Power-law vein-thickness distributions and positive feedback in vein

Power-law vein-thickness distributions and positive feedback in
vein growth
M. Brooks Clark*
Susan L. Brantley
Donald M. Fisher
Department of Geosciences, Pennsylvania State University, University Park, Pennsylvania 16802
ABSTRACT
The thicknesses of veins of a single generation from the Kodiak accretionary complex
show a power-law distribution with an exponent of D 5 1.33. This value (D > 1) indicates
that a significant portion of the vein-related extension is accommodated by the thinnest
veins. Thick vein segments are typically confined to southeast-dipping brittle-ductile shear
zones. Textural observations of veins within these zones indicate a progressive evolution
from crack-seal to euhedral growth of quartz with an increase over time in the duration that
fractures remained open. Model simulations of vein-thickness distributions show that a
constant growth model results in a negative-exponential distribution and that it is necessary to assume size-proportional growth to produce a power-law distribution. Textures and
power-law thickness distributions for the Kodiak veins are thus consistent with an increasing time-averaged growth rate as vein thickness increased, comprising a positive
feedback. The positive feedback resulted in the progressive evolution from fractures to
brittle-ductile shear zones and ultimately to faults.
INTRODUCTION
Many biological, sociological, economic,
and geologic phenomena display size distributions that are best approximated by positively skewed frequency distributions. For
example, sizes of cities and business firms
obey a power-law distribution (Ijiri and Simon, 1977), and sizes of crystals in rocks are
often negative exponential (Marsh, 1988).
The size distribution of objects is controlled
by the birth (nucleation) rate, the growth
rate, and the death (annihilation) rate
(Marsh, 1988). A constant birth, growth, and
annihilation rate yields a negative-exponential distribution (Marsh, 1988); a constant
birth rate plus a proportional growth rate
yields a power-law distribution (Ijiri and
Simon, 1977).
Power-law distributions are scale invariant and have a power-law relation between
the size of an object (L) and the number of
objects of that size or larger (N):
N~L!}L2D.
(1)
Thus, power-law distributions are fractal
(Mandelbrot, 1983), where D is the fractal
dimension. Many geologic phenomena, including topography, earthquakes, and oilfield sizes, are fractal (e.g., Turcotte, 1992).
Alternatively, negative-exponential distributions are not scale invariant and reveal an
exponential relation:
N~L!}e2lL,
(2)
where l equals the mean and standard deviation. Power-law and negative-exponen*Present address: Exxon Production Research Company, P.O. Box 2189, Houston,
Texas 77252-2189.
tial fracture- and vein-spacing distributions
have been reported (La Pointe and Hudson,
1985; Manning, 1994), but lognormal distributions are also common (Narr and Suppe,
1991; McCaffrey et al., 1993). Lognormal
distributions result where distributions grow
from constant birth, proportional growth,
and constant death rate (Ijiri and Simon,
1977).
Because veins do not disappear after formation (i.e., death rate 5 0), different thickness distributions reflect differences in the
nucleation and growth rates. We analyze a
system of veins from the Kodiak accretionary complex (Fisher and Brantley, 1992) and
relate observed size-frequency distributions
to vein textures.
VEIN GEOMETRIES AND TEXTURES
FROM THE KODIAK FORMATION
The Kodiak Formation is a turbidite sequence that was underplated along the
Alaska margin during Late Cretaceous to
early Tertiary subduction. Deformation involved folding, imbrication, formation of a
subhorizontal slaty cleavage, and development of a regionally extensive (90 km 3 15
km) network of subvertical quartz veins in
conjunction with metamorphism at lower
greenschist facies. During vein formation, a
regionally pervasive subhorizontal cleavage
within the Kodiak Formation acted as a barrier to upward fluid flow. Consistent with
this, textures and wall-rock Si concentrations indicate that most vein Si was derived
from local transport from matrix to vein,
rather than from fluid advection (Fisher et
al., 1995).
The vein network is composed of nearly
vertical, meter-long, closely spaced (0.5–3
Geology; November 1995; v. 23; no. 11; p. 975–978; 5 figures.
cm) thin veins with median thicknesses of
#1 mm. These thin veins widen into thick
segments (median thickness of ;1 cm) that
are arranged into dominantly southeast-dipping brittle-ductile shear zones (Fig. 1). The
tips of veins are approximately parallel for
both southeast-dipping and less abundant
northwest-dipping en echelon sets. In some
cases, faults have developed along zones of
en echelon vein arrays.
Vein thicknesses and spacings were measured along 24 scan lines from various localities. Veins thinner than ;1 mm were
noted, but the thicknesses could not be measured accurately and are not included in the
analysis. Figure 2 shows the cumulative frequency of vein thicknesses in log-log and
semilog space; the plot is linear in log-log
space, indicating a power-law distribution
(D 5 1.33). The value of D reflects the relative importance of thin and thick parts of
veins, where D . 1 indicates that a significant portion of the extension is accommodated by the thinner vein segments (Scholz
and Cowie, 1990; Fisher et al., 1995).
Three distinct microtextures have been
recognized in the Kodiak veins, including
continuous crack-seal, discontinuous crackseal, and euhedral growth textures (Fig. 1A;
Fisher and Byrne, 1990; Fisher and Brantley, 1992; Fisher et al., 1995). Continuous
crack-seal veins have laterally continuous,
planar bands of phyllosilicate inclusions and
are interpreted to reflect periodic cracking
followed by complete sealing (Fisher and
Brantley, 1992). Continuous crack-seal textures are present in all veins, but dominate
the vein segments between shear zones
where from 1 to 50 crack-seal bands with a
median thickness of ;7 mm are observed
(Fig. 1B). Fisher et al. (1995) reported that
inclusion spacing generally decreased over
time in the thin vein segments.
Vein textures in en echelon arrays are
also of the continuous crack-seal type in the
oldest parts of the veins, but are dominated
successively by discontinuous crack-seal and
euhedral growth textures (Fisher and Brantley, 1992) (Fig. 1). Discontinuous crack-seal
bands have a median spacing of ;10 mm and
reflect partial sealing of the fracture by
quartz grains favorably oriented for rapid
growth (Fig. 1C). The prismatic terminations in quartz veins with euhedral growth
textures are interpreted to indicate longer975
Figure 1. A: Schematic diagram showing geometry of D2
Kodiak veins. Veins are generally not bed-limited and are
oriented roughly perpendicular to subhorizontal cleavage.
Dashed lines indicate boundaries between domains with
different textural characteristics. B–D: Thin lines represent bands of phyllosilicate
inclusions in quartz grains,
and thick black lines represent collapse features composed primarily of insoluble
residue. Most quartz shows
unidirectional growth in individual veins, but direction of
growth varies from vein to
vein. Histograms of (E) inclusion-band spacing in continuous crack-seal veins, (F)
discontinuous crack-seal
veins, and (G) spacing of collapse features in euhedral
growth veins.
duration growth into open, fluid-filled cracks
without high-frequency sealing events. However, insoluble pressure-solution residues
trapped within and adjacent to euhedral
quartz grains reflect periodic collapses of
the fractures. The spacing between collapse
features in these veins is much larger
(millimetres) than inclusion band spacing
(Fig. 1D).
Thus, the observed evolution from continuous crack-seal to euhedral growth textures reflects an increase through time in the
duration that fractures remained open. This
interpretation is consistent with a model
whereby nearly vertical thin veins developed, followed by localized shearing and
faster vein growth in the en echelon fracture
arrays (type II sets; Beach, 1975). The thick
vein segments in en echelon sets developed
because segments of fractures remained as
open, fluid lenses for long periods of time.
Periodically, fluid was expelled and fractures collapsed, perhaps due to fracture
linkage (Fig. 3). Crack-seal textures in distal
portions of the veins are interpreted to reflect cracking without linkage of en echelon
arrays, followed by sealing, which forced
fluid back into the localized shear zones
(Fig. 3; Fisher et al., 1995).
STOCHASTIC MODELS
We can investigate size-frequency distributions for veins by simulating vein growth
with stochastic models. For the simplest
model, we assume (1) new veins with minimum thickness L0 are introduced at a constant rate, (2) the probability P of any given
vein refracturing and growing thicker is the
same (i.e., P } 1 4 number of veins), and (3)
once a vein refractures, it continues to grow
for only one time step. The model steps
through time, either introducing a new fracture (vein) or allowing refracturing and
growth of an existing vein during each time
step. A constant ratio, a, of the number of
birth events (nucleation of new veins) to refracturing events is maintained. The instantaneous growth rate of a vein always occurs
by zeroth-order kinetics:
]L
5 k.
]t
Figure 2. Cumulative frequency diagram for D2 vein thicknesses in Kodiak Formation showing
total number of veins (N) of thickness L or larger (measured normal to vein walls) vs. L. Measurements were made along scan lines oriented normal to dominant trend of veins. Plotted in
(A) log-log space and (B) semilog space. D is slope of least-squares best-fit line, and r2 is
regression coefficient.
976
(3)
Once the size distribution remains unchanged over multiple time steps (.;1000
veins), the results are plotted as cumulative
size-frequency distributions (Figs. 4 and 5).
For such a constant-growth-rate model, the
size-frequency distribution is negative expoGEOLOGY, November 1995
Figure 3. Schematic diagram illustrating (A–C) crack-seal and (D) collapse events. A: At time 0,
veins are completely sealed in distal portions, and fluids (white) are stored in thick, en echelon
vein segments. B: Cracking of thin vein segments and injection of fluid into thin fractures
(dashed-line arrows show direction of fluid migration). C: Sealing of thin vein segments and
migration of fluid back into en echelon vein arrays. D: Linkage of fractures allows fluid expulsion, causing collapse and growth of collapse features. Fluids may be expelled either vertically
or horizontally along fractures.
nential (Fig. 3). This model may apply to
crystal growth (Randolph and Larson, 1971;
Marsh, 1988) or to some vein distributions
(Johnston, 1993), but it is not applicable to
the Kodiak veins.
As pointed out by Ijiri and Simon (1977),
a size-frequency distribution becomes a
power-law distribution when the growthrate function is proportional to size. Therefore, our second model assumes that larger
veins grow thicker at a rate proportional to
thickness. Textural evidence suggests that
thicker vein segments found in en echelon
arrays remained fluid-filled (allowing quartz
growth) longer than thinner vein segments.
Therefore, our proportional growth model
differs from the first model in only the third
assumption: once a vein refractures, it continues to grow as many time steps as needed
until it has grown to some proportion b of its
thickness. After refracturing, then, each
vein of thickness L grows to thickness (1 1
b)L. For this model, the thicknesses show a
power-law distribution as predicted, where
the slope shallows with decreasing a and increasing b, reflecting the greater thickness
of the larger veins. For example, if all time
steps are birth events (a 5 1/0), all veins
have the same thickness and the slope is
vertical. If all time steps are growth events
(a 5 0), D approaches 0.
Although more complicated models incorporating time-dependent nucleation and
growth rates could explain observed sizefrequency distributions, the simplest explanation is that negative-exponential distributions reflect size-independent growth, and
power-law distributions result from a positive-feedback process where time-averaged
growth rates are proportional to size.
CONCEPTUAL MODEL FOR VEIN
GROWTH AND SHEAR-ZONE
NUCLEATION
Microtextures within thick veins record a
progressive evolution from crack-seal
growth, with a median aperture of ;7 mm,
to euhedral growth into open, fluid-filled
reservoirs. Initially, fracturing occurred
throughout the argillaceous layers, and the
Figure 4. Comparison of vein-thickness distributions assuming time-averaged growth rate that is (A–C) independent of size and (D–F) proportional to size. Size-independent growth results in (A) negative-exponential distribution; cumulative frequencies are (B) concave down in log-log
space and (C) linear in semilog space. (D) Size-independent growth results in power-law distribution; cumulative frequencies are (E) linear in
log-log space and (F) concave up in semilog space.
GEOLOGY, November 1995
977
ACKNOWLEDGMENTS
Supported by National Science Foundation
(NSF) grant EAR-9305101 (to Fisher and Brantley) and by the David and Lucile Packard Foundation and NSF grant EAR-8657868 (Brantley).
We thank T. Engelder, C. Manning, M. Brandon,
and R. Marrett for reviews, and B. Wambold, M.
Everett, and E. Rufe.
Figure 5. Cumulative frequency diagrams for modeled vein-thickness distributions assuming
that growth is proportional to size. Vein thicknesses show power-law distribution with slope
dependent on ratio of birth events to growth events (a) and the amount of growth per refracturing episode (1 1 b)L.
fracture porosity was distributed over numerous small vertical fractures. Fluid— derived from metamorphic dehydration or
transported upward from depth—accumulated within fractures. Initially, each fracture
sealed with quartz from crack tip back to
crack center and accumulated fluid at its
central widest point where the fracture did
not seal or sealed incompletely (Fig. 3). As
fluid accumulated, veins refractured, allowing a new crack-seal band to precipitate
along the fracture. However, sealing of
crack tips segregated fluid back into the widest portion of each fracture. Precipitation of
locally derived silica therefore concentrated
along these sites of fluid accumulation.
Thus, the thickness along a given vein segment is a measure of the cumulative amount
of fluid at that position.
The interaction between neighboring
fluid lenses produced zones of weakness
that localized strain as brittle-ductile shear
zones. Progressively, fluid was stored in the
shear zones at the expense of the small fractures between shear zones. As larger fractures remained open longer and served as
fluid reservoirs, less fluid entered the thin
crack-seal fractures, and these fractures
opened less over time, explaining the decrease in inclusion spacing (see Fig. 6 in
Fisher et al., 1995). Veins in the shear zones
remained open longer and were sinks for
978
both fluid and silica; thus, vein growth responded to a positive feedback. Strain softening and fluid segregation into closely
spaced fractures has in some cases led to
fault nucleation along en echelon sets, which
could have drained fluid and caused the collapse recorded by pressure-solution selvages
(Fig. 1).
CONCLUSIONS
Subvertical D2 veins from the Kodiak accretionary complex are arranged in southeast-dipping en echelon vein segments that
show a power-law thickness distribution.
Microtextures of veins in these shear zones
indicate that these veins grew by multiple
events of cracking and sealing, with an increase in the duration that fractures remained open. Stochastic models show that a
power-law size distribution can be produced
by a constant birth plus a proportional growth
rate, where the time-averaged growth rate is
proportional to size. Both microtextures and
the power-law thickness distribution therefore suggest that growth of the Kodiak veins
was a positive-feedback process, where the
veins grew thicker as fluids segregated into
shear zones. In the extreme case, the positive-feedback process may have led to fault
nucleation along the en echelon sets.
Printed in U.S.A.
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Manuscript received February 27, 1995
Revised manuscript received August 9, 1995
Manuscript accepted August 14, 1995
GEOLOGY, November 1995