Poliakov Nato 93

8/17/2019 Poliakov Nato 93

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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


4/21

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


9/21

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


11/21

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).


12/21

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.


13/21

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


14/21

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).


15/21

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.


16/21

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.


17/21

..

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


18/21

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


19/21

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.


20/21

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


21/21

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.

8 References

Biot, M.A. 1965) Mechanics of Incremental Deformations, John Wiley Sons, New York.

Chery, J., Vilotte, J P and Daignieres, M. 1991) Thermornechanical evolution of a thinned

continental lithosphere under compression: Xmplicadons for the Pyrenees. J. Geophys. Res.

96,

4385-4412.

Cundall, P.A 1982) Adaptive density-scaling for time-explicit calculations. In

4th

Int. Cod

Numerical Methods in Geomechanics, Edmonton, Canada, Vol. 1, pp. 23-26.

Cundall,

P A

1989) Numerical experiments

on

localization in frictional materials.

Ingenieur-Archiv 59, 148-159.

Cundd, P.A and Board,

M

1988)

A

microcomputer program for modelling largestrain

plasticity problems. In

G.

Swoboda ed.), Numerical Methods in Geomechanics. Balkema,

Rotterdam, pp. 2101-2 108.

Last, N.C. 1988) Deformation of a sedimentary overburden on a slowly creeping substratum.

In

G.

Swoboda ed.), Numerical Methods in Geomechanics. Balkema, Rotterdam.

Marti, J. and Cundall, P A 1982) Mixed discretization procedure for accurate modelling of

plastic collapse. Int. J. for Num. and Anal. Meth in Geomech. 6, 129-139.

Melosh,

H.J.

and

Raefsky, A 1980) The dynamical origin of subduction zone topography.

Geophys. J.R. Astr. Soc. 60, 333-354.

Poliakov, A.N. and Podladchikov, Yu.Yu. 1992) Diapirism and topography. Geophys.

J.

Int.

in press).

Weijermars, R. and Schrneling, H. 1986) Scaling of Newtonian and non-Newtonian fluid

dynamics without inertia for quantitative modelling of rock flow due to gravity including

concept of rheological similarity). Phys. Earth Planet. Inter. 43,

316-330.

Weinberg,

R B

and Schrneling, H. 1992) Polydiapirs: Multiwavelength gravity structures.

J

Struct. Geol. in press).

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