Understanding of Relationship between the Average Mass Transport Rate and the Moments of Permeability
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predict the shape of the breakthrough curve from the three moments of the borehole data. INTRODUCTION
Generally, borehole data of permeability are well known as one of the most significant hydraulic properties about geologic media including innumerable cracks or fast flow paths. However, we cannot always obtain sufficient measured data to understand the spatial variability of permeability for the whole or concerned regions of the disposal site including the accessible environment (e.g., [1-4]). In such a situation, we need general understandings for the mass transport rates and the permeability distribution. The available data of permeability at least give us the moments for the distribution. In a previous paper [5], the authors investigated the relation between the moments and the average mass transport rates, applying the conventional advection-dispersion model to a two-dimensional region and assuming Bernoulli trials or a lognormal distribution of the permeability. The calculated results showed good agreement of the mass transport rates for Peclet number=1O, when the permeability distributions had the same values of the three respective moments; the arithmetic mean, the standard deviation and the skewness. The purpose of this work is to get a more general understanding of the relationship between the moments and the average mass transport rates, using probability density functions other than the ones previous examined. The approach is to compare the average mass transport rates through the medium having various types of permeability distribution with the same arithmetic mean, standard deviation and skewness. The process conducted is to examine the relationship between the three 751
Mat. Res. Soc. Symp. Proc. Vol. 556 © 1999 Materials Research Society
moments and the specific parameters which describe each probability density function. As a result, upper and lower bounds in the skewness are derived as a function of the arithmetic mean and the standard deviation. Using these relations, we discuss the dependencies of the mass transport rates on the spatial variability of permeability. PERMEABILITY DISTRIBUTION AND THE MOMENTS We assume a medium with a spatial distribution of permeability, k (M 2), having an arithmetic mean, kA (m 2). If we divide each observation, k, by kA, we get dimensionless permeability values, K, from which we can obtain the dimensionless arithmetic mean, KA(=1), the dimensionless standard deviation, cT, and the skewness, SK. These are generally defined by 0r2 SK
F(K)(K- KA)2dK
f
,
f~o F(K)(K - KA) 3dK
(1) (2)
where F(K) means a probability density function describing the measured permeability data. However, since we can not specify the type of F(K), this study investigates the following discrete or continuous probability density functions (PDFs); the Bernoulli trials, the truncated normal distribution, the beta distribution, the triangle distribution and the log-normal distribution. These functions are frequently used as uncertainty distributions. The specific parameters describing
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