An analytical model of hanging-wall and footwall deformation at ramps on normal and thrust faults

Z-ecton~physics,

153

163 (1989) 153-168

Elsevier Science Publishers

B.V.. Amsterdam

An analytical

- Printed

in The Netherlands

model of hanging-wall and footwall deformation at ramps on normal and thrust faults BILL KILSDONK

forTectonophysics

Center

and Departments (Received

and ~Y~OND

C. FLETCHER

of Geology and Geophysics. Texas A&M

January

University,

4, 1988; revised version accepted

October

College Station,

TX 77843 (U.S.A.)

6, 1988)

Abstract Kilsdonk,

B. and Fletcher,

normal

and thrust

We determine developed surface

R.C., 1989. An analytical

faults.

stress, strain,

on both normal with ramp

(n-hanging

configuration

Sliding down a normal This conforms

evolution

ramps)

separates

are planar,

and regional

ramp causes vertical

extension

extension

Both a hanging-wall

syncline

+ n’)], where V is the sliding velocity.

in the model ramp,

Sliding vertical syncline

may correspond

up a thrust

shortening form

inhibiting

ramp

in both

passively.

fracturing.

anticline

stresses,

and horizontal

Although

the deviatoric

ramp migrates

stress

in magnitude

B.V.

is due

Both

favoring

has

by the ratio y/q’_

shape. The continuous zone in brittle local faulting

sign. Horizontal

a hanging-wall

anticline

so is the mean

on the adjacent

wall and the

in front of the rising footwall

or chaos

of the opposite

at the ramp,

stress is lowered

anticlines and duplexes form at thrust ramps. In an extensional terrane, the structure analogous to a thrust ramp is a ramp on a low-angle normal fault, called a normal ramp (Fig. 1). Although normal ramps have only recently been recognized (Wernicke and Burchfiel, 1982), the tapering forms of extensional wedges imply that normal ramps may be as important here as thrust ramps in fold-and-thrust wedges. Wernicke and Burchfiel 0 1989 Elsevier Science Publishers

faulting.

is raised

duplex

stress is raised,

rates, and strains

viscosity

Deformation

The stress in the footwall

But the ramp does not change

of an extensional

of different

in both the hanging

ramps.

form. The normal

Fold-and-thrust wedges taper towards the foreland, and their basal detachments climb section on thrust ramps. Major structures, such as ramp

0040-1951/89/$03.50

shortening rate is reduced

with normal

the mean compressive

half-spaces

sliding

is ignored.

models of normal

and deviatoric

strain

walls is consistent

However,

on

ramps

a frictionless

are locally rounded.

strain

to the creation

stress is lowered

causes

at ramps

angle (10-20”)

In the model,

viscous

corners

or shortening

wall but the footwall

and a footwall

mean compressive

deformation

of moderate

ramps).

isotropic,

with reverse faults seen in experimental

a rate P& = V[n/(n

model

faults (thrust

but ramp-flat

as that in the hanging

motion

wall and footwall

for an analytical

and thrust

the same magnitude

normal

of hanging

two incompressible,

Ramps

across the ramp,

model

163: 153-168.

and structural

Caults (normal

wall and n’-footwall).

solely to translation footwall.

Tectonoph.ysics,

rocks.

At a

and fracture. extension and

and

a footwall

compressive

flats, favoring

at

ramp

fracturing

stress, there.

(1982) and Gibbs (1984) have proposed that slip down a normal ramp produces complex structures, such as chaos zones and extensional duplexes. Martin and Bartley (1986) have described a normal ramp in the Worthington Mountains, Nevada. Rarely, normal ramps may also occur along thrust faults (Knipe, 1985). M&lay and Ellis (1987) studied the kinematics of normal ramps in sand box models. Reverse faults form over the ramp (Fig. 2). The present study provides a quantitative picture of deformation, stress, and structural evolution at a ramp by means of an analytical model. Analytical models have been

used to study

deformation

at a thrust

154

Normal

ing layer. It may also be applied to the study of a ramp-like irregularity in the shape of a fault of any orientation.

Ramp

The model

Thrust

Ramp

1 ramp width 1 Id

_

*

-

To qu~titatively model tectonic structures, one generally adopt relatively simple models to obtain tractable boundary-value problems. In addition, to obtain physical insight, the behavior of these simple models must be understood before passing to more complex models which may fit certain features of the observed structures. Berger and Johnson (1980) formulated a simple mechanical model for the deformation and structural evolution caused by sliding over a thrust ramp. In their model, a linear viscous, incompressible half-space slides on a frictionless surface above a rigid footwall. The sliding surface has the form of a smoothly-curving thrust ramp joining two adjacent flats. The maximum slope of the thrust ramp is 15 to 20 O. Because the hanging wall is a half space, the results apply only to the case where the hanging-wall thickness is greater than the ramp width. By chasing both a linear viscous rheology and a low ramp slope, an adequate solution for the deformation is obtained by superposing the linearly independent solutions for the Fourier components in the shape of the sliding surface. This is the same standard technique used by Sanford (1959) to obtain stress distributions in an elasticity deforming rock mass subject to a complicated distribution of basal displacements. In our model, the rigid footwall of Berger and Johnson is replaced by a deformable viscous half space. The sliding surface itself then becomes deformable. Generally, the viscosity of the footwall will differ from that of the hanging wall. Deformation is due solely to sliding past the ramp, and any regional shortening or extension is ignored. A regional deformation could be included provided it were the same in both the hanging wall and the footwall. The weak sliding surface will be treated in the limiting case as frictionless. The sliding surface, which forms the interface between the two media, is cylindrical with generator normal to the direction of slip. Deformation is therefore plane. The

must _

+

-

*

Fig. 1. Ramps developed on both a thrust fault (thrust ramp) and a normal fault (normal ramp). Rarely normal ramps may occur on thrust faults and thrust ramps may occur on normal faults.

ramp both when the thickness of the hanging wall is much greater than the ramp width (Berger and Johnson, 1980, 1982) and when it is much less than the ramp width (Wiltschko, 1979a, b; 1981). In these models, the footwall is rigid, and the ramp has a smoothly curved shape. Our model differs in two ways: (1) to address the footwall deformation implied by the formation of structures such as duplexes at a foreword-stepping thrust ramp, or chaos zones at a back-stepping normal ramp, both the hanging wall and the footwall can deform; and (2) the ramp has a more realistic planar form. Like the models of Berger and Johnson (1980,1982) and of Wiltschko (1979a, b; 1981), our solution is approximate and is only valid for a ramp of modest dip, no greater than 15 to 20”. Because both media are treated as halfspaces, our model applies to the case where the ramp width is less than the thickness of the overly-

20% EX’rE:NSIoN

Fig. 2. Sand-box model of deformation over a ramp in a normal fau’ft (normal ramp), Thrust faults form over the ramp. From McClay and Ellis (1987).

155

a-LFig. 3. Local

coordinates

n, normal

sinusoidal

to, and

s tangent

to a

sliding surface.

height of the sliding surface above its mean plane, z = 0, is z = C(.% f)

(1)

Let n and s denote coordinates locally normal and tangent to the sliding surface (Fig. 3). At the sliding surface, both normal and shear stresses are continuous, and, because the fault is frictionless, the latter vanishes: Q”,(X, 0 = a;,(%

5)

(2)

fl,,(& S) = e,l,(x, l) = 0 where a prime denotes a quantity in the lower medium. At the scale of interest, the two media rn~nt~n perfect contact during sliding. This requirement is satisfied if the normal component of the velocity is continuous: u,(x, r> = u;(x, S)

(3)

Form of the sliding surface

The sliding surface of chief interest is an alternating series of normal ramps and thrust ramps,

separated by long flats (Fig. 4a). Although there is a weak, large-scale flow away from thrust ramps and toward normal ramps, because the width of a ramp, w, is much less than the width of a flat, L/2 - w, the ramps are effectively isolated structures. Because the sliding surface contains both normal ramps and thrust ramps, it allows for comparison of the two. An alternate form (Fig. 4b), consisting solely of normal ramps separated by long, very gently sloping “flats”, also was treated and was found to give the same local deformation at a ramp. The analysis is valid only for ramps of modest slope, up to 20 O, and is carried out to first-order in ramp slope. To this degree of appro~mation, the flow for an arbitrary sliding surface configuration is the sum of the flows for the Fourier components in that configuration. Consequently, much physical insight is provided by the elementary solution for a sinusoidal sliding surface. Analysis

With the modifi~tions noted above, our analysis follows that of Berger and Johnson (1980). Further details on this kind of first-order analysis are given in Fletcher (1977). We use the convention that a positive stress is tensile and a negative stress is compressive. Pressure, the mean compressive stress, is the opposite of the mean stress. Consider a basic state, in which the sliding surface is perfectly plane. Two half-spaces slide N_oormalRamp

Fig. 4. a. Portion normal (t/2

ramps

of the sliding

separated

- w)/w

surface,

by long

= 20. Direction

flats.

I(x). Ramp

The sliding

surface

width = W, ramp

of slip is as indicated

by arrows.

comprises

a periodic

height = 2h, and b. Alternate

array flat

sliding surface

(wavelength

length = L/2 containing

L) of short - W. In our

thrust

and

calculations

only one kind of ramp.

156

past each other with no internal we specify velocities The velocity

relative

components

deformation.

Here,

The quantities sliding

to the lower medium.

surface,

Expansion

are:

to first-order

is, = V

(7) and (8) die off away from the as required. of boundary

conditions

in the slope (Fletcher,

(2) and (3) 1977), where,

from (6): u: = 0

(4)

al/ax

= - (XA)

sin( Ax)

(9)

u, = 6; = 0 yields: where the bars denote denotes

a quantity

the basic state, and a prime in the lower

medium.

The

stresses in the basic state are: a,, = CY;:,= CYz; = i?,,

(5)

a,, = Ciz = 0 The vertical ignored. Now sinusoidal

gradient

in lithostatic

stress (pgh)

is

&(x,

0) =4:(x,

&(x,

0) = C?iz(x, 0) = 0

u,(x,

0) + V(U)

let the

sliding

surface

be altered

to a

shape:

with amplitude

(10)

sin(Ax)

= u:(x,

0)

Substitution of the expressions (7) and (8) into (10) leads to four algebraic equations in the arbitrary constants

whose solutions

are:

a = b = - VAXq’/( TJ+ 7’) c=

C(x) =A cos(Xx)

-d=

(11)

VAh~/(q+q’)

(6)

A and wavelength

L = 2a/X.

The

Periodic array of ramps

velocity and stress components (4) and (5) no longer satisfy boundary conditions (2) and (3) applied

0)

to this surface.

We obtain the solution

In order to satisfy boundary conditions (2) and (3) to the desired order of approximation, we add a perturbing flow. A suitable form for the components of velocity and stress, which satisfy the equations governing the flow of a linear viscous fluid (Fletcher, 1977) is, for the upper half-space

i&= -[a+b(Xz-l)]

eCX’cos(Xx)

6X,,= 29A [a + b( AZ - l)] e-x’ cos( Ax)

for a single array

ramp

of short

from ramps

separated by long flats (Fig. 4a). If the ratio of ramp width to flat length is small, the interference between ramps is negligible. The periodic array in Fig. 4a is represented the finite series:

by

N lx=

fiz = [a + b( Xz)] eeXZ sin( Xx)

the results for a periodic

c k=l

YJk4Akx)

(12)

where: (7)

yk = {sin([2k-

l]a/2N)}/([2k

A,=

k sin[(2k

- 1]~/2N)

&X,x= 27~X[a + b( AZ - 2)] eCX’ sin( Ax) gzz = -217X[a+~(Az)]

eChz sin(Ax)

-8/z-l)

- l)Aw/2]

/[ wX7r(2k - l)‘] and for the lower half-space:

h, = (2k - 1)A

Ci = [c + d( AZ)] ehr sin( hx) 5: = [c + d( Xz + l)]

u“:,=2n’X[c+d(Xz+l)] -l u = -217’X[c+d(Xz+2)] 67=2n’A[c+d(Xz)]

Here, array.

exz cos( hx) eX’cos(Xx)

(8)

e”‘sin(Xx) exz sin( Ax)

where a, b, c, and d are arbitrary constants to be fixed by application of the boundary conditions.

L = 2~7/X

is the

repeat

distance

of the

The value of N determines the shape of the sliding surface. For N = 10, the ramp shape is broadly rounded; for N = 100, the comers are rather sharp; in the limit N + cc, the comers are perfectly sharp. In the Fourier series expansion of the limiting sharp-cornered form A, is the Fourier coefficient. Because this form has discontinuities

157

in slope, its truncated haved.

The

Gibbs’

Fourier

series is not well-be-

factor,

yk (Lanczos,

improves

the

convergence

and

properties

of the representation,

and it allows us

to treat the case of a locally rounded, shape. only

Without smoothly

the Gibbs’ curving

factor

ramps

flows

array of ramps from

velocities, obtained

and coefficients

where

ramp treat

to that

flow

by summing

its components,

by replacing

planar we could

similar

Berger and Johnson (1977). We solve for the perturbing periodic

1961),

differentiability

from

of

the

A-B

the

the separate stresses,

for each component

are

A by A,, A by A,, and

a,

b, c, and d by ak, b,, ck, and d, in (7), (8), and

Fig. 5. Streamlines rigid, sinusoidal right.

from

sliding

The perturbing

downward inflection

points

flow over a frictionless,

The basic flow is from left to

flow is upward

over lee slopes. Vertical of the sinusoidal

of the approximation

(11).

perturbing

surface.

used

over stoss

streamlines sliding surface.

the portion

shown below the diagram

slopes

and

occur over the To the order

of the slip surface

is the horizontal

line AE.

Kinematics and structural development at a normal ramp

The perturbing streamlines motion over stoss slopes

Velocity field for a sinusoidal sliding surface

over lee slopes (Fig. 5). The cells are separated by vertical streamlines at the inflection points. The

Since flow at any low angle ramp is the sum of the elementary flows at its Fourier components, the study of a single component provides much

perturbing velocity at the sliding surface is vertical and the vertical velocity component decays with

insight into ramp processes. The lee and stoss slopes of the sinusoid are rough analogs to normal and thrust ramps, respectively. Moreover, the more complex behavior that occurs when the footwall deforms is most easily understood in terms of the sinusoidal sliding surface. The velocity field is graphically represented by plotting streamlines, which are contours of the stream function #(x, z), where: u, = aq/az vZ=

03)

-aq/ax

The velocity vector is tangent to the streamline, and its magnitude is inversely proportional to the streamline spacing for a fixed contour interval. First consider the case of a rigid footwall. The stream function for the basic flow is: I//= -vz

height. The horizontal at the sliding surface, L/27,

and then decays

form cells with upward and downward motion

velocity component reaches a maximum with height.

is zero at z =

The perturb-

ing velocity adds to the basic velocity and subtracts from it over troughs.

over crests,

Streamlines for the total velocity field for a sliding surface with maximum slope of about 7O (A/L = 0.02) are shown in Fig. 6. The basal streamline coincides with the sliding surface. Because they are fixed in the reference frame, the streamlines are paths along which particles continue to move. Now consider those modifications when both media are deformable.

which occur At symmetri-

(14)

and streamlines are horizontal and uniformly spaced. The stream function for the perturbing flow is: 4 = VA( XL + 1) e-&’ cos( Xx)

Fig. 6. Streamlines

(15)

from the total (basic + perturbing)

a frictionless,

rigid, sinusoidal

sliding surface.

flow over

158

equation

for evolution

(Fletcher, al/at

of the sliding

surface shape

1977):

= &;(x, s> - I&(x, S>(X/iM

is common

to both

of the normal

surfaces,

velocity

that the right-hand the two surfaces

06)

since the continuity

component,

(3)

sides evaluated

implies

separately

are equal. We replace

sion (6) by one in which an arbitrary

for

the expresphase angle,

+, is inserted: {=A

cos(Xx-+)

Substituting either

the

making Fig. 7. Streamlines sliding sinusoidal

from

between

two

interface.

used

shown below the diagram Perturbing

viscosity flow

slopes. Vertical

flow resulting

media

on

a

the portion

is the central

is away sinusoidal

from

stoss

or lower

of

(11)

medium,

together

to first-order

with

in

(7) or (8) (17)

and

in AA, yields:

dA/dt=O

(I8a)

and :

of the slip surface

horizontal

(7’) is twice the hanging-wall

streamlines

upper

components

from

frictionless

Sense of shear is right lateral. To the order

of the approximation footwall

the perturbing

deformable

into (16), the velocity

use

evaluating

(17)

slopes

line AB. The viscosity

and

occur at the inflection

toward

(n). lee

points of the

sliding surface.

dWdt=XVh/(v+d)l The

(18b)

amplitude

remains

of the

constant,

surface

but the surface

tive to the footwall K=

sliding

therefore

translates

rela-

at a velocity:

[17/(77-tV’)l~

(19)

Since the sliding

surface

shape is constant,

it is

tally equivalent points, the perturbing streamlines in the lower medium (Fig. 7) have a form that is

convenient to use it, rather than the footwall, as a reference frame. The sliding surface then appears

complementary to those in the upper medium. In this example, the viscosity of the footwall is twice

fixed

that of the perturbing (TJ/TJ’) that of sliding,

and

each

medium

moves

Streamlines

are

stationary

with

relative respect

to it. to this

hanging wall, and the magnitude of the velocity in the lower medium is one-half in the upper medium. For a fixed rate I’, the perturbing velocity in the upper

medium scales with n’/(q + 7’) and is therefore less here than in the case of a rigid footwall. It is useful here to consider two reference frames. In the first, the mean velocity of the footwall is zero. Since the footwall deforms, the sliding surface also deforms, and the streamlines in Fig. 7 are not stationary relative to the footwall. In this case, the perturbing streamline field moves in harmony with the sliding surface. We first show that the locus of particles on the sliding surface, [(x, t), translates at a uniform velocity, but does not change its shape. The sliding surface is the common locus of the surfaces bounding the upper and lower half-spaces. While these surfaces slide past each other, the

I-

i

lJ’=ZTl

Fig

8. Streamlines

resulting stationary face

from

sliding

frictionless

is stationary

hanging basic

from

the total

between sinusoidal

with

flow component.

interface.

respect

wall and the footwall

flow

media

Because

to the reference viscosity

viscosity

(n).

(n’)

on a

the interframe,

each has a mean velocity

Footwall

hanging-wall

(basic+perturbing)

two deformable

the and a

is twice

the

159

Fig.

9. Deformed

hanging-wall placement

grid

distortion

illustrating caused

on a sinusoidal

grid is not shown. causes an increment

the

small

by a small

slip surface.

The undeformed

The same small increment of distortion

by the viscosity ratio (n/n’)

increment

increment

square

of displacement

in the footwall

and reflected

of

of dis-

that is scaled

across the interface.

coordinate system and particles move along them (Fig. 8). The hanging wall translates at V[ n’/( n + n’)] to the right, and the footwall translates at - V[r,/(q + n’)] to the left. Deform~ti5n

shown in Fig. 10. For this example, the ratio between the ramp width and the flat width is w/D = 0.05, 7’ = 2~, and N = 100. As in the case of a sinusoidal sliding surface, the large scale flow converges towards the normal ramp, on the right, and diverges from the thrust ramp, on the left. The streamline patterns are symmetrical across the sliding surface, but the velocity magnitude in the footwall is one-half that in the hanging wall. The large-scale perturbing flow between ramps, which results from the alternation of normal and thrust ramps, has negligible effect on the local deformation at the ramp, represented by the convergent pattern of streamlines shown in Fig. 11. A series of normal ramps separated by gently rising “flats” (Fig. 4b) has a different pattern of large-scale flow, but the same local deformation. Because the rate of translation of a sinusoidal perturbation is independent of wavelength (19), all

at a sinusoidal sliding surface

The deformation of an initially square grid, after a small increment of sliding, is shown for a portion of the hanging wail (Fig. 9). Strain and rotation arise solely from the perturbing flow, and the increment of horizontal translation is not shown. The deformation of the lower medium is given by the mirror image of the grid in Fig. 9, but the magnitude is scaled by the reciprocal of the viscosity. The center half shows the deformation on the lee side of a sinusoidal sliding surface, analogous to the normal ramp. Material extends vertically and shortens horizontally at the inflection point, both above and below the sliding surface. Elements at the frictionless sliding surface rotate, but do not strain. Although it is difficult to see in the figure, the strain rate is maximum at z = L/2a, and decays with height above that. The grid shows the local deformation of an element as it translates relative to the sliding surface. An element deforms periodically, and after a wavelength of translation, its net defo~ation is zero.

11'

11’=2q

Fig. 10. Streamlines between

two deformable

Fig. 1. The number

Velocity field and deformation

for a normal ramp

The sense of shear lateral.

The perturbing streamlines for a sliding surface with alternating thrust and normal ramps are

Footwall ramps.

flow caused by sliding

media on the frictionless

of terms in the series summation from

viscosity

ity (7). Perturbing normal

from the perturbing

the basic

of

(N ) = 100.

flow (not shown)

is right

(n’) is twice the hanging-wall

viscos-

flow is away from thrust

Vertical

interface

streamlines ramps.

ramps

and toward

occur at the mid points

of

160

Fig.

13. Deformed

hanging-wall placement

Fig. 11. Streamlines

from

right.

To the order

horizontal

the perturbing

flow in the hanging

ramp. The basic flow (not shown) of the approximation

line AB is the portion

used,

is to the

of distortion

ratio (f/q’)

increment

increment

The undeformed

of

of dis-

square

of displacement

in the footwall

and reflected

grid is causes

that is scaled by the

across

the interface.

N =

100.

the bottom

of the slip surface

shown

below the diagram.

flow (Fig. 11). Flow is fastest of ramp-flat

of the sinusoidal periodic array

ramp.

the small

by a small

The same small increment

an increment viscosity

illustrating caused

on a normal

not shown.

wall over a normal

grid

distortion

components that compromise the translate at the same velocity,

corners

sides of corners. After a small

on the convex

and slowest

increment

sides

on the concave

of sliding

above

the

without changing amplitude, and therefore so does the ramp form. Thus we can show the total velocity field, in the convenient stationary-ramp refer-

normal

ence frame (Fig. 12). The velocities in the two media increase on the approach to a normal ramp, decrease over the ramp, and increase again after

The deformation of the lower medium is given by the mirror image of this grid, but the magnitude is

exiting

the ramp, corresponding

to the perturbing

ramp,

an initially

square

grid deforms

as

shown in Fig. 13. The correspondence of this to the deformed grid shown in Fig. 9 is apparent.

scaled by the reciprocal of the viscosity. Above and below a normal ramp, material stretches vertically and shortens horizontally. In the hanging wall, elements on the right margin of the ramp undergo positive shear while elements on the left margin undergo negative shear. The deformation diffuses

laterally

and

decreases

in intensity

increasing height above the ramp. If the same pattern of deformation to take place in brittle materials, might accommodate the horizontal

I

I

Fig. 12. Streamlines by sliding

from between

the total

tionary

normal

ramp.

respect

to the

reference

footwall

each have a mean velocity

nent. Footwall

viscosity

(basic+perturbing)

two deformable

Because

the interface

frame,

media wall

and

with the

and a basic flow compo-

(q’) is twice the hanging-wall (a).

flow on a sta-

is stationary

the hanging

is imagined thrust faults shortening, as

seen in the sand-box model (Fig. 2). In this model, the normal faulting to the left of the ramp would be attributed to “regional” extension. The normal

ll’=Zl-

caused

with

viscosity

faulting to the right of the ramp may be associated with the effect of the right-hand boundary of the model. At a thrust ramp, normal faults would accommodate the horizontal extension associated with sliding. The instantaneous state of flow at a sharpcornered, planar, normal ramp is equivalent to the flow in a medium above a block of rigid basement that is descending between two vertical fault surfaces

to form

a graben.

This is suggested

im-

161

mediately

by the deformation

increment

Fig. 13. The equivalence

requires

between

upper

the deformable

rigid basement continuity within

be frictionless.

of the normal

the accuracy

tion of a uniform the ramp comers, faults,

vertical

that bound

The

is equivalent the basement

zones. Below a normal

and

In both

velocity velocity

the

cases, the

is equivalent,

local

along a strip at

rounding

to replacing block

by

of the the faults

by narrow

same

shear

footwall

is

of equivalence

applies

to a

but in this case the rigid basement

block rises as a horst to deform

the hanging

wall.

Displacement past a normal ramp causes local omission of section and forms two passive folds, a syncline

and a footwall

14a). The folds grow in inverse viscosity (1982),

ratio. these

As noted folds

wall syncline

anticline

proportion

by Berger

are nearly

case of a frictionless

sliding

and Johnson

symmetric surface.

is seen in the sand-box

(Fig. to the in the

A hangingmodel shown

in Fig. 2. Because

ramp, a deformable

kind

ramp,

hanging-wall

to the applica-

The strip is bounded

The thrust

in one case, and by the vertical

in the other.

ramp comers

medium

of the analysis,

the base of the medium.

shown in

that the contact

horizontal

shortening

fault but reaches a maximum folds, initially vertical planes

is zero

on the

in the cores of both warp in a way that

subject to a uniformly upward vertical velocity, and the velocity field in this medium is the mirror image of that in the hanging wall. The magnitudes

could be interpreted in terms of fault drag. At a normal ramp, the sense of drag is normal on the

of the vertical velocities applied to each medium equal the respective velocities of translation, in the

back-limbs limbs.

stationary

sliding

surface

reference

frame,

times

Motion

of the folds, and reverse on the thrust

ramp

on the front

duplicates

section

the ramp slope. Having noted this, it is sufficient to keep in mind the equivalence in the deformation of the upper medium alone.

to form a familiar hanging-wall anticline (Rich, 1934), and a footwall syncline (Fig. 14b). Material

This equivalence applies to the state of stress in the deformable medium, and, excluding translations, to the instantaneous velocity fields, but it does not extend to finite deformations. In the graben case, the material stays above the down-

the deformed grid shown in Fig. 14b. Surprisingly, then, horizontal extension occurs in the cores of these folds. Joints, veins, and normal faults which strike perpendicular to the direction of motion

dropping block, while, in the ramp case, it translates past the strip of vertical velocity. Thus different patterns of finite deformation arise.

extends

horizontally

at the ramp, as is well seen in

have been observed in thrust sheets (Kilsdonk and Wiltschko, 1988), and in experimental models of thrust ramps (Chester et al., in press). Initially vertical planes show normal drag on the front

Fig. 14. Deformation after 1.5 ramp widths of right lateral fault displacement for: (a) a normal ramp, (b) a thrust ramp. Footwall viscosity (q’) is twice the hanging-wall viscosity (q). N = 100.

162

limbs limbs.

of the folds, but reverse-drag Although

bedded

this drag

rocks, it could

might

a foliated

at a large angle to its planar

or gneissic

the ratio of strain

to that in the footwall

not be seen in rock

the footwall

are of equal viscosity,

In the case of the sinusoidal

sliding

tensor

from

surface,

(Al/)( h.4). This

a component

the velocity

=

Notice

of rounding

[et,+<:,I 1’2 41

values

rate

rate for sliding

shapes

have

over

variable

may be read off of Fig. 15, of maximum stress

of the contour

factor depends

width,

w, to the

profile,

L; in the present

product

(20)

that in this case the maximum

whose

of a characteristic

numerical

(ww)h’/(71’+ XKW exd-WI

the strain

wall sliding over a

shearing

stress

u*. The corre-

units

are, for the

C* = a*/217 = 0.19[9’/($ + hanging wall, n)](hV/w2) and for the footwall c*‘=u*/~v’= The 0.19[ 71/( 9’ + n)]( h V/w 2), respectively.

surface is:

( 1*y2 =

amounts

sponding

from

(11). Obtained in this way, the maximum shear strain rate in the hanging wall above a sinusoidal sliding

ramps

in units

components

for the constants

shear strain

isolated

which shows contours

of the

to

wall and

at the same velocity.

The maximum

(7) or (8) and substituting

If the hanging

in each is half that of a hanging rigid footwall

strain-rate

q’/n.

wall

proportional

ratio,

Model strain rates

can be seen by generating

is inversely

the viscosity

fabric.

strain rates scale with the product

rate in the hanging

fea-

be seen in ramp-like

tures on a fault cutting

petted,

on the back-

on the ratio of the ramp

repeat

hV/w2

distance

ramp The

has the dimensions

rate. It may be convenient slope of the ramp.

shear strain

of the

case w/L = l/42.

of a strain

to identify

2/z/w

as the

To estimate the magnitude of the strain rate associated with sliding across a ramp, we suppose the hanging vall is weaker than the footwall and

rate is a function only of height above the plane z = 0. The final factor takes a maximum value of l/e at Xz = 1, or z = 1/2m. As would be ex-

/ / -- \ \\ 0/\ - /\ \ \D- / -

/

/

\

\

I

I

\

-

\

Fig. 15. Contours

of normalized

lines) in both the hanging AB is the portion

deviatoric

of the slip surface

N = 100. In (c). four extreme in the footwall.

With the ramp dip direction

shown

values of fi

The sense of shear is right lateral and c’/z*’

stress (G/e

wall and the footwall

of a normal

The strain

below each contour

reversed

rate contour

diagram.

the J”;

Ramp

also represent

are normal

are as shown:

in the hanging

(a) N = 10, (b) N = 32, (c)

just over and under

values of normalized

units are E* = e*/2n

stress directions

shapes

= 100 * contours,

the ramp is a thrust ramp. The contour

compressive

/

*) and-for (a) and (c)-maximum compressive stress directions (short ramp. To the order of the approximation used, the central horizontal line

= 200 * lie within

as shown. The contours

/

/

strain

the flat-ramp

wall and c*’ = c*/2q’

values are the same for a thrust

to the short lines shown here.

corners.

rate, c/r * in the hanging

wall

in the footwall.

ramp, but the principal

163

seek to estimate viscosity

term

6 *. The maximum is unity.

Consider

of the

ramps with a slope of 20 O, so that h/w =

similar

0.182. This is approximately the Pine Mountain

equal to the angle of

thrust ramp (Wiltschko,

The value of e* then depends sliding

to the

ramp

sliding

velocity

were

could

value

geometrically

be increased

creasing

width, held

1979a).

V/w. fixed,

to large values

Now,

if the

the strain simply

size of the ramp might typically

rate

deformation

as the basal

forward

and a new frontal

forms.

In this case, the analogy

ous translation

of the sliding

detachramp

with the continu-

surface in the present

model is convincing. and likely

Stress distribution

scale

or area of the fault, and with the

sliding velocity.

Then,

would be independent

of the absolute scale of the ramp. Choosing, arbitrarily, a sliding velocity of 1 cm/yr, and a width of 1 km, V/w = 3 X lOpi3 SK’. Combining f* z 10-14 s-1,

Stress distribution, stress concentrations, fracture patterns

by de-

with the length

V/w

undergo

on the ratio of the

the width of the ramp. More realistically,

the absolute

rocks

ment propagates

values,

Since regional strain rates might lie in the range of lo-l5 s-’ to lo-i4 ss’, the strain rates due solely to sliding at a ramp might be typically of the same order, and in some cases, larger. The observation of extensional features in rocks which travelled over a thrust ramp, alluded to above, suggests that in some cases the deformation associated with travel across ramp locally dominates over the regional deformation.

Contour

plots

of the mean

(Fig. 16) and the maximum

compressive

stress

shear stress (Fig. 15)

with the orientation of the maximum principal stress (Fig. 15), illustrate the stress distribution. Tensile stress is taken as positive. Both in the hanging wall and tudes scale linearly (I

* = 0.38[ hV,‘w*]

Again,

in the footwall, with (I *, where:

stress

[TJ~‘/(TJ + q’)]

the numerical

constant

depends,

magni-

(21) in part, on

the choice of w/L. Clearly, u* = 2qe* = 217/c*‘. Stress magnitudes in the footwall are equal to those in the hanging wall, as indicated by the equality both of the characteristic stress in the two media and of the contours in Figs. 14, 15, and 16. Stress magnitudes are symmetrical across the slid-

Footwall deformation

ing surface, regardless of the viscosity ratio. Moreover, the stress distribution in the footwall is a

Both Gibbs (1984) and Wernicke and Burchfiel (1982) infer extensional duplexes and chaos zones below low-angle normal faults. In both structures, the active fault plane migrates in the direction of hanging-wall displacement. The migration takes place by failure of hanging-wall rock and accre-

mirror image, across the mean sliding surface, of that in the hanging wall. This symmetry holds to the accuracy of the approximation. To this degree of accuracy, perturbing quantities on the sliding surface are equal to their values on the mean plane, z = 0, and in the figures the sliding surface

tion of the failed rock onto the footwall. In contrast to our model, this kind of fault migration is not continuous and occurs by addition to, rather than deformation of, the footwall. However, it seems plausible that this process is the analog, in brittle rocks, of the continuous ramp translation

The stress distribution broad-scale features, at

in the model. This poorly-characterized process in the extensional regime may be compared with the observationally well-characterized process of duplex formation in thrust terranes. It clearly involves the forward jumping of the active flat-ramp-flat sliding surface. Moreover, it is clear that the footwall

width, and small-scale features, which depend sensitively on the sharpness of the ramp corners and represent the stress concentrations there. The latter depend on the ratio of the smallest wavelength in the series expansion, L,, to the ramp width, w, L,/w = (L/w)(2N - l), where, for the present model, L/w = 42. For N = 10, 32, and 100, for

is drawn

as a plane.

approximation ing the model tions.

This

consequence

of the

must be bourne in mind in applyresults to interpret field observanear a planar ramp has the scale of the ramp

A

A

A_! (I

B

B

c

B

Fig. 16. Contours of normalized pressure (normalized mean compressive stress = -It/o * ) in both the hanging wall and the footwall of a normal ramp. To the order of the approximation used, the central horizontal line AB is the portion of the slip surface shown below each contour diagram. Ramp shapes are as shown; (a) N = 10, (b) N = 32, (c) N = 100. In (c), two extreme values of -I, P - 400 * lie on the slip surface, within the - Jt = - 200 * contours. The sense of shear is right lateral. With the ramp dip direction reversed such that it is a thrust ramp the diagrams are contour plots of normalized mean stress (1,/o * ) near a thrust ramp.

which the stress dist~bution is shown in each figure, L,/w = 2.2, 0.67, and 0.21, respectively. As the corners are sharpened, the broader-scale contours are modified only slightly, while higher contours appear at the comers, representing the stress concentrations there. The stress concentrations increase as the comers sharpen, and a rough estimate of the effect may be obtained by comparing the maximum contour value with the inverse measure of sharpness, L,/w. The product of these is about equal, impl~ng a linear relations~p. Perfectly sharp comers imply stress singularities, no matter how low the ramp angle. In actual examples, other mechanisms of deformation such as brittle fracture, will occur at sharp corners, bounding the value of the stress concentrations and effectively smoothing the comers. Stress magnitudes depend on the viscosities through the factor [qn’/(n + $)I. If the footwall viscosity is twice that of the hanging wall, and the latter is n = 1020 Pa s, the unit of the stress contour, for E* = lo-r4 s-l, is 2 MPa. Although not shown by contours, the maximum stresses obtained in the model are 20 times (deviatoric stress, Fig. 1%) and 40 times (pressure or negative mean stress, Fig. 16~) this amount. Thus, with these assumptions, a normal ramp with locally

rounded corners may reduce the mean regional compressive stress (pressure) by over 80 MPa and raise the maximum deviatoric stress by over 40 MPa. Although the value of the differential stress is the same at a thrust ramp as at a normal ramp, the maximum pressure reduction is only about half as large and differs in location. While near a normal ramp the pressure is reduced directly over, and under, the ramp, near a thrust ramp the pressure is reduced over, and under, the adjacent parts of the flats. Because the sliding surface cannot support a tensile stress, the normal stress on the sliding surface, which is the sum of basic-state and perturbing values, u~,,(x, h) = Cz, + I?~,,, must be negative (compressive). In the model, the deviatoric stress vanishes at the sliding surface, so that the normal stress on the fault is equal to the mean stress. For no separation, the magnitude of the compressive vertical stress must exceed the maximum, tensile, mean stress (~nimum pressure). The pressure is reduced at a normal ramp and enhanced on the adjacent parts of the flats (Fig. 16). The opposite relation holds at a thrust ramp. In both cases the pressure changes signs at the flat-ramp corners. In addition to the ramp-scale

165

pressure distribution, strong local pressure concentrations arise as the comers become sharp (Fig. 16). Necessarily, the same sign changes are seen in these. The same kind of ramp-scale and near comer features occur in the ma~mum shearing stress (Fig. 15). The ramp-scale maximum lies near the center of the ramp, but if the comers are sharp enough, this single maximum is replaced by maxima near the comers. The magnitude of the deviatoric stress is the same for both normal and thrust ramps. The greatest (least compressive) and least (most compressive) principal stress directions switch between normal and thrust ramps. In the perturbing flow, material is pushed away from a thrust ramp in all directions in the plane of flow, and the maximum compressive stress directions form a radial pattern. At a normal ramp they form a concentric pattern. The local pattern near sharp corners causes deviations from this ramp-scale pattern. Faulting and tensile fracture As is commonly done (Hafner, 1951; Sanford, 1959), the stress distribution from a simple elastic

Fig. 17. Contours

of normalized

maximum

compressive

horizontal

line AB is the portion

maximum

stress near a thrust

or viscous model may be interpreted in terms of brittle fracture. Regions of high stress in the model are likely to experience fracture enhancement in an actual example, and the orientations and magnitudes of the principal stresses in the model may be used to estimate the o~entation and nature of shear or extension fractures. This procedure is only quantitatively accurate in estimating initial yield. As the regions of brittle fracture grow to a significant fraction of the ramp scale, the viscous solution will no longer give a good approximation to the stress distribution. Tensile fracture will occur in regions of maximum (tensile) stress, provided the pore pressure is large enough. Maximum stress is contoured near two normal ramps of different shapes in Fig. 17. These plots also represent the ~~rnurn stress near thrust ramps of the same shape. Fracture orientations would parallel the minimum stress directions. Regardless of the ramp shape, the maximum tensile stress always lies on the sliding surface, If, then, the only stresses are due to slip past a ramp, and if the rock everywhere resists fracture at least as well as the fault resists separation, then no extension fractures will form. But if a stress field resulting from a regional extension

tensile stress near a normal ramp).

of the slip surface

Tensile

ramp

stress is positive.

shown below each contour N=lOO.

(or, if the ramp dip direction To the order diagram.

is reversed,

of the approximation

Ramp

shapes

normalized

used, the central

are as shown;

{a) n = 10, (b)

166

is superposed, as might be expected at a normal ramp, fracture will occur first where the maximum stress direction is nearly horizontal. A yield criterion for shear fracture can be written: sin@

&-~cos@=p

122)

where I- is the cohesion, Q, is the angle of internal friction, and p is the opposite of the mean stress, or the pressure. In terms of basic-state and perturbing quantities, with perturbing terms on the left-hand side, the yield criterion is: iz--p

sintP=rcos@++ij

sin@

(23)

If the region of interest is sufficiently small, the gradient of lithostatic pressure (pgh) near the ramp can be ignored, and the right-hand side is a constant. This may be specified for a particular example from the rock strength and the lithostatic stress at the depth of the ramp. The value of the left hand side, normalized by the characteristic stress u* is contoured for a normal ramp in Fig. 18. For small values of u*, the left-hand side will be less than the right-hand side and yield will not occur. At some value of u* the yield condition (23) will be met at the isolated maximum or maxima of the left-hand side. As (T* increases

Fig. 18. Inferred failure. contour

shear fracture

orientations

To the order of the approximation diagram.

Ramp

(short lines) near a normal used, the central

shapes are as shown;

values lie just inside of the flat-ramp

comers,

shear fractures.

horizontal

further, the left-hand side will exceed the righthand side in regions bounded by the contour satisfying the yield condition (23). In reality, this is impossible, and both the stress distribution and flow consistent with the yield condition (22) will differ from the estimate of the model in four ways: (1) Within the region undergoing brittle deformation, the stresses will satisfy the yield condition; (2) the principal directions will differ; (3) the positions of the bounding surface of the yielding region will be different; and (4) the stresses outside the yielding region will also differ. The premise of our inte~retation, and of previous workers, is that these differences may be relatively modest, and that the method may still provide reasonably accurate estimates of the yielded regions and the orientations of faults. Near a normal ramp deviatoric stress is high and pressure is low, favoring localized generation of shear fractures (Fig. 18). Because the stress field is symmetrical across the fault, the faulting wiIl occur in both the footwall and the hanging wall. Fractures will first develop in regions just inside of the ramp-flat corners, slightly above and below the fault. In close agreement are the observations by McClay and Ellis (1987) that all reverse faults in their sand-box model nucleated at a fixed point

ramp and contours

of the l~elihood

line AB is the portion

(a) N = 10, (b) N = 100. The sense of shear is right lateral. both above and below the sliding surface,

of focal Mob-Coulomb

of the slip surface

and contain

In both (a) and (b), the first faults to form are thrusts.

shown below each

In (b), the highest contour

the locations

most apt to develop

167

above the ramp’s upper hinge and that the earliest fault was a reverse fault. The orientations of conjugate shear fractures, inferred from principal stress directions, are shown as line segments in Fig. 18. At a normal ramp, thrust faults form first, at a lower value of u* then do normal faults. However, a superposed regional extension would both shrink regions of thrust faulting and expand regions of normal faulting. A significantly large regional tension would suppress thrust faulting all together. By considering this interaction between the stresses associated with sliding across a ramp and the regional stress, it might be possible to estimate the relative values of the characteristic stress u* and the regional stress. However, it should be noted that our model assumes a linear stress to strain-rate relation, and only grossly estimates the consequences of brittle fracture. Accurate treatments would necessitate numerical modeling. The left-hand side of (23) is contoured in Fig. 19 for a thrust ramp. Because pressure (mean

Fig. 19. Inferred and contours

shear fracture

orientations

near a thrust ramp

of the Likelihood of iocal Mohr-Coulomb

To the order of the approbation

used, the central

failure. horizontal

line AB is the portion

of the slip surface

shown

contour

shape is as shown;

iV = 32. The hori-

diagram.

Ramp

zontal

line AB represents

shown.

The sense of shear is right lateral.

values he just outside

the portion

of the flat-ramp

below the sliding surface, develop shear fractures.

and contain

below

of the sliding

the

surface

The highest contour

comers,

both above and

the locations

most apt to

The first faults to form are thrusts.

compressive stress) is reduced on flats adjacent to a thrust ramp, shear fractures will form there first. Thrust faults form in these locations. Normal faulting above and below a thrust ramp requires larger values of u * and will be suppressed by regional compressive stress. Summary and conclusions We modeled the deformation associated with sliding across an isolated ramp, and emphasized the normal-ramp configuration. The model treats both the footwall and the hanging wall as deformable, linear-viscous media, and the weak fault is treated in the limiting case as a frictionless sliding surface. The ramp width is sufficiently small relative to its burial depth that the presence of a free surface can be ignored. In the model, the equality of stresses in the footwall with those in the hanging wall supports the notion of footwall deformation. The planar shape of the ramp is more realistic than those treated in previous studies, and leads to strong concentrations of stress and deformation at the ramp-flat corners. Translation down a normal ramp causes vertical elongation and horizontal shortening, consistent with reverse faults produced in experimental models. Structures such as thrust faults or sub-horizontal extension cracks or veins are expected in natural examples. Conversely, translation up a thrust ramp causes horizontal extension and vertical shortening, suggesting normal faulting and sub-vertical extension crack or veins over, and under, the ramp. However, the stress field on the flats flanking a thrust ramp suggest localized thrust faulting there. A hanging-wall syncline and a footwall anticline form passively at a normal ramp. The material in the cores of both folds is vertically stretched and horizontally shortened. At a thrust ramp, the folds are a hanging-wall anticline and a footwall syncline. The material in the cores of these folds is ho~zontally stretched and vertically shortened. Maximum stress magnitudes depend largely on the sharpness of the ramp-flat corners. The stress magnitude scales linearly with slip velocity and the tangent of the ramp dip.

At a normal ramp pressure is low and deviatoric stress is high. Localized fracturing will occur first in the ramp region. Deviatoric stress is high near a thrust ramp and pressure is low at its adjacent flats, favoring localized fracturing in the regions flan~ng the ramp. In the footwall of either kind of ramp, the stress field due to sliding is a mirror image of that in the hanging wall. Associated fracture patterns should be similarly related.

Gibbs,

A.D.,

margins. Halfner, Kilsdonk.

B. and Wiltschko,

tain Block, Tennessee. Knipe,

1985. Footwafl

Struct. Lanczos, Martin,

M.W.

deformation

of

Tectonophysics, Berger,

a

thrust

P. and Johnson,

structures

J. Struct.

the Rocky Perry

a

ramp.

of blind

Appalachian

blind

Geoi., 4: 343-353.

fault rock models

Mountain

(Editors).

of passive layers

near terminations

to the central

J.S., Spang, J.H. and Logan,

son of thrust

over

of

Foreland.

Thrust

J.M., in press, Compari-

to basement-cored

folds in

In: C.J. Schmidt

and W.J.

Belt-Foreland

Interation.

Geol.

Sot. Am., Spec. Pap. Fletcher,

R.C.. 1977. Folding

infinitesimal-amplitude 593-606.

of a single viscous solution.

layer:

Tectonophysics,

exact 39:

mecha-

of the Pine Moun-

Geol. Sot. Am. Bull., 100: 653-664.

geometry

and

Mountains,

McClay,

and the rheology

Analysis.

Prentice

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J.

Hall, Englewood

Bartley,

J.M.,

1986.

Nevada.

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Meet. Expo., Abstr.

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systems

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K.R. and Ellis, P.G., 1987. Geometries

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overthrust

Geology,

15: 341-344. Rich, J.L., 1934. Mechanics illustrated

by Cumbertand

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Gil Field, Grand

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Banks, New-

Pet. Geol. Bull., 71: 1210-1232. B.C., 1982. Modes

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This work was supported by NSF grant EAR8708326. We thank Richard Groshong, Jr., Mel Friedman, and an anonymous reviewer for thier helpful suggestions.

evolution

Am. Bull.. 62: 373-398.

normal

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K.R. Nappe