Abstract
To physically investigate permeability upscaling, over 13,000 permeability values were measured with four different sample supports (i.e., sample volumes) on a block of Berea Sandstone. At each sample support, spatially exhaustive permeability datasets were measured, subject to consistent flow geometry and boundary conditions, with a specially adapted minipermeameter test system. Here, we present and analyze a subset of the data consisting of 2304 permeability values collected from a single block face oriented normal to stratification. Results reveal a number of distinct and consistent trends (i.e., upscaling) relating changes in key summary statistics to an increasing sample support. Examples include the sample mean and semivariogram range that increase with increasing sample support and the sample variance that decreases. To help interpret the measured mean upscaling, we compared it to theoretical models that are only available for somewhat different flow geometries. The comparison suggests that the nonuniform flow imposed by the minipermeameter coupled with permeability anisotropy at the scale of the local support (i.e., smallest sample support for which data is available) are the primary controls on the measured upscaling. This work demonstrates, experimentally, that it is not always appropriate to treat the local-support permeability as an intrinsic feature of the porous medium, that is, independent of its conditions of measurement.
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REFERENCES
Ababou, R., and Wood, E. F., 1990, Comment on “Effective groundwater model parameter values: Influence of spatial variability of hydraulic conductivity, leakance, and recharge by J. J. Gomez-Hernandez and S. M. Gorelick”: Water Resources Res., v. 26, no. 8, p. 1843–1846.
Bear, J., and Bachmat, Y., 1990, Introduction to modeling of transport phenomena in porous media: Kluwer Academic Publishers, Boston, 553 p.
Bracewell, R. N., 1986, The Fourier transform and its applications: McGraw-Hill, New York, 474 p.
Clark, I., 1977, Regularization of a semi-variogram: Computers & Geosciences, v. 3, p. 341–346.
Cushman, J. H., 1984, On unifying the concepts of scale, instrumentation, and stochastics in the development of multiphase transport theory: Water Resources Res., v. 20, no. 11, p. 1668–1676.
Dagan, G., 1981, Analysis of flow through heterogeneous random aquifers by the method of embedding matrix 1. Steady flow: Water Resources Res., v. 17, no. 1, p. 107–121.
Dagan, G., 1989, Flow and transport in porous formations: Springer-Verlag, New York, 465 p.
Desbarats, A. J., 1992a, Spatial averaging of hydraulic conductivity in three-dimensional heterogeneous porous media: Math. Geology, v. 24, no. 3, p. 249–267.
Desbarats, A. J., 1992b, Spatial averaging of transmissivity in heterogeneous fields with flow toward a well: Water Resources Res., v. 28, no. 3, p. 757–767.
Deutsch, C. V., 1989, Calculating effective absolute permeability in sandstone/shale sequences: SPE Formation Evaluation, v. 4, no. 3, p. 343–348.
Deutsch, C. V., and Journel, A. G., 1997, GSLIB: Geostatistical Software Library and User's Guide: Oxford University Press, New York, 368 p.
Durlofsky, L. J., 1991, Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media: Water Resources Res., v. 27, no. 5, p. 699–708.
Gelhar, L. W., and Axness, C. L., 1983, Three-dimensional stochastic analysis of macrodispersion in aquifers: Water Resources Res., v. 19, no. 1, p. 161–180.
Goggin, D. J., Thrasher, R. L., and Lake, L. W., 1988, A theoretical and experimental analysis of minipermeameter response including gas slippage and high velocity flow effects: In Situ, v. 12, no. 1–2, p. 79–116.
Gutjahr, A. L., Gelhar, L. W., Bakr, A. A., and Macmillan, J. R., 1978. Stochastic analysis of spatial variability in subsurface flows, 2. Evaluation and application: Water Resources Res., v. 14, no. 5, p. 953–960.
Indelman, P., and Abramovich, B., 1994, Nonlocal properties of nonuniform averaged flows in heterogeneous media: Water Resources Res., v. 30, no. 12, p. 3385–3393.
Journel, A. G., and Huijbregts, C. J., 1978, Mining geostatistics: Academic Press, New York, 600 p.
Kitanidis, P. K., 1990, Effective hydraulic conductivity for gradually varying flow: Water Resources Res., v. 26, no. 6, p. 1197–1208.
Kossack, C. A., Aasen, J. O., and Opdal, S. T., 1990, Scaling up heterogeneities with pseudofunctions: SPE Formation Evaluation, v. 5, no. 3, p. 226–232.
Marle, C.-M., 1967, Ecoulements monophasiques en milieu poreux: Rev. Inst. Fr. Pet., v. 22, p. 1471–1509.
Matheron, G., 1967, Elements pour une theorie des milieux poreux: Maisson et Cie, Paris, 166 p.
Neuman, S. P., 1994, Generalized scaling of permeabilities: Validation and effect of support scale: Geophys. Res. Lett., v. 21, no. 5, p. 349–353.
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 Resources Res., v. 29, no. 2, p. 341–364.
Pepper, J. F., De Witt, W., Jr., and Demarest, D. F., 1954, Geology of the Bedford Shale and Berea Sandstone in the Appalachian Basin: U.S. Geol. Survey Prof. Paper 259, U.S. Geological Survey.
Rubin, Y., and Gomez-Hernandez, J. J., 1990, A stochastic approach to the problem of upscaling of conductivity in disordered media: Theory and unconditional numerical simulations: Water Resources Res., v. 26, no. 4, p. 691–701.
Sanchez-Vila, X., 1997, Radially convergent flow in heterogeneous porous media: Water Resources Res., v. 33, no. 7, p. 1633–1641.
Tidwell, V. C., and Wilson, J. L., 1997, Laboratory method for investigating permeability upscaling: Water Resources Res., v. 33, no. 7, p. 1607–1616.
Tidwell, V. C., and Wilson, J. L., 1999a, Upscaling experiments conducted on a block of volcanic tuff: Results for a bimodal permeability distribution: Water Resources Res., in press.
Tidwell, V. C., and Wilson, J. L., 1999b, Heterogeneity, permeability patterns, and permeability upscaling: Physical characterization of a block of Massillon Sandstone exhibiting nested scales of heterogeneity: SPE Reservoir Evaluation and Engineering, submitted.
Tidwell, V. C., Gutjahr, A. L., and Wilson, J. L., 1999, What does an instrument measure? Empirical spatial weighting functions calculated from permeability data sets measured on multiple sample supports: Water Resources Res., v. 35, no. 1, p. 43–54.
White, C. D., and Horne, R. N., 1987, Computing absolute transmissivity in the presence of fine-scale heterogeneity: Presented at the SPE Symposium on Reservoir Simulation, San Antonio, Texas, paper SPE 16011.
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Tidwell, V.C., Wilson, J.L. Permeability Upscaling Measured on a Block of Berea Sandstone: Results and Interpretation. Mathematical Geology 31, 749–769 (1999). https://doi.org/10.1023/A:1007568632217
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DOI: https://doi.org/10.1023/A:1007568632217