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

Abstract

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.

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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

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