AN EXPLICIT INERTIAL METHOD FOR THE SIMULATION OF VISCOELASTIC
FLOW: AN EVALUATION OF ELASTIC EFFECTS ON DIAPIRIC FLOW IN TWOAND THREE- LAYERS MODELS
A.N.B. POLIAKOV~
HLRZ,KFA-Jglich
Postfach 1913,D-5170 Jiilich, Germany
P.A. CUNDALL
Itasca Consulting Group Inc.
1313 5th Street, Minneapolis, MN 55414, USA
Y .Y. PODLADCHIKOV
Institute of Experimental Mineralogy, Chernogolovka
Moscow District, 142 432, Russia
V.A.LYAKHOVSKY
Department of Geophysics and Planetary Sciences
Tel-Aviv University, Ramat-Aviv
69978 Tel Aviv, Israel
ABSTRACT. The explicit finite-differenceapproach used in the FLAC (Fast Lagrangian Analysis of Continua) algorithm is combined with a marker technique for solving multi-component problems. A 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 and influence of inertia are expIained,
Analytical and numerical examples of Rayleigh-Taylor instability with different Deborah, and Poisson's
ratios are given. A three-layer model with a high viscous upper layer representing the lithosphere has been
studied. Amplification of stresses in the upper layer due to unrelaxed elastic stresses and topography elevation for different De number and viscosity contrasts is calculated.
1. Introduction
Modelling of viscoelastic flow in geophysics i s still a very difficult problem which i s 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 a dependence of the stress on the history of loading. Some finite-element models of
viscoelastic behavior are shown by Melosh and Raefsky (1980) for a fluid with a nonNewtonian viscosity, and by Chery et al. (1991) for coupled viscoelastic and plastic behavior.
lPresent address: Hans Ramberg Tectonic Laboratory Institute of Geology, Uppsala University, Box 555,
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.
O 1993 Kluwer Academic Publishers. Printed in the Netherlands.
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 time 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 nonlinear 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 sirnulate
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 that id'lc Dcbarilh
number D e (equal to the ratio of the Maxwell relaxation time of viscoclastlc material to the
characteristic time of viscous flow) and Poisson's ratio v control different types of viscae1ar;tic
behavior. De number is also equal to the ratio of thc stress magnitude to the shear moduli G
for Maxwell type rheology. We note that the Maxwell rclaxation time~,,,z
(ratio of viscosity
q to shear moduli G) affects only the time scale over which flow occurs and docs not affect
the qualitative behavior,
Wc show that the RT instability grows faster as the Deborah number incrcascs 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 De = I O ~ - I Oand
-~
viscoelastic bchavior for De > 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 De = 10-210-3.
2. Numerical Method
2.1. THECONCEPTUAL BASIS OF U
C
The method used in FLAC (Fast Lagrangian Analysis of Continua) employs an explicit, timemarching 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 gridpoint in the body
whcre vi is velocity and Fi is force applied to a node of mass 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 next 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 clcmcnt shape functions Lk can be defined as follows (e.g. Zienkiewicz,
1989)
LI, = a k + P k x l + ~ h x 2 , k : = 1,3
(3)
rid coordinates. These shape functions are
where Uk,pk and yk arc constants and ( ~ 1 x 2are
)
used to linoarly interpolate thc nodal velocities v p ) within each triangular element (el. This
yields tho fallowing equation for velocity vik) at any point (x,y) within an element
k=1
This formula enables the calculations of the strain increments A$]
in each triangle (e) as
178
where
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 are computed invoking a constitutive law
where the operator M is the specified constitutive model, and Sj are state variables which vary
with constitutive models.
) calculatcd by
When the stresses in each triangle are known, the forces at node n, ~ i ( "are
projecting the stresses from all elements surrounding that node. The projection of stresses
adjacent triangles onto the n'th node is given by
where nj 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 minert is inertid mass of the node which can vary during calculations (see Section 2.31,
and a is a damping parameter.
If a body is at mechanical equilibrium. the net force ~i(")on 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 a1 ~ ~ ( sign(vi)
~ ' 1 is proportional to the accelerating
(out-of-balance) force and a sign opposite to velocity to ensure the dissipation o f cncrgy.
This term vanishes for the system in steady-state.
New coordinates of the grid nodes can be computed by
and then calculations are repeated for new conflguration.
This method has an advantage over implicit methods because it is computationally inexpensive for each time step and it is memory cfficicnt because matrices storing the system of
equations arc: not ~ q u i r e d ,
2,s.TIME %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 dtcrir is the minimum 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,,can be treated as relaxation parameter, and can be adjustcd during a calculation in
arder 16 obtain a cle%slrecieffcct.
If we msurna reasonable walucs for the density and elastic moduli thcn the time step At will
be very emnlX and equal to only a few seconds far typical geophysical problems. Therefore,
the sirnutatfan of creeping flow, which occurs over hundrcds of thousands ycars, will require
too many time 8tcpg for a fill simulation,
One means o f circumventing this problem i s the adaptive density scaling (Cundall, 1982).
For quasiestatic:pmblcms, oJxo acceleration sf the system i s nearly zero. Thus, it is possible to
lnescase the: value o f Inertid density, providing that inertial forces minerl 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~~it Ginerl
(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 it 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 a follows
where V and L are the characteristic velocity and length and T ) is viscosity of creeping flow.
This number is estimated in at each time step cycle and constrains the growth of inertial
density. We will show below how this parameter affects the dynamics of the simulations.
2.4. METHOD OF MARKERS AND REMESHING
There are many problems in geophysics which require simulating the dynamics of several
phases with different material 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 the mesh deforms with the fluid, are very fast and are easier to
implement than other methods. However, this approach fails when the mesh becomes too
distorted.
Fixed Eulerian meshes combined with the method of markers (Hirt and Nichols, 1981;
Weinberg and Schmeling, 1992) avoid this problem. This method is robust for finitedifference algorithms on rectangular grids but requires a lot of computational time for nonregular triangular meshes.
The combination of a moving Lagrangian mesh and a method of markers was found to be
optimum (Poliakov arid Podladchikov, 1992). The idea of this technique can be explained as
follows. At 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 are converted to local coordinates (area coordinates for triangular elements).
At each time step the grid nodes are then updated according to eq. 11. This Lagrangian
movement is very fast because it is only necessary to move the mesh nodes with known nodal
velocities.
When the mesh becomes too deformed, It is necessary to remesh. Since the local coordinates 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 steps and only few remeshing procedures. Another essential advantage of this method is that there are only substantial derivatives on time in constitutive laws
(compared to the partial derivatives in space and time required on 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 in undesirable acceleration and oscillations of nodes.
Damping of these non-physical oscillations causes the loss of the history of loading. In other
words, stresses and velocities will have jumps 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.
I
d
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 Eij tensors are decomposed to their deviatoric sq, i$
and isotropic parts Oii, Eii as follows
1
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,
0;;
=
~ I < E ~ ~
(18)
where K is the elastic bulk modulus.
Equations 17-18 are solved at 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 are
~ 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 semiimplicit approximation of stress in the term corresponding to the viscous strain increment.
Thus the deviatoric and volumetric stresses are updated as
1
sij
AtG
= ( s i j . (1 - -)+
AtG
2Gaa e di j ) / ( l + -1
277
277
a:;= 0;; 3 K ~ i i
+
and then full stress tensor will be
I
I
aij = oii
+
+ sij.
I
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 E~~ = 0 but q3must be calculated during the calculations.
4. Dimension Analysis for Maxwell Rheoiogy
In order to ensure rheological and dynamic similarity between numerical models and simulated 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). Equations 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, t = Tt', 7 = 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 - -.
Tvisc
GO
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 only 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
where Apo and L are scaling density and length factors defined as follows,
Using these relations, the Deborah nwnber can be rewritten as
This definition will be used in the present work. Weijermars and Schmeling (1986) performed
the scaling analysis for the momentum 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
and its natural analog are dynamically similar since they can be described by the same nondimensional 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 similarity).
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 scale.
5. Analytical Analysis of the Bottom Boundary Layer with Viscoelastic Rheology: Long
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 twolayer 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 layers with the same viscosity and elastic
moduli.
Figure 1: 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
Vl. Displacements on the
wavelength L is applied on the interface with displacements U1,
bottom boundary are zero U 2= V2 = 0. Hence the problem may be formulated entirely in
terms of the two displacement components U1, Vl on the 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(pt), where p is the growth rate factor. The
displacement field ui and stresses og are then written
, are functions only coordinates xi, while the time t appears only
where the amplitudes u ~ ( x )crij
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 viscoelastic 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
where a@) and E! are the deviatoric stress and strain which are only functions of position.
For the case of pure viscous flow, Biot (1965) gives the expression of growth factor p as
where a is a nondimensional parameter that depends on boundary conditions and the wavelength of applied perturbation.
Following Biot's derivation and making the following substitution
P'I ==+
PV
P r ~ e ~ 3a z1
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 nonlinear and for fluids with a high viscosity contrast.
6. Numerical Modelling of Viscoelastic Diapirisrn
6.1.INFLUENCE OF INERTIA, TUlE STEP AND GRID SIZE: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 all 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).
Because we 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
shown that using the adaptive density scaling the time step could be increased by increasing
the inertial density (see eq. 13).
This relation shows that 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 Re = 0.001 - 1. The 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 De = 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.
0.0141
De =
I
I
v = 0.25, A P / ~
I
1
I
=
0.1
I
Re = 0.001
(dt < 0.5 T / G )
- ................ Re = 0.01
o.o,,- - - - - - - Re = 0.01+-0.1(dt = 0.5 r ) / G )
Re=O.l
h
5
- ,.,.,.
0,010 -
I
-
Re = 1 .
0
+
E 0.008 l
i
E 0.006 $
z
-
-
.r(
-
-
-
.-__-+.
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 / ~It. 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 x 21 grid yields
satisfactory results.
6.2. Influence of the De Number and Poisson's Ratio
In 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 thicknesses 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
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 De = 10-3 - 10-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 from 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 vary from v = 0.25 for sedimentary and 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.
-
-0.25 -
.k
t 4
n
cd
;f; -0.30
w
0
-3
fb.& -0.35
G
-
- FEM
(Poliakov & Podladchikov, 1992)
De = 0.001
- - - . De = 0.01
-.-.-. De = 0.1
-...-.
-
-
I
--
-
-0.40 -
-
-0.45
1
0
10
,
+ , , , , , , , I , , , , , , , , ~ I , , , , , , , , , .
20
30
40
TIME
Figure 4: Height of the viscoelastic diapir for different values of Deborah number (De =
~ ~ ~ l = S/G).
a J For
~ comparison
~ i ~ ~ with a pure viscous simulation of RT instability the FEM
calculations are shown (Poliakov and Podladchikov, 1992).
D e = loe2, A p / p = 0.1, R e = 0.01
- o . z o i ~ ' ' " ' " " ' " " " " " " ' ' " r ~ ' ' l ~ ~ ' l ' ~ '//.//I
-~
.
7
-0.25 -
-...-.
---.
-.,.-.
FEM
v =0.25
v = 0.40
v = 0.45
./:/' -,.;'
/
/
;f'
.k
r(
G4
cd
rw
0
-0.30 --
-
4
%
.,-I
-0.35
-
-
-0.40 -
-0.45
0
10
20
30
40
TIME
Figure 5: 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-LAYER MODEL Wll'Ii HIGH RELAICATIONITMEFOR UPPER LEVEL: THE INTERACI1:ON OFTHE
D W I R WlTH THE HIGH VISCOUS LJTHOSPHERE
I
I
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.
This choice makes the calculations much faster because 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 is limited by lower viscosity as shown above. Therefore the time of calculations
is proportional to the viscosity contrast between two 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 the 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/q2and De number on the growth
of the diapir, topography and stress evolution.
1119
-
p1.G. h, '
or pure elastic
h2
1 1 2 9 ~2.G9
113-
93
vG9
h3
Figure 6: A model representing the interaction of a 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 equal 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 in 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,/q21 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 flow field and also
the dimensions of the diapir cell.
..------=
-- -
CCCCC
C
-
5.6e-03 Time =
31.
Velmax = 1.3e-02 Time =
60.
Velmax
Velmax = 5.0e-03 Time
=
Velmax = 8.5e-03 Time =
38.
77.
F i g u ~7: Velocity field evolution for a three-layer system for De =
Apl/pz = 0.1 and pl
= 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.This effect can be explained only 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 the nondimensional time interval A 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.
0
6
15
10
20
25
0
6
TIME
-0.740
-0.730
-0.720 -0.710
Height or diapir
16
20
25
TIME
De =
D e = 10"
-0.750
10
-0.700
-0.760 -0.740
-0.730
-0.720 -0.710 -0.700
Height of diapir
Figure 8: The influence of the D e number and viscosity contrast q l h 2on 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: 1 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 q1/q2< lo2, elastic behavior ( ~ e (>>
; ~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 De 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 dependence 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/q2a 1o4 and De
0 the 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
*
q 1 / q 2= lo2 and q 1 / q 2= 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
above 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
Maximum stress for each case is shown at
Figure 9: Principle deviatoric stresses at De =
Thick lines represent
the bottom of each picture. Scaling stress for a l l pictures is 1.1 x
= lo2 on the left and
compressing stresses. Note the distribution is nearly viscous for
non-relaxed elastic for q1/q2= 104 on the right.
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 q1/q2because 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 qlh2>
lo2 and v i s c o u for vl/qz< lo2. Stress is viscoelastic at the viscosity contrast 1o2 - 103. Initially the magnitude of stress grows rapidly because of the rapid
elastic response of layer 1 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 low 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 nonmoving Eulerian mesh may help to solve problem.
Numerical simulations and analytical estimations for the initial stages of the RayleighTaylor 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
From our results this implies that influence of elasticity on
diapirism yields De =
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 extensional stresses on the surface above the rising diapir up to one order of magnitude.
De = lom2
De =
0 + 2 6 . z ~ q 8 r T - . n ~ - . ~ v ~0 . v2 5 ~. 0 ~r . ~
q ~ ~
7 . ~
r ~ .1 0 ~n * ~u 1 - r r v v 1 . , a .
71/% = 1
0.20 -..*.*..
--
.-
B
0.15 :. -- -- -*-.'*
2
s
0
5
0.1,
B
0.05 L
-
r11/77* = 10"
71/778 = go3
r11/77* = to4
ql/?ht =
-
loa
/-=+
111
9
0.20 -
,JI"'
0
U
-
111
't:
-
0.10 1
m
7
-1
5
e----
-
P
0.15
/*'
/-
0.05 1
_----.
0.00
4
I
b
b
. - . m _ _ _ _ . _ . _ _ . . . - * - - * - - -
:
-0.05
-
*
:-. --.
-.-.-..
--...***..I....-.....
.
:
-O.IO~.-------0.16..
0
8
.
.
I
6
, . ,
. I , .
10
. .15 .
TIME
I . .
, I . ,
20
, ,
25
a)
-0.16:.
0
-
. . . 6, . . . .10. . . . 15. . . .20, . . .
I
I
TIME
.i
26
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).
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 Hemnan is greatly thanked for the discussion of
the model and providing the excellent HLRZ facilities during completion of the programming, calculations, and preparing the manuscript. We thank Dave Yuen 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 his visit
to Uppsala University.
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Zienckiewicz, O.C. (1989) The Finite Element Method, 4th edition. McGraw-Hill.