The classic crustal strength-depth profile based on rock mechanics predicts a brittle strength s1 � s3 ¼kðrgz� Pf Þ that increases linearly with depth as a consequence of [1] the intrinsic brittle pressuredependence k plus [2] an assumption of hydrostatic pore-fluid pressure, Pf ¼ rwgz. Many deep boreholestress data agree with a critical state of failure of this form. In contrast, fluid pressures greater thanhydrostatic rgz> Pf > rwgz are normally observed in clastic continental margins and shale-rich mountainbelts. Therefore we explore the predicted shapes of strength-depth profiles using data from over-pressured regions, especially those dominated by the widespread disequilibrium-compaction mecha-nism, in which fluid pressures are hydrostatic above the fluid-retention depth zFRD and overpressuredbelow, increasing parallel to the lithostatic gradient rgz. Both brittle crustal strength and frictional faultstrength below the zFRD must be constant with depth because effective stress ðrgz� Pf Þ is constant, incontrast with the classic linearly increasing profile. Borehole stress and fluid-pressure measurements inseveral overpressured deforming continental margins agree with this constant-strength prediction, withthe same pressure-dependence k as the overlying hydrostatic strata. The role of zFRD in critical-taperwedge mechanics and jointing is illustrated. The constant-strength approximation is more appropriatefor overpressured crust than classic linearly increasing models.
The classic Brace and Kohlstedt (1980) strength-depth graph,based on experimental rock mechanics, predicts an initial linearincrease in brittle frictional strength with increasing pressure,followed by an exponential decay in ductile strength withincreasing temperature (Fig. 1A). This first-order prediction hasbeen remarkably successful, for example in predicting the tem-perature of the brittle-ductile transition in common rock types.The intricacies of this graph in the lower crust and upper mantlecontinue to be widely discussed and are even controversial in thecase of the proposed “jelly sandwich” of weak lower continentalcrust above a presumed stronger mantle (e.g. Chen and Molnar,1983; Suppe, 1985, p. 188e189; Jackson, 2002; Burov andWatts, 2006; Bürgmann and Dresen, 2008). In contrast, thepredicted linear increase in brittle strength is generally assumedwithout discussion and agrees with deep borehole stress data incrystalline rock, showing that the crust is commonly close to acritical state of brittle failure (e.g. Townend and Zoback, 2000;Zoback, 2007), for example the deep KTB borehole in Germany(Fig. 1B).
td. This is an open access article un
1.1. Classic brittle strength-depth relationship
The predicted linear increase in brittle strength s1 � s3 withdepth is a consequence of [1] the intrinsic pressure dependence k ofbrittle strength, plus [2] an assumption that pore-fluid pressure ishydrostatic and therefore linearly increasing Pf ¼ rwgz. Ignoringcohesionwe approximate brittle crustal strength very simply as thevertical effective stress rgz� Pf , times the pressure dependence k.
s1 � s3 ¼ k�rgz� Pf
�; (1a)
(Suppe, 1985, p. 185), where r is the mean bulk density of theoverlying rock. Eq. (1) also can be written in terms of the classicHubbert-Rubey (1959) fluid-pressure ratio l ¼ Pf =rgz
s1 � s3 ¼ kð1� lÞrgz; (1b)
where (1 � l) may be considered the fractional fluid-pressureweakening. In submarine cases l is measured with respect to thesea bottom (Davis et al., 1983). For the hydrostatic case (1a)becomes.
s1 � s3 ¼ kðr� rwÞgz: (2)
These crustal-strength equations express the fact that brittlepressure-dependent strength is a function of effective stress
der the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
Fig. 1. (A) The classic strength-depth graph with linearly increasing brittle strength,which is based on the assumption of hydrostatic pore-fluid pressures (Eq. (2)). (B) Thisbrittle-hydrostatic model agrees well with stress data from the deep KTB borehole inGermany where fluid pressures are hydrostatic (Brudy et al., 1997; Zoback and Harjes,1997; Grawinkel and St€ockhert, 1997), as well as data elsewhere (see Townend andZoback, 2000; Zoback, 2007).
J. Suppe / Journal of Structural Geology 69 (2014) 481e492482
rgz� Pf and not the solid pressure rgz or even depth, which is adistinction that becomes more important in the case of fluidoverpressures.
The pressure dependence k can be expressed in terms of thefamiliar Coulomb internal friction (m ¼ tan4) and varies betweenkc ¼ 2sin4 (1 � sin4) in pure-thrust compression and ke¼2sin4(1 þ sin4) in pure normal-fault extension, with strike-slip stressstates (s2 vertical) lying between these limits, as can be shown fromthe Mohr diagram.
1.2. Success of the classic brittle-hydrostatic assumption
Brace and Kohlstedt (1980) assumed that brittle crustal strengthis controlled by an optimally oriented static fault friction ofm¼ 0.6e0.85 based on lab static frictionmeasurements from awidevariety of rock types (Byerlee, 1978) (kc ¼ 2.1e3.7, ke ¼ 0.68e0.79).Townend and Zoback (2000) showed that deep borehole stressesare typically close to a critical state of failure consistent withm ¼ 0.55e1 (kc ¼ 1.9e4.8, ke ¼ 0.65e0.83) and an assumed hydro-static fluid pressure, implying that stress z regional strength inmany regions (Zoback, 2007). Therefore the Brace and Kohlstedt(1980) brittle-hydrostatic proposal (Eq. (2)) is in good agreementwith available observations, as is illustrated by stress data from thedeep KTB borehole in Germany (Fig. 1B).
We have expressed brittle crustal strength (Eq. (1a)) in terms ofthe vertical effective stress because this quantity can be directlycalculated from commonly available borehole data or estimatedfrom seismic velocities in clastic sedimentary sections that have notundergone uplift and erosion (e.g. Fertl, 1976; Magara, 1978;Swarbrick et al., 2002; Zoback, 2007). We will use Eq. (1a) tocompare the observed vertical effective stress rgz� Pf with in situborehole stress measurements of s1 � s3 to determine if they areconsistent with a constant value of the pressure dependence k overthe entire depth range of the data from actively deforming sedi-mentary basins. This strategy provides a more explicit test of therole of pore-fluid pressures on crustal strength and avoidsassuming what processes may control crustal brittle strength,which is an open question. For example, the regional-scale pressuredependence of the crust k may be controlled by a combination of
faulting, off-fault fracturing, and folding processes, some of whichmay be coseismic and may be controlled by dynamic frictionalprocesses in earthquakes (e.g. Di Toro et al., 2011).
2. Implications of fluid overpressures for regional strength
In contrast with the classic view of linearly increasing crustalstrength dominated by hydrostatic pore-fluid pressures, it is wellestablished from petroleum boreholes and seismic velocity analysisthat fluids are overpressured in deeper parts of thick, fine-grainedclastic sedimentary basins and in deforming shale-rich plate-boundary mountain belts, largely due to disequilibrium compac-tion from stratigraphic and tectonic loading, but with additionaleffects including vertical and lateral pressure redistribution and gasgeneration (Bredehoeft and Hanshaw, 1968; Fertl, 1976; Magara,1978; Hart et al., 1995; Swarbrick and Osborne, 1996; Yardley andSwarbrick, 2000; Tingay et al., 2009a). The basins of most interestfor crustal strength and large-scale tectonics are continental mar-gins built on oceanic or highly-thinned continental crust that havevast deforming clastic-rich volumes that approach a significantfraction of crustal thickness, even spanning the brittle-ductiletransition (e.g. Gulf of Mexico, Niger delta, Bangladesh/Myanmar,Sumatra, Makran, Gulf of Alaska, New Zealand, Nankai trough,Barbados/Trinidad, and offshore southwestern Taiwan). Here weillustrate typical observed fluid-pressure/depth relationships fromboreholes. We then compute the expected strength-depth profiles(Eq. (1a)) and compare them with borehole stress data in activelydeforming regions, where stress is expected to be close to regionalstrength, or at least limited by Eq. (1a) as an upper bound.
2.1. Disequilibrium-compaction mechanism and crustal strength
The fluid-pressure depth relationship for the Yinggehai basinoffshore south China (Fig. 2A) (Luo et al., 2003) is typical of thedisequilibrium-compaction mechanism (e.g. Swarbrick andOsborne, 1996; Magara, 1978), which has also been called“leaky overpressure” (Crans and Mandl, 1980; Mandl and Crans,1981). Pressures are hydrostatic down to a critical depth of~1.8 km, the fluid-retention depth zFRD, below which fluid pres-sures increase parallel to the lithostatic gradient (Fig. 2A),because permeability becomes sufficiently low that the incre-mental increases in sedimentary or tectonic load are supportedby the pore fluid, rather than by incremental compaction of thesolid-grain framework. Fluid pressures under the disequilibrium-compaction mechanism are a simple function of the fluid-retention depth, representing the sum of the hydrostatic andlithostatic contributions (Fig. 2A).
Pf ¼ rwgz for z � zFRD (3a)
Pf ¼ rwgzFRD þ rgðz� zFRDÞ for z � zFRD (3b)
It follows that the Hubbert-Rubey fractional fluid-pressureweakening (1 � l) (see Eq. (1b)) is a simple function of the fluid-retention depth zFRD
ð1� lÞ ¼ ½1� ðrw=rÞ�zFRD=zz0:6zFRD=z for z � zFRD (4)
We make use of Eq. (4) in the discussion section of this paper.Because of the observed parallelism of fluid pressures to the
lithostatic gradient, the vertical effective stress rgz� Pf is approx-imately constant below the fluid-retention depth (Fig. 2A) andequal to the effective stress at the fluid-retention depthðrgz� Pf Þ ¼ ðr� rwÞgzFRD. Therefore pressure-dependent strength
Fig. 3. (A) Typical fluid pressures in the active western Taiwan thrust belt, Chingtsaohu(CTH) anticline, with (B) strength computed from Eq. (1a), using a pressure-dependence to match in situ stress data in the TCDP borehole from Hung et al.(2009). Point data are shut-in pressure tests from permeable strata, and the solidcurve is pressure in shale computed from sonic log (Suppe and Wittke, 1977;Yue, 2007; Yue and Suppe, 2014).
Yinggehai basin, ChinaLD3011
0 50 100 MPa
1
0
2
3
4
5 km
1
0
2
3
4
5 km
0 25 50 MPa
hydrostatic ρ
w gz
ZFRD
exte
nsio
nal κ
e=0.
65
com
pres
sion
al κ
c=1.
86
μ = 0.55
ZFRDZFRD
strength σ1−σ3
predictedA. B.
lithostatic ρgz
fluid pressure Pf
effective stressρgz – Pf
ρgz +
FRD
w
ρ g(z–z )FRD
Fig. 2. (A) Typical fluid pressures dominated by the disequilibrium compactionmechanism, which is characterized by hydrostatic pressures down to a critical depth,the fluid-retention depth zFRD, below which fluid pressures increase parallel to thelithostatic gradient, because permeability becomes sufficiently low that incrementalincreases in sedimentary or tectonic load are supported by the pore fluid, rather thanby incremental compaction of the solid-grain network. Data are from well LD3011,Yinggehai basin, offshore south China (Luo et al., 2003). Depths and fluid pressures aremeasured with respect to a sea bottom datum in this and other figures (cf. Davis et al.,1983). (B) The predicted brittle strength in compression and extension is computedfrom Eq. (1a) and is approximately constant below zFRD because effective stress rgz�Pf is approximately constant. The pressure dependence is assumed (m ¼ 0.55).
J. Suppe / Journal of Structural Geology 69 (2014) 481e492 483
in regions dominated by disequilibrium compaction is predicted tobe constant below the fluid-retention depth
s1 � s3 ¼ ðr� rwÞkgzFRD for z � zFRD (5)
but above the zFRD strength follows the classic linearly increasinghydrostatic prediction (Eq. (2)). Thus the predicted strength-depthprofile in regions dominated by disequilibrium compaction, shownin Fig. 2B, is radically different from the classic linearly increasingstrength-depth profile of Fig. 1.
The strength of the Yinggehai basin in compression and exten-sion in Fig. 2B is calculated from Eq. (1a) based on the observedfluid pressures and an arbitrary assumed pressure dependence(m ¼ 0.55). However, the actual pressure dependence of regionalcrustal strength has to be calibrated with stress and fluid pressuremeasurements in deforming regions. A similar fluid-pressure depthpattern is found throughout the active western Taiwan thrust beltwith zFRD z 3e3.5 km (Fig. 3A) (Yue, 2007; Yue and Suppe, 2014),with the predicted strength approximately constant below thefluid-retention depth (Fig. 3B). Here borehole stress measurementsat ~1 km (Hung et al., 2009) in the hydrostatic zone are used to fixthe pressure dependence at mz 0.45 (kc z 1.4). A regional analysisof wedge mechanics in western Taiwan (Suppe, 2007) indicates asimilar pressure dependence of m z 0.35e0.40 (kc z 1) with themajor thrust faults much weaker (mf z 0.03e0.09), indicating thatoff-fault deformation associated with bending of thrust sheets,rather than the strength of major faults, dominates the crustalstrength.
3. Testing the constant-strength prediction
We now test the constant-strength prediction in an area ofactive normal faulting in the Gulf of Mexico where fluid pressureand borehole stress data exist over a substantial depth interval. Inregions of active extensional tectonics, s3 can be constrained byleakoff and fracture-initiation tests, which are hydrofracture tests
done for engineering purposes in petroleum boreholes (e.g. Whiteet al., 2002; Zoback, 2007).
The Brazos or Corsair region of offshore Texas is a region ofactive normal faulting abovemajor lystric detachments as shown inFig. 4 (Xiao et al., 1991; Worrall and Snelson, 1989). Many of thefaults are observed to cut near-surface seismic reflectors (Fig. 4).Fluid pressures are typical of disequilibrium compaction (Fig. 5A)with a well-defined fluid-retention depth at ~1.7 km, and deeperfluid pressures increasing parallel to the lithostatic gradient. Thestrength predicted from Eq. (1a) is approximately constant becauseeffective stress is close to constant below the fluid-retention depth.Leakoff test data from Xiao et al. (1991) show increasing strengthabove the fluid-retention depth and approximately constant below(Fig. 5B), in agreement with our prediction.
The form of the strength-depth curve is fixed by the fluid-pressure depth curve, whereas the magnitude of the strength isfixed by the borehole stress data (Fig. 6). The pressure dependencerequired for Eq. (1a) to match the stress data corresponds tom z 0.25 (ke z 0.39). This very low intrinsic pressure dependencemay imply that stress is controlled by very weak major normalfaults whose strength is dominated byweak clay-rich smear gouges(mf z 0.2e0.3; e.g. Brown et al., 2003; Numelin et al., 2007), withlittle contribution of off-fault deformation to regional strength, incontrast with Taiwan.
The combined fluid pressure and borehole stress data from theactively deforming Brazos area confirms the prediction of constantcrustal strength below the fluid-retention depth. The stress-depthrelationship is consistent with a critical limiting state of brittlefailure and a constant pressure-dependence k in both the hydro-static and overpressured strata.
4. Strength in areas of complex overpressures
In addition to the disequilibrium-compaction mechanism, pro-files of fluid pressure as a function of depth can be dominated orsubstantially modified by vertical and lateral pressure redistribu-tion within permeable strata or along faults within the over-pressured zone, by seals, and by pressure increases caused by gas
Fig. 4. Seismic section of the Brazos/Corsair extensional province offshore Texas, showing the location of OCS-G-1757 borehole (Fig. 5). Normal faults that cut near-surface reflectorsare indicated. For section location see Xiao et al. (1991).
J. Suppe / Journal of Structural Geology 69 (2014) 481e492484
generation (e.g. Yardley and Swarbrick, 2000; Swarbrick et al.,2002; Tingay et al., 2009a). In spite of such complexities of thefluid-pressure controlling mechanisms, we show for two suchcomplex overpressured regions that the observed borehole stressesare consistent with a critical limiting state of brittle failure and aconstant pressure-dependence k in both the hydrostatic and over-pressured strata. These examples are the Tertiary Brunei deltaoffshore northern Borneo, and the Mesozoic Sable basin offshoreNova Scotia, eastern Canada.
Permeable stratigraphic intervals that are isolated within over-pressured volumes of low permeability shale show vertical gradi-ents in fluid pressure that are approximately parallel to thehydrostatic gradient, rather than parallel to the lithostatic gradient,as shown schematically in Fig. 7A. The top of an isolated permeablelayer commonly has a fluid pressure higher than the overpressuredshale gradient, whereas the base of the permeable layer will havepressure lower than the shale gradient (points 1 and 2 Fig 7A). Thecorresponding strengths are lower at the top (point 1, Fig 7A) and
Fig. 5. (A) Fluid pressures from the well OCS-G-1757 along section Fig. 4, Brazos/Corsair extensional province offshore Texas. (B) The predicted brittle strength inextension is computed from Eq. (1a) and is approximately constant below zFRD at~1.7 km depth because effective stress is approximately constant. Stress data from leak-off tests (Xiao et al., 1991) show increasing strength above the fluid-retention depthand approximately constant below, in agreement with the prediction of Eq. (1a).
higher at the bottom (point 2, Fig 7A). Short intervals of isolatedpermeable strata can be seen in the Yinggehai and Taiwan data(Fig. 7B, ~2.8e3.2 km; Fig 3, ~5e5.15 km), but this phenomenon ismore pronounced in the Brunei data (Fig. 7C, ~1.8e2.35 km,2.6e2.8 km). In the case of folded strata with dipping, isolatedpermeable layers, the crests of anticlines will be weakened by thismechanism and the synclines strengthened, which leads to afocusing of deformation into structurally high locations (e.g.Krueger and Grant, 2011).
Typical effects of isolated permeable sands can also be seen inthe fluid pressure from the crest of the Chingtsaohu anticline(CTH) in Taiwan (Fig 3A), which is composed of two data types.The point measurements are borehole static shut-in pressureswithin permeable sands, whereas the continuous red line is thefluid pressure in surrounding shales and mudstones computed
μ = 0.55μ = 0.1 μ = 0.25
hydrostatic ρw
gz
0
1
2
3
4
5 km
0
1
2
3
4
5 km
0 50 100 MPa 0 10 20 MPa
A. B.strength σ1−σ3
in situ stress &fluid pressure Pf
Brazos/Corsair OCS-G-1757
ZFRDZFRD
lithostatic ρgz
effective stresseffective stressρgz – gz – Pf
effective stressρgz – Pf
leak-offdata
computed from Pf
Fig. 6. Predicted crustal strength for different values of the pressure dependence forthe Brazos/Corsair province offshore Texas. The predicted strengths are from Eq. (1a),based on the observed fluid pressures and several assumed depth-constant pressuredependences ke. Note that the form of the predicted strength-depth curve is the samefor different pressure dependences, differing only in magnitude. The stress data fromleakoff tests (Xiao et al., 1991) are consistent with a pressure dependence of ke z 0.39(m z 0.25).
Fig. 7. (A) Simple schematic model of the effect of heterogeneous fluid pressures onstrength. Permeable horizons imbedded within low-permeability shale-rich over-pressured crust show fluid-pressure gradients that are parallel to the hydrostaticgradient, whereas the surrounding shales show lithostatic-parallel gradients. Thepredicted strength-depth gradients alternate between linearly increasing hydrostatic-parallel gradients in the permeable layers, and constant strength within the imper-meable shale-rich intervals. Note that the base of a permeable zone (point 2) is pre-dicted to be stronger than the top (point 1). (B) Example of Yinggehai Basin offshoreChina from Fig. 2. (C) Example of Brunei delta from Fig. 9.
J. Suppe / Journal of Structural Geology 69 (2014) 481e492 485
from sonic-log data. Note that below the fluid-retention depthmany of the shut-in pressures are substantially higher thanpressures within the surrounding shales. This reflects the factthat the sands are permeable and are of large vertical and lateral
extent down the flanks of the anticline. Therefore the nominalcomputed strengths are lower for these higher-pressure strati-graphic intervals (Fig. 3B).
This vertical pressure redistribution, in some cases combinedwith effects of hydrocarbon generation, can lead to failure by ten-sile fracture and slip on faults, with leakage along faults or high-permeability pipes (e.g. Finkbeiner et al., 2001; Krueger andGrant, 2011; Leduc et al., 2013). Permeable fault zones can chargeshallow structures above the fluid-retention depth with high fluidpressures given suitable local seals, as is the case in the Yinggehaibasin (Luo et al., 2003). These processes may be viewed as redis-tributing fluid overpressure that is generated by disequilibriumcompaction or other mechanisms, such as the generation of hy-drocarbons. Tingay et al. (2009a) argue for very large-scale redis-tribution within the Brunei delta, charging already compactedstrata with high fluid pressures that do not show low porositiesconsistent with classic disequilibrium compaction.
4.1. Brunei delta, north Borneo
The Champion region of the Brunei delta, offshore north Borneo,is characterized by active normal faulting above major lystric de-tachments (Fig. 8A), which overlie a deep accretionary complex ofthe South China Sea (James, 1984; Morley et al., 2008). The fluid-pressure data as a function of depth (Tingay et al., 2009a,b) aresubstantially more complex than our previous examples but typicalof many regions with interlayered permeable and impermeableformations (cf. Swarbrick and Osborne, 1996). Hydrostatic fluidpressures are observed in the Champion Deep well (Fig. 9A) above afluid-retention depth of zFRDz 1.2e1.5 kmwhere extensional stressstates exist based on analysis of failure of deviated boreholes(Tingay et al., 2009b). The top of overpressures is at ~1.7 km and ismarked by a discontinuity in permeability, with a shale-rich sectionbelow. Abundant stress data exist in this well; minifrac hydro-fracture data provide the best estimate of strength, whereas theleakoff tests provide an effective lower bound. A single pressure-dependence k is required for Eq. (1a) to match a limiting critical-failure envelope to the stress data within both the hydrostaticand overpressured zones above ~2900 m (Fig. 9B). The best fittingpressure dependence is ke ¼ 0.45 (m ¼ 0.3). Deeper leakoff andminifrac pressures approximate the overburden pressure, sug-gesting s3 may be vertical, which may represent loading within thedeep Borneo compressional wedge from which the Brunei deltaextensional system detaches (Tingay et al., 2009b; Morley et al.,2008; see Fig. 9B below ~2900 m).
Finally, we note that in spite of these complexities of permeablezones and pressure redistribution in Brunei, the overall strengthbelow the top of overpressures (~1.7 km) can be approximated asconstant with depth, with a mean value of ~7e8 MPa (Fig. 9B), incontrast with the classic linearly increasing hydrostatic predictionof Fig. 1.
4.2. Scotia Shelf, eastern Canada
The 10e18 km thick Mesozoic Sable basin offshore Nova Scotia,eastern Canada (Fig. 8B), shows an abrupt stratigraphic transitionwith depth from hydrostatic to overpressured conditions, as in theGlenelg J-48 well (Fig. 10A). A regionally consistent pattern ofborehole breakout directions implies a regional state of stress closeto extensional failure (Fig. 8C) (Yassir and Bell, 1994). The Glenelgfield is near the shelf edge and lies within a growth normal faultsystem that was mildly active in the late Cenozoic (Williamson andSmyth, 1992). The Glenelg well contains breakouts over much of itsdepth (1181e4442 m; Bell, 1990). Modeling of the Glenelg fieldsuggests a complex fluid-pressure history that is dominated by
KK
K K
JJJJ
CC
C
salt
20 km20 km
s.l.
sectionGlenelg J-48
Sable Island
breakout direction
100km
shelf edge
500m500m500m
10 km
ChampionDeep
ChampionMain
1.6ma3.8ma
top Miocene
5 sec
s.l.
5 sec
s.l.
B. Scotia Shelf, offshore eastern CanadaC. Borehole breakouts Scotia Shelf
A. Brunei delta, offshore northern Borneo
500m500m500m
top of overpressure
Fig. 8. (A) Cross section of the Brunei delta extensional province showing the location of the Champion Deep borehole (Fig. 9) and top of overpressures from regional well datacompiled from Morley et al. (2008); James (1984). (B) Cross section of the Scotia Shelf, offshore eastern Canada with (C) location map of section and Glenelg-J-48 borehole (Fig. 10),and minimum horizontal stress directions from borehole breakouts (Yassir and Bell, 1994).
J. Suppe / Journal of Structural Geology 69 (2014) 481e492486
disequilibrium compaction with minor contribution of gas gener-ation (Williamson and Smyth, 1992). Borehole stress data from theGlenelg J-48 well (Fig. 10B) can be fit reasonably well to a limitingstrength envelope based on the observed fluid pressures and asingle pressure-dependence k within both the hydrostatic andoverpressured zones (ke z 0.39, m z 0.25). In contrast with ourprevious examples, there are insufficient data below the top ofoverpressures (~4 km) in the Scotia Shelf to constrain or predict theform of the strength-depth relationship within this very deep basin(Fig. 8B).
Fig. 9. Fluid pressure and stress from the Champion Deep structure, Brunei deltaextensional province (data from Tingay et al., 2009a,b). The predicted strengths arefrom Eq. (1a), based on the observed fluid pressures and the best-fitting constantpressure dependence k that forms a limiting envelope for the stress data. See text fordiscussion.
5. Discussion
In summary, for the three overpressured examples for which wehave stress and fluid-pressure data over a substantial depth interval(Figs. 5, 9 and 10), we are able to reasonably fit the stress data to alimiting strength envelope that is computed from observed fluidpressures by Eqs. (1) and (2), using a single pressure dependence k
in both the hydrostatic and overpressured zones. These examplesalso show regionally consistent stress directions as indicated byborehole breakouts, natural hydrofractures and active normal faulttrends. Therefore we conclude that they are close to a regional state
κe= 0.39μ = 0.25
Scotia Shelf, CanadaGlenelg J-48 well
0 50 100 MPa0
1
2
3
4
5 km
0
1
2
3
4
5 km
0 10 20 MPa
strength computedfrom Pf & Eqn.1
A. B.
lithostatic ρgz
hydrostatic ρw
gz
strength σ1−σ3
in situ stress &fluid pressure Pf
IF stressleakoff stress
Fig. 10. Fluid pressure and stress from the Glenelg J-48 well, Scotia Shelf offshoreeastern Canada (data from Bell, 1990). The predicted strengths are from Eq. (1a), basedon the observed fluid pressures and the best-fitting constant pressure dependence k
that forms a limiting envelope for the stress data. See text for discussion.
J. Suppe / Journal of Structural Geology 69 (2014) 481e492 487
of pressure-dependent failure approximated by Eqs. (1) and (2).Finally, these examples give similar low pressure dependences ofke z 0.39e0.45 (m z 0.25e0.3), consistent with the frictionalstrengths of clay-smear fault gouges on normal faults (e.g. Brownet al., 2003; Numelin et al., 2007).
In light of this success in predicting the form and magnitude ofthe limiting strength-depth envelope from fluid pressure data, wesketch two regional scale tectonic applications.
5.1. Application to regional upper crustal deformation
Our concern in this paper is exploring the first-order effects offluid overpressures on regional upper crustal strength. We brieflydiscuss two regional applications: [1] the expected role of regionalfluid overpressures in the critical-taper mechanics of accretionarywedges and fold-and-thrust belts, and [2] the possible effects offluid overpressures on crustal strength in very thick continentalmargins and shale-rich deforming mountain belts, which in somecases are thick enough to approach the brittle-ductile transition.
For these regional applications the fluid-retention depth can bereasonably estimated in many actively deforming regions to suffi-cient accuracy based on borehole data and seismic velocity analysis.In contrast, predicting pore-fluid pressures and their hetero-genieties at the level required in petroleum exploration, productionand engineering is more technically and observationallydemanding and is beyond our present tectonic scope. Nevertheless,at the more local scale of individual structures, fluid pressuresappear to play very significant roles of interest to structural geol-ogy, for example see applications of Krueger and Grant (2011) to theNiger delta deep-water thrust belt, and Finkbeiner et al. (2001) inthe northern Gulf of Mexico, South Eugene Island.
5.1.1. Role of fluid overpressures in critical-taper wedge mechanicsand Hubbert-Rubey fault weakening
There has been a long history of considering the role of fluidoverpressures in the mechanics of thrust faults, accretionarywedges and thrust belts (e.g. Hubbert and Ruby, 1959; Davis et al.,1983; Dahlen, 1990). Fluid pressures generally enter these the-ories as the Hubbert-Rubeyweakening coefficient (1� l), which forsimplicity generally has been assumed constant, implying a linearlyincreasing crustal strength (Eq. (1)). Unfortunately constant (1 � l)is unrealistic except in the hydrostatic case of l ¼ rwgz=rgz. In theoverpressured case (1 � l) is everywhere vertically variable, as wehave seen (Figs. 2, 3, 5, 9 and 10). However, in the case of classicdisequilibrium compaction we can express the vertical variation in(1 � l) as a simple function of fluid-retention depth zFRD given byEq. (4), (1�l) z 0.6zFRD/z. This relationship has been applied tothrust and accretionary wedgemechanics by Yue and Suppe (2014).Here we summarize the basic roles of overpressure in mechanics ofthrust belts, in light of the results of this paper. For the purposes ofillustration, we make use of some data from the deep-water Nigerdelta thrust belt (Fig. 11).
There are two common but contrasting ways of expressing roleof fluid pressure in brittle behavior: [1] in effective coefficients offriction m*f , internal friction m* and pressure dependence k* and [2]in absolute fault st and crustal (s1 � s3) strengths. Up to this pointwe have been considering the effects of overpressure on absolutecrustal strength [2]; in what follows we see that critical-taperwedge mechanics depends on the effective-friction coefficients[1]. These effective coefficients are controlled by the ratio of fluidpressure to solid overburden pressure, l ¼ ðPf =rgzÞ, whereas theeffect of fluid pressure on absolute strength is controlled by theirdifference, the effective stress ðrgz� Pf Þ. The effective internalfriction, pressure dependence and fault friction coefficients are
m* ¼ mh1�
�Pf.rgz
�i¼ mð1� lÞ (6a)
k* ¼ kh1�
�Pf.rgz
�i¼ kð1� lÞ (6b)
m*f ¼ mf
h1�
�Pf.rgz
�i¼ mf
�1� lf
�(6c)
In contrast, the absolute crustal and fault strengths are
ðs1 � s3Þ ¼ k�rgz� Pf
�(7a)
st ¼ mf
�sn � Pf
�(7b)
where the effective fault-normal stress (sn � Pf) varies with faultorientation. For example, in the case of horizontal compression(sn � Pf) varies between the values for horizontal and vertical faultsðrgz� Pf Þ � ðsn � Pf Þ � kcðrgz� Pf Þ.
If we apply these basic relationships (Eqs. (6) and (7)) to thefluid-pressure distributions predicted by the classic disequilibriumcompaction model, then fluid pressures increase parallel to thelithostatic gradient below the fluid-retention depth, such thatðrgz� Pf Þ is constant with depth (Fig. 11A). This leads to constantabsolute crustal and fault strengths (Eqs. (5) and (7)) below thefluid-retention depth as shown in Fig. 11C. (We note that constantfault strength with depth has beenwidely assumed in fault-rupturedynamical models, e.g. Rice, 1992). In contrast, the effective frictioncoefficients continuously decrease (Eq. (6)) because l ¼ ðPf =rgzÞcontinuously increases below the fluid-retention depth, as illus-trated in Fig. 11B. In each case (Fig. 11B and C), the effect of fluidpressure on brittle strength is the same for both the crust and thefaults, only differing by the magnitudes of their intrinsic pressuredependences (mf, m, k).
Active accretionary wedges and thrust belts deform to a criticaltaper (a þ b), where a is the surface slope and b is the detachmentdip (e.g. Davis et al., 1983). This critical taper is controlled bydetachment strength F ¼ st=rgH and wedge strengthW ¼ ðs1 � s3Þ=rgH, where H is the local depth of the detachment
aþ b ¼h1�
�rf
.r�i
bþ Fh1�
�rf
.r�i
þW(8)
and rf is the density of the fluid overlying the wedge, air or water(Suppe, 2007). Taper angles a and b are in radians. Awide variety ofbrittle and ductile processes operating at different time and spatialscales potentially may control F andW (Dahlen, 1990). For example,in some wedges the detachment may be activated in large-slipearthquakes with dynamical processes dominating the faultstrength, during which wedge taper might be established by off-detachment deformation (e.g. Suppe et al., 2009). However, in thefollowing we assume cohesionless brittle frictional static strengthsfor both the wedge and the detachment (Eqs. (6) and (7)) to illus-trate the predicted role of static pore-fluid pressures controlled bydisequilibrium compaction (Eq. (4)).
Yeh and Suppe (2014) show for the case of a general heteroge-neous wedge of Dahlen (1990) that, if vertical variation in fluidpressure can be approximated by our simple fluid-retention depthzFRD formulation (Eq. (4)), then the effect of fluid overpressures onwedge mechanics is very simply described by zFRD/H. Here weillustrate this for the approximation of constant zFRD and anotherwise homogeneous and cohesionless brittle wedge, for which
deep water Niger delta
0 50 100 MPa
1
s.b.
2
3
4
5
6
7 km
hydrostatic ρ
w gz
ZFRD
fluid pressureA.
lithostatic ρgz
ρgz +
FRD
w
ρ g(z–z )FRD
5
6
1
s.b.
2
3
4
7 km
ZFRD
effective friction & pressure dependence
= 1.86κc
= 0.2μf = 0.55μ
μ f 0
.6(Z
/ Z)
FRD
κ c 0
.6(Z
/ Z)
FRD
fμ* cκ*μ*
μ 0
.6(Z
/ Z)
FRD
κc1 + (Z /H)FRD
μβα + β = f + (Z /H)FRD
στ horiz. fault
στ vert. fault
ZFRD
0 40 6020 80 MPa
2
3
4
5
6
7 km
absolutecrustal & fault strength
C.
crus
tal s
treng
thσ 1−
σ3
κρ c
= 0
.6gZ
FRD
μ ρf =0.6 =0.6 gZgZFRDFRDμ ρf =0.6 gZFRD
wedge taper predicted by static fluid pressures
0 2 4 6 8 10 km0.0°
0.3°
0.6°
0.9°
1.2°
1.5°
depth to detachment H
ZFRD overpressured wedge surface slope α
hydrostatic wedge surface slope α
wed
ge s
urfa
ce s
lope
α&
det
achm
ent d
ip β
detachment dip β = 1.5°
D.
horiz. scale100km
D
H
FRDzα
β
σ σ1 3
FRDz
z
sea level
overpressured wedgesea level
hydrostatic wedgeD
H
α
β
σ σ1 3
z
E. F.
Fig. 11. Contrasting roles of fluid overpressure on effective friction and absolute strength and their effect on critical-taper thrust mechanics, illustrated for the Niger delta. (A) Fluid-pressures from a well in the deep-water Niger delta thrust belt (Krueger and Grant, 2011). (B) Predicted decay of effective friction (m*f , m*) and pressure dependence k* with depth(Eq. (6)), based on the decay of Hubbert-Rubey weakening below the fluid-retention depth (1 � l) z 0.6zFRD/z. (C) Predicted constant absolute fault and crustal strength below thefluid-retention depth zFRD (Eq. (7)), based on constant effective stress ðrgz� Pf Þ. (D) Predicted wedge taper for the deep-water Niger delta thrust belt, based on the assumption thatstrength is controlled by static regional fluid pressure (Eqs. (9) and (11)). Because both fault strength and wedge strength would be affected in identical proportion by regional fluid-pressure, through zFRD/H, regional overpressures are incapable of explaining the order-of-magnitude contrast between wedge strengthW and fault strength F in the Niger delta andmany other thrust belts (see discussion). Nevertheless, overpressures produce some reduction in taper relative to the already very low taper predicted for a hydrostatic wedge. (E &F) Schematic hydrostatic and overpressured wedges, showing contrasting strength-depth profiles.
J. Suppe / Journal of Structural Geology 69 (2014) 481e492488
the detachment and wedge strengths become simply the effectivestrength coefficients of Eq. (6) written in terms of zFRD/H
F ¼ mf ð1� lHÞ ¼ mf ½1� ðrw=rÞ�zFRD.H (9a)
W ¼ kcð1� lHÞ ¼ kc½1� ðrw=rÞ�zFRD=H (9b)
where lH is the regional l at the detachment depth H. Thus we seethat the effect of fluid overpressures on critical taper under the
disequilibrium compaction model is a very simple function of zFRD/H, which affects fault strength and wedge strength identically
aþ b ¼h1�
�rf
.r�i
bþ mf ½1� ðrw=rÞ�zFRD.H
h1�
�rf
.r�i
þ kc½1� ðrw=rÞ�zFRD.H
(10)
In the case of the Niger delta, the variation of zFRD/H predicts a veryslowly decreasing taper with increasing H (Fig. 11D).
J. Suppe / Journal of Structural Geology 69 (2014) 481e492 489
In the case of a submarine wedge, rf ¼ rw, the expression fortaper becomes further simplified to
aþ b ¼bþ mf zFRD
.H
1þ kczFRD=H(11a)
and for a hydrostatically pressured submarine wedge (zFRD � H) thetaper is constant
aþ b ¼ bþ mf
1þ kc(11b)
We see from the overpressured wedge-taper equation (Eq.(11a)) that it is the contrast in crustal and fault strength coefficientskc and mf that dominates the contrast in wedge and fault strengthsW and F and hence the taper magnitude. It is only in cases in whichzFRD/H is small that fluid overpressures begin to dominate wedgetapers.
The contrast between F and W is quite large for the Niger delta,F/W < 0.1, based on regional magnitudes of F ¼ 0.04 and W ¼ 0.7that were computed from the covariation of surface slope a withdetachment dip b, without any strong assumptions of strength-controlling mechanisms (Suppe, 2007, using Niger delta taperdata of Bilotti and Shaw, 2005). This large contrast between F andWis typical of many accretionary wedges and thrust belts; a typicalvalue of F/W z 0.1 is indicated based on the observed narrow ta-pers of accretionary wedges (Suppe et al., 2009). Narrow wedgetapers cannot be dominated by regional fluid overpressures exceptin cases of small zFRD because fluid pressures affect the wedge andfault strengths by the same proportions in Eqs. 9 and 11. Othermechanisms of fault weakening relative to wedge strength willgenerally be required, for example the diverse mechanisms ofdynamical weakening in large earthquakes (e.g. Di Toro et al., 2011).In other tectonic environments quasistatic mechanisms of faultweakening are indicated, such as evaporite detachments orintrinsically weak clay gouges (e.g. Brown et al., 2003; Numelinet al., 2007).
Nevertheless, static regional fluid overpressures are predicted tohave two well-defined effects on wedge tapers. [1] First, over-pressures reduce the taper relative to what it would be for anequivalent hydrostatic wedge (Eqs. 11a and b). In the case of theNiger delta well (Fig. 11A) the predicted effect of zFRD/H is a ~30%reduction of an already very small predicted hydrostatic surfaceslope of ~1.1� to an overpressured slope of ~0.8� (Fig.11D) (based onan ~1.5� detachment dip b, zFRD ¼ 2.3 km, and a detachment depthH¼~7 km). The predicted effect will be larger for wedges with avery shallow fluid-retention depth relative to detachment depth.[2] Second, given a relatively constant zFRD we expect a decreasingwedge taper towards the wedge interior based on a regionaldecrease in zFRD/H. In the case of the Niger delta well this predictedeffect is quite subtle (Fig. 11D). The surface slope would decreaseonly ~0.1� over 100 km, given the very gentle detachment dip of~1.5�, which is difficult to verify given widespread wedge-topdeposition. In contrast, decreasing surface slopes away from thetoes of submarine accretionary wedges are widely observedwithout any associated change in detachment dip (e.g. Zhao et al.,1986). In the case of the convex toe of the Barbados accretionarywedge, Yeh and Suppe (2014) show that the decrease in taper is afluid-pressure effect, dominated by the decrease in zFRD/H towardsthe wedge interior and a very shallow fluid-retention depth.
5.1.2. Strength of very thick overpressured crustHere we briefly consider the implications of fluid overpressures
for crustal-scale strength-depth profiles. Borehole observations inthe range 1e9 km suggest a widespread limiting critical state of
pressure-dependent failure controlled by effective stress underboth hydrostatic and overpressured conditions (Figs. 1e3, 5, 9, 10;Eqs. (1) and (2)). Extrapolating to greater depths, Brace andKohlstedt (1980) argued that hydrostatic pore-fluid pressureswould be typical of crystalline crust, based on expected high per-meabilities. Townend and Zoback (2000) argued that continuedfaulting is required to provide high-permeability conduits thatkeep fluid pressures near hydrostatic in crystalline crust, especiallyin plate interiors (Zoback and Townend, 2001). These consider-ations have led to the classic strength-depth profile (Fig. 1).
Here we consider the possibility of overpressured conditionsexisting within substantial thicknesses of the crust. Thick sedi-mentary basins on continental margins built on oceanic or highly-thinned continental crust have vast deforming clastic-rich volumesthat approach a significant fraction of crustal thickness, and mayeven span the brittle-ductile transition (e.g. Gulf of Mexico, Nigerdelta, Sumatra, Nankai trough). These thick clastic stratigraphicaccumulations may deform into shale-rich plate boundary moun-tain belts (e.g. Bangladesh/Myanmar, Barbados/Trinidad, Makran,Gulf of Alaska, offshore SW Taiwan, New Zealand). In the activecompressional Southern Alps of New Zealand there is deepgeophysical evidence for near lithostatic pore-fluid pressures overmuch of the width of the mountain belt, extending to depths of20e30 km, based on seismic P-wave velocity analysis, local to-mography and magnetotelluric data (Stern et al., 2001; Eberhart-Phillips and Bannister, 2002; Wannamaker et al., 2002; Scher-wath et al., 2003). This is a region of the crust interpreted to becomposed of clastic sedimentary rocks undergoing progrademetamorphism to schist and is expected to extend beyond thebrittle-ductile transition. Very deep (~40 km) near lithostatic fluidpressures are widely regarded as a central ingredient of slow-slipearthquake phenomena (e.g. Peng and Gomberg, 2010).
Furthermore, petrologic observations suggest that fluid pres-sures may be near lithostatic in fine-grained crust undergoing thetransition from shale to phyllite and schist, dominated by pressuresolution. It is widely argued that pore-fluid pressures are close tolithostatic, based on the existence of some types of crack-seal andopen-void veins (e.g. Etheridge, 1983; Etheridge et al., 1984; Fisherand Brantley, 1992; Hilgers et al., 2006; Van Noten et al., 2011), andbased on studies of fault-related mesothermal gold-quartz veinsystems, which are suggestive of fault-valve phenomena near thebase of the seismogenic zone (e.g. Sibson et al., 1988). Fluid-inclusion studies within ore-hosting conduits indicate large oscil-lations between near-lithostatic and possibly near-hydrostaticpressures. These observations are suggestive of a dual-permeability system with a fine-grained matrix at near-lithostaticpressure, and fault conduits that are alternately broken and sealedby vein formation with associated fluid pressure and stress fluc-tuations of the seismic cycle (e.g. Robert et al., 1995;Micklethewaite, 2008).
Studies by Van Noten et al. (2011) of the stress-state and fluid-pressure evolution in the High-Ardenne slate belt of Germany,based on analysis of quartz veins and fluid inclusions, indicate nearlithostatic pore-fluid pressures under very low grade metamorphicconditions of ~250�C, and an assumed depth of ~7 km based onstratigraphy. They estimated a limiting compressive crustalstrength s1 � s3 during vein formation of ~35e45 MPa, which isconsistent with a nominal fluid-retention depth of 1e3 km, nearthe top of the Lower Devonian siliciclastic section. In contrast, acrustal strength of 100e200 MPa would be expected at ~7 kmdepth under the classic brittle-hydrostatic model. The super-lithostatic fluid pressures suggested for vein formation mayreflect transient vertical redistribution (cf. Fisher et al., 1995).Furthermore, the low ambient background crustal stress statesrequired for widespread joint and vein formation are expected at
J. Suppe / Journal of Structural Geology 69 (2014) 481e492490
great depth in overpressured crust, in contrast with the classichydrostatic strength-depth profile (see Fig. 12).
If we interpolate our borehole observations from overpressuredcontinental margins with the geophysical and petrologic evidencefor near-lithostatic fluid pressures, and hence very low strength,near the brittle-plastic transition, then constant strength may be areasonable first-order strength-depth model for fine-grained,clastic-dominated crust, as shown in Fig. 13B. This constantstrength model is in contrast with the classic Brace and Kohlstedt(1980) hydrostatic strength-depth profile that is well justified forcrystalline crust (Figs. 1 and 13A). That said, the strength, fluid
ΔPf
Z=8km
Z=Z =2.4kmFRD
effective stress Z>ZFRD
cohesionless regional strength
cohesive rock strength (T = 10MPa)
o
cohesionless regional strength
cohesive rock strength (T = 10MPa)
o
o
tensile strength T
oTo2T
effective stressfor regionaljointing
στ
στ
(σ – P )n f
(σ – P )n f
100MPa
–100MPa
–100MPa
50 100100100 200MPa200MPa200MPa
50 100100100 200MPa200MPa200MPa
100MPa
Z=2.4kmZ=2.4kmZ=2.4km
A.
B.
30°
30°
hydrostatic regional strength
overpressured regional strength
σ1 vertical
Fig. 12. Effective stresses required for regional joint formation and the predictedregional stress states for (A) hydrostatic and (B) overpressured crust at a critical state offailure, illustrated with Mohr diagrams. Effective stresses at great depth in over-pressured crust are small and therefore relatively favorable for regional joint forma-tion, in contrast with hydrostatic crust. Note that for all depths greater than the fluid-retention depth zFRD, the effective stress state is identical to that at the fluid-retentiondepth, therefore only a modest increase in fluid pressure DPf would allow the effectivestresses needed for joint formation by tensile fracture, even at very great depth.
12
14
20
Z
12
14
20
Z
plastic strength plastic strength
brittle strength
possible stress states
Fig. 13. Contrasting hydrostatic and overpressured crustal strength-depth curves. (A)The classic strength-depth graph showing linearly increasing brittle strength, whichrequires hydrostatic pore-fluid pressures (Eq. (2)). This model agrees with much deepborehole data (Fig. 1). (B) The constant-strength model for overpressured crust, basedon borehole data from this paper combined with geophysical and petrologic evidencefor near-lithostatic overpressures and low strength in fine-grained rocks at very lowmetamorphic grade near the brittle-plastic transition.
pressure and state of stress through the brittle-ductile transition inshale-rich mountain belts remains a significant research frontierbeyond the scope of this paper. However, current viscous-plasticmodeling of disequilibrium compaction suggests that matrixweakening by pressure solution becomes a dominant mechanismin generating near-lithostatic fluid pressures at great depths(Morency et al., 2007).
6. Conclusions
The classic view of upper crustal strength is a linearly increasingpressure-dependent brittle strength, which requires a hydrostaticpore-fluid pressure. This model has been confirmed by muchborehole data in crystalline crust to depths reaching 8e9 km(Fig.1). In contrast, clastic sedimentary basins and deforming shale-rich mountain belts display pore fluids that are overpressuredbelow some critical depth, based on much borehole data andseismic velocity analysis. Therefore we have explored the predictedshapes of strength-depth curves based on fluid-pressure data in anumber of overpressured regions, using the same assumption usedin the hydrostatic case: that a limiting critical strength is controlledby effective stress times a pressure dependence s1 � s3 ¼kðrgz� Pf Þ. We have successfully tested this prediction for threecases inwhich we have both fluid pressure data and in situ boreholestress data over a substantial depth interval (Gulf of Mexico, Bruneidelta, and Scotia Shelf, eastern Canada).
Fluid overpressures that are dominated by the disequilibriumcompaction mechanism are of particular interest for regional-scaletectonics because they are very widely observed, have straightfor-ward implications for crustal strength, and are predictable based onobservablequantities. Themost straightforwardsignatureof regionaldisequilibrium compaction is hydrostatic fluid pressures at shallowdepths above the fluid-retention depth zFRD, and overpressuredbelow it, increasing parallel to the lithostatic gradient rgz. Thereforeeffective stress ðrgz� Pf Þ is constant below the zFRD, which impliesthat brittle strengthwill be also constant s1� s3z 0.6kgzFRD. Crustalstrength profiles in areas of regional disequilibrium compaction are
J. Suppe / Journal of Structural Geology 69 (2014) 481e492 491
radically different from the classic linearly-increasing hydrostaticstrength-depth profile (e.g. Figs. 1, 2 and 13).
Fluid-retentiondepth zFRD is particularly important because it canbe observed based on commonly available borehole data and onseismic velocity analysis in non-eroded siliciclastic sedimentarysections. In addition, fossilfluid-retentiondepths canbepreserved inuplifted and eroded strata and identified by a variety of means(Magara, 1978; Yue and Suppe, 2014). Once we know the fluid-retention depth we can estimate the Hubbert-Rubey weakening(1 � l) z 0.6zFRD/z everywhere within the overpressured zone (Eq.(4)), which allows us to immediately know the vertical variation ineffective friction (m*f , m*, k*) and absolute crustal (s1 � s3) and staticfault strength st (Eqs. (6) and (7), Fig. 11). Therefore, fluid-retentiondepth zFRD should be considered an important observable param-eter in crustal mechanics.
Observations of fluid-retention depth zFRD are of particularimportance for critical-tapermechanics of accretionarywedges andthrust belts. Yeh and Suppe (2014) have incorporated the rela-tionship between Hubbert-Rubey weakening and fluid-retentiondepth (1 � l) z 0.6zFRD/z into the general heterogeneous wedgetheory of Dahlen (1990). This allows us to directly incorporateobservations of fluid-retention depth in applications of wedgemechanics, obtaining geologically reasonable fluid pressure andstrength distributions. Yeh and Suppe (2014) have applied this todata from the Barbados accretionary wedge, and show that theconvex geometry of its toe, which is typical of many classic accre-tionary wedges, is dominated by regional fluid overpressures asdescribed by wedge equations that incorporate fluid-retentiondepth zFRD and other observable parameters. In the present paperwe apply a simplified version of the Yeh and Suppe (2014) theory toillustrate the effects of fluid overpressures onwedge taper (Eq. (11),Fig. 11). This analysis suggests that the very low tapers of manyaccretionary wedges and fold-and-thrust belts, which imply wedgestrengths much higher than basal fault strengths F/W z 0.1, arereduced but not dominated by regional overpressure effects,because static fluid overpressures affect wedge strength and basalfault strength by the same proportions (Eqs. (9) and (11)).
We have briefly discussed some geophysical and petrologicevidence for near-lithostatic overpressures and low strength atdeeper levels of the crust, at very low metamorphic grade near thebrittle-ductile transition. Combining this deeper evidence with theborehole evidence of this paper, we suggest that a first-orderconstant strength model is more appropriate for overpressuredcrust than the classic linearly-increasing strength-depth profile(Fig. 13).
Acknowledgements
I am grateful to collaborators and organizations for their supportand encouragement over an extended period, including Li-Fan Yue,En-Chao Yeh, the Chinese Petroleum Corporation (CPC Corp), Tex-aco USA, National Taiwan University, Ludwigs Maximillian Uni-versity, Alexander von Humboldt Foundation, and ROC NationalScience Council grant NSC102-2116-M-002-002.
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