluid flows. Annu. Rev. Fluid Mech. 30, 579–612.CrossRefGoogle Scholar
24. Holt, J. K., Park, H. G., Wang, Y., Staderman, M., Artyukhin, A. B., Grigoropoulos, C. P., Noy, A. & Bakajin, O. 2006 Fast mass transport through sub-2-nanometre carbon nanotubes. Science 312, 1034–1037.CrossRefGoogle ScholarPubMed
25. Hsia, Y. T. & Domoto, G. A. 1983 An experimental investigation of molecular rarefaction effects in gas lubricated bearings at ultra-low clearances. Trans. ASME: J. Lubr. Technol. 105, 120–130.Google Scholar
26. Hsieh, P. A., Tracy, J. V., Neuzil, C. E., Bredehoeft, J. D. & Silliman, S. E. 1981 A transient laboratory method for determining the hydraulic properties of ‘tight’ rocks. Part 1. Theory. Intl J. Rock Mech. 18, 245–252.CrossRefGoogle Scholar
27. Javadpour, F. 2009 Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone). J. Can. Petrol. Technol. 48 (8), 16–21.CrossRefGoogle Scholar
28. Javadpour, F., Fisher, D. & Unsworth, M. 2007 Nanoscale gas flow in shale sediments. J. Can. Petrol. Technol. 46 (10), 55–61.CrossRefGoogle Scholar
29. Javadpour, F., Moravvej-Farshi, M. & Amrein, M. 2012 Atomic-force microscopy: a new tool for gas-shale characterization. J. Can. Petrol. Technol. 51 (4), 236–243.CrossRefGoogle Scholar
30. Jeans, J. H. 1954 The Dynamical Theory of Gases. Dover.Google Scholar
31. Jones, S. C. 1997 A technique for faster pulse decay permeability measurements in tight rocks. SPE J. Reservoir Eval. Engng 12, 19–25.Google Scholar
32. Kilislioglu, A. & Bilgin, B. 2003 Thermodynamic and kinetic investigations of uranium adsorption on amberlite IR-118H resin. Appl. Radiat. Isot. 58 (2), 155–160.CrossRefGoogle ScholarPubMed
33. Klinkenberg, L. J. 1941 The permeability of porous media to liquids and gas: drilling and production practice. Drilling Production Practice 200–213.Google Scholar
34. Koplik, J. & Banavar, J. R. 1995 Continuum deductions from molecular hydrodynamics. Annu. Rev. Fluid Mech. 27, 257–292.CrossRefGoogle Scholar
35. Lasseux, D., Jolly, P., Jannot, Y. & Omnes, E. S. B 2011 Permeability measurement of graphite compression packings. Trans. ASME: J. Press. Vessel Technol. 133, 041401.Google Scholar
36. Loucks, R. G., Reed, R. M., Ruppel, S. C. & Hammes, U. 2012 Spectrum of pore types and networks in mudrocks and a descriptive classification for matrix-related mudrock pores. AAPG Bull. 96 (6), 1071–1098.CrossRefGoogle Scholar
37. Majumder, M., Chopra, N., Andrews, R. & Hinds, B. J. 2005 Enhanced flow in carbon nanotubes. Nature 438, 44.CrossRefGoogle ScholarPubMed
38. Mao, Z. G. & Sinnott, S. B. 2001 Separation of organic molecular mixtures in carbon nanotubes and bundles: molecular dynamics simulations. J. Phys. Chem. 105, 6916–6924.CrossRefGoogle Scholar
39. Mehrotra, S. 1992 On the implementation of a primal–dual interior point method. SIAM J. Optim. 2 (4), 575–601.CrossRefGoogle Scholar
40. Mills, A. F. 2001 Mass Transfer. Prentice Hall.Google Scholar
41. Mitsuya, Y. 1993 Modified Reynolds equation for ultra-thin film gas lubrication using 1.5-order slip-flow model and considering surface accommodation coefficient. Trans. ASME: J. Tribol. 115, 289–294.CrossRefGoogle Scholar
42. Nie, X. B., Chen, S. Y. & E, W. N. 2004 A continuum and molecular dynamics hybrid method for micro- and nano-fluid flow. J. Fluid Mech. 500, 55–64.CrossRefGoogle Scholar
43. Poling, B. E., Prausnitz, J. M. & O’Connell, J. P. 2001 The Properties of Gases and Liquids, fifth edition. McGraw-Hill.Google Scholar
44. Roy, S., Raju, R., Chuang, H. F., Cruden, B. A. & Meyyappan, M. 2003 Modelling gas flow through microchannels and nanopores. J. Appl. Phys. 93, 4870–4879.CrossRefGoogle Scholar
45. Ruthven, D. M. 1984 Principles of Adsorption and Adsorption Processes. Wiley.Google Scholar
46. Tartakovsky, D. M. 2000 Real gas flow through heterogeneous porous media: theoretical aspects of upscaling. Stoch. Environ. Res. Risk Assess. (SERRA) 133, 109–122.CrossRefGoogle Scholar
47. Trimmer, D., Bonner, B., Heard, H. C. & Duba, A. 1980 Effect of pressure and stress on water transport in intact and fractured gabbro and granite. J. Geophys. Res. 85, 7059–7071.CrossRefGoogle Scholar
48. Walder, J. & Nur, A. 1986 Permeability measurement by the pulse-decay method: effect of poroelastic phenomena and nonlinear pore pressure diffusion. Intl J. Rock Mech. Min. Sci. Geomech. Abstr. 23 (3), 225–232.CrossRefGoogle Scholar
49. Wu, L. 2008 A slip model for rarified gas flows at arbitrary Knudsen number. Appl. Phys. Lett. 93, 2531–03.CrossRefGoogle Scholar
50. Xiao, J. & Wei, J. 1992 Diffusion mechanism of hydrocarbons in zeolites. Part 1. Theory. Chem. Engng Sci. 47 (5), 1123–1141.CrossRefGoogle Scholar
51. Yamada, S. E. & Jones, A. H. 1980 A review of a pulse technique for permeability measurements. SPE J. 20, 357–368.Google Scholar
Supplementary material: PDF

Darabi Supplementary Material

Appendix

Download Darabi Supplementary Material(PDF)
PDF 446.9 KB