Fluid Overpressures and Strength of the Sedimentary Upper Crust

Journal of Structural Geology 69 (2014) 481e492

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Journal of Structural Geology

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Fluid overpressures and strength of the sedimentary upper crust

John Suppe

Department of Geosciences, National Taiwan University, Taipei 106, Taiwan article info abstract

Article history: The classic crustal strength-depth profile based on rock mechanics predicts a brittle strength s1 s3 ¼ Available online 1 August 2014 kðr Þ gz Pf that increases linearly with depth as a consequence of [1] the intrinsic brittle pressure dependence k plus [2] an assumption of hydrostatic pore-fluid pressure, Pf ¼ rwgz. Many deep borehole Keywords: stress data agree with a critical state of failure of this form. In contrast, fluid pressures greater than Crustal strength hydrostatic rgz > P > r gz are normally observed in clastic continental margins and shale-rich mountain fl f w Pore- uid overpressures belts. Therefore we explore the predicted shapes of strength-depth profiles using data from over- Fluid-retention depth pressured regions, especially those dominated by the widespread disequilibrium-compaction mecha- Critical-taper wedge mechanics fl fl Jointing nism, in which uid pressures are hydrostatic above the uid-retention depth zFRD and overpressured r Strength-depth profile below, increasing parallel to the lithostatic gradient gz. Both brittle crustal strength and frictional fault ðr Þ strength below the zFRD must be constant with depth because effective stress gz Pf is constant, in contrast with the classic linearly increasing profile. Borehole stress and fluid-pressure measurements in several overpressured deforming continental margins agree with this constant-strength prediction, with the same pressure-dependence k as the overlying hydrostatic strata. The role of zFRD in critical-taper wedge mechanics and jointing is illustrated. The constant-strength approximation is more appropriate for overpressured crust than classic linearly increasing models. © 2014 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

1. Introduction 1.1. Classic brittle strength-depth relationship

The classic Brace and Kohlstedt (1980) strength-depth graph, The predicted linear increase in brittle strength s1 s3 with based on experimental rock mechanics, predicts an initial linear depth is a consequence of [1] the intrinsic pressure dependence k of increase in brittle frictional strength with increasing pressure, brittle strength, plus [2] an assumption that pore-fluid pressure is followed by an exponential decay in ductile strength with hydrostatic and therefore linearly increasing Pf ¼ rwgz. Ignoring increasing temperature (Fig. 1A). This first-order prediction has cohesion we approximate brittle crustal strength very simply as the r k been remarkably successful, for example in predicting the tem- vertical effective stress gz Pf , times the pressure dependence . perature of the brittle-ductile transition in common rock types. s s ¼ k r ; The intricacies of this graph in the lower crust and upper mantle 1 3 gz Pf (1a) continue to be widely discussed and are even controversial in the case of the proposed “jelly sandwich” of weak lower continental (Suppe, 1985, p. 185), where r is the mean bulk density of the crust above a presumed stronger mantle (e.g. Chen and Molnar, overlying rock. Eq. (1) also can be written in terms of the classic fl l ¼ =r 1983; Suppe, 1985,p.188e189; Jackson, 2002; Burov and Hubbert-Rubey (1959) uid-pressure ratio Pf gz Watts, 2006; Bürgmann and Dresen, 2008). In contrast, the s s ¼ kð1 lÞrgz; (1b) predicted linear increase in brittle strength is generally assumed 1 3 without discussion and agrees with deep borehole stress data in where (1 l) may be considered the fractional fluid-pressure crystalline rock, showing that the crust is commonly close to a weakening. In submarine cases l is measured with respect to the critical state of brittle failure (e.g. Townend and Zoback, 2000; sea bottom (Davis et al., 1983). For the hydrostatic case (1a) Zoback, 2007), for example the deep KTB borehole in Germany becomes. (Fig. 1B). s s ¼ kðr r Þ : 1 3 w gz (2)

These crustal-strength equations express the fact that brittle E-mail address: [email protected]. pressure-dependent strength is a function of effective stress http://dx.doi.org/10.1016/j.jsg.2014.07.009 0191-8141/© 2014 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). 482 J. Suppe / Journal of Structural Geology 69 (2014) 481e492

A.σ −σ B. in situ stress σ −σ faulting, off-fault fracturing, and folding processes, some of which strength 1 3 1 3 0 500MPa 0 200MPa may be coseismic and may be controlled by dynamic frictional 0 0 processes in earthquakes (e.g. Di Toro et al., 2011). possible stress states

4 2 2. Implications of ﬂuid overpressures for regional strength

hydrostatic In contrast with the classic view of linearly increasing crustal 8 brittle strength 4 strength dominated by hydrostatic pore-fluid pressures, it is well established from petroleum boreholes and seismic velocity analysis that fluids are overpressured in deeper parts of thick, fine-grained 12 6 clastic sedimentary basins and in deforming shale-rich plate- boundary mountain belts, largely due to disequilibrium compaction from stratigraphic and tectonic loading, but with additional 16 8 effects including vertical and lateral pressure redistribution and gas plastic strength Germany generation (Bredehoeft and Hanshaw, 1968; Fertl, 1976; Magara, KTB borehole 1978; Hart et al., 1995; Swarbrick and Osborne, 1996; Yardley and 20km 10km Swarbrick, 2000; Tingay et al., 2009a). The basins of most interest for crustal strength and large-scale tectonics are continental mar- Fig. 1. (A) The classic strength-depth graph with linearly increasing brittle strength, gins built on oceanic or highly-thinned continental crust that have which is based on the assumption of hydrostatic pore-fluid pressures (Eq. (2)). (B) This fi brittle-hydrostatic model agrees well with stress data from the deep KTB borehole in vast deforming clastic-rich volumes that approach a signi cant Germany where fluid pressures are hydrostatic (Brudy et al., 1997; Zoback and Harjes, fraction of crustal thickness, even spanning the brittle-ductile 1997; Grawinkel and Stockhert,€ 1997), as well as data elsewhere (see Townend and transition (e.g. Gulf of Mexico, Niger delta, Bangladesh/Myanmar, Zoback, 2000; Zoback, 2007). Sumatra, Makran, Gulf of Alaska, New Zealand, Nankai trough, Barbados/Trinidad, and offshore southwestern Taiwan). Here we illustrate typical observed fluid-pressure/depth relationships from r r boreholes. We then compute the expected strength-depth profiles gz Pf and not the solid pressure gz or even depth, which is a distinction that becomes more important in the case of fluid (Eq. (1a)) and compare them with borehole stress data in actively overpressures. deforming regions, where stress is expected to be close to regional The pressure dependence k can be expressed in terms of the strength, or at least limited by Eq. (1a) as an upper bound. familiar Coulomb internal friction (m ¼ tan4) and varies between kc ¼ 2sin4 (1 sin4) in pure-thrust compression and ke¼2sin4 2.1. Disequilibrium-compaction mechanism and crustal strength (1 þ sin4) in pure normal-fault extension, with strike-slip stress states (s2 vertical) lying between these limits, as can be shown from The fluid-pressure depth relationship for the Yinggehai basin the Mohr diagram. offshore south China (Fig. 2A) (Luo et al., 2003)istypicalofthe disequilibrium-compaction mechanism (e.g. Swarbrick and 1.2. Success of the classic brittle-hydrostatic assumption Osborne, 1996; Magara, 1978), which has also been called “leaky overpressure” (Crans and Mandl, 1980; Mandl and Crans, Brace and Kohlstedt (1980) assumed that brittle crustal strength 1981). Pressures are hydrostatic down to a critical depth of is controlled by an optimally oriented static fault friction of ~1.8 km, the fluid-retention depth z , below which fluid pres- m ¼ 0.6e0.85 based on lab static friction measurements from a wide FRD sures increase parallel to the lithostatic gradient (Fig. 2A), variety of rock types (Byerlee, 1978)(k ¼ 2.1e3.7, k ¼ 0.68e0.79). c e because permeability becomes sufficiently low that the incre- Townend and Zoback (2000) showed that deep borehole stresses mental increases in sedimentary or tectonic load are supported are typically close to a critical state of failure consistent with by the pore fluid, rather than by incremental compaction of the m ¼ 0.55e1(k ¼ 1.9e4.8, k ¼ 0.65e0.83) and an assumed hydro- c e solid-grain framework. Fluid pressures under the disequilibrium- static fluid pressure, implying that stress z regional strength in compaction mechanism are a simple function of the fluid- many regions (Zoback, 2007). Therefore the Brace and Kohlstedt retention depth, representing the sum of the hydrostatic and (1980) brittle-hydrostatic proposal (Eq. (2)) is in good agreement lithostatic contributions (Fig. 2A). with available observations, as is illustrated by stress data from the deep KTB borehole in Germany (Fig. 1B). ¼ r Pf wgz for z zFRD (3a) We have expressed brittle crustal strength (Eq. (1a)) in terms of the vertical effective stress because this quantity can be directly ¼ r þ r ð Þ calculated from commonly available borehole data or estimated Pf wgzFRD g z zFRD for z zFRD (3b) from seismic velocities in clastic sedimentary sections that have not It follows that the Hubbert-Rubey fractional fluid-pressure undergone uplift and erosion (e.g. Fertl, 1976; Magara, 1978; weakening (1 l) (see Eq. (1b)) is a simple function of the fluid- Swarbrick et al., 2002; Zoback, 2007). We will use Eq. (1a) to retention depth z r FRD compare the observed vertical effective stress gz Pf with in situ s s ð lÞ¼½ ðr =rÞ = z : = borehole stress measurements of 1 3 to determine if they are 1 1 w zFRD z 0 6zFRD z for z zFRD (4) consistent with a constant value of the pressure dependence k over the entire depth range of the data from actively deforming sedi- We make use of Eq. (4) in the discussion section of this paper. mentary basins. This strategy provides a more explicit test of the Because of the observed parallelism of fluid pressures to the fl r role of pore- uid pressures on crustal strength and avoids lithostatic gradient, the vertical effective stress gz Pf is approx- assuming what processes may control crustal brittle strength, imately constant below the fluid-retention depth (Fig. 2A) and which is an open question. For example, the regional-scale pressure equal to the effective stress at the fluid-retention depth k ðr Þ¼ðr r Þ dependence of the crust may be controlled by a combination of gz Pf w gzFRD. Therefore pressure-dependent strength J. Suppe / Journal of Structural Geology 69 (2014) 481e492 483

A. B. predicted fluid pressure Pf strength σ1−σ3 0 50 100 MPa 0 25 50 MPa 0 0 Yinggehai basin, China μ = 0.55 LD3011 1 1 ZFRD Z FRD ZFRD 2 2 effective stress ρgz – Pf

hydrostatic

ρ =1.86

3 w 3 c

gz + κ

lithostatic =0.65

FRD e κ

ρ 4 g(z–z ) 4

ρ ρ

w gz gz FRD compressional 5 km 5 km extensional

Fig. 2. (A) Typical fluid pressures dominated by the disequilibrium compaction mechanism, which is characterized by hydrostatic pressures down to a critical depth, the fluid-retention depth zFRD, below which fluid pressures increase parallel to the Fig. 3. (A) Typical fluid pressures in the active western Taiwan thrust belt, Chingtsaohu lithostatic gradient, because permeability becomes sufficiently low that incremental (CTH) anticline, with (B) strength computed from Eq. (1a), using a pressure- increases in sedimentary or tectonic load are supported by the pore fluid, rather than dependence to match in situ stress data in the TCDP borehole from Hung et al. by incremental compaction of the solid-grain network. Data are from well LD3011, (2009). Point data are shut-in pressure tests from permeable strata, and the solid Yinggehai basin, offshore south China (Luo et al., 2003). Depths and fluid pressures are curve is pressure in shale computed from sonic log (Suppe and Wittke, 1977; measured with respect to a sea bottom datum in this and other figures (cf. Davis et al., Yue, 2007; Yue and Suppe, 2014). 1983). (B) The predicted brittle strength in compression and extension is computed from Eq. (1a) and is approximately constant below zFRD because effective stress rgz m ¼ Pf is approximately constant. The pressure dependence is assumed ( 0.55). done for engineering purposes in petroleum boreholes (e.g. White et al., 2002; Zoback, 2007). in regions dominated by disequilibrium compaction is predicted to The Brazos or Corsair region of offshore Texas is a region of be constant below the fluid-retention depth active normal faulting above major lystric detachments as shown in Fig. 4 (Xiao et al., 1991; Worrall and Snelson, 1989). Many of the s s ¼ðr r Þk 1 3 w gzFRD for z zFRD (5) faults are observed to cut near-surface seismic reflectors (Fig. 4). Fluid pressures are typical of disequilibrium compaction (Fig. 5A) but above the zFRD strength follows the classic linearly increasing with a well-defined fluid-retention depth at ~1.7 km, and deeper hydrostatic prediction (Eq. (2)). Thus the predicted strength-depth fluid pressures increasing parallel to the lithostatic gradient. The profile in regions dominated by disequilibrium compaction, shown strength predicted from Eq. (1a) is approximately constant because in Fig. 2B, is radically different from the classic linearly increasing effective stress is close to constant below the fluid-retention depth. strength-depth profile of Fig. 1. Leakoff test data from Xiao et al. (1991) show increasing strength The strength of the Yinggehai basin in compression and exten- above the fluid-retention depth and approximately constant below sion in Fig. 2B is calculated from Eq. (1a) based on the observed (Fig. 5B), in agreement with our prediction. fluid pressures and an arbitrary assumed pressure dependence The form of the strength-depth curve is fixed by the fluid- (m ¼ 0.55). However, the actual pressure dependence of regional pressure depth curve, whereas the magnitude of the strength is crustal strength has to be calibrated with stress and fluid pressure fixed by the borehole stress data (Fig. 6). The pressure dependence measurements in deforming regions. A similar fluid-pressure depth required for Eq. (1a) to match the stress data corresponds to pattern is found throughout the active western Taiwan thrust belt m z 0.25 (ke z 0.39). This very low intrinsic pressure dependence with zFRD z 3e3.5 km (Fig. 3A) (Yue, 2007; Yue and Suppe, 2014), may imply that stress is controlled by very weak major normal with the predicted strength approximately constant below the faults whose strength is dominated by weak clay-rich smear gouges fl uid-retention depth (Fig. 3B). Here borehole stress measurements (mf z 0.2e0.3; e.g. Brown et al., 2003; Numelin et al., 2007), with at ~1 km (Hung et al., 2009) in the hydrostatic zone are used to fix little contribution of off-fault deformation to regional strength, in the pressure dependence at m z 0.45 (kc z 1.4). A regional analysis contrast with Taiwan. of wedge mechanics in western Taiwan (Suppe, 2007) indicates a The combined fluid pressure and borehole stress data from the similar pressure dependence of m z 0.35e0.40 (kc z 1) with the actively deforming Brazos area confirms the prediction of constant major thrust faults much weaker (mf z 0.03e0.09), indicating that crustal strength below the fluid-retention depth. The stress-depth off-fault deformation associated with bending of thrust sheets, relationship is consistent with a critical limiting state of brittle rather than the strength of major faults, dominates the crustal failure and a constant pressure-dependence k in both the hydro- strength. static and overpressured strata.

3. Testing the constant-strength prediction 4. Strength in areas of complex overpressures

We now test the constant-strength prediction in an area of In addition to the disequilibrium-compaction mechanism, pro- active normal faulting in the Gulf of Mexico where fluid pressure files of fluid pressure as a function of depth can be dominated or and borehole stress data exist over a substantial depth interval. In substantially modified by vertical and lateral pressure redistribu- regions of active extensional tectonics, s3 can be constrained by tion within permeable strata or along faults within the over- leakoff and fracture-initiation tests, which are hydrofracture tests pressured zone, by seals, and by pressure increases caused by gas 484 J. Suppe / Journal of Structural Geology 69 (2014) 481e492

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 reflectors are indicated. For section location see Xiao et al. (1991). generation (e.g. Yardley and Swarbrick, 2000; Swarbrick et al., higher at the bottom (point 2, Fig 7A). Short intervals of isolated 2002; Tingay et al., 2009a). In spite of such complexities of the permeable strata can be seen in the Yinggehai and Taiwan data fluid-pressure controlling mechanisms, we show for two such (Fig. 7B, ~2.8e3.2 km; Fig 3,~5e5.15 km), but this phenomenon is complex overpressured regions that the observed borehole stresses more pronounced in the Brunei data (Fig. 7C, ~1.8e2.35 km, are consistent with a critical limiting state of brittle failure and a 2.6e2.8 km). In the case of folded strata with dipping, isolated constant pressure-dependence k in both the hydrostatic and over- permeable layers, the crests of anticlines will be weakened by this pressured strata. These examples are the Tertiary Brunei delta mechanism and the synclines strengthened, which leads to a offshore northern Borneo, and the Mesozoic Sable basin offshore focusing of deformation into structurally high locations (e.g. Nova Scotia, eastern Canada. Krueger and Grant, 2011). Permeable stratigraphic intervals that are isolated within over- Typical effects of isolated permeable sands can also be seen in pressured volumes of low permeability shale show vertical gradi- the fluid pressure from the crest of the Chingtsaohu anticline ents in fluid pressure that are approximately parallel to the (CTH) in Taiwan (Fig 3A), which is composed of two data types. hydrostatic gradient, rather than parallel to the lithostatic gradient, The point measurements are borehole static shut-in pressures as shown schematically in Fig. 7A. The top of an isolated permeable within permeable sands, whereas the continuous red line is the layer commonly has a fluid pressure higher than the overpressured fluid pressure in surrounding shales and mudstones computed shale gradient, whereas the base of the permeable layer will have pressure lower than the shale gradient (points 1 and 2 Fig 7A). The corresponding strengths are lower at the top (point 1, Fig 7A) and A. B. in situ stress & fluid pressure Pf σ −σ strength 1 3 0 50 100 MPa 0 10 20 MPa 0 0 Brazos/Corsair OCS-G-1757 leak-off data 1 1 Z ZFRD FRD

2 2 effectiveeffective stressstress ρgzgz – Pf 3 3

hydrostatic lithostatic

4 4

ρ computed from P ρ f gz w gz 5 km 5 km μ = 0.1 μ = 0.25 μ = 0.55

Fig. 6. Predicted crustal strength for different values of the pressure dependence for Fig. 5. (A) Fluid pressures from the well OCS-G-1757 along section Fig. 4, Brazos/ the Brazos/Corsair province offshore Texas. The predicted strengths are from Eq. (1a), Corsair extensional province offshore Texas. (B) The predicted brittle strength in based on the observed ﬂuid pressures and several assumed depth-constant pressure extension is computed from Eq. (1a) and is approximately constant below zFRD at dependences ke. Note that the form of the predicted strength-depth curve is the same ~1.7 km depth because effective stress is approximately constant. Stress data from leak- for different pressure dependences, differing only in magnitude. The stress data from off tests (Xiao et al., 1991) show increasing strength above the ﬂuid-retention depth leakoff tests (Xiao et al., 1991) are consistent with a pressure dependence of ke z 0.39 and approximately constant below, in agreement with the prediction of Eq. (1a). (m z 0.25). J. Suppe / Journal of Structural Geology 69 (2014) 481e492 485

extent down the flanks of the anticline. Therefore the nominal computed strengths are lower for these higher-pressure stratigraphic intervals (Fig. 3B). This vertical pressure redistribution, in some cases combined with effects of hydrocarbon generation, can lead to failure by tensile fracture and slip on faults, with leakage along faults or high- permeability pipes (e.g. Finkbeiner et al., 2001; Krueger and Grant, 2011; Leduc et al., 2013). Permeable fault zones can charge shallow structures above the fluid-retention depth with high fluid pressures given suitable local seals, as is the case in the Yinggehai basin (Luo et al., 2003). These processes may be viewed as redis- tributing fluid overpressure that is generated by disequilibrium compaction or other mechanisms, such as the generation of hy- drocarbons. Tingay et al. (2009a) argue for very large-scale redistribution within the Brunei delta, charging already compacted strata with high fluid pressures that do not show low porosities consistent 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 detachments (Fig. 8A), which overlie a deep accretionary complex of the South China Sea (James, 1984; Morley et al., 2008). The fluid- pressure data as a function of depth (Tingay et al., 2009a,b)are substantially more complex than our previous examples but typical of many regions with interlayered permeable and impermeable formations (cf. Swarbrick and Osborne, 1996). Hydrostatic fluid pressures are observed in the Champion Deep well (Fig. 9A) above a fluid-retention depth of zFRD z 1.2e1.5 km where extensional stress states exist based on analysis of failure of deviated boreholes (Tingay et al., 2009b). The top of overpressures is at ~1.7 km and is marked by a discontinuity in permeability, with a shale-rich section below. Abundant stress data exist in this well; minifrac hydrofracture data provide the best estimate of strength, whereas the leakoff 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 hydrostatic and overpressured zones above ~2900 m (Fig. 9B). The best fitting pressure dependence is ke ¼ 0.45 (m ¼ 0.3). Deeper leakoff and minifrac pressures approximate the overburden pressure, sug- gesting s3 may be vertical, which may represent loading within the deep Borneo compressional wedge from which the Brunei delta extensional 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 permeable zones and pressure redistribution in Brunei, the overall strength below the top of overpressures (~1.7 km) can be approximated as constant with depth, with a mean value of ~7e8 MPa (Fig. 9B), in contrast with the classic linearly increasing hydrostatic prediction of Fig. 1.

4.2. Scotia Shelf, eastern Canada Fig. 7. (A) Simple schematic model of the effect of heterogeneous fluid pressures on strength. Permeable horizons imbedded within low-permeability shale-rich over- e pressured crust show fluid-pressure gradients that are parallel to the hydrostatic The 10 18 km thick Mesozoic Sable basin offshore Nova Scotia, gradient, whereas the surrounding shales show lithostatic-parallel gradients. The eastern Canada (Fig. 8B), shows an abrupt stratigraphic transition predicted strength-depth gradients alternate between linearly increasing hydrostatic- with depth from hydrostatic to overpressured conditions, as in the parallel gradients in the permeable layers, and constant strength within the imper- Glenelg J-48 well (Fig. 10A). A regionally consistent pattern of meable shale-rich intervals. Note that the base of a permeable zone (point 2) is predicted to be stronger than the top (point 1). (B) Example of Yinggehai Basin offshore borehole breakout directions implies a regional state of stress close China from Fig. 2. (C) Example of Brunei delta from Fig. 9. to extensional failure (Fig. 8C) (Yassir and Bell, 1994). The Glenelg field is near the shelf edge and lies within a growth normal fault from sonic-log data. Note that below the fluid-retention depth system that was mildly active in the late Cenozoic (Williamson and many of the shut-in pressures are substantially higher than Smyth, 1992). The Glenelg well contains breakouts over much of its pressures within the surrounding shales. This reflects the fact depth (1181e4442 m; Bell, 1990). Modeling of the Glenelg field that the sands are permeable and are of large vertical and lateral suggests a complex fluid-pressure history that is dominated by 486 J. Suppe / Journal of Structural Geology 69 (2014) 481e492

A. Brunei delta, offshore northern Borneo Champion Champion Deep Main s.l. s.l.

1.6ma 3.8ma

top Miocene

top of overpressure 10 km

5 sec 5 sec

B. Scotia Shelf, offshore eastern Canada C. Borehole breakouts Scotia Shelf s.l. C C K C breakout direction K K m K Sable Island 500m500 J J J J salt shelf edge

00m 20 km section500m5 20 km 100km Glenelg J-48

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 data compiled 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).

disequilibrium compaction with minor contribution of gas gener- 5. Discussion ation (Williamson and Smyth, 1992). Borehole stress data from the Glenelg J-48 well (Fig. 10B) can be fit reasonably well to a limiting In summary, for the three overpressured examples for which we strength envelope based on the observed fluid pressures and a have stress and fluid-pressure data over a substantial depth interval single pressure-dependence k within both the hydrostatic and (Figs. 5, 9 and 10), we are able to reasonably fit the stress data to a overpressured zones (ke z 0.39, m z 0.25). In contrast with our limiting strength envelope that is computed from observed fluid previous examples, there are insufficient data below the top of pressures by Eqs. (1) and (2), using a single pressure dependence k overpressures (~4 km) in the Scotia Shelf to constrain or predict the in both the hydrostatic and overpressured zones. These examples form of the strength-depth relationship within this very deep basin also show regionally consistent stress directions as indicated by (Fig. 8B). borehole breakouts, natural hydrofractures and active normal fault trends. Therefore we conclude that they are close to a regional state

computed A. B. in situ stress & fluid pressure Pf strength σ −σ A. B. 1 3 fluid pressure Pf strength σ −σ 0 50 100 MPa 0 10 20 MPa 1 3 0 0 0 50 100 MPa 0 10 20 MPa 0 0 Brunei Champion Deep minifrac stress Scotia Shelf, Canada leakoff stress Glenelg J-48 well IF stress leakoff stress 1 1 1 1 strength computed strength computed from P & Eqn.1 2 2 f from Pf & Eqn.1 lithostatic

2 2 lithostatic 3 3 hydrostatic hydrostatic ρ gz σ vert κ1 e=0.45 ρ 4 4 3 km gz 3 km κ =0.81 ρ c w σ vert ρ gz 3 w gz κ = 0.39 μ= 0.3 e 5 km 5 km μ = 0.25

Fig. 9. Fluid pressure and stress from the Champion Deep structure, Brunei delta extensional province (data from Tingay et al., 2009a,b). The predicted strengths are Fig. 10. Fluid pressure and stress from the Glenelg J-48 well, Scotia Shelf offshore from Eq. (1a), based on the observed fluid pressures and the best-fitting constant eastern Canada (data from Bell, 1990). The predicted strengths are from Eq. (1a), based pressure dependence k that forms a limiting envelope for the stress data. See text for on the observed fluid pressures and the best-fitting constant pressure dependence k discussion. that forms a limiting envelope for the stress data. See text for discussion. J. Suppe / Journal of Structural Geology 69 (2014) 481e492 487 h . i of pressure-dependent failure approximated by Eqs. (1) and (2). * m ¼ m 1 P rgz ¼ mð1 lÞ (6a) Finally, these examples give similar low pressure dependences of f k z 0.39e0.45 (m z 0.25e0.3), consistent with the frictional e h . i strengths of clay-smear fault gouges on normal faults (e.g. Brown k* ¼ k r ¼ kð lÞ et al., 2003; Numelin et al., 2007). 1 Pf gz 1 (6b) In light of this success in predicting the form and magnitude of the limiting strength-depth envelope from fluid pressure data, we h . i m* ¼ m r ¼ m l sketch two regional scale tectonic applications. f f 1 Pf gz f 1 f (6c)

In contrast, the absolute crustal and fault strengths are 5.1. Application to regional upper crustal deformation ðs s Þ¼k r Our concern in this paper is exploring the first-order effects of 1 3 gz Pf (7a) fluid overpressures on regional upper crustal strength. We briefly discuss two regional applications: [1] the expected role of regional fl s ¼ m s uid overpressures in the critical-taper mechanics of accretionary t f n Pf (7b) wedges and fold-and-thrust belts, and [2] the possible effects of fluid overpressures on crustal strength in very thick continental where the effective fault-normal stress (sn Pf) varies with fault margins and shale-rich deforming mountain belts, which in some orientation. For example, in the case of horizontal compression cases are thick enough to approach the brittle-ductile transition. (sn Pf) varies between the values for horizontal and vertical faults For these regional applications the fluid-retention depth can be ðrgz P Þðsn P Þkcðrgz P Þ. fi f f f reasonably estimated in many actively deforming regions to suf - If we apply these basic relationships (Eqs. (6) and (7)) to the cient accuracy based on borehole data and seismic velocity analysis. fluid-pressure distributions predicted by the classic disequilibrium fl In contrast, predicting pore- uid pressures and their hetero- compaction model, then fluid pressures increase parallel to the genieties at the level required in petroleum exploration, production lithostatic gradient below the fluid-retention depth, such that and engineering is more technically and observationally ðr Þ gz Pf is constant with depth (Fig. 11A). This leads to constant demanding and is beyond our present tectonic scope. Nevertheless, absolute crustal and fault strengths (Eqs. (5) and (7)) below the fl at the more local scale of individual structures, uid pressures fluid-retention depth as shown in Fig. 11C. (We note that constant fi appear to play very signi cant roles of interest to structural geol- fault strength with depth has been widely assumed in fault-rupture ogy, for example see applications of Krueger and Grant (2011) to the dynamical models, e.g. Rice, 1992). In contrast, the effective friction Niger delta deep-water thrust belt, and Finkbeiner et al. (2001) in fi l ¼ð =r Þ coef cients continuously decrease (Eq. (6)) because Pf gz the northern Gulf of Mexico, South Eugene Island. continuously increases below the fluid-retention depth, as illustrated in Fig. 11B. In each case (Fig. 11B and C), the effect of fluid 5.1.1. Role of fluid overpressures in critical-taper wedge mechanics pressure on brittle strength is the same for both the crust and the and Hubbert-Rubey fault weakening faults, only differing by the magnitudes of their intrinsic pressure m m k There has been a long history of considering the role of fluid dependences ( f, , ). overpressures in the mechanics of thrust faults, accretionary Active accretionary wedges and thrust belts deform to a critical a þ b a b wedges and thrust belts (e.g. Hubbert and Ruby, 1959; Davis et al., taper ( ), where is the surface slope and is the detachment 1983; Dahlen, 1990). Fluid pressures generally enter these the- dip (e.g. Davis et al., 1983). This critical taper is controlled by ¼ s =r ories as the Hubbert-Rubey weakening coefficient (1 l), which for detachment strength F t gH and wedge strength ¼ðs s Þ=r simplicity generally has been assumed constant, implying a linearly W 1 3 gH, where H is the local depth of the detachment increasing crustal strength (Eq. (1)). Unfortunately constant (1 l) h . i l ¼ r =r r r b þ is unrealistic except in the hydrostatic case of wgz gz. In the 1 f F overpressured case (1 l) is everywhere vertically variable, as we a þ b ¼ h . i (8) r r þ have seen (Figs. 2, 3, 5, 9 and 10). However, in the case of classic 1 f W disequilibrium compaction we can express the vertical variation in (1 l) as a simple function of fluid-retention depth zFRD given by and rf is the density of the fluid overlying the wedge, air or water Eq. (4),(1l) z 0.6zFRD/z. This relationship has been applied to (Suppe, 2007). Taper angles a and b are in radians. A wide variety of thrust and accretionary wedge mechanics by Yue and Suppe (2014). brittle and ductile processes operating at different time and spatial Here we summarize the basic roles of overpressure in mechanics of scales potentially may control F and W (Dahlen, 1990). For example, thrust belts, in light of the results of this paper. For the purposes of in some wedges the detachment may be activated in large-slip illustration, we make use of some data from the deep-water Niger earthquakes with dynamical processes dominating the fault delta thrust belt (Fig. 11). strength, during which wedge taper might be established by off- There are two common but contrasting ways of expressing role detachment deformation (e.g. Suppe et al., 2009). However, in the of fluid pressure in brittle behavior: [1] in effective coefficients of following we assume cohesionless brittle frictional static strengths m* m k friction f , internal friction * and pressure dependence * and [2] for both the wedge and the detachment (Eqs. (6) and (7)) to illus- in absolute fault st and crustal (s1 s3) strengths. Up to this point trate the predicted role of static pore-fluid pressures controlled by we have been considering the effects of overpressure on absolute disequilibrium compaction (Eq. (4)). crustal strength [2]; in what follows we see that critical-taper Yeh and Suppe (2014) show for the case of a general heteroge- wedge mechanics depends on the effective-friction coefficients neous wedge of Dahlen (1990) that, if vertical variation in fluid [1]. These effective coefficients are controlled by the ratio of fluid pressure can be approximated by our simple fluid-retention depth l ¼ð =r Þ fl pressure to solid overburden pressure, Pf gz , whereas the zFRD formulation (Eq. (4)), then the effect of uid overpressures on effect of fluid pressure on absolute strength is controlled by their wedge mechanics is very simply described by zFRD/H. Here we ðr Þ difference, the effective stress gz Pf . The effective internal illustrate this for the approximation of constant zFRD and an friction, pressure dependence and fault friction coefficients are otherwise homogeneous and cohesionless brittle wedge, for which 488 J. Suppe / Journal of Structural Geology 69 (2014) 481e492

A. effective friction C. absolute fluid pressure & pressure dependence crustal & fault strength 0 50 100 MPa 0406020 80 MPa s.b. s.b. deep water Niger delta μf = 0.2 μ = 0.55 1 1 κc = 1.86 Z Z FRD FRD ZFRD 2 2 2 μ* μ* κ*

f c FRD lithostatic vert. fault ) στ 3 3 3 gZ

/ Z ρ )

ρ c w / Z gz + κ FRD ) ρ FRD Z 4 gz ( 4 4 FRD

f hydrostatic Z ρ ( = 0.6 μ 3 g(z–z ) FRD μ horiz. fault Z/ Z στ ( 0.6 c −σ ==0.60.6μ ρgZgZ 1 0.6 FRDFRD 5 FRD κ 5 f

5 σ

0.6

w 6 gz 6 6

7 km 7 km 7 km crustal strength D. 1.5° α wedge taper predicted by static fluid pressures detachment dip β = 1.5°

1.2° p hydrostatic wedge surface slope α 0.9° Z FRD overpressured wedge surface slope 0.6° α β + μ (Z /H) α + β = f FRD horiz. scale100km 0.3° 1 + & detachment di κc(Z FRD /H) wedge surface slope 0.0° 0 2 4 6 8 10 km depth to detachment H

E. sea level F. sea level σ σ hydrostatic wedge 13 overpressured wedge D D σ13 σ

α α zFRD zFRD

β H z β H z

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- m* m k pressures from a well in the deep-water Niger delta thrust belt (Krueger and Grant, 2011). (B) Predicted decay of effective friction ( f , *) and pressure dependence * 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 the fl ðr Þ uid-retention depth zFRD (Eq. (7)), based on constant effective stress gz Pf . (D) Predicted wedge taper for the deep-water Niger delta thrust belt, based on the assumption that strength 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 strength W and fault strength F in the Niger delta and many 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.

the detachment and wedge strengths become simply the effective disequilibrium compaction model is a very simple function of zFRD/ strength coefﬁcients of Eq. (6) written in terms of zFRD/H H, which affects fault strength and wedge strength identically . h . i . F ¼ m ð1 l Þ¼m ½1 ðr =rÞz H (9a) f H f w FRD r r b þ m ½ ðr =rÞ 1 f f 1 w zFRD H a þ b ¼ h . i . (10) r r þ k ½ ðr =rÞ ¼ k ð l Þ¼k ½ ðr =rÞ = 1 f c 1 w zFRD H W c 1 H c 1 w zFRD H (9b) where lH is the regional l at the detachment depth H. Thus we see In the case of the Niger delta, the variation of zFRD/H predicts a very that the effect of ﬂuid overpressures on critical taper under the slowly 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 for pressure-dependent failure controlled by effective stress under taper becomes further simplified to both hydrostatic and overpressured conditions (Figs. 1e3, 5, 9, 10; . Eqs. (1) and (2)). Extrapolating to greater depths, Brace and b þ m fl f zFRD H Kohlstedt (1980) argued that hydrostatic pore- uid pressures a þ b ¼ (11a) would be typical of crystalline crust, based on expected high per- 1 þ k z =H c FRD meabilities. Townend and Zoback (2000) argued that continued and for a hydrostatically pressured submarine wedge (z H) the faulting is required to provide high-permeability conduits that FRD fl taper is constant keep uid pressures near hydrostatic in crystalline crust, especially in plate interiors (Zoback and Townend, 2001). These consider- b þ m ations have led to the classic strength-depth profile (Fig. 1). a þ b ¼ f (11b) Here we consider the possibility of overpressured conditions 1 þ kc existing within substantial thicknesses of the crust. Thick sedi- We see from the overpressured wedge-taper equation (Eq. mentary basins on continental margins built on oceanic or highly- fi (11a)) that it is the contrast in crustal and fault strength coef cients thinned continental crust have vast deforming clastic-rich volumes k m c and f that dominates the contrast in wedge and fault strengths that approach a significant fraction of crustal thickness, and may W and F and hence the taper magnitude. It is only in cases in which even span the brittle-ductile transition (e.g. Gulf of Mexico, Niger fl zFRD/H is small that uid overpressures begin to dominate wedge delta, Sumatra, Nankai trough). These thick clastic stratigraphic tapers. accumulations may deform into shale-rich plate boundary moun- The contrast between F and W is quite large for the Niger delta, tain belts (e.g. Bangladesh/Myanmar, Barbados/Trinidad, Makran, < ¼ ¼ F/W 0.1, based on regional magnitudes of F 0.04 and W 0.7 Gulf of Alaska, offshore SW Taiwan, New Zealand). In the active a that were computed from the covariation of surface slope with compressional Southern Alps of New Zealand there is deep b detachment dip , without any strong assumptions of strength- geophysical evidence for near lithostatic pore-fluid pressures over controlling mechanisms (Suppe, 2007, using Niger delta taper much of the width of the mountain belt, extending to depths of data of Bilotti and Shaw, 2005). This large contrast between F and W 20e30 km, based on seismic P-wave velocity analysis, local to- is typical of many accretionary wedges and thrust belts; a typical mography and magnetotelluric data (Stern et al., 2001; Eberhart- z value of F/W 0.1 is indicated based on the observed narrow ta- Phillips and Bannister, 2002; Wannamaker et al., 2002; Scher- pers of accretionary wedges (Suppe et al., 2009). Narrow wedge wath et al., 2003). This is a region of the crust interpreted to be fl tapers cannot be dominated by regional uid overpressures except composed of clastic sedimentary rocks undergoing prograde fl in cases of small zFRD because uid pressures affect the wedge and metamorphism to schist and is expected to extend beyond the fault strengths by the same proportions in Eqs. 9 and 11. Other brittle-ductile transition. Very deep (~40 km) near lithostatic fluid mechanisms of fault weakening relative to wedge strength will pressures are widely regarded as a central ingredient of slow-slip generally be required, for example the diverse mechanisms of earthquake phenomena (e.g. Peng and Gomberg, 2010). dynamical weakening in large earthquakes (e.g. Di Toro et al., 2011). Furthermore, petrologic observations suggest that fluid pres- In other tectonic environments quasistatic mechanisms of fault sures may be near lithostatic in fine-grained crust undergoing the weakening are indicated, such as evaporite detachments or transition from shale to phyllite and schist, dominated by pressure intrinsically weak clay gouges (e.g. Brown et al., 2003; Numelin solution. It is widely argued that pore-fluid pressures are close to et al., 2007). lithostatic, based on the existence of some types of crack-seal and fl Nevertheless, static regional uid overpressures are predicted to open-void veins (e.g. Etheridge, 1983; Etheridge et al., 1984; Fisher fi have two well-de ned effects on wedge tapers. [1] First, over- and Brantley, 1992; Hilgers et al., 2006; Van Noten et al., 2011), and pressures reduce the taper relative to what it would be for an based on studies of fault-related mesothermal gold-quartz vein equivalent hydrostatic wedge (Eqs. 11a and b). In the case of the systems, which are suggestive of fault-valve phenomena near the Niger delta well (Fig. 11A) the predicted effect of zFRD/H is a ~30% base of the seismogenic zone (e.g. Sibson et al., 1988). Fluid- reduction of an already very small predicted hydrostatic surface inclusion studies within ore-hosting conduits indicate large oscil- slope of ~1.1 to an overpressured slope of ~0.8 (Fig. 11D) (based on lations between near-lithostatic and possibly near-hydrostatic b ¼ an ~1.5 detachment dip , zFRD 2.3 km, and a detachment depth pressures. These observations are suggestive of a dual- ¼ H ~7 km). The predicted effect will be larger for wedges with a permeability system with a fine-grained matrix at near-lithostatic fl very shallow uid-retention depth relative to detachment depth. pressure, and fault conduits that are alternately broken and sealed [2] Second, given a relatively constant zFRD we expect a decreasing by vein formation with associated fluid pressure and stress fluc- wedge taper towards the wedge interior based on a regional tuations of the seismic cycle (e.g. Robert et al., 1995; decrease in zFRD/H. In the case of the Niger delta well this predicted Micklethewaite, 2008). effect is quite subtle (Fig. 11D). The surface slope would decrease Studies by Van Noten et al. (2011) of the stress-state and fluid- only ~0.1 over 100 km, given the very gentle detachment dip of pressure evolution in the High-Ardenne slate belt of Germany, fi ~1.5 , which is dif cult to verify given widespread wedge-top based on analysis of quartz veins and fluid inclusions, indicate near deposition. In contrast, decreasing surface slopes away from the lithostatic pore-fluid pressures under very low grade metamorphic toes of submarine accretionary wedges are widely observed conditions of ~250C, and an assumed depth of ~7 km based on without any associated change in detachment dip (e.g. Zhao et al., stratigraphy. They estimated a limiting compressive crustal 1986). In the case of the convex toe of the Barbados accretionary strength s1 s3 during vein formation of ~35e45 MPa, which is wedge, Yeh and Suppe (2014) show that the decrease in taper is a consistent with a nominal fluid-retention depth of 1e3 km, near fl uid-pressure effect, dominated by the decrease in zFRD/H towards the top of the Lower Devonian siliciclastic section. In contrast, a fl the wedge interior and a very shallow uid-retention depth. crustal strength of 100e200 MPa would be expected at ~7 km depth under the classic brittle-hydrostatic model. The super- 5.1.2. Strength of very thick overpressured crust lithostatic fluid pressures suggested for vein formation may Here we briefly consider the implications of fluid overpressures reflect transient vertical redistribution (cf. Fisher et al., 1995). for crustal-scale strength-depth profiles. Borehole observations in Furthermore, the low ambient background crustal stress states the range 1e9 km suggest a widespread limiting critical state of required for widespread joint and vein formation are expected at 490 J. Suppe / Journal of Structural Geology 69 (2014) 481e492 great depth in overpressured crust, in contrast with the classic A.Hydrostatic crust B. Overpressured crust hydrostatic strength-depth profile (see Fig. 12). strength σ −σ strength σ −σ If we interpolate our borehole observations from overpressured 1 3 1 3 0 500 MPa 0 500 MPa continental margins with the geophysical and petrologic evidence 0 0 for near-lithostatic fluid pressures, and hence very low strength, hydrostatic brittle strength hydrostatic brittle strength near the brittle-plastic transition, then constant strength may be a Z reasonable first-order strength-depth model for fine-grained, 4 4 FRD clastic-dominated crust, as shown in Fig. 13B. This constant strength model is in contrast with the classic Brace and Kohlstedt (1980) hydrostatic strength-depth profile that is well justified for 8 8 crystalline crust (Figs. 1 and 13A). That said, the strength, fluid overpressured brittle strength Z km 12 Z km 12

A. στ hydrostatic regional strength 14 possible stress states 14

100MPa plastic strength plastic strength 20 20

Fig. 13. Contrasting hydrostatic and overpressured crustal strength-depth curves. (A) The classic strength-depth graph showing linearly increasing brittle strength, which requires hydrostatic pore-ﬂuid pressures (Eq. (2)). This model agrees with much deep borehole data (Fig. 1). (B) The constant-strength model for overpressured crust, based 2To on borehole data from this paper combined with geophysical and petrologic evidence To 50 100100 2200MPa 00MPa for near-lithostatic overpressures and low strength in ﬁne-grained rocks at very low (σ ) metamorphic grade near the brittle-plastic transition. n – Pf Z=2.4kmZ=2.4km

pressure and state of stress through the brittle-ductile transition in 30° cohesionless regional strengthZ=8km shale-rich mountain belts remains a signiﬁcant research frontier cohesive rock strength beyond the scope of this paper. However, current viscous-plastic modeling of disequilibrium compaction suggests that matrix –100MPa weakening by pressure solution becomes a dominant mechanism (T = 10MPa) o in generating near-lithostatic ﬂuid pressures at great depths (Morency et al., 2007).

6. Conclusions B. στ overpressured regional strength The classic view of upper crustal strength is a linearly increasing 100MPa pressure-dependent brittle strength, which requires a hydrostatic σ vertical 1 pore-fluid pressure. This model has been confirmed by much borehole data in crystalline crust to depths reaching 8e9km ΔP (Fig.1). In contrast, clastic sedimentary basins and deforming shale- f rich mountain belts display pore fluids that are overpressured effective stress Z>Z tensile FRD below some critical depth, based on much borehole data and strength To seismic velocity analysis. Therefore we have explored the predicted 50 100100 2200MPa 00MPa shapes of strength-depth curves based on fluid-pressure data in a (σ ) number of overpressured regions, using the same assumption used n – Pf in the hydrostatic case: that a limiting critical strength is controlled Z=ZFRD =2.4km by effective stress times a pressure dependence s1 s3 ¼ effective stress kðr Þ gz Pf . We have successfully tested this prediction for three for regional 30° cohesionless regional strength jointing cases in which we have both fluid pressure data and in situ borehole cohesive rock strength stress data over a substantial depth interval (Gulf of Mexico, Brunei delta, and Scotia Shelf, eastern Canada). –100MPa Fluid overpressures that are dominated by the disequilibrium (T = 10MPa) o compaction mechanism are of particular interest for regional-scale tectonics because they are very widely observed, have straightforward implications for crustal strength, and are predictable based on

Fig. 12. Effective stresses required for regional joint formation and the predicted observable quantities. The most straightforward signature of regional regional stress states for (A) hydrostatic and (B) overpressured crust at a critical state of disequilibrium compaction is hydrostatic fluid pressures at shallow failure, illustrated with Mohr diagrams. Effective stresses at great depth in over- depths above the fluid-retention depth zFRD, and overpressured pressured crust are small and therefore relatively favorable for regional joint forma- below it, increasing parallel to the lithostatic gradient rgz. Therefore tion, in contrast with hydrostatic crust. Note that for all depths greater than the fluid- effective stress ðrgz P Þ is constant below the z , which implies retention depth z , the effective stress state is identical to that at the fluid-retention f FRD FRD s s z k depth, therefore only a modest increase in fluid pressure DPf would allow the effective that brittle strength will be also constant 1 3 0.6 gzFRD. Crustal stresses needed for joint formation by tensile fracture, even at very great depth. strength 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 hydrostatic Bredehoeft, J.D., Hanshaw, B.B., 1968. On the maintenance of anomalous fluid strength-depth profile (e.g. Figs. 1, 2 and 13). pressures: I. Thick sedimentary sequences. Geol. Soc. Am. Bull. 79, 1,097e1,106. Fluid-retention depth zFRD is particularly important because it can Brown, K.M., Kopf, A., Underwood, M.B., Weinberger, J.L., 2003. Compositional and be observed based on commonly available borehole data and on fluid pressure controls on the state of stress on the Nankai subduction thrust: a weak plate boundary. Earth Planet. Sci. Lett. 214, 589e603. seismic velocity analysis in non-eroded siliciclastic sedimentary € fl Brudy, M., Zoback, M.D., Fuchs, K., Rummel, F., Baumgartner, J., 1997. Estimation of sections. In addition, fossil uid-retention depths can be preserved in the complete stress tensor to 8 km depth in the KTB scientific drill holes: im- uplifted and eroded strata and identified by a variety of means plications for crustal strength. J. Geophys. Res. 102, 18,453e18,475. (Magara, 1978; Yue and Suppe, 2014). Once we know the fluid- Bürgmann, R., Dresen, G., 2008. Rheology of the lower crust and upper mantle: evidence from rock mechanics, geodesy and field observations. Annu. Rev. Earth retention depth we can estimate the Hubbert-Rubey weakening Planet. Sci. 36, 531e567. (1 l) z 0.6zFRD/z everywhere within the overpressured zone (Eq. Burov, E.B., Watts, A.B., 2006. The long-term strength of continental lithosphere: (4)), which allows us to immediately know the vertical variation in “jelly sandwich” or “creme brûlee ”? GSA Today 16 (1), 4e10. m* m k s s Byerlee, J.D., 1978. Friction of rocks. Pure Appl. Geophys. 116, 1189e1198. effective friction ( f , *, *) and absolute crustal ( 1 3) and static Chen, W.P., Molnar, P., 1983. Focal depths of intracontinental and intraplate earth- fault strength st (Eqs. (6) and (7), Fig. 11). Therefore, fluid-retention quakes and their implications for the thermal and mechanical properties of the e depth zFRD should be considered an important observable param- lithosphere. J. Geophys. Res. 88, 4,183 4,214. eter in crustal mechanics. Crans, W., Mandl, G., 1980. On the theory of growth faulting Part II(a): genesis of the “unit”. J. Pet. Geol. 3, 209e236. Observations of fluid-retention depth zFRD are of particular Dahlen, F.A., 1990. Critical taper model of fold-and-thrust belts and accretionary importance for critical-taper mechanics of accretionary wedges and wedges. Annu. Rev. Earth Planet. Sci. 18, 55e99. thrust belts. Yeh and Suppe (2014) have incorporated the rela- Davis, D., Suppe, J., Dahlen, F.A., 1983. Mechanics of fold-and-thrust belts and accretionary wedges. J. Geophys. Res. 88, 1153e1172. tionship between Hubbert-Rubey weakening and fluid-retention Di Toro, G., Han, R., Hirose, T., De Paola, N., Nielsen, S., Mizoguchi, K., Ferri, F., depth (1 l) z 0.6zFRD/z into the general heterogeneous wedge Cocco, M., Shimamoto, T., 2011. Fault lubrication during earthquakes. Nature e theory of Dahlen (1990). This allows us to directly incorporate 471, 494 498. http://dx.doi.org/10.1038/nature09838. fl Eberhart-Phillips, D., Bannister, S., 2002. Three-dimensional crustal structure in the observations of uid-retention depth in applications of wedge Southern Alps region of New Zealand from inversion of local earthquake and mechanics, obtaining geologically reasonable fluid pressure and active source data. J. Geophys. Res. 107 http://dx.doi.org/10.1029/2001JB000567. strength distributions. Yeh and Suppe (2014) have applied this to Etheridge, M.A., 1983. Differential stress magnitudes during regional deformation and metamorphism: upper bound imposed by tensile fracturing. Geology 11, data from the Barbados accretionary wedge, and show that the 231e234. convex geometry of its toe, which is typical of many classic accre- Etheridge, M.A., Wall, V.J., Cox, S.F., Vernon, R.H., 1984. 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