Understanding of Relationship between the Average Mass Transport Rate and the Moments of Permeability


To estimate the transport rate of radionuclides in the geosphere, we must consider the spatial variability of permeability. However, the borehole data of permeability are limited and we can not determine the type of probability density function, though the measurement data reflect the most significant hydraulic properties about geologic media including innumerable cracks or fast flow paths. While the recent models describing radioactive nuclide transport in near/far-field have assumed a certain probability density function (typically a lognormal distribution) as a permeability distribution, we cannot always obtain sufficient measurement data to define the function. However, the available data of permeability at least give us the moments such as the arithmetic mean, the standard deviation and the skewness for the distribution.

The purpose of this paper is to get an understanding of the general relationship between the average mass transport rates and the moments. Using various types of probability density functions and pseudo random-numbers, hypothetical permeability distributions are generated. With these distributions, this paper obtains the average transport rates described as the numerical impulseresponse based on the advection-dispersion model for a two-dimensional region. The calculated results show that, for the dimensionless standard deviation up to around 1, the three moments are enough to characterize the permeability distribution for the purposes of the nuclide transport prediction.

In this work, for five specified probability density functions, the upper and lower bounds of skewness are derived as a function of the dimensionless arithmetic mean and standard deviation. The obtained upper and lower bounds explicitly show that the Bernoulli trials (a discrete probability density function) yield the widest range in the skewness against the standard deviation. Since the response has lower peak and longer tail as the skewness goes to the lower bound value, we can predict the shape of the breakthrough curve from the three moments of the borehole data.

This is a preview of subscription content, access via your institution.


  1. 1.

    N.A. Chapman and I.G. McKinley, The Geological Disposal of Nuclear Waste, (JOHN WILEY & SONS, Chichester, 1987, pp.108–132.

    Google Scholar 

  2. 2.

    P. Grindrod, T. McEwen, P. Robinson and D. Savege, Chapter 5 The far-field, in The Scientific and Regulatory Basis for the Geological Disposal of Radioactive Waste, edited by D. Savege, JOHN WILEY & SONS, Chichester, 1995, pp. 119–183.

    Google Scholar 

  3. 3.

    K.J. Clark, T. Ikeda, M.D. Impey, T. McEwen and M. White: An Analysis of Bias in Groundwater Modelling due to the Interpretation of Site Characterization Data, in Scientific Basis for Nuclear Waste Management XX, edited by W.J. Gray and I.R. Triay ( Mater.Res. Soc. Proc., 465, Pittsburgh, PA 1997), pp. 1125–1132.

    CAS  Google Scholar 

  4. 4.

    X.H. Wen and J.J. Gómez-Hernádez: Numerical modeling of macrodispersion in heterogeneous media: a compariosn of multi-Gaussian and non-multi-Gaussian models, J. Contaminant Hydrology, 30, 129–156 (1998).

    CAS  Article  Google Scholar 

  5. 5.

    Y. Niibori, O. Tochiyama and T. Chida: Estimation of the radionuclide Transport by Applying the mean, the standard deviation and the skewness of permeability, in Scientific Basis for Nuclear Waste Management XX, edited by W.J. Gray and I.R. Triay (Mater.Res. Soc. Proc., 465, Pittsburgh, PA 1997), pp.917–924.

    CAS  Google Scholar 

  6. 6.

    Y.W. Tsang, C.F. Tsang, I. Neretnieks and L. Moreno: Flow and Tracer Transport in Fractured Media: A Variable Aperture Channel Model and Its Properties, Water Resour. Res., 24, 2049–2060 (1988).

    Article  Google Scholar 

  7. 7.

    L. Moreno, Y.W. Tsang, C.F. Tsang, F.V. Hale and I. Neretnieks: Flow and Tracer Transport in a Single Fracture: A Stochastic Model and Its Relation to Some Field Observations, Water Resour. Res., 24, 2033–2048 (1988).

    CAS  Article  Google Scholar 

  8. 8.

    L. Smith and F.W. Schwartz: Mass Transport, 1. A Stochastic Analysis of Macroscopic Dispersion, Water Resour. Res., 16, 303–313 (1980).

    Article  Google Scholar 

  9. 9.

    C.F. Tang, Y.W. Tsang and F. Hale: Tracer Transport in Fractures: Analysis of Field Data Based on a Variable-Aperture Channel Model, Water Resour. Res., 27, 3095–3106 (1991)

    Article  Google Scholar 

  10. 10.

    L. Moreno, C.F. Tsang, Y. Tsang and I. Neretnieks: Some anomalous features of flow and solute transport arising from fracture aperture variability, Water Resour. Res., 26, 2377–2391 (1990)

    CAS  Article  Google Scholar 

  11. 11.

    Y. Niibori and T. Chida: The Use of Standard Deviation and Skewness for Estimating Apparent Permeability in Two-Dimensional, Heterogeneous Medium, Transport in Porous Media, 15, 1–14 (1994)

    Article  Google Scholar 

  12. 12.

    Y. Niibori, H. Ogura and T. Chida: Identification of Geothermal Reservoir Structure Analyzing Tracer Responses Using The Two-Fractured-Layer Model, Geothermics, 24, 49–60 (1995).

    Article  Google Scholar 

  13. 13.

    O. Levenspoel, Chemical Reaction Engineering, 2nd Edn., JOHN WILEY & SONS, New York, 1972, pp.282–283.

    Google Scholar 

  14. 14.

    Y. Niibori, H.C. Xie and T. Chida: Tracer response in a vertical, two-dimensional, porous medium accompanied by heat transfer–Comparison of the laboratory experiments with the numerical model, J. Groundwater Hydrology, 36, 123–132 (1994) (Japanese with English abstracts).

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Y. Niibori.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Niibori, Y., Tochiyama, O. & Chida, T. Understanding of Relationship between the Average Mass Transport Rate and the Moments of Permeability. MRS Online Proceedings Library 556, 751 (1998). https://doi.org/10.1557/PROC-556-751

Download citation