Higgins M.D. Quantitative textural measurement in igneous and metamorphic petrology

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processes. Obviously this can only be investigated completely in three dimen-

sions. Unfortunately, with most 3-D analytical methods it is difficult to separate

touching crystals and hence there has been no work on this subject. If the surface

of contact between two grains is not parallel to the section plane, as is usually the

case, then some, but not all, touching grains will be identified in sections.

Clusters of touching grains are frequently observed in volcanic rocks. In

some cases the touching grains are sub-parallel (synneusis). Although it is

possible to quantify this effect by looking at grain edge distance against

differences in orientation, this has yet to be done.

Some rocks contain a touching network of plagioclase crystals that are

visible in sections as a collection of crystal chains (Philpotts et al., 1998). In

three dimensions these chains must be projections of foam-like surfaces of

crystals, but it is easier to continue the use of the term chain. The crystal chains

are revealed most clearly when a block of basalt is melted: the block retains its

shape even when 70% is liquid. Philpotts et al.(1998) identified the chains

manually from reconstructed serial sections and X-ray tomography. There are

many aspects of the chains that can be quantified, but so far the most useful

has been the number density of the chains (Philpotts et al., 1999). This was

determined by counting the number of chains intersected along various ran-

dom straight lines in sections. The number density can be transformed easily

into the mean chain spacing, also called the network mesh size. The orientation

of such chains is discussed in Section 5.8.3.1.

–1.0

–0.5

0.5

Sphene

residual distance

1.0

10 20 30 40

cluster frequency (%)

50 60 70 80 90 100

Figure 6.4 Complete linkage cluster analysis of sphene in a granulite sample

(Jerram & Cheadle, 2000). Here the distance between the crystal centres is

compared to that expected for a random distribution (residual distance). The

dotted line is the random variation tolerance limit. Sphene crystals are more

closely spaced than expected in the frequency 5–15%, which corresponds to

cluster size of 1.5 to 5.1 mm.

6.3 Methodology 209

6.3.3.4 Wavelet transform and Fourier analysis

The wavelet transform is a powerful method to identify ordered structures in

rocks (Gaillot et al., 1997, Gaillot et al., 1999). It can be used to identify the

orientation and spacing of ordered structures and has been discussed in the

section on grain orientation (see Section 5.4.3.3). Fourier analysis of grain

sections has been used to look at grain distribution in sandstones (Prince et al.,

1995). Both these methods examine an image of the distribution of a mineral or

grains, not individual grains.

6.3.4 Extraction of mineral association parameters

6.3.4.1 Mineral point correlations

Morishita and Obata (1995) proposed a rather different way of looking at the

spatial distribution of grains and the relationship between grains of different

minerals. It is based on the frequency of occurrence of pairs of points in a

section with the same or different minerals versus distance. It uses a classified

mineral map, but it is not necessary to define individual mineral grains. The

method can only be applied where there is more than one phase in the rock.

The method could also be applied in three dimensions.

We will consider first a homogeneous, bimineralic rock, for simplicity. If

two points are randomly selected then the unconditional probability that both

are mineral 1 is V

, where V

is the volumetric abundance of phase 1.

Similarly, the probability that both are mineral 2 is V

. The probability that

we have chosen a mixed pair is 2V

. We next consider the conditional

probability of occurrence of mineral pairs at a specified distance D. These

depend on the grain size and spatial distribution of the grains and are denoted

(D) for a pair of points of mineral 1, P

(D) for a pair of points of mineral 1

and mineral 2, etc. For D ¼ 0 the points are coincident and the probability is

equal to the volumetric abundance: P

(D) ¼ V

. Similarly the probably that

the points are different is zero: P

(D) ¼ 0. Where D is very large the probabil-

ities become equal to the unconditional probability, that is P

(D) ¼ V

and

(D) ¼ 2V

. The difference between the unconditional and conditional

probabilities expresses the grain size, degree of order and degree of association

of the mineral pairs. It is useful to define textural parameters that express these

differences and extend the treatment to more than two minerals. For each

mineral i a parameter s

(D) is defined for a distance D:

ðDÞ¼

ðDÞV

 V

210 Grain spatial distributions and relations

For each mineral pair i, j a parameter t

(D) is defined for a distance D:

ðDÞ¼

ðDÞ

At D ¼ 0 all s ¼ 1 and t ¼ 0; at D ¼1all s ¼ 0 and t ¼ 1.

Morishita and Obata (1995) examined a granite and a hornfels using these

methods. They found that s values tend to decrease exponentially with dis-

tance and the slope is a measure of mean crystal size. If the rock was ordered

strongly then the s values should peak at that distance, but this behaviour has

not been recorded. They also found that t values tend to increase monotoni-

cally with distance. Some mineral pairs rise to a peak and then go down,

expressing a spatial association of those two minerals and giving a typical

mineral spacing.

Morishita (1998) developed these measurements further by comparing

s and t values of an ‘ideal rock texture’. In this all crystals are randomly

( uniformly) distributed in the rock. He showed that if a number of assump-

tions are made then the crystal size distribution can be calculated from the

variation in the probability that a point lies within a crystal against the distance

from the crystal centre. However, it is not clear if this technique has any practical

use, as it is not obvious how close natural rocks lie to this ideal texture.

This technique could certainly give interesting information on mineral

associations, data which are not given by other methods. However, the lack

of a readily available computer program to calculate the textural parameters

has hampered the application of the method.

6.3.4.2 Box-counting method

Mineral associations can also be detected using a box-counting method

(Saltzer et al., 2001). This starts with a classified mineral image of a section

of a rock. The image is first divided into four square areas each with an area

of 2

pixels where n is the largest integer possible allowed by the image size.

The value of n is decreased to zero, where each box contains one pixel. For

each value of n the numbers of boxes dominated by two minerals are

tabulated for each mineral pair. This is compared with a synthetically

derived random distribution. A significant difference in the distributions

indicates that the mineral pair is associated. This method does not quantify

the correlation, but just indicates significant associations. The method

does not seem to have any advantages over the method of Morishita and

Obata (1995) and suffers from the same lack of an easy tool for doing the

measurements.

6.3 Methodology 211

6.4 Typical applications

One of the problems with this subject is that there is a large amount of

theoretical and modelling work that produces patterns that resemble rocks

(e.g. Ortoleva, 1994, Jamtveit & Meakin, 1999), but few actual measurements

of the rocks themselves. Therefore, commonly, it has not been possible to

verify if the proposed pattern-making processes actually occurred. In addition,

many of the techniques presented in papers have not been taken up by other

workers; hence the only example is the original paper. In some cases this is

clearly a result of a lack of an easy-to-use computer program. A clear exception

to this trend is the ‘R’ parameter of Jerram et al.(1996).

6.4.1 Metamorphic rocks

Kretz (1966a) examined both the size distribution and position of garnet and

biotite in a schist, and phlogopite, pyroxene and titanite in a marble. He

measured the 3-D position of the crystals by disintegrating the samples with

a small chisel. He found that the spatial distribution of garnet crystals was

random and their size did not correlate with the distance to their nearest

neighbour. Three years later he examined a section of pyroxene–scapolite–

titanite granulite in more detail (Kretz, 1969). He divided up the section into

sub-areas and measured the number density of crystals in each sector. By

comparing the data to the expected abundance he was able to use a chi-squared

test to show that the crystals were distributed randomly. He also showed that

titanite was not associated preferentially with any of the major phases. He

concluded that nucleation of all phases was random. The same section was

re-analysed by Jerram and Cheadle (2000) using their cluster-analysis method.

They found that the larger crystals of scapolite and pyroxene act as ‘domains

of order’ – that is, there are fewer small nearest-neighbour distances than

expected (see Figure 6.4). This method also showed that titanite is clustered.

6.4.2 Igneous rocks

Jerram et al.(2003) have reviewed data on clustering using a graph of their R

parameter against per cent porosity (Figure 6.5). They find that crystals from a

rhyolite laccolith are distributed randomly (see below; Mock et al., 2003), but

that all other examples are clustered. They established a field on this graph for

touching frameworks by the inspection of samples. A good number of clus-

tered examples with known touching frameworks have been used to define this

region, but few non touching frameworks have been analysed, so perhaps

212 Grain spatial distributions and relations

more data are needed to establish this field more securely. Ikeda et al.(2002)

examined two samples of different granites using the method of Jerram et al.

(1996) and concluded that all minerals were clustered, except for biotite in one

sample.

Boorman et al.(2004) examined the positions of orthopyroxene and plagio-

clase crystals in mafic rocks from the lower part of the Bushveld complex using

the nearest-neighbour approach of Jerram et al.(1996). The R against porosity

diagram was developed for spheres, but despite this limitation Boorman et al.

(2004) considered that it could be applied to tabular crystals, such as those

measured here (Figure 6.6). The vector defined by their samples could be the

result of sorting, but does not correlate with the aspect ratio of the grains or the

slope of the CSD, as would be expected. Instead, Boorman et al.(2004) suggest

that this vector represents deformational compaction, similar to that proposed

by Meurer and Boudreau (1998).

The spatial distribution of plagioclase, orthoclase and quartz in a rhyolite

laccolith were examined by Mock et al.(2003). The more tabular feldspars

seemed to show greater variation than the more equant quartz (Figure 6.5).

There was a trend to clustering towards the top of the laccolith, and internal

pulses of magma were distinguished.

0.7

0.9

1.1

1.3

1.5

1.7

1.9

0 20 40 60 80 100

Belingwe komatiite (Joe’s Flow)

Alexo

Kambalda

Vammala

Campbell et al. (1978) experiment

Holyoke Colonnade

Holyoke Entablature

RSDL

Belingwe komatiite

Alexo

Kambalda

Vammala

Campbell

Entablature

Touching framework

CLUSTERED

ORDERED

Non-touching crystals

(random to ordered

away from the RSDL)

Non-touching crystals

(random to clustered

away from the RSDL)

porosity (φ) (% melt)

Belingwe basal chill (Joe’s Flow)

Belingwe

chill

Komatiites

Mock et al. (2003) data

RSDL = Random sphere

distribution line

Holyoke Colonnade

Figure 6.5 Porosity (%) or percent age melt against R value (from Jerram

et al., 2003).

6.4 Typical applications 213

The more complex cluster analysis method of Jerram and Cheadle ( 2000)

was applied by them to two samples. They showed that in a komatiite sample

olivine crystals are clustered with diameters of 0.3–2.5 mm. In the scapolite–

pyroxene–titanite rock examined by Kretz (1969) they found that titanite was

clustered on a scale of 1.8–5.1 mm.

Chains of plagioclase crystals appear to be an important petrological struc-

ture in the thick Holyoke basalt lava flow and other slowly cooled bodies of

similar composition (Philpotts et al., 1999). The chains are a few crystals wide

and consist of crystals up to 0.5 mm long attached in random orientations.

They have been traced by hand from thin sections (Figure 6.7). They have been

quantified by counting the number of intercepts per unit length and appear to

be most numerous at the base of the flow and diminish upwards. The chains

appear to originate in the upper part of the flow as groups of plagioclase

crystals embedded in and surrounding pyroxene oikocrysts (Philpotts &

Dickson, 2000). These crystal ensembles descend by convection to the lower

part of the flow where the pyroxene exsolves and recrystallises into masses of

granular pyroxene surrounded by the plagioclase chains. An investigation of

the chains with a view to determining anisotropy produced by compaction is

discussed in Section 5.8.3.1 (Grey et al., 2003).

Igneous layering is commonly observed and has been the subject of numer-

ous theoretical and qualitative studies (Parsons, 1987, Naslund & McBirney,

1996). However, so far there has been little quantitative work on crystal

% porosity

R-value

Better

sorting

Mechanical

compaction

Deformational

compaction

Overgrowth

RSDL

Aging

Ordered

Clustered

100500

Group 1 opx

Group 4 plag

Group 4 opx

Group 2 opx

Group 3 opx

Figure 6.6 Spatial distribution patterns of orthopyroxene (squares) and

plagioclase (circles) in the lower part of the Bushveld complex (Boorman

et al., 2004). Shaded region is for touching frameworks.

214 Grain spatial distributions and relations

positions in layered plutonic rocks. A good example is the inch-scale layering

in the Stillwater intrusion. Boudreau (1987, 1995) has done a fine job of

modelling the origin of the layering by self-organisation, but there are no

quantitative studies of the grain positions and sizes in the rock.

There have been few studies of mineral associations: all are by the authors

that originated the method and all just indicate significant associations with-

out giving a value to this parameter. The spatial association of garnet and

clinopyroxene in a sample of mantle peridotites was considered to be produced

by mineral breakdown: orthopyroxene reacted to form garnet and clinopyr-

oxene (Saltzer et al., 2001). This was used to reconstruct a pressure–temperature

history for these samples. Morishita and Obata (1995) examined two samples

of hornfels and granite and found that biotite and quartz were spatially asso-

ciated in the granite.

Flow base

Centre of 174 m-thick flow

No chains present in first few metres

2 cm

74.2 m

58.7 m

40.9 m

19.9 m

Increasing mesh size

compacted dilated

Vertical sections

Figure 6.7 Plagioclase crystal chains in the Holyoke basalt flow (Philpotts

et al., 1999). Chains were traced by hand from thin sections.

6.4 Typical applications 215

Textures of fluid-filled pores

7.1 Introduction

Many igneous and metamorphic rocks contain a fluid component or evidence

of the former existence of such a fluid.

Fresh lavas and pumice blocks

commonly contain vesicles, formed by the exsolution of gas from the silicate

fluid. Metamorphic rocks may preserve evidence of partial melt pockets. The

term ‘pore’ is used here to indicate space filled, or formerly filled, with fluid

surrounded by a more viscous fluid (e.g. silicate melt, glass) or crystals. Pores

may be considered as the fluid equivalent of grains or crystals in solid phases.

Fluid inclusions in crystals are excluded here as their textures have yet to be

studied quantitatively.

Pores do not always stay filled with their original fluid during lithification.

For instance, vesicles may be filled with other minerals to become amygdules.

In partially molten silicate rocks the fluid may quench to a glass, or a mixture

of minerals, whose textures pseudomorph the original pores. Hence the por-

osity of a rock, as discussed here, is the original volume proportion of fluid at

the time of interest. For example, the porosity of a lava sample is the volu-

metric proportion of vesicles at the moment of solidification; the porosity of a

partially molten rock is the volumetric proportion of fluid when the texture is

preserved and such a melt may have gas bubbles. Clearly, it is important to

specify what feature is being examined. The porosity is characterised at a first

approximation by the total volumetric proportion of fluid, but like grain size,

it is also possible to quantify the size distribution of pores.

Surfaces are important in many geological processes, as sites of chemical or

mechanical processes. The pores define an internal surface area in a rock. The

outside of grains defines the external surface area, which may be significant for

loose materials. Clearly some materials, such as pumiceous pyroclastic depos-

its will have significant internal and external contributions to the total surface

216

area. In natural materials surfaces are complex, hence the actual value of the

total surface area depends partly on the scale of the analysis and the measure-

ment method (Brantley et al., 1999, Brantley & Mellott, 2000).

Another important associated property is the permeability of the rock. In

many ways it is much more important than the porosity, as this parameter

measures how readily a fluid can flow through a rock. The total permeability

can be measured, in addition to the size distribution of pore channels and

‘throats’ that control fluid flow. Rocks with a high porosity tend to be highly

permeable, but the porosity–permeability relationships diverge as the porosity

goes down.

7.2 Brief review of theory

7.2.1 Permeability

The permeability of a rock is used qualitatively to describe the ability to

conduct fluid: a more permeable rock conducts fluid more easily that a less

permeable rock. Hence the permeability expresses the connectedness of the

pores. The original development of the subject considered the flow of water in

a rock and is expressed as Darcy’s law:

DP=l ¼ m=k

where DP is the pressure drop over a length l,  is the specific discharge or filter

velocity, m is the dynamic viscosity of the fluid and k is the permeability. The

permeability is sometimes expressed in Darcy units, which are equal to about

8

. The permeability of a rock is not necessarily the same in all

directions.

Darcy’s law is only applicable in low Reynolds number (ratio of inertial

forces to viscous forces) flow where energy loss is due to viscous effects. At

higher flow rates the fluid inertia becomes important and another term must be

added to Darcy’s law (see review in Rust & Cashman, 2004):

DP=l ¼ m=k

þ 

r=k

where k

is the viscous (Darcian) permeability, r is the fluid density and k

the inertial (non-Darcian) permeability.

7.2.2 Magmas and partially molten rocks

Movement of fluid with respect to solids is a fundamental process in the

solidification of igneous rocks and the melting of metamorphic rocks. It may

7.2 Brief review of theory 217

occur by compaction (McKenzie, 1984), porous-media convection or in

response to shear. It is especially important in the production of monominera-

lic rocks, such as anorthosite or dunite. The final porosity is commonly

expressed as the ‘trapped liquid’ component and can be determined by chemi-

cal or textural methods. The key property is clearly the variation in perme-

ability during solidification or melting: if the permeability is finite for all

values of porosity, then all the silicate fluid can leave the rock. However, it is

commonly proposed that there is a percolation threshold, below which the

permeability falls to zero. The actual value of such a threshold may be partly

controlled by the interfacial angles of the fluid with the solid phases, which in

turn express differences in interfacial energy (see Section 4.2.3).

Cheadle et al.(2004) have reviewed and investigated the relationship

between textural equilibrium, dihedral angles and permeability using a

geometrically simple numerical model. The simplest models are for settled or

random rigid shapes. These are termed non equilibrated as their shape is

unchanged after accumulation. For these materials the percolation threshold

varied with crystal shape – it was about 3% for spheres and 10% for

cubes (Figure 7.1). This means that there will always be a minimum of 3 to

10% melt trapped in the cumulate. For texturally equilibrated samples, with

dihedral angles less than 408, the percolation threshold was close to zero.

That is, all interstitial fluid (silicate liquid) can escape and monomineralic

rocks can form.

porosity (%)

permeability, k, (mm

–2

)

0102030

–14

–12

–10

–8

(

)

Figure 7.1 Melt permeability against porosity for a crystal–liquid system.

For the settled and random models the porosity is the space betw een crystals

represented as simple geometrical form s. The crystals are allowed to interact

in the equilibrated model, giving dihedral angles, , of less than 408.From

Cheadle et al.(2004).

218 Textures of fluid-filled pores