Melt segregation and matrix compaction: closed governing equation set, numerical models, applications
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irk-123456789-1013882016-06-04T03:01:43Z Melt segregation and matrix compaction: closed governing equation set, numerical models, applications Khazan, Ya. Aryasova, O. 2010 Article Melt segregation and matrix compaction: closed governing equation set, numerical models, applications / Ya. Khazan, O. Aryasova // Геофизический журнал. — 2010. — Т. 32, № 4. — С. 62-65. — Бібліогр.: 9 назв. — англ. 0203-3100 http://dspace.nbuv.gov.ua/handle/123456789/101388 en Геофизический журнал Інститут геофізики ім. С.I. Субботіна НАН України |
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Khazan, Ya. Aryasova, O. Melt segregation and matrix compaction: closed governing equation set, numerical models, applications Геофизический журнал |
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Khazan, Ya. Aryasova, O. |
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Khazan, Ya. |
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Melt segregation and matrix compaction: closed governing equation set, numerical models, applications |
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Melt segregation and matrix compaction: closed governing equation set, numerical models, applications |
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Melt segregation and matrix compaction: closed governing equation set, numerical models, applications |
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Melt segregation and matrix compaction: closed governing equation set, numerical models, applications |
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Melt segregation and matrix compaction: closed governing equation set, numerical models, applications |
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melt segregation and matrix compaction: closed governing equation set, numerical models, applications |
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Інститут геофізики ім. С.I. Субботіна НАН України |
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2010 |
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Melt segregation and matrix compaction: closed governing equation set, numerical models, applications / Ya. Khazan, O. Aryasova // Геофизический журнал. — 2010. — Т. 32, № 4. — С. 62-65. — Бібліогр.: 9 назв. — англ. |
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Геофизический журнал |
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AT khazanya meltsegregationandmatrixcompactionclosedgoverningequationsetnumericalmodelsapplications AT aryasovao meltsegregationandmatrixcompactionclosedgoverningequationsetnumericalmodelsapplications |
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2025-07-07T10:51:48Z |
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2025-07-07T10:51:48Z |
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1836985105843748864 |
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Melt segregation and matrix compaction: closed governing
equation set, numerical models, applications
Ya. Khazan1, O. Aryasova1, 2010
Institute of Geophysics, National Academy of Sciences of Ukraine, Kiev, Ukraine
ykhazan@gmail.com; oaryasova@gmail.com
Partially molten systems are commonly mode-
led as an interpenetrating flow of two viscous liquids
and are therefore described in terms of fluid me-
chanics [Drew, 1983; McKenzie, 1984; Nigmatulin,
1990]. In the gravitational field a liquid filling a vis-
cous permeable porous matrix is in mechanical
equilibrium only if its pressure gradient is equal to
the hydrostatic one, and the pressures of the liquid
and matrix are the same. If the liquid and matrix
densities differ, with the liquid forming an intercon-
nected network, the two conditions cannot be sa-
tisfied simultaneously, and the liquid segregates from
the matrix while the latter compacts. The averaged
momentum and mass conservation equations for a
multi-phase medium are formulated separately for
every phase. Considering the energy conservation
equation, this results in 4N+1 equations for a N-
phase medium while the number of unknowns is 5N
(3N velocity components, N pressures, temperature,
and N–1 independent phase fractions). Therefore,
for the problem to be fully determined it becomes
necessary to add N–1 coupling equations. Khazan
(2010; on review at GJI) and Khazan and Aryasova
(Rus. Earth Phys., 2010, in press) derived a gener-
al equation (the mush continuity equation, MCE)
closing the governing equation set. Its simplified 1D
form valid for a two-phase system in the case of
low-melt-fraction mush and linear matrix rheology,
together with equation describing the rate of the ine-
lastic porosity change [Scott, Stevenson, 1986], con-
stitute a closed subset of governing equation set:
pl pmm g
z
pl pk
z
)( ) (
,
pl pm
t
, (1)
where pl, and pm are the melt and matrix pressure,
respectively, is the melt fraction or porosity,
k( ) n is the matrix permeability, = m– l is the
difference between the matrix, m, and melt, l, den-
sities, and are matrix and melt viscosities, cor-
respondingly, n=2 to 3, g is acceleration due to gra-
vity; Z axis points upward. Let L be the thickness of
the partially molten zone, and 0 be the maximum
of the initial porosity distribution. In terms of dimen-
sionless coordinate =z/L, time =t gL/ , melt
overpressure �����pl–pm)/ gL, and porosity = / 0
the equations may be written as
01 c
n , ,
where
2
2
c
L
c , k( 0 )
0
c , (2)
c and c are referred to as the compaction/segrega-
tion parameter and length, respectively. If coordinate
is normalized by the compaction length, the first of
Eqs. (2) does not contain c [Grègoire et al., 2006] but
it appears instead in the boundary conditions.
In what follows two characteristic situations re-
ferred to as segregation and compaction are con-
sidered. The former is a model of the evolution of a
bounded partially molten zone. Its outer boundary
coincides with solidus where the porosity and pres-
sure difference vanish. The boundary and initial con-
ditions are ( ,0) ( ,1) 0 , (0, ) 4 (1 ) .
For compaction (of bottom sediments, e. g.), it is
assumed that the bottom is impermeable, and po-
rosity at =0 is the same throughout the layer, so
that: ( ,0)/ 1 , ( ,1) 0 , (0, ) 1 .
The solutions to Eqs. (2) are shown in Fig. 1 for
segregation, and Fig. 2 for compaction. One may
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Fig. 1. Evolution of porosity � �� ��at segregation: a — c = 10–2, b — c = 102, c — ��� c Eqs. (2) reduce to � ,
�
��
n
�with
being a formally introduced time variable ����� c . Note that the first and the second waves at ������� � (b,
inset) contain 57 and 13 % of the melt, respectively, with the rest of the melt residing in the tail.
Fig. 2. Compaction of the bottom sediments at: a — c = 0,
b — c = 1, c — c = 102, d — c = 104 (n = 3).
Fig. 3. Characteristic time of compaction c and segregation
s vs. c.
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see from Fig. 1 that at low ãc practically all the melt
segregates to the roof with amplitude of the porosi-
ty growing with time (Fig. 1, a). The pressure gra-
dient remains hydrostatical in the enriched layer and
evolves to zero in the low-melt fraction tail. At high
c (Fig. 1, b) a wave structure develops with the
dimensionless pressure being generally low and sigh
reversible, and a significant part of the melt remai-
ning trapped in the tail. The case c = (Fig. 1, c)
corresponds to � �(or pl = pm). The porosity ampli-
tude remains the same (i. e. no segregation occurs),
and it takes
a finite time to reach the –8 im-
plying a numerical instability, which is absent from
the finite c models. The variation of porosity while
a liquid is expulsed from a compacting porous layer
(Fig. 2) is similar to that at segregation indicating
that a property to generate a wave-like structure
becomes more pronounced with increasing c, i. e.
at high melt viscosity. A large pluton layering [Wa-
ger, Brown, 1968] may be due to this wave struc-
ture, which is supported by the observation [Brown,
1973] that layered intrusions are commonly intru-
sions of a tholeiitic basalt while those of the alkali
basalt parentage are rare, which may be due sim-
ply to about an order higher viscosity of the tholeitic
magmas.
The dependence of characteristic compaction,
c and segregation, s, times on c , is shown in
Fig. 3. In dimensional variables an approximate fit
to these results may be written as follows:
D
c V
L
gL
t 49.2 ,
D
s V
L
gL
t 25.05,19 ,
0
0 )(gkVD , (3)
where VD is the Darcy’s velocity. The L–1 scaling of
the compaction/segregation times at low c effec-
tively constrains the thickness of compacting po-
rous sediments as well as the maximum possible
thickness of the partially molten zone. Really, if the
mushy layer thickness increases gradually, due to,
for instance, sedimentation with a rate of rd then the
steady sediment thickness, Ld, may be estimated
as Ld/rd = tc wherefrom grL dd 7.1 . Similar-
ly, let a protokimberlite melt result from a decom-
pression melting of a diapir floating at a velocity Vd
with its temperature varying along an adiabate. The
melting starts when the diapir top reaches the inter-
section level of the adiabate and solidus, and ma-
ximum possible thickness of the molten zone, Ls,
may be estimated as Ls 4.5 Vd / g . The
estimates of Ld and Ls are valid if c >> 2 for the
compaction problem or c>>80 for a segregation
one. Also, Darcy’s velocity is to be large, namely
VD >> rd at compaction and VD>>0.25Vd at segrega-
tion. These estimates relate to an evolution of a
porous layer filled with a low viscosity liquid, and
may be used to estimate, for instance, a steady
thickness of porous marine sediments, or a maxi-
mum possible thickness of a partially molten zone
filled with a low viscosity magma (kimberlite, car-
bonatite) at the moment of segregation. To illustrate
the latter case, adopt the following parameter va-
lues: = 1019 Pas, = 100 kg m 3, Vd = 3 cm y 1,
k( )=a2 3/270 [Wark et al., 2003], grain size
a = 1 mm, 0 = 0.01, = 0.1 Pas. Then compaction/
segregation length c = 6 km, Vd = 30 cm y 1,
ts = 0.2 My, Ls = 8 km, c = 1.7. As soon as melt seg-
regates, new partially molten zone grows, and the
sequence of the events repeats until the whole dia-
pir passes by the melting level. A diapir size, D,
can be estimated based upon diameters D = 20 to
80 km of low-amplitude uplifts known to correlate
with kimberlite fields [Kaminsky et al., 1995]. There-
fore, the decreasing dependence of the segregation
time on a mushy layer thickness implies formation
of clusters of D/Ls = 3 to 10 low volume eruptions of
almost the same age and composition. An attrac-
tive feature of the model is that it relates the kim-
berlite origin to a localized incipient melting.
Brown P. E. A layered plutonic complex of alkali basalt
parentage; the Lilloise Intrusion, East Greenland /
/ J. Geol. Soc. London. — 1973. — 129. — P. 405—
418.
Drew D. A. Averaged field equations for two-phase
flow // Ann. Rev. Fluid Mech. — 1983. — 15. —
P. 261—291.
Grégoire M., Rabinowicz M., Janse A. J. A. Mantle
mush compaction: A key to understand the mecha-
nisms of concentration of kimberlite melts and ini-
tiation of swarms of kimberlite dykes // J. Petrol. —
2006. — 47. — P. 631—646.
References
McKenzie D. The generation and compaction of par-
tially molten rock // J. Petrol. — 1984. — 2. —
P. 713—765.
Nigmatulin R. I. Dynamics of multi-phase media. —
New York: Hemisphere, 1990.— 1. — 507 p.
Scott D. R., Stevenson D. J. Magma ascent by po-
rous flow // J. Geophys. Res. — 1986. — 91. —
P. 9283—9296.
Wager L. R., Brown G. M. Layered Igneous Rocks.
— Edinburgh, London: Oliver, Boyd, 1968. — 588 p.
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�� !� "#$�%&'( )#�&)*�$+,$�(!$)'#�' �'-$ �.�&)*�! *$//()0
Wark D., Williams C., Watson E., Price J. Reassess-
ment of pore shapes in microstructurally equili-
brated rocks, with implications for permeability of
upper mantle // J. Geophys. Res. — 2003. —
108(B1). — DOI:10.1029/2001JB001575.
Kaminsky F. V., Feldman A., Varlamov V., Boyko A.,
Olofinsky L., Shofman I., Vaganov V. Prognosti-
cation of primary diamond deposits // J. Geochem.
Explor. — 1995. — 53. — P. 167—182.
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