Abstract
Steady-state radial flow in three-dimensional heterogeneous media is investigated using a geostatistical approach. The goal of the study is to develop a model of the relationship between corescale hydraulic conductivities measured at the wellbore and the conductivity of the surrounding drainage region as measured by a larger scale flow experiment such as a pump test. Conductivity at the point or core-scale is modeled as a stationary and multivariate lognormal spatial random function. Conductivity of the drainage region is obtained by a weighted nonlinear spatial average over the point-scale values within. This empirical spatial averaging process is shown to yield excellent approximations of true effective drainage region conductivities calculated using a numerical flow model. The geostatistical model for point-scale conductivity and the spatial averaging process are used to determine the first and second order ensemble moments of drainage region conductivity. In particular, an expression is derived for the conditional expectation of drainage region conductivity given point-scale values measured at the wellbore. The results are illustrated in a case study of a well from a sandstone oil reservoir where both core and transient-test conductivity data from the same interval are available for comparison.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- α :
-
mean log-conductivity [−]
- b :
-
formation thickness [L]
- Δ :
-
grid spacing [L]
- g :
-
acceleration of gravity [L/T 2]
- K(x):
-
point hydraulic conductivity [L/T]
- K E :
-
effective conductivity [L/T]
- K v :
-
volume-averaged conductivity [L/T]
- K w :
-
average conductivity at the well [L/T]
- λ :
-
integral range of correlation [L]
- ω :
-
averaging exponent [−]
- φ :
-
piezometric head or potential [L]
- Q :
-
pumping rate [L 3/T]
- r w :
-
well-bore radius [L]
- r e :
-
external radius of drainage region [L]
- r(x):
-
distance fromx to well-bore axis [L]
- ρ :
-
fluid density [M/L 3]
- σ(h):
-
log-conductivity covariance [−]
- σ 2 :
-
log-conductivity variance [−]
- μ :
-
fluid dynamic viscosity [M/T.L]
- V :
-
averaging volume for drainage region [L 3]
- x :
-
spatial location vector [L]
- Y(x):
-
log-conductivity [−].
References
Ababou, R., 1991, Identification of Effective Conductivity Tensor in Randomly Heterogeneous and Stratified Aquifers,in S. Bachu (Ed.), Proceedings of the 5th Canadian/American Conference on Hydrogeology: Parameter Identification and Estimation for Aquifer and Reservoir Characterization, National Well Water Ass., Dublin, Ohio, p. 155–157.
Aitchison, J., and Brown, J. A. C., 1957,The Lognormal Distribution: Cambridge University Press, London, 176 p.
Alabert, F. G., 1989, Constraining Description of Randomly Heterogeneous Reservoirs to Pressure Test Data: A Monte Carlo Study, SPE paper 19600, presented at the 64th Ann. Tech. Conf. of the SPE, San Antonio, Texas, Oct. 8–11, 1989.
Anderson, T. W., 1958,An Introduction to Multivariate Statistical Analysis Wiley, New York, 374 p.
Bennion, D. W., and Griffiths, J. C., 1966, A Stochastic Model for Predicting Variations in Reservoir Rock Properties: SPEJ Trans. AIME, v. 237, p. 9–16.
Butler, J. J., Jr., 1988, Pumping Tests in Nonuniform Aquifers: The Radially Symmetric Case: J. Hydrol., v. 101(1/4), p. 15–30.
Butler, J. J., Jr., and McElwee, C. D, 1990, Variable Rate Pumping Tests for Radially Symmetric Nonuniform Aquifers: Water Res. Res., v. 26, n. 2, p. 291–306.
Butler, J. J., Jr., and Wenzhi Liu, 1993, Pumping Tests in Nonuniform Aquifers: The Radially Asymmetric Case: Water Res. Res., v. 29, n. 2, p. 259–270.
Cardwell, W. T., and Parsons, R. L., 1945, Average Permeabilities of Heterogeneous Oil Sands: Trans. AIME, v. 160, p. 34–42.
Craft, B. C., and Hawkins, M. F., 1959,Applied Petroleum Reservoir Engineering: Prentice-Hall, Englewood Cliffs, NJ, 437 p.
Desbarats, A. J., 1987, Numerical Estimation of Effective Permeability in Sand-Shale Formations: Water Res. Res., v. 23, n. 2, p. 273–286.
Desbarats, A. J., 1992a, Spatial Averaging of Transmissivity in Heterogeneous Fields with Flow Towards a Well: Water Res. Res., v. 28, n. 3, p. 757–767.
Desbarats, A. J., 1992b, Spatial Averaging of Hydraulic Conductivity in Three-Dimensional Heterogeneous Porous Media: Math. Geol., v. 24, n. 3, p. 249–267.
Deutsch, C., 1989, Calculating Effective Absolute Permeability in Sandstone/Shale Sequences, SPE Form. Eval., Sept., p. 343–348.
Dimitrakopoulos, R., and Desbarats, A. J., 1993, Geostatistical Modeling of Gridblock Permeabilities for 3D Reservoir Simulators: SPE Res. Eng., v. 8, n. 1, p. 13–18.
Haldorsen, H. H., 1986, Simulator Parameter Assignment and the Problem of Scale in Reservoir Engineering,in L. W. Lake and H. B. Carroll (Eds.),Reservoir Characterization: Academic Press, Orlando, p. 293–340.
Journel, A. G., and Huijbregts, C., 1978,Mining Geostatistics: Academic Press, London, 600 p.
Kumar, A., and Ramey, H. J., 1974, Well Test Analysis for a Well in a Constant-Pressure Square, Soc. Pet. Eng. J., v. 14, n. 2, p. 107–116.
Luster, G. R., 1985, Raw Materials for Portland Cement: Applications of Conditional Simulation of Coregionalization, Ph.D. dissertation, Dept. of Appl. Earth Sci., Stanford Univ., Stanford, CA, 531 p.
Matheron, G., 1967, Composition des Perméabilités en Milieu Poreux Hétérogène: Méthode de Schwydler et Règles de Pondération, Rev. Inst. Fr. Pet., v. 22, n. 3, p. 443–466.
Naff, R. I., 1991, Radial Flow in Heterogeneous Porous Media: An Analysis of Specific Discharge: Water Res. Res., v. 27, n. 3, p. 307–316.
Neuman, S. P., and Orr, S., 1993, Prediction of Steady State Flow in Nonuniform Geologic Media by Conditional Moments: Exact Nonlocal Formalism, Effective Conductivities, and Weak Approximation: Water Res. Res., v. 29, n. 2, p. 341–364.
Oliver, D. S., 1990, The Averaging Process in Permeability Estimation from Well-Test Data: SPE Form Eva., v. 5, n. 3, p. 319–324.
Oliver, D. S., 1993, The Influence of Nonuniform Transmissivity and Storativity on Drawdown: Water Res. Res., v. 29, n. 1, p. 169–178.
Peaceman, D. W., 1978, Interpretation of Well-Block Pressures in Numerical Reservoir Simulation: Soc. Pet. Eng. J., v. 18, n. 3, p. 183–194, Trans. AIME 265.
Peaceman, D. W., 1983, Interpretation of Well-Block Pressures in Numerical Reservoir Simulation with Nonsquare Grid Blocks and Anisotropic Permeability: Soc. Pet. Eng. J., v. 23, n. 3, p. 531–543.
Reinson, G. E., Clark, J. E., and Foscolos, A. E., 1988, Reservoir Geology of Crystal Viking Field, Lower Cretaceous Estuarine Tidal Channel-Bay Complex, South-Central Alberta: AAPG Bull., v. 72, n. 10, p. 1270–1294.
Vandenberg, A., 1977, Pump Testing in Heterogeneous Aquifers: J. Hydrol., v. 34, n. (1/2), p. 45–62.
Warren, J. E., and Price, H. S., 1961, Flow in Heterogeneous Media: Soc. Pet. Eng. J., v. 1, p. 153–169.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Desbarats, A.J. Spatial averaging of hydraulic conductivity under radial flow conditions. Math Geol 26, 1–21 (1994). https://doi.org/10.1007/BF02065873
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02065873