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The use of standard deviation and skewness for estimating apparent permeability in a two-dimensional, heterogeneous medium

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Abstract

In this theoretical study, characteristic or effective permeabilities (referred to as ‘apparent permeability’) of a radial/parallel flow system in a heterogeneous medium are calculated by the Monte Carlo method and the finite element method. The permeability distribution in the radial and parallel flow systems are not the same, as going from Cartesian to cylindrical coordinates changes the probability measure. The Bernoulli trials, the normal distribution or the log-normal distribution, is assumed to be the probability density function of permeability. The results are summarized as follows: (1) when the skewness of the distribution function is equal to zero or nearly equal to zero (that is, when the permeability distribution is regarded to be symmetric), the apparent permeability depends on the standard deviation, but not on the kind of distribution function, (2) when the skewness is not equal to zero, the apparent permeability depends not only on the standard deviation, but also on the skewness, (3) the above facts appeared in the radial and the parallel flow systems.

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Abbreviations

K :

dimensionless permeability,K=k/k A

K ap :

dimensionless apparent permeability,K ap=k ap/k A

¯K ap :

average value of apparent permeabilities calculated by the Monte Carlo method (defined by Equation (18))

k :

permeability [m2]

m :

sub-section number inz-direction

N :

frequency in the Bernoulli trial

n :

sub-section number inx-direction

n r :

sub-section number inr-direction

n t :

trial number of computation of apparent permeability

n θ :

sub-section number inθ-direction

P :

dimensionless pressure,P=(p−p out )/(P in−p out)

p :

pressure [Pa]

Q :

volume flow rate per unit length [m2/s]

q :

flow flux [m/s]

R :

dimensionless variable inr-direction,R=r/r in

r, θ :

directions in the cylindrical coordinate system

S k :

skewness (defined in Equation (19))

x, z :

directions in the rectangular coordinate system

X 1 :

dimensionless medium length ofx-direction,X 1=x 1/x *

x * :

characteristic length

Z 1 :

dimensionless medium length ofz-direction,Z 1=z 1/x *

Μ :

arithmetic mean of ln(K) in the log-normal distribution

Μ f :

viscosity of fluid [Pa s]

Μ 3 :

central moment of third-order

σ :

standard deviation

σ * :

standard deviation of ln(K) in the log-normal distribution

A:

arithmetic

ap:

apparent

in:

inlet

L:

larger value in the Bernoulli trials

out:

outlet

S:

smaller value in the Bernoulli trials

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Authors and Affiliations

  1. Department of Resources Engineering, Faculty of Engineering, Tohoku University, Aoba-ku, 980, Sendai, Japan

    Y. Niibori & T. Chida

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  1. Y. Niibori
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  2. T. Chida
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Niibori, Y., Chida, T. The use of standard deviation and skewness for estimating apparent permeability in a two-dimensional, heterogeneous medium. Transp Porous Med 15, 1–14 (1994). https://doi.org/10.1007/BF01046155

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  • DOI: https://doi.org/10.1007/BF01046155

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