Mathematical Geology, Vol. 29, No. 5, 1997
Curved Scanline Theory 1 Ken G r o s s e n b a c h e r , 2 Kenzi Karasaki, 2 a n d D o v B a h a t 3
This work develops the theory of measuring fracture frequency with curved scanlines, as a direct development of work done by others on straight scanlines. Various possible shapes for curved scanlines range between triangular and rectangular, with circular as a reasonable preliminary selection. The discrepancy among different selections decreases with increasing roughness amplitude of the scanline. Analytic solutions for average fracture frequency are given for circular scanlines through single and multiple fracture sets. Results for single fracture sets are plotted. The analytic solution for the general situation of any shape scanline through multiple fracture sets is given. Analytic solutions are given and plotted for circular scanlines through a fracture fabric ellipsoid. A circular scanline spanning 180 degrees yields a global fracture frequency of statistical significance. KEY WORDS: interval, scan line, fracture, statistics.
INTRODUCTION Networks of fractures occur in virtually all bodies of rock. Characterizing such networks has been of importance and, with the recent concern about waste migration in fractured rock, has become more essential. The challenge lies in how to best characterize such networks in a systematic manner so that results are comparable from location to location. Furthermore, results are sought which apply directly to such matters as rock strength, stability, and fracture permeability. This study focuses on characterizing networks on the basis of fracture intervals and its inverse, fracture frequency. Fracture intervals are the distances between fractures along any given line. This differs from fracture spacing which 1Received 27 November 1995; revised 4 November 1996. This work was carded out under U.S. Department of Energy Contract No. DEAC03-76SF00098 for the Director, Office of Civilian Radioactive Waste Management, Office of External Relations, and was administered by the Nevada Operations Office, U.S. Department of Energy. This work also was supported partially by the Power Reactor and Nuclear Fuel Development Corporation of Japan. 2Earth Sciences Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720. e-mail: kkarasaki @ lbl.gov 3Department of Geology and Mineralogy, Ben Gurion University of the Negev, P.O. Box 653, Beer Sheva 84 105, Israel. 629
0882-8121/97/0700-0629512.50/I 9 1997InternationalAssociation for MathematicalGeology
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Grossenbacher,Karasaki, and Bahat
is the distance between two parallel fractures. As fracture intervals pertain to both parallel and nonparallel fractures, they are more applicable universally than spacing. Also, average intervals are of direct use in rock mechanics and essential also for understanding hydrologic flow paths. Fracture intervals are measured along a specified path, the scanline. This study develops the theory for a curved or kinked scanline. This theory can be applied to characterize fractures on a curved artificial surface, such as a tunnel wall. Alternately, it can be applied to characterize fractures at arbitrarily shaped outcrops. The principle contribution of this work is that, although there is extensive literature and methodology for characterizing fracture networks with straight scanlines, all of those methods require a flat surface or straight borehole to work. These constraints limit applications to those rare outcrops that are flat, or to the long axis only of tunnels. This work extends those methods to handle curved scanlines such as the circular arc perpendicular to the tunnel axis, or to curved or kinked lines following the exact shape of a rough outcrop.
Straight Scanline Theory Background Much work has been done on fracture intervals on straight scanlines, starting with Priest and Hudson (1976). Further studies focus on several issues with respect to average spacing, and look at error (Sen and Kazi, 1984) and precision (Priest and Hudson, 1981). More advanced uses include characterizing orientation of sets (i.e., La Pointe and Hudson, 1985). Much work also has been done on characterizing fracture interval distributions with scanlines (Baecher, 1983; Boadu and Long, 1994; Gillespie and others, 1993; and Wallis and King, 1990). One key assumption inherent to most such works is that fractures occur in sets of primarily parallel fractures, and that each set has a characteristic average density. We initially assume this as well, but later discuss relaxing this assumption.
Objectives and Approach The principle limitation for all these works is that they require data from a straight scanline. This works fine where such data are available, such as from a borehole, straight tunnel wall, or natural pavement. The problem however, is, that it may be desirable to characterize fracture networks where no such circumstances prevail. In fact, the general situation for surface outcrops is that they are characterized by irregularly shaped surfaces. Furthermore, fracture networks differ spatially to such an extent that one cannot extrapolate with confidence from those few and sparse naturally flat outcrops that are available. Accordingly, there exists a need for a comprehensive method to characterize fracture networks at irregularly shaped outcrops. We present such a method here, motivated by several specific objectives.
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631
The principle objective is to develop a mathematically sound theory for fracture density on a curved or kinked scanline. Such a method must consider the effect of differently shaped rock surfaces, and must provide usable solutions for at least some likely shapes. The approach should be able to utilize all of the extensive work which has been done on straight scanlines. It must consider the presence of multiple, poorly defined fracture sets as well as single discrete sets. It must yield an analytical method which can discriminate among different hypotheses as to type of fracture networks. The approach used is to analyze the effect of different scanline shapes. First, a circular scanline shape is selected. Average fracture spacing is calculated for different incident sets. Finally, the effect of superimposing multiple sets is considered.
THEORY
We formulate the theory for circular scanlines. Prior to doing so however, we discuss the issue of scanline shape and why a circular path is reasonable to analyze. We then extend established theory for straight scanlines to curved scanlines, both for single and multiple fracture sets. Scanline Path
Natural rock typically has a rough and irregular surface that must be rendered suitable for analysis. For a tunnel wall, the transformation is straightforward. If the scanline is in a plane perpendicular to the tunnel axis, the scanline forms a circular arc. If it is oblique to the tunnel axis, the intersection of the plane with the tunnel wall forms a scanline which comprises a section of an elipse. Consider a cross section through an outcrop of arbitrary shape, with the scanline comprising the intersection of the perimeter of the outcrop with a plane. The scanline then lies entirely within the plane of the cross section (here we assume that the scanline all lies within a single plane, although in principle this need not be the situation). The first step is to transform the actual shape of the real scanline into one suitable for analysis (Fig. 1). This requires several assumptions. We assume that the entire scanline lies within a single fracture domain, that is the fracture density is homogeneous statistically within the region the scanline transects. Furthermore, we assume for a natural surface that the scanline is straight between points of intersection with fractures (reasonable for the approach used). These assumptions allow us to immediately make an important transformation. The scanline can be considered as a number of short segments joined together in sequence, forming a set. However, as our analysis considers only
Grossenbacher, Karasaki, and Bahat
632
Outcrop / ~
S
Tu n
~
PossiblCharact e erizations ~'~
CirculaArc r
~ ~
Parabola Ellipse
/L I
I Fartherest frombaseline SawTooth
}
1lCIosest tobaseline
Square
Figure I. Schematic characterization of scanline into simple geometric forms. Scanline from outcrop has many possible characterizations, where S is scanline path and D is baseline. Scanline from tunnel (perpendicular to axis) comprises circular arc. Saw tooth and square profiles yield scanlines which respectively, are the farthest and closest to baseline.
the properties of each element of that set (i.e., the magnitude of the interval), the order of the elements is irrelevant. The assumption that the outcrop lies within a single fracture domain implies that the only control on measured density is the orientation and length of the scanline, and independent of position within the scanline of any particular segment. Accordingly, we can transpose the segments in any way we please without detriment. Therefore, we can view any scanline path as a single curve with a monotonic first derivative, that is a line which curves in one direction only and does not wiggle back and forth as a natural scanline might. Note, with this method, some fractures may be counted twice. This "double counting" is accounted for by what we seek to measure is the fractures per length of scanline. Accordingly, this "double counting" is implicit in all formulations within this paper. One concern is that the exact shape of the curve is unknown, and will differ
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633
1 $1 f
~'
I S
2
/' 'J i I
[ ~
~
Baseline
A
is1 i2B
h3
1 r
B Figure 2. Effect of scanline shape on number of fractures counted. S~, $2, and $3 constitute (respectively) square, halfcircle, and saw tooth scanline shapes, h~, h2, and h 3 illustrate respective maximum heights of these shapes. A, Set is perpendicular to baseline. Shape of scanline curve has no effect on measured fracture frequency; B, fracture set is parallel to baseline and perpendicular to plane which contains scanline. Shape of curve has maximum possible effect on observed fracture frequency. Number of fractures counted is proportional to h.
somewhat from outcrop to outcrop and from scanline to scanline. However, certain limiting constraints prevail; if we assume that we are trying to quantify fracture sets of specific orientations, then we can consider the maximum error that would result from uncertainty in the exact shape of the scanline. Figure 2 shows the effect of curve shape on (h), the m a x i m u m distance the scanline departs from the baseline. (Here, we define baseline as the line connecting the endpoints o f the scanline; see caption o f Fig. 3.) Variation in h then affects the number of fractures encountered as illustrated. M a x i m u m error in counting fractures occurs when the pole o f the fracture set is perpendicular to the baseline as shown in Figure 2. In this orientation, any change in h will affect proportionately the number o f fractures counted (see Fig. 3). Note that this error is reduced to zero as fracture orientation changes from parallel to the baseline to perpendicular to the baseline, and differs as the cosine of the angle between the fracture pole and the vector in the direction o f h.
634
Grossenbacher, Karasaki, and Bahat 180 160
--l--
140
hmax
---O-- hmin
120
h max/min
1O0
% error med value
80 6O
"%
40 20 0 0
1
2
3
4
5
6
7
9
8
10
\
10,.
9.!
\
8.:.
7! 6!
"'t,.
6!
3.! 2,! 0 .i 0
F
2
3
4
5
6
7
8
9
1
O
s/d
Figure 3. Effect of scanline shape on en-or in predicting fracture densities for fractures parallel to baseline and perpendicular to plane of scanline. Lower diagram is expansion of upper one, s/d is ratio of scanline length to baseline axis. Note that in both diagrams, vertical axis is dimensionless, so no units are required.
To calculate the maximum value of this error, the following approach is taken. For a given scanline length, the curves yielding minimum and maximum h respectively are a square curve and a sawtooth curve. Therefore, all actual curves must lie between these end members, h then can be calculated for a square wave (hmax) and a sawtooth wave (hmin) as a function o f the ratio o f scanline length (s) to baseline length (d) with s and d as given by S and D in Figure 1. This yields:
(s 2 - d2) 1/2 hmax -
2
(1)
and
s-d hmi n -
2
(2)
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635
These two equations are simplified by setting d = 1, and are plotted in Figure 3. Also plotted is their ratio and the maximum percentage error (in fractures counted) if using a median curve with h midway between the two extremes. The error is calculated as one half of the percentage difference between hma• and hmin. This error decreases with increase in s/d, and declines to 50% at s/d = 1.7, and 10% at s/d = 5.5. Many practical situations would have a small s/d (i.e., less than 5), so we must consider some type of assumption as to the shape of the curve. To provide the magnitude of this error for a number of situations, Figure 4 shows the possible relative error (in comparing fractures counted by a circular scanline to those counted by either a sawtooth or square scanline) for a circular scanline as a function of s/d, and relates this to the angle of arc which the scanline traverses. Thus far, we have discussed error in fracture counting solely as a function of scanline shape, and not as a function of fracture orientation. This view is justified by the fact that we are looking at maximum error, which assumes that the fractures are parallel to the baseline. Hence, this error is proportional directly to changes in h (the maximum distance the scanline departs from the baseline). Here, we derive only this upper bound, and then utilize the consequence that
4
a (radians) hcircular
3.5
hmax hmin
3
-[2-- h rnax/min
2.5
3rd4
/
2
1.5
1-
/
;,t/2
i
~4
0.5.
O-
-
11
1.2
1.3
1.4
1.5
0
16
s/d
Figure 4. Effectof scanline shape on error in predicting fracture density for small
s/d for circle. Note that left-hand vertical axis is dimensionless in that it measures magnitudes of ratios, and therefore has no units.
Grossenbacher, Karasaki, and Bahat
636
any actual error must be less. In this work, we assume a path that is a circular arc. We do so because this selection combines several desirable features: (1) it allows direct and rigorous application to the curving wall of a tunnel; (2) it is treated easily analytically and illustrates the procedure; and (3) it is a reasonable first approximation to applying this method to studies of arbitrarily shaped outcrops. Later, we discuss other possible selections of path shape. Mention must be made here as to why we assume the scanline path lies within a plane. The reasons are twofold. First, analysis is easier to do and simpler to understand initially. Furthermore, once the principle is established, the analysis can be extended easily to a three-dimensional path shape. Second, it is expected that data will be available readily from field studies in progress which is gathered by stretching a tape measure over a curved outcrop, and recording fracture intersections from the tape, which lies within a plane.
Circular Path In analyzing a circular scanline path, we consider a single set of fractures. We will start with the equations of Hudson and Priest (1983):
Xs = X Icos o~1
(3)
where Xs is the fracture frequency along the scanline, X is the fracture frequency in the maximal direction (perpendicular to the fracture plane), and ot is the angle between the scanline and the fracture pole. This equation yields frequency loci as depicted in Figure 5, and forms the basis for the established theory for straight scanlines. We next consider the path as a circular arc and integrate Equation (3) in order to determine average fracture frequency. For simplicity, let us consider
pole to fracture set , scanline direction
plane of fractures
Figure 5. Loci of X, fracture frequency as function of direction (Hudson and Priest, 1983). Also shows measured seanline fracture frequency, X,.
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637
Scanline, lengths
20
Baseline length d
B Figure 6. Path shape for circular arc scanline. Scanline spans arc of 20. Baseline has measured fracture frequency of Xs. A, Idealized circular shape of scanline; B, range of scanline arc with respect to fracture frequency loci.
the arc as less than or equal to half a circle, that is, s/d _< 7r/2. We define 0 as the angle the arc makes with the baseline (see Fig. 6). So, the scanline covers a length of s, with an orientation uniformly differing with respect to the base line from - 0 to +0. Simple trigonometric relations, as applied to Figure 6, yield 0 implicitly as a function of s and d: s d
0 sin 0
(4)
This allows explicit integration of Equation (3) as follows: =
O=o-0 ]cos c~I dot
(5)
where is ~s the average frequency along the scanline and o~o is the angle between the baseline of the scanline (i.e., the chord of the arc) and the pole of the fracture set. This yields: Xs = ~0 ([sin
Ot]~=Min(t~0+0'Tr/2)=~O_0 + [ - s i n OLl~176
(6)
where Min( ) and Max( ) are, respectively, the minimum and maximum of terms in the parenthesis. The results of Equation (6) are given in Figure 7. The plot is normalized with respect to maximum fracture frequency (X), so it covers all situations for single fracture sets. It shows average spacing as measured along a circular scanline for different scanline arcs and different angles of fractures to baseline.
Grossenbacher, Karasaki, and Bahat
638
Therefore, it can be used to interpret data for all circular scanlines for single fracture sets. At this stage, it is important to point out that straightforward inversion from average fracture frequency to fracture spacing and orientation is not necessarily the most imoprtant use for Equation (6) or Figure 7. Rather, they show useful trends which allow more versatile interpretation and collection of fracture data. Principally, they show that for alarge set of examples (i.e., angular span > 160 ~ or c~o between 35 ~ and 60~ apparent fracture frequency is on the order of 2/3 (between 0.5 and 0.8) of the maximum fracture density. This will be delved into further in the discussion section but, for now, we can consider it an indication that measured average spacing should be fairly constant in a given fracture domain. We now move to the more general situation of n sets of fractures, each with a different orientation. Again, Hudson and Priest (1983) provide a solution for a straight scanline: n
ks = ~
hi [cos c~il
(7)
i=1
Again, we integrate this equation along a circular path yielding: ~'
:
i= l
~/([sin
oL
1~ = Min(~i
iJa:=oti--Oi
+ 0i,'x/2)
+ [-sin
O( ] a = Max(o'i + 0i,w/2)) /Jet = 7r/2
(8)
This equation rigorously predicts the average fracture frequency of a circular scanline through any number of sets of fractures of known orientation and average density. It can be used as part of an inversion scheme.
Theory of General Curves Established literature is concerned with straight scanlines, and our extension of that describes the situation of circular scanlines. This brings up the important question of the general situation, that is the situation of a scanline of any shape. In principal, if the shape of the scanline is known, and the fracture sets are known, the average fracture density along the scanline is fixed and should be determinable. We present the method for doing so. Our effort focuses on developing the theory. The goal is to develop a general formulation of the problem which later can be used in other studies for numerical inversion of fracture data from complex scanline paths. Consider the situation of n straight scanlines, each of length l,, each with an average fracture frequency of ~,,. The number of fractures in each nth scanline would be the product simply of fracture frequency and length, 3,, 1,. The total number of fractures counted then would be the sum of the fractures in each
Curved Scanline Theory
Figure 7. Average fracture frequency as function of fracture orientation and angular span of circular scanline. A, Perspective view shows region of constant average frequency; B, plan view, showing contours of equal fracture frequency.
639
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Grossenbacher, Karasaki, and Bahat
scardine. (Note that some fractures may be counted more than once, but this is acceptable, because we am measuring fracture frequency and not trying to count the actual number of fractures.) Clearly, the total length is the sum of the individual lengths, and so the average combined fracture frequency, Xs, can be calculated;
~'
Xs = i=
(9)
Z li
i=l
This quantity is given a different notation from the average frequency along a straight scanline (X~) because for a curved line, the frequency differs with position on the scanline. We must translate this to integral form. To aid in doing this, we use vector notation for fracture density (Priest, 1993): ~s = m 9 s
(10)
where m is the vector of fracture density defined as follows: D
mx =
~
i=1
~kinix
D
my = ~a ~kiniy i=1
(11)
D
m z = ~_a i=I
)kiniz
where the coefficients na, niy, n~z are the direction cosines for the normal to each ith fracture set, and s is the vector of the scanline and can be parameterized with respect to (t) as follows: s = Sx(t)i + Sy(t)j + sz(0k
(12)
with i, j, k as unit vectors of the principal axis, and t as the path variable. The coordinate system x, y, z is taken to be an arbitrary fixed set, such as north, east, and vertical. Now, let us consider an arbitrary curved path of differing orientation. We can consider this path as a series of straight, infinitesimally short scanlines. Integrating along this path will yield the average frequency along the scanline, The task becomes to integrate Equation (10) with respect to position (t) on the scan line. This can be done by looking at the limit of Equation (9) as I goes to zero. To do this, it is helpful to parameterize l with respect to t as follows (after Kreyszig, 1979):
Curved Scanline Theory
641
I(t) =
i
t
a
(13)
~/r 9 t d t
where dr
t = --
(14)
dt
with r as the vector function representing the scanline, and t as the variable of integration. We can combine Equations (10), (12), (13), and (14) to yield the general equation for fracture frequency for a curved scanline: ib (m 9 s(t))~]i'(t) -- i'(t) d t
x,
Q
=
b
(15)
I a 4r (t) - e(t) at Note that s and r are the same vector functions, separate notation is maintained to facilitate understanding the origin of the separate components of Equation (15). This equation applies to any continuously differentiable function s. Should s not be continuously differentiable, or not even continuous, then each continuously differentiable segment may be substituted into Equation (12) as follows: n
ibi
j=
ai
=Z
L =
(m 9 s(t))dr(t) - r(t)
. ~,I
j=
b a
dt (16)
4i,(t)-
i'(t)dt
If desired for substitution into Equation (16), the Cartesian components for s and m are (following those by Priest, 1993): sx
= cos a s (t) cos/3, (t)
sy
= sin ~s(t) cos fl~(t)
(17)
s z = sin fls(t) and P
rn~ ~- ~
i=1
)ki COS
Odni COS ~3hi
P my = ~ Xi sin Olni COS /3ni i=1 P
mz
~---
E ~i sin i=l
~ni
(18)
642
Grossenbacher, Karasaki, and Bahat
with D the total number of fracture sets. %,/3s are the trend and plunge of the scanline, and er [~ni a r e the trend and plunge of the pole to the ith fracture set. The general formulation, as presented by Equation (16), constitutes the principle result of this section. It provides the basis for developing numerical inversion schemes to determine orientation and spacing for fracture sets. As such, we leave further utilization to other studies. Fracture Frequency Ellipsoids As more and more fracture sets are present, the loci of frequency (for the two-dimensional situation) starts to resemble an ellipse. This concept is evidenced by work presented in Priest and Samaniego (1983) and discussed in Priest (1993). It is illustrated schematically in Figure 8. This change manifests for two reasons. As more and more fracture sets are added together, the overall shape of the superimposed arcs becomes smoother. This effect manifests in several of the figures in Hudson and Priest (1983). Furthermore, they show quantitatively for the situation of evenly distributed uniform sets that cusps disappear as the number of sets goes to infinity. The other smoothing effect is that of randomness. Random variation in fracture orientation about the average normal for that set (ubiquitous in natural fractures) round cusps to an increasing extent with increase in standard deivation (i.e., Priest and Samaniego, 1983). Thus, in general, the more sets there are, and the more randomness in orientation, the more valid is the approximation of the fracture frequency loci as an
Figure 8. Schematic illustration of effect of increasing numberof fracture sets and randomized on fracture frequency loci.
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643
ellipse in two dimensions. The extension of this argument to an ellipsoid in three dimensions is intuitive. So, one could argue that where fracture sets are few and with little random component, the method for discrete sets outlined in the previous section is an appropriate method of analysis. However, in situations where many sets are present, or there is a strong random component to orientation, analysis of a fracture frequency ellipsoid becomes attractive. Next, we discuss such a method and provide its implementation for both straight and curved scanlines. Assuming a fracture frequency ellipsoid to exist, we look at how to predict fracture frequency within it. Let the ellipsoid be defined as: x'2 y~2 zt2 a 2 + -~- + ~-2 = 1
(19)
where x', y', and z' are in the coordinate system aligned with the principle axis of the ellipsoid. They are related to the external coordinate system (x, y, z) by the standard rotation matrix:
lxl [lmlnlEl y'
=
Z'
12 m2
n2
13
n3
m3
(20)
where Ii, rnl, and nl; l z, m 2, and n2;/3, m3, and n 3 are, respectively, the direction cosines of the a, b, and c axis of the ellipsoid. The fracture frequency for a straight scanline is simply the length of a semidiameter in the direction of the scanline. To determine this, direction cosines of the scanline are substituted into the right-hand side of Equation (20) to yield direction cosines l', m', n' in the x ' , y', z' system. This result then is substituted into the equation for length of a semidiameter (Bloss, 1961), yielding: 1
X, =
,
(21)
sin2 0 cos2 ~b sin2 O s in2 q5 c0s2_._.__s a2 + b2 + c2 with: O = acos(n') 0_-a
cos
(22)
Thus, we have a method of predicting fracture frequency, given the fracture frequency ellipsoid.
644
Grossenbacher, Karasaki, and Bahat
Accordingly, we are ready to apply the circular scanline, so as to model a scanline through a fracture network with an elliptical loci. We define the fracture density ellipsoid in terms of coordinate axis x", y', and z ", with x " parallel to the baseline, y" parallel to the plane of the scanline and perpendicular to the baseline, and z" perpendicular to the plane of the scanline. (Summarizing the different coordinate systems introduced thus far, x, y, z refers to an arbitrary external reference frame; x ' y', z' refers to the fracture density ellipsoid, and x", y", z" refers to the scanplane.) To implement this we use the following transformation matrix:
Y"I
=
LImnillxl 12 m2 n2
y'
l3
Z'
(23)
1
Z"3
m3
rl3
where l I , ml, and nl;/2, mz, and n2; 13 m3, and n 3 are, respectively, the direction cosines of the x", y", z" coordinate axis in the x', y', z' coordinate system [those are determined by substituting the directions cosines of the x", y", z" axis in the x, y, z coordinate system into Equation (20)]. This yields an equation in the general form for an ellipsoid: a(x") 2 + b(y") 2 + C(Zn)2 "Jr-dx"y" + ex"z" + f y " z "
(24)
+ gx" + hy" + kz" + l = 0
where a, b, c, d, e, f, g, h, i, j, k, l are all constants. We now determine the trace of this ellipsoid within the plane of the scanline by setting z equal to zero, yielding: A(x") z + 2Bx"y" + C(y") 2 + 2Dx" + 2Ey" + F = 0
(25)
where A, B, C, D, E, and F are all constants. We can express this ellipse in standard form in fracture frequency coordinates by using the transformation equations (Peterson, 1960): x' = x" cos 3' + y" sin 3' (26)
y' = - x " sin 3" + y" cos 3" with: (C - A) + 4(C
3" = a tan
2B
-
A) z +
4B2"~
)
(27)
where the arbitrary sign in (27) is selected to make 3" positive. Note that 3' is the angle the base line makes with the minor axis at the ellipse. We can formulate
Curved ScanlineTheory
645 X'
minor
a D
9
.
yt
Elliptical section of fracture fabdc ellipsoid
Figure 9. Elliptical section of fracture fabric ellipsoid which contains plane of scanline. 3, is angle baseline makes with minor axis of section, 0 is angle circular scanline makes with baseline.
fracture frequency as a function of the changing orientation of the scanline. This is derived by simplifying Equation (21) to the two-dimensional situation: Xs
1
(28)
/
[COS2 ot sin2 o~ %/1a---Y - + b ~ where o~ is the angle of the scanline to the minor axis of the ellipse, and a is a length of the minor axis and b is the length of the major axis. This yields the elliptical section depicted in Figure 9. We now let 0 equal the departure of the scanline from the base line (as before). Therefore, the range of ot is: a=~_+0
(29)
We integrate Equation (28) over this interval, yielding:
I "+~ Xs = ~ 0 ~,-0
ff _ /cos2ol -§
(30) sinE a
Thus, we now have a method of predicting fracture density for a circular scanline through rock possessing a fracture fabric ellipsoid. Equation (30) can be put into standard form, by normalizing the ellipse with respect to a, the minor axis (i.e., letting a equal 1), which yields:
31, 1-
1 - ~
sin2c~
646
Grossenbacher, Karasaki, and Bahat
This equation yields average fracture density for a circular scanline path through an elliptical fracture frequency loci. Clearly, a family of surfaces can be generated for ellipses of differing aspect ratios. However, before doing so, we must examine first the issue of which reasonable aspect ratios might be appropriate. Here, aspect ratio of the ellipse is the ratio of relative frequencies of individual fracture sets. First, we must determine the maximum 'and minimum aspect ratios possible. Clearly the minimum aspect ratio is 1, for the isotropic situation where the loci is a circle. The maximum aspect ratio is less obvious. To determine it, we start with the isotropic situation and see what happens if we gradually change it to the extreme anisotropic situation, that of a single fracture set. As the locus changes, the length of the maximum locus dimension expands relative to the minimum dimension. As it does so, the elliptical locus drapes itself about the locus for a single set, as depicted in Figure 10. Calculations which set the radius of curvature at the focal point equal to the circular radius for the single fracture set locus show that the smallest true ellipse which can enclose these circles has an aspect ratio of 1.156. Thus, the truly elliptical situation is limited, only allowing a slight degree of anisotropy. However as
CircleDrap~
Figure 10. Possible ellipses. Different possibilities for elliptical loci for fracture frequency are shown. Limiting situations are circle (i.e., isotropic situation) can maximally draped "'ellipse" (not true ellipse). Largest possible aspect ratio is 2.
Curved Scanline Theory
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Figure 10 shows, an elliptical shape does approximate the portion of the locus which lies within about 45 degrees of the minor axis. Accordingly, justification for plotting Equation (31) becomes more evident. Also apparent is the need to interpret such plots judiciously. We must keep in mind that "ellipses" of aspect ratio greater than 1.156 may have rigorous application only over part of the possible scanline path. This does not necessarily negate their utility, as the scanline path can be selected deliberately so that it lies entirely within the useful region of Figure 10. (Concern is warranted, however, as this would be perpendicular to the pole of the most frequent fractures.) In further support for considering "ellipses" of aspect ratios greater than 1.156 is the effect of randomized orientation of fractures. If the fracture sets have a slightly randomized component, then a larger aspect ratio can exist, as shown in figure 4 of Priest and Samaniego (1983).
Figure 11. Averagescanline fracture frequency for elliptical fracture fabrics (normalized with respect to minor axis of ellipse), for aspect ratios of 2 and 5.0 is 1/2 angular span of scanline, and ~' is angle of baseline to minor axis.
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Grossenbacher, Karasaki, and Bahat Frecuency 2.2 2 1.8 1.6 1.4 1.2 Aspect Ratio 2
4
6
8
10
Figure 12. Globalaverage fracture frequencyfor ellipses of differentaspect ratios.
Armed with this perspective, we can examine plots of Equation (31), which are given in Figure 11. Shown are plots for aspect ratios of 2 and 5. These are similar in character to the plots for a single fracture set, except that the lower limit on fracture frequency is 1 rather than 0 and the upper limit is equal to the aspect ratio of the ellipse. Also, the measured average fracture frequency for a scanline which spans 180 ~ is constant for each ellipse, but is a function of the aspect ratio. Specifically, this average frequency is the square root of b. This suggests a new measure of fracture frequency, that is measured (or predicted) for a 1/2 circle scanline. The benefit of this is that its measured value should be fairly constant within a given fracture domain, regardless of outcrop shape. Let us term this the Global Average Frequency (GAF). Note that such a designation requires specifying the orientation of the plane which contains the scanline. A plot for GAF as a function of aspect ratio is given in Figure 12. Note that for the most likely aspect ratios (i.e., less than 2) the GAF is less than 1.4. DISCUSSION Several matters suggest themselves for discussion. They divide into two principle areas, the issue of scanline shape, and the uses for our results. Subsidiary matters include geological context and method of data collection. Scanline Shape With regard to scanline shape, the principle requirement is justification for selecting a circular path. Until additional data becomes available, we must rely on the heuristic arguments presented earlier. However, our results add a signif-
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icant impetus for seeking such circular paths. In particular, if data can be gathered along a circular path which spans about 180 degrees, we can potentially acquire a most significant parameter, the GAF. Accordingly, this suggests that studies should be planned which can utilize such a scanline.
Application This leads us to how best to use the formulation herein. Recall that our definition of GAF requires that the orientation of the plane be specified. Accordingly, our formulation should apply directly to any fracture situation where a uniaxial anisotropy prevails, such as a columnar jointed basalt flow. Here, fractures are typically isotropic within the horizontal plane. In this situation, any scanline within the horizontal plane should yield identical results. Another, more usual situation is the circumstance where the fractures of interest are all subvertical. Such a situation prevails with bedding-perpendicular joints in sedimentary rock formations. Here, any 180 degree scanline in the horizontal plane would yield the horizontal GAF. Perhaps the greatest utility, however, involves complex fracture networks evident in irregularly shaped outcrops exposed at the Earth's surface. One can construct three sets of scanlines, each set defining the orientation of a plane perpendicular to the other sets. These scanlines then can be cast over a rock outcrop. If these scanliens are 1/2 circles, they would each yield a GAF. Our formulation suggests that any of the fracture frequencies measured must lie within a factor of two of all the other measurements for that outcrop. Furthermore, although the calculation is omitted, one can envision that averaging several GAFs from three mutually perpendicular planes would increase this precision significantly. Accordingly, we can construct a statistically meaningful GAF for any outcrop. Hence, we can compare fracture frequencies among outcrops where no fiat surface exists. Related to the concept of global fracture frequency, is the matter of block volume or block density. This is of great interest in the field of rock mechanics, and has been studied extensively and related to fracture frequency for straight scanlines (i.e., Hudson and Priest, 1979; Kazi and Sen, 1985; PalmstrCm, 1985). Likewise, such methods can be developed for circular scanlines. Specific locations perhaps most suitable for direct application of circular scanlines include large tunnels under construction. Many data are available at some such tunnels (Nance, Lauback, and Dutton, 1994), but successful analysis needs new techniques. Other tunnels which have data available for them include the Exploratory Shaft Facility at Yucca Mountain, Nevada. Circular scanlines could be used readily at such locations to either acquire global fracture density, or to test various hypotheses of fracture network geometry.
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Grossenbacher, Karasaki, and Bahat Inversion
Scanline theory can be applied in two principle ways: forward and reverse modeling. We have outlined the principles of forward modeling for straight and circular scanlines in rocks with either discrete fracture sets or fracture fabric ellipsoids. This provides a test for different hypotheses with fracture network geometry. A more sophisticated use involves the method of numerical inversion. Such an algorithm has been developed for inverting fracture frequency as measured by straight scanlines (Karzulovic and Goodman, 1985), and producing principle fracture frequencies for the sets present, an extension of this or a similar algorithm is required to invert data from circular scanlines. Constructing such an algorithm lies beyond the scope of the present work. However, a brief discussion of this matter is in order. The motivation for such inversion is clear, as the entire field of fracture hydrology depends on obtaining the best possible understanding of fracture network geometry. Accordingly, strong impetus exists to push forward along these lines. To achieve significant results probably would involve constructing numerical simulations of known synthetic networks and imposing scanlines of different geometries upon them. CONCLUSIONS As evidenced by the preceding discussion section, we leave our analysis of curved scanline theory with useful results, but also with potential for further development. The theory as it stands provides a new and easily applied tool for acquiring the Global Average Frequency of fractures at specified locations. Such locations can be irregularly shaped outcrops or cliff faces. What now is needed is an actual application of this procedure in the field. Such a study is currently underway in the Sierra Nevada Foothills of California. The results will be reported in a separate paper. The formulation developed herein focuses on circular scanlines. However, the general approach can apply to other characteristic shapes and procedure new equations and plots accordingly. As additional data becomes available for shapes of different surfaces (and the scanlines which cross them), these additional characteristic shapes can be selected appropriately. ACKNOWLEDGMENTS We thank Bo Bodvarsson for his persistent encouragement. We thank Maria Fink, Jackie Gamble, and Carol "l;aliaferro for help with the figures and editing. We thank Kevin Hestir for reviewing the equations and Robert Zimmerman for discussion of elliptical integrals. This work was carried out under U.S. Department of Energy Contract No. DEAC03-76SF00098 for the Director, Office of Civilian Radioactive Waste Management, Office of External Relations, and
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w a s a d m i n i s t e r e d b y t h e N e v a d a O p e r a t i o n s Office, U . S . D e p a r t m e n t o f E n e r g y . T h i s w o r k also w a s s u p p o r t e d partially b y t h e P o w e r R e a c t o r a n d N u c l e a r F u e l Development Corporation of Japan. REFERENCES Baecher, G. B., 1983, Statistical analysis of rock mass fracturing: Math. Geology, v. 15, no. 2, p. 329-347. Bloss, F. D., 1961, An introduction to the methods of optical crystallography: Holt, Rinehart and Winston, New York, 294 p. Boadu, F. K., and Long, L. T., 1994, The fractal character of fracture spacing and RQD: Intern. Jour. Rock Mechanics and Mining Sciences and Geomechanics Abstracts, v. 31, no. 2, p. 127-134. Gillespie, P. A., Howard, C. B., Walsh, J. J., and Watterson, J., 1993, Measurement and characterization of spatial distributions of fractures: Tectonophysics, v. 226, no. 1-4, p. 113-141. Hudson, J. A., and Priest, S. D., 1979, Discontinuities and rock mass geometry: Intern. Jour. Rock Mechanics and Mining Sciences and Geomechanics Abstracts, v. 16, no. 6, p. 339-362. Hudson, J. A., and Priest, S. D., 1983, Discontinuity frequency in rock masses: Intern. Jour. of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, v. 20, no. 2, p. 73-89. Karzulovic, A., and Goodman, R. E., 1985, Technical note: determination of principal joint frequencies: Intern. Jour. Rock Mechanics and Mining Sciences and Geomechanics Abstracts, v. 22, no. 6, p. 471-473. Kazi, A., and Sen, Z., 1985, Volumetric RQD: an index of rock quality, in Stephansson, O., ed., Proc. Intern. Syrup. Fundamentals of Rock Joints (Bjorldiden, Sweden). Centek Publ., Lulea, Sweden, p. 95-102. Kreyszig, E., 1979, Advanced engineering mathematics (4th ed.): John Wiley & Sons, New York, 939 p. La Pointe, P. R., and Hudson, J. A., 1985, Characterization and interpretation of rock mass joint patterns: Geol. Soc. America, Spec. Paper 199, 37 p. Nance, H. S., Laubach, S. E., and Dutton, A. R., 1994, Fault and joint measurements in Austin Chalk, Superconducting Super Collider Site, Texas: Trans. Gulf Coast Assoc. Geol. Societies, v. 44, p. 521-532. PalmstrCm, A., 1985, Application of the volumetric joint count as a measure of rock mass jointing, ,!n Stephansson, O., ed., Proc. Intern. Symp. Fundamentals of Rock Joints (Bjorkliden, Sweden): Centek Publ., Lulea, Sweden, p. 103-110. Peterson, T. S., 1960, Calculus with analytic geometry: Harper and Brothers, New York, 586 p. Priest, S. D., 1993, Discontinuity analysis for rock engineering: Chapman & Hall, New York, 473 p. Priest, S. D., and Hudson, J. A., 1976, Discontinuity spacings in rock: Intern. Jour. Rock Mechanics and Mining Sciences and Geomechanics Abstracts, v. 13, no. 5, p. 135-148. Priest, S. D., and Hudson, J. A., 1981, Estimation of discontinuity spacing and trace length using scanline surveys: Intern. Jour. Rock Mechanics and Mining Sciencs and Geomechanics Abstracts, v. 18, no. 3, p. 183-197. Priest, S. D., and Samaniego, J. A., 1983, A model for the analysis of discontinuity characteristics in two dimensions: Proc. 5th ISRM Congress (Melbourne): ISRM, p. FI99-F207. Sen, Z., and Kazi, A., 1984, Discontinuity spacing and RQD estimates from finite length scanlines: Intern. Jour. Rock Mechanics and Mining Sciences and Geomechanics Abstracts, v. 21, no. 4, p. 203-212. Wallis, P. F., and King, M. S., 1990, Discontinuity spacings in a crystalline rock: Intern. Jour. Rock Mechanics and Mining Sciences and Geomechanics Abstracts, v. 17, no. 1, p. 63-66.
