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Acta Mechanica Sinica

, Volume 32, Issue 1, pp 38–53 | Cite as

Effect of quadratic pressure gradient term on a one-dimensional moving boundary problem based on modified Darcy’s law

  • Wenchao LiuEmail author
  • Jun Yao
  • Zhangxin Chen
  • Yuewu Liu
Research Paper

Abstract

A relatively high formation pressure gradient can exist in seepage flow in low-permeable porous media with a threshold pressure gradient, and a significant error may then be caused in the model computation by neglecting the quadratic pressure gradient term in the governing equations. Based on these concerns, in consideration of the quadratic pressure gradient term, a basic moving boundary model is constructed for a one-dimensional seepage flow problem with a threshold pressure gradient. Owing to a strong nonlinearity and the existing moving boundary in the mathematical model, a corresponding numerical solution method is presented. First, a spatial coordinate transformation method is adopted in order to transform the system of partial differential equations with moving boundary conditions into a closed system with fixed boundary conditions; then the solution can be stably numerically obtained by a fully implicit finite-difference method. The validity of the numerical method is verified by a published exact analytical solution. Furthermore, to compare with Darcy’s flow problem, the exact analytical solution for the case of Darcy’s flow considering the quadratic pressure gradient term is also derived by an inverse Laplace transform. A comparison of these model solutions leads to the conclusion that such moving boundary problems must incorporate the quadratic pressure gradient term in their governing equations; the sensitive effects of the quadratic pressure gradient term tend to diminish, with the dimensionless threshold pressure gradient increasing for the one-dimensional problem.

Keywords

Quadratic pressure gradient term Threshold pressure gradient Porous media Numerical solution Moving boundary 

Notes

Acknowledgments

The authors would like to acknowledge the funding by the project (Grant 51404232) sponsored by the National Natural Science Foundation of China, the National Science and Technology Major Project (Grant 2011ZX05038003), and the China Postdoctoral Science Foundation project (Grant 2014M561074). In particular, Wenchao Liu would also like to express his deepest gratitude to the China Scholarship Council for its generous financial support of the research. Special thanks go to Dr. Yongfei Yang, Dr. Lili Xue, and Dr. Lei Zhang for their tremendous help in improving the writing and wording of the paper.

References

  1. 1.
    Huang, Y.Z., Yang, Z.M., He, Y., et al.: An overview on nonlinear porous flow in low permeability porous Media. Theor. Appl. Mech. Lett. 3, 022001 (2013)CrossRefGoogle Scholar
  2. 2.
    Monteiro, P.J.M., Rycroft, C.H., Barenblatt, G.I.: A mathematical model of fluid and gas flow in nanoporous media. Proc. Natl. Acad. Sci. USA 109, 20309–20313 (2012)CrossRefGoogle Scholar
  3. 3.
    Balhoff, M., Sanchez-Rivera, D., Kwok, A., et al.: Numerical algorithms for network modeling of yield stress and other non-Newtonian fluids in porous media. Transp. Porous Media 93, 363–379 (2012)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Yu, R.Z., Bian, Y.N., Li, Y., et al.: Non-Darcy flow numerical simulation of XPJ low permeability reservoir. J. Pet. Sci. Eng. 92–93, 40–47 (2012)Google Scholar
  5. 5.
    Yu, R.Z., Bian, Y.N., Zhou, S., et al.: Nonlinear flow numerical simulation of low-permeability reservoir. J. Cent. South Univ. Technol. 19, 1980–1987 (2012)CrossRefGoogle Scholar
  6. 6.
    Guo, J.J., Zhang, S., Zhang, L.H., et al.: Well testing analysis for horizontal well with consideration of threshold pressure gradient in tight gas reservoirs. J. Hydrodyn. 24, 561–568 (2012)CrossRefGoogle Scholar
  7. 7.
    Luo, W.J., Wang, X.D.: Effect of a moving boundary on the fluid transient flow in low permeability Reservoirs. J. Hydrodyn. 24, 391–398 (2012)CrossRefGoogle Scholar
  8. 8.
    Yao, J., Liu, W.C., Chen, Z.X.: Numerical solution of a moving boundary problem of one-dimensional flow in semi-infinite long porous media with threshold pressure gradient. Math. Probl. Eng. 2013, 384246 (2013)MathSciNetGoogle Scholar
  9. 9.
    Zeng, B.Q., Cheng, L.S., Li, C.L.: Low velocity non-linear flow in ultra-low permeability reservoir. J. Pet. Sci. Eng. 80, 1–6 (2012)CrossRefGoogle Scholar
  10. 10.
    Liu, W.C., Yao, J., Wang, Y.Y.: Exact analytical solutions of moving boundary problems of one-dimensional flow in semi-infinite long porous media with threshold pressure gradient. Int. J. Heat Mass Transf. 55, 6017–6022 (2012)CrossRefGoogle Scholar
  11. 11.
    Liu, W.C., Yao, J., Chen, Z.X., et al.: Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability. Acta Mech. Sin. 30, 50–58 (2014)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Zhu, W.Y., Song, H.Q., Huang, X.H., et al.: Pressure characteristics and effective deployment in a water—bearing tight gas reservoir with low-velocity non-Darcy flow. Energy Fuels 25, 1111–1117 (2011)CrossRefGoogle Scholar
  13. 13.
    Beygi, M.E., Rashidi, F.: Analytical solutions to gas flow problems in low permeability porous media. Transp. Porous Media 87, 421–436 (2011)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Cai, J.C., Yu, B.M.: A discussion of the effect of tortuosity on the capillary imbibition in porous media. Transp. Porous Media 89, 251–263 (2011)CrossRefGoogle Scholar
  15. 15.
    Wang, X.W., Yang, Z.M., Qi, Y.D., et al.: Effect of absorption boundary layer on nonlinear flow in low permeability porous media. J. Cent. South Univ. Technol. 18, 1299–1303 (2011)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Jing, W., Liu, H.Q., Pang, Z.X., et al.: The investigation of threshold pressure gradient of foam flooding in porous media. Pet. Sci. Technol. 29, 2460–2470 (2011)Google Scholar
  17. 17.
    Xu, Q.Y., Liu, X.G., Yang, Z.M., et al.: The model and algorithm of a new numerical simulation software for low permeability reservoirs. J. Pet. Sci. Eng. 78, 239–242 (2011)CrossRefGoogle Scholar
  18. 18.
    Yao, Y.D., Ge, J.L.: Characteristics of non-Darcy flow in low-permeability reservoirs. Pet. Sci. 8, 55–62 (2011)CrossRefGoogle Scholar
  19. 19.
    Civan, F.: Porous Media Transport Phenomena. JohnWiley & Sons Press, Inc, Hoboken (2011)CrossRefGoogle Scholar
  20. 20.
    Song, F.Q., Wang, J.D., Liu, H.L.: Static threshold pressure gradient characteristics of liquid influenced by boundary wettability. Chin. Phys. Lett. 27, 024704 (2010)CrossRefGoogle Scholar
  21. 21.
    Daprà, I., Scarpi, G.: Unsteady simple shear flow in a viscoplastic fluid: comparison between analytical and numerical solutions. Rheol. Acta 49, 15–22 (2010)CrossRefGoogle Scholar
  22. 22.
    Xie, K.H., Wang, K., Wang, Y.L., et al.: Analytical solution for one-dimensional consolidation of clayey soils with a threshold gradient. Comput. Geotech. 37, 487–493 (2010)CrossRefGoogle Scholar
  23. 23.
    Yue, X.A., Wei, H.G., Zhang, L.J., et al.: Low pressure gas percolation characteristic in ultra-low permeability porous media. Transp. Porous Media 85, 333–345 (2010)CrossRefGoogle Scholar
  24. 24.
    Yun, M.J., Yu, B.M., Lu, J.D., et al.: Fractal analysis of Herschel-Bulkley fluid flow in porous media. Int. J. Heat Mass Transf. 53, 3570–3574 (2010)CrossRefzbMATHGoogle Scholar
  25. 25.
    Li, Y., Yu, B.M.: Study of the starting pressure gradient in branching network. Sci. China Technol. Sci. 53, 2397–2403 (2010)CrossRefzbMATHGoogle Scholar
  26. 26.
    Zhao, Y.S., Kumar, L., Paso, K., et al.: Gelation behavior of model wax-oil and crude oil systems and yield stress model development. Energy Fuels 26, 6323–6331 (2012)CrossRefGoogle Scholar
  27. 27.
    Fossen, M., Øyangen, T., Velle, O.J.: Effect of the pipe diameter on the restart pressure of a gelled waxy crude oil. Energy Fuels 27, 3685–3691 (2013)CrossRefGoogle Scholar
  28. 28.
    Papanastasiou, T.C., Boudouvis, A.G.: Flows of viscoplastic materials: models and computation. Comput. Struct. 64, 677–694 (1997)CrossRefzbMATHGoogle Scholar
  29. 29.
    Prada, A., Civan, F.: Modification of Darcy’s law for the threshold pressure gradient. J. Pet. Sci. Eng. 22, 237–240 (1999)CrossRefGoogle Scholar
  30. 30.
    Nedoma, J.: Numerical solution of a Stefan-like problem in Bingham rheology. Math. Comput. Simul. 61, 271–281 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Chen, M., William, R., Yannis, C.Y.: The flow and displacement in porous media of fluids with yield stress. Chem. Eng. Sci. 60, 4183–4202 (2005)CrossRefGoogle Scholar
  32. 32.
    Wang, S.J., Huang, Y.Z., Civan, F.: Experimental and theoretical investigation of the Zaoyuan field heavy oil flow through porous media. J. Pet. Sci. Eng. 50, 83–101 (2006)CrossRefGoogle Scholar
  33. 33.
    Song, F.Q., Jiang, R.J., Bian, S.L.: Measurement of threshold pressure gradient of microchannels by static Method. Chin. Phys. Lett. 24, 1995–1998 (2007)CrossRefGoogle Scholar
  34. 34.
    Hao, F., Cheng, L.S., Hassan, O., et al.: Threshold pressure gradient in ultra-low permeability reservoirs. Pet. Sci. Technol. 26, 1024–1035 (2008)CrossRefGoogle Scholar
  35. 35.
    Yun, M.J., Yu, B.M., Cai, J.C.: A fractal model for the starting pressure gradient for Bingham fluids in porous media. Int. J. Heat Mass Transf. 51, 1402–1408 (2008)CrossRefzbMATHGoogle Scholar
  36. 36.
    Wang, F., Yue, X.A., Xu, S.L., et al.: Influence of wettability on flow characteristics of water through microtubes and cores. Chin. Sci. Bull. 54, 2256–2262 (2009)CrossRefGoogle Scholar
  37. 37.
    Xiong, W., Lei, Q., Gao, S.S., et al.: Pseudo threshold pressure gradient to flow for low permeability reservoirs. Pet. Explor. Dev. 36, 232–236 (2009)CrossRefGoogle Scholar
  38. 38.
    Cai, J.C., Yu, B.M., Zou, M.Q., et al.: Fractal analysis of invasion depth of extraneous fluids in porous Media. Chem. Eng. Sci. 65, 5178–5186 (2010)CrossRefGoogle Scholar
  39. 39.
    Cai, J.C., Hu, X.Y., Standnes, D.C., et al.: An analytical model for spontaneous imbibition in fractal porous media including gravity. Coll. Surf. A 414, 228–323 (2012)CrossRefGoogle Scholar
  40. 40.
    Darcy, H.: Les Fontaines Publiques de La Ville de Dijon [The Public Fountains of the Town of Dijon]. Dalmont, Paris (1856) (in French)Google Scholar
  41. 41.
    Cai, J.C.: A fractal approach to low velocity non-Darcy flow in a low permeability porous medium. Chin. Phys. B 23, 044701 (2014)CrossRefGoogle Scholar
  42. 42.
    Pascal, H.: Nonsteady flow through porous media in the presence of a threshold pressure gradient. Acta Mech. 39, 207–224 (1981)CrossRefzbMATHGoogle Scholar
  43. 43.
    Wu, Y.S., Pruess, K., Witherspoon, P.A.: Flow and displacement of Bingham non-Newtonian fluids in porous Media. SPE Reserv. Eng. 7, 369–376 (1992)CrossRefGoogle Scholar
  44. 44.
    Song, F.Q., Liu, C.Q., Li, F.H.: Transient pressure of percolation through one dimension porous media with threshold pressure gradient. Appl. Math. Mech. 20, 27–35 (1999)CrossRefzbMATHGoogle Scholar
  45. 45.
    Zhu, Y., Xie, J.Z., Yang, W.H., et al.: Method for improving history matching precision of reservoir numerical simulation. Pet. Explor. Dev. 35, 225–229 (2008)CrossRefGoogle Scholar
  46. 46.
    Feng, G.Q., Liu, Q.G., Shi, G.Z., et al.: An unsteady seepage flow model considering kickoff pressure gradient for low-permeability gas reservoirs. Pet. Explor. Dev. 35, 457–461 (2008)CrossRefGoogle Scholar
  47. 47.
    Marshall, S.L.: Nonlinear pressure diffusion in flow of compressible liquids through porous media. Transp. Porous Media 77, 431–446 (2009)CrossRefMathSciNetGoogle Scholar
  48. 48.
    Bai, M., Ma, Q.G., Roegiers, J.C.: A nonlinear dual-porosity model. Appl. Math. Modell. 18, 602–610 (1994)CrossRefzbMATHGoogle Scholar
  49. 49.
    Cao, X.L., Tong, D.K., Wang, R.H.: Exact solutions for nonlinear transient flow model including a quadratic gradient term. Appl. Math. Mech. 25, 102–109 (2004)CrossRefzbMATHGoogle Scholar
  50. 50.
    Tong, D.K., Zhang, H.Q., Wang, R.H.: Exact solution and its behavior characteristic of nonlinear dual-porosity model. Appl. Math. Mech. 26, 1277–1283 (2005)CrossRefzbMATHGoogle Scholar
  51. 51.
    Chakrabarty, C., Farouq, A.S.M., Tortike, W.S.: Effects of the nonlinear gradient term on the transient pressure solution for a radial flow system. J. Pet. Sci. Eng. 8, 241–256 (1993)CrossRefGoogle Scholar
  52. 52.
    Odeh, A.S., Babu, D.K.: Comparison of solutions of the nonlinear and linearized diffusion equations. SPE Reserv. Eng. 3, 1202–1206 (1988)CrossRefGoogle Scholar
  53. 53.
    Finjord, J., Aadnoy, B.S., Rogaland, R.C.: Effects of the quadratic gradient term in steady-state and semisteady-state solutions for reservoir pressure. SPE Form. Eval. 4, 413–417 (1989)CrossRefGoogle Scholar
  54. 54.
    Wang, Y., Dusseault, M.B.: The effect of quadratic gradient terms on the borehole solution in poroelastic Media. Water Resour. Res. 27, 3215–3223 (1991)CrossRefGoogle Scholar
  55. 55.
    Chakrabarty, C., Farouq, A.S.M., Tortike, W.S.: Analytical solutions for radial pressure distribution including the effects of the quadratic-gradient term. Water Resour. Res. 29, 1171–1177 (1993)Google Scholar
  56. 56.
    Braeuning, S., Jelmert, T.A., Vik, S.A.: The effect of the quadratic gradient term on variable-rate well-tests. J. Pet. Sci. Eng. 21, 203–222 (1998)CrossRefGoogle Scholar
  57. 57.
    Li, W., Li, X.P., Li, S.C., et al.: The similar structure of solutions in fractal multilayer reservoir including a quadratic gradient term. J. Hydrodyn. 24, 332–338 (2012)Google Scholar
  58. 58.
    Dewei, M., Ailin, J., Chengye, J., et al.: Research on transient flow regulation with the effect of quadratic pressure gradient. Pet. Sci. Technol. 31, 408–417 (2013)CrossRefGoogle Scholar
  59. 59.
    Nie, R.S., Ge, F., Liu, Y.L.: The researches on the nonlinear flow model with quadratic pressure gradient and its application for double porosity reservoir. In: Flow in porous media: from phenomena to engineering and beyond: 2009 International Forum on Porous Flow and Applications. Wuhan (2009)Google Scholar
  60. 60.
    Yao, Y.D., Wu, Y.S., Zhang, R.L.: The transient flow analysis of fluid in a fractal, double-porosity reservoir. Transp. Porous Media 94, 175–187 (2012)CrossRefMathSciNetGoogle Scholar
  61. 61.
    Nie, R.S., Jia, Y.L., Yu, J., et al.: The transient well test analysis of fractured-vuggy triple-porosity reservoir with the quadratic pressure gradient term. In: Latin American and Caribbean Petroleum Engineering Conference. Cartagena de Indias (2009)Google Scholar
  62. 62.
    Crank, J.: Free and Moving Boundary Problems. Clarendon Press, Oxford (1984)zbMATHGoogle Scholar
  63. 63.
    Gupta, R.S., Kumar, A.: Treatment of multi-dimensional moving boundary problems by coordinate transformation. Int. J. Heat Mass Transf. 28, 1355–1366 (1985)CrossRefzbMATHGoogle Scholar
  64. 64.
    Méndez-Bermúdez, A., Luna-Acosta, G.A., Izrailev, F.M., et al.: Solution of the eigenvalue problem for two-dimensional modulated billiards using a coordinate transformation. Commun. Nonlinear Sci. Numer. Simul. 10, 787–795 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  65. 65.
    Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Brooks/Cole, West Lafayette (2010)Google Scholar
  66. 66.
    Poularikas, A.D.: The Handbook of Formulas and Table for Signal ‘Processing, the Electrical Engineering Handbook Series. CRC Press LLC and IEEE Press, New York (1999)Google Scholar
  67. 67.
    McCollum, P.A., Brown, B.F.: Laplace Transform Tables and Theorems. Holt Rinehart and Winston, New York (1965)Google Scholar
  68. 68.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Wenchao Liu
    • 1
    • 2
    Email author
  • Jun Yao
    • 2
  • Zhangxin Chen
    • 3
  • Yuewu Liu
    • 1
  1. 1.Institute of MechanicsChinese Academy of SciencesBeijingChina
  2. 2.School of Petroleum EngineeringChina University of Petroleum (Huadong)QingdaoChina
  3. 3.Department of Chemical and Petroleum Engineering, Schulich School of EngineeringUniversity of CalgaryCalgaryCanada

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