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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, F01014, doi:10.1029/2006JF000714, 2008
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Modeling weathering pathways and processes of the
fragmentation of salt weathered quartz-chlorite schist
T. Wells,1 G. R. Willgoose,1 and G. R. Hancock2
Received 19 October 2006; revised 5 August 2007; accepted 9 November 2007; published 29 February 2008.
[1] The relationship between the rate of rock breakdown and environmental and
geological factors must be understood in order to establish the conditions under which
weathering limits erosion. In this study qualitative and quantitative models of the rock
fragmentation process are fitted to previously published data obtained from laboratory salt
weathering trials of quartz-chlorite schist. Weathering was modeled as a combination of
(1) a fragmentation event that fragments the parent particle into a number of daughter
particles while preserving mass, and (2) a fracture probability, that determines the
probability that a fragmentation event will occur in a given time period. We show that
observations of the complex breakdown of salt weathered schist are consistent with model
assumptions of a simple fracture geometry model and an increase in fracture probability
with time. For the fragmentation geometry the best fit to the experimental data was
achieved by assuming that each fragmentation event involves splitting of the parent
particle into two daughter fragments of equal volume. For the fragmentation rate the data
could best be described with a fracture probability, and hence the weathering rate, that
increased linearly with time. This paper shows that it is possible to use a physically based
fragmentation model to infer the process of fragmentation for individual particles using a
time evolving particle size distribution for the weathering rock fragments.
Citation: Wells, T., G. R. Willgoose, and G. R. Hancock (2008), Modeling weathering pathways and processes of the fragmentation
of salt weathered quartz-chlorite schist, J. Geophys. Res., 113, F01014, doi:10.1029/2006JF000714.
1. Introduction
[2] The erodibility of the surface of newly constructed,
man-made landforms can change dramatically in just a few
decades [Moliere, 2001]. However, we are unable to predict
with certainty the rate, or in some cases the dominant
processes and pathways, of soil erodibility and soil production over design lifetimes (typically 100-1000 years). This is
in large part due to the fact that current physically based
erosion models assume that the surface properties do not
evolve over time, or with cumulative erosion and deposition.
Recent theoretical work with landscape evolution models has
highlighted that weathering of the coarser, non-transportable
rock material can act as the rate limiting step in the erosion
process particularly over the medium to long-term (100s to
1000s of years) [Willgoose and Sharmeen, 2006; Sharmeen
and Willgoose, 2006, 2007]. By adapting river armoring
models [e.g., Parker and Klingeman, 1982] for erosion on
hillslopes Willgoose and Sharmeen showed that hillslope
erosion rates decline over time as fine material is selectively
removed from the land surface until eventually a stable
armor of coarse particles is formed. Thereafter the fine
1
School of Engineering, The University of Newcastle, Callaghan, New
South Wales, Australia.
2
School of Environmental and Life Sciences, The University of
Newcastle, Callaghan, New South Wales, Australia.
Copyright 2008 by the American Geophysical Union.
0148-0227/08/2006JF000714$09.00
material generated by weathering of this armor is the only
sediment available for transport so that erosion becomes
weathering-limited.
[3] To improve our predictions of the evolution of soil
properties and weathering-limited erosion, we need to
estimate not only the rate at which fine, transportable
material is produced from the weathering of larger, nontransportable particles but also the resultant fragment size
distribution and how that distribution evolves over time.
This quantitative analysis of the rock weathering process
represents a significant shift in emphasis from previous,
more qualitative, work which concentrates on the pathways
of breakdown rather than the rates of breakdown and the
physical properties of the weathering products [McGreevy
and Smith, 1982].
[4] Many studies [e.g., Cooke, 1979; Sperling and Cooke,
1985; Williams and Robinson, 1998, 2001; Rivas et al.,
2003] utilize the change in mass of the largest (or collection
of suitably large) fragment(s) to characterize the extent of
the weathering process. Other studies [e.g., Davison, 1986]
report only the total mass of debris produced. None of these
studies provide information on the particle size distribution
(PSD) of the weathering products.
[5] Other studies examine the potential of salt weathering and other processes as possible sources of loess and
silt sized particles, [Goudie et al., 1979; Pye and Sperling,
1983; Wright and Smith, 1993; Wright et al., 1998;
Wright, 2001], and have presented a more detailed picture
of the size distributions of rock debris produced. Fahey
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WELLS ET AL.: MODELING ROCK WEATHERING PATHWAYS
[Fahey, 1983, 1985; Fahey and Dagesse, 1984] reports
particle size distribution data for debris produced through
the salt, freeze/thaw and wet/dry weathering of a variety
of rock and aggregate materials including sandstone,
dolomite, shale and schist. Unfortunately these studies
report the fragment size distribution at only a few discrete
times, such as at the beginning and end of weathering.
Goudie [1986] and Wright et al. [1998] provide more
detail with snapshots of sediment size distribution produced from sandstones samples after 40 and 60 salt
weathering cycles. To develop fragmentation models suitable for input into erosion models, detailed PSD data have
recently been collected by our research group for quartzchlorite schist cubes and fine aggregates undergoing salt
weathering in a simulated tropical environment [Wells et
al., 2006, 2007].
[6] The data presented in these latter studies are difficult
to interpret directly because they involve the evolution of
numerous interrelated sizes fractions. The change in one
size fraction represents the net effect of population depletion
due to breakdown into smaller size fractions and population
increase due to the breakdown of larger size fractions. To
allow the PSD evolution data to be used in landscape
evolution and soil production models, such as SIBERIA
[Willgoose et al., 1991] and ARMOUR [Willgoose and
Sharmeen, 2006] we must model the evolution of the
PSD. In this paper we will fit a physically based fragmentation model to the evolving PSD. The model we discuss
here characterizes both the kinetics of the weathering
process, (the probability of any single fragment fracturing
per unit time, Pfr, or fragmentation rate), and the size
distribution of the daughter particles produced by the
fragmentation event, (i.e., the fragmentation geometry).
[7] We determine if the fragmentation of schist due to salt
weathering [Wells et al., 2006, 2007], can be successfully
modeled using a range of simple fragmentation models. We
also examine relationships for fracture probability with
time. These models provide insights into the nature of the
weathering process which are otherwise obscured by the
complexity of the evolving size distribution data.
2. Rock Fragmentation Data
[8] The rock fragmentation data we use were obtained
from the salt weathering of quartz-chlorite schist specimens
recovered from the Ranger Uranium Mine waste rock
dumpsite located in the tropical monsoonal Kakadu region
of the Northern Territory of Australia. The full details of the
experimental procedures, chemical and mineralogical composition of the schist, and measured fragmentation data are
reported elsewhere [Wells et al., 2005, 2006, 2007; hereafter
collectively Wells].
[9] This study focuses on three data sets. We use data
from Wells’ most aggressive salt weathering experiments
because they produced a more complete picture of the
weathering process than the less aggressive experiments
which were terminated before complete conversion of the
starting material to fines (particles of <1% of the original
sample mass). This enabled a more rigorous testing of the
different fragmentation models and comparison of competing weathering hypotheses to be undertaken. The experimental results may or may not be indicative of the dominant
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weathering process and/or post-weathering grading of less
aggressive experiments or field behavior, but they are
adequate for the purposes of this paper, that is, demonstrating that we can fit a physically based fragmentation model
to an evolving particle size distribution, and that this
evolving particle size distribution is sufficient to determine
which weathering processes are responsible for the observed particle size distribution. Moreover, Wells et al.
[2006] suggests, subject to the limited duration of their
experiments, that the particle size distributions of the less
and more aggressive experiments were indistinguishable,
provided their timescale was adjusted to reflect the higher
breakdown rates experienced by samples subjected to the
harsher weathering regimes.
[10] Two of the data sets pertain to the fragmentation of
1 cm schist cubes over 60 cycles of aggressive salt weathering; daily immersion in a 250 g/l MgSO4 solution coupled
with either a 60° or 100°C temperature cycle. The critical
temperatures for hydration of MgSO4 salts [Mellor, 1960]
are 48.2°C for conversion from MgSO 4 .6H 2 O to
MgSO4.7H2O (12% volume change) and 67.5°C for conversion from MgSO4.6H2O to MgSO4.H2O (135% volume
change). As a result salt weathering is more aggressive over
the 100°C weathering cycle which includes both hydration
steps. We individually weighed each resulting fragment,
grouping the mass distribution data into 5 categories; fragments >50% of the original block mass, 25– 50%, 10– 25%,
1 – 10% and <1%. Salt located both within and on the
surface of the fragments was included in our weight
determinations, however the salt contribution was found
to be typically c. 5% by mass and consequently the trends
indicated by the salt inclusive weights are indicative of the
fragmentation processes [Wells et al., 2007]. The fragment
mass distributions for the 9 replicates at 100°C (Figure 1)
and 3 replicates at 60°C were then averaged to obtain the
evolving fragment mass distribution.
[11] The third data set relates to the salt weathering of a
sub 500 mm powder generated by the crushing of the schist
from the same parent material used for the cube experiments. Here we use data from one of the more aggressive
salt weathering regimes; 45 cycles of daily immersion in
250 g/l MgSO4 solution with a 100°C temperature cycle. A
laser particle sizer was employed to determine the PSD over
time (Figure 2).
[12] The rock samples used by Wells were chosen to
reflect the typical rock type, and size, found on the surface
of the Ranger Uranium Mine (RUM) waste rock dump. The
RUM ore body lies in an area dominated by quartz-chlorite
schists and the waste rock dump surface gravels are
‘. . .dominated by schist fragments, platy in shape, and
largely 1 to 2 cm in diameter’ [Riley, 1994]. Wells used
as their source material two quartz-chlorite schist samples
collected from newly mined material deposited on the
surface of the dumpsite. The two rock samples were
retrieved within 10 m of each other at the dump site, and
reflect the same mined stratigraphy and locality. Salt weathering was carried out on schist cubes cut from the interior of
the two parent rocks well away from any obvious weathered
rind or internal cracks. The chemical and mineralogical
compositions of the parent rocks were determined by X-ray
fluorescence and semi-quantitative X-ray diffraction [Wells
et al., 2006]. XRD analysis revealed that the two samples
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breakdown and we believe that they may be indicative of
field behavior [Wells et al., 2006].
3. Description of Rock Fragmentation Models
[14] The size and quantity of fragments produced by
weathering is a product of the fragmentation geometry
and rate of breakdown. We use four fragmentation models
that we have computationally studied previously [Sharmeen
and Willgoose, 2006]. Each of these fragmentation models
consists of (1) a fragmentation geometry function describing
the morphology (size and number) of fragments produced
from a parent particle and, (2), a weathering rate expression,
parameterized by a time varying probability of fracture per
unit time for the parent particle.
3.1. Weathering Geometry
[15] The geometry of the weathering process governs the
grading of the (fine) transportable material generated, and
the subsequent rate of erosion on the landform surface being
weathered. Sharmeen and Willgoose [2006] showed, for
instance, that a mass of rock which weathers via the
granular disintegration of fine surface particles will produce
a markedly different erosion history from one where parent
fragments split into two equal sized daughter fragments. In
our study four simple weathering geometries were investigated, three involving the fracturing of the rock along a
cracked surface, and one involving the flaking of fine
particles from a thin surface layer (Figure 3). These models,
summarized below, are discussed in detail in Sharmeen and
Willgoose [2006, 2007].
3.1.1. Symmetric Fragmentation
[16] A parent fragment, volume V, fractures into Nfr
daughter particles of equal volume (Model 1 in Figure 3)
such that:
Vfr ¼
Figure 1. Fragment mass distribution profiles of schist
samples subjected to an aggressive salt weathering regime.
[Daily immersion in 250 g/l MgSO4 solution and 100°C
drying cycle; average of 9 samples. Error bars represent
±1 standard deviation].
were mineralogically similar, both being quartz-chlorite
schists. Thin section analysis indicated that both source
rocks were originally quartz-biotite schists that had been
strongly altered by infiltration of a low temperature hydrothermal fluid, which converted the original biotite content to
chlorite and a white mica.
[13] The aim of this study was primarily to test the
validity and applicability of using simple physically based
rock fragmentation models to simulate observed rock fragmentation. The question of how representative the laboratory fragmentation results are of field behavior is not the
central focus of this paper. However, the analysis that
follows will highlight the advantages and limitations of
laboratory weathering data as a means of parameterizing a
physically based model of rock fragmentation. Accordingly,
data from more aggressive weathering regimes were used.
These regimes produced the greatest level of fragment
V
Nfr
ð1Þ
Figure 2. Particle size distribution of unweathered schist
powder (.) and schist powder after 45 cycles of daily
immersion in 250 g/l MgSO4 followed by drying at 100°C (6).
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Figure 3. The four fragmentation geometries examined in this study.
[17] The number of daughter particles, Nfr, can be constant or a function of parent size [Lerman, 1979]:
Nfr ¼
d
dmin
b
¼
V
Vmin
b=3
ð2Þ
where d is the diameter of the parent particle, and dmin and
Vmin are the minimum threshold parent fragment diameter
and volume (in this paper nominally fixed at 0.1% of the
initial particle volume). Constant b was varied between
0.005 and 1.
3.1.2. Asymmetric Fragmentation
[18] The parent particle splits into two daughter particles
of unequal volumes (Model 2 in Figure 3):
Nfr ¼ 2
Vfr;1 ¼ aV ; Vfr;2 ¼ ð1 aÞV
ð3Þ
where a is the ratio of the daughter fragment volumes, and
Vfr,1, Vfr,2 are the volumes of the two daughter fragments.
Values of a ranging from a = 0.5 (i.e., symmetric
fragmentation model, Nfr = 2) to a = 0.01 (i.e., granular
surface disintegration) were examined. In all cases a was
assumed to be independent of time and parent fragment size.
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3.1.3. Whitworth Fragmentation
[19] A parent particle is broken into Nfr daughter particles
with a volume distribution proposed by W.A. Whitworth
[Lerman, 1979] (Model 3 in Figure 3):
Nfr
V X
1
or
Vfr;j j¼1;Nfr ¼
Nfr j¼1 Nfr j þ 1
V 1 V
1
1
;...;
;
þ
Nfr Nfr Nfr Nfr Nfr 1
V
1
1
1
þ
þ ...:
...: þ 1
Nfr Nfr Nfr 1
Nfr i þ 1
ð4Þ
[20] Fragmentation numbers of Nfr = 2, 3, 4 were used for
the rock cube data, while a range from 2 to 10 was used for
the powder.
3.1.4. Granular Surface Disintegration
[21] A thin surface layer encompassing the parent particle
breaks down to produce a collection of equal sized spherical
daughter particles of diameter equal to the layer thickness,
and a single large daughter particle with a diameter equal to
the parent particle diameter less twice the layer thickness
(Figure 3). The number of fine particles generated was
calculated from volume continuity (Model 4 in Figure 3):
Nfr;fine
6d 2 12d
¼8þ 2 dr
dr
ð5Þ
where Nfr,fine is the number of fine daughter particles spalled
from each parent particle, dr is the surface layer thickness,
and once again d represents the parent particle diameter. For
small parent particles (d < 2dr) the parent particle splits into
2 particles of equal volume.
[22] The surface layer thickness, dr, was assumed to be
either (a) a constant or (b) vary with time as [Bland and
Rolls, 1998]:
1
dr ðnÞ ¼ kr nr
ð6Þ
where kr and r are constants (r 1), and n is the cumulative
number of elapsed weathering cycles (a surrogate for
elapsed time).
3.2. Weathering Rate
[23] The weathering rate is parameterized by the fracture
probability, Pfr, the likelihood of a parent fragment fracturing per unit time. Our unit of time is one weathering cycle.
Three fracture probability models were considered:
3.2.1. Constant Rate
[24] The probability of fragment fracturing in a single
weathering cycle is constant and independent of time or
fragment size.
3.2.2. Rate as a Function of Volume
[25] The fracture probability decreases with particle volume. The relationship was derived from Weibull statistics of
brittle failure. Weibull proposed that the survival probability, Ps, for a specimen of volume, V, under tensile stress, s,
is [McDowell and Bolton, 1998]:
m V
s
Ps ¼ exp Vref
sT
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where s is the stress applied to the fragment in the current
weathering cycle, sT is the tensile stress that causes failure
in 63% of rock samples, Vref is the reference particle volume
at which 63% of the samples fail, and m is the Weibull
modulus which decreases with increasing variability in
tensile strength. For chalk, stone pottery and cement m 5
[McDowell and Bolton, 1998]. We assumed that the applied
stress, s, (in our experiments caused by hydration and/or
crystallization during the salt weathering process), is
independent of particle size. Lee [1992, as cited by
McDowell and Bolton, 1998] observed that the mean tensile
strength of the rock particles, sM, was related to particle
diameter, d, in the following fashion:
sM / d c
ð8Þ
where c is an exponent varying with rock type. Lee
observed values of c from 0.35 to 0.42 for some sands
and limestones. Assuming that the fragment is spherical
with a density independent of size, sT / sM and s is
constant, then equation (7) can be re-written as:
Pfr ¼ 1 exp k1 V k2
ð9Þ
where Pfr is the probability of fracture, Pfr = (1 Ps), k1 is a
1
, and is therefore depenfragmentation constant (/ smVref
dent on the rock material and environmental conditions),
and k2 is an exponent (=1 mc/3). In this study we fit k1
and k2. Using the data of Lee, and McDowell and Bolton k2
lies in the range 1.55 – 1.70, but for generality in this study
k2 is restricted to a range of 1 – 2.
3.2.3. Rate as a Function of Time
[26] Wells et al. [2006] observed that weathered cubes
were increasingly susceptible to fracture as weathering
progressed. To model this affect we assumed that each rock
fragment possesses a fixed number of sites, S, which ensure
the structural integrity of the rock sample, but which are
vulnerable to weathering. After n cycles a fraction, (wS), of
those sites have failed. If the rate at which vulnerable sites
are weathered is proportional to the number of S sites
remaining then
d ðwS Þ
¼ k3 ðS wS Þ
dn
ð10Þ
or
dw
¼ k3 dn
ð1 wÞ
solving for w while noting that w = 0 when n = 0 leads to:
w ¼ 1 expðk3 nÞ
where k3 parameterizes the rate of decrease of vulnerable
sites.
[27] If it is assumed that the probability of fragmentation
is linearly related to the fraction of weathered sites then:
ð7Þ
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Pfrn¼n ¼ Pfrn¼0 þ 1 Pfrn¼0 w
ð11Þ
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WELLS ET AL.: MODELING ROCK WEATHERING PATHWAYS
Figure 4. A comparison of the evolving fragment mass
distribution from (1) an analytical solution and (2) a Monte
Carlo simulation (200 simulations). Symmetric fragmentation, Nfr = 2, Pfr = 0.5. —: Analytical prediction; Monte
Carlo Prediction: 6, >50%; &, 25– 50%; 4, 10 –25%; u,
1 – 10%; 5, <1%.
n=n
where Pn=0
are the probabilities of fracture at the
fr , Pfr
commencement, and cycle n, of the weathering sequence
respectively. The fracture probability can then be expressed
as the following function of elapsed weathering cycles:
Pfrn ¼ Pfrn¼0 þ 1 Pfrn¼0 ð1 expðk3 nÞÞ
ð12Þ
[28] In this study both k3 and w were fit.
[29] It should be noted that this model parameterizes
weathering rate as a function of time by one particular
mechanism. Other physical mechanisms that lead to a time
varying weathering rate, (e.g., cumulative build up of
crystallized salt in the rock fragment), can be postulated
and they may yield different trends with time. We have not
investigated alternative mechanisms for temporal trends
because we believe that our data are insufficient to differentiate competing hypotheses for time varying weathering
rates.
4. Fragmentation Model Validation Procedure
4.1. Cube Experiments
[30] For the simpler weathering geometries and fracture
probability scenarios, the mass distributions of fragments
generated in the rock cube experiments were determined
directly from analytical expressions (Appendix A, Figure 4).
For more complex models, the rock material was partitioned
into 400 size classes and a Monte Carlo simulation performed to determine an average temporal evolution of
fragmentation. Figure 4 shows a comparison between one
of the analytic solutions and a Monte-Carlo simulation. At
the commencement of each weathering cycle a uniformly
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distributed random number (Prand = U(0, 1)) was used to
decide whether a fragment would be fractured (i.e., when
Prand < Pfr). After processing all volume fractions, the new
volume distribution was recalculated. The process was
repeated for each weathering cycle. The average of 200
Monte-Carlo simulations was used to determine the ‘‘average’’ fragmentation. Comparison of analytical profiles with
the averaged numerical simulations suggested that 200
simulations were sufficiently accurate (Figure 4).
[31] The fragment mass distribution profiles for the
different breakdown models were then compared to the
experimental data. There were insufficient data to do a
statistical comparison, so we qualitatively assessed each
model’s ability to predict the following key features:
[32] 1. The dominance of large (>25%) fragments in the
first 10 cycles.
[33] 2. The lack of fines (<1%) production during the first
20 cycles.
[34] 3. The successive ‘‘waves’’ of the intermediate
volume fractions (25 – 50%, 10– 25% and 1 –10%) in the
middle cycles.
[35] 4. The peaking of the intermediate volume fraction at
approximately 30– 60% of the original rock volume.
[36] 5. The presence, or otherwise, of the ‘S’ shaped fines
curve in the latter part of the trial.
[37] 6. The relative positions of the coarse (>50%),
and fine (<1%), fragment curves (important for erosion
modeling).
4.2. Powder Experiments
[38] The unweathered PSD (Figure 2) was used as the
initial PSD in the powder weathering simulations. The
experimental PSD profiles (Figure 2) show a depletion of
particles in the 183– 224 mm and larger size categories, and
a corresponding increase for particles smaller than 183 mm.
Maximum depletion occurred in the 332 – 404 mm size
fraction and maximum increase in the 124 – 151 mm size
fraction. To improve model accuracy the experimental data
were interpolated into 2000 finer size fractions. At the
commencement of each weathering cycle the probability
of fracture, (Pfr), of material in each of the size classes was
determined, and a fraction of the material equal to Pfr was
fragmented. Once each size fraction was calculated the new
PSD was recalculated and used as input for the next cycle.
After the final cycle the volume distribution was aggregated
into the original experimental size fractions classes.
[39] Each fragmentation model was assessed qualitatively
by their ability to:
[40] 1. Match the form, location, and width of the PSD
profile of the coarse material (>100 mm).
[41] 2. Reproduce the ‘debris tail’ (PSD < 50 mm).
[42] 3. Match the cumulative volume distribution curve.
[43] 4. Not produce bimodal, or higher modalities, which
were absent in the experimental data.
5. Results
5.1. Cube Experiments
5.1.1. Symmetric Fragmentation
[44] Table 1 summarizes the performance of each fragmentation model. The best fit to the experimental data was
obtained using symmetric fragmentation with a low fracture
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Table 1. Goodness of Fit Ratings Achieved by the Various Rock Breakdown Modelsa
Probability of Fracture Function
Geometry
1.Symmetric fragmentation
a) Nbr = 2,4,6,8
b) Nfr = fn(size)
2. Asymmetric fragmentation
a = 0.005-0.5
3. Whitworth fragmentation
Nfr = 2,3,4
4. Surface disintegration
a) dr = constant
b) dr = fn (time)
Pf = 1
Pf = constant (<1)
Pf = F (size)
Pf = F (time)
very poor
very poor
very good
poor
poor
not tested
excellent
good
very poor
very good
poor
excellent
poor
poor
poor
very good
very poor
very poor
very poor
very poor
very poor
very poor
very poor
very poor
a
Each model was evaluated by assessing the number of criteria listed in the fragmentation model validation procedure which
are achieved by the model: Very Poor: No criteria achieved; Poor: 1 to 2 criteria achieved; Good: 3 criteria achieved; Very
Good: 4 to 5 criteria achieved; Excellent: all criteria achieved.
number, Nfr = 2, coupled with a fracture probability that
increased with time (Figure 5). The model was able to
simultaneously fit both the decline of the coarse (>50%)
fraction, and the rise of the fine (<1%) fraction. The position
and magnitude of the intermediate size fractions was also
superior to the other models. Higher fracture numbers, (up
to Nfr = 8), produced less realistic simulations. Simulations
with Nfr > 4 did not yield any material in the 25– 50% size
fraction and overestimated early levels of fines (Figure 5).
[45] Less satisfactory fits were obtained when using a
constant fracture probability term. Best fits occurred when
Pfr = 0.20 – 0.25 (Figure 5), however the timing of large
fragment breakdown and the appearance of the intermediate
size fractions was poorly modeled.
[46] Symmetric fragmentation models with a size dependent fracture number Nfr, (equation (2)), also yielded
unsatisfactory fits. The relative percentage of the coarse
and fine fraction curves could not be reproduced without
predicting unrealistically high values for the intermediate,
(particularly the 1 – 10%), size fractions.
[47] The inclusion of a size dependent fracture probability
(equation (9)) resulted in poorer fits, with late fragmentation
rates being drastically reduced (Figure 5) as particles
became finer.
5.1.2. Asymmetric Fragmentation
[48] This model was assessed for a values from a = 0.5,
(equivalent to symmetric fragmentation, Nfr = 2), to a =
0.01, (more akin to surface disintegration). The best match
to the data was obtained for a = 0.5 (Figure 6), indicating
that symmetric fragmentation (Nfr = 2) was a more appropriate model. As for symmetric fragmentation, both constant
and size dependent fracture probability produced poor fits.
Best results were obtained with a fracture probability that
increased with time.
5.1.3. Whitworth Fragmentation
[49] Fracture numbers, (Nfr), of 2, 3 and 4 were investigated. Fragmentation distributions were dominated by the
smaller size fractions (1 – 10% and <1%, Figure 6). Increasing the fracture number increased both the weathering rate
and the dominance of finer fractions.
[50] Low fracture number values and a fracture probability increasing with time again produced the best fit. The fit,
however, was poorer than that observed for the symmetric
and asymmetric fragmentation. While the coarse and fine
fractions were well reproduced, there was a tendency for the
10– 25% and 25 –50% fractions to be under predicted.
5.1.4. Granular Surface Disintegration
[51] All surface disintegration models performed poorly.
Mass distribution profiles were characterized by the early
development of a large amount of fine material (Figure 6)
with intermediate size fractions almost completely absent.
Figure 5. Fit of symmetrical fragmentation model to the
experimental fragmentation data. —: Averaged experimental data; —: Time dependent fracture probability (Nfr = 2,
Pfrn=0 = 0.03, k3 = 0.0105) – excellent fit. ——: Constant
fracture probability, high fracture number (Nfr = 8, Pfr =
0.1) – poor fit. — —: Constant fracture probability, low
fracture number (Nfr = 2, Pfr = 0.22) – good fit. >>>>:
Size dependent fracture probability (Nfr = 2, k1 = 0.1,
k2 = 1) – poor fit.
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[54] A constant fracture probability, Pfr = 0.03, resulted in
the correct positioning of the particle distribution peak
(183– 224 mm). Increasing the fracture probability simply
shifted the peak to lower particle sizes at any given time. A
size dependent fracture number increased the proportion of
finer material but also created multiple peaks (Figure 7). A
time dependent fracture probability, which improved the
modeling of the cube weathering, did not improve the fit to
the powder data.
5.2.2. Asymmetric Fragmentation
[55] The performance of the asymmetric model, (for
0.01 a 0.5), was similar to the symmetric model
with the best fit achieved when 0.2 < a < 0.5. As in the
case of symmetric fragmentation, the main peak was
well modeled while the tail was under predicted. Size
and time dependent fracture probabilities did not produce
any significant improvement.
5.2.3. Whitworth Fragmentation
[56] The main deficiency of the above models was an
under prediction of the fine debris tail. Employing a high
fracture number, Nfr, in equation (4) generates a large range
of particle sizes, and hence should perform better. For
instance Nfr = 10 produces particles ranging in size from
0.3% to 97.5% of the volume of the parent particle. Nfr
values from 2 to 10 were examined. In general a high
fracture number gave a better fit to the fines data (<20 mm),
however the intermediate size particles were over predicted
and coarse particles under predicted (Figure 7).
Figure 6. Fit of asymmetric, Whitworth fragmentation and
granular surface disintegration models to the experimental
fragmentation data. —: Averaged experimental data. —:
Asymmetric model with time dependent fracture term
(Pfrn=0 = 0.03, k3 = 0.010) for various a values, (a = 0.5).
The case for a = 0.4 is also shown (— . —) – excellent fit in
both cases. — —: Whitworth fragmentation model with
time dependent fracture probability, (Nfr = 2, Pfrn=0 = 0.03,
k3 = 0.01) – good fit. . . . .: Granular surface disintegration
model, (Pfr = 1, dr/d0 = 0.025) – poor fit.
5.2. Powder Experiments
[52] No model fulfilled all the criteria for fitting the
powder data although several satisfied 3 out of the 4
conditions. As observed above for modeling cube breakdown, the three cracking models (symmetric, asymmetric
and Whitworth models) all performed better than the surface
disintegration model.
5.2.1. Symmetric Fragmentation
[53] We examined fracture numbers, Nfr, from 2 to 8.
Symmetric fragmentation scenarios with constant fracture
probabilities and low fracture numbers (2 to 3) reproduced
the weathered fragment PSD > 100 mm with a slightly
broader peak width, however the fine debris distribution
was not reproduced (Figure 7). To reproduce the finer
particle population, higher fragmentation numbers were
required. However, multiple peaks were then predicted, a
feature not observed in the experimental data.
Figure 7. A comparison of experimentally derived PSD
—) with that predicted from several the fragmentation
(—
model. — + —: Symmetric fragmentation model with
constant fracture probability (Nfr = 2, Pfr = 0.03). .....:
Symmetric fragmentation model with size dependent Nfr
term (b = 0.9, Vmin = 10, Pfr = 0.01). — . —: Whitworth
fragmentation model with constant probability of fracture
(Nfr = 10, Pfr = 0.03). — —: Granular surface disintegration
model with time dependent fracture probability (dr = 10 um,
Pfr = 0.01).
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WELLS ET AL.: MODELING ROCK WEATHERING PATHWAYS
5.2.4. Granular Surface Disintegration
[57] Once again the surface disintegration models performed poorly (Figure 7). Surface disintegration generated
a localized peak of fine material at 10 mm with very little
material in the 10– 100 mm range. Again, this reflects the
propensity of the surface disintegration mechanism to create
weathered daughter particles in a very narrow size range that
is determined by the depth of the disintegrating surface layer.
6. Discussion
[58] The results of this study show that the evolution of
the fragment mass distribution produced during the salt
weathering of a quartz-chlorite schist can be well represented using a simple fracture geometry and probability
model. The geometry model indicates the mechanics of the
fragmentation process. The probability term indicates how
the weathering rate varies over time. Characterizing rock
weathering in this fashion allows us to test the applicability
of various rock weathering models using complex PSD
data, which otherwise would be difficult to interpret.
6.1. Cube Experiments
[59] The best fit was achieved by simply assuming that
the rock fragments split in half, (symmetric fragmentation,
Nfr = 2). The schistose nature of the samples [Wells et al.,
2005, 2006] governs the breakdown geometry. Consequently
the results obtained in this study may not be applicable to
other rock types. Salt weathering of rocks types exhibiting
more granular structures, (e.g., sandstones for example),
might be better modeled with granular surface disintegration
or Whitworth fragmentation geometries.
[60] Two conclusions can be drawn. First, the entire
fragmentation process, from weathering of the initial rock
sample to the final production of fines, was satisfactorily
described using a single fragmentation geometry, hinting at
scale invariant behavior. From the limited study here it is
impossible to say whether this is a general finding or
specific to our rock and weathering regime. Data over a
greater range of materials and weathering conditions are
needed to draw more general conclusions. Secondly, the
fragmentation geometry which best described the fracture
process was one of the simplest examined; splitting the parent
fragment into two daughter particles of equal volume. This is
not simply a result of the average daughter particle size for
Nfr = 2 being half that of the parent particle. We used the best
fit parameters for the 50/50 split and simulated the case where
the ratio a randomly varied from 0.1 to 0.9 at any time, while
maintaining a long term average value of 0.5. This simulation
gave a poorer fit to the data than when a constant value of
a = 0.5 was employed.
[61] We can also draw conclusions as to how the rate of
weathering changes over time. Like Goudie [1986] and
Davison [1986], we observed salt weathering rates accelerating with time, possibly as a result of increasing surface
roughness, increasing pore volume, improved access for salt
bearing fluids to the rock interior, and/or salt build up within
the rock particles. In contrast, Goudie and Viles [1995]
noted many factors that may decelerate the fragmentation
rate. Wright et al. [1998], found that >90% of debris
produced during a 60 cycle sandstone weathering trial was
liberated in the first 40 cycles. A slowdown over time of the
F01014
deepening of tafoni [Matsukura and Matsuoka, 1991], and
the weathering of coastal bedrock [Mottershead, 1989], has
also been reported. Rivas et al. [2003] and Williams and
Robinson [1998] witnessed a linear reduction over time of the
largest remaining fragment mass, after an initial period with
little change. The deceleration of weathering observed in
these studies may be attributable to pore plugging or preferential removal of more easily weathered rock in the early
stages of the weathering process. Finally, Cooke [1979]
observed a three stage salt weathering disintegration curve:
(1) a initial phase with little or no observable change, (2) a
longer period of rapid disintegration; and (3) a final phase in
which the rate of weathering slows appreciably.
[62] Wells et al. [2006] qualitatively concluded that acceleration/deceleration depended in part on the criteria used
to characterize the weathering process. In Figure 1 the
largest mass fraction (>50%) decreases linearly with time,
while the production of fines (<1%) shows an initial period
of acceleration (0-30 cycles) followed by a deceleration
(30 cycles onwards). Consequently, in this paper we have
used a more general approach to describe the weathering
rate, characterizing the weathering rate by the fracture
probability, Pfr. When modeling the schist fragmentation
data shown in Figure 1, an adequate fit could only be
obtained when the fracture probability increases with time
(equation (12)). One explanation for this behavior arises
from the Wells experimental procedure. In the Wells experiments salt solution was added after each drying phase,
possibly leading to a build up of salt crystals in the particle.
There was evidence of surface efflorescence on some of the
larger particles in the laboratory as well as in the field,
however smaller particles did not show this behavior. Another explanation is that the detrimental effects of previous
weathering of the particle structure outweighed the increased
resistance to fragmentation assumed for smaller particles. In
both cases weathering history is important.
[63] Finally, the modeling methodology in this paper can
also be used to analyze the effect of changes in environmental conditions on the fragmentation process. For instance, Wells et al. [2006] showed that lowering the
temperature cycle from 100°C to 60°C significantly slowed
the fragmentation of schist rock cubes. Geometry parameters obtained from the modeling of the 100°C data in
conjunction with a reduced weathering rate term were used
= 0.03 and k3 =
to model the 60°C data, (Nfr = 2, Pn=0
fr
0.001; Figure 8). The good fit achieved suggests that while
the fracture probability was lower for the 60°C weathering
cycle, the fragmentation geometry was unaffected by the
change in temperature cycle. This result hints at the tantalizing conclusion that an increase in the temperature range
may be a viable method of accelerating weathering experiments if all that is desired is the PSD produced by the
weathering process. The weathering rate, however, would
still need to be determined by other means. This is consistent with the qualitative conclusions of Wells et al. [2006].
6.2. Comparing Cube and Powder Experiments
[64] As noted earlier, the material used in the rock cube
and powder experiments originated from the same source
material and were weathered under similar conditions.
Particle volumes examined in the two studies however
differed by many orders of magnitude, (rock cube fragments
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WELLS ET AL.: MODELING ROCK WEATHERING PATHWAYS
F01014
an additional (second order) fragmentation process is operating which produces a range of very fine debris (perhaps
via a surface disintegration process) when the finer schist
powder is weathered.
[66] A final point of contrast between the two studies is
the role of fracture probability. For the rock cube fracturing,
the initial fracture probabilities of about 0.03 increased over
time to approximately 0.4 after 45 100°C weathering cycles.
Fracture probabilities for the schist powders were of the
order 0.01 – 0.03 and remained constant over 45 cycles. This
represents about a 10 fold decrease in fragmentation rates
for the powder. This reduction is most likely a consequence
of the smaller particle size (equation (9)). The large disparity in particle sizes examined in the two studies (3 to 12
orders of magnitude) relative to the weathering rate (1 order
of magnitude) suggests however that any size dependence
of the fracture probability term is weak, which would
explain why a size dependent fracture probability term did
little to improve the fit of the fragmentation models. A more
definitive conclusion would require further experimental
data for size ranges intermediate between those of the cubes
and powders examined here.
Figure 8. Fit of symmetric fragmentation model to
experimental fragmentation data obtained at lower drying
temperature (60°C max). —: Averaged experimental data.
— — —: Symmetric fragmentation model with time dependent fracture probability (Nfr = 2, Pfrn=0 = 0.03, k3 = 0.001).
0.03 to 3.3 cm3; powder fragments 6 1014 to 6 105 cm3). Despite this large difference, symmetric fragmentation with low fragmentation number (Nfr = 2) was
found to provide the best model of the fragmentation
process in both cases. This suggests that the dominant
weathering mechanism was a scale invariant cleavage.
[65] The symmetric fragmentation model however, was
unable to characterize the <100 mm particle population for
the powders. This may simply be a consequence of the
dramatically differing size scales and measurement techniques employed in the two experimental studies. The powders were sized with a laser diffraction device while
throughout the cube experiments the individual fragment
masses were determined manually. One problem with laser
sizing is that it returns the diameter of an equivalent
spherical particle, so that it can be unreliable for non
spherical particles such as the platy particles generated in
the weathering of schist. It is also possible that very fine
(<100 mm) material was generated in the cube experiments
but was not observed or recorded. Finally, it is possible that
6.3. Implications for Future Work
[67] A key finding of this study is that the rate of
weathering, (modeled by the probability of fragmentation),
and the grading of the weathering products, (modeled by the
fragmentation geometry), can be separated. This result is
consistent with the qualitative conclusions reached by Wells
et al. [2006] who suggested that two sets of his experiments
were consistent, one done using a field temperature range
and one using a more extreme temperature range. The more
extreme experiments produced more rapid weathering,
however the PSD at early times was similar to that which
occurred at later times in the lower temperature experiment.
This tantalizing result suggests it may be possible to
perform accelerated experiments in the laboratory to yield
data on the evolution of the particle size distribution of the
fragmentation products. Sharmeen and Willgoose [2006]
showed that the particle size distribution was essential to the
correct modeling of the impact of weathering on erosion.
[68] The rate of weathering, which is difficult to determine
in the laboratory even with realistic experimental conditions,
might then be determined from field studies of weathered
products (e.g., old mine waste dumps). This would yield a
snapshot of weathering products at a given time rather than
the evolution through time. This combination of laboratory
and field data may eliminate the need to mimic field conditions in the laboratory to determine weathering rates.
[69] We do not believe that all weathering model parameters can be determined from field data. Different models can
yield the same end-weathering product, (i.e., exhibit equifinality), but have different temporal evolution. This suggests
that field data from a material of one age of weathering
exposure is unlikely to provide sufficient data to calibrate the
weathering model. Thus both laboratory and field data are
required to fully characterize the weathering process.
7. Conclusions
[70] This study has shown that it is possible to model the
breakdown of schist rock subjected to an aggressive salt
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WELLS ET AL.: MODELING ROCK WEATHERING PATHWAYS
weathering regime using a simple fragmentation geometry
function coupled with a time dependent fracture probability.
The approach adopted by the authors of using a fragmentation geometry coupled to a fracture probability term
enabled some insight into the fragmentation process to be
gained which would otherwise be lost in the complexities of
the particle size distribution data.
[71] Application of the model to the schist fragmentation
data suggests the following:
[72] (1) The fragmentation geometry is best described by
assuming that each fragment splits into two daughter fragments of equal volume.
[73] (2) Smaller particles, currently unidentified, with
diameters of the order of the schist layer thickness or less,
weather via another mechanism.
[74] (3) The rate of weathering, (as represented by the
probability of fragmentation), increased over time for the
weathering of larger particles.
[75] (4) The fracture probability showed a very weak
scale dependence, with deceases in particle diameter of 3 –
12 orders of magnitude producing approximately 1 order of
magnitude decrease in fracture probability.
[76] (5) The rate of weathering, (in this study represented
by the probability of fragmentation), and the grading of the
weathering products, (modeled using the fragmentation
geometry), can be separated. This suggests that it may be
possible to determine PSD evolution from accelerated
laboratory experiments while using field studies of weathered products to determine rates of weathering.
[77] As stated earlier, the experimental results, (and hence
the model outcomes), may or may not be indicative of less
aggressive experiments, weathering involving other rock
types or field behavior. The main purpose of this paper is to
demonstrate a methodology for quantification of the effects
of weathering in erosion. We have shown that it is possible
to use these data to interpret and develop a systematic model
of the fragmentation process. Sharmeen and Willgoose
[2006] showed how this information can then be used to
examine the relative effect of weathering, armouring and
erosion, as well as the effect of the fragmentation process on
erosion rate, and finally the possible erosion characteristics
of a weathering slope.
[78] There remain significant unresolved issues. These
include: (1) the effect of environmental conditions on the
weathering rate, and (2), the extent to which fragmentation
characteristics are rock type dependent. Nevertheless, we
feel that the approach described here points the way forward
in assessing the long term evolution of the surface erodibility of disturbed hillslopes (e.g., rehabilitated mine sites) and
engineered covers for encapsulation of waste (e.g., hazardous and nuclear waste dumps).
F01014
volume, Vfr. If the rock population begins with a single
rock fragment of volume, Vn=0 then the following analytical
expressions are applicable.
[108] Case 1. 0 < Pfr = constant < 1, Nfr = constant
[109] When both the fracture probability, Pfr, and the
break up number, Nfr, are constant, symmetric fragmentation of a solitary initial rock sample will produce fragments
of n + 1 different individual volumes after n cycles of
weathering. The volumes of the different fragments produced after n weathering cycles (Vnj ), and the total volume,
Wnj , of fragments in each of the n + 1 volume fractions are:
V n¼0
Vjn ¼ j ; j ¼ 0 ! n
Nfr
Wjn ¼
nj n¼0
n! j V ;j ¼ 0 ! n
Pfr 1 Pfr
ðn iÞ!
ðA2Þ
[110] Case 2. Nnfr,j = fn(Vnj ), 0 < Pfr = constant < 1
[111] Where the fracture number is a function of particle
volume, (equation (2)), the volumes of the fragments
produced and the total volume of each volume fraction are:
d
Vjn ¼ V n¼0 ðVmin Þe ; j ¼ 0 ! n
where d = (1 3c)j and e =
Wjn ¼
c
3
j1
P
k¼0
ðA3Þ
(1 3c)k
nj n¼0
n! j V
Pfr 1 Pfr
ðn jÞ!
ðA4Þ
A2. Model 2. Asymmetric Fragmentation
[112] A rock fragment of volume, V, fractures along a
cracked surface to produce 2 sub-fragments of volumes, aV
and (1 a)V.
[113] Case 2. 0 < Pfr = constant < 1, a = constant
[114] When both the fracture probability, Pfr, and the
fracture ratio, a, are constant, asymmetric fragmentation of
j¼nþ1
P
a solitary initial rock sample will produce fragments of
j
j¼1
different individual volumes after n cycles of weathering.
The volumes of the different individual fragments are:
Vjn ¼ ai ð1 aÞj V n¼0 ; j ¼ 0 ! n; i ¼ 0 ! ðn jÞ
ðA5Þ
with the total volume of fragments in each volume fraction
being
nji ðnjÞ n¼0 n!
;
a
V
1 Pfr
ðn j iÞ!j!i!
j ¼ 0 ! n; i ¼ 0 ! ðn jÞ
Wjn ¼
Appendix A
[106] For simpler fragmentation geometries we directly
determined the fragment distribution after a number of
weathering cycles, n, using analytic expressions.
ðA1Þ
Notation
A1. Model 1. Symmetric Fragmentation
[107] A rock fragment of volume, V, fractures along a
cracked surface to produce Nfr sub-fragments of equal
11 of 12
a ratio of daughter fragment volumes in asymmetric fragmentation model (1)
b fitting exponent in symmetric fragmentation
model, (1)
c exponent used to determine particle tensile
strength (1)
d diameter of parent particle (L3)
ðA6Þ
F01014
WELLS ET AL.: MODELING ROCK WEATHERING PATHWAYS
dmin minimum particle diameter (L)
dr layer thickness for surface disintegration (L)
kr coefficient in granular surface disintegration
model (L)
k1 fitting coefficient used in determining fracture
probability as a function of particle volume
(L3)
k2 fitting coefficient used in determining fracture
probability as a function of particle volume (1)
k3 fitting coefficient used in determining fracture
probability as a function of time (1)
m Weibull modulus (1)
Nfr number of particles formed when parent particle
fractures (1)
n cumulative number of elapsed weathering
cycles (1)
Pfr probability of a particle fracturing (1)
Ps probability of a particle surviving intact (1)
r exponent in granular surface disintegration
model (1)
S number of potential failure sites within particle
(1)
V parent particle volume (L3)
Vfr fractured particle volume (L3)
Vmin minimum particle volume (L3)
Vref reference particle volume at which 63% fracture
(L3)
W total volume of particles possessing a given
individual volume (L3)
w fraction of potential failure sites which have
failed (1)
s stress applied to particle (M L1 T2)
sM mean tensile strength of particles (M L1 T2)
sT tensile stress required to fracture 63% of particle
(M L1 T2)
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