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Exact Solutions for One-Dimensional Transient Gas Flow in Porous Media with Gravity and Klinkenberg Effects

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Abstract

Attempts have been made to find exact solutions for the one-dimensional transient gas flow equation in porous media. By introducing a traveling wave variable, a traveling wave solution of the gas flow equation has been found. The traveling wave solution is presented in an explicit form of the space and time variables, and it takes into account both gravity and Klinkenberg effects (pressure-dependent permeability). We investigated the properties of the traveling wave solution and the effect of some parameters such as the Klinkenberg coefficient. A numerical study has been carried out, which confirms the stability of the traveling wave solution. The traveling wave solution is then used to derive two benchmark solutions defined over the semi-infinite domain. The first one assumes uniform initial gas pressure and non-uniform boundary condition, and the second assumes uniform boundary condition and non-uniform initial distribution of the gas pressure. The benchmark solutions are easy to use and are useful for validating numerical solutions. Two illustrative examples are presented in order to compare the benchmark solutions with the numerical solutions. The results show good agreements between the solutions.

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References

  • Adzumi, H.: Studies on the flow of gaseous mixtures through capillaries: II. The molecular flow of gaseous mixtures. Bull. Chem. Soc. Jpn. 12, 285–291 (1937)

    Article  Google Scholar 

  • Al-Hussainy, R., Ramey, H.J., Crawford, P.B.: The flow of real gases through porous media. J. Petrol. Technol. 18(5), 624–636 (1966)

    Article  Google Scholar 

  • Aronofsky, J.S.: Effect of gas slip on unsteady flow of gas through porous media. J. Appl. Phys. 25(1), 48–53 (1954)

    Article  Google Scholar 

  • Baehr, A.L., Hult, M.F.: Evaluation of unsaturated zone air permeability through pneumatic tests. Water Resour. Res. 27(10), 2605–2617 (1991)

    Article  Google Scholar 

  • Baehr, A.L., Joss, C.J.: An updated model of induced air-flow in the unsaturated zone. Water Resour. Res. 31(2), 417–421 (1995)

    Article  Google Scholar 

  • Beygi, M.E., Rashidi, F.: Analytical solutions to gas flow problems in low permeability porous media. Transp. Porous Media 87(2), 421–436 (2011)

    Article  Google Scholar 

  • Dake, L.P.: Fundamentals of Reservoir Engineering, Development in Petroleum Science, 8. Elsevier, Amsterdam (1978)

    Google Scholar 

  • DiGiulio, D.C., Varadhan, R.: Development of Recommendations and Methods to Support Assessment of Soil Venting Performance and Closure (EPA/600/R-01/070), p. 394. Environmental Protection Agency (EPA), Massachusetts (2001)

  • Falta, R.W.: Analytical solutions for gas flow due to gas injection and extraction from horizontal wells. Groundwater 33(2), 235–246 (1995)

    Article  Google Scholar 

  • Hindmarsh, A.C., Gresho, P.M., Griffiths, D.F.: The stability of explicit Euler time-integration for certain finite difference approximations of the multi-dimensional advection-diffusion equation. Int. J. Numer. Methods Fluids 4(9), 853–897 (1984)

    Article  Google Scholar 

  • Ho, C.K., Webb, S.W.: Gas Transport in Porous Media, Series: Theory and Applications of Transport in porous Media, vol. 20. Springer, New York (2006)

    Book  Google Scholar 

  • Ikoku, C.U.: Natural Gas Reservoir Engineering, p. 503. Wiley, New York (1984)

    Google Scholar 

  • Johnson, P.C., Kemblowski, M.W., Colthart, D.J.: Quantitative analysis for the cleanup of hydrocarbon-contaminated soils by in-situ soil venting. Groundwater 28(3), 413–429 (1990)

    Article  Google Scholar 

  • Katz, D.L., et al.: Handbook of Natural Gas Engineering. McGraw-Hill, New York (1959)

    Google Scholar 

  • Kidder, R.F.: Unsteady flow of gas through a semi-infinite porous medium. J. Appl. Mech. 24, 329–332 (1957)

    Google Scholar 

  • Klinkenberg, L.J.: The Permeability of Porous Media to Liquids and Gases, Drilling and Production Practice. American Petroleum Inst., pp. 200–213 (1941)

  • Li, J., Zhan, H., Huang, G.: Applicability of the linearized governing equation of gas flow in porous media. Transp. Porous Media 87(3), 815–834 (2011)

    Article  Google Scholar 

  • Massmann, J.W.: Applying groundwater-flow models in vapor extraction system-design. J. Environ. Eng. ASCE 115(1), 129–149 (1989)

    Article  Google Scholar 

  • Massmann, J.W., Shock, S., Johannesen, L.: Uncertainties in cleanup times for soil vapor extraction. Water Resour. Res. 36(3), 679–692 (2000)

    Article  Google Scholar 

  • McWhorter, D.B.: Unsteady radial flow of gas in the vadose zone. J. Contam. Hydrol. 5(3), 297–314 (1990)

    Article  Google Scholar 

  • Morrison, F.A.: Transient gas flow in a porous column. Ind. Eng. Chem. Fundam. 11(2), 191–197 (1972)

    Article  Google Scholar 

  • Muskat, M.: The Flow of Homogeneous Fluids through Porous Media. J. W. Edwards, Ann Arbor (1946)

    Google Scholar 

  • Pruess, K.: TOUGH2—A General-Purpose Numerical Simulator for Multiphase Fluid and Heat Flow, Report LBL-29400. Lawrence Berkeley Laboratory, Berkeley (1991)

    Google Scholar 

  • Scanlon, B.R., Nicot, J.P., Massmann, J.W.: Soil gas movement in unsaturated system. In: Warrick, A.W. (ed.) Soil Physics Companion, pp. 297–341. CRC Press, Boca Raton (2002)

    Google Scholar 

  • Shan, C., Falta, R.W., Javandel, I.: Analytical solutions for steady state gas flow to a soil vapor extraction well. Water Resour. Res. 28(4), 1105–1120 (1992)

    Article  Google Scholar 

  • Shan, C.: An analytical solutions for transient gas flow in a multiwell system. Water Resour. Res. 42(10), W10401 (2006). doi:10.1029/2005WR004737

    Article  Google Scholar 

  • Vanderborght, J., Kasteel, R., Herbst, M., Javaux, M., Thiéry, D., Vanclooster, M., Mouvet, C., Vereecken, H.: A set of analytical benchmarks to test numerical models of flow and transport in soils. Vadose Zone J. 4(1), 206–221 (2005)

    Article  Google Scholar 

  • Wu, Y.-S., Pruess, K., Persoff, P.: Gas flow in porous media with Klinkenberg effects. Transp. Porous Media 32(1), 117–137 (1998)

    Article  Google Scholar 

  • White, M.D., Oostrom, M.: STOMP Subsurface Transport Over Multiple Phases Theory Guide, Variously Paginated. Pacific Northwest National Laboratory, Richland (1996)

    Book  Google Scholar 

  • Zhang, Y., Oldenburg, C.M., Benson, S.M.: Vadose zone remediation of carbon dioxide leakage from geologic carbon dioxide sequestration sites. Vadose Zone J. 3(3), 858–866 (2005)

    Article  Google Scholar 

  • Zlotnik, V.A., Wang, T., Nieber, J.L., Šimunek, J.: Verification of numerical solutions of the Richards equation using a traveling wave solution. Adv. Water Resour. 30(9), 1973–1980 (2007)

    Article  Google Scholar 

  • Zyvoloski, G.A., Robinson, B.A., Dash, Z.V., Trease, L.L.: Models and Methods Summary for the FEHM Application. Los Alamos National Lab, Los Alamos (1996)

    Book  Google Scholar 

Download references

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  1. Groundwater Protection and Waste Disposal, AF-Consult Switzerland Ltd, Täfernstrasse 26, 5405, Baden, Switzerland

    Mohamed Hayek

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  1. Mohamed Hayek
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Hayek, M. Exact Solutions for One-Dimensional Transient Gas Flow in Porous Media with Gravity and Klinkenberg Effects. Transp Porous Med 107, 403–417 (2015). https://doi.org/10.1007/s11242-014-0445-x

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