Download A perturbation solution for dyke propagation in an elastic medium

Geophys. J. Int. (2007) 169, 348–356
doi: 10.1111/j.1365-246X.2006.03297.x
A perturbation solution for dyke propagation in an elastic medium
with graded density
Zuan Chen,1,2 Z.-H. Jin2 and S. E. Johnson3
1 Institute
of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China
of Mechanical Engineering, University of Maine, Orono, ME 04469, USA. E-mail: [email protected]
3 Department of Earth Sciences, University of Maine, Orono, ME 04469, USA
2 Department
GJI Volcanology, geothermics, fluids and rocks
Accepted 2006 November 14. Received 2006 November 13; in original form 2006 June 7
SUMMARY
We present a perturbation method to investigate the steady-state propagation of a dyke from
an over pressured source (e.g. a magma chamber) into a semi-infinite elastic solid with graded
mass density. The non-linear dyke propagation/magma transport problem is reduced to a series
of linear problems using a perturbation technique with the small non-dimensional parameter
ε = 12ηV /(H 2 ρ 0 g), where η is the magma viscosity, V the propagation velocity, ρ 0 the
difference between the densities of the host rock and magma at the dyke tail, g the gravitational
acceleration and H a parameter on the order of maximum dyke thickness. In general, the
perturbation method is applicable to mafic dyke propagation at a relatively low propagation
velocity wherein ε remains small (e.g. less than 0.1). We describe an integral equation approach
to obtain the stress intensity factor at the dyke tip and the separation displacement of the two
dyke surfaces. Numerical examples are presented to examine the effects of various physical
parameters, for example, buoyancy, density gradation, propagation velocity and overpressure
on the dyke propagation behaviour. It is found that dyke propagation could reach a steady-state
in some specific ranges of growth for a given density gradient of the host rock. The propagation
tends to decelerate when the dyke tip approaches the level of neutral buoyancy.
Key words: buoyancy, density gradation, dyke propagation, magma transport, overpressure,
perturbation method.
1 I N T RO D U C T I O N
A magma chamber in the crust or upper mantle undergoes pressure increase induced by the inflow of magma from partially molten regions
to the side or below the chamber. When the magma pressure exceeds the lithostatic stress, the host rock may fracture if the stresses caused by
the overpressure satisfy an appropriately established failure criterion. Once a fracture forms, it will be filled by the magma to form a dyke.
Subsequent dyke propagation is driven by the magma chamber overpressure and the buoyancy force due to the density difference between
the host rock and magma. Tectonic stresses may also play an important role in dyke propagation. Magma transport via dyke propagation
significantly influences the geological evolution of Earth’s crust and volcanism on Earth’s surface (e.g. Pollard 1987; Spence et al. 1987;
Gudmundsson 1990; Rubin 1995a).
Studies of dyke propagation generally involve fluid mechanics of magma flow and solid mechanics of rock fracture. Significant progress
has been made in understanding fluid-driven fracture in general and dyke propagation in particular. Weertman (1971) considered the problem of a
liquid-filled crack in a horizontal elastic plate subjected to a tensile stress and gravity-induced hydrostatic pressure. He showed that a sufficiently
large buoyant force can cause the crack to propagate upwards. Secor & Pollard (1975) studied buoyancy-driven fracture as a mechanism for
magma transport. Spence & Sharp (1985) obtained a series of similarity solutions for fluid-driven fracture in an infinite solid fed by a specified
source flux without consideration of buoyancy. Spence et al. (1987) derived a steady state solution for buoyancy-driven propagation of a
semi-infinite fluid-filled crack in an elastic solid. They used lubrication theory to describe the viscous fluid flow in the fracture and an integrodifferential equation approach to describe the elastic deformation. Lister (1990) considered material toughness effects of buoyancy-driven
crack propagation fed by a constant flux of fluid. He obtained a solution of crack surface displacement that shows a bulbous head and a constant
width tail for the dyke, a feature also obtained by Spence et al. (1987). Lister & Kerr (1991) described the fluid-solid coupling effects in dyke
propagation. They showed that the transport of magma in feeder dykes is governed by a local balance between buoyancy forces and viscous
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pressure drop, and that elastic forces play a secondary role except near the dyke tip. Rubin (1993) obtained a self-similar solution of dyke growth
with a given source pressure. Bonafede & Olivieri (1995) presented a crack model for magma emplacement within a shallow vertical dyke in a
half space having vanishing tensile strength. Rubin (1995a) wrote an excellent review and covered various aspects of dyke propagation including
observations, physical processes and future directions. Meriaux & Jaupart (1998) considered dyke propagation in an elastic plate of finite
thickness driven by buoyancy and a specified overpressure. Bonafede & Rivalta (1999) used a dislocation model to study the stress and
displacement fields around dykes. Besides theoretical/computational modelling, some experimental investigations of dyke propagation have
been carried out in recent years. For example, Takada (1990), Menand & Tait (2002) and Ito & Martel (2002) have used liquid-filled fractures
in gelatin to experimentally model dyke propagation. Recently, Roper & Lister (2005) studied the propagation of a liquid-filled crack from an
overpressured source into a semi-infinite elastic solid. They obtained both early (dyke length buoyancy length) and late (dyke length buoyancy length) time solutions and found that the combination of buoyancy and overpressure leads to significantly different behaviour from
buoyancy or overpressure alone.
While various solutions for dyke propagation in the mantle and crust have been derived over the past two decades, the lithosphere has
been considered as a homogeneous medium with a constant density in most studies. It is well known that the mass density of lithosphere is
not a constant but varies with depth (Ryan 1994; Turcotte & Schubert 2002). Lister (1991) considered steady-state magma flow in a feeder
dyke through a density-stratified lithosphere under a constant magma flow rate at the base of the dyke. He demonstrated a transition from
vertical flow to lateral flow directed along the level of neutral buoyancy. The non-linear nature of the coupled fluid/solid mechanics equations
for describing magma flow/dyke propagation makes the solution approach significantly complicated.
In this paper, we propose a perturbation method to solve the non-linear equations of the coupled fluid/solid mechanics problem for
describing magma flow and dyke propagation in an elastic medium with graded density under a constant overpressure at the base of the
dyke. As pointed out by Rubin (1995a), the specified source overpressure seems more realistic geologically than the specified source flux
as adopted by Lister (1991). A small perturbation parameter is identified when the dyke propagation velocity is relatively low and the
magma viscosity is typical of mafic magma. To simplify the analysis, we consider steady-state (constant velocity) dyke propagation from an
overpressured source into a semi-infinite solid. The non-linear integral equation of the magma transport/dyke propagation problem is reduced
to a series of linear integral equations using the perturbation method. Numerical examples are given to illustrate and discuss the effects
of various physical parameters, for example, buoyancy, density gradation, propagation velocity and overpressure on the dyke propagation
behaviour.
2 M O D E L A N D F O R M U L AT I O N
Consider a dyke that propagates vertically from a magma chamber into the host rock. We assume that the vertical dyke length is smaller than
the horizontal size of the magma chamber. The dyke thus may be approximately considered as a magma filled, plane strain edge crack in a
half plane, as shown in Fig. 1, where a = a(t) denotes the crack length and t is time. This 2-D dyke propagation model has also been adopted
by Roper & Lister (2005).
Figure 1. Propagation of a dyke into a semi-infinite space.
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2.1 Fluid pressure
Dyke propagation is driven by the net pressure on its surfaces due to magma flow, lithostatic stress and tectonic stress. The lubrication theory
of fluid mechanics has been used to describe magma flow in dykes (Spence et al. 1987; Lister & Kerr 1991; Rubin 1995a). In the lubrication
theory, the fluid flux q and the fluid pressure p follow the Poiseuille law
δ3 ∂
( p + ρm g Z ),
12η ∂ Z
where η is the magma viscosity, ρ m the magma density, δ the separation of the two dyke surfaces, g the gravitational acceleration and Z
upward coordinate. The continuity equation is
∂q
∂δ
=−
.
∂t
∂Z
Substituting eq. (2) into eq. (1) yields
∂δ
1 ∂
∂
=
δ3
( p + ρm g Z )
∂t
12η ∂ Z
∂Z
q=−
(1)
the
(2)
(3)
The fluid pressure p may be expressed as
p = pe − (ρs g Z + σ ),
(4)
where p e = p e (Z ) is the pressure due to elastic deformation of the host rock, ρ s the density of the host rock and σ the tectonic stress
perpendicular to the dyke plane (Rubin 1995a; Roper & Lister 2005). Roper & Lister (2005) assumed that σ could vary linearly in the Z
direction and thus could be absorbed into the lithostatic stress using an effective (or modified) rock density defined by
d
(ρs g Z + σ )
(5)
ρeff g =
dZ
Substituting eqs (4) and (5) into eq. (3), we have
∂δ
∂
1 ∂
=
δ3
( pe − ρg Z ) ,
(6)
∂t
12η ∂ Z
∂Z
where ρ = ρ eff − ρ m .
Consider a special case of steady-state magma flow (Spence et al. 1987). Introducing a moving coordinate with the origin at the dyke
tip, we have
d
∂
= −V ,
(7)
dt
∂z
where z = Z − a is the moving coordinate, as shown in Fig. 1, and V is the dyke propagation velocity. In the moving coordinate system eq. (6)
becomes
∂δ
1 ∂
∂
−V
=
δ 3 [ pe − ρg(z + a)]
(8)
∂z
12η ∂z
∂z
or
1
∂
Vδ = −
δ 3 [ pe − ρg(z + a)] .
(9)
12η
∂z
Integrating the above equation and noting that the elastic pressure p e tends to the overpressure P at the dyke tail z = −a(t), we obtain the
elastic pressure along the dyke surface
z
pe = P + ρg(z + a) − 12ηV
δ −2 dz.
(10)
−a
2.2 Fracture mechanics formulation
We use a singular integral equation method to attack the dyke propagation problem. Because the dyke propagation velocity is much smaller
than the wave speed of the host rock, the inertia effect can be ignored and the problem is quasi-static. The basic integral equation for the plane
strain edge crack problem of a semi-infinite space has the following form (Gupta & Erdogan 1974)
E 0
1
1
)
ϕ(z ) dz = pe (z), −a ≤ z ≤ 0,
(11)
+
k(z,
z
2π −a z − z
a
where E = E/(1 − ν 2 ), E is Young’s modulus, ν is Poisson’s ratio, φ(z) is the unknown dislocation density function
∂u x ϕ(z) =
,
∂z (12)
x=0
with u x (z, x) being the displacement perpendicular to the crack, and k(z, z ) is a non-dimensional kernel given by
k(z, z ) =
2a(z + a)
a
4a(z + a)(z + a)
−
+
.
2
z + z + 2a
(z + z + 2a)
(z + z + 2a)3
(13)
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We note that the damage and cavity zones around the crack tip are not considered in eq. (11). The damage zone may be treated as a
cohesive zone which is characterized by a cohesive law, or a relationship between the cohesive traction, σ and the separation displacement,
δ, of the two cohesive surfaces, as follows
σ = f (δ).
(14)
When the damage zone size is much smaller than the dyke length, the classical stress intensity factor approach will be applicable with the
effective critical energy release rate equal to the intrinsic Griffith energy release rate plus the cohesive energy density defined by
δ0
0 =
f (δ) dδ,
(15)
0
where δ 0 denotes the critical separation at which the cohesive traction vanishes. The cohesive zone length scales with (Hillerborg et al. 1976)
E0
a0 = 2 ,
(16)
σ0
where σ 0 is the peak cohesive traction. For rock undergoing microcracking, crack face grain bridging and other forms of damages at the
microstructural level, 0 is on the order of 100 J m−2 . Using E = 50 GPa and σ 0 = −10 MPa yields a cohesive zone length on the order
of 0.05 m. Damages at the tip of a long dyke, however, may occur at a much larger length scale, for example, as large bridges (Hoek 1994,
1995). In these cases, 0 could reach the order of 104 J m−2 (Li 1987). Using this cohesive energy density will yield a cohesive zone length
on the order of 5 m. For a dyke that has propagated more than 1000 m, the cohesive zone can still be considered small because the dominant
zone of stress intensity factor usually exceeds one percent of the crack length. Hence, a standard stress intensity approach may be employed.
In the same spirit of the small damage zone, the crack tip cavity may also be ignored if the cavity zone is much smaller than the dyke length.
However, it should be kept in mind that the applicability of classical linear elastic fracture mechanics also requires that the magma flow should
not reach the cohesive zone which occupies—a 0 < z < 0 with a 0 a.
The integral eq. (11) may be rewritten in the following form using eq. (10)
E 1
1
a
1
+ K (r, s) ϕ(s) ds = P + ρg × (1 + r )
2π −1 s − r
2
2
r
(17)
δ −2 ds, |r | ≤ 1,
− 6aηV
−1
where r and s are related to z and z by
a
a
z = (−1 + r ), z = (−1 + s)
(18)
2
2
and K (r , s) = k(z, z ).
In general, the density of the host rock gradually increases from the surface towards the inner part of the Earth due to compositional
changes and pressure gradient. The density difference ρ thus decreases from a maximum value at the dyke tail (z = − a) with increasing
distance from the magma chamber. We assume that ρ varies with Z according to
ρ = ρ0 (1 − Z /L)n = ρ0 [1 − (z + a)/L]n ,
(19)
where ρ 0 is the density difference at Z = 0, L denotes a length at which magma and the host rock have the same density and n is the
exponent. With appropriately selected L and n, eq. (19) could, for example, be used to describe the density–depth relation reported in Ryan
(1994).
Because the pressure on the dyke surface due to magma flow will not reach the dyke tip (r < 1 in eq. 17) and material inhomogeneities
do not alter the crack tip singularity characteristics (Jin & Noda 1994), a standard square root singularity exists at the dyke tip and the integral
eq. (17) has the following form of solution
ψ(r )
ϕ(r ) = √
,
(20)
1−r
where ψ (r) is continuous and bounded. Once the solution of the integral eq. (17) is obtained, the stress intensity factor at the dyke tip can be
calculated from
√
E
πa
KI = −
ψ(1).
(21)
(1 − ν 2 ) 2
The dyke face separation displacement (dyke thickness) is
1
ϕ(s) ds.
(22)
δ(r ) = −a
r
In the present study, we adopt the following two assumptions regarding the dyke propagation in a density stratified medium: (i) steadystate dyke propagation and magma flow prevail, and (ii) the propagation velocity is known a priori. Magma flow and dyke propagation are
generally transient processes during which the dyke propagation velocity is time dependent. A solution based on the first assumption may
be approximately applicable to some specific stages of dyke propagation during which the effects of density gradation and overpressured
source are traded off. The propagation velocity V (t) is generally determined using the dyke propagation condition that K I = K Ic , where K Ic
is the fracture toughness of the host rock. Here an inverse approach is used based on the second assumption, which is approximately valid
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when the calculated stress intensity factor remains constant. Finally, steady-state dyke propagation may theoretically take place only after the
dyke becomes semi-infinitely long. Quasi-steady dyke propagation may be achieved with a long but finite-length dyke as considered here.
3 P E RT U R B AT I O N S O L U T I O N
Eq. (17) is a non-linear, singular integral equation for the dislocation density function ϕ(r). Here, we present a perturbation method to solve
the non-linear integral equation. To this end, we first introduce the following normalized dislocation density function ϕ̃ and dyke surface
separation displacement δ̃
ϕ(r ) = ϕ̃(r )
1 − ν2
ρ0 ga,
E
(23)
1 − ν2
(24)
ρ0 ga.
E
Using the above normalized quantities and eq. (19), the integral eq. (17) is reduced to
1
1
1
2π
1+r a n
+ K (r, s) ϕ̃(s) ds =
P + π(1 + r ) 1 −
2
ρ0 ga
2 L
−1 s − r
2
r
HE
(25)
− πε
ds, |r | ≤ 1,
δ̃(s)(1
−
ν 2 )ρ0 ga
−1
where H is a parameter on the order of the dyke tail separation and ε is given by
12ηV 1
.
(26)
ε=
H 2 ρ0 g
For some magma flow and dyke propagation problems, ε could be a small parameter. For example, for mafic dyke propagation with typical
values of H = 1 m, ρ 0 = 300 kg m−3 , g = 10 m s−2 , η = 50 Pa s and V = 0.5 m s−1 , ε = 0.1.
The non-linear integral eq. (25) may be solved using a perturbation approach under small ε conditions with the basic unknown expressed
as
δ(r ) = δ̃(r )
ϕ̃(r ) = ϕ̃0 (r ) + εϕ̃1 (r ) + ε 2 ϕ̃2 (r ) + ε 3 ϕ̃3 (r ).
(27)
The normalized dyke surface separation displacement becomes
δ̃(r ) = δ̃0 (r ) + εδ̃1 (r ) + ε 2 δ̃2 (r ) + ε 3 δ̃3 (r ),
with
δ̃i (r ) = −a
1
ϕ̃i (s)ds,
(28)
i = 0, 1, 2, 3.
(29)
r
Substituting the above expressions into eq. (25) and collecting the terms of the same order of ε yield the following linear integral equations
for ϕ̃i (r )(i = 0, 1, 2, 3)
n
1
1
1+r a
2π
1
+ K (r, s) ϕ̃0 (s) ds = −
P − π(1 + r ) 1 −
,
2
ρ0 ga
2 L
−1 s − r
2 r 2
1
1
E
1
H
+ K (r, s) ϕ̃1 (s) ds = π
ds,
2
(1 − ν 2 )ρ0 ga
δ̃0
−1 s − r
−1
2 2 r 1
1
δ̃1 H
1
E
−2
ds,
+ K (r, s) ϕ̃2 (s) ds = π
2
(1 − ν 2 )ρ0 ga
δ̃0 δ̃0
−1 s − r
−1
2 r 2 2
1
1
E
δ̃2
1
H
δ̃1
+ K (r, s) ϕ̃3 (s) ds = π
−
2
3
ds.
(30)
2
(1 − ν 2 )ρ0 ga
δ̃0
δ̃02
δ̃0
−1 s − r
−1
Once the solutions of the above integral equations are obtained, the stress intensity factor at the dyke tip can be computed from
√
ρ0 ga πa KI = −
ψ̃0 (1) + εψ̃1 (1) + ε 2 ψ̃2 (1) + ε 3 ψ̃3 (1) ,
2
where
√
ψ̃i (r ) = 1 − r ϕ̃i (r ), i = 0, 1, 2, 3.
(31)
(32)
4 N U M E R I C A L R E S U LT S
This section presents numerical examples to illustrate effects of various physical parameters on the stress intensity factor and dyke surface
profile. In all calculations, we use the following typical properties for the host rock and basaltic magma (Rubin 1995a): E = 50 GPa, ν =
0.25, ρ 0 = 300 kg m−3 , η = 50 Pa s and g = 9.8 m s−2 .
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Figure 2. The stress intensity factor distribution for different overpressure and dyke length [V = 0.5 m s−1 , no density gradation (n = 0)].
We first examine the convergence of the perturbation series of stress intensity factor K I in eq. (31). When the density gradation is not
considered (n = 0 in eq. 19) and for typical values of overpressure P = 3.0 MPa, dyke propagation velocity V = 0.5 m s−1 and dyke length
a = 2000 m, K I = 523.12 MPa m1/2 and the first four terms in the perturbation solution are 585.22, −53.84, −6.80 and −1.46, respectively. It
can be seen that the fourth term is insignificant and the converged result is obtained using the first three terms. When the density gradation is
considered (n = 1 and L = 5000 m in eq. 19) and for typical values of P = 3.0 MPa, V = 0.5 m s−1 and a = 5000 m, K I = 671.86 MPa m1/2
and the first four terms in the perturbation solution are 712.03, −37.70, −2.24 and −0.23, respectively. Now both the third and fourth terms
are insignificant. We note that the convergence of the perturbation series depends on the smallness of the perturbation parameter ε in eq. (26).
Generally speaking, for basaltic magma with a viscosity around 50 Pa s, the perturbation series converges rapidly when the dyke propagation
velocity is lower than 0.5 m s−1 .
Fig. 2 shows the stress intensity factor K I as a function of overpressure P and dyke length a. The dyke propagation velocity is taken
as V = 0.5 m s−1 . The density gradation is not considered in this case, i.e. n = 0 in eq. (19). It can be seen from the figure that for a given
overpressure P, K I increases with increasing dyke length. The increasing stress intensity factor implies that the steady-state propagation
assumption is violated. To keep a constant K I for steady-state propagation, the overpressure P will have to decrease with dyke propagation.
Generally speaking, the overpressure P will decrease with dyke propagation if magma supply to the magma chamber from the partial molten
region can not compensate the magma loss transported to the propagating dyke. It is thus possible for K I to remain approximately constant if
the overpressure P and dyke length follow a specific relationship.
Fig. 3 shows the stress intensity factor K I versus dyke length for various values of overpressure P. The density gradation is included
in this case with n = 1 and L = 5000 m in eq. (19). The propagation velocity is still taken as 0.5 m s−1 . An exponent of n = 1 corresponds
to a linear variation of the density difference ρ. Ryan (1994) has reported the in situ density–depth relations for the East Pacific Rise which
may be approximately described by eq. (19) with the above n and L. A linear decrease in ρ was also used by Lister (1991). For P = 3.0
MPa, it is found from the figure that K I is nearly a constant of about 725 MPa m1/2 over a range of dyke length from 4000 m to 5000 m
Figure 3. Stress intensity factor versus dyke length for various values of overpressure [V = 0.5 m s−1 , with density gradation (n = 1, L = 5000 m)].
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indicating steady-state dyke propagation. For the case of a reduced overpressure of P = 2.0 MPa, K I also remains nearly constant but
the value is about 600 MPa m1/2 over a range of dyke length from 4000 to 4600 m. Finally, for a further reduced overpressure of P =
1.0 MPa, K I remains nearly constant with a value of about 470 MPa m1/2 over a range of dyke length from 4000 to 4500 m. These results
demonstrate that steady-state propagation may be achieved when the dyke has propagated into some specific regions near the level of neutral
buoyancy (LNB) where the magma density equals that of the host rock. Dyke propagation tends to decelerate when the dyke tip approaches
the LNB characterized by the decreasing K I . It is possible that dyke propagation is arrested at this stage or deflected towards the horizontal
direction as indicated by Lister (1991) although the stress intensity factor was not a constant in his study. We note from eqs (17)–(21) that K I
is proportional to overpressure P, buoyancy force ρg a and magma pressure drop (proportional to −aηV ). When the dyke propagates in
a region near the LNB, the reducing buoyancy force and magma pressure drop will cause K I to decrease whereas P renders K I to increase
with increasing dyke length. The compensating nature of these effects makes it possible for K I to keep approximately constant at a specific
stage of dyke propagation.
Fig. 4 shows K I versus propagation velocity at a dyke length of a = 1000 m for a given overpressure of P = 3.0 MPa. It is seen that K I
decreases significantly with increasing propagation velocity. For example, K I is approximately 315 MPa when the dyke propagation velocity
is 0.1 m s−1 and reduces to about 165 MPa at V = 0.5 m s−1 . The decrease in the stress intensity factor is caused by the lower magma pressure
on the dyke surfaces associated with higher flow velocities.
Fig. 5 shows the dyke surface profile when the dyke has propagated 5 km for various values of overpressure P. The propagation velocity
is 0.5 m s−1 and the density gradation is included with n = 1 and L = 5000 m in eq. (19). It is seen from the figure that at a given location
along the dyke surface, the dyke thickness decreases with decreasing overpressure as expected. The dyke thickness at the tail (z = −5 km) is
about 2.8 m for P = 3.0 MPa. The thickness reduces to about 1.6 m when P = 1.0 MPa. These dyke thickness values are in agreement
with the data of field observations (Rubin 1995a) for dykes of kilometers long.
Figure 4. Stress intensity factor versus dyke propagation velocity [P = 3.0 MPa, a = 1000 m, no density gradation (n = 0)].
Figure 5. Dyke surface profile [V = 0.5 m s−1 , with density gradation (n = 1, L = 5000 m)].
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5 C O N C LU D I N G R E M A R K S
We have obtained a perturbation solution for the propagation of a dyke in a semi-infinite elastic medium with graded density. The stress
intensity factor at the dyke tip and the dyke surface profile are calculated using various combinations of physical parameters, for example,
overpressure, propagation velocity and density gradation characteristics. The stress intensity factor is found to range from 100 to 800 MPa m1/2
for the selected parameter combinations. While these values are significantly higher than the experimental results of fracture toughness obtained
at atmospheric pressure, they agree with previously reported modelling results of 60–1000 MPa m1/2 under mantle pressure conditions (Heimpel
& Olson 1994). Moreover, these stress intensity factor values also include the effect of increased fracture energy due to dyke tip damages. The
perturbation method is a convenient and efficient approach to solve the non-linear coupled fluid-solid mechanics equations for the magma
transport/dyke propagation problem.
The numerical results show that the density gradation has a profound effect on the stress intensity factor and dyke propagation, especially
when the dyke tip reaches the level of neutral buoyancy wherein the magma density equals that of the host rock. While a continuously decreased
overpressure in the magma chamber could result in steady-state dyke propagation, density gradation is essential for achieving the steady-state
propagation because it greatly influences the buoyancy force on the dyke surfaces. Density gradation will also play an important role in dyke
propagation arrest and deflection.
The present solution of steady-state propagation has some limitations when it is used to study magma transport via dyke propagation
towards the surface of Earth. Magma flow and dyke propagation are generally transient processes during which the dyke propagation velocity
is time dependent. Dyke propagation may also be affected by magma solidification (see e.g. Lister 1994a,b; Rubin 1995b; Bolchover & Lister
1999) which is not considered in this study. Moreover, a complete understanding of dyke propagation requires 3-D analyses. The present
solution may be approximately applicable to some specific stages of dyke propagation during which the density gradation characteristics and
overpressure reduction behaviour permit steady-state propagation.
AC K N OW L E D G M E N T S
This work is supported by the University of Maine. ZC is partly supported by the National Natural Science Foundation of China (40574017
and 40574046). The authors would also like to thank Professor Peter Koons and two reviewers for helpful discussions and comments.
REFERENCES
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