poliakov nato 93
TRANSCRIPT
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AN EXPLICIT INERTIAL METHOD
FOR
THE SIMULATIONOF VISCOELASTIC
FLOW: N
EVALUATION OF ELASTIC EFFECTS
ON
DIAPIRIC FLOW IN
TWO
AND
THREE LAYERS
MODELS
A.N.B.
POLIAKOV~
HLRZ
FA-Jglich
Postfach 191 3 D- 517 0 Jiilich Germany
P.A. CUNDALL
Itasca Consulting GroupInc.
1313 5th Street Minneapolis M N
55414,
USA
Y .Y.
PODLADCHIKOV
Institute of Experimental Mineralog y Che rnog olovk a
Moscow District
142
432, Russia
V.A.
LYAKHOVSKY
Department of Geophysics and Planetary Sciences
Tel-Aviv University Ramat-Aviv
69 978 Tel Aviv Israel
ABSTRACT. The explicit finite-difference approach used in the FLAC (Fast
Lagrangian
Analysis of Con-
tinua) algorithm is combined with
a
marker technique for solving multi-component problems. remeshing
procedure is introduced in order to follow the viscoelastic flow when
a
Lagrangian mesh is
too
distorted.
Dimension analysis for the case of Maxwell rheology is made. The adaptive density scaling for increasing
time step of explicit scheme nd influence of inertia
are
expIained,
Analytical and numerical examples of Rayleigh-Taylor instability with different Deborah, and Poisson s
ratios are given. three-layer model with a high viscous upper layer representing the lithospherehas been
studied. Amplification of stresses in the upper layer due to unrelaxed elastic stresses and topography eleva-
tion for different
De
number and viscosity contrasts is calculated.
1. Introduction
Modelling
of
viscoelastic
flow
in geophysics is still
a
very difficult problem which is quite
different from classic numerical modelling
of
purely viscous or purely elastic media. The
fundamental problem in modelling viscoelastic flow is the mixed rheological properties which
result in dependence of the stress on
the
history of loading. Some finite-element models of
viscoelastic behavior are shown b y Melosh and Raefsky
(1980)
for a fluid with a non-
Newtonian viscosity, and by Chery et al. (1991)
for
coupled viscoelastic and plastic behavior.
lPresent address:
ans
Ramberg Tectonic Laboratory Institute of Geology, Uppsala University, Box 5 5 5
751 22 Uppsala, Sweden
D.
B.
Stone and S.
K
Runcorn eds.},
Flow and Creep in the Solar System: Observations, Modeling and Tlzeory
175 195.
1993
Kluwer Academic Publishers. Printed in the Netherlands.
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Both techniques are very powerful, but they simulate relatively small deformations and thus
are limited by the distortion of the Lagrangian grid.
Therefore, it is important to introduce new non-traditional methods to model complex
rheologies over long periods of time and which are convenient for remeshing. Numerical
methods using the explicit form of the constitutive relation between stress and strain are most
appropriate for these purposes (Cundall and Board, 1988). Explicit methods have very short
time increments which are chosen to be small enough that perturbations can not physically
propagate from one element to the next within one t ime step. However, the computational
effort per time step is very small due to the fact that no system of equations
necds to be
formed and solved. By performing many short time steps, it is easy to model flows with non-
linear rheologies provided that certain stability criteria are satisfied.
However, the simulation of viscoelastic systems requires modelling processes an both short
time scales (elastic behavior) as well as on very long ones (viscous flow) simultaneously, Thus
explicit methods require a large nurnber of time steps for full simulation. It is important,
therefore, to maximize time step during the calculations. As shown by Cundall (1982), this
can be accomplished via a density scaling, provided that inertial forces are negligible.
Furthermore, a typical difficulty for the large strain problems is that deformable
Lagrangian mesh is required. At some point in the simulation, the distortion of the mesh is so
great that it is impossible to continue calculations. A combination of marker tracem which are
moving with the grid is found most preferable and fast (Poliakov and Podladchikov,
1992).
Markers are used during remeshing for interpolation of physical properties of the syxtcm
wilh
sharp material discontinuities. However the problem of interpolating the stress stale
in during
the remeshing procedure remains open.
A
combination of the
l AC
technique
and
a remcshlng procedure allows us to sirnul te
thc Rayleigh-Taylor
RT)
instabilities in viscoelaslic media, but with somc limitations
explained in a later section.
On the basis
of
dimensional analysis and numerical calculations wc show
th t
id lc Dcbarilh
number
e
(equal to
the
ratio of the Maxwell relaxation time
of
viscoclastlc
m teri l
to the
characteristic time of viscous flow) and Poisson s ratio control
different
types of viscae1ar;tic
behavior. e number is also equal to
the
ratio of thc stress magnitude to the shear moduli
for Maxwell
type
rheology.
We
note that the Maxwell rclaxation time
z
(ratio o viscosity
q to shear moduli G affects only the time scale over which flow occurs nd docs not affect
the qualitative behavior,
Wc show that the
RT
instability grows faster as the Deborah number ncrcascs and Poisson s
ratio dccreascs,
The estimated limits for thc Deborah number of the upper crust arc 104-10-~and 10 3 10 2
for thc upper mantle.
We
show that thc flow exhibits viscous behavior for e = I O ~ - I O - ~nd
viscoelastic bchavior for e >
10 2.
We
show that the elasticity has a strong influence on the time evoluiion of
topography
and
stress in the lithosphere. This occurs when the viscosity contrast between the
lithosphere
and
the
underlying mantle is greater
than lo4 for
e =
10 210 3.
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2
Numerical Method
2.1.
TH
ONCEPTUAL BASIS OFU
The method used in FLAC Fast Lagrangian Analysis of Continua) employs
n
explicit, time-
marching solution of the full equations of motion Cundall and Board,
1988;
Cundall,
1989 .
The general procedure basically involves solving a force balance equation for each
grid-
point in the body
whcre
vi
is velocity and
i
is force applied to a node of m ass m. Or in its general form,
where
p
is density,
gi
is acceleration due to gravity, and
ojj
is
the
stress tensor.
Solution of the equations of motion provide velocities at each of the gridpoints which are
used to calculate intcrnal clement strains. These strains are used in the constitutive relation to
provide clemcnt stresses and equivalent gridpoint forces. These forces are the basic input
necessary for the solution of the equations of motion on the nex t calculation cycle.
Although the dynamic motion equation is implemented, the mechanical solution
is
limited
to equilibrium or stcady condition through the use of damping to extract oscillation energy
from the
system,
2.2 GEUE3BAL NUMERICALPROCEDUIZE
The
cornputationd mesh consists
of
quadrilateral elements, which
are
subdivided
into
pairs of
constant-strain triangles, with different diagonals. This overlay scheme ensures symmetry of
the solution by averaging results obtained on two meshes Cundall and Board,
1988 .
Lincar triangular l m nt shape functions Lk can be defined as follow s e.g. Zienkiew icz,
1989
LI a k P k x l ~ h x 2 , k : =1 3
3 )
where Uk and y arc constants and
~ 1 x 2 )
re
rid coordinates. These shape functions are
used to linoarly interpolate thc nodal velocities v p within each triangular elemen t el. This
yields
tho
fallowing equation for velocity
vik
at any poin t x,y) within
an
element
k=1
This formula enables the calculations of the strain increments A ] in each triangle e ) as
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178
wh r
At this stage, a mixed discretization scheme is applied in order to overcome the mesh
locking problem associated with the satisfying incompressibility condition of viscous or
plastic flow
Marti
and Cundall,
1982).
The isotropic part
of
strain is averaged ovcr cach pair
of triangles, while the deviatoric components are treated separately for each triangle. This
procedure decreases the number of incompressibility constraints by two times and prcvcnis
the mesh from locking.
Element stresses
re
computed invoking
a
constitutive law
where
the operator
M
is the specified constitutive model, and Sj are state variables
which
vary
with constitutive models.
When the stresses in each triangle are known, the forces at node n,
~ i ( )
re calculatcd by
projecting
the
stresses from
all
elements surrounding that node. The projection of
stresses
adjacent triangles onto the n th node is given by
where
n
is j s component of the unit vector normal to the each of two
elerncnt
s i b s
adjacent
to node n. The length of each side is denoted by
Af
Thc minus sign is a consequence of
Newton's Third Law. After the stresses are
projcctcd the
gravitational force acting
an each
node is determined
and
the force on each node is updated
as
follows
where m n) is an equivalent mass
of
node n ) obtained
by
distributing continuous
density
ncld
to discrete nodes.
Once the forces are known, new velocities are computed by integrating aver
a
givcn time
step
A t
where
min rt
is inertid mass of the node
which
can vary during calculations
(see
ection
2.31,
and is a damping parameter.
If
a
body is at mechanical equilibrium. the net force ~ i ( ) n each node is zero;
othcrwlsc.
the
node is accelerated. This scheme allows the solution of quasi-static problems by damping
the
oscillation energy. The damping term
a ~ ( ~ 1
ign vi) is proportional to
the accelerating
(out-of-balance) force and a sign opposite to velocity to ensure the dissipation o cncrgy
This term vanishes for the system in steady-state.
New coordinates
of
the grid nodes can be computed
by
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and
then
calculations are repeated for new conflguration.
This method
has
an advantage over implicit methods because it is computationally inex-
pensive for e ch time step nd it is memory cfficicnt because matrices storing the system of
equations arc: not ~qui red ,
2 s. IME
T P
ANX ADAFrZTVQDENSITY SCASMU
The choice of the
prspcr
time step fur the time-dependent; calculations is a crucial point for
st;ntaility, precision and
run
timc of the calculations, The time step must be chosen in such a
way
that infarmatian cannot physically propagate from one element to another during one
cdalculatian cycle, For clastic and viscoelastic models the critical time step dtcriris the mini-
mum
of
the Maxwell relaxation t h e and propagation of the elastic compression wave across
a distance equal to local
grid
llrpacing Ax This statement can be written as follows
where K and 3 we bulk and shcar elastic moduli, q is shcar viscosity, The inertial density
piner
an be treated as relaxation parameter, and can be adjustcd during a calculation
in
arder
16
obtain
a
cle slreci ffcct.
If
we
msurna
reasonable walucs for the density and elastic moduli
thcn
the time step A t will
be
very
emnX
and
equal to
only a
few
seconds
far
typical geophysical problems. Therefore,
the
sirnutatfan o f creeping
flow
which occurs over hundrcds of thousands ycars, will require
too
many time 8tcpg for a fill simulation,
ne
means
o f
circumventing this problem
s
the adaptive
density
scaling Cundall,
1982).
For
quasiestatic:
pmblcms,
o xo
acceleration
sf
the
system
s
nearly zero. Thus, it
is
possible to
lnescase
the:
value
o f
Inertid density, providing that inertial forces
mi ner l
9i are small
comptlX Ca to
the other forces in the system I. ;, gravitstianal
bady
force).
From eq.
12 we can
see that
c i ~ ~ i t i n e r l
(13)
and
therefore in order to
increase
the timc step it
is
necessary to scale inertial density
properly,
preserving the
stability
OI
the scheme.
Note
that
pinerr
is different from the density
used
for calculation
of the
gravitational body
force. The
aIgorithm
is designed in such a way
that
i f tke accelerating (i,e, out-of-balance} forces are
smaller
then a certain value,
then
time
step
ntnd
lnartiat
density
are Increased
Cunddl, 1982).
For creeping flow simulations i t is necessary to ensure that inertial forces remain small
compared to viscous forces Last, 1988). The Reynolds
number
is a measure of
the
ratio of
these
zwa forces.
We:
choose to write the hynolds numbcr ollows
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where V and are th e characteristic velocity nd length and T is viscosity of creeping flow.
This number s estimated in at each time step cycle and constrains the growth of inertial
density. W e will show below ho w this param eter affects the dynamics of the simulations.
2 4
METHOD
O M RKERS
ND
R M SHING
There are
many
problems in geophysics which require simulating the dynamics of several
phases with different m aterial properties
and
rheologies simultaneously. For example, the case
of Rayleigh-Taylor instability when a lower density fluid rises up and displaces another fluid
of higher density.
Lagrangian methods, where th e mesh deform s with
the
fluid, are very fast
and
are easier to
implement
th n
other methods. However, this approach fails when the mesh becomes too
distorted.
Fixed Eulerian mesh es comb ined with the metho d of markers Hirt and Nichols, 198
1;
Weinberg and Schmeling, 1992 avoid this problem. This method is robust for finite-
difference algorithms
on
rectangular grids but requires a lot of computational time for non-
regular triangular meshes.
The combination of a moving Lagrangian mesh and a m ethod of markers was found to
be
optimum Poliakov arid Pod ladch ikov ,
1992 .
The idea of this technique can be explained s
follows. t the initial stage, material properties of
the
different layers are assigned to each
element and to the markers. Also, within each element the Cartesian coordinates of the
markers ar e converted to local coordinates area coordinates for triangular elements).
At each time step the grid nodes are then updated according to eq. 11. Th is L agrangian
movement is very fast because i t is only necessary to move the mesh nodes with known nodal
velocities.
When the mesh becomes too deformed, It s necessary to remesh. Since the local coordi-
nates of
the
markers remain unchanged during the Lagrangian movement of the mesh, the
Cartesian coordinates of the markers can be obtained
by
simple interpolation from the nodes
of the elements. Only at this stage is it necessary to interpolate from the markers to the array
containing material properties of each element. This is in contrast to the Eulerian method
where this interpolation must be performed at every time step.
The advantage of this procedure becomes very important in the case of explicit methods
which require many time s teps and only few re mesh ing procedures. Another essential advan-
tage of this method i s
that
there are only substantial derivatives on time in constitutive laws
compared to the partial derivatives in space and ti m e required o n a Eulerian mesh).
In the case of
the
viscoelastic rheology we face the additional problem of interpolating the
stress field during remeshing. For triangular elements, stresses are piece-wise discontinous
across elements. Thus considerable interpolation error can occur after remeshing which can
lead to unbalanced stresses within the system. Because of the strong elastic response of
the
system these unbalanced stresses result i n un desirable acceleration and oscillations of nodes.
Damping of these non-phy sical oscillations cau ses t h e loss of the history of loading. In other
words, stresses and velocities w ill have jump s and oscillations after each remeshing. Therefore,
the results of this paper are partly based on calculations where remeshing is delayed as long
as possible to characterize the initial response.
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3 Algorithm for the Simulation
of
Maxwell Behavior
It
is convenient to study the response of a viscoelastic material in shear and dilatation
separately. Thus the stress
Oij
and strain Ei j tensors are decomposed to their deviatoric sq,
i
and isotropic parts
Oii i i
as follows
The rheological constitutive relations are also separated into their deviatoric and volumetric
parts. Recalling that for a linear Maxwell viscoelastic material elastic
and
viscous strains add
and stress components are identical, the constihltive relation for the deviators is
where
G
is the elastic shear modulus and q
is
the shear viscosity. Due to the fact that bulk
viscosity does not play an important role and rocks respond elastically in dilatation the
constitutive law between isotropic stresses and strains is purely elastic,
;; ~ I E ~ ~ 18)
where is the elastic bulk modulus.
Equations
17-18
are solved t each time step i.e. stresses are updated from the previous
time step) as follows.
First, the isotropic and deviatoric components of the initial stress oij and strain increments
A E ~re calculated for the current time step using eq.
5
and 16.
The finite-difference discretization in time of eq. 17-18 gives
where prime accent
( )
denotes the variables at the end of the time step
At
Note the semi-
implicit approximation of stress in
the
term
corresponding
to the viscous strain increment.
Thus the deviatoric
nd
volumetric stresses are updated as
AtG tG
1 = s i j . 1 -)+ 2Ga e i j ) / l + -1
s i j
277 277
a ;=
;;
3 K ~ i i
and then
ful l
stress tensor will be
I
i j
=
oii sij.
When the stresses in each element are known,
the
net forces acting
on
each node, updated
velocities and coordinates are calculated as described
by
eq.
8-11
following to
the
general
FLAC
algorithm Section
2.2).
This algorithm is applied for the plane strain formulation,
therefore
~~ = 0
but q ust be calculated during the calculations.
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4. Dimension Analysis for
Maxwell Rheoiogy
In order to ensure rheological
nd
dynamic similarity between numerical models
and
simu-
lated natural phenomena, the correct scaling of the constitutive rheological model (eq. 17
and momentum equation (eq. 2 is required e.g. Weijermars and Schmeling,
1986 .
Equa
tions are reduced to a convenient form containing scaling parameters or non-dimensional
numbers equal to the
numbers
estimated
from
nature.
To nondirnensionalize the rheological law (eq.
17
we scale stress, time and physical
properties as follows
S ; j
SSL
Tt , 709 , K
Kol< ,
G GoG
where a prime indicates non-dimensional quantities, and S,T are some characteristic values,
which will be introduced below. Substituting these expressions into eq. 17-18 gives
Here we need to choose a characteristic time scale
T
and there are two possibilities, either to
choose a characteristic viscous time where
or Maxwell relaxation time
If we choose the viscous time scale then eq. 25-26 become
where De is the Deborah number, equal to the ratio of the viscoelastic to the viscous
characteristic time
T ~ e l a a : S
De =z
.
visc
O
It is interesting to note that the scaling viscosity factor qo is excluded from eq. 30
and
affects only the time scale
of
the process but not its qualitative behavior. In other words, the
rheological behavior of two Maxwell bodies with two different scaling viscosity factors is
similar and differs on y in the time scale.
Since variations in density often drive the flow in geophysical problems, the characteristic
stress S is chosen to be the hydrostatic pressure
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where
Apo
and are scaling density and length factors defined as follows,
Using these relations, the Deborah nw nber can be rewritten as
This definition will be used in the present work. Weijermars
and
Schmeling 1986) performed
the scaling analysis fo r themomentum eq. 2 and showed that if the characteristic time scale is
chosen to
be
viscous (eq. 27 then
where R e is the Reynolds number (see eq. 14). For a system with low inertia (Re c
1
in
nature) the left-hand side of eq. 35 can
be
neglected. Then the remaining part of the equation
does not contain
any
scaling parameters or non-dimensional numbers. It means the model
nd its natural analog are dynamically similar since they can
be
described by
th
sam e non-
dimensional equation of motion (eq. 35 and the same density field.
Finally we arrive at the conclusion that the numerical solution for a viscoelastic fluid is
similar to the modelling of natural phenomena under the same boundary conditions and
geometry of the modelling object if:
1)
inertia forces in the model are small compared to
other forces dynamic similarity),
2) De
numbers are equal and distribution of elastic moduli,
viscosity
and
density field are similar rheological sim ilarity).
Note that two viscoelastic bodies with different relaxation times will behave similarly, if
their ratio of stresses to elastic moduli are equal. In this case the difference in relaxation time
will affect only the time sca le.
5 Analytical Analysis of the Bottom Boundary Layer with Viscoelastic Rheology Lon g
Wavelength Case
In this section we will consider the behavior of a viscoelastic medium with buoyantly driven
flow because it is a common mechanism of flow in the Earth. The analysis of a simple two-
layer system can help us to understand the physical behavior of the viscoelastic media in
a
gravity field.
Biot
1965)
performed an analytical stability analysis for layered viscous and viscoelastic
media. One of the cases he considered is the stability of
a
low density layer overlain by an
infinite viscous layer of higher density. A generalization of his analysis can easily
be
done for
a viscoelastic rheology, but only for the case of two la ye rs w ith the sam e viscosity nd elastic
moduli.
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Figure : Viscoelastic layer with rigid base lying under an infinite viscoelastic fluid in
a
gravity field.
Fig.
1
shows the geometry
and
variables used in our analysis following Biot,
1965). The
bottom layer of density p 6p , thickness h lies under a semi-infinite fluid of density p. Both
layers have viscosity q and elastic shear moduli G. A small sinusoidal perturbation of
wavelength L is applied on the interface with displacements
U1 l.
Displacements on the
bottom boundary are zero
U V 0
Hence the problem may be formulated entirely in
terms of
the
two displacement components
U1 Vl
on
th
top of the layer, and reduced to
a
system of differential equations.
The characteristic solutions for stresses and displacements of differential equations are
proportional to the same exponential factor, exp p t ) , where
p
is the growth rate factor.
The
displacement field ui and stresses og are then written
where the amplitudes u ~ x ) ,r i j are functions only coordinates xi, while the time appears only
in the
exponential factor.
These characteristic solutions are obtained by substituting eq. 36 into the rheological Eq
17
and the momentum equation and the application of boundary conditions for the visco-
elastic medium. Because the equations are homogeneous, the exponential term is factored
out.
An important advantage of this approach is that
the
characteristic equation is obtained
immediately by treating the derivatives as algebraic quantities.
Substituting the stresses and displacements into the constitutive eq. 17 for a Maxwell body,
we obtain
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where a@ and
E
are the deviatoric stress and strain which are on y functions of position.
For
th
case of pure viscous flow, Biot 1965) gives the expression of
growth
factorp as
where
a
s a nondimensional parameter that depends on boundary conditions
and
the wave-
length of applied perturbation.
Following Biot s derivation and making the following substitution
P I ==
V
P r ~ e ~ a z
we obtain the following expression for p
The difference in growth factors
p
between the viscous and the viscoelastic cases is
which means that if
G
goes to infinity
then
the influence of elasticity will be excluded and the
system will have a purely viscous behavior.
If
this ratio increases then
the
instability will
grow
faster than for a simply viscous instability. In other words, the elasticity term accelerates the
instability. Theoretically
a
resonance can be reached when this ratio
equals
a This case is
irrelevant geophysically
Note this analysis only applies in the
small
deformation limit and for the isoviscous case.
Therefore the influence of elasticity can be much higher when the deformations become non-
linear and for fluids with a high viscosity contrast.
6 Numerical Modelling of Viscoelastic Diapirisrn
6.1. NFLUENCE
OF
NERTIA
TUl
TEP AND GRIDSIZE:TWO LAYER CASE
A
viscoelastic analog of the classic Rayleigh-Taylor instability is the problem of a viscoelastic
layer overlain
by a
viscoelastic layer of greater density. The model geometry is shown in Fig.
2 The two layers are described by their density, viscosity, shear modulus and thickness;
p ,
q
G
v
and h respectively. Along ll sides the free-slip condition was chosen, The aspect ratio of
the box is equal
to
one. An initial sinusoidal perturbation of magnitude
0.05(hl
h2 is
superimposed on the boundary between layers. The physical properties of the upper layer
were chosen as scaling parameters (eq. 24, 32,
33).
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Because w use the inertial method for studying non-inertial systems it is necessary to show
the influence
of
inertia on our results. It is always desirable to increase the time step in
explicit simulations because of the strong limitation imposed by the stability criteria. It was
s own hat using
the
adaptive density scaling the time step could be increased by increasing
the inertial density (see eq. 13).
This relation shows th t the influence
of
inertial forces can be controlled by limiting the
Reynolds number
see
eq. 14). In our calculations we define a maximum Reynolds number
that limits the maximum time step and keeps the magnitude of the inertial forces relatively
low compared to the viscous forces.
The time-evolution of the vertical velocity on the perturbated interface between two
Maxwell layers is shown in Fig. 3 for
e
0.001 1 he thicknesses of the two layers are
equal
h,
h,
0.5).
The
viscosities and elastic moduli are the same for both layers. The
Poisson s ratio is v 0.25, the density contrast is Ap pl
0.1
and
e
0.01.
As can
be
seen in Fig.
3
velocities at a given time become smaller as the Reynolds number
is increased because it takes more time to overcome inertial effects.
height
Figure 2: Description of the model for the viscoelastic RT instability. In this section, we
consider a model where the rheological parameters of the two layers are equal but the
densities are different.
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De
=
=
0.25
A P / ~
=
0.1
0.014
I I
1
I
-
Re = 0.001
dt <
0.5 T / G
-
................
Re = 0.01
o.o - - - - - - -
Re = 0.01 -0.1
dt = 0 5
r / G
-
R e = O . l
-
, Re = 1
-
5 0,010
-
-
-
0.008
-
-
-
-
.r
-
li
E
0.006
-
-
-
-
0.004
-
.-__- .
-
__--.----
_L__._..---- -
-
20
Time
Figure
3:
Growth rate of
RT
instability versus time for viscoelastic fluid at different
Re
=
p i n e r t V L / ~ t can be seen that the solutions with Reynolds number close to 0.01 show little
inertial effect. Physical parameters of the system are shown on the top of the box.
Increasing inertial effects cause the time step to increase
and
to become close to the
relaxation time. The time step exceeds the relaxation time for Re >
0.1.
Thus there are two
limitations on the time step a maximum Re number and the relaxation time). Depending on
the problem either of these limits is more strict
and
controls the time step.
A
comparison of solutions at various
grid
spacings indicates that a
21
21 grid yields
satisfactory results.
6.2. nfluenceof the
e
Number and Poisson s Ratio
n
eq. 40 we showed analytically the influence of elastic moduli on the growth of
RT
instability. This equation predicts that the incompressible
RT
instability will grow faster in
a
viscoelastic medium than in a purely viscous one. However, our analytical solutions assumed
that the thickness of the upper layer was infinite. We also assumed that the medium was
incompressible. Therefore our numerical solutions with a two-layer system of finite thick-
nesses and with finite compressibility can not be directly compared
with
our analytical
formulas. Through our analysis
of
the nondimensional equations we found that the behavior
of a viscoelastic body depends upon the Deborah number see
eq.
31), the density contrast
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Ap/pl,
and
Poisson s ratio v. This analysis indicated that the instability grows faster at high
Deborah numbers and has a viscous limit at
De
=
0.
This effect is demonstrated in our
numerical calculations for
e
= 10 3 0 I in Fig. 4. The parameters of this numerical model
were chosen the same as in the previous section.
In order to compare these results with those
rom
purely viscous fluid we show the curve
computed by a finite element code that solves the Stokes equation for incompressible flows
(Poliakov and Podladchikov,
1992).
Because the Poisson s ratio can
v ry
from
v
= 0.25 for sedimentary
nd
up to
v
= 0.4 for
ultramafic rocks, it is interesting to see the influence of Poisson s ratio on the dynamics of
instability.
Thus we performed calculations where
all
parameters were fixed except for Poisson s ratio.
Our results are shown in Fig. 5. As we increase Poisson s ratio our calculations approach the
viscous
FE
calculations. The
RT
instability grows faster as the compressibility of the material
increases (at low v . Thus both bulk compressibility and shear elasticity accelerate the diapir
because they provide additional mechanisms of deformation (compared to a purely viscous
and incompressible diapir).
As
an
additional comment on the behavior of compressional systems we consider
an
unstable compressible system with two layers of
the
same thickness
and
an open upper
boundary. The bottom layer has a larger uncompressed volume than the upper layer because
it is compressed more than upper layer due to hydrostatic pressure. Therefore, during an
overturn the volume of the bottom layer will expand and the volume of the upper layer will
contract. When overturning is completed the total volume of the system will increase.
-
EM
Poliakov
Podladchikov,
1992
-
-
-...-.
e
= 0.001
-
-
- -
- .
e
=
0.01
0.25
- -.-.-.
e
= 0.1
-
. -
n
d
;f; -0.30
w
-
-
3
-
f -0.35
-
-
-
-
-0.40 -
-
-
-
0.45
1
, ,
, I ~ I .
0
10
2 30
40
TIME
Figure 4: Height of the viscoelastic diapir for different values of Deborah number D e
=
~ ~ ~ l a ~ ~ i ~ ~
S / G .
For comparison with a pure viscous simulation of RT instability the
FEM
calculations are shown (Poliakov and Podladchikov,
1992).
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e =
loe2
A p / p = 0.1
R e
= 0 01
- o . z o i ~ ' ' ' ' ' ' r ~ ' ' l ~ ~ ' l ' ~ '
-~
I
//.
FEM
/
:/ -
/
./
-...-. =0 25
-
- - - .
=
0 40
; f ,
-0.25
-
-. .-.
=
0 45
G
d
-0.30
-
-
rw -
-
4 -
0.35
-
. -I -
-
-0.40
-
-
-
-
-0.45
0 10
20
30
40
TIME
Figure : Height of the diapir as
a
function of time. Each curve corresponds to a solution with
a different Poisson's ratio v. Note the convergence of the results to the incompressible FEM
calculations with increasing v
6 3
THREE-LAYERMODEL
Wll'Ii
HIGHRELAICATION
ITME
FOR
UPPER LEVEL:
THE
NTERACI1:ON
OFTHE
D W I R
WlTH
THE
HIGH
VISCOUS
LJTHOSPHERE
In this section a three-layer model with a highly viscous upper layer was studied. This third
layer can approximate a more viscous lithosphere overlying two gravitationally unstable
layers (see Fig.
6).
For simplicity the viscosities of two lower layers are chosen to be equal.
Th is choice makes the calculations mu ch faster b ecause
the
critical time step in
both
layers is
the same. If there were a viscosity contrast between two lower layers then the characteristic
velocity in the region would be controlled
by
the layer
with
the highest viscosity. However,
the time step i s limited by lower viscosity as shown above. Therefore the time of calculations
is proportional to
the
viscosity contrast betw een tw o layers.
In
contrast the viscosity of the upper layer does not influence the characteristic velocity of
the diapir and has little effect on the speed of calculations.
Thus
the viscosity of the
upper
layer can be greatly increased compared to th e viscosity of
the
bottom layer (up to six orders
of magnitude in our calculations) with almost
no
change in the computational time. In this
section we examine the influence of the viscosity contrast
q1 q2 nd
e number on the rowth
of the diapir, topography
and
stress evolution.
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9 p1 G h,
or pure elastic
2 9
~2 G9
113 93
v G h3
Figure 6:
A
model representing the interaction of diapir with viscoelastic lithosphere with
high relaxation time q /G >>
qJG
The geometry of this problem is shown in Fig.
6.
The three layers are described by their
densities, viscosities, shear moduli
and
thicknesses p, q, G,
v
and h respectively. The upper
boundary is stress free and the other sides are free-slip. The aspect ratio is equal to one and
an initial sinusoidal
perturbation
of magnitude
0.05 hl h2
h3 was superimposed on the
boundary between layers 2 and 3.
The
physical properties of the intermediate layer were
chosen as scaling parameters eq.
24, 32,
33
Poisson s ratio
was
set
equ l
to
0.25
for all models and
p1
=
p2.
Fig. 7 shows the evolution of the velocity field for the following two cases:
q1/q2= 1
(left
column) and 771 772=
100
(right column). According to our simulations,
the
velocity field
does not significantly depend on the De number. Calculations with viscosity contrast
ql/qz
greater than
102
show velocities which are very similar (compare Fig. 7a and
7b).
This effect
can be observed i n Fig, 8 a,b). The evolution of the diapiric growth does not strongly
depend on the De number. At the same time we
can see that
curves representing 77,/q2 100
are very similar to
each
other and are very distinctive from the case q1/q2=
1.
For q1/q2
>
100
the top layer is effectively rigid and has little participation in the overall flow. The
presence of this layer changes the effective boundary conditions on the f low field nd also
the dimensions of the diapir cell.
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..
Velmax
=
5.6e 03 Time
=
31
Velmax
=
1.3e 02 Time
=
60.
Velmax
= 5.0e 03
Time
=
38.
Velmax
= 8.5e 03 Time =
77
F i g u ~
:
Velocity field evolution for a three-layer system for De
=
Ap l / p z
=
0.1
and
p
= p2
a) q1/q2
1. b q 1 / q 2=
100.
Note
that the high viscosity upper layer is excluded
from
the diapiric cell in the right column.
A different dependence was found
for
topography. which drastically changes only at
the
qlIqz
> 104
his effect can
be
explained
on y by
the high relaxation time of the upper layer.
Unrelaxed elastic stresses in the upper layer resist
the growth of
topography in this
case.
This observation
may be
supported
by
comparing
the
elastic and viscous terms in
the
rheological equation
30
for the
upper
layer. Using
th
nondimensional time interval
t
equal
to
25
(taken
from
Fig.
8)
as
a
characteristic viscous
time,
we can derive an effective Deborah
number in layer
1
where G' is
taken
to
be unity
and 17
~~1772
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0 6
10 15
20 25 6
10 16 20 25
TIME TIME
e = 10 e =
0.750 0.740 0.730 0.720 0.710 0 .700
0.760
0.740
0.730 0.720 0.710 0.700
Height
r diapir
Height of diapir
Figure
8:
The influence of the D e number and viscosity contrast q l h 2 n the evolution of the
diapir
and topography. The evolution of diapiric height with time a,b) and evolution of the
topography above diapir versus height of diapir c,d). The topography and the diapiric growth
rates decrease with increasing viscosity contrast. Note the drastic change at lo2 q /g
104
for the topography whereas for growth rate:
c
q1 q2 lo2).
Substituting the
De
number (defined for
the
whole model) equal to the
gives us
viscous-like behavior De ;/n 1) when ql q2
z 104 and viscoelastic behavior for intermediate viscosity contrast.
The topography increases as the
De number
increases for fixed ~7 117 see Fig. 8 c,d). A
higher
e
number assumes that
the
elastic modulus is softer (other parameters being fixed).
Thus elastic deformations are greater when the Deborah number is lower. The same depen-
dence was outlined analytically
and
numerically in Section 5 for a two-layer model. A
perturbation grows faster and elastic deformations are larger for higher De number. In the
limits
q1 q2 o
and
De
he topography goes to zero (rigid upper layer) which is
consistent with the observed
numerical
dependencies.
The difference between nearly viscous and nearly elastic behavior can be demonstrated by
the two-dimensional distribution
of
the principal stresses as well (Fig.
9).
Strong differences
in the magnitudes and orientations of the principal stresses are observed between models with
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8/17/2019 Poliakov Nato 93
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q1/q2
lo2
and q1/q2=
104
Stresses in the two bottom layers for both cases are
approximately the same and differences are observed only in the upper layer. There are two
contributions to this difference, one due to viscous stresses
and
one to unrelaxed elastic
stresses. The elastic component
can
be seen
from
stress distribution in the upper layer directly
bove
the diapir on the right column. At the bottom of the upper layer the principal stresses
change directions because of the effect of bending of an elastic plate. This change is up to
90
degrees at
the
left boundary.
Again,
as
for
topography, these two examples represent two
types of the mechanical behavior of the upper layer: viscous for q1/q2
lo2
and elastic
type for q1/q2 lo4 .
01 Time =
95.
94.
Shear
m a x
=1.6e-01 Time =
1.3e
Shear rnax
=1.2e 00
Time =
1.3e
Figure 9: Principle deviatoric stresses
at
e =
aximum
stress or each case is shown
at
the bottom of each picture. Scaling stress for
a l l
pictures is
1.1
x
Thick lines represent
compressing stresses. Note the distribution is nearly viscous for = lo on the left and
non-relaxed elastic for
q1/q2 1 4
on the right.
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For geophysical applications it is important to know the magnitude of the stresses on
the
surface for the determination of
the
different tectonic mechanisms. In Fig. 10
the
evolution
of
the horizontal surface stress above the center of the diapir is shown. Magnitude of
the
surface stresses is higher for the higher viscosity ratio q 1 q 2because the relaxation time is
longer in the upper layer. It is interesting to see again the transition between two types of the
behavior: elastic for
qlh >
o2
and v i scou
for
vl/qz<
lo2.
Stress is viscoelastic at the vis-
cosity contrast
1
-
103.
Initially the magnitude of stress grows rapidly because of the rapid
elastic response of layer on the upwelling diapir
and
then the stress expanentially relaxes.
7. Conclusions
We shaw how the explicit inertial technique
FLAC can
be applied to geophysical problems
with l ow inertia. This method can easily simulate phenomena with a viscoelastic rheology.
The problem which
remains open is remeshing for large strains. It occurs due to
problematic interpolating discontinuous stress field from one mesh to another. Combining
Eulerian
and
Lagrangian meshes at the same time and accumulating solution on
the
non-
moving Eulerian mesh may help to solve problem.
Numerical simulations and analytical estimations for the initial stages of the Rayleigh-
Taylor instability
show
that for higher
De
numbers instability grows faster than for purely
viscous where De
=
0 . Estimates of the Deborah number for sedimentary basins and salt
diapirism yields De =
- rom
our results
this
implies that influence of elasticity on
the diapirism in the crust is insignificant for isoviscous cases. If we estimate De for mantle
diapirism, for example diapirs from the 670
krn
boundary, then we arrive
De = 10-3
-
10-2.
According to our results the elasticity plays a considerable role in the interaction of the
lithosphere
and
underlying mantle and can decrease the surface elevation and increase exten-
sional stresses on the surface above t h rising diapir up to one order of magnitude.
Figure 10: Evolution of the horizontal deviatoric stress at the surface above the center of the
diapir for
De =
a) and for
De =
10-3b .
0 + 2 6 . z ~ q 8 r T - . n ~ - . ~ v ~ v ~ ~ ~ ~ ~ . ~ ~ -
De =
l om2
0 . 2 5 . 0 r . q ~ 7 . r ~ 1 0 n u 1 r r v v 1 . , a .
De =
0.20
-..* *..
71/
=
1
r11/77
=
10 - 0.20 -
.- -
-
71/778
=
go3
0.15 - - - -
-
. - - * . *
r11/77
=
to4
ql/?ht =
loa 0.15
-
0.1, s
-
t: 0.10
111 /-= m
/
/*
B
9
0.05
,JI ' - 1
0.05
-
e 5 _ - - - - .
-
P
0.00
b I4
b
. - . m _ _ _ _ . _ . _ _ . . . - * - - * - - -
-0.05
-0.16..
I
, . . I , . . . I . . . I . ,
:
:-.
.
-
--.
.
.
..
- . . . . . I . . . . - . . . . .
- O . I O ~ . - - - - - - - :
0
6
1 15
20
25a
-0.16:. . 6
.
. . .10. . . I . . .
.
.
i
TIME
15
20 26
TIME
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The study of the morphology of diapiric flow including high viscosity lid shows the
decomposition of the diapiric cell into two parts upper rigid lithosphere and inner diapir
cell) for viscosity contrast greater
than
100.
Decomposition in terms of stresses urnelaxed
stresses in the lithosphere and viscous-like distribution in the diapiric cell) occurs
only
for
viscosity contrast higher than
lo4
Acknowledgments A Poliakov thanks David Stone and
the
NATO travel fund
for
making it
possible to take part in the meeting.
Hans emnan
is greatly thanked for the discussion of
the model and providing the excellent
HLRZ
facilities during completion of the program-
ming, calculations, and preparing the manuscript. We thank Dave uen for
the
review.
Christopher Talbot is thanked for the helpful discussion. We are grateful to the Mimesota
Supercomputer Institute and Dave Yuen who supported A. Poliakov during writing of the first
version of the code. Matthew Cordery and Ethan Dawson are greatly thanked for patient
revising of the English of our manuscript. Without Catherine Thoraval and Valentina
Podladchikova it would be impossible to complete
this
work.
Y. Podladchikov was greatly supported by the Swedish Academy of Science during
is
visit
to Uppsala University.
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