Earth and Planetary Science Letters 276 (2008) 152–166
Contents lists available at ScienceDirect
Earth and Planetary Science Letters
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e p s l
Post-emplacement melt flow induced by thermal stresses: Implications for
differentiation in sills
Ingrid Aarnes ⁎, Yuri Y. Podladchikov, Else-Ragnhild Neumann
Physics of Geological Processes, Postbox 1048 Blindern, N-0316 Oslo, Norway
a r t i c l e
i n f o
Article history:
Received 12 November 2007
Received in revised form 5 September 2008
Accepted 14 September 2008
Available online 28 October 2008
Editor: C.P. Jaupart
Keywords:
thermal stresses
differentiation
melt flow
sill
D-shape
a b s t r a c t
We present the first steps of a new explanation model for differentiation in sills, using a combination of
geochemical data and field observations, numerical modeling and dimensional analysis. Geochemical data from a
saucer-shaped dolerite sill intruded into the Karoo basin, South Africa reveal a differentiation process which causes
D-shaped profiles. The geometry name is based on the variation in whole-rock Mg-number (Mg# =Mg/(Mg+ Fe))
from floor to roof in a sill; the D-shaped geochemical profiles represent sheet-intrusions with the most primitive
composition (i.e. high Mg#) in its center, and progressively more evolved composition (i.e. low Mg#) towards the
upper and lower margins. The differentiation is reversed compared to the normal differentiation produced by
fractional crystallization (C-shaped profiles). C-shaped profiles are believed to be formed by segregation of crystals
from the magma. We propose that the opposite, the D-shaped profile, may result from melt segregation from the
crystal mush. This is achieved by porous melt-flow through a consolidated crystal network after the main phase of
emplacement, and before complete solidification. We show that a significant flow is feasible under natural
occurring conditions. An underpressure of magnitude 108 Pa develops at the cooling margins due to volume
reduction of the crystallizing porous melt. The resulting pressure gradient is the driving force for the melt-flow
towards cooling margins considered in this work. As a result the margins will be enriched in the incompatible
elements associated with the melt phase, while the center will be depleted. We show that the amount of flow is
primarily a function of viscosity of the melt and permeability of the crystal network, which in turn is a transient
phenomenon dependent on a number of parameters. Diagrams have been constructed to evaluate the feasibility of
substantial melt extraction in terms of these poorly constrained parameters. Data from the Golden Valley Sill and
many other natural occurrences of D- and I-shaped geochemical profiles show a reasonable agreement with our
predictions of melt flow potential, and are thus well explained by the presented model. We conclude that in order
to fully understand differentiation processes occurring in sheet intrusions, we need to account for post emplacement
segregation of melt from crystals, and not only segregation of crystals from melt.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Extensive sill and dyke complexes emplaced in sedimentary basins
are common in large igneous provinces found in several places around
the world, like the Karoo basin, South Africa, and the North Atlantic
igneous province (e.g. Chevallier and Woodford, 1999; Du Toit, 1920;
Planke et al., 2005). Detailed geochemical studies of such intrusive
complexes reveal the complexity of the processes associated with
cooling and crystallization of magmatic sheet-intrusions (Galerne
et al., 2008). Profiles sampled from the floor to the roof of sills
show variations in geochemistry which cannot be explained by the
existing theories for the emplacement and crystallization processes of
intrusive bodies. Some geochemical profile geometries are
repeatedly observed in sills and referred to as D-, I and S-shaped
profiles (Fig. 1) (e.g. Marsh, 1996; Latypov, 2003a and references
⁎ Corresponding author.
E-mail address: ingrid.aarnes@fys.uio.no (I. Aarnes).
0012-821X/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2008.09.016
therein). The geometry-notations are based on the variations in the
whole-rock Mg-number (Mg# = 100 ⁎ Mg/(Mg + Fetotal)) from floor to
roof of a sill. The whole rock Mg# is used because magnesium is
preferentially partitioned into early-forming crystals such as olivine
and pyroxene as compared to iron. A C-shaped profile shows the most
evolved composition (e.g. lowest Mg#) in the center of the sill (Fig. 1). A
D-shaped profile, in contrast, shows the most evolved composition at
the sill-margins and progressively less evolved compositions (e.g.
increasing Mg#) towards the center of the sill. The I-shaped profile
shows no variation throughout the sill and is in general believed to be a
result of closed-system crystallization (e.g. Mangan and Marsh, 1992;
Marsh, 1996). The S-shaped profile is an intermediate shape, with an Ior C-shaped upper part and a D-shaped lower part within the same sill,
resembling a “beer belly”. One of the main problems with the
formation mechanism of D-shaped profiles is that they cannot be
explained by classical fractional crystallization theory as developed for
large plutons. Plutons are known to have mafic margins and felsic cores
(normal zoning), forming C-shaped geochemical profiles, like the
I. Aarnes et al. / Earth and Planetary Science Letters 276 (2008) 152–166
153
Fig. 1. A schematic drawing of a sill at one time-step during crystallization. The end- member geochemical profiles are I, D, C and S.
Skaergaard intrusion (Wager and Brown, 1968; Naslund, 1984). The
formation of C-shaped profiles is interpreted as the result of magmatic
differentiation by fractional crystallization (Wager and Brown, 1968).
The presence of D-shaped profiles in many sills imply that we cannot
directly apply theories developed for large magma chambers (3D
structures of several kilometers thickness) to sheet-intrusions like sills
(essentially 2D structures of commonly 10–200 m thickness).
Several models have been proposed for the formation of D-shaped
profiles, such as crystal settling and convection (e.g. Wager and Brown,
1968), multiple injections (e.g. Gibb and Henderson, 1992; Gibb and
Henderson, 2006), flow differentiation causing phenocryst to concentrate in the center of the sill (e.g. Simkin, 1967; Richardson, 1979; Marsh,
1996), Soret fractionation in combination with in situ crystallization
(Latypov, 2003a,b) or melt-flow into the margins (Cherepanov et al.,
1982; this study). There is no general agreement of one particular model.
We will use our data to discuss the feasibility of these mechanisms.
We here present the first step in developing an alternative formation
mechanism of D- shaped geochemical profiles. The model also explains
Fig. 2. The Karoo basin, South Africa. The Golden Valley Sill Complex chosen for this study is located close to Queenstown in the Beaufort Group.
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I. Aarnes et al. / Earth and Planetary Science Letters 276 (2008) 152–166
the formation of I- and S-shaped profiles, and has the potential of
explaining other common phenomena observed in sills. Differentiation
in a magma may occur when crystals and melt are separated. C-shaped
profiles are believed to be formed by segregation of crystals from the
magma. We propose that the opposite, the D-shaped profile, may result
from melt segregation from the crystal mush. Melt can segregate from its
equilibrium crystals by post-emplacement porous flow, thus causing
differentiation by advection. We explore the possibility for such flow to
be induced by thermal stresses associated with cooling and crystallization of an intrusion. We have developed a numerical model to
constrain the conditions under which a substantial flow may occur. We
assume that a stationary crystal network is formed during cooling and
crystallization, and that melt may flow through this porous network. We
also assume that the rigidity of the crystal network only allows a
negligible volume change from the density change associated with
crystallization of the interstitial melt. The prevented volume change is
accommodated by an underpressure and subsequent flux of melt into
the zones of fastest crystallization along the sill margins.
This paper has three main sections: 1) Presentation of geochemical
and petrological data from a representative D-shaped profile, sampled
from a tholeiitic sill in the Karoo igneous province, South Africa. The
data are provided as a basis for the proposed differentiation model
and are, along with similar profiles reported in the literature, used to
constrain the validity of the numerical model. 2) Presentation of the
numerical model developed to evaluate the feasibility of postemplacement flow under natural conditions. 3) Discussion of the
potential formation mechanisms of D- and I-shaped profiles based on
dimensional analysis, numerical modeling and geochemical data.
2. Geochemistry and petrography of a D-shaped profile
2.1. Geological setting
A large volume of mafic magma was emplaced as sills and dykes in
sedimentary strata during the Karoo volcanism (183 ± 1 Ma ago;
(Duncan et al., 1997). The Karoo intrusive event is a part of the Karoo–
Ferrar large igneous province and has affected nearly two thirds of
southern Africa (e.g. Marsh et al., 1997; Chevallier and Woodford,
1999; Le Gall et al., 2002). Extensive erosion has removed much of the
extrusives and revealed the underlying sill intrusions. The Karoo basin
has not been affected by major tectonic activity following emplacement and the sills have thus kept their original geometry.
The diameter and geometry of the sills are related to the
stratigraphic level of intrusion. The sills forming the larger, subhorizontal structures are intruded at the base of the Karoo sequence,
while the smaller, sills (b10 km in diameter) intruded the upper part
of the sequence (i.e. the Beaufort Group; Fig. 2) and they are also
commonly saucer-shaped (Chevallier and Woodford, 1999). A group of
saucer-shaped sills, the Golden Valley Sill Complex, located near
Queenstown, South Africa, was chosen for this study (Fig. 2).
2.2. Sampling
The Golden Valley Sill Complex was sampled in detail (Galerne
et al., 2008). Geochemical profiles display varying shapes, including
D-, S- and I-shapes. We chose to focus on the profile showing the most
prominent D-shape. This profile is sampled on the western inclined
sheet of the Golden Valley Sill (Fig. 3a). The Golden Valley Sill is the
main sill of the complex. It is saucer-shaped, consisting of an inner,
flat sill which continues into sheets that transgress radially upwards
with a 5–10° angle (Fig. 3a). The diameter across the shortest axis of
the sill is ~ 10 km and the transgressive sheets are on average 100 m
thick, but the thickness varies up to tens of meters along the
transgressive sheets. The sampling of a vertical wall ~ 100 m thick
provided some difficulties. The sampling of the profile was therefore
done in a gully, which made us able to climb and sample with fairly
uniform intervals (Fig. 3b). A sampling interval of 10–15 m was
judged to be sufficient to display the general trend. The thickness of
the inclined sheet in the gully was approximately 70 m. The erosional
level of the inclined sheet is on average less than 1 m. However, at the
profile location some part of the sill roof (less than 10 m) was eroded
away due to the gully-effect. The upper sample of the profile is
therefore located only 65 m above the lower contact of the sill and the
total number of samples is six.
The chemical variations of the presented profile from the Golden
Valley Sill are identical to other reported D-shaped profiles (e.g.
Richardson, 1979; Marsh, 1996; Gibb and Henderson, 1992 among
others). We therefore regard the Golden Valley profile as representative
Fig. 3. a) The Golden Valley Sill with latitudinal and longitudinal coordinates. The Golden Valley Sill is approximately 10 km across in the shortest direction. The profile was sampled
on the west inclined sheet of the sill. b) The profile was sampled in an eroded gully. The upper contact between the sill and the sediments has been eroded away and could not be
sampled. Sampling points are shown with stars.
I. Aarnes et al. / Earth and Planetary Science Letters 276 (2008) 152–166
of D-shaped profiles in general, although we were not able to sample
the very top of the sill. The profile is representative of the process
we want to study, i.e. the mechanism responsible for more primitive
composition in the center of a sill relative to the margins. Furthermore,
the Golden Valley profile makes a good basis for developing and
constraining the numerical model because we have access to a
complete set of information from mapping of the area, collecting the
samples and doing textural studies of thin-sections, along with the
geochemical data.
It is important to note that although D-shaped profiles have been
reported in the literature from sills in several areas, the study of the
Golden Valley Sill Complex shows that a variety of different profile
shapes occur in nature and may be found within a single sill.
Furthermore, our data suggest that D- and I- shapes are less common
(6 and 4 profiles, respectively) than S-shapes or other forms of
“zigzag” profiles (N16 profiles) (data from Galerne et al., 2008).
2.3. Analytical methods
The samples discussed in this paper were analyzed for whole-rock
major and trace element compositions by XRF at the University of
Bergen. The XRF is a sequential spectrometer, Philips PW1404, with
LIF200, LIF220, PE, GE, PX1 and PX5 crystals and a sample changer
with 12 positions, using the wavelength-dispersive system. Major
element compositions are given in weight percent (wt.%) and trace
elements are given in parts per million (ppm). The detection limit for
most of the trace elements is approximately 5 ppm. The accuracy of
the XRF analysis was measured using the USGS reference rock
standards W2 (a basaltic rock) and BCR1 (a monzonitic rock).
Electron microprobe (EMP) analyses of minerals were conducted at
the University of Oslo using a Cameca SX100 with integrated energy
dispersive spectrometer and 5 wavelength-dispersive crystal spectrometers. The focused beam is 2 μm in diameter. Three measurements
were done within a small area at each point chosen for analysis, and
the average was used. For each analyzed mineral in each sample at
least two different grains were analyzed. The detection limit of the
analyses is on average 0.05 wt.%.
155
samples actually have slightly less content of Na2O (Fig. 4b). K2O shows
a clear C-shape (Fig. 4b). The relative abundance of K2O compared to
Na2O is higher at the margins than in the center. K2O behave like an
incompatible element, while Na2O is preferred over K2O in the
plagioclase structure. Ni and Cr, which are strongly compatible
elements with olivine and the pyroxenes, respectively, show identical
D-shaped profiles (Fig. 4c). Ni is ranging from 63 and 69 ppm at the
upper and lower margins, and 84 ppm in the center. Cr is ranging from
269 and 296 ppm to 384 ppm. Zr is a strongly incompatible element,
and thus shows the opposite curvature, with 101 and 96 ppm at the
upper and lower margins, and 87 ppm in the center (Fig. 4d). Y (Fig. 4d)
is also strongly incompatible and shows the same curvature as Zr,
although the curvature is sharper.
Olivine shows no zoning in individual crystals, suggesting that
this mineral has re-equilibrated with the melt over time. The Mg# of
olivine shows a D-shape with the samples at the upper and lower
margins having Mg# of 33.5 and 33.0 respectively, and the central
samples 39.3–41.5 (Fig. 4e). In the olivine profile the central samples
have similar Mg# while the two margin samples deviate strongly.
The pyroxenes show normal zoning from core to rim of individual
crystals although the average compositions vary little through the
profile. The plagioclase (measured in the core of the crystals) tends
to be more anorthitic in the center and more albitic at both margins
(Fig. 4e).
3. Numerical modeling of post-emplacement melt flow
3.1. Presentation of the numerical model
Before we can start to argue whether post-emplacement flow can
cause differentiation or not, we need to test the feasibility of such flow
occurring under natural conditions. In order to carry out the feasibility
study, we have developed a numerical model using the Finite Element
Method in Matlab. The numerical model couples three main
processes: (1) Sill cooling and crystallization; (2) development of
pressure anomalies due to cooling; (3) melt flow driven by pressure
anomalies (Fig. 5). We have chosen a simple case of sill cooling in
2.4. Geochemistry and petrography
The profile samples are medium to fine grained dolerite of
tholeiitic composition. The main phases are plagioclase, clinopyroxene
and orthopyroxene, with some olivine and late-stage oxides. Microscope analysis indicates that the profile has an overall homogeneous
modal composition through the profile. However, small differences in
modal compositions cannot be excluded. Plagioclase is found both as
large crystal aggregates (~ 2 mm) and small groundmass laths
(~ 0.5 mm). Olivine occurs as sub-hedral, single crystals (~ 0.5 mm)
or clusters of generally smaller crystals (~0.2 mm). The pyroxenes are
found mostly as large oikicrysts (~ 2 mm), enveloping and filling the
spaces between olivine and plagioclase laths. The oxides are only
found as oikicrysts (~0.5 mm). Olivine and plagioclase were the first
minerals on the liquidus, preceding the onset of crystallization of
pyroxenes; the oxides were latest. The textures and grain sizes are
overall homogeneous through the profile, although there is a slight
tendency for larger plagioclase groundmass laths towards the center.
Compositional variations along the profile are presented in Tables 1
and 2, and plotted for some elements (Fig. 4). Both strongly
incompatible (i.e. TiO2, K2O, V, Cu, Y and Zr) and strongly compatible
(i.e. MgO, CaO, Cr and Ni) elements show the most primitive
composition in the center and the most evolved compositions at the
sill margins. Elements which are not favored by either the melt or
crystal phases show no particular trends. Mg# varies from 51.9 and
52.2 at the upper and lower margins to 54.8 in the center, giving a Dshape (Fig. 4a). CaO (Fig. 4a) shows a similar trend. Na2O shows a
tendency for C-shape in the four central samples, while the two margin
Table 1
Whole-rock XRF data for the presented profile
Meter above contact
K04
AA-17
K04
AA-16
K04
AA-15
K04
AA-14
K04
AA-13
K04
AA-12
1
%
STD
18
30
39
47
65
Major elements (wt.%)
51.82
SiO2
1.01
TiO2
15.87
Al2O3
Fe2O3 total
11.66
MnO
0.18
MgO
6.44
CaO
10.64
2.18
Na2O
0.68
K2O
0.15
P2O5
L.O.I
0.15
Sum
100.78
52.03
0.96
16.09
11.16
0.18
6.58
10.76
2.26
0.64
0.15
0.09
100.90
51.57
0.95
15.36
11.67
0.18
7.13
10.81
2.12
0.61
0.13
0.02
100.55
51.44
0.93
15.57
11.41
0.18
6.98
10.89
2.14
0.60
0.13
0.00
100.27
52.02
0.92
16.20
10.99
0.17
6.65
10.86
2.23
0.64
0.15
0.11
100.94
52.31
0.97
15.28
11.80
0.19
6.42
10.37
2.18
0.71
0.16
0.19
100.58
0.09
0.08
0.44
0.11
0.00
0.35
0.15
0.40
0.00
6.29
0.00
0.06
Trace elements (ppm)
V
Cr
Co
Ni
Cu
Zn
Rb
Sr
Y
Zr
Nb
286
341
46
76
91
89
15
206
24
88
6
286
384
49
84
95
92
16
196
22
87
8
274
362
48
81
92
89
15
198
21
89
6
273
346
45
77
83
85
14
207
24
90
7
280
269
48
63
98
94
16
198
26
101
7
1.80
1.70
1.20
1.26
0.86
1.65
4.09
0.80
4.25
1.68
6.42
277
296
45
69
102
90
17
204
25
96
7
156
I. Aarnes et al. / Earth and Planetary Science Letters 276 (2008) 152–166
Table 2
EMP analysis giving the composition of the main mineral phases through the presented profile
Sample
K04 AA-17
K04 AA-16
K04 AA-15
K04 AA-14
K04 AA-13
K04 AA-12
K04 AA-17
K04 AA-16
K04 AA-15
K04 AA-14
K04 AA-13
K04 AA-12
Meter above contact
1
18
30
39
47
65
1
18
30
39
47
65
50.93
0.42
0.77
26.44
0.53
16.02
4.60
0.06
0.00
0.02
0.03
99.85
49.52
0.32
0.38
32.74
0.68
13.27
2.27
0.02
0.01
0.01
0.03
99.25
49.74
0.18
0.35
32.23
0.73
14.01
1.76
0.02
0.01
0.01
0.03
99.08
50.08
0.66
1.36
15.52
0.33
13.21
16.90
0.22
0.00
0.14
0.02
98.45
51.07
0.55
1.44
15.21
0.36
13.29
17.32
0.23
0.00
0.16
0.04
99.67
50.32
0.84
1.62
16.20
0.36
12.93
16.84
0.21
0.01
0.04
0.02
99.40
49.78
0.77
1.15
17.44
0.35
12.30
16.12
0.19
0.00
0.03
0.04
98.16
32.99
0.04
0.00
48.09
0.64
17.33
0.12
0.00
0.00
0.00
0.04
99.26
32.94
0.03
0.03
48.26
0.60
17.52
0.15
0.01
0.01
0.02
0.05
99.62
33.62
0.02
0.01
47.28
0.58
18.24
0.11
0.00
0.00
0.00
0.04
99.91
32.41
0.03
0.00
51.91
0.72
14.51
0.12
0.01
0.00
0.01
0.03
99.76
Orthopyroxene crystal-core (wt.%)
SiO2
TiO2
Al2O3
FeO
MnO
MgO
CaO
Na2O
K2O
Cr2O3
NiO
Total
51.67
0.33
0.71
24.73
0.51
18.16
3.61
0.04
0.01
0.02
0.03
99.83
49.73
0.30
0.43
31.50
0.67
14.58
2.07
0.02
0.01
0.01
0.01
99.32
34.65
0.18
0.56
15.18
0.31
12.97
2.70
0.04
0.00
0.02
0.04
66.66
Orthopyroxene crystal-rim (wt.%)
52.41
0.25
0.64
25.58
0.53
18.86
2.41
0.03
0.00
0.02
0.02
100.75
50.36
0.43
0.48
30.04
0.65
15.23
2.65
0.03
0.00
0.02
0.03
99.90
51.42
0.34
0.65
25.05
0.58
17.91
3.47
0.04
0.01
0.02
0.02
99.50
52.64
0.33
1.63
8.47
0.22
17.24
18.62
0.22
0.01
0.47
0.02
99.86
52.49
0.37
1.69
8.90
0.25
16.83
18.79
0.19
0.01
0.29
0.02
99.83
51.92
0.38
1.59
9.31
0.24
16.89
18.48
0.20
0.00
0.13
0.01
99.16
Clinopyroxene crystal-core (wt.%)
SiO2
TiO2
Al2O3
FeO
MnO
MgO
CaO
Na2O
K2O
Cr2O3
NiO
Total
52.76
0.27
1.12
10.01
0.30
15.58
20.00
0.14
0.01
0.08
0.05
100.32
47.18
0.67
3.20
13.44
0.29
12.52
16.11
5.74
0.03
0.21
0.04
99.43
51.34
0.50
1.49
10.98
0.29
15.39
18.58
0.19
0.01
0.19
0.03
98.99
49.19
0.27
0.29
33.25
0.68
13.22
1.61
0.01
0.02
0.00
0.04
98.58
50.03
0.39
0.64
29.59
0.62
14.67
3.46
0.03
0.00
0.01
0.00
99.44
Clinopyroxene crystal-rim (wt.%)
Olivine crystal-core (wt.%)
SiO2
TiO2
Al2O3
FeO
MnO
MgO
CaO
Na2O
K2O
Cr2O3
NiO
Total
50.85
0.31
0.62
28.02
0.61
15.88
3.18
0.03
0.00
0.01
0.02
99.54
50.56
0.79
1.21
18.34
0.37
11.89
16.28
0.20
0.00
0.01
0.01
99.66
49.99
0.63
0.96
17.68
0.41
11.58
17.57
0.22
0.02
0.01
0.02
99.10
Olivine crystal-rim (wt.%)
32.49
0.03
0.01
51.36
0.73
14.47
0.27
0.01
0.00
0.01
0.02
99.41
32.55
0.02
0.00
47.56
0.62
18.19
0.18
0.00
0.00
0.01
0.05
99.19
32.86
0.03
0.01
48.17
0.63
17.53
0.21
0.00
0.01
0.00
0.05
99.50
33.14
0.02
0.01
48.26
0.60
17.56
0.16
0.02
0.00
0.01
0.03
99.81
33.84
0.02
0.01
47.19
0.61
18.76
0.21
0.00
0.01
0.00
0.06
100.71
32.21
0.02
0.01
52.01
0.72
14.38
0.18
0.02
0.00
0.01
0.06
99.62
52.36
0.05
29.04
0.69
0.01
0.13
12.71
4.13
0.28
0.01
0.01
99.43
52.46
0.06
28.61
0.60
0.01
0.04
12.60
4.29
0.31
0.01
0.03
99.02
49.23
0.02
30.87
0.52
0.02
0.03
15.10
2.94
0.18
0.01
0.02
98.93
50.88
0.05
30.19
0.57
0.01
0.04
13.98
3.58
0.22
0.01
0.01
99.56
51.75
0.05
29.76
0.47
0.02
0.04
13.66
3.72
0.20
0.00
0.00
99.66
52.29
0.04
29.53
0.69
0.02
0.02
13.09
4.08
0.19
0.01
0.01
99.97
32.53
0.03
0.01
52.30
0.72
13.81
0.16
0.02
0.01
0.02
0.05
99.66
32.68
0.01
0.00
46.71
0.58
18.42
0.15
0.00
0.01
0.02
0.06
98.66
Plagioclase (wt.%)
SiO2
TiO2
Al2O3
FeO
MnO
MgO
CaO
Na2O
K2O
Cr2O3
NiO
Total
Detection limit (wt.%)
SiO2
TiO2
Al2O3
FeO
MnO
MgO
CaO
Na2O
K2O
Cr2O3
NiO
opx
cpx
olivine
plag
0.02
0.02
0.02
0.06
0.06
0.03
0.03
0.02
0.02
0.05
0.08
0.02
0.02
0.02
0.07
0.06
0.03
0.03
0.03
0.02
0.05
0.08
0.03
0.02
0.01
0.06
0.06
0.03
0.03
0.03
0.02
0.05
0.08
0.02
0.02
0.02
0.06
0.05
0.02
0.03
0.03
0.02
0.06
0.08
I. Aarnes et al. / Earth and Planetary Science Letters 276 (2008) 152–166
157
Fig. 4. Whole rock and olivine compositions showing the D-shaped profile. a) D-shape recognized by Mg#. CaO (5⁎wt.%) is superimposed to show that it follows approximately the
same trend. b) Na O (wt.%) and K2O (3.5⁎wt.%) show a similar trend for the central samples, but deviates at the margin samples where the relative amount of K O is higher. c) The
2
2
trace elements Ni (ppm) and Cr (ppm/4.5) show clear, and similar, D-shaped trends through the profile. d) The incompatible elements Zr (ppm) and Y (4⁎ppm) show opposite trends
of the D-shape, this is in agreement with less evolved melt in the center than at the margins. e) Mg# of olivine and Ca#/2 of plagioclase from the crystal cores. The olivine
compositions in the center are similar (Mg# around 40), but have a strong deviation at both margins (Mg# around 33). Plagioclase is more albitic at the margins and more anorthitic
in the center. Both are following an approximate D-shaped trend.
order to visualize and explain the key aspects of the model. We
consider a 2D sill which may be infinitely long in the lateral direction.
All physical properties are non-dimensionalized and the procedure is
described below. Description of physical parameters can be found in
Table 3. The model and its applications are discussed further in
Section 4 (Discussion).
The model is used to test whether melt displacement may arise
from thermal stresses associated with melt-to-crystal transition, and
to investigate factors controlling the maximum segregation of melt
from crystals. In the model, cooling progresses inwards from the upper
and lower margins, causing a vertical melt-pressure gradient from the
center to the margins. Thus the melt segregation is calculated for
vertical flow in the sill. We would like to point out that in more
complex systems, e.g. large sills, there can be several other possibilities
for melt to flow than just towards the margins, and that in a dynamic
system there may be other driving forces like e.g. buoyancy promoting
lateral or upward moving melt (e.g. Galerne, 2008; Tait and Jaupart,
1992). Heat (step 1) and pressure (step 2) are solved using conduction
type equations for both sill and host-rock. Porous melt flow is solved
using the Darcy equation, and is only solved for within the sill at the
condition of 55–90% crystals by volume (Fig. 1). The maximum melt
segregation potential, or cumulative displacement, is calculated by
integrating vertical melt flow velocity over total time (step 3). In
Section 4 we present maximum melt displacements resulting from
several runs as a function of the most important parameters.
3.2. Equations and non-dimensionalization
The equations are solved on a 2D square grid with a resolution of
100 by 400 elements. We solve the equations only before total
solidification. The boundaries (no-flux condition) are located far away
from the intrusion to assure that they are not influencing our results.
Step 1 in the model is computed using the heat diffusion equation,
AT
¼ KTeff r2 T;
At
ð1aÞ
where T is the temperature and Keff is the thermal diffusivity
coefficient. The del-operator ▿ indicates that the equations can be
solved for all spatial dimensions, however in this study we focus on
Fig. 5. Schematic setup for the numerical model. Cooling and crystallization (1) is linked to development of pressure anomalies (2) which in turn drives a porous flow through the
partly crystallized network (3). The boxes illustrate the processes at one time- step during cooling of the sill.
158
I. Aarnes et al. / Earth and Planetary Science Letters 276 (2008) 152–166
Table 3
Description of physical parameters
Symbol
Description
Units
X
d
P
Tm
Thr
TS
TL
ΔT
t
KT
kheat
Cp
ρ
KH
Χ
µ
β
α
Utotal
ϕ
D
L
Ste
System size
Sill thickness
Pressure
Initial temperature of melt
Initial temperature of host rock
Solidus temperature of melt
Liquidus temperature of melt
Initial difference; Tm − Thr
Time
Thermal diffusivity (kheat/Cp/ρ)
Thermal conductivity
Specific heat capacity
Density
Hydraulic diffusivity (χ/μ/β)
Permeability of crystal network
Viscosity of melt
Isothermal compressibility
Thermal expansion coefficient
Total melt displacement
Porosity (T − TS) / (TL − TS)
Grain size
Latent heat of crystallization
Stefan number L/Cp/ρ / (TL − TS)
m
m
Pa
K
K
K
K
K
s
m2 s− 1
J m− 1 K− 1 s− 1
J kg− 1 K− 1
kg m− 3
m2 s− 1
m2
Pa s
Pa− 1
K− 1
m
%
m
J/m3
–
Note that porosity is a linear function of temperature, where T = TL represents 100% melt,
and correspondingly T = TS represents 100% crystals.
the vertical y-direction. The effective thermal diffusivity accounts for
the latent heat of fusion:
KTeff ¼
KT
for ðTs bTbTL Þ
1 þ Ste
ð1bÞ
KTeff ¼ KT for ðTs NT Þ
Non-dimensional ratio quantifying the effect of the latent heat is
the Stefan number, Ste, given by
Ste ¼
L
ðTL −Ts ÞρCP
where KH is the hydraulic diffusivity, P the pressure, α the volumetric
coefficient of thermal expansion and β the isothermal compressibility.
The first part on the right hand side of Eq. (2b) describes the pressure
diffusion (similar to heat conduction Eq. (1a)); the second part
describes the development of pressure anomalies due to changes in
temperature. The initial pressure is zero because the flow only
depends on the developed pressure anomalies.
In step 3 we use the developed pressure gradients to calculate melt
flow, following Darcy's law for flow in porous media,
Y
v
χ
¼−
rP ¼ −KH rP
β
μβ
ð3Þ
→
where v is the average velocity of the melt and β the isothermal
compressibility. Note that the standard equation is divided by β on both
sides to reduce the uncertainty of having permeability over viscosity (χ/
μ) as a separate parameter in the model. The Darcy velocity is integrated
over time into total length of melt-displacement, U,
U
¼
β
Z
t
0
jvy j
dt
β
ð4Þ
As a consequence of Eq. (3), the left hand side of Eq. (4) is also
divided by β.
The chosen values of physical parameters for Eqs. (1)–(4) (Table 3),
along with initial conditions, will greatly influence the magnitude of
the calculated flow. We introduce dimensional analysis to reduce the
number of variables needed to solve the system of equations, and
thus reduce the uncertainties of the model. The main goal of the
analysis is to identify the key parameters that influence the calculated
total displacement, Utotal.
To non-dimensionalize the equations we choose a set of independent new units (Table 4), ΔT = Tm − Thr , KH (hydraulic diffusion), d (sill
thickness) and α/β (expansion coefficients). These independent parameters are used to non-dimensionalize all other (dependent)
parameters used in the following equations (Table 4). By introducing
ð1cÞ
where ρ is melt density, CP heat capacity and L the latent heat of
fusion per volume (note that KT also include ρ and CP). The initial
conditions are: 1) The sill has an initial higher temperature, Tm, than
the host rock, Thr. 2) The upper and lower margins of the sill (0.2% of
the sill thickness) are initially set to a temperature intermediate
between the initial temperature in the sill and that of the host-rock in
order to account for chilled margins formed during emplacement
(Galushkin, 1997). 3) The thermal diffusivities are unequal for the sill
and the host-rock, due to their different thermal properties. In Fig. 6
we show how the numerical model can be used to solve the heat
conduction equation on the 2D grid for a dimensional case. The initial
values used in this example are of little importance, as it is just one out
of many possible scenarios.
In step 2, Eq. (1) is coupled with pressure through thermal stresses,
dP ¼
α
dT
β
ð2aÞ
as described by e.g. Turcotte and Schubert (2002), p. 172, assuming
isochoric conditions for crystallization. Taking the partial derivative of
Eq. (2a) with respect to time, our hydraulic equation becomes
AP
α AT
¼ KH r2 P þ
At
β At
ð2bÞ
Fig. 6. Typical evolution of temperature in a 2D simulation in a dimensional version. The
boundary conditions are the same as in Fig. 8.
I. Aarnes et al. / Earth and Planetary Science Letters 276 (2008) 152–166
Table 4
Parameters used for non-dimensionalization, and the resulting non-dimensional
parameters
Scale
Independent dimensional parameters
Temperature
Diffusion
Expansion
Length
Latent heat⁎
ΔT = Tm − Thr, initial temperature difference
KH, hydraulic diffusion coefficient
α/β, expansion coefficients
D, sill thickness
Ste, ⁎already non-dimensional
Scale
Dependent non-dimensional parameters
Temperature
System lengths
Diffusion
Time
Pressure
Velocity
Displacement
T′ = T/ΔT
∇′ = ∇/d (i.e. x′ = x/d, y′ = y/d)
K′ = KT/KH
t′ = tKH/d2
P′ = P/(α/βΔT)
v′ = v · d/KH
U′ = U /d
For symbol descriptions see Table 3.
the non-dimensional parameters T′ = T / ΔT, ∇′ = ∇/ d, K′ = KT / KH and
t′ = tKH/d2, Eq. (1a) becomes
AT 0
2
¼ K 0r0 T 0
At 0
ð5Þ
Similarly, by introducing P′ = P/(α/βΔT), Eq. (2) becomes
AP 0
2
AT 0
¼ r0 P 0 þ 0
At 0
At
159
3.3. Assumptions
The numerical model is focused on the main processes associated
with post-emplacement melt flow, and does not attempt to describe
the full natural system. The model cannot predict the geochemical Dshape directly because it is not coupled to chemical evolution of the
system.
The equations are based on well defined physical principles, and
the only major assumption in our model is isochoric condition. In
other words, we assume that thermal contraction or mechanical
compaction of the crystal network is negligible for the area where we
calculate melt flow. The flow is calculated where the crystals form a
stationary, solid network with interstitial melt, i.e. 55–90% crystals by
volume. It is now recognized that magma chambers can be crystallized to a rigid, but not completely solid, crystal mush. A rheological
“locking-point”, i.e. a stationary crystal network, forms when the
volume fraction of crystals exceeds 55% (Marsh, 1996), although for a
plagioclase-rich system a considerable strength of the network can be
reached with as low as 30% crystals (Philpotts and Carroll, 1996). The
stationary crystal network is an essential criterion for the geochemical
differentiation; if the crystals move along with the melt, no
differentiation by flow occurs.
The Peclet number is set to 0, i.e. no feedback to the main
temperature profile due to advection of fluids in the contact aureole or
internal advection of melt. It would be interesting to investigate
further implications of substantial melt-advection on the cooling and
ð6Þ
and by introducing v′ = v · d/KH Eq. (3) becomes
v0 ¼ −r0P 0
ð7Þ
Finally, the maximum total displacement is calculated by introdu→
cing U′ = U /d into Eq. (4)
U0 ¼
Z
t0
0
jv0y jdt 0
ð8Þ
The dimensional analysis gives that the only important parameters
determining the system is K′ = KT/KH and an unknown function of Ste.
Thus, the total displacement becomes
U KH
f ðSteÞ
¼
d KT
ð9aÞ
Varying the ratio KT/KH we find that displacement is a linear function
of KH/KT (i.e. flipped) in the pressure domain (KH dominates) (Fig. 7a).
Flow is expected when pressure build-up is dominating the system, and
cooling is slow, i.e. the pressure domain. The Ste has an influence on
displacement as long as we are in the regime dominated by temperature
(i.e. KT). Varying Ste corresponds to varying proportion of initial crystal
content and thus varying release of latent heat of crystallization. At the
crossover, where pressure becomes the governing factor (and flow is
expected), the data collapses and the coefficient replacing f (Ste) is
approximately 0.06 (Fig. 7b). This gives us the formula,
U KT
¼ f ðSteÞ≈0:06
d
d KH
ð9bÞ
Or
U≈0:06
KH
d
KT
which will be elaborated on in the discussion.
ð9cÞ
Fig. 7. a) Varying the ratio KT/KH gives a linear relationship of KH/KT (note that the ratio
is flipped) with displacement, resulting in Eq. (9a). The range of possible combinations
of parameters is wide, and does not necessarily correspond to conditions occurring in
nature. b) Plotting of Eq. (9a) for different values of Ste gives a data-collapse for values
where pressure (i.e. KH) is more dominant than temperature (i.e. KT). We get the
relationship f(Ste) ~ 0.06 (Eq. (9b)). Ste = 0 corresponds to results with no latent heat.
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I. Aarnes et al. / Earth and Planetary Science Letters 276 (2008) 152–166
Table 5
Values used in numerical calculation (new basic parameters are all 1)
Nondimensional numbers
Dimensional values assumed
Values
(KT/KH)melt
(KT/KH)host-rock
(TS − Thr)/ΔT
(TL − Thr)/ΔT
Length in y-direction
Length in x-direction
Ste
Total time
1 × 10− 6/2.0 × 10− 6
1 × 10− 6/2.2 × 10− 6
(900 − 100) / 1100
(1200 − 100) / 1100
2.5 × sill thickness
5 × sill thickness
300 / (1200 − 900)
1/2.0 × 10− 6
0.5
0.45
0.72
1
2.5 × d
5×d
1
1/KH
crystallization history of sills. However, this study is concentrated on
the implications of melt flow on chemical differentiation. Gravity is
neglected because it is not a triggering mechanism for the flow.
Gravity may, nevertheless, affect the flow pattern in natural systems.
Eq. (1) assumes simplified linear dependency of melt fraction with
temperature in order to calculate the effect of the latent heat of
crystallization. The effect of latent heat is important because it slows
down the cooling and crystallization. Otherwise the cooling would be
at least twice as fast, which would imply less time for melt flow to
occur. Nevertheless we show that for a realistic range of Ste, (where
Ste = 0 corresponds to no latent heat) the effect of latent heat (Ste = 1)
on calculated displacement is negligible (Fig. 7a). The effect of latent
heat varies in the temperature domain where cooling dominates over
flow (Fig. 7b). Ste has a constant value of ~0.06 in the pressure domain
where dominant flow is expected. For simplicity we excluded
temperature dependency on the material properties, i.e. the diffusivity
ratio for the host-rock is constant with exception of the interval of
released latent heat. The validity of these assumptions is discussed in
Section 4.1.
4. Discussion
4.1. Validity of model assumptions
The assumption of isochoric condition is obviously not true for all
stages in the cooling process of a sill. Columnar jointing is, for
example, a strong indication of thermal contraction. The reasoning
behind the assumption of isochoric condition is that the density
change due to temperature changes in a stationary, solid crystal
network is negligible compared to the density change caused by meltto-crystal transition in the interstitial melt due to the cooling. As
crystallization continues in the interstitial melt, the latter “wants” to
contract much more that the rigid crystal network will allow, because
no voids can develop. Since the volume is constant due to the rigidity
of the network, an underpressure develops. To allow for the crystal
network to contract thermally will probably give a minor correction to
the developed underpressure, but would not change the main results
of the calculations. The implication of the isochoric condition for the
host-rock is that pore-fluids want to expand much more than the
surrounding rock due to the increased temperature. The result will be
a hydrostatic overpressure. Evidence for overpressure in the contact
aureole of sills can be observed, for example in vent-structures
associated with sill intrusions (Svensen et al., 2006; Svensen et al.,
2007).
Linearity of Ste is a crude assumption, especially for eutectic
crystallization. However, the effect on cooling time by using a linear
Ste is negligible for melts of mafic composition in comparison with
more elaborated models for the Stefan number (Podladchikov and
Wickham, 1994; Turcotte and Schubert, 2002). Latent heat of crystallization is known to be one of the main processes related to sill cooling
and crystallization. However, we show that processes like release of
latent heat, with only minor contributions compared to conduction,
will have negligible influence on our final predictions. This strengthens our assumption that we have included the most important
processes which will affect porous melt-flow by thermal stresses.
Constant diffusivity independent on temperature is obviously not
true in nature. We could, for example, have made the permeability
evolve with porosity (melt fraction). Melts tend to maintain an
interconnected network even though porosity decreases (e.g. Shirley,
1986). Also, hot inflow of melt may dissolve crystals on its way in order
to maintain local equilibrium thermodynamics, thus sustaining a local
high permeability (Tait and Jaupart, 1992). Thus, the relationship
between permeability and porosity may not be trivial, and by
including it we would have had to include additional assumptions.
Fig. 8. Numerical results of a vertical cross-section of the sill intrusion. a) Temperature profile: the sill is cooling, while host-rock is heating according to the heat diffusion equation,
accounting for latent heat of fusion. b) Melt percentage profile: the phase transition from melt to crystals is progressing inwards as the sill cools. c) Pressure anomaly profile: an
underpressure develops where the sill cools, while the heating of the host-rock results in a positive pressure, according to the constitutive equation of thermal stresses. d) Melt
displacement profile: because the underpressure is larger at the margins (faster cooling) than in the center (slower cooling), melt flows towards both margins.
I. Aarnes et al. / Earth and Planetary Science Letters 276 (2008) 152–166
Likewise, the effective thermal expansion coefficient should vary,
taking into account the density change of crystallization. Instead of
having parameters changing with temperature we solve the problem
by varying our restricted set of key constants systematically by several
orders of magnitude to capture all possible combinations of e.g.
permeabilities and expansion coefficients.
4.2. Application of the numerical model
The main aim of the numerical study is to find under which
conditions we can expect substantial flow and insignificant flow,
respectively. D-shaped profiles can only result from a porous meltflow if our calculations show that substantial flow in sills is possible.
The modeling of one case study uses the values listed in Table 5. It
is important to remember that the results for one chosen set of
parameters is not sufficient to decide whether or not this happened in
the Golden Valley Sill, because the details of the natural occurring
parameters and their time evolution is not well enough constrained by
our data.
The first stage of the modeling is characterized by cooling of the sill
and heating of the host rock as shown in Fig. 8a. With progressive
cooling the crystallization front moves from the contacts towards the sill
center (Fig. 8b). The second stage is characterized by the development of
pressure anomalies, P, due to changes in temperature (Fig. 8c).
Where the sill is cooling an underpressure develops. Where the host
rock is heated (in the contact aureole), an overpressure develops. The
non-dimensional pressure P′ = 0.1 corresponds to a physical pressure of
P ~ 108 Pa, for the chosen values of α = 10− 5 K− 1, β = 10− 11 Pa− 1 and
ΔT = 103 K (see Section 3.2). The expansion coefficients used are typical
values for rocks (e.g. Turcotte and Schubert (Turcotte and Schubert, 2002,
p. 172). The pressure anomaly is of the same magnitude as other studies
associated with magmatic processes (Bachmann and Bergantz, 2006;
Voight et al., 2006). The uniaxial compressive strength of fresh basalt
exceeds 250 MPa which is sufficient to support our estimates of pressure
anomalies (~100 MPa) (Hoek and Brown, 1997). This suggests that for
the system to re-equilibrate from the large developed thermal-stress
anomalies, it is favorable to induce melt-flow as opposed to deform the
rigid network. However, we cannot exclude some degree of compaction
in the partly crystallized regions.
There is a strong gradient going from the heated host-rock into the
low-pressurized sill margins, but we only consider flow within the sill.
The cooling in the central parts of the sill is minor compared to that at
the margins, thus there is little or no anomaly developing in the
center. The melt is “sucked” into the low-pressure margins from the
less affected central parts as the cooling front moves inwards. The
resulting displacement, U/d, is shown in Fig. 8d. The maximum total
displacement corresponds to 20 m in a 100 m thick sill.
Twenty meters of displacement is a measure of the potential
change in concentration. Advection can only change the composition
if there is a gradient in the melt composition, which is likely to be
present due to the thermal gradient and different degrees of
crystallization. We evaluate the instantaneous rate of melt extraction
(Eq. (7)) and integrate over time in order to get maximum possible
melt displacement from a given point (Eq. (8)). We therefore do not
trace the exact trajectories of the melt patches, but evaluate extraction
potential of melt at the given point. The sensitivity of extraction
potential to the variations in the input variables (e.g. the host-magma
heat diffusivity ratio) that vary within one order in magnitude is not
considered here.
Commonly, in studies dealing with magma crystallization, melt
pressure is assumed lithostatic and thermal stresses are not
considered. There are exceptions, however, where non-lithostatic
and coupled models for pressure evolution are used in e.g. numerous
studies of fluid circulation in contact aureoles (Delaney, 1982; Barton
et al., 1991; Podladchikov and Wickham, 1994; Ingebritsen and
Sandford, 1998) among others. Some studies document significant
161
transient pressure anomalies, which are comparable in magnitude to
the lithostatic pressure. In the specific context of dike or sill intrusions
these were discussed by e.g. Litvinovski et al. (1990); Tommasini et al.
(1997) and Jamtveit et al. (2004). Litvinovski et al. (1990) documented
quartz grain crushing and melting of the host-rock, combined with
contact melt injecting into intruded magma, as evidence for increased
pressure in the contact aureole. Tommasini and Davies (1997) also
argue for pressure rise in order to explain contact melting. Jamtveit
et al. (2004) focus on explaining vent overpressure associated with sill
intrusions. All these previous studies are focused on transient pressure
anomalies developing in the contact aureoles. In this study we
examine pressure anomalies developing within the crystallizing
magma.
4.3. Model feasibility using dimensional analysis
To test the feasibility of the model, we have to relate the relationship
obtained in the dimensional analysis to physical parameters within a
feasible range of values. Rewriting Eq. (9c), the relationship between
displacement and the diffusivity ratio becomes
U
χ
≈0:06
d
μβKT
ð10Þ
Two of the least constrained parameters in Eq. (10) are the
permeability of the crystal network, χ, and the viscosity of the melt,
μ (parts of KH, Table 3). These parameters indeed may vary by
several orders of magnitude within one magmatic system, depending on factors like temperature and crystal size. We choose the
values for thermal diffusivity (K) of 10− 6 ms2 s− 1 (Delaney, 1988),
isothermal compressibility (β) of 10− 11 Pa− 1, and put them into Eq.
(10) to get
U
χ
¼ 6 1015
d
μ
ð11Þ
which shows how total melt displacement responds to different
values of viscosity (μ) and permeability (χ) in the calculations. This
relationship is plotted in Fig. 9, where the contours represent the
magnitude of the displacement. We set a rough transition zone
between the D-shape and I-shape regimes at above 10% of the sill
thickness (U/d N 0.1).
The results (Fig. 9) suggests that in a melt of basaltic viscosity (i.e.
~102 Pa s), we have significant flow when the permeability of the crystal
network is above 10− 14 m2, corresponding to ~99% crystallinity of a
basaltic rock (Hersum et al., 2005) showing flow is feasible for natural
occurring conditions.
S-shaped profiles, or other intermediate shapes, probably form in the
transition zone between the D-shape and the I-shape regime, requiring
less melt flow than D-profiles. There is also a possibility that the Sshaped, or beer belly, profiles are a combination of porous melt
segregation with some other processes, like compaction. In general,
other formation mechanisms for the S-shaped profile cannot be
excluded by the results in this study. However, only the classical and
well documented process for magma evolution in large plutons, that is,
gravitational settling, is needed in addition to the process advocated in
this study to explain the formation of more complex shapes. Indeed
downward settling combined with D-shape-porous flow would result in
a beer belly-profile.
4.4. Formation mechanisms of D-shaped geochemical profiles
The D-shaped and I-shaped geochemical profiles are end members
used to categorize observed profiles. The D-shaped profile represents
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I. Aarnes et al. / Earth and Planetary Science Letters 276 (2008) 152–166
Fig. 9. The response in melt displacement for different viscosities and permeabilities. A
melt of basaltic viscosity requires a permeability of the crystallizing network having a
magnitude above 10− 14 m2 to be able to create a substantial flow.
the process we aim at understanding, i.e. why a sill can have the least
evolved composition in the center and the most evolved composition
at its upper and lower margins.
Petrographic observations from the Golden Valley Sill show no
evidence for larger concentrations of phenocrysts in the center of the
sill as compared to its margins. The occurrence of D-shaped profiles in
phenocryst-poor melts has previously been pointed out as a weakness
of the flow differentiation model (e.g. Latypov, 2003a,b). In addition
Barriere (1976) has shown that crystal segregation is practically
inoperative for sills with thickness of 100 m or more. Olivine is only a
minor phase, so the phenocryst assemblage consists mostly of
plagioclase. Plagioclase does not partition MgO from FeO and can
therefore not be responsible for the observed Mg# profile. Thus, flow
segregation is unlikely to be the major differentiation mechanism,
although some variations in the phenocryst distribution cannot be
excluded.
There is no evidence for a second intrusion in the Golden Valley
Sill. The samples are all homogeneous, and the geochemical profiles
show no abrupt changes. The field observations strongly suggest one
intrusion causing the Golden Valley Sill. Where the Golden Valley Sill
is in direct contact with another sill there is clear evidence of two
separate intrusions, both by observations and geochemistry (Galerne,
in press). We cannot exclude the possibility of continuous infill,
although this requires a source of continuously more primitive magma
in order to explain the D-shape. However, D-shapes are occurring in
lava flows where there is clearly one single pulse (Latypov, 2003b and
references therein).
Soret fractionation (thermal diffusion) is a process that causes the
heavy components (e.g. Fe) to migrate toward the colder end of a
thermal gradient, and the lighter components (e.g. Mg) to migrate
towards the hotter end. The geochemical data on the Golden Valley Sill
show strong evidence against Soret fractionation. If heavy elements
migrate towards the cooling margins, we would expect the same trend
for all elements of differing molecular weight. In contrast, Ca increases
towards the center of the sill and Na towards the margin, although Ca
is heavier than Na. Also Ni and Cr are enriched in the center although
the molecular weights are within the same range as Fe (Fig. 4).
The presented model of post-emplacement flow explains well the
symmetrical behavior of the geochemical data, and strengthens the
assumption that gravity is not an important factor. Gravity settling is
expected to result in more asymmetric profiles (e.g. Gibb and
Henderson, 1992). The overall efficiency of gravity settling as a
differentiation mechanism in sills is largely questioned (Shirley, 1987;
Marsh, 1988, 1989). In large magma chambers, however, settling of
crystals behaving as effective dense plumes may occur (Brandeis and
Jaupart, 1986).
It is widely recognized that the main cause of melt differentiation
occurs when melt and crystals are segregated, for example by crystal
settling, or by compositional convection (Kerr and Tait, 1986; Tait and
Jaupart, 1992; Jaupart and Tait, 1995). Post-emplacement melt-flow
has the potential of causing differentiation by advective transport
(Fig. 10a). An outwards flux of melt from the central parts of the sill can
increase the content of compatible elements in the center, and the
content of incompatible elements in the distal parts. The resulting
profile of whole-rock geochemistry will be D-shaped (Fig. 10b).
Normal zoning observed in the pyroxenes suggests that fractionation of the melt occurs continuously throughout the whole magma
body implying that melt becomes increasingly enriched in incompatible elements compared to the crystals. This favors the presence of a
compositional melt-gradient from the center to the margins. For
example, plagioclase is more anorthitic in the center and more albitic
in the margins. The proportion of olivine may be too small to influence
the whole-rock Mg# significantly, although the olivine crystals also
Fig. 10. a) A schematic drawing of the flow process at one time-step during crystallization. Early forming crystals are mainly plagioclase and do not affect the Mg#. Melt flows from the
central parts and into the margins, thus depleting the center and enriching the margins in incompatible and less incompatible elements, and vice versa for the compatible elements.
The resulting geochemical profile will be according to b).
I. Aarnes et al. / Earth and Planetary Science Letters 276 (2008) 152–166
show the D-shaped trend with the largest changes towards the
margins. This is in agreement with the pressure gradients being
largest at the margins where cooling is most rapid, thus giving the
strongest driving force for the melt flow. Assuming that we start with
a homogeneous intrusion, no flow will result in an I-shaped profile
(closed system crystallization).
Melt can ultimately be drawn from areas that have less than 55 vol.
% crystals, and mechanical compaction or inflow from other parts of
the sill will make sure no voids develop where the last melt extraction
occurs. If the entire sill, contemporaneously and in all directions, has a
crystal-content above the limit of mechanical compaction, the sill will
have a relatively flat cooling gradient. Hence, there will be no strong
internal pressure gradients. In such settings post-emplacement flow
should not be expected.
There are several studies suggesting that reactive-advective meltflow will cause changes in geochemistry (McKenzie, 1984; Navon and
Stolper, 1987; Spiegelman, 1996; Hauri, 1997; Spiegelman et al., 2001;
Steefel et al., 2005). Weinberg (2006) stresses that segregation
processes are very important for our understanding of the physical
processes that lead to chemical and mineralogical differentiation in
magmas. A study by Johnson et al. (2003) suggests that also the
thermal, compositional and rheological gradients in the host-rocks
may influence directions of pluton expansion. Rabinowicz et al. (2001)
ask for physical processes that are capable of inducing strong
variations in the melt/rock ratio.
4.5. Application to the Golden Valley Sill
The result of the numerical modeling (Fig. 9) can be applied to the
Golden Valley Sill, and other natural systems in terms of measurable
quantities. The permeability, χ, of the crystal network is a function of
grain size and porosity (i.e. melt percentage), which in turn are
functions of cooling time. To get a rough measure of permeability
expressed in average grain size we use the Carman (1956) relationship
2
χ¼
D
/3
114 ð1−/Þ3
ð12Þ
—
where 114 is a fit parameter (Hersum et al., 2005), D is mean crystal
length in meters and ϕ is porosity of the crystal network (melt
fraction). We set the porosity to 10%, i.e. 90% crystals. Viscosity is a
function of the chemical composition of the melt. We use the model of
Giordano and Dingwell (2003) to describe the viscosity of multicomponent silicate melts. Viscosity is linked to composition by
log10 μ ¼ c1 þ
c2 c3
c3 þ SM
163
A representative I-shaped profile from another part of the Golden
Valley Sill has an average grain size of ~ 0.5 mm and a SM number of
~25 (Galerne et al., 2008). A D-shaped profile from a sill underlying
the Golden Valley Sill has an average grain size of ~0.75 mm and an
average SM number of ~ 32 (Galerne et al., 2008). The local variations
from sills that are overall homogeneous seem to be sufficient to
separate between substantial flow and insignificant flow. Thus, the
model can explain occurrences of different profiles within one sill by
varying degrees of flow.
In Fig. 11 we have also plotted examples of D- and I-shaped profiles
reported in the literature. Richardson (1979) reports a D-shape from
the Tandjiesberg sill (80–110 m thick), South Africa. The SM number
~26 is calculated on basis of bulk compositions and average of solid
(from Table 5 in the paper Richardson, 1979). Grain sizes vary from 0.2
at the margins to 2–3 mm in the center. An average of ~1 mm is used.
Gunn (1966) gives data from a D-shape (with some tendencies
towards S-shape) from the Lake Vanda sill (332 m thick), Antarctica.
The SM number varies from ~ 25–32 (Table 8 in the paper of Gunn,
1966), and the average is used. Reported grain sizes are 1–4 mm. We
have used 1.25 mm. Gunn (1962) gives data for a profile with limited
variation (I-shape) from the Peneplain sill (330 m thick), Antarctica.
The SM number is ~23 (at the center). Crystal sizes vary from 0.4–
1 mm. An average of 0.7 mm is used. Gibb and Henderson (2006)
published data from the Shiant Isles Main Sill, Scotland, composed of
multiple intrusions. One batch of crinanite shows an approximate Ishape, while a lower batch of picrite shows a nice D-shape. The SM
numbers of the two batches are clearly different, the I-shape having a
SM number of ~27 and the D-shape having a SM number of ~ 41. The
crinanite has an average grain size of b0.75 mm (Gibb and Henderson,
1996). We have visualized the crinanite data for a grain size of
~0.6 mm. The grain size for picrite is reported as “large”, hence a
dashed star is used at ~ 0.8 mm, as a conservative value.
The natural examples fit very well with the predictions of the
model. The examples were all from relatively low-viscosity melts of
mafic compositions. A viscous granitic melt with higher SiO2-content
(lower SM) is less likely to segregate from the crystal network,
although evidence for segregation in granitic bodies has been reported
(Weinberg, 2006). It is not obvious that our transition zone between
substantial and insignificant flow is correct. The boundary between
the D- and I-regime should therefore be shifted if other observations
show a significant misfit between observed data and our model. The
ð13Þ
where SM = Σ(Na2O + K2O + CaO + MgO + MnO + FeOtot/2). Lower SM
number corresponds to higher viscosity. c1, c2 and c3 represent
adjustable parameters,
c1 ¼
−17:80106 þ 0:018708103T ð-CÞ
1−2:2869 10−3 T ð-CÞ
c2 ¼ 0:02532 þ 2:5124expð−6:3679 10−3 T ð-CÞ
þ40:4562 10−6 T ð-CÞ−1
c3 ¼
:
−3
1−1:6569 10 T ð-CÞ
0:017954−63:90597 10−6 T ð-CÞ
The result of these conversions is plotted in Fig. 11, and can be used
directly to compare with natural cases. The samples in this study have an
average grain size of 1 mm, and an average SM of ~31. Thus, the profile
plots within the regime of significant flow where D-shape is feasible.
Fig. 11. Permeability as a function of grain size and viscosity as a function of
composition. The probability of getting a significant flow increases with longer
cooling time and lower silica content. For 10% porosity the various occurrences of D- and
I-shaped profiles plot within the domains of significant and no flow, respectively,
showing a good agreement with our predictions.
164
I. Aarnes et al. / Earth and Planetary Science Letters 276 (2008) 152–166
Table 6
Calculations of bulk composition in the margin and center using Eqs. (14a) and (14b)
Method
Element
Xinitial
Xmargin
Xcenter
Calculated 14a
Calculated 14b
Data
Calculated 14a
Calculated 14b
Data
Zr
Zr
Zr
Y
Y
Y
92
92
101
101
101
26
26
26
87
87
87
21
23
21
24
24
10% influx to the margins and 5% outflux of the center correspond very well to the data
from the Golden Valley.
model predicts that flow is likely to occur under natural conditions.
However, prediction of the geochemical effects of such flow in nature
is not trivial. Such prediction can only be made when a proper closed
set of equations coupling flow, mass balance, thermodynamic
equations and element partitioning has been established.
Some recent progress has been made in the direction of developing
a fully coupled set of equations solving the scenario of differentiation
due post-emplacement melt flow (Tantserev, 2008), but this is still
work in progress. We will therefore provide simple mass balance
calculations for the incompatible elements Zr and Y in order to
illustrate the possibility of melt flow causing chemical differentiation.
We calculate the bulk concentration X of the element by weighting the
fraction F of the phase and concentration C of the element in that phase
Xmargin ¼ Fc Cc þ Fm Cm þ Fin Cin
Xcenter ¼ Fc Cc þ Fm Cm −Fout Cout
ð14aÞ
with indexes c, m, in and out indicating crystals, melt, influxed and
outfluxed melt, respectively.
For comparison we adapt the equations of specifically developed
for differentiation by infiltration by Jellinek and Kerr (2001),
Xmargin ¼ ðFin e þ 1ÞCm
Xcenter ¼ ð/ þ ð1−/ÞDÞð1−Fout eÞCm
ð14bÞ
where ε is enrichment factor of the moving melt and ϕ is melt
fraction.
The results and data are shown in Table 6. We let the average
compositions represent the bulk composition of the melt before
differentiation (I-shape), Cm = Xinitial. We use an influx of 10% melt to
the margins, Fin = 0.1, and 5% outflux from the center Fout = 0.05. We
assume that the elements are perfectly incompatible in the crystal
network of plagioclase and olivine, i.e. Cc = 0 and D = 0. We use an
enrichment factor of 1ε = 1, and ϕ = 1, i.e. pure melt, as a conservative
value.
Our results fit the data very well, and show that the compositional
trends can readily be explained by 10% net influx of melt at the
margins, and 5% net outflux in the center. For fluxes of more enriched
melt, ε N 1, which is to be expected in nature, the differentiation trend
is calculated to be more pronounced than what the data shows.
However, a retardation factor keeping the local thermodynamic
equilibrium would slow down the differentiation and requires more
melt to see the same trends as observed. It is important to note that
these calculations are not showing the full “truth” of what is actually
going on. The estimates show that a feasible amount of melt
segregation is needed to explain our data.
An interesting implication of the formation of pressure anomalies
in and around sills is that this may explain the occurrence of
sandstone-dykes intruding into the sill roof of the Golden Valley Sill,
a phenomenon described by Planke et al. (2000) and Van Biljon and
Smitter (1956). The heating results in a large overpressure in the
contact aureole, and combined with the underpressure developing in
the sill roof, there is a strong driving force for sucking fluidized
sediments into the sill as soon as a crack develops.
There are still unresolved problems associated with the cooling
and crystallization of sheet-intrusions exemplified by the variety of
profile-shapes within the limit of one sill. The proposed postemplacement flow-model may explain some of these varieties by
varying degrees of melt displacement, and thus differentiation. In our
numerical model we have demonstrated that thermal stresses
develop in the cooling margins of the sill (Fig. 8), but the margins
are not necessarily the only domains where thermal stresses can
develop in an extensive intrusion like the Golden Valley Sill. A several
kilometers wide saucer-shaped sill can have local differences in
degree of cooling and crystallization both in lateral (or sub-lateral)
and vertical directions. Melt may also be transported laterally or sublaterally, e.g. by flow of melt from hotter parts closer to the feeder
towards cooler parts of the sill (Fig. 12). Dissimilar flow patterns in
different parts of a sill open for the possibility that different
geochemical profiles may develop in different parts of a single sill.
Different chemical profiles are indeed found in different parts of the
Golden Valley Sill and in the York Haven sheet intrusion (Mangan
et al., 1993; Galerne, pers. comm.).
Fig. 12. a) Scenario of lateral flow. More melt is leaving from the center than the margins due to longer cooling time. b) More differentiation is occurring in the center due to more flow.
Margin samples correspond to the original composition in this scenario.
I. Aarnes et al. / Earth and Planetary Science Letters 276 (2008) 152–166
Post-emplacement melt flow may explain geochemical variations
in several magmatic intrusions were former models have failed. With
very few assumptions we have shown that post-emplacement flow is
feasible under natural conditions, and also compatible with geochemical observations from several sills and the formation of D-shaped
profiles.
165
grain sizes); 5) data from this study combined with other studies
coincide well with the predicted conditions for substantial and
insignificant flow. Our results strongly suggest that post-emplacement
melt flow is an important, and previously overlooked, factor in
differentiation of sill intrusions.
Acknowledgements
4.6. Geological significance
Zones adjacent to the basal contact of large intrusions are
expected to be fully crystallized in situ before the onset of
compositional convection. If negligible mass transport occurs, the
same final rock composition should be found at all levels (I-shape).
The basal zone of the Stillwater Complex (Page, 78, Jaupart and Tait,
1995) becomes progressively enriched in pyroxene and olivine up
from the contact (D-shape). This progression may be interpreted as
recording the advection of fresh melt towards more crystallized
cooling margins (see Jaupart and Tait, 1995). For large-scale
differentiation to be efficient a large amount of melt is expected to
“flush” through the system. An extensive influx of melt towards the
margins would require a corresponding outflux. Such outflow is
expected to be localized into narrow zones (Jaupart and Tait, 1995).
Vertical or subvertical segregation veins, pegmatitic structures and
evolved pipe-like bodies are indeed found to crosscut the horizontal
layering in several sills, like the Bushveld intrusion, South Africa
(McKenzie, 1984; Hauri, 1997), the Stillwater Complex, Montana
(Navon and Stolper, 1987), the Peneplain Sill, Antarctica (Spiegelman,
1996), and also in the upper and lower part of 1959 Kilauea Iki lava
lake (Spiegelman et al., 2001; Steefel et al., 2005; Weinberg, 2006), or
as “sandwich horizon” of incompatible elements in the Palisades Sill
(Shirley, 1987). Flow induced by thermal stress, combined with local
buoyancy effects from either composition or thermal differences,
may help to explain these occurrences in the ca.100 m basal zones of
large intrusions. These are not directly linked to formation of Dshape, but confirm that segregation processes are highly present in
sills.
Modal mineral zoning is the ultimate degree of compositional
segregation that can be achieved by reactive melt flow. Chromatographic columns are commonly found in sills (e.g. Jaupart and Tait,
1995; Latypov, 2003b, and references therein), and can also display Dshaped geochemical profiles. The modal layers in such sills can be
regarded as metasomatic columns where the chemical fronts are
formed due to a differential movement of melt relative to crystals
(Guy, 1993; Korzhinskii, 1973). Thermal stress induced by cooling is a
strong driving force with the potential of segregating large amounts of
melts from its equilibrium solids. Thus, post-emplacement flow can
readily explain D-shaped profiles in sheet-intrusions of variable types
and sizes.
5. Conclusions
D-shaped geochemical profiles in sills are formed by a differentiation
process which causes the upper and lower margins to have a more
evolved composition than the center of the sill. Where no differentiation
occurs we get I-shaped profiles. We propose that the differentiation is
caused by a separation of melt from crystals by a porous melt flow
through a rigid crystal network. Thermal stresses associated with the
cooling and crystallization of the sill is a feasible driving force for
substantial porous-flow. Numerical modeling and dimensional analysis
have shown that 1) cooling of sills leads to large underpressure within
the sill, while heating of the host-rock leads to large over-pressure in the
contact aureole; 2) the underpressure can drive a substantial melt flow
after the main emplacement episode of a sheet-intrusion and 3) a
substantial melt flow is feasible under natural occurring conditions, like
the case of the Golden Valley Sill; 4) the main factors controlling the flow
are the viscosity of the melt (i.e. geochemistry) and the cooling time (i.e.
This study was supported by the Norwegian Research Council
(grant 159824/V30, “Emplacement mechanisms and magma flow in
sheet intrusions in sedimentary basins”) through PGP, a Center of
Excellence at the University of Oslo. We thank professor Julian
(Goonie) Marsh, Rhodes University, Department of Geology in
Grahamstown, South Africa, for supplying field equipment, the
farmers whose land we were accessing, and K. Haaberg, S. Polteau
and C. Galerne for assistance and collaboration during and after field
work. H. Svensen, O. Galland, G. Gisler and B. Jamtveit are thanked for
assistance and discussions. We would also like to thank Richard Kerr
and two anonymous reviewers for constructive comments which
led to substantial improvement of the manuscript. Thanks to
M. Erambert for electron microprobe analytical support and R. Latypov
for discussions and useful references.
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