Tectonophysics, 228 (1993) 189-198
Elsevier Science Publishers B.V., Amsterdam
Numerical models of complex diapirs
Yu. Podladchikov a?*, C. Talbot a and A.N.B. Poliakov
a?b
Hans Ramberg Tectonic Laboratory, Institute of Earth Sciences, Uppsala University, Norbyuagen 18B, S-752 36 UppsaEa, Sweden
HLRZ, KFA -Jiilich, Postfach 1913, 0-5170 Jiilich, Germany
(Received January 31,1993; revised version accepted July 8,1993)
ABSTRACT
Numerically modelled diapirs that rise into overburdens with viscous rheology produce a large variety of shapes. This
work uses the finite-element method to study the development of diapirs that rise towards a surface on which a
diapir-induced topography creeps flat or disperses ("erodes") at different rates. Slow erosion leads to diapirs with
L ' m ~ ~ h r ~shapes,
~ m l moderate
y
erosion rate to "wine glass" diapirs and fast erosion to "beer glassn- and "columnm-shaped
diapirs. The introduction of a low-viscosity layer at the top of the overburden causes diapirs to develop into structures
resembling a "Napoleon hat". These spread lateral sheets.
Introduction
I
1
1
There are still large discrepancies between the
shapes of natural salt diapirs and the shapes of
diapirs simulated in laboratory experiments or
numerical models. Natural salt structures have a
rich diversity of shape which includes pillows,
walls, columns, mushrooms and horizontal sheets
(Jackson and Talbot, 1991). The style of deformation of the surrounding overburden also varies,
from horizontal layers pierced by vertical salt
stocks, to complex structures smoothly conformable with the shape of salt bodies. Current
numerical models reproduce only a few of these
features. Most previous numerical modelling
studies (e.g., Woidt, 1980; Romer and Neugebauer, 1990) only simulated diapirs with "balloon
on a string" geometries; these are rare in nature,
or at least, rarely recognized. Only recently have
Zaleski and Julien (1992) modelled diapirs that
were asymmetric because the starting configurations were asymmetric (e.g., a tilted basement).
Poliakov et al. (1994) modelled a new set of
deformation styles by adding to the controlling
parameters the redistribution at different rates of
*
Present adress: Institute of Earth Sciences, Vrije Universiteit, 1081 HV Amsterdam, The Netherlands.
a potential diapir-induced surface topography
during continuous sedimentation (downbuilding).
Their downbuilt diapir shapes included "columns" which extruded onto the surface above
"turtlebacks" in the deep overburden, and shallow allochthonous sheets that reactivated a second generation of smaller diapirs.
This papers aims to model numerically some
of the wide range of diapir shapes known in
nature and to make genetic statements about
their origin. We used a finite-element method
(Poliakov and Podladchikov, 1992; Poliakov et al.,
1994) to model the development of salt diapirs
with viscous overburdens. Diapirs in overburdens
which have constant viscosity and very large erosion coefficients have already been studied by
Poliakov et al. (1994). Here we study the effects
of smaller erosion coefficients and different viscosity profiles. Our results include some new
diapir shapes (such as "wine" and "beer glass")
and we simulate the asymmetric lateral spread of
sheets of salt in low-viscosity subsurface layers.
Influence of erosion rates and viscosity profiles
on salt diapirism
Poliakov et al. (1994) demonstrated the strong
influence of erosion and sedimentation rates on
0040-1951/93/$06.00 O 1993 - Elsevier Science Publishers B.V. All rights reserved
SSDI 0040-1951(93)E0175-T
YU.PODLADCHIKOV ET AL.
the style of salt diapirism. They simulated redistribution of a potential topography by erosion
and redeposition of surface sediments, using a
one-dimensional diffusion equation:
where h is topographic height, t is time, K is
transportation coefficient and x is horizontal dist ance.
Redistribution of material from topographic
highs changes the effective forces acting on the
salt. This completely changes, not only the shapes
and growth rates of diapirs, but also the geometry
of the layering in the intervening overburden.
The main effects of rapid erosion rates are:
(1) diapirs grow 10-100 times faster with erosion than without;
(2) diapirs may rise above their level of neutral buoyancy and extrude;
(3) diapirs are "finger" like rather than
"mushroom" shaped;
(4) overburden layers are nearly horizontal and
only steepen close to the diapir.
Poliakov et al. (1994) studied only the end
members in a time spectrum of diapir development by considering examples with a coefficient
of erosion K that was either zero or extremely
high. Natural diapirs are likely to lie somewhere
between these two extremes and it is, therefore,
instructive to study how diapirs develop with intermediate erosion rates.
Another unexplored field of parameters is the
rheology of the overburden. Previous numerical
models approximated the overburden as fluids of
constant viscosity for a number of reasons that
include simplicity (Poliakov et al., 1994). Overburdens with a constant viscosity of about 1021-22
Pa s simulated diapirs with realistic shapes in
reasonable times. There are not many experimental data on the rate at which sedimentary rocks
creep in natural conditions. To model the dependence of the 'effective viscosity of sediments with
depth, we assume here that the viscosity of the
overburden increases with depth as does compaction: synchronously and exponentially with increasing pressure. A possible simplification is that
the effective viscosity of the overburden increases
abruptly at some critical depth (e.g., due to cementation or a change in lithology). We numerically explore these two possibilities by comparing
the evolution of diapirs developing in overburdens with two different types of viscosity profiles,
where (1) viscosity increases exponentially with
depth or (2) viscosity increases sharply at a particular depth.
Many salt diapirs in the Gulf of Mexico are
strongly asymmetric in profile. As far as we know,
asymmetric diapirs have not been simulated numerically before. We therefore follow the analogue models of Talbot (1977) and begin here the
numerical modelling of diapirs which develop
asymmetric lateral sheets along sloping interfaces.
Model geometry
This section describes our computational
model for Rayleigh-Taylor (RT) instability in
viscous fluids, where the thickness of the upper
layer (the overburden) continuously increases with
time. As the overburden downbuilds, we also
redistribute the potential topography induced at
the accumulating surface by the diapirs rising
beneath it (Biot and Ode, 1965). The upper surface is flattened by depositing in topographic
lows the material eroded from topographic highs
that tend to form above active diapirs. This redistribution of surface overburden is modelled using
eqn. (1). The geometry and the boundary conditions of the model used for the calculations are
shown in Figure 1.
Free surface + Erosion and redeposition
No slip
Fig. 1. Geometry and boundary conditions for viscous
Rayleigh-Taylor instability in a continuously thickening overburden of compacting sediments with a potential topography
at the upper boundary diffused according to eqn. (1).
NUMERICAL MODELS OF COMPLEX DIAPIRS
The bottom of the box is a non-slip boundary,
while free slip is allowed along the left and right
sides. The upper surface is taken as a free surface
and eqn. (1)is solved after every time step.
Poliakov et al. (1994) justify the following material properties and initial parameters for modelling the overburden:
r1= 1020-21 p a s . h i1n i t = l o 3 m;
p , = 2490 - 590 exp(ay) - ayexp( p y )
7
/
!
Velmax = 188.8(m/Ma)
Time =149.6(Ma)
= 101.3(rn/Ma)
Time =115.7(Ma)
Velmax = 97.4-(m/Ma)
Time = 71.3(Ma)
Velrnax
101
where a = 5.9 x 10-' rn-', P = 1.6 x l ~ . . " 'm-I.
a = 5 x lo-' kg/mJ and - y is the depth in
meters (from Biot and Ode, 1965).
For the salt:
772 -
IO'~-'' Pa s; h , = lo3 rn;
p, = 2.2 x
lo3 kg/m3
where hyit is the initial thickness of the overburden. The horizontal length of the model, 20 x 10'
Fin 2 Development of salt diapirs for g = 1 0 6 rn2/Ma. "Mushroom" and "baloon on the string'' shapes. Sedimentstion rate is
500 m/Ma. Viscosities of overburden and salt are constant (ql = lo2' Pa s, 1 2 = lol"a
s).
YU. PODLADCHIKOV ET AL.
m, contains 1.5 periods of a cosine perturbation
(L =: 13.3 km) with an initial amplitude of
Ah ,,,, = 30 rn. The overburden attains to a
final thickness of 5000 m in 8 Ma when sedimentation stops. For the given density-depth relation
(Biot and Ode, 1965) the level of neutral buoyancy for salt is approximately 1000 m beneath the
surface.
To solve the viscous Stokes problem (flow of
non-inertial incompressible fluid) we combine a
Velmax = 278.4(m/Ma)
Influence of erosion rate
Figures 2-4 show how model diapirs develop
for three different rates of erosion. The sedimen-
Time = 83.8(h/Ia)
I
I
Velmax = 118.6(m/Ma)
Lagrangian finite-element code with markers developed by Poliakov and Podladchikov (1992) and
adapted for problems involving sedimentation,
erosion and redeposition by Poliakov et al. (1994).
Time = 58.9(Ma)
Fig. 3. Development of salt diapirs for K
= lo7 m2/Ma.
All other parameters are as in previous model. Note "wine glass" shape in
top picture.
h
NUMERICAL MODELS O F COMPLEX DIAPIRS
193
L
1
Velmax = 176.O(m/Ma)
Time = 75.7(Ma)
Velmax = 656.8(m/Ma)
Time = 66.5(Ma)
I
I
Velmax = 131.9(m/Ma)
Time = 39.9(Ma)
Fig. 4. Development of salt diapirs for K = 10" mL/Ma. Note "Bayreuth beer glass" shape in middle picture.
tation rate, viscosities and transport coefficients
are given in the captions for each figure.
Very slow erosion leads to model diapirs with
"bulb" or "mushroom" shapes, as in earlier numerical studies (Fig. 2). This shape corresponds
to the classical Rayleigh-Taylor instability with
high viscosity contrast in a box with either free
slip or free stress at the upper boundary. Increas-
ing the transport coefficient K accelerates erosion of the overburden (and any surface salt)
from topographic highs above growing diapirs
and resedimentation in adjacent topographic lows.
This redistribution of surface sediments is equivalent to applying an additional force at the surface
which promotes diapiric ascent. Consequently,
initiated instabilities grow faster and diapirs sur-
-
--
YU. PODLADCHIKOV ET AL.
face earlier as K increases. As diapirs reach the
upper surface by buoyant flow, they begin to vent
by channel or pipe flow (Weijermars et al., 1993).
Viscosity of s e d i m e n t s
*
1017 ( P a s)
Fast redistribution leads to "column"-shaped
diapirs (Fig. 4) Intermediate erosion coefficients
lead to a mixture of "bulb" and "column7' shapes
Viscosity of s e d i m e n t s
0
l
"
'
l
*
"
1017 (Pa s )
r
l
'
Velmax = 4.8e+02(m/Ma)
Time =
20.(Ma)
Velmax = 3.8e+03(m/Ma)
Time =
5.8(Ma)
Velmax = 8.5e.t02(m/Ma)
Time =
12.(Ma)
Velmax = 3.8e+03(m/Ma)
Time =
3.7(Ma)
'?.O(Ma)
I
Velmax = 3.3e+03(m/Ma)
Time =
3.2(Ma)
Time =
0.91(Ma)
I
I
Velmax = 2.0e+03(m/Ma)
Time =
I
I
Velmax = 1.0e+03(m/Ma)
Time =
I
I
I
2.7(Ma)
Velmax = l,Be+03(m/Ma)
Fig. 5. Development of salt diapirs in overburdens with different viscosity profiles (shown at the top of each column). In (a) the
viscosity increases exponentially with depth, from 10" Pa s at the surface, to lo2' Pa s at 5 km deep. In (b) there are two layers
with viscosities of 1017 Pa s from 0 to a depth of 1000 m (light grey) and lo2' Pa s below 1000 rn.NBL is the neutral buoyancy
level.
NUMERICAL MODELS OF COMPLEX DIAPIRS
(Fig. 3) that can be referred to as a "wine glass"
(G. Ranalli, pers. commun., 1992). The structure
in the middle of Figure 4 resembles a "Bayreuth
beer glass" (D. Tetlev, pers. commun., 1992).
We know of no natural example of salt structures with these profiles, but we suspect that this
reflects, not their definite absence, but more our
incapacity to image natural diapirs with these
shapes. Poliakov et al. (1993) obtained structures
similar to naturally occuring diapirs in Texas and
in the Gulf of Mexico, by modelling diapirism at
infinitely fast erosion rate.
of a diapiric sheet may occur at an abrupt step in
the viscosity profile. Triangular profiles of salt
diapirs like that in Figure 6b (from Fiduk et al.,
1989) have been attributed to lateral extension of
stiff overburden (e.g., Vendeville and Jackson,
1992). However, although the triangular shape of
the diapir in our model is certainly attributable to
an overburden that is stiff (but highly viscous and
thus incapable of faulting), no lateral extension
was involved.
Influence of overburden viscosity
Many of the diapirs seismically imaged on the
continental slope of the Gulf of Mexico are
strongly asymmetric (Wu et al., 1989). The upper
surface of the floor of the Gulf is not horizontal,
but slopes gently away from land. This slope may
strongly influences the development of diapirs
growing beneath it, ensuring their asymmetry
(e.g., Talbot, 1977, 1992). We model this situation
by keeping the slope of the upper surface constant over time. The set of parameters is the
same as in the previous model with a low-viscosity
layer at the top (Fig. 5b) and differs only by a
constant slope (only 0.1") of the upper overburden. Figure 7a shows the modeled asymmetric
diapirs which are similar to the asymmetric structures observed in nature (Fig. 7b).
We consider two viscosity profiles likely to
occur in sedimentary sequences: exponential and
step-like. The results of diapir ascent in overburdens of each of these types are presented in
Figure 5a and b. To emphasize the differences,
our numerical experiments were carried out for
large rates of dispersion of topography on the
surface of an overburden with constant thickness.
Our erosion removed and dispersed any salt, at
the surface as well as the overburden.
For an overburden with exponential viscosity
profile (Fig. 5a), the decrease in viscosity at shallow depths helped a mushroom-shaped diapir to
spread laterally along its level of neutral buoyancy. By contrast, in an overburden with step-wise
viscosity profile, the high viscosity contrast between the salt and the overburden below the step
retarded lateral spreading. Instead, the diapir
rose with a triangular shape until it reached the
low-viscosity level. Here it passed through a
"Napoleon Hat" stage and then rapidly spread as
a thin sheet of allochthonous salt in the lowviscosity layer (Fig. 5b). In practise, our numerical
code cannot handle viscosity contrasts greater
than ten thousand. This means that triangular
and "Napoleon hat" diapirs (Fig. 5b) are currently the closest we can reach to simulating the
intrusion of salt sills under air, water or unconsolidated sediments (Nelson, 1991).
Figure 6 compares two such numerical models
with seismic profiles of two diapirs in the Gulf of
Mexico. The obvious similarity between Figure 6a
and c suggests that the sudden sideways intrusion
Diapirism on a continental slope
Conclusions
Some of the diapiric shapes illustrated here
have been simulated numerically for the first
time. Whereas Poliakov et al. (1994) demonstrated the erosion effect first discussed by Biot
and Ode (1965), here we have emphasised how
sensitive is the shape of diapirs to: (1) the rate of
redistribution of surfacial sediments over the upper surface; (2) the viscosity profile of the overburden; and (3) any slope in the system. The
main results follow.
(1) Increasing the rate of diffusion of a potential diapir-induced topography changes the shapes
of diapirs from "bulbs" (at low rates of surface
sediment dispersal) through "wine glass" shapes
(for intermediate rates) to "beer glass7' or "column" shapes (at high dispersal rates). Diapiric
(b)
... ?MILE
--
- -
--
Velmax = 3.5e+03(m/Ma)
Time
=
I
5.6(Ma)
NBL
Velmax = 3.3e+03(m/Ma)
Time =
3.2(Ma)
Fig. 6. Comparison between seismic profiles from the Gulf of Mexico after (a) Nelson (1991) and (b) Fiduk et al. (1989) with numerical models of diapirism with a low-viscosity layer
(and a high rate of surface dispersion). Both the triangular diapirs (b and d) and the "Napoleon hat" diapir in (a and c) are probably due to the step-wise viscosity profile in the
overburden. Neither of these models involved any lateral extension.
4
C
g
NUMERICAL MODELS OF COMPLEX DIAPIRS
(a)
Velmax = 4.5e+03(m/Ma)
Time =
Tie Figure 58
I
3.8(Ma)
SSE
,,
PRE-MIDDLE JURASSIC
10
10
Fig. 7. Numerical model (a) shows a diapir which has risen through a high-viscosity overburden (dark) and then spread as an
allochthonous sheet asymmetrically in a layer with a viscosity a thousand times smaller (pale). The diapiric sheet is asymmetric
because of a constant 0.1" slope on the top. Interpretation of seismic profile (b) shows a salt diapir with similar asyrnmetly under
the Gulf of Mexico (from Wu et al., 1989).
stems neck to shallower levels as the surface
redistribution rate decreases. The deeper the
necking stem, the more difficult it is to image
seismically.
(2) If the viscosity increases step-wise with
depth, diapirs are triangular at depth because the
viscosity contrast there is large; no faults are
associated with the crest and there is no need to
invoke lateral extension. Rapid spreading of
"Napoleon hat" diapirs in the low-viscosity layer
results in wide allochthonous sheets, like many in
the Gulf of Mexico.
(3) Very low topographic slopes (e.g., 0.1") at
the top of the sedimentary overburden results in
model diapirs with allochthonous sheets that are
strongly asymmetric.
Acknowledgements
Ronald van Balen kindly helped us with his
explanations of erosion processes. Hans Herrman
and Dave Yuen are thanked for their support
and generous hospitality to Alexei Poliakov and
Yuri Podladchikov during their visits to Jiilich
and Minnesota. We also gratefully acknowledge
the Minnesota Supercomputer Institute, and the
YU. PODLADCHIKOV ET AL.
WLRZ at KFA Jiilich for use of their excellent
computing facilities. Alexei Poliakov is very grateful to Francis Lucazeau and Jean Chery at Centre Giologique et G60physique7 Montpellier for
help during the completion of the paper. Catherine Astier is thanked for assistance and hospitality provided to A. Poliakov in Montpellier. This
work is part of the project of Russian-Swedish
research collaboration sponsored by the Swedish
Royal Academy of Sciences, who financed Yu.
Podladchikov's year at Uppsala University.
References
Biot, M.A. and Ode, H., 1965. Theory of gravity instability
with variable overburden and compaction. Geophysics,
30(2): 215-227.
Fiduk, J.C., Behrens, E.W. and Buffler, R.T., 1989. Distribution and movement of salt on the Texas-Louisiana continental slope, Garden Banks and Eastern East Breaks
areas, Gulf of Mexico. In: Gulf of Mexico Salt Tectonics,
Associated Processes and Exploration Potential. Tenth
Annual Research Conference, pp. 39-47.
Jackson, M.P.A. and Talbot, C.J., 1991. A glossary of salt
tectonics. Bur. Econ. Geol. Univ. Tex. Austin, Geol. Circ.
91-4.
Nelson T.H., 1991. Salt tectonics and listric normal faulting.
In: A. Salvador (Editor), The Gulf of Mexico Basin (The
Geology of North America, J.). Geol. Soc. Am., Boulder,
CO, pp. 73-89.
Poliakov, A.N.B. and Podladchikov, Yu. Yu., 1992. Diapirism
and Topography. Geophys. J. Int., 109: 553-564.
Poliakov, A.N.B., Van Balen, R., Podladchikov, Y., Daudre,
B., Cloething, S. and Talbot, C.J., 1994. Numerical analysis
of how sedimentation and redistribution of surficial sediments affects salt diapirism. Tectonophysics (in press in
special volume on Sedimentary Basin Evolution).
Romer, M. and Neugebauer, H.J., 1991 The salt dome problem: A multilayered approach. J. Geophys. Res., 96(B2):
2389-2396.
Talbot, C.J., 1977. Inclined and asymmetric upward moving
gravity structures. Tectonophysics, 42: 159-181.
Talbot, C.J., 1992. Centrifuge models of Gulf of Mexico
profiles. Mar. Pet. Geol., 9: 412-432.
Talbot, C.J. and Jarvis, R.J., 1984. Age, budget and dynamics
of an active salt extrusion salt in Iran. J. Struct. Geol., 6:
521-524.
Vendeville, B.C. and Jackson, M.P.A., 1992. The rise of
diapirs during thin-skinned extension. Mar. Pet. Geol., 9:
354-371.
Woidt, W.D., 1978. Finite element calculation applied to salt
dome analysis. Tectonophysics, 50: 369-386.
Weijermars, R., Jackson, M.P.A. and Vendeville, B., 1993.
Rheological and tectonics modelling of salt provinces.
Tectonophysics, 217: 143- 174.
Wu, S., Cramez, C., Bally, A.W. and Vail., P.R., 1989. Evolution of allochtonous salt in the Mississipi Canyon area. In:
Gulf of Mexico Salt Tectonics, Associated Processes and
Exploration Potential. Tenth Annual Research Conference, Abstr. Progr., pp. 161-165.
Zaleski, S. and Julien, Ph., 1992 Numerical simulation of
Rayleigh-Taylor instability for single and multiple salt
diapirs. Tectonophysics, 206: 55-69.