Knudsen-Like Scaling May Be Inappropriate for Gas Shales
- Tadeusz W. Patzek (Ali I. Al-Naimi Petroleum Engineering Research Center, Physical Sciences and Engineering Division King Abdullah University of Science and Technology)
- Document ID
- Society of Petroleum Engineers
- SPE Annual Technical Conference and Exhibition, 9-11 October, San Antonio, Texas, USA
- Publication Date
- Document Type
- Conference Paper
- 2017. Society of Petroleum Engineers
- 2.4 Hydraulic Fracturing, 5.8.2 Shale Gas, 5 Reservoir Desciption & Dynamics, 1.2.3 Rock properties, 2.4 Hydraulic Fracturing, 3 Production and Well Operations, 5.1 Reservoir Characterisation, 5.1 Reservoir Characterisation, 1.10 Drilling Equipment, 5.8 Unconventional and Complex Reservoirs, 1.10 Drilling Equipment
- microfracture, crack, slip flow, scaling, sorption, permeability enhancement
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- 210 since 2007
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We assert that a classification of gas flow regimes in shales that is widely accepted in the petroleum industry, may be inconsistent with the physics of high-pressure gas flow in capillaries. This classification follows from the 1946 work by Brown et al. (1946) that deals with the flow of gases in large industrial metal pipes, elbows and orifices under vacuum, with gas pressures of the order of 1 mm Hg or less. In another pioneering paper that year, Tsien (1946) analyzed the hypersonic flight of rockets in the thermosphere (above 50 miles of altitude), and established the widely accepted Knudsen flow regimes for the high-Reynolds, high-Mach flow of rarified gases. We show why both these papers are not quite applicable to flow of compressed gas in the hot, high-pressure shale pores with rough surfaces. In addition, it may be inappropriate to use the capillary tube metaphor to describe shale micropores or microcracks, simply because each is fed with gas by dozens or hundreds of intricately connected nanopores, which themselves may be slits rather than circular cylinders, and are charged with the dense, liquid-like gas.
In the small-scale, low-velocity flows of gases, failure of the standard Navier-Stokes description (the standard Darcy law in petroleum engineering) can be quantified by the Knudsen number, ratio of the mean free path, λ, of gas molecules at the reservoir pressure and temperature to the characteristic pore radius, R. We carefully enumerate the multiple restrictive conditions that must hold for the slip-flow boundary condition to emerge. We also describe the dependence of the slip correction factor on the gas pressure and temperature, as well as the median pore size and rock roughness. In the derivation, we revisit the original approaches of Helmholtz and von Piotrowski (1860) and Maxwell, Niven (1890), which were somehow lost in the multiple translations from physics to petroleum engineering.
For example, in Barnett mudrocks, naturally occurring pores are predominantly associated with organic matter and pyrite framboids. In organic matter, the median pore length is 100 nm, Loucks et al. (2009), and the pore radii are likely to be between 1 and 10 nm, Clarkson et al. (2013). Other thermally mature mudrocks may be similar, Ross and Bustin (2009), or not, Clarkson et al. (2013). With R = 50 nm, the ratio of λ/R is less than 0.1 for pressures exceeding 60 bars. When we compare the actual slip-flow correction with the accepted classification of gas flow regimes, there is an order of magnitude discrepancy. It appears that our new classification is conservative for pores larger than 5 nm in radius. Therefore, unless the fraction of gas molecules that are bounced off diffusively from the rough pore walls is very low, slip flow is unlikely to dominate in shales.
The generally accepted "Knudsen-diffusion" in shales is based on a mistranslation of the flow physics and may give theoretically unsound predictions of the increased permeability of shales to gas flow. This increase of permeability is real, and it comes from the micropores, fine-scale microfractures and cracks. The nanopores in shales provide gas storage by sorption and capillary condensation of heavier gas components. In the smallest nanopores even methane molecules are increasingly ordered and resemble more liquid than gas. These nanopores feed the macroscopic flow paths in ways that are not captured well by the generally accepted equations.
|File Size||4 MB||Number of Pages||24|
Adesida A. G., I. Y. Akkutlu, D. E. Resasco, and C. S. Rai, Kerogen pore size distribution of Barnett Shale using DFT analysis and monte carlo simulations, in paper SPE 147397 presented at the Society of Petroleum Engineers Annual Technical Conference … Exhibition, Denver, Colorado; October 30 - November 2, 2011.
Brown G. P., A. DiNardo, G. K. Cheng, and T. K. Sherwood, The flow of gases in pipes at low pressures, Journal of Applied Physics, 17(10), 802–813, doi:http://dx.doi.org/10.1063/1.1707647, 1946.
Christopher R. H., and S. Middleman, Power-Law Flow through a Packed Tube, Industrial … Engineering Chemistry Fundamentals, 4(4), 422–426, doi:10.1021/i160016a011, 1965.
Clarkson C., N. Solano, R. Bustin, A. Bustin, G. Chalmers, L. He, Y. Melnichenko, A. Radlmski, and T. Blach, Pore structure characterization of north american shale gas reservoirs using usans/sans, gas adsorption, and mercury intrusion, Fuel, 103, 606–616, doi:https://doi.org/10.1016/j.fuel.2012.06.119, 2013.
Graham T., On the law of the diffusion of gases, Journal of Membrane Science, 100(1), 17–21, doi: http://dx.doi.org/10.1016/0376-7388(94)00228-Q, 1995.
Hadjiconstantinou N. G., The limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics, Physics of Fluids, 18(11), doi:http://dx.doi.org/10.1063/1.2393436, 2006.
Knudsen M., Die gesetze der molekularströmung und der inneren reibungsströmung der gase durch röhren, Annalen der Physik, 333(1), 75–130, doi:10.1002/andp.19093330106, 1909.
Knudsen M., The laws of molecular flow and of inner friction flow of gases through tubes, Journal of Membrane Science, 100(1), 23–25, doi:http://dx.doi.org/10.1016/0376-7388(94)00299-E, 1995.
Kundt A., and E. Warburg, Ueber reibung und wärmeleitung verdünnter gase, Annalen der Physik, 232(10), 177–211, doi:10.1002/andp.18752321002, 1875.
Marder M., C.-H. Chen, and T. Patzek, Simple models of the hydrofracture process, Phys. Rev. E, 92, 062,408, doi: 10.1103/PhysRevE.92.062408, 2015.
Mehmani A., M. Prodanovic, and F. Javadpour, Multiscale, multiphysics network modeling of shale matrix gas flows, Transport in Porous Media, 99(2), 377–390, doi: 10.1007/s11242-013-0191-5, 2013.
Niven W. D. (Ed.), The scientific papers of James Clerk Maxwell, vol. II, Dover Publications, New York, https://strangebeautiful.com/other-texts/maxwell-scientificpapers-vol-ii-dover.pdf, 1890.
Patzek T. W., F. Male, and M. Marder, Gas production in the Barnett Shale obeys a simple scaling theory, Proceedings of the National Academy of Sciences, 110(49), 19,731-19,736, doi:10.1073/pnas.1313380110, 2013.
Prausnitz J. M., and P. R. Benson, Effective collision diameters and correlation of some thermodynamic properties of solutions, AIChE Journal, 5(3), 301–303, doi:10.1002/aic.690050310, 1959.
Ross D. J., and R. M. Bustin, The importance of shale composition and pore structure upon gas storage potential of shale gas reservoirs, Marine and Petroleum Geology, 26(6), 916–927, doi:https://doi.org/10.1016/j.marpetgeo.2008.06.004, 2009.
Roy S., R. Raju, H. F. Chuang, B. A. Cruden, and M. Meyyappan, Modeling gas flow through microchannels and nanopores, Journal of Applied Physics, 93(8), 4870–4879, doi: 10.1063/1.1559936, 2003.
Toledo P. G., L. E. Scriven, and H. T. Davis, Pore-Space Statistics and Capillary Pressure Curves From Volume-Controlled Porosimetry, SPE Formation Evaluation, March, 46–54, doi:doi:10.2118/19618-PA, 1994.
Wenski T. E., T. T. Olson, C. T. Rettner, and A. L. Garcia, Simulations of air slider bearings with realistic gas-surface scattering, ASME. J. Tribol, 120(3), 639–641, doi:doi:10.1115/1.2834599, 1998.
Yuan H. H., Pore-Scale Heterogeneity From Mercury Porosimetry Data (includes associated papers 22051 and 22052), SPE Formation Evaluation, June, 233–243, doi:doi:10.2118/19617-PA, 1991.
Yuan H. H., and B. F. Swanson, Resolving Pore-Space Characteristics by Rate-Controlled Porosimetry, SPE Formation Evaluation, March, 17–24, doi:doi:10.2118/14892-PA, 1989.