REGRESSION ANALYSIS TO DETERMINE THE PERMEABILITY TENSOR A. Introduction

In the previous chapter directional permeability measurements were simply plotted and the resemblance of these plots to ellipses was noted in a qualitative way. Appropriate quantification of the best-fit

ellipse and a measure of the goodness of fit of the data to the best-fit ellipse would be very useful for comparing the behavior of different fracture systems. This chapter gives a regression technique for calcu-lating the components of the best-fit permeability tensor and the error

.

associated with using the best-fit tensor as opposed to the actual meas-ured values of directional permeability.

Directional permeability measurements of random fracture systems are made as previously described. From the measured values of direc-tional permeability, Kg(a), we wish to determine the three components of the permeability tensor, Kij' which best fit these measurements. We also wish to determine the principal values (eigenvalues) and principal axes (eigenvectors) of the permeability tensor. Further, we wish to find a quantitative measure of the difference between the measured val-ues, Kg(a), and the best-fit values.

Recall that for an ideal anisotropic homogeneous porous medium directional permeability, Kg, measured in the direction of the gradient a, is defined by the following equation:

q.n.

=

^K J,

1 1 g (V-1)

where ni is a unit vector in the direction of the gradient, J is the magnitude of the gradient, and qi is the specific flux. Solving Darcy's

law for qi and substituting this into V-1 gives K .. J.n.

=

^{K J,}

lJ J 1 g and since Jj/J

=

nj we have

Kg

=

Kijnin j , or

where n1 and nZ are direction cosines and n1

=

cos a, nZ

=

sin a.

(V-Z)

(V-3)

(V-4)

If 1/1Kg is plotted in the direction a (the direction of the gradi-ent), then n1

=

^{cos a}

=

xlKg and nZ

=

^{sin a}

=

^ylKg. Equation V-4

becomes

1=K .. x.x.

lJ 1 J where xi

= {; } •

( V-5)

(V-6)

(V-7) Equation V-7 is the quadratic form of the permeability ellipse equation.

If each measurement of Kg(a) can be considered an independent meas-urement of the value of Kij' then methods of statistics can be used to estimate the parameters K11, K1Z and KZZ. The statistical technique can be used on measurements of Kg(a) from different, equally incremented directions on one fracture pattern or the combined measurements from any number of fracture pattern realizations.

B. Distribution of Kg(a)

In a random fracture pattern, the measured values of Kg(a) will not all plot on a single, unique ellipse (Figure V-1). In order to use all of the individual measurements to derive a single, most representative set of parameters for the permeability tensor, we must assume that each measurement is independent and similarly distributed. Figure V-1 shows an example of a set of measurements, Kg(a), and a possible ellipse with parameters K11, K12 and K22. Each measurement can be assumed to be distributed about a different mean which is a point on the ellipse determined by a. Therefore, the value of the mean for each measurement depends on a. Thus, each Kg(a) is considered to be distributed with the same form but each has a different or shifted mean. The variance of each Kg(a) is assumed to be identical. In this way, all the measure-ments are considered as one population.

It would be very useful to be able to define a likely distribution function for Kg(a), but this is not easily done. The normal distribu-tion does not match the data because Kg(a) can never be less than zero.

A lognormal is not proper because the probability of Kg(a)

=

0 is finite, not zero. Exponential, Gamma and Beta distributions also are not suit-able. A normal distribution truncated at Kg(a)

=

0 is a likely choice.

Unfortunately, assumption of this distribution leads to a contradiction with the basic assumption that all the measurements are members of the same distribution. At each angle a, the mean value of the distribution is different. However, since all the distributions are truncated at zero, the difference between the mean value and the truncation limit is different for each value of a. This means that each measurement must be

l'

1

~---~~--~---~+-~x

x---1 = K'jXjXj . /

Figure V-1.

I I

X

I ^X

XBL 817-3316 Example of a set of directional

permea-bility measurements plotted as 1/~ in polar-coordinates.

a member of a different, truncated normal distribution and not just a shifted one as required in the original assumptions. Since a simple, likely, distribution form for Kg(a) which conforms to the basic assump-tions cannot be identified, a least squares regression technique must be used to derive estimates of the parameters K11, K12 and K22.

c.

Regression Techniques

Two regression techniques are discussed here. They are based on a technique, discussed by Scheidegger (1954), which will also be briefly described. The most direct approach to finding the best-fit ellipse is to find the ellipse which minimizes the sum RI, where

2

/ K. _{IJ 1 1}

~n.n.

^{) , (V-B)}

where N is the number of measurements made, either on one fracture

pattern or all measurements on all realizations of the fracture pattern.

The difference between 1/(/Kg(an») and 1/(/Kijninj) ⁱⁿ each case is the actual distance on the permeability ellipse plot between the plotted measurement and the ellipse (Figure V-1). Note that this distance is not the perpendicular distance, but rather the distance along the ray inclined at the angle aO. To minimize R we solve for K11, K12 and K22 in the following equations:

N aaRKI

= '\

2 [(K

g(a

n))-1/2- (K .. n.n .)-1/2](K .. n.n .)-3/2cos 2a

=

0, (V-10)

11

L

^{IJ 1 J IJ 1 J} n

n=1

-1/2J( )-3/2

(K .. n.n.) K .. n.n. cosa sina

IJ 1 J IJ 1 J n n ^{= 0,}

(V-11)

N aaRKI =

~

²

r

(K

g(a

n))-1/2_ (K .. n.n.)-1/2J(K .. n.n.)-3/2sin 2a = O. (V-12)

22 ~

L

IJ 1 J IJ 1 J n

n=1

These nonlinear equations are difficult to solve. An iterative method would have to be employed.

The second regression technique is to minimize the function RII:

= ~~

_~ ^{[K (a ) - (K .. n. n . )] 2.}

g n IJ 1 1 ^(V-13)

n=1

In this case we are not directly regressing to the best-fit ellipse. We are finding the parameters K11, K12, and K22 which best fit the data ex-pressed by Kg(on) in equation V-5. Figure V-2 illustrates the type of three parameter curve which is fitted to the data. This curve is less easily visualized than the ellipse is but it is much simpler mathe~

matically. There is no reason to expect the two techniques to give the same answer, but there is also no obvious physical reason to expect one technique to give a better answer than the other. As such, the second technique is pursued here.

A similar technique was used by Scheiddeger (1954). Scheiddeger minimized the function

(V-14) Although not stated by Scheiddeger, this regression techniques applies

K

Measured values of Kg(a)

x/ /3

parameter curve: K=

~jninj

I

x-i :

^{x x x 1}

t

Kg(a)-Kijnini

x

x

o

30 60 90 120 150 180 210 240 270 300330360

aO

Figure V-2.

XBL817-3317

Example of a set of directional permeability measurements plotted in Cartesian coordinates.

to measurements of permeability Kf(a) made in the direction of flow.

Thus, mi is a unit vector in the direction of flow. To see this recall that permeability in the direction of flow is defined by

1 J.m.

1 1

r=--

_{f q}

where q is the specific flux and Ji is the gradient.

Substituting Darcy's law we have

1 1 m.q.

( )

-

~

Kf

=

^{qi Kij q}

-K¹

=

(K .. )-1m.m .•

f lJ J 1

(V-15)

(V-16)

( V-17)

So equation V-14 is effectively the same as equation V-13, except that in V-14, Kij becomes the inverse of the permeability tensor.

D. Solution of the Regression Equations

The solution of the regression equations is exactly the same as the solution given by Scheidegger (1954) and is repeated here only for com-pleteness. The equations are

aRU ^N

Now, if for each fracture mesh, measurements are made at equal angle

N

E. Principal-Permeabilities and Directions

Knowing the values of K11, K12, and K22 the values and directions

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