Finite extension and low-angle normal faulting

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Finite extension and low-angle normal faulting Donald W. Forsyth Department of Geological Sciences, Brown University, Providence, Rhode Island 02912

ABSTRACT Andersonian theory for infinitesimal strains predicts that normal faults will form at high angles to the surface in extensional terranes. In contrast, low-angle normal faults are sometimes observed on which tens of kilometres of slip has occurred, and moderate-angle faults with slip of 5 to 15 km are common. A simple extension of the theory to account for the stresses required to drive finite deformation shows that low angles are preferred for large extension on individual faults. The formation of low-angle faults may require inhomogeneities in stress or material properties, but once slip begins on a low-angle fault it is likely to continue, whereas slip on high-angle faults is likely to end after modest slip, to be replaced by the formation of new high-angle faults. INTRODUCTION Major, low-angle normal faults are found in extensional terranes such as the Basin and Range province. These faults dip at angles of less than 30° to subhorizontal and have displacements of up to several tens of kilometres; sometimes called "detachments" or "decollements," in many places they form the boundary between low-grade, "upper-plate" rocks and the underlying, tectonized, high-grade metamorphic rocks of metamorphic core complexes (Crittenden et al., 1980). The dynamics of low-angle normal faulting has been the subject of much discussion, because simple models based on rock mechanics theory predict that normal faults should form at dip angles of - 6 0 ° . Although less controversial, major blockbounding normal faults in extensional regimes such as the Basin and Range province also depart significantly from the ideal angle; displacements of 5 to 15 km are common on planar faults dipping at about 45° (Stein et al., 1988; Wernicke and Axen, 1988). The purpose of this paper is to point out the effects of finite displacement on the conditions for continued slip on a single fault. As slip proceeds on a fault in response to regional extension, bending stresses within the lithosphere and stresses due to the weight of the topography both accumulate and tend to oppose further slip. Slip is expected to terminate when extension can occur with lower applied regional stress, either by forming a new high-angle fault or by reactivating a lower-angle fault. The stresses created by finite deformation are much smaller for low-angle faults, so once slip begins on a low-angle fault, very large displacements can accumulate before slip ends. Thus, although most normal faults will form at high angles, the few low-angle faults that do form due to stress or strength inhomogeneities within the lithosphere may play an important tectonic role in accommodating extension. We can assess the conditions for slip on faults of various dips by considering the regional stress required to accomplish extension as the sum of the stress required to drive slip on the fault itself and the stress required to uplift and deform the lithosphere. Both of these processes can be represented to a first approximation by very simple models. MODEL The maximum principal compressive stress is assumed to be vertical and equal to the weight of the overburden. The least principal stress is horizontal and differs from the vertical stress by an applied regional stress, ox (Fig. 1A). The lithosphere is assumed to be elastic except where it yields along a fault. In an extensional environment such as the Basin and Range province, characterized by high heat flow, the lithosphere is limited GEOLOGY, v. 20, p. 27-30, January 1992

Regional

Extension I H I Cr = Of =

HpgH/2 + c _ H sin 2 9 + sine c o s e

B Equilibrium

level

tz

'O-AX

Flexure

_ Or = C(

IE (p e gH e ) 3 / 3(1 -v 2 )) 1 / 4 AX tan 2 6

4H

Figure 1. Schematic model of lithospheric extension accomplished by slip on fault. A: Regional stress, o R , drives extension, causing failure of plate at dip angle 6. Regional stress, a,, required to overcome friction is function of dip, cohesion, c, and normal stress on fault. Vertical principal stress is assumed to be lithostatic, averaging pgH/2, where p is density, g is gravitational acceleration, and H is thickness of brittle layer. Horizontal principal stress is assumed to be equal to vertical principal stress minus regional stress. B: Finite extension on planar fault causes deflection of plate from equilibrium isostatic level. C: In response to gravitational forces, plate flexes. Regional stress needed for finite deformation is sum of a f and stress required to drive uplift and flexure, which is function of dip, effective elastic thickness, H e , effective density, p e , amount of uplift or extension, Ax, and elastic properties (Young's modulus, £, and Poisson's ratio, v).

to the upper crust. The lower crust is probably hot enough to deform in a ductile fashion at low stresses and is modeled here as an inviscid fluid. For simplicity, I consider only planar normal faults that cut through the entire lithosphere, which in this case means only to the depth of the brittle-ductile transition within the crust. Extension in response to the applied regional stress is accomplished by slip on the planar fault. In finite extension, the regional stress works on the system to overcome frictional resistance on the fault and to deform the lithosphere. Traditional models of the angle of dip of the fault consider only the frictional resistance (Fig. IB). The shear stress, r, required for slip on an existing fault is of the form r = y.on + c (Byerlee, 1978), where an is the normal stress, y. is the coefficient of friction, and c is the cohesion. The Mohr-Coulomb criterion for failure and formation of a new fault has the same form, shear failure occurring once the shear or cohesive strength and internal friction are exceeded on some plane within the material. Formulated in terms of a regional stress argument, the angle of dip of the fault plane, 6, can be predicted by finding the angle that minimizes the regional stress required to accomplish infinitesimal extension, Ax, in the horizontal direction (Anderson, 1942). The angle 0 is about 60° for a cohesionless fault with a typical coefficient of friction of 0.6 to 0.7. The regional stress, 27

Downloaded from geology.gsapubs.org on September 15, 2015 ay, required to overcome friction as a function of dip is shown in Figure 2. Note that although the absolute minimum is reached at about 60°, it is a broad minimum, so that in a realistic medium with other sources of stress variations and variable physical properties, a wide range of dip angles might be expected. This analysis of the regional stress required for slip on a fault clearly ignores the stress required to displace mass vertically against the attraction of gravity. As illustrated in Figure IB, the footwall is displaced above the equilibrium isostatic level and the hanging wall is displaced below the equilibrium level. Work is required both to lift the footwall and to depress the hanging wall. If there were no deformation of the lithosphere, an infinite amount of work would be required to accomplish finite extension; i.e., if the only driving force is the applied regional stress distributed over a finite thickness plate, an infinite stress would be required. The plate bends in response to gravity, distributing the work required for vertical displacements between elastic strain energy associated with bending and potential energy associated with displacement from the equilibrium level (Fig. 1C). Modeling in terms of local stresses is more difficult, but it is clear that both the topographic loads and the flexure induce stresses that, acting alone, would tend to reverse the sense of slip on the fault: the elastic stresses will tend to restore the plate to its original unflexed position, and the topographic loads will tend to restore the plate to local isostatic equilibrium. The applied regional stress must increase as displacement on the fault increases to overcome these uplift-induced stresses. To estimate the regional stresses required, I represent the lithosphere on either side of the fault as a broken, thin elastic plate floating on an inviscid fluid layer with the same density as the plate (Fig. IB). I approximate the work required to deform the lithosphere as the work required to displace vertically the ends of each of the thin plates by an amount equal to half of the total vertical displacement on the fault. The vertical deflection from the isostatic equilibrium level is assumed to be equal in amplitude but opposite in sign for the footwall and the hanging wall, because this minimizes work or regional stress if there is no density contrast at the base of the plate. Some isostatic models have predicted that the vertical displacements of the hanging wall and footwall would be different and their relative amplitude dependent on dip angle (Heiskanen and Vening Meinesz, 1958; Jackson and McKenzie, 1983), but a more realistic treatment of this isostatic problem (Weissel and Karner, 1989) shows that the displacements would be equal in the case of no density contrast. The vertical displacement of either side, Az, is given by Az = tanfl Ax/2. The vertical force required to displace the end of a thin plate vertically by a given

Figure 2. Regional stress required to drive extension as function of dip of fault. Solid line—infinitesimal extension on fault; dashed line—after 2 km of extension has already been accommodated by slip on fault; dotted line—stress required to begin slip on new fault with cohesion of 20 MPa. All calculations assume m = 0.65 and c = 0 for active faults. Arrows mark optimum angles that minimize stress required for further slip. 28

amount can be derived easily from an expression given by Turcotte and Schubert (1982, equation 3-140) for vertical deflection under a line load at the end of a plate. Integrating this force through the vertical deflection gives the work. Differentiating the work with respect to horizontal displacement gives that part of the regional stress required to deform the lithosphere (Fig. 1C). The regional stress, or, is assumed to be the sum of the regional stress required to deform the lithosphere and the regional stress needed to overcome friction on the fault, neglecting any modification of the normal stress on the fault associated with the finite deformation. An important assumption is that the dip and length of the fault remain constant as deformation proceeds. This is an excellent approximation as long as displacement is small compared to length of the fault and flexural wavelength is large compared to fault length. For large displacements, the fault geometry may change, but the assumption may still be a good approximation if the brittle layer is thin enough and displacement slow enough for thermal diffusion to maintain a relatively constant geothermal gradient along the fault surface. Rotation of the fault will be small unless the flexural rigidity is low, and it should always be rotating toward shallower dip angles (Spencer, 1984; Buck, 1988). O P T I M U M D I P ANGLE The optimum dip angle is defined as the angle of dip of the fault that minimizes the regional stress needed to drive continued slip on the fault. With increasing extension, the optimum dip angle decreases. Figure 2 compares the regional stress required for slip on a newly formed cohesionless fault of varying dip with the regional stress required for slip on the same fault after 2 km of extension has been accomplished by slip on the fault. Two primary effects are apparent. First, the dip angle that requires the least stress for an incremental increase in extension shifts from more than 60° to about 40° for this example. Second, the size of the increase in regional stress suggests that finite extension should be very effective in terminating slip on a steeply dipping fault, more effective than flexural rotation to an unfavorable angle. For a fault initially dipping at 60°, the effects of stresses associated with vertical displacement accompanying even a 200 m extension would be equivalent to the increase in regional stresses required to overcome frictional resistance on the same fault rotated by 19° to a dip of 41°. Any rotation of the fault plane to lower angles accompanying flexure caused by slip on the fault actually makes it easier for slip to continue on the fault. For the quantitative illustrations herein, the lithosphere is assigned a thickness of 10 km. Earthquakes in the Basin and Range province extend to a depth of 10 to 15 km (Smith and Bruhn, 1984), as does coseismic deformation associated with major normal-faulting earthquakes (Barrientos et al., 1987). The 10 km thickness is also roughly consistent with estimates of the apparent thickness of the elastic layer based on studies of the flexural isostatic response in this area. On a gross scale, the effective elastic thickness of the Basin and Range province, based on a coherence study of gravity and topography, is 4 to 5 km, but this probably represents a compromise between the regional isostatic response of relatively stiff individual crustal blocks and the local response accomplished by slip on major faults (Bechtel et al., 1990). There is clearly a significant stiffness to the plate, because individual basins and ranges are not locally isostatically compensated (Eaton et al., 1978). The effective elastic thickness based on response to the Lake Bonneville load is 20 to 25 km (Nakiboglu and Lambeck, 1983; Bills and May, 1987), whereas the deformation associated with sediment loading in the vicinity of the Lost River normal fault in Idaho and the Cricket Mountain fault in Utah yields estimates of 2 to 4 km (Stein et al., 1988). As slip on a major normal fault increases, the stresses involved in flexure may exceed the yield strength of the crust, thus decreasing the effective elastic thickness in the vicinity of the fault (Buck, 1988; Wernicke and Axen, 1988). If the elastic plate thickness and the seismogenic thickness are held equal, then decreasing the plate thickness increases GEOLOGY, January 1992

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the importance of the finite deformation relative to the friction. If the elastic plate thickness is less than the seismogenic thickness, then the frictional term becomes more important. These calculations can accommodate the effective elastic thickness being less than the thickness of the layer in which frictional slip on the fault occurs by simply using different values in the frictional and deformational terms (Fig. 1, A and C, respectively). The optimum dip angle that minimizes the regional stress required for an infinitesimal increment of extension is shown in Figure 3 as a function of the coefficient of friction and the amount of extension previously accommodated on the fault. After extension of a few kilometres is accomplished by slip on a single fault, it clearly requires less regional stress to continue slip on a fault dipping 30° to 40° than to continue slip on a fault dipping at the steep dips that are predicted for infinitesimal extension. Displacement of tens of kilometres is accommodated most easily on lowangle normal faults dipping less than 30°. ACTIVATION O F LOW-ANGLE FAULTS Given that large displacements on single faults are most easily achieved if the fault dips at low angles, there are two primary questions remaining. First, how does slip begin on low-angle faults or how do low-angle faults form if infinitesimal slip tends to occur on high-angle faults? Second, if finite displacement tends to terminate slip on steep normal faults, will they simply be replaced by new high-angle faults? The keys to these questions within the framework of rock mechanics lie with the coefficient of friction, the shear strength or internal cohesion of unfractured rocks, and heterogeneities within the lithosphere. Some normal faults clearly rotate to lower angles, either domino-style (Proffett, 1977; Wernicke and Burchfiel, 1982; Davis, 1983) or by flexural response to unroofing (Spencer, 1984; Buck, 1988; Wernicke and Axen, 1988). However, the large across-strike extent of major low-angle faults extending through the upper crust (Crittenden et al., 1980) makes it unOPTIMUM DIP ANGLE

likely that these originated as high-angle faults and subsequently rotated into position. The absence of earthquakes on normal faults with dips of less than 30° (Jackson, 1987) leads to the argument that slip occurs on faults only at steep angles within the brittle upper crust, but that subsequent flexural rotation of a plate with very low effective rigidity rotates the inactive fault surface to low angles (Buck, 1988; Wernicke and Axen, 1988). There is evidence, however, that slip continues on low-angle faults (von Tish et al., 1985; Reynolds and Spencer, 1985; Miller and John, 1988), albeit aseismically. Although normal faults are undoubtedly flattened by isostatic rebound, there must be a mechanism by which normal faults begin at low angles. The broad minimum in the regional stress required to drive slip on faults of different dip (Fig. 2) means that it is relatively easy for the system to be perturbed so that slip begins at shallower angles than the optimum angle. One possibility is that there are stress inhomogeneities within the upper crust that locally perturb the orientation of the principal stress axes so that low-angle faults form, although these would also facilitate the formation of high-angle faults in adjacent areas. Another possibility is that heterogeneity or anisotropy in material properties provide an easier pathway for extension than high-angle faulting. One example of this mechanism is reactivation of low-angle thrust faults as normal faults. The conditions for fault reactivation were discussed by Sibson (1985); an alternative representation is presented in Figure 4. The curved lines map combinations of dip and coefficient of friction that require equal regional stress to drive extension. For a cohesionless plane, about the same stress is required to drive a fault dipping 30° with a coefficient of friction of 0.4 as that needed to drive a fault dipping at the optimum angle of about 62° with a coefficient of friction of 0.65. Thus, a plane dipping 30° with fx 6 0 ° ) , h o w e v e r , slip m a y t e r m i n a t e b e f o r e yielding thins the effective elastic thickness significantly b e l o w the seismog e n i c thickness. A l t h o u g h the limits a r e e x p a n d e d , the basic principle r e m a i n s t h e same: high-angle faults c a n a c c o m m o d a t e only limited exten-

Journal of Geophysical Research, v. 92, p. 11,493-11,508. Buck, W.R., 1988, Flexural rotation of normal faults: Tectonics, v. 7, p. 959-973. Byerlee, J.D., 1978, Friction of rocks: Pure and Applied Geophysics, v. 116, p. 615-626. Crittenden, M.D., Jr., Coney, P.J., and Davis, G.H., eds., 1980, Cordilleran metamorphic core complexes: Geological Society of America Memoir 153,490 p. Davis, G.H., 1983, Shear-zone model for the origin of metamorphic core complexes: Geology, v. 11, p. 342-347. Eaton, G.P., Wahl, R.R., Prostka, H.J., Mabey, D.R., and Kleinkopf, M.D., 1978, Regional gravity and tectonic patterns: Their relation to late Cenozoic epeirogeny and lateral spreading in the western Cordillera, in Smith, R.B., and Eaton, G.P., eds., Cenozoic tectonics and regional geophysics of the western Cordillera: Geological Society of America Memoir 152, p. 51-91. Heiskanen, W.A., and Vening Meinesz, F.A., 1958, The Earth and its gravity field: New York, McGraw-Hill, 470 p. Jackson, J.A., 1987, Active normal faulting and crustal extension, in Coward, M.P., Dewey, J.F., and Hancock, P.L., eds., Continental extensional tectonics: Geological Society of London Special Publication 28, p. 3-17. Jackson, J., and McKenzie, D., 1983, The geometrical evolution of normal fault systems: Journal of Structural Geology, v. 5, p. 471-482. King, G., and Ellis, M., 1990, The origin of large local uplift in extensional regions: Nature, v. 348, p. 689-692. Miller, J.M.G., and John, B.E., 1988, Detached strata in a Tertiary low-angle normal fault terrane, southeastern California: A sedimentary record of unroofing, breaching and continued slip: Geology, v. 16, p. 645-648. Nakiboglu, S.M., and Lambeck, K., 1983, A réévaluation of isostatic rebound of Lake Bonneville: Journal of Geophysical Research, v. 88, p. 10,439-10,447. Proffett, J.M., Jr., 1977, Cenozoic geology of the Yerington District, Nevada, and implications for the nature and origin of Basin and Range faulting: Geological Society of America Bulletin, v. 88, p. 247-266. Reynolds, S.J., and Spencer, J.E., 1985, Evidence for large-scale transport on the Bullard detachment fault, west-central Arizona: Geology, v. 13, p. 353-356. Sibson, R.H., 1985, A note on fault reactivation: Journal of Structural Geology, v. 7, p. 751-754. Smith, R.B., and Bruhn, R.L., 1984, Intraplate extensional tectonics of the eastern Basin-Range: Inferences on structural style from seismic reflection data, regional tectonics and thermal-mechanical models of brittle-ductile deformation: Journal of Geophysical Research, v. 89, p. 5733-5762. Spencer, J.E., 1984, Role of tectonic denudation in warping and uplift of low-angle normal faults: Geology, v. 12, p. 95-98. Stein, R.S., King, G.C.P., and Rundle, J.B., 1988, The growth of geological structures by repeated earthquakes, 2. Field examples of continental dip-slip faults: Journal of Geophysical Research, v. 93, p. 13,319-13,331. Turcotte, D.L., and Schubert, G., 1982, Geodynamics: Applications of continuum physics to geological problems: New York, John Wiley & Sons, 450 p. von Tish, D.B., Allmendinger, R.W., and Sharp, J.W., 1985, History of Cenozoic extension in central Sevier Desert, west-central Utah, from COCORP seismic reflection data: American Association of Petroleum Geologists Bulletin, v. 69, p. 1077-1087. Weissel, J.K., and Karner, G.D., 1989, Flexural uplift of rift flanks due to mechanical unloading of the lithosphere during extension: Journal of Geophysical Research, v. 94, p. 13,919-13,950. Wernicke, B., and Axen, G.J., 1988, On the role of isostasy in the evolution of normal fault systems: Geology, v. 16, p. 848-851. Wernicke, B., and Burchfiel, B.C., 1982, Modes of extensional tectonics: Journal of Structural Geology, v. 4, p. 105-115. ACKNOWLEDGMENTS Supported in part by National Science Foundation Grant OCE-8911326. I thank Doug Wiens for providing the opportunity for concentrated thought about this problem and Roger Buck and Ray Fletcher for constructive reviews.

sion b e f o r e t h e y a r e replaced b y n e w faults, w h e r e a s low-angle faults c a n Manuscript received April 1, 1991 Revised manuscript received September 6, 1991 Manuscript accepted September 18,1991

a c c o m m o d a t e very large displacements.

REFERENCES CITED Anderson, E.M., 1942, The dynamics of faulting: London, Oliver and Boyd, 183 p. Barrientos, S.E., Stein, R.S., and Ward, S.N., 1987, Comparison of the 1959 Hebgen Lake, Montana and the 1983 Borah Peak, Idaho earthquakes from geodetic observations: Seismological Society of America Bulletin, v. 77, p. 784-808. Bechtel, T.D., Forsyth, D.W., Sharpton, V.L., and Grieve, R.A.F., 1990, Variations in effective elastic thickness of the North American lithosphere: Nature, v. 343, p. 636-638. Bills, B.G., and May, G.L., 1987, Lake Bonneville: Constraints on lithospheric thickness and upper mantle viscosity from isostatic warping of Bonneville, 30

Reviewer's

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S h o u l d p r o v o k e . . . substantial m e n t a l activity in t h e r e a l m of structural geology-tectonophysics.

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GEOLOGY, January 1992

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Geology Finite extension and low-angle normal faulting Donald W. Forsyth Geology 1992;20;27-30 doi: 10.1130/0091-7613(1992)0202.3.CO;2

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