Abstract
Pressure transient testing during water injection is undertaken to evaluate the injection potential of a well. If properly interpreted, it can yield information such as effective mobilities of fluids, wellbore damage, and residual oil saturation. This is best done by the simultaneous use of downhole flow-rate and pressure measurements.
Analytical solutions obtained under various assumptions for pressure response of an injection well are investigated. For a constant downhole flow rate, it is demonstrated that exact solutions may be obtained for an infinite reservoir during both the injection and the falloff periods. Due to the inherent nonlinearity of the problem, the constant rate solutions are not readily extended for the general case of varying flow rates. Therefore, we have employed an approximate technique. This technique can be used with an altered form of convolution and permits calculation of the pressure response for arbitrary rate data. The range of parameters under which this method may be used are also identified. Computational methods related to convolution are presented.
The numerical techniques developed in this paper are sufficiently general that they may be applied to similar well-testing problems involving single-phase flow.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- c :
-
compressibility
- E :
-
error
- E n :
-
exponential integral
- H :
-
Heaviside function
- h :
-
formation thickness
- I n :
-
modified Bessel functions of the first kind
- k :
-
permeability
- K n :
-
modified Bessel functions of the second kind
- M :
-
number of flow rate jumps
- N :
-
number of data points
- p :
-
pressure
- P :
-
pressure scale
- r :
-
radius
- R :
-
characteristic radius of pressure propagation
- q :
-
flow rate
- s :
-
Laplace transform variable
- S :
-
skin factor
- T :
-
characteristic time (total injection time)
- t :
-
time
- W n :
-
well function defined by Equation (C3)
- x :
-
selection criterion
- α :
-
wellbore storage constant
- δ :
-
annulus correction
- Β S :
-
jump in saturation
- ε :
-
parameter defined by Equation (16)
- ε ′ :
-
parameter defined by Equation (25)
- \(\hat \in \) :
-
parameter defined by Equation (A7)
- ζ :
-
location of the front
- η :
-
similarity variable
- θ :
-
defined by Equation (C10)
- λ :
-
mobility
- λ :
-
parameter
- Μ :
-
viscosity
- Ν :
-
dummy variable
- σ:
-
constant
- Τ :
-
slow-time variable
- Υ :
-
parameter defined by Equation (25)
- Φ :
-
porosity
- χ :
-
mobility ratio
- χ z :
-
zero error mobility ratio
- ψ :
-
diffusivity ratio
- a :
-
annulus
- b :
-
slope
- D :
-
dimensionless
- f :
-
formation
- fi :
-
fictitious
- h :
-
Horner
- M :
-
measured
- p :
-
unit pulse response
- q :
-
rate
- s :
-
surface
- t :
-
time
- w :
-
well or wellbore
- i :
-
invaded
- u :
-
uninvaded
- -:
-
Laplace transform of
- ^:
-
excess or shifted
References
Abbaszadeh, M. and Kamal, M., 1989, Pressure transient testing of water injection wells,SPE Res. Engng. 4, 115–124.
Agarwal, R. G., Al-Hussainy, R., and Ramey, H. J. Jr., 1970, An investigation of wellborne storage and skin effect in unsteady liquid flow: II. Finite difference treatment,Soc. Pet. Eng. J. 10, 291–297.
Bixel, H. C. and Van Poollen, H. K., 1967, Pressure drawdown and buildup in the presence of radial discontinuities,Soc. Petrol. Engng. J. 7, 301–309.
Boley, B. A., 1961, A method of heat conduction analysis of melting and solidification problems.J. Math. Phys. 40, 300–313.
Bratvold, R. B. and Horne, R. N., 1988, Analysis of pressure falloff tests following cold water injection, in63rd SPE Annual Technical Conference and Exhibition, SPE 18111. Houston, TX.
Carlslaw, H. S. and Jaeger, J. C., 1959,Conduction of Heat in Solids, Clarendon Press, Oxford.
Crump, K. S., 1976, Numerical inversion of Laplace transforms using a Fourier series approximation,J. Assn. Comp. Mach. 23, 89–96.
Davies, B. and Martin, B., 1979, Numerical inversion of the Laplace transform: a survey and comparison of methods,J. Comp. Phys. 33, 1–32.
Dubner, H. and Abate, J., 1968, Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform,J. Assn. Comp. Mach. 15, 115–123.
Earlougher, R. C., 1977,Advances in Well Test Analysis, SPE of AIME, New York.
Kazemi, H., Merrill, L. S., and Jargon, J. R., 1972, Problems in interpretation of pressure falloff tests in reservoirs with and without fluid banks,J. Petrol. Technol. 24, 1147–1156.
MacDonald, J. R., 1964, Accelerated convergence, divergence, iteration, extrapolation and curve fitting,J. Appl. Phys. 35, 3034–3041.
Matthews, C. S. and Russell, D. G., 1967,Pressure Buildup and Flow Tests in Wells, SPE of AIME, New York.
Merrill, L. S., Kazemi, H., and Gogarty, W. B., 1974, Pressure falloff analysis in reservoirs with fluid banks,J. Petrol. Technol. 26, 809–818.
Moore, D. J. H. and Parker, D. J., 1973, On nonlinear filters involving transformation of the time variable,IEEE Trans. Information Theory IT-19, 415–422.
Nayfeh, A. H., 1981,Introduction to Perturbation Techniques. Wiley, New York.
Rubenstein, L. I., 1971,The Stefan Problem. Amer. Math. Soc., Providence, RI.
Sosa, A., Raghavan, R., and Limon, T. J., 1981, Effect of relative permeability and mobility ratio on pressure falloff behavior,Soc. Petrol. Engng. J. 21, 1125–1135.
Stehfest, H., 1969, Numerical inversion of Laplace transforms,Comm. ACM 13, 368-P1–368-P2.
Van Everdingen, A. F. and Hurst, W., 1949, The application of the Laplace transformation to flow problems in reservoirs,Trans. AIME 186, 305–324.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ramakrishnan, T.S., Kuchuk, F.J. Pressure transients during injection: Constant rate and convolution solutions. Transp Porous Med 10, 103–136 (1993). https://doi.org/10.1007/BF00617004
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00617004