Abstract
It is becoming widely recognized that natural earth materials are rather heterogeneous in terms of their hydrologic properties. Figure 1 shows observations of permeability and porosity in a sandstone aquifer in Illinois. These data are based on laboratory analysis of cores collected from deep boreholes in the Mt. Simon aquifer in central Illinois. The variation of permeability is seen to be rather large; the values range over four orders of magnitude. Porosity on the other hand shows a somewhat smaller degree of variability but still a significant percentagewise change. Figure 2 is the second example of the variability of permeability in earth materials. These data show observed infiltration rates of surface soils along a transect some 700 m long in an irrigated area near Socorro, New Mexico. Here the saturated hydraulic conductivity is seen to vary over two orders of magnitude. The observations in Figure 1 and 2 also illustrate that the variation of hydraulic conductivity or permeability is not completely disordered in space. There is some spatial structure which should be taken into account when considering the effects of such heterogeneity. Many other observations from different geologic environments show similar kinds of natural variability of permeability and other properties.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bakr, A.A.M. Stochastic Analysis of the Effects of Spatial Variations of Hydraulic Conductivity on Groundwater Flow. Ph.D. Dissertation, New Mexico Institute of Mining and Technology, Socorro, New Mexico (1976).
Bakr, A.A., L.W. Gelhar, A.L. Gutjahr, and J.R. MacMillan. Stochastic Analysis of Spatial Variability in Subsurface Flows 1. Comparison of One- and Three-dimensional Flows. Water Resour. Res., 14 (1978) 263–271.
Bear, J. Dynamics of Fluids in Porous Media (American Elsevier Publishing Company, Inc., New York, 1972 ).
Binsariti, A.A. Statistical Analyses and Stochastic Modelling of the Contaro Aquifer in Southern Arizona. Ph.D. Dissertation, Dept. of Hydrology and Water Resour., Univ. of Arizona, Tucson (1980).
Brieman, L. Probability and Stochastic Processes: With a View-point toward Applications (Houghton-Miflin, 1969 ).
Chirlin, G.R. and G. Dagan. Theoretical Head Variograms for Steady Flow in Statistically Homogeneous Aquifers. Water Resour. Res. (1980) 1001–1015.
Dagan, G. Models of Groundwater Flow in Statistically Homogeneous Porous Formations. Water Resour. Res. 15 (1979) 47–63.
Dagan, G. Stochastic Modeling of Groundwater Flow by Unconditional and Conditional Probabilities, Part 1: Conditional Simulation and the Direct Problem. Water Resour. Res. 18 (1982a) 813–833.
Dagan, G. Stochastic Modeling of Groundwater Flow by Uncon-ditional and Conditional Probabilities. Part 2: the Solute Transport. Water Resour. Res. 18 (1982b) 835–848.
Dagan, G. Analysis of Flow through Heterogeneous Random Aquifers 2: Unsteady Flow in Confined Formations, Water Resour. Res. 18 (1982c) 1571–1585.
Delhomme, J.P. Kriging in the Hydrosciences. Advances in Water Resour. 1 (1978) 251–266.
Delhomme, J.P. Spatial Variability and Uncertainty in Ground-water Flow Parameters: A Geostatistical Approach. Water Resour. Res. 15 (1979) 269–280.
Dettinger, M.D. and J.L. Wilson. First Order Analysis of Un-certainty in Numerical Models of Groundwater Flow, 1, Mathematical Development, Water Resour. Res. 17 (1981) 149–161.
Freeze, R.A. A Stochastic-Conceptual Analysis of One-Dimen- sional Groundwater Flow in Nonuniform Homogeneous Media. Water Resour. Res. 11 (1975) 725–741.
Gaillard, G., A. Lallemand-jBarres, J. Molinari, and P. Peaudecerf. Etude Méthodologique des Caractéristiques de Transfert de Substances Chimiques dans les Nappes. Centre National de la Recherche Scientifique. SARR/GARTHI/76-07/JM-BG Orleans, France, 43 pp. (1976).
Gelhar, L.W. Stochastic Analysis of Phreatic Aquifers. Water Resour. Res. 10 (1974) 539–545.
Gelhar, L.W. Effects of Hydraulic Conductivity Variations on Groundwater Flows. Proceedings Second International Symposium on Stochastic Hydraulics, Lund, Sweden, in Hydraulic Problems Solved by Stochastic Methods, Water Resources Publications, Fort Collins, CO (1977) 409–431.
Gelhar, L.W., A.A. Bakr, A.L. Gutjahr and J.R. MacMillan. Comments on a Stochastic-Conceptual Analysis of One-Dimension- al Groundwater Flow in Nonunifom Homogeneous Media by R.A. Freeze. Water Resour. Res. 13 (1977) 477–479.
Gelhar, L.W., C.J. Duffy, and G.W. Gross. Stochastic Methods of Analyzing Groundwater Recharge. Proceeding IAHR Symposium on Hydrology of Areas of Low Precipitation, Canberra, IAHS-AISH Pub. No. 128 (1979) 313–321.
Gelhar, L.W., A.L. Gutjahr and R.L. Naff. Stochastic Analysis of Macrodispersion in a Stratified Aquifer. Water Resour. Res. 15 (1979) 1387–1397.
Gelhar, L.W. and C.L. Axness. Stochastic Analysis of Macro- dispersion in Three-Dimensionally Heterogeneous Aquifers, New Mexico Institute of Mining and Technology, Hydrology Report H-8 (1981).
Gelhar, L.W. and C.L. Axness. Three-Dimensional Stochastic Analysis of Macrodispersion in Aquifers, Water Resour, Res. 19 (1983) 161–180;
Gelhar, L.W., A.L. Gutjahr and R.L. Naff. Reply to Comments on Stochastic Analysis of Macrodispersion in a Stratified Aquifer, Water Resour. Res. 17 (1981) 1739–1740.
Gutjahr, A.L., L.W. Gelhar, A.A. Bakr and J.R. MacMillan. Stochastic Analysis of Spatial Variability in Subsurface Flow. Part II: Evaluation and Applications. Water Resour. Res. 14 (1978) 953–959.
Gutjahr, A.L. and L.W. Gelhar. Stochastic Models of Subsurface Flow: Infinite versus Finite Domains and Stationarity. Water Resour. Res. 17 (1981) 337–350.
Heller, J.P. Observations of Mixing and Diffusion in Porous Media. Proceedings of the Second Symposium on Fundamentals of Transport Phenomena in Porous Media, IAHR-ISSS, Vol. 1, Guelph, Canada (1972) 1–26.
Jenkins, G.M. and D.G. Watts, Spectral Analysis and Its Applications. ( Holden-Day, San Francisco, 1968 ).
Koopman, L.H. The Spectral Analysis of Time Series. (Academic Press, 1974 ).
Lumley, J.L. and H.A. Panofsky. The Structure of Atmospheric Turbulence ( John Wiley, New York, 1967 ).
Matheron, G. Composition des Permeabilities en Milieu Poreux Hétérogène Methode de Schwydler et Régies de Pondération. Rev. Inst. Fr. Petrole (1967) 443–466.
Matheron, G., and G. DeMarsily. Is Transport in Porous Media Always Diffusive? A Counter Example. Water Resour. Res. 16 (1980) 901–917.
Mizell, S.A., L.W. Gelhar and A.L. Gutjahr. Stochastic Analysis of Variability in Two-Dimensional Groundwater Flow with Implications for Observation-Well-Network Design. Hydrology Research Report No. H-6, New Mexico Institute of Mining and Technology (1980).
Mizell, S.A., A.L. Gutjahr, and L.W. Gelhar. Stochastic Analysis of Spatial Variability in Two-Dimensional Steady Groundwater Flow Assuming Stationary and Nonstationary Heads. Water Resour. Res. 18 (1982) 1053–1067.
Neuman, S.P. A Statistical Approach to the Inverse Problem-of Aquifer Hydrology 3. Improved Solution Method and Added Perspective. Water Resour. Res. 16 (1980) 331–346.
Perlmutter, N.M., and M. Lieber. Dispersal of Plating Wastes and Sewage Contaminants in the Ground Water and Surface Water, South Farmingdale-Massapequid Area, Nassau County, New York. U.S. Geological Survey Water-Supply Paper 1879-G (1970).
Robson, S.G. Application of Digital Profile Modeling Technique to Ground-Water Solute Transport at Barstow, Calif. U.S. Geological Survey Water Supply Paper 2050 (1978).
Sagar, B. Galerkin Finite Element Procedure for Analyzing Flow through Random Media. Water Resour. Res. 14 (1978) 1035–1044.
Smith, L. and R.A. Freeze. Stochastic Analysis of Steady State Groundwater Flow in a Bounded Domain, 1, One-Dimensional Simulations, Water Resour. Res. 15 (1979a) 521–528.
Smith, L. and R.A. Freeze. Stochastic Analysis of a Steady State Groundwater Flow in a Bounded Domain 2. Two-Dimensional Simulations. Water Resour. Res. 15 (1979b) 1543–1559.
Smith, L., and F.W. Schwartz. Mass Transport 1. A Stochastic Analysis of Macroscopic Dispersion. Water Resour. Res. 16 (1980) 303–313.
Smith, L. Spatial Variability of Flow Parameters in a Stratified Sand, Mathematical Geology 13 (1981) 1–21.
Sudicky, E.A., J.A. Cherry, and E.O. Frind. Hydrogeological Studies of a Sandy Aquifer at an Abandoned Landfill: 4, A Natural Gradient Dispersion Test. Journal of Hydrology 63 (1983) 81–108,
Vieira, S.R., D.R. Nielsen, and J.W. Biggar. Spatial Vari-ability of Field-Measured Infiltration Rate. Soil Sci. Soc. Am. J. 45 (1982) 1040–1048.
Warren, J.E., and F.F. Skiba. Macroscopic Dispersion. Trans. AIME, 231 (1964) 215–230.
Warrick, A.W., and D.R. Nielsen. Spatial Variability of Soil Physical Properties in the Field. in Hillel, D. Applications in Soil Physics. Academic Press (1980) 319–344.
Yeh, T.-C. Stochastic Analysis of Effects of Spatial Vari-ability on Unsaturated Flow. Ph.D. Dissertation, New Mexico Institute of Mining and Technology (1982).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 Martinus Nijhoff Publishers, Dordrecht
About this chapter
Cite this chapter
Gelhar, L.W. (1984). Stochastic Analysis of Flow in Heterogeneous Porous Media. In: Bear, J., Corapcioglu, M.Y. (eds) Fundamentals of Transport Phenomena in Porous Media. NATO ASI Series, vol 82. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-6175-3_14
Download citation
DOI: https://doi.org/10.1007/978-94-009-6175-3_14
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-6177-7
Online ISBN: 978-94-009-6175-3
eBook Packages: Springer Book Archive