# Energetics of normal earthquakes on dip-slip faults **Authors:** David Dempsey, Susan Ellis, Rosalind Archer, Julie Rowland **Published in:** Geology (2012) **DOI (version of record):** https://doi.org/10.1130/g32643.1 **This copy:** Accepted manuscript (peer-reviewed, author's version). Text extracted from the PDF at https://ddempsey.github.io/papers/2012-dempsey-energetics-normal-earthquakes-dip-slip/dempsey2012-energetics-normal-earthquakes.pdf; equations, tables and figure placement may be imperfect, so cite the DOI. **Notice:** Accepted manuscript. The version of record was published by the Geological Society of America in Geology: https://doi.org/10.1130/g32643.1 --- - 1 The energetics of normal earthquakes on dip-slip faults - 2 **David Dempsey****1** **, Susan Ellis****2** **, Rosalind Archer****1** **, and Julie Rowland****3** - 3 _1Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland Mail_ - 4 _Centre, Auckland, New Zealand_ - 5 _2GNS Science, 1 Fairway Drive, Lower Hutt, New Zealand_ - 6 _3School of Environment, University of Auckland, Private Bag 92019, Auckland Mail Centre,_ - 7 _Auckland, New Zealand_ # 8 **ABSTRACT** 9 That earthquakes release vast quantities of energy is widely accepted, however the most 10 commonly experienced component, radiated seismic energy, is a minor contribution to the total 11 energy budget. The elastic rebound model for earthquakes recognizes that elastic strain energy 12 does work displacing, deforming and accelerating the crust, as well as causing frictional heating. - 13 In this paper an energy budget for dip-slip fault rupture in an extensional tectonic regime is 14 presented. A computational model of an elastic-plastic-viscous crust hosting a single fault, 15 modeled as two surfaces in frictional contact, demonstrates contrasting energy flows between the - 16 hangingwall and footwall fault blocks. Our analysis suggests that in the period leading up to an 17 earthquake, the total strain energy contained within the crust decreases, although a local increase - 18 within the footwall at mid-crustal depths is observed. During an earthquake, the footwall is - 19 subject to an elastic rebound, whereupon uplift of the fault scarp is caused by a mid-crustal stress - 20 drop and elastic expansion that releases strain energy. In contrast, gravitational potential energy - 21 released from a subsiding hangingwall does work compressing the wider crust, particularly in the - 22 mid-crust at the fault tip. This has the unusual consequence of increasing strain energy - 23 throughout much of the upper crust during an earthquake. These counter-intuitive energy flows - 24 suggest that extensional deformation is caused by stored gravitational potential and elastic strain - 25 energy, and not by the external tectonic forcing. # 26 **BACKGROUND** 27 The conventional view of earthquakes asserts that elastic strain energy (Reid, 1911; 28 Sibson, 1974) accumulates within the crust during an interseismic period of fault dormancy prior 29 to its release through fault rupture. This description of the seismic cycle implies that work done 30 by tectonic forces stores elastic energy (EE), which in turn is expended to do work accelerating 31 and deforming the crust during an earthquake via elastic rebound. The elastic rebound model was 32 developed for strike-slip systems (Reid, 1911) where gravitational forces do not contribute to the 33 work budget. However, geodetic observations of normal earthquakes on dip-slip faults record a 34 significant drop in elevation of the hangingwall (e.g., Barrientos et al., 1987; Koukouvelas and 35 Doutsos, 1996), which implicates gravitational potential energy (GPE) as a contributor to 36 coseismic deformation in extensional tectonic environments (e.g., McKenzie, 1978; Stewart, 37 1980). - 38 Recent examinations of the energy budget for thrust fault systems (Cooke and Murphy, 39 2004; Del Castello and Cooke, 2007) indicate a response consistent with elastic rebound, i.e., 40 elastic strain energy is expended to do work against friction on the fault and against gravity 41 deforming the surface, with a smaller component of radiated seismic energy. Here, we consider 42 an energy budget derived from a 2-D computational model of a dip-slip fault within an extending - 43 elastic-plastic-viscous crust. Our analysis contrasts the ability of stored EE and GPE to do work - 44 against friction and gravity (Tanimoto et al., 2002), and cause brittle or ductile deformation - 45 during the seismic cycle. Radiated seismic energy, which is a minor component of the work of - 46 faulting at low seismic efficiencies (e.g., McGarr, 1994; Kanamori, 2001), is not accounted for in - 47 the static analysis presented. # 48 **MODEL DESCRIPTION** 49 Following Ellis et al. (2006), a 2-D crust, 60 km wide and 25 km deep (Fig. 1A), is 50 modeled under plane-strain conditions using the finite element software Abaqus/Standard. 51 Boundary-parallel free-slip boundary conditions are applied at the base and sides of the model 52 while the top surface is free to deform. Extension of 12 mm yr-1 is divided equally between the 53 two lateral boundaries and a linear temperature gradient of 35o C km-1 is imposed throughout the 54 crust. 55 56 Deformation of the crust is governed by linear, isotropic elasticity, limited by frictional plastic Mohr-Coulomb yield giving way to thermally-controlled power-law creep with increasing 57 depth. Frictional yield incorporates a modified strength to account for effects of hydrostatic pore 58 fluid pressure (Byerlee, 1978; Townend and Zoback, 2000). The transition to power law 59 dislocation creep is determined by the model for ambient temperatures, pressures and strain60 rates; we delineate “upper” and “lower” regions of the crust based on the depth to the brittle61 ductile transition, which occurs at temperatures greater than ca. 350o C for the imposed bulk 62 strain rate of 6x10-15 s-1 . - 63 The crust hosts a single fault, modeled as two surfaces in frictional contact (e.g., Ellis et 64 al., 2006; Nüchter and Ellis, 2010), dipping at 50o to a depth of 15 km. The seismic cycle is 65 approximated by a 250 year period of dormancy, in which fault creep is prohibited by an infinite 66 friction coefficient along the fault plane. The static effects of an earthquake are simulated by - 67 changing the friction coefficient to a slip-weakened value of 0.2 (e.g., Scholz and Engelder, - 68 1976) for a 60 second period. Note that dynamic effects such as wave propagation and rate-state - 69 friction are neglected, which precludes estimation of brittle afterslip or radiated seismic energy. - 70 The stress state is “spun-up” (e.g., Ellis et al, 2006; Hetland and Hager, 2006) over multiple - 71 earthquake cycles. Self-similar patterns of stress and deformation emerge after 10 cycles, - 72 indicating a state of dynamic equilibrium has been achieved, and energy results presented are 73 derived after the 12th cycle. 74 The model produces coseismic vertical surface displacements (Fig. 1B) that are 75 qualitatively similar to geodetic observations following the MW = 6.9 1983 Borah Peak 76 earthquake, Idaho, and the MW = 7.3 1959 Hegben Lake earthquake, Montana (Barrientos et al, 77 1987). Both historical earthquakes exhibit well resolved vertical displacement profiles and are - 78 inferred to have occurred on fault planes dipping at 50o . Comparison between these profiles 79 shows that the model reproduces uplift of the footwall block and collapse of the adjacent 80 hangingwall, with a total vertical offset at the surface of 1.2 m. For a 25 km long surface rupture 81 this offset corresponds to an M = 5.8 earthquake (Hanks and Kanamori, 1979). Magnitudes and 82 offsets for the historical earthquakes are larger than that in the model, which suggests either a 83 greater stress buildup before rupture for those faults, or some difference in the elastic properties 84 of the crust, e.g., Young’s modulus (Gudmundsson, 2004). The comparatively smaller fault 85 ruptures modeled here indicate that unrealistically large stress states are not attained. 86 # **EXTENSION, STRESS AND STRAIN ENERGY** 87 A consequence of lithostatic loading is that, with the exception of the near surface in - 88 extensional regimes, normal stresses within the crust are always compressive. Thus, a volume of - 89 rock brought to the surface from depth expands in response to the removal of lithostatic forces. - 90 Such physics underpins the violent and rapid expansion of rock associated with well-bore - 91 breakout (Zoback et al, 1985) and rock-burst (e.g., Cook et al., 1966) phenomena. It also attests - 92 to the vast reservoirs of EE that exist within the crust and are accessible via changes in the stress - 93 state. - 94 Adopting the convention that stresses and strains are positive in compression, we - 95 consider a crustal volume in which a small increment of tectonic extension occurs. This - 96 extension is associated with a small, negative change in horizontal stress. As the existing 97 horizontal stress state is compressive, and thus positive, the addition of a negative stress change 98 moves the crust incrementally closer to zero stress. The corresponding changes in elastic strain, 99 and thus elastic strain energy, are then negative. The reduction of internal energy indicates that 100 work has been done by, not on, the crustal volume, and implies that extension might be caused 101 by the crust itself, rather than imposed by some external force. The validity of these concepts is 102 tested in the next section by considering energy flows within a faulted crust undergoing 103 extension. 104 # **ENERGY CONSERVATION AND WORK BALANCE** 105 106 107 A static energy budget (e.g., Cooke and Murphy, 2004; Savage and Cooke, 2010) for the model is comprised of strain energy (EE), energy dissipated by viscous (VE) or frictional plastic (PE) means, energy dissipated by frictional slip along the fault surface (FE), and work done on - 108 the model by external forces (W). External work includes work against gravity (thus W includes 109 changes in GPE) and work done by tractions at the model boundaries. Strain energy, EE, is - 110 further decomposed into volumetric, EEVOL, and distortional, EEDIS, components, representing - 111 energy flow due to changes in volume and shape respectively. Conservation of energy for the - 112 systems is then expressed as W = EE + VE + PE + FE. Details of the calculation of these energy - 113 terms can be found in Cooke and Murphy (2004) and Popov and Sobolev (2008). - 114 Fig. 2A presents the evolution of each energy budget component summed over the model - 115 domain over several seismic cycles from a common origin. Energy dissipation terms, i.e., PE, - 116 VE and FE, plot in the positive domain and are always increasing due to their irreversible nature. - 117 Due to its recoverable nature, EE is permitted to plot in either domain. - 118 The application of far-field tectonic forces over many seismic cycles evolves an - 119 extensional deviatoric stress, i.e., σx< σz and σx, σz > 0, such that the continuous accumulation of - 120 horizontal elastic strain is limited by the Mohr-Coulomb yield criterion. Deformation post-yield, - 121 termed “frictional plastic”, corresponds physically to slip on pre-existing fracture planes - 122 accompanied by the dissipation of EE as heat. This energy dissipation is widespread and - 123 consistent in time, except for a brief post-seismic period (PE in Fig. 2A) in which the upper - 124 crustal stress state is below yield. The reduction in the rate of accumulation of PE is due to a - 125 coseismic increase in horizontal compression, which suppresses failure. This effect is reversed - 126 by continued tectonic extension following an earthquake. - 127 Viscous deformation in the lower crust dissipates deviatoric stresses, and thus EEDIS, that - 128 are generated by extension. This occurs at an almost constant rate that is insensitive to the - 129 earthquake cycle. Combined with the frictional plastic deformation that operates in the upper - 130 crust these mechanisms ensure that the EE profile trends downward during the interseismic - 131 period. Summing PE, VE and EE components during the interseismic period produces the - 132 external work profile, W, and its decline indicates that work is being done by, not on, the crust - 133 between earthquakes. In contrast, models of thrust faulting (Del Castello and Cooke, 2007; Cruz - 134 et al., 2010) demonstrate that external work is done on the crust in these systems. - 135 Relative to the interseismic period, an earthquake is a brief event during which boundary - 136 displacements are negligible. Any change in W then represents work done by gravity alone, i.e., - 137 a negative change in GPE. During an earthquake energy flow is limited to a decrease in GPE, - 138 and increases in EE and FE, (Fig. 2A) as frictional plastic and viscous deformation operate too - 139 slowly to have an appreciable effect. These changes are an aggregate of three concurrent - 140 processes; a localized reduction in stress in the mid-crust that does work against friction and - 141 gravity to uplift the footwall; gravitational collapse of the hangingwall, which compresses the - 142 mid-crust and creates EE; and widespread horizontal compression of the upper crust, which - 143 creates EE and temporarily suppresses frictional plastic deformation. While the final two - 144 processes differ in scale only, their consequent effects on surficial processes warrant distinction. - 145 Coseismic components of the energy budget are resolved within each fault block by - 146 extending the fault plane to the base of the model and summing components either side. Changes - 147 in these components (Fig. 2B) indicate that mechanisms of deformation differ across the fault - 148 plane. For instance, energy flow within the footwall is characteristic of an elastic rebound. - 149 Approximately 0.85 TJ of EE are expended to produce 0.43 TJ of GPE, manifesting as surface - 150 uplift, and 0.14 TJ frictional heating along the fault. The EE expended is mostly volumetric with - 151 a small distortional component, and represents a reduction in stress that drives an expansion of - 152 the mid-crust and vertical movement of the footwall. The remaining EEVOL, approximately 0.28 - 153 TJ, is used to do work on the hangingwall block, resulting in a normal displacement of the fault - 154 plane of up to 20 cm at mid-crustal depths. This work, termed FW, is internal and not - 155 encapsulated by W in Fig. 2A. - 156 Energy changes within the hangingwall attest to a contrasting mechanism of deformation. - 157 In this block 1.0 TJ of GPE, 0.1 TJ of distortional elastic energy, i.e., EEDIS, and 0.28 TJ of - 158 internal work done by the footwall, i.e., FW, is converted to 1.3 TJ of EEVOL and 0.14 TJ of - 159 frictional heating along the fault. In physical terms, the GPE reduction that occurs due to a loss - 160 of elevation is used to do work to compress the crust, thereby creating EE. Work done to cause - 161 this compression is derived primarily from a loss of GPE rather than an elastic ‘bounce-back’ of - 162 the fault block. In this sense the elastic response of the hangingwall differs fundamentally to that - 163 of the footwall. - 164 **ENERGY FLOW DURING THE SEISMIC CYCLE** - 165 Further insight into footwall elastic rebound and hangingwall collapse is gained by - 166 identifying those regions from which energy is given up and to which energy flows. Dissipation 167 of heat by fault friction and internal work (FE and FW in Fig. 2B) are both restricted to the fault - 168 plane, however, changes in EEVOL, EEDIS and GPE occur throughout the crust. - 169 Distortional changes in strain energy are consistent with the elastic rebound - 170 interpretation, i.e., EEDIS accumulates in the upper crust (shallower than 8 km) during the - 171 interseismic period, and is subsequently released during fault rupture (Fig. 3A). A particularly - 172 large release of EEDIS occurs within the footwall close to the fault plane and is associated with - 173 up-dip displacement of the fault block. Concurrently the near fault region at mid-crustal depths - 174 (8-12 km) experiences an increase in EEDIS, principally within the hangingwall. This is - 175 associated with an increase in stress at the maximum gradient in fault slip (see inset Fig 3A). - 176 This anomaly is largely dissipated by viscous deformation in the interseismic quiescence - 177 following an earthquake (Ellis et al., 2006). - 178 Changes in EEVOL and GPE are approximately an order of magnitude greater than EEDIS - 179 and are thus primary drivers of coseismic deformation. In the footwall, at a depth of 8-10 km, - 180 there is a large decrease in EEVOL representing a rapid decrease in mid-crustal stress and - 181 associated expansion of the crust. This energy, which had accumulated in the preceding - 182 interseismic period, does work against gravity resulting in uplift of the footwall (Fig. 1B) and the - 183 creation of GPE (Fig. 3C). Vectors of displacement direction show that footwall uplift occurs in 184 the wider context of block rotation. - 185 In the hangingwall, coseismic displacement is down-dip or directed horizontally away - 186 from the fault. The crust immediately adjacent to the fault subsides by up to 1 m at the surface - 187 (Fig. 1B) and a reduction in GPE from this region is observed (Fig. 3C). A concurrent increase in - 188 EEVOL occurs at depths of 6-10 km corresponding to the cessation of slip on the fault plane. - 189 In addition to these localized elastic changes, there is also a widespread, low level - 190 increase in EEVOL throughout the majority of the upper crust. This increase is approximately 50% - 191 larger than the simultaneous reduction in EEDIS (Fig. 3A), and thus the net change in EE is - 192 positive during an earthquake. The increase in EEVOL is due to horizontal compression of the - 193 crust as fault blocks withdraw from each other and may be linked to the expulsion of fluids at the - 194 surface due to fracture closure (Muir-Wood, 1994). - 195 **DISCUSSION** - 196 Energy flow during frictional plastic failure can be regarded as analogous to that - 197 described for faults, but at a smaller scale. A small increment of dip-slip on an optimally oriented - 198 fracture plane results in a small loss of GPE, dissipation of FE, and increase in EEVOL. Similarly - 199 a local increase in horizontal stress ensures the region remains below the Mohr-Coulomb failure - 200 limit until the next episode of failure on that fracture plane. - 201 Coseismic energy flows in the hangingwall occur in the opposite direction to those - 202 attributed to elastic rebound of the footwall. Fig. 4 presents a spring-block schematic accounting - 203 for coseismic displacement, stress and energy changes within the hangingwall for an extending - 204 crust. Dip-slip of the fault block occurs when the sum of the shear components of the vertical - 205 lithostatic stress, a gravitational force directed down-dip, and the horizontal stress, an elastic - 206 force directed up-dip, exceeds the static frictional threshold. Tectonic extension, in addition to - 207 decreasing EE, causes a reduction in the up-dip shear stress, while leaving the gravitational shear - 208 component unchanged. An earthquake occurs after a sufficient reduction of the up-dip shear - 209 stress, at which time gravitational forces cause the hangingwall to shift down-dip, expending - 210 GPE. Horizontal displacement of the fault block recompresses the wider crust, creating EE, and - 211 increasing the up-dip shear force on the fault such that the balance of forces is restored. - 212 In contrast to the steady accumulation of EE required for an elastic rebound, GPE - 213 released during a collapse of the hangingwall does not accumulate during the interseismic period. - 214 Instead, GPE is ‘always there’ but is only made available to do work through the removal of - 215 elastic and frictional constraints, i.e., by imposing a material discontinuity (a fault) and then - 216 permitting slip to occur on its surface. Far-field tectonic extension is not directly responsible for - 217 loading the crust in the classical sense, i.e., by increasing its potential to do work. Rather, it alters - 218 the balance of stresses on the fault plane until failure of an ‘already loaded’ crust occurs. - 219 **ACKNOWLEDGMENTS** - 220 We thank T. Wright, M. Reyners and C. Williams for reading of the manuscript. This work is - 221 funded by the Tertiary Education Commission, New Zealand, through the Bright Futures Scheme - 222 and by an Energy Education Trust of New Zealand Doctoral Scholarship. - 223 **REFERENCES CITED** - 224 Barrientos, S. E., Stein, R. S., and Ward, S. N., 1987, Comparison of the 1959 Hebgen Lake, - 225 Montana and the 1983 Borah Peak, Idaho, earthquakes from geodetic observations, Bulletin - 226 of the Seismological Society of America, v. 77, p. 784-808. - 227 Byerlee, J., 1978, Friction of rocks, Pure and Applied Geophysics, v. 116, p. 615-626. - 228 Cook, N. G. W., Hoek, E., Pretorius, J. P. G., Ortlepp, W. D., and Salamon, M. D. G., 1966, - 229 Rock mechanics applied to the study of rockbursts, Journal of the South African Institute for 230 Mining and Metallurgy, v. 66, p. 435-528. - 231 Cooke, M. L., and Murphy, S., 2004, Assessing the work budget and efficiency of fault systems - 232 using mechanical models, Journal of Geophysical Research, v. 109, - 233 doi:10.1029/2004JB002968. - 234 Cruz, L., Malinski, J., Wilson, A., Take, W. A., and Hilley, G., 2010, Erosional control of the - 235 kinematics and geometry of fold-and-thrust belts imaged in physical and numerical sandbox, - 236 Journal of Geophysical Research, v. 115, doi:10.1029/2010JB007472. - 237 Del Castello, M., and Cooke, M. L., 2007, Underthrusting-accretion cycle: Work budget as - 238 revealed by the boundary element method, Journal of Geophysical Research, v. 112, - 239 doi:10.1029/2007JB004997. - 240 Ellis, S., Beavan, J., Eberhart-Phillips, D., and Stöckhert, B., 2006, Simplified models of the - 241 Alpine Fault seismic cycle: stress transfer in the mid-crust, Geophysical Journal 242 International _,_ v. 166, p. 386-402. - 243 Gudmundsson, A., 2004, Effects of Young’s modulus on fault displacement, Comptes Rendus - 244 Geosciences, v.336, p. 85-92. - 245 Hanks, T. C., and Kanamori, H., 1979, A moment magnitude scale, Journal of Geophysical - 246 Research, v. 84, p. 2348-2350. - 247 Hetland, E. A., and Hager, B. H., 2006, The effects of rheological layering on post-seismic - 248 deformation, Geophysical Journal International, v. 166, p. 277-292. - 249 Isacks, B., Oliver, J., and Sykes, L. R., 1968, Seismology and the new global tectonics, Journal - 250 of Geophysical Research, v. 73, p. 5855-5899. - 251 Kanamori, H., 2001, Energy budget of earthquakes and seismic efficiency, in Teisseyre, R., and - 252 Majewski, E., eds., Earthquake thermodynamics and phase transformations in the Earth’s 253 interior, New York, Academic Press, p. 293-305. - 254 Koukouvelas, I. K., and Doutsos, T. T., 1996, Implications of structural segmentation during - 255 earthquakes: the 1995 Egion earthquake, Gulf of Corinth, Greece, Journal of Structural 256 Geology, v. 18, p. 1381-1388. - 257 McGarr, A., 1994, Some comparisons between mining-induced and laboratory earthquakes, Pure 258 and Applied Geophysics, v. 142, p. 467-489. - 259 McKenzie, D., 1978, Active tectonics of the Alpine-Himalayan belt: the Aegean Sea and - 260 surrounding regions, Geophysical Journal of the Royal Astronomical Society, v. 55, p. 217261 254. - 262 Muir-Wood, R., 1994, Earthquakes, strain-cycling and the mobilization of fluids, in Parnell, J., - 263 ed., Geofluids: Origin, Migration and Evolution of Fluids in Sedimentary Basins, Geological 264 Society Special Publication, v. 78, p. 85-98. - 265 Nüchter, J. –A., and Ellis, S., 2010, Complex states of stress during the normal faulting seismic - 266 cycle: Role of midcrustal postseismic creep, Journal of Geophysical Research, v. 115, doi: 267 10.1029/2010JB007557. - 268 Popov, A. A., and Sobolev, S. V., 2008, SLIM3D: A tool for three-dimensional - 269 thermomechanical modeling of lithospheric deformation with elasto-visco-plastic rheology, - 270 Physics of the Earth and Planetary Interiors, v. 171, p. 55-75. - 271 Reid, H. F., 1911, The elastic-rebound theory of earthquakes, Bulletin of the Department of - 272 Geology, University of California Publications, v. 6, p. 413-444. - 273 Savage, H. M., and Cooke, M. L., 2000, Unlocking the effects of friction on fault damage zones, - 274 Journal of Structural Geology, v. 32, p. 1732-1741. - 275 Scholz, C. H., and Engelder, J. T., 1976, The role of asperity indentation and ploughing in rock - 276 friction - I. Asperity creep and stick slip, International Journal of Rock Mechanics and - 277 Mining Sciences & Geomechanics Abstracts, v. 13, p. 149-154. - 278 Sibson, R. H., 1974, Frictional constraints on thrust, wrench and normal faults, Nature, v. 249, p. 279 542-544. - 280 Stewart, J. H., 1980, Regional tilt patterns of late Cenozoic basin-range fault blocks, western 281 United States, Geological Society of America Bulletin, v. 91, p. 460-464. - 282 Tanimoto, T., Okamoto, T., and Terra, F., 2002, Tectonic signatures in coseismic gravitational - 283 energy change, Geophysical Journal International, v. 149, p. 490-498. - 284 Townend, J., and Zoback, M. D., 2000, How faulting keeps the crust strong, Geology, v. 28, p. - 285 399-402. - 286 Zoback, M. D., Moos, D., Mastin, L., and Anderson, R. N., 1985, Well bore breakouts and in situ - 287 stress, Journal of Geophysical Research, v. 90, p. 5523-5530. - 288 **FIGURE CAPTIONS** - 289 Figure 1: A: Model dimensions and boundary conditions applied during the stress spin-up phase - 290 and seismic cycling, superimposed on model mesh. Approximately 10 000 elements are used, - 291 ranging in size from 1000 m at the base to 250 m on the fault. B: Coseismic vertical surface - 292 displacements produced by the model (solid line) and for historical earthquakes at Borah Peak - 293 (BP), Idaho (1983, M = 6.9) and Hegben Lake (HL), Montana (1959, M = 7.3). Note that error - 294 bars for the historical data smaller than the marker dimension. - 295 - 296 Figure 2: Energy budget during seismic cycle. A: Evolution over several seismic cycles of the - 297 energy budget components, i.e., W=EE+PE+VE+FE, relative to their initial values, Eo, and - 298 summed over the model domain. VE = energy dissipated by viscous strain; PE = energy - 299 dissipated by frictional plastic strain; FE = energy dissipated by frictional slip along the fault; W - 300 = work done by external forces including gravity and tractions at the boundaries; EE = changes - 301 in elastic energy. Note that W includes changes in GPE as indicated for the final coseismic - 302 period. Earthquakes (EQ) are marked by dashed lines. B: Coseismic energy changes within the - 303 hangingwall (solid) and footwall (hatched) blocks. Note that EE has been divided into - 304 volumetric, EEVOL, and distortional, EEDIS, components. - 305 - 306 Figure 3: Contours of coseismic change in energy density throughout the crust for A: EEDIS. B: - 307 EEVOL C: GPE. Vectors on C indicate displacement direction. Contour levels for A are an order 308 of magnitude smaller than B and C, and units for energy density are dimensionally equivalent to - 309 stress. As the model is in dynamic equilibrium, changes in EEDIS and EEVOL during the - 310 interseismic period are equal in distribution but opposite in sign to those in A and B respectively. - 311 Dip-slip profiles for the footwall (dotted) and hangingwall (solid) are inset in A. Vertical scale is - 312 the same as the contour plot and down-dip slip is negative. - 313 - 314 Figure 4: A spring (of stiffness k) and block model accounting for observed displacements, - 315 stress, and EE and GPE changes in the hangingwall. A: The unloaded spring, equivalent to the - 316 absence of gravity. B: Gravitational loading, FG, compresses the crust vertically and induces a - 317 horizontal spring force, FE. The sum of opposing shear components of FG and FE is below the - 318 threshold for frictional slip. C: An increment of tectonic extension, Δx, decreases EE and reduces - 319 the spring force until the threshold for friction slip is exceeded. D: Dip-slip of the hangingwall - 320 recompresses the spring, creating EE at the cost of GPE, and restores the balance of forces on the - 321 fault. Figure 1: A: Model dimensions and boundary conditions applied during the stress spin­up phase and seismic cycling, superimposed on model mesh. Approximately 10 000 elements are used, ranging in size from 1000 m at the base to 250 m on the fault. B: Coseismic vertical surface displacements produced by the model (solid line) and for historical earthquakes at Borah Peak (BP), Idaho (1983, M = 6.9) and Hegben Lake (HL), Montana (1959, M = 7.3). Note that error bars for the historical data smaller than the marker dimension. Figure 2: Energy budget during seismic cycle. A: Evolution over several seismic cycles of the energy budget components, i.e., W=EE+PE+VE+FE, relative to their initial values, Eo, and summed over the model domain. VE = energy dissipated by viscous strain; PE = energy dissipated by frictional plastic strain; FE = energy dissipated by frictional slip along the fault; W = work done by external forces including gravity and tractions at the boundaries; EE = changes in elastic energy. Note that W includes changes in GPE as indicated for the final coseismic period. Earthquakes (EQ) are marked by dashed lines. B: Coseismic energy changes within the hangingwall (solid) and footwall (hatched) blocks. Note that EE has been divided into volumetric, EEVOL, and distortional, EEDIS, components. Figure 3: Contours of coseismic change in energy density throughout the crust for A: EEDIS. B: EEVOL C: GPE. Vectors on C indicate displacement direction. Contour levels for A are an order of magnitude smaller than B and C, and units for energy density are dimensionally equivalent to stress. As the model is in dynamic equilibrium, changes in EEDIS and EEVOL during the interseismic period are equal in distribution but opposite in sign to those in A and B respectively. Dip­slip profiles for the footwall (dotted) and hangingwall (solid) are inset in A. Vertical scale is the same as the contour plot and down­dip slip is negative. Figure 4: A spring (of stiffness k) and block model accounting for observed displacements, stress, and EE and GPE changes in the hangingwall. A: The unloaded spring, equivalent to the absence of gravity. B: Gravitational loading, FG, compresses the crust vertically and induces a horizontal spring force, FE. The sum of opposing shear components of FG and FE is below the threshold for frictional slip. C: An increment of tectonic extension, Δx, decreases EE and reduces the spring force until the threshold for friction slip is exceeded. D: Dip­slip of the hangingwall recompresses the spring, creating EE at the cost of GPE, and restores the balance of forces on the fault. Formulation Parameters _Linear, isotropic elasticity_ εij strain tensor σij stress tensor  _ij_ = _E_ 1 [( 1 +) _ij_ − _kk_ ] E Young’s modulus [10 GPa] ν Poisson’s ratio [0.25] _Mohr-Coulomb failure criterion (frictional plasticity)_ σ1,3 maximum, minimum principal stress p pressure  1 − 3  _p_ .sin(') + _C_ '.cos(') φ rock angle of friction [30o (19o )] C cohesion [1 MPa (0.91 MPa)] _Thermally activated creep_  _v_ viscous strain rate [6.3×10-15 s-1 ] A pre-exponential const. [4×10-34 Pa-4  _v_ = _A_ ( 1 − 3 ) _n_ exp(− _Q_ / _RT_ ) n s-1 ] Q power-law exponent [4] R activation energy [135 kJ mol-1 ] |||T|gas constant [8.314]| |---|---|---|---| ||||temperature| ||_Fault friction_||| |||τ|fault shear stress| |_d_
_n_


||μd|dynamic friction coefficient [0.2]| |||σ_n_|fault normal stress| ||_Other_||| |||ρ|density [2800 kg m-3]| |||g|gravity [9.81 m s-2]| **Table 1:** Summary of model methods and rheologies. Prescribed parameter values given in brackets. Note that we have used adjusted values (in parentheses) for rock angle of friction, φ’, and cohesion, C’. These values reflect the reduced strength of a crust saturated by pore fluids at hydrostatic pressure. Parameters for thermally activated creep are derived from Paterson and Luan (1990). # **REFERENCES CITED** Paterson, M. S., and Luan, F. S., 1990, Quartzite rheology under geological conditions, in Knipe, R. J., and Rutter, E. H., eds., Deformation Mechanisms, Rheology and Tectonics, Geological Society Special Publications, v. 54, p. 299-307.