Rough-draft manuscript written for discussion in GE277 Caltech January 2006 and
Lithospheric Dynamics Seminar Ludwig Maximillians Univ., Munich June 2006
Not ready for wider distribution or quotation
GE277LMUSuppeWedgeDraft_7.doc
Absolute fault and upper crustal strength from wedge tapers
John Suppe
Department of Geosciences
Princeton University
Princeton NJ 08544 USA
ABSTRACT
The strengths of mountain belts and of major faults have been notoriously difficult to
constrain and their controlling mechanisms and associated stress magnitudes continue
to be debated. Here we show several ways by which the large-scale strengths of
active thrust belts and their basal detachments can be fully determined or strongly
constrained directly from their critical tapers, independent of assumptions about the
specific underlying strength-controlling mechanisms. We then apply these methods
to the active Taiwan mountain belt and Niger delta thrust belt and find that their basal
detachments are exceedingly weak relative to most laboratory friction (Byerlee’s law
µb = 0.85 ); expressed as a normalized basal-shear traction (effective coefficient of
friction) " b #gH = 0.04 for the Niger delta, 0.08 for the Taiwan mountain belt and
!
0.06-0.1 for the basal detachment of the Taiwan M=7.6 Chi-Chi earthquake. In
contrast, these wedges are moderately strong internally; expressed as a normalized
!
deviatoric stress, ("1 # " 3 ) $gH = 0.6-0.7. These large-scale strengths, which are
measured over a horizontal scale of ~50-100 km, are within the range of pressuredependent strengths determined in deep boreholes. The observed ratio of fault
!
strength to wedge strength " b (#1 $ # 3 ) is 0.06-0.13, which raises again the
fundamental question of why faults are so weak and yet the deformed upper crust is
so strong.
!
INTRODUCTION
The question of what is the large-scale strength of the crust and of the faults within it
has been an outstanding question for a century and a half. George Ary recognized in
his 1855 paper on isostasy that the large-scale strength of mountain belts must be less
than the strength of intact rock, but the extent and specific mechanisms of large-scale
weakening remain difficult to constrain and even controversial. Also it has been long
recognized that the existence of thin intact thrust sheets, some of which exceed ~50100 km in length, imply very weak detachments relative to their internal strength,
which is the classic thrust-fault problem addressed by Hubbert & Rubey (1959). Lowtaper accretionary wedges and fold-and-thrust belts also require weak detachments
relative to crustal strength within these wedges (Davis et al., 1983). However the
causes and absolute magnitudes of such apparent fault weakness remain controversial
and this controversy extends to wrench faults such as the San Andreas and Sumatran
faults (Brune et al., 1969; Lachenbruch and Sass, 1980; Mount and Suppe, 1987,
1992; Zoback et al., 1987; Scholz and Hanks, 2004; Fulton et al., 2004) and to lowangle normal faults (Xiao et al., 1991; Axen, 2004). A central problem is that we have
limited knowledge of in-situ conditions and theories contain a number of large-scale
parameters that are difficult to observe. Furthermore, the few available constraints on
stress from deep boreholes indicate stresses that are a significant fraction of the smallsample laboratory frictional strengths of Byerlee’s law (1978), suggesting that the
upper crust is relatively strong (e.g. Brusy et al., 1997; Townend and Zoback, 2000).
However it is not the purpose of this paper to review further the multifaceted issue of
strength and state of stress of the crust and of faults. Here we show that it is possible
to recast critical-taper wedge mechanics (Davis et al., 1983; Dahlen, 1990) into a
simple form that allows us to directly determine or strongly constrain absolute largescale strength and stress simply from the observed geometries of active critical-taper
wedges, given appropriate geologic circumstances. Furthermore we do this in a way
that is largely free of assumptions about the strength-controlling mechanisms in the
crust. We find in specific examples exceedingly weak basal detachments and
relatively high large-scale wedge strengths, similar to deep bore-hole stress
measurements.
ABSOLUTE STRENGTH FROM WEDGE TAPERS
A central premise of critical-taper wedge mechanics (Davis et al., 1983; Dahlen,
1990) is that actively deforming fold-and-thrust belts and accretionary wedges are
simultaneously at large-scale failure internally and along their base, which is
reasonable since these wedges are constructed by deformation. It is this assumption
that large-scale average stress equals the large-scale strength that allows wedge theory
to form relationships based on static equilibrium between the critical taper " + # of a
wedge and the strength of the wedge and its base ( " is the surface slope and " is the
dip of the detachment, Fig. 1a). These strengths potentially may be controlled by a
!
variety of mechanisms operating at various
of the rock
! scales, even with much!
remaining undeformed. Furthermore the strength-controlling mechanisms may be
quasi-static or dynamical. For example the large-scale strength of some basal
detachments is the average regional stress necessary for dynamical ruptures to
propagate in brief great earthquakes that are separated in time by centuries, such as
the 2004 Sumatra M=9.0 earthquake, whereas other detachments have strengths
controlled by static or quasistatic processes, such as the continuous creep of
detachments in salt.
Wedge theory traditionally has been used to infer the range of possible values of
certain strengths-controlling parameters that are required to satisfy the observed
tapers, for example internal and basal friction coefficients ( µ = tan " , µb ) or
normalized pore-fluid pressures ( " = Pf / #gz ). This approach to critical-taper wedge
theory requires rather complex equations that contain
! of average large-scale
! a number
fault and crustal strength parameters and physical properties about which we would
!
like to know much more, but unfortunately have little direct constraints in many
actively deforming regions. An example is the elegant general weak-base theory of
Dahlen (1990), from which we take the special case of a mechanically homogeneous
wedge (Dahlen eq. 99) as our starting point, but recasting it in an equivalent much
simpler form in terms of stresses at failure rather than mechanical properties
"+# $
# (1% ( & f / & )) + F
(1% ( & f / & )) + W
[1]
where the fault strength F is the normalized basal shear traction determined by the
!
large-scale failure processes F " # b / $gH and the wedge strength W is the
normalized deviatoric stress W " (# xx $ # zz ) / %gH (see Dahlen 1990, eqs. 88, 90, 91,
97).
!
!
!
Before proceeding we note that equation [1] as originally derived by Dahlen (1990,
eq. 99) was written in terms of mechanical properties with a fault strength
F = µb (1" #b ) + Sb $gH that is the sum of a pressure-dependant fault strength,
expressed as a coefficient of friction µb and a Hubbert-Rubey normalized pore-fluid
!
pressure "b = Pf / #gz on the detachment, and a normalized nonpressure-dependent
basal strength such as a cohesion
or a viscous strength Sb / "gH , where H is the
!
wedge thickness. Similarly the wedge strength was written in terms of mechanical
!
sin $
properties as W = 2(1" #) 1"sin
$ + C %gH which is the sum of
!the pressure-dependent
!
strength, expressed by an internal-friction angle ϕ and a Hubbert-Rubey normalized
[
]
fluid-pressure ratio " = Pf / #gz , and a depth-normalized nonpressure-dependent
!
strength C / "gH .
!
This simplified wedge equation [1] raises the expectation that we may be able to
!
usefully constrain F and W from appropriate observations of wedge shape (α, β).
This seems plausible because the only remaining term in the equation (1" # f / # )
contains the ratio of the density of the overlying fluid (seawater or air) to the mean
density of rock and is thus 1 for subaerial wedges and ~0.4 for submarine wedges.
!
The critical surface slope " in a mechanically homogeneous wedge is linearly related
to the dip of the detachment " , as we see by rearranging [1]
F
W
[2]
" # " $ = 0 % s$ =!
%
$
(1% ( & f / &)) + W (1% ( & f / & )) + W
!
Therefore if we have an active homogeneous wedge with a variable dip to the basal
!
detachment we can determine the slope and intercept from linear regression of
suitable data (Fig. 1b). From [2] we find that the wedge strength W is a very simple
function of the slope of the regression
W =
!
s
(1" ( # f / #))
1" s
[3]
and the fault strength F is simply the regression intercept "# = 0 (Fig. 1b) times the
wedge strength
[4]
!
Notice in addition that knowing the dip of the detachment of a critical wedge of zero
F = "# = 0W
!
!
surface slope "# = 0 immediately gives us the ratio of fault strength to wedge strength
"# = 0 = F /W = $ b /(% xx & % zz ) , as pointed out previously by Dahlen (ref).
!
APPLICATION TO ACTIVE WEDGES
We consider two active compressive wedges that may approximate the assumption of
large-scale homogeneity, Taiwan and the Niger delta, because they are rather thick (H
= 5-10 km) and therefore their strength properties are less likely to be changing
laterally rapidly than at the toes of active accretionary wedges such as Nankai and
Barbados where H < ~1 km. Carena et al. (2002) present a set of taper measurements
for central Taiwan (Fig. 2a) that show a quasi-linear relationship of negative slope, as
predicted for homogeneous wedges by eq. [2]. Similarly, Bilotti and Shaw (2005)
present large-scale (~100 km) measurements of taper in the deep-water thrust belt of
the toe of the Niger delta that also show a quasi-linear relationship with a negative
slope (Fig. 2b).
We compute the normalized wedge strength W = (" xx # " zz ) $gH based of the
regression slopes (eq. 3) and obtain similar results for both wedges. Taiwan gives W
= 0.6 and the Niger delta W = 0.7. The normalized basal shear traction F = " b #gH ,
!
which is like an effective coefficient of friction, is F = 0.08 for Taiwan and F = 0.04
for the Niger delta. (Carena et al. computed identical F and W for Taiwan.) Finally,
!
the observed ratio of fault strength to wedge strength F /W = " b (#1 $ # 3 ) is 0.13 for
Taiwan and 0.06 for the Niger delta. Therefore the basal detachments are exceedingly
weak relative to the wedge strengths.
!
If we assume that the strengths of these wedges are dominated by their pressuredependent strengths, which is likely because of the substantial wedge thickness, then
we can estimate the large-scale pressure dependence µ = tan " from W if we can
constrain the fluid-pressure ratio " . In the case of the Niger delta (Bilotti and Shaw,
!
!
2005) borehole measurements give " = 0.55 , which yields a large-scale cohesionless
pressure dependence of µ = tan " = 0.5 , which is a significant fraction of laboratory
measurements on small samples.
Borehole fluid-pressure measurements in the
!
western margin of the Taiwan mountain belt are in the range " = 0.4-0.7 with large
!
drops in fluid pressure toward hydrostatic ( " = 0.4) associated with uplift and erosion
(Yue and Suppe, 2006). Therefore it may be reasonable to expect fluid pressures to
!
be generally close to hydrostatic within the mountain belt. In any case, if " = 0.4-0.7,
!
we estimate a large-scale cohesionless pressure dependence to the strength of the
Taiwan mountain belt of µ = tan " = 0.3-0.33.
!
COMPARISON WITH DEEP BOREHOLE DATA
!
We compare the large-scale wedge strengths W from Taiwan and Niger delta with
recent stress measurements from several scientific boreholes, the German KTB deep
borehole (Brusy et al., 1997) and the California SAFOD pilot hole (Hickman and
Zoback, 2004). In the KTB borehole " 2 is vertical, whereas in compressive wedges
" 3 is vertical, therefore we represent the KTB stress data as W * = ("1 # " 3 ) " 3 ,
!
which is directly comparable to W with "1 vertical. Furthermore the W* is relatively
!
constant as a function of depth indicating that the KTB region is dominated by
!
pressure-dependent strength, with W*=1.0±0.2 to a depth of 8 km (Fig. 3).
!
In contrast W * = W in the SAFOD pilot hole shows a strong decrease with increasing
depth (Fig. 3), suggesting that the measurements, which are at a depth of 1-2 km in
granite, are still within the cohesive boundary layer in which cohesion dominates (cf.
!
Dahlen et al., 1984). The large-scale cohesive strength C=~46MPa is given by linear
regression of the stress data "1 = 46MPa + 0.059z . Note that this large-scale C is a
factor of four less than the small-scale cohesion estimated for the SAFOD pilot hole
at 197-212 MPa (Hickman and Zoback, 2004). Knowing the large-scale C we obtain
!
the pressure-dependent component of the stress of (W " (C / #gz)) =~ 0.5 for the
SAFOD hole, as shown in Fig. 3.
!
Therefore the large-scale wedge strength W the Taiwan and Niger delta at depths of 5-
15 km lies within the range given by the pressure-dependent component of recent
borehole-stress measurements at 2-8 km. We use these indications of the range of
upper-crustal strength to estimate fault strength from a single taper measurement in
the following section.
THE CONSTRAINT OF A SINGLE TAPER
In many cases we do not have extensive taper data sets such as those from Taiwan
and the Niger delta (Fig. 2). We now address the question of what constraint on F
and W is provided by a single taper observation (α, β), still making the important
assumption that mechanical properties are homogeneous. As illustrated in Fig. 4a
there are a number of possible pairs of (F, W) that satisfy a single taper,
corresponding to a set of straight lines of negative slope s < 1 passing through the
point (α, β). By rearranging eq. [2] we obtain
[6]
!
which shows that the set of all possible (F, W) corresponds to a line whose slope is
F " # (1$ ( % f / % )) + (# + & )W
!
the taper and whose intercept is the surface slope times the buoyancy " (1# ( $ f / $ )) ,
as shown in Fig. 4b. Therefore the set of possible strengths is substantially
constrained by a single taper observation. In particular if we have independent
!
constraints on the possible magnitude of the wedge strength W, for example from
general experience such as deep-borehole data (Fig. 3) or wedges like Taiwan or
Niger delta, we can immediately constrain significantly the fault strength F.
As an example we estimate the strength of the Pliocene Chinshui Shale detachment
associated with the 1999 M=7.6 Chi-Chi earthquake, which is the basal detachment of
the active thrust belt in central western Taiwan, as shown in Fig. 5a (Yue et al., 2005).
The largest slip in this complex earthquake was on several branches of the Chelungpu
thrust, which is a thrust ramp stepping up from the Chinshui Shale detachment (Yue
et al., 2005). Other thrusts step up from this detachment and are active, including the
frontal Changhua thrust to the west and the Tiechenshan thrust (south of Fig. 5a)
(Simões, 2005, Yue et al., 2005; Yue and Suppe, 2006). The Shangtung thrust to the
east also steps up from this detachment, but does not appear to be highly active. We
also note that this Chinshui Shale detachment at a depth of 5-6 km is substantially
shallower than the Taiwan Main detachment of the previous analysis, which is near
the base of the Eocene at a depth of 10-12 km (Carena et al., 2002; Yue et al., 2005).
Therefore the Chinshui Shale detachment could have a different fault strength F,
which we here estimate.
The present shape of the wedge above the Chinshui Shale detachment is " = ~1.3° and
" = ~3.3° (Fig. 5a). The set of all F and W that is are consistent with this taper is
shown in Fig. 5b, based on eq. [5], which is a line of slope (" +!# = 4.6°) and
intercept " = 1.3° in radians.
!
!
We can now estimate F if we can place constraints on W. If we assume that Byerlee’s
!
Law ( µ = 0.85 ) with hydrostatic pore-fluid pressures ( " = 0.4 ) might be an upper
bound on large-scale upper crustal strength, then W " 2.2 and F " 0.2 . If the
!
!
!
Chinshui wedge strength is similar to the KTB borehole W * = 1.0 then F " 0.1. If we
!
use the observed Taiwan wedge strength of W = 0.6 then the fault strength is
!
!
F = 0.07 . If the wedge strength is as low as the pressure-dependent strength in the
!
!
SAFOD pilot hole (W=0.5) then F = 0.06 . Therefore we conclude that the Chinshui
!
Shale detachment is exceedingly weak, with our best estimate in the range
F = " b #gH $ 0.1-0.06. This weakness is especially striking in light of the
!
observation that the regional pore-fluid pressures surrounding the Chinshui Shale
detachment are hydrostatic ( " = 0.4 ) (Yue and Suppe, 2005b; Yue et al., 2006).
Therefore the static ambient Hubbert-Rubey fluid-pressure hypothesis is not the cause
of the weakness of the Chinshui Shale detachment.
!
DISCUSSION
The theory and examples show that under the special geologic circumstance of
mechanically homogeneous wedges we can immediately determine or strongly
constrain the absolute large-scale wedge and fault strengths from suitable
measurements of wedge geometry. Under these circumstances the wedge-taper data
lie along a line of negative slope "s = W /[W " (1" ( # f / # ))], which is therefore a
direct measure of absolute wedge strength W. The fault strength F is simply the
wedge strength times the intercept "# = 0 (Fig. 1). Data from Taiwan and the toe of the
!
!
Niger delta approximate this condition (Fig. 2) and demonstrate large-scale wedge
strengths that are about an order of magnitude higher than their large-scale fault
strengths F/W=0.06-0.13, with exceedingly low large-scale absolute fault strengths
" b #gH = 0.04-0.08 relative to normal laboratory friction measurements (Byerlee,
1978). The wedge strengths are within the range of deep borehole stress
!
measurements and of upper-crustal stresses from some lithospheric modeling (e.g.
Flesh et al., 2001). These results raise again the fundamental questions of why major
faults are so weak and yet the deformed crust containing these faults is so strong.
The present analysis assumes mechanically homogeneous wedges, which does not
include many of the geologically interesting examples. If significant lateral gradients
in wedge strength exist, which is expected in thin accretionary wedges, then these
gradients substantially effect the estimation of F and W from wedge tapers and the
present analysis no longer holds (see Fletcher, 1989; Dahlen, 1990, eq. 103). As
expected, taper data from thin accretionary wedges typically do not lie along lines of
negative regression slope s < 1, in contrast with the thick wedges of Taiwan and deepwater Niger delta.
!
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a)
b)
+
α
β
α
α β=0
W=
s ( (
1– ρf /ρ))
(1–s)
F = β α=0 W
–s
β α=0
0
β
mechanically
homogeneous wedge
−
Fig.1
a)
4°
Taiwan tapers
2°
α
W = 0.6
F = 0.13
–s = –0.37
βα=0 = 7.7°
0°
5°
10°
15°
β
20°
-2°
α
β
-4°
2°
b)
W = 0.7
F = 0.04
–s = –0.55
α
1°
βα=0 = 3.35°
Niger delta tapers
0°
0°
1°
2°
3°
β
Fig. 2
4°
W* = (σ1−σ3)/σ3
0
0
0.5
1.0
2
SAFOD
non-cohesive
4
2.0
SAFOD total
Niger Delta
KTB
Taiwan
6
1.5
8 km
Fig. 3
a)
observed taper
α
possible F & W
0°
β
b)
set of all possible
strengths F & W
α+β
α(1-(ρf /ρ))
F
F = α(1-(ρf /ρ)) + (α+β)W
0
W
Fig. 4
a)
WByerlee
b)
0.2
Chi-Chi taper
F=0.20
*
WKTB
F
0.1
WTaiwan
(W–(C/ρgz))
SAFOD
F=0.10
F=0.07
F=0.06
0
0
0.5
1.0
1.5
2.0
W
Fig 5