Transport in Porous Media

, Volume 29, Issue 1, pp 27–45 | Cite as

Well Test Solutions for Vertically Fractured Injection Wells

  • J. B. Spath
  • R. K. M. Thambynayagam


The transient behavior of a vertically fractured pressure response due to the presence of an infinite-conductivity vertical fracture is determined by solving the diffusivity equation in elliptical coordinates. The solution is then extended to a composite elliptical system to provide for the different fluid banks present during water injection. The validity of the analytical solutions presented is demonstrated by comparing limiting forms with those available elsewhere in the literature. Computational issues which became evident during the verification stage of our work are also discussed. The solutions have been developed in the Laplace domain to facilitate the addition of fissures and variable rate production (i.e. wellbore storage).

A pressure transient test for tracking the advancement of a water front during the early stages of waterflooding is described. We utilize the composite elliptical model developed herein to provide for two distinct regions in which the flow behavior resulting from an induced fracture is elliptical rather than radial. A relationship between the increasing elliptical distance to the waterfront and the resulting change in the apparent (total) skin factor is obtained. Through the analysis of successive falloff tests, this relationship may be used to monitor the advancement of the front provided the cumulative volume of injected water is known. The fluid saturations and the mobilities of the swept and unswept regions are assumed unknown and are obtained from the test analysis.

Finally, we present methods for computing the Mathieu functions necessary in solving the diffusivity equation in elliptical coordinates. Mathieu functions are utilized in many applications involving elliptical geometry and we feel the efficient evaluation of these functions is an important contribution of this work.

injection well elliptical flow waterflooding Mathieu functions well testing transient analysis water injection 


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  1. 1.
    Hazebroek, P., Rainbow, H. and Mathews, C. C.: Pressure fall-off in water injection wells, Trans. of AIME (1958), 250-260.Google Scholar
  2. 2.
    Kazemi, H., Merrill, L. S. and Jargon, J. R.: Problems in interpretation of pressure fall-off tests in reservoirs with and without fluid banks, J. Petrol. Tech. (1972), 1147-1156.Google Scholar
  3. 3.
    Merrill, L. S., Jr., Kazemi, H. and Gogarty, W. B.: Pressure falloff analysis in reservoirs with fluid banks, Journal of Petroleum Technology, (1979), 809-818.Google Scholar
  4. 4.
    Woodward, D. K., Thambynayagam, R. K. M.: Pressure buildup and falloff analysis of water injection tests, paper SPE 12344, 1983.Google Scholar
  5. 5.
    Thambynayagam, R. K. M.: Analytic solutions for pressure buildup and falloff analysis of water injection tests of partially penetrating wells: Non-unit mobility ratios, paper SPE 12965, 1983.Google Scholar
  6. 6.
    Kuchuk, F. and Brigham, E.W.: Transient flow in elliptical systems, Soc. Petrol. Eng. J. (1979) 401-410.Google Scholar
  7. 7.
    Obut, S. T. and Ertekin, T.: A composite system solution in elliptic flow geometry, SPEFE (1987), 227-236.Google Scholar
  8. 8.
    Stanislav, J. F., Easwaran, C. V. and Kokal, S. L.: Analytical solutions for vertical fractures in a composite system, J. Canad. Petrol. Technol. 1987.Google Scholar
  9. 9.
    Abbaszadeh, M. and Kamal, M.: Pressure-transient testing of water injection wells, SPERE (1989), 115-124.Google Scholar
  10. 10.
    Ramakrishnan, T. S. and Kuchuk, F.: Transient testing in injection wells: constant rate solutions and convolution, SDR Research Note, 1989.Google Scholar
  11. 11.
    U.S. Bureau of Standards, Tables Relating to Mathieu Functions, Vol. II. Washington, DC: US Govt. Printing Office, 1965.Google Scholar
  12. 12.
    Gringarten, A. C., Ramey, H.J., Jr and Raghavan, R.: Unsteady-state pressure distribution created by a well with a single infinite-conductivity vertical fracture, Soc. Petrol. Eng. J. (1974), 347-360.Google Scholar
  13. 13.
    Muskat, M.: Physical Principles of Oil Production, McGraw-Hill, New York, 1949.Google Scholar
  14. 14.
    Benson, S. M. and Bodvarrsson, G.: A pressure transient method for front tracking. Paper SPE 12130 presented at the 58th Annual Technical Conference and Exhibition, San Francisco, CA., 1983.Google Scholar
  15. 15.
    McLachlan, N.W.: Theory and Application of Mathieu Functions, London: Oxford University Press., 1947.Google Scholar
  16. 16.
    Peskin, Edward: Periodic Differential Equations.Boston Technical Publishers, Inc., Cambridge, Mass., 1965.Google Scholar
  17. 17.
    Smith, J. J.: A method of solving Mathieu's equation, J. AIEE, 1955.Google Scholar
  18. 18.
    Rengarajan, S. R. and Lewis, J. E.: Mathieu functions of integral orders and real arguments, IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-28, No. 3, 1980.Google Scholar
  19. 19.
    Fruchting, H.: Fourier coefficients of Mathieu functions in stable regions, J. Res. Nat. Bureau of Standards, 73B(1) (1969).Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • J. B. Spath
    • 1
  • R. K. M. Thambynayagam
    • 1
  1. 1.LafayetteU.S.A.

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