Buoyancy tectonic models of uplift and subsidence along high-angle faults in extensional and compressional tectonic regimes

TECTONOPHYSICS ELSEVIER

Tectonophysics 246 (1995) 71-95

Buoyancy tectonic models of uplift and subsidence along high-angle faults in extensional and compressional tectonic regimes Marshall Reiter New Mexico Bureau of Mines and Mineral Resources, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

Received 17 February 1994; accepted in revised form 30 January 1995

Abstract This study models the potential uplift and subsidence of brittle, upper-crustal trapezoidal blocks which move along pre-existing high-angle faults in extensional tectonic regimes; similar models for uplift in compressional tectonic regimes are also presented. A force balance is assumed to exist between the weight of the trapezoidal block, the buoyancy force derived from the lower-ductile crust, the supporting forces from the neighboring blocks, and the frictional resistance to motion across the fault. Generally, the weight of the brittle crustal block is effectively opposed by the buoyancy force, making vertical movement quite dependent on tectonic forces redirected along the high-angle bounding faults. The analysis relates the state of stress in the brittle-upper crust (and other critical parameters such as structure width, thickness of the brittle-upper crust, frictional resistance along faults, mobility of the brittle-ductile transition which is related to slow and fast geologic movement, different lower crustal densities, and deposition and erosion) to possible uplift or subsidence. Greater vertical displacement is generally compatible with narrower crustal blocks, more deviatoric (tectonic) stress, lower fault friction, a thicker brittle crust, and similar upper and lower crustal densities. Without deposition or erosion, approximately equal amounts of vertical movement appear possible for trapezoidal blocks with similar conditions; however, deposition can typically induce far more subsidence than erosion can induce uplift. Although large amounts of uplift appear possible in both extensional and compressional regimes, the effects of erosion and the higher stresses possible with tectonic compression should produce larger amounts of uplift in compressional vs. extensional regimes. Slow or fast geologic movement can be defined as whether the brittle-ductile transition remains at a constant depth or is displaced; if the brittle-ductile transition remains at a constant depth (slow movement), uplift is enhanced in both extensional and compressional regimes. Low resistance to motion along faults is necessary for vertical movement and is consistent with several other recent studies. The models suggest that much of the observed relief on the earth's surface (up to several tens of kilometers width) can be caused by the dynamics in the earth's crust, however related to deeper earth processes; i.e. much topography can be analyzed in terms of stresses in the brittle crust and the buoyancy effect of the lower crust, however the stresses in the brittle crust may be generated by deeper and more global processes. I. Introduction T h e p u r p o s e of this study is to c o n s i d e r a m o d e l that can explain the d e v e l o p m e n t of at

least some of the observed t o p o g r a p h y o n the e a r t h u p to a few tens of kilometers in width. I n the p r e s e n t m o d e l I a t t e m p t to i n t e g r a t e buoyancy tectonics, with tectonically derived horizon-

0040-1951/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0040-1951(95)00015-1

72

M. Reiter / Tectonophysics 246 (1995) 71-95

tal crustal stresses and fault friction, in order to consider the potential for vertical movement of trapezoidal blocks in the brittle crust. It appears that considerable relief up to several tens of kilometers in width is possible if rigid crustal blocks move along high angle pre-existing faults in response to horizontal tectonic stresses, lsostatic compensation of this movement is proposed to occur in the ductile (middle or lower) crust. The derivation of the tectonic stresses in the crust may be different for different regions (e.g. plate interactions, lithospheric stretching caused by convection mechanisms, thermal expansion of the lithosphere, etc.); these different processes should influence brittle crustal thickness, lower crustal rheology and long wavelength topography. However, the vertical movement of brittle crustal blocks having appropriate widths may depend largely on the local crustal dynamics, whatever the derivation; i.e. the local tectonic stresses in the brittle crust and the flow response of the middle a n d / o r lower crust to upper crustal movement. Vening Meinesz (1950) and Heiskanen and Vening Meinesz (1958) first developed the model of trapezoidal graben blocks bounded by high angle normal faults. Initial subsidence of the crust along a normal fault, creating an asymmetric graben, was proposed to develop bending stresses in the crust which ultimately caused a second complimentary high-angle fault. The trapezoidal block bounded by these high-angle faults subsided and isostatic compensation occurred in the mantle, as the high-angle normal faults penetrated the entire crust. Bott (1976,1981) and Bott and Mithen (1983) did further studies on the trapezoidal model. Bott and Mithen (1983) proposed that crustal faults penetrated only the brittle crust and the trapezoidal graben subsided only into the lower crust. The possible movement of graben blocks and neighboring horsts was determined by considering the potential energy of the system, and the work necessary to move the graben and horst blocks and the ductile material in the crust. Tectonic tension was assumed constant with depth and local mass conservation was assumed in the lower crust. They considered the effects of deviatoric tension, fault friction, graben

width and sediment loading on graben subsidence. Barroll and Reiter (1987) approached the subsidence of a trapezoidal crustal block by balancing the weight of the block with the crustal tectonic forces and the buoyancy force of the lower crust, allowing isostatic compensation in the lower crust. Conservation of lower crustal mass was suggested to occur by changes in crustal thickness a n d / o r extension beyond the model. Models of both uplift and subsidence along appropriately oriented high-angle faults are investigated in the present study for extensional tectonic regimes. For compressional tectonic regimes, I consider only uplift along high-angle faults. Maximum subsidence and uplift are calculated for symmetrical trapezoidal blocks using a static balance of buoyancy forces, weight, crustal forces acting across faults, and fault friction. Subsidence and uplift are considered in terms of crustal tectonic stresses, elastic crustal thickness, block width, fault angle, effective fault friction, lower crustal density changes, mobility of the brittle-ductile boundary which can be related to rate of movement, and erosion and sedimentation. Although some of these parameters have previously been considered in studies of graben subsidence, calculations regarding uplift have not been previously done. Strains in the upper crust are often accommodated along parallel and planar, deeply penetrating normal faults, resulting in parallelogramshaped blocks or half-grabens (Jackson, 1987; Lister and Davis, 1989). The rotation of these half-grabens has been analyzed in terms of crustal stresses, elastic crustal thickness, graben width, and fault friction by Reiter et al. (1992). Although the development of symmetrical, non-rotational horst-graben structures is relatively rare (Bally, 1982; Rosendahl, 1987), such studies do provide insight into the first-order parameters that interact to control graben subsidence. Examples of horst-graben features developed during extensional tectonic regimes are present in many rift structures (e.g. the Rio Grande rift) and in the Basin and Range Provinces of the western United States, where during the past ~ 10 .my considerable topography has developed largely due to normal movement along high-angle faults

M. Reiter / Tectonophysics 246 (1995) 71-95

regularly spaced 30-40 km apart (e.g. Stewart, 1978; Brown et al., 1979). The deep grabens offshore of Atlantic Canada can also be used for comparison with results obtained in the present study. The Zuni Mountains in west-central New Mexico and the central core of the northern Wind River Mountains in Wyoming are two proposed examples of significant uplift occurring along high-angle reverse faults in compressional environments (Mita et al., 1988; Chamberlin and Anderson, 1989). I have been unable to find examples of basins formed in compressional regimes, and bounded on both sides by high-angle reverse faults (although the southwestern part of the Wind River Basin and the western part of the Denver Basin are two examples of basins which may be bounded on one side by high-angle reverse faults (Bieber, 1983; Mita et al., 1988). Therefore, the present analysis is limited to uplifts in compressional environments; although it could be straightforwardly extended to subsidence if such structures are observed.

2. Model basics and background In the present study we perform a static force analysis assuming that grabens and horsts will subside or rise when vertical forces do not balance. Subsidence or uplift of symmetrical trapezoidal blocks is considered in terms of the weight of the blocks, the buoyancy forces of the lower (ductile) crust, the forces developed from stresses in the brittle-upper crust, and the resistance to motion along the faults (effective fault friction, discussed further below). The high-angle faults bounding the trapezoids are assumed to be preexistent and to penetrate the upper-brittle crust from the surface to the ductile-lower crust. The bounding faults are suggested to be planar and to dip at angles of 80 ° to 50 ° from the horizontal (10 ° to 40 ° from the vertical). It is proposed that the lower crust deforms and accommodates the vertical movement of the brittle, upper-crustal blocks. Justification for most of these assumptions can be given. Strehlau and Meissner (1987) suggest that the lower crust is a mobile shear zone de-

73

forming by both broadly distributed and localized flow. They cite observations of banded gneisses as demonstrating broad ductile flow at conditions appropriate for the lower crust and they define strain rates of 10 - 1 4 S-1 as representing rapid deformation of crustal blocks (or ductile behavior). The rheology of the lower crust may depend on water content as well as temperature and composition (Meissner and Strehlau, 1982; Strehlau and Meissner, 1987); confined water in the lower crust could significantly enhance creep processes (Turcott, 1987). Gans (1987) suggests that a relatively uniform crustal thickness in the east-central Basin and Range Provinces indicates lower crustal flow is accommodating upper crustal brittle extension; and Jackson and White (1989) point out that flow in the lower crust may be necessary to balance crustal scale cross sections. Seismic reflection characteristics of the lower crust also appear to indicate ductile deformation (Percival and Berry, 1987). The nature of the brittle-ductile transition is also being investigated. Listric faults in the upper crust typically sole out in the lower crust and rarely offset the Moho (Percival and Berry, 1987). Scholz (1988) suggests that the deformation of mylonites and other experimental studies indicate a broad transition between zones of upper crustal brittle behavior and lower crustal bulk-plastic flow; he also points out that there may be a specific depth limit of earthquake nucleation but rupture can propagate across the transition zone. Interestingly, Hartse (1991) demonstrates that the bottom of the seismogenic zone near Socorro, New Mexico, is quite distinct at about 12 km, a depth compatible with temperatures where quartz begins to flow ductitely. Thus for the present study, the depth of bounding high-angle faults may be thought of as approximately coincident with the depth of the brittle-ductile transition, defining the initial base of the upper crustal trapezoidal blocks. In addition, a few studies have combined geodetic, earthquake, and vertical deformation data to indicate the planar nature of several high-angle faults in Montana and Idaho, USA, where large earthquakes have occurred (Stein and Barrientos, 1985; Barrientos et al., 1987).

74

M. Reiter / Tectonophysics 246 (1995) 71-95

In the present study the u p p e r crustal blocks are thought to move rigidly with little internal flexing. It is therefore necessary to estimate the approximate width that might satisfy this coherency condition. At least two rather different approaches yield approximately consistent estimates of possible trapezoidal width. As discussed above, Vening Meinesz (1950) and Heiskanen and Vening Meinesz (1958) proposed a graben forming model where an initial high angle fault allows the bending of the crustal plate; the location of the complimentary graben fault depends on the flexural rigidity of the crustal plate. Using the analysis given by Vening Meinesz and Heiskanen along with values of E (Young's modulus) = 70 GPa, u (Poisson's r a t i o ) = 0.25, and p = 2800 k g / m 3, graben widths of 24-41 km are predicted for elastic or brittle upper crustal thicknesses of 10-20 km. Alternatively, calculating the maxim u m shear strength required to support basement topography, Jackson and White (1989) estimate that blocks up to about 30 km width may rotate coherently without exceeding stresses greater than the 1-10 MPa stress drops observed in earthquakes. Their straightforward analysis implies that the width of rotating half-grabens depends on the strength and thickness of the brittle-upper crust. These two estimates are in approximate agreement with the 30-40 km wide spacing of block-faulted domains in the G r e a t Basin of the western US (Burchfiel et al., 1992); and therefore we shall pursue models with trapezoidal widths up to about 40 km. Models are examined in which the brittleductile transition either moves with the initial bottom boundary of the trapezoidal block or is stationary at a constant depth. If the brittleductile transition moves with the base of the u p p e r crustal block, then the vertical movement should occur quicker than the t e m p e r a t u r e change which would accompany a change in theology (from brittle to ductile or vise versa). If on the other hand the brittle-ductile transition remains at a constant depth, then sufficient temperature change has occurred for rheological change to take place in the region from the base of the trapezoid to the depth of the brittle-ductile transition. I shall attempt to relate the possible time

of temperature change to rates of uplift and subsidence; however, the uncertainties of thermal boundary conditions and the dependence of rheology on temperature will make such estimates ambiguous. Rates of uplift in the mountains along the Rio Grande rift and in the Southern Rocky Mountains are estimated to be on the order of 100 m per million years (Kelley et al., 1992); and maxim u m rates of deposition in the Albuquerque Basin are estimated to be almost 200 m per million years (Reiter et al., 1986). These rates are only a local average estimate of vertical movement and are dependent on the time interval being considered. Over time intervals of the order of a million years, deposition rates between several meters and several thousand meters have been estimated; if the time interval is shortened, orders of magnitude increase in the deposition rate are possible (Enos, 1991). We can compare these rates with a simple temperature diffusion model. Assuming an instantaneous temperature change at the trapezoidal base, and maintaining the new temperature, 97% of the temperature change will propagate about 250 m during 1 million years (using the one-dimensional analysis given in Turcott and Schubert (1982), and thermal diffusivity of 1 mmZs - l ) . If this temperature change corresponds to a rheology change (from brittle to ductile or vise versa) then vertical movement of the order of 100 m per million years (and less) probably corresponds to a relatively constant brittle-ductile transition; however, movement rates much greater than 100 m per million years may see appreciable change in the depth of the brittle-ductile transition. This is only an approximate rule of thumb and will be sensitive to thermal boundary conditions and other model parameters.

3. Models--description and results 3.1. Subsidence in extensional-stress regimes The first models to be presented are those concerning graben subsidence in extensionalstress regimes. Although basic models of graben

M. Reiter / Tectonophysics 246 (1995) 71-95

75

tional force will always oppose motion (negative for uplift, positive for subsidence, Eq. 1). We assume in our two-dimensional analysis that the subsiding or uplifting structures are much longer along strike than along width, and that the volumes relevant to Eq. (1) are one unit in length (horizontal-third dimension along strike). Therefore

subsidence have been analyzed by previous investigators, as discussed above, the present study expands on these analyses. The concepts developed for the subsiding graben will also be used for models of uplift. Fig. 1 shows the various forces acting on a simple subsiding trapezoidal block. For the trapezoid to drop the weight must exceed the sum of the buoyancy force, the forces transmitted from the adjacent crustal blocks, and the frictional force or resistance along the fault planes. The general form for the governing static equation which will be used for all models in the present study is

where p is density (crustal a n d / o r sediment) and g is gravity acceleration. For example, if there are no sediments and crustal density is constant then

weight - buoyancy force + (or - )

the weight of the trapezoid per unit strike length

the weight of the trapezoid per unit strike length (graben) = p g area

vertical component of forces

(2a)

=pcgl(W2 + W3)*(h -s)

transmitted bounding crustal

The weight of sediments (between A and B, Fig. 1) will be included to consider effects of sedimentation (Fig. 1). Discussion in the next paragraph will further explain Eq. (2b). From Fig. 1, it may be noticed that sediment and crustal rock densities are somewhat different; however, in Fig. 1 the brittle upper crust and the ductile lower crust are shown with the same density as well as different densities. This point de-

blocks = ( _+) vertical component of frictional force (or resistance)

(1)

Because of the fault angle, the force from the bounding crustal blocks will either oppose the weight of the trapezoidal block (for extensional subsidence and compressional uplift) or compliment the weight (for extensional uplift). The fric-

Surface

B

Sediments, Ps = 2650 kg/m3 ]

Brittle (Elastic) Upper Crust, Pc = 2750 kg/m3 Weight

Crustal Thickness

Ff,

(h)

FN,V

BrittleDuctile Transition Pc =

2750 kg/m3 (2950 kg/m3)

"Ductile" Lower Crust

(2b)

I'\

~ \. Buoyancy// \\,_ . .Force .. z/

FN

/

T Subsidence (S)

~Wl

Fig. 1. Diagram of subsiding trapezoidal block showing forces relevant for equilibrium equation.

M. Reiter / Tectonophysics 246 (1995) 71-95

76

DNDROP 1 (B-DT MOVES)

(a)

(e)

UPLIFT1 ( B-DT MOVES)

Pc = 2750 kg/m 3

J

Ducte g/m3

7_ transition

~)-= 2750 kg/m3 c (2950 kg/m~) I

W1 I

- ~- - - - "

),;

~ ,Bl~rtlalsidiuctile e ' Pc = 2750 kg/m 3 (2950 kg/m3)

(t3 (b)

wl

~'.........

; crust

......

I

UPLIFT 2 (B-DT STATIONARY)

DNDROP 1 (B- DT STATIONARY)

\

I \

!

Wl

(c)

(g)

DNDROP 2 (B-DT) MOVES 3 A /Sediments, ~)=2650kg/m

... ~ ~ ' ._

• . f

....

_

'

~

/ ....

_

s _

L_

w]

(d)

DNDROP 2 (B-DT STATIONARY) A []

!

w1

~

UPLIFT 3 (B-DT MOVES) ed~aterial _

i1"

~~

"~\ -- ~" - - - !

1

M. Reiter / Tectonophysics 246 (1995) 71-95 serves c o m m e n t . T h e u p p e r crust m a y b e slightly less d e n s e t h a n t h e l o w e r crust; this w o u l d b e c o n s i s t e n t with p r e v i o u s g r a b e n m o d e l s o f a l i g h t e r crust floating on a d e n s e r s u b s t r a t e (e.g. B a r r o l l a n d R e i t e r , 1987). If t h e lower crust is c o m p o s e d l a r g e l y o f g r a n u l i t e s (Percival a n d Berry, 1987) a n d t h e u p p e r crust l a r g e l y o f grano d i o r i t e s , t h e n it is p o s s i b l e t h e y have t h e s a m e densities; a l t h o u g h t h e r a n g e o f d e n s i t i e s for g r a n u l i t e e x t e n d s to g r e a t e r v a l u e s t h a n for grano d i o r i t e s (e.g. t h e d e n s i t y r a n g e o f g r a n o d i o r i t e s is 2 . 6 7 - 2 . 7 9 g / c m 3, a n d for g r a n u l i t e s is 2.67-3.1 g / c m 3 ; J o h n s o n a n d O l h o e f t , 1984). A l s o , n e a r S o c o r r o , N e w Mexico, t h e b r i t t l e - d u c t i l e transition at 12 k m d o e s n o t c o i n c i d e with t h e m i d crustal p wave velocity i n c r e a s e at ~ 20 km ( T o p p o z a d a a n d S a n f o r d , 1976; O l s e n et al., 1979; S a n f o r d et al., 1983; H a r t s e , 1991); implying a c o m p o s i t i o n a l c o n s i s t e n c y across t h e b r i t t l e d u c t i l e t r a n s i t i o n at this locale. I have t h e r e f o r e g e n e r a l l y c o n s i d e r e d m o d e l s with t h e s a m e upp e r - a n d l o w e r - c r u s t a l densities, which will r e s u l t in a m a x i m u m e s t i m a t e o f vertical m o v e m e n t for t h e crustal blocks. H o w e v e r , I will c o m p a r e t h e s e results with e s t i m a t e s o f crustal b l o c k m o v e m e n t b a s e d u p o n g r e a t e r l o w e r - c r u s t a l densities. If lower- a n d u p p e r - c r u s t a l d e n s i t i e s a r e the same, then the same value of weight minus buoyancy force will result for b o t h t h e s u b s i d e n c e m o d e l of a constant depth brittle-ductile transition, and for t h e m o d e l with a m o b i l e b r i t t l e - d u c t i l e transition in which t h e l o w e r g r a b e n i n d e n t s into t h e l o w e r crust (this h e l p s e x p l a i n Eq. 2b). A s such, for t h e case o f c o n s t a n t crustal density, total

77

s u b s i d e n c e s h o u l d b e i n d e p e n d e n t of t h e r a t e of subsidence because the movement of the brittled u c t i l e t r a n s i t i o n with t h e b a s e o f t h e g r a b e n is n o t critical. T h e e x p r e s s i o n u s e d for the b u o y a n c y force in Eq. (1) is t h e b u o y a n c y force p e r unit strike l e n g t h =pcgh(W3)

(3)

w h e r e h is t h e t h i c k n e s s o f the b r i t t l e - u p p e r crust a n d W 3 is the w i d t h o f t h e g r a b e n b a s e at a c o n s t a n t b r i t t l e - d u c t i l e t r a n s i t i o n d e p t h (Fig. 1). F o r c e s d e r i v e d f r o m r e g i o n a l stresses act on t h e g r a b e n to b a l a n c e the forces t h e g r a b e n exerts across t h e fault p l a n e (Fig. 1). B e c a u s e t h e coefficient o f friction a l o n g t h e fault is n o t known, t h e p r e s e n t a p p r o a c h will be to b a l a n c e t h e norm a l forces across t h e fault p l a n e . This a p p r o a c h has b e e n discussed by B a r r o l l a n d R e i t e r (1987), w h e r e t h e n o r m a l force across t h e fault was derived f r o m t h e h o r i z o n t a l stress acting in t h e crust. A m o r e g e n e r a l a p p r o a c h a p p l i c a b l e to b o t h high a n d low angle faults is t a k e n in t h e p r e s e n t study, w h e r e t h e n o r m a l c o m p o n e n t s o f b o t h t h e vertical a n d h o r i z o n t a l crustal forces acting on t h e fault p l a n e , d e r i v e d f r o m the vertical a n d h o r i z o n t a l crustal stresses, respectively, a r e a d d e d to e s t i m a t e t h e total n o r m a l force, F N (Fig. 1). T o c a l c u l a t e t h e crustal forces a c t i n g along t h e fault o n e m u s t e s t i m a t e crustal stresses. T h e vertical stress at d e p t h m a y b e a p p r o x i m a t e d as the w e i g h t o f t h e o v e r b u r d e n ( M c G a r r a n d Gay, 1978); which a p p e a r s to b e t h e m o s t r e a s o n a b l e

Fig. 2. Illustration of trapezoidal blocks undergoing subsidence and uplift in extensional tectonic environments. (a) Subsidence without sedimentation, trapezoidal base extends into ductile lower crust, i.e. brittle-ductile transition (B-DT) moves with base of graben (fast geologic movement). Model is DNDROP1. (b) Subsidence without sedimentation, trapezoidal base does not intrude ductile lower crust, i.e. brittle-ductile transition (B-DT) is at a constant depth (slow geologic movement). Model is also DNDROP1 because net vertical force does not change (if density of upper and lower crust equal). (c) Subsidence with sedimentation, trapezoidal base extends into lower crust, i.e. brittle-ductile transition (B-DT) moves with base of graben (fast geologic movement). Trapezoidal weight increased by deposition between A and B. Model is DNDROP2. (d) Subsidence with sedimentation, trapezoidal base does not intrude ductile lower crust, i.e. brittle-ductile transition (B-DT) is at a constant depth. Trapezoidal weight increased by deposition between A and B. Model is also ONOROP2(for subsidence, net vertical force does not change if density of upper and lower crust equal). (e) Uplift without erosion, trapezoid base and brittle-ductile transition (B-DT) move upward (rapid geologic movement). Model is UPLIFrl. (f) Uplift without erosion, trapezoid base and brittle-ductile transition (B-DT) stay at constant depth (slow geologic movement). Model is UPLIVT2. (g) Uplift with erosion as indicated, trapezoidal base and brittle-ductile transition (B-DT) move upward (rapid geologic movement). Model is UPLIFr3.

78

M. Reiter / Tectonophysics 246 (1995) 71-95

state of stress for all three coordinates in the absence of applied tectonic stresses (i.e. lithostatic stress where ¢rx = Cry = ~rz = pgh; McGarr, 1988). Although the horizontal stress field in the crust is not well known (Hanks and Rayleigh, 1980), data from the upper crust (1-5 km depth) suggest that the horizontal stress appears to vary with depth as a fraction of overburden, f (McG a r r and Gay, 1978). Therefore, the vertical stress orv, and the horizontal stress orH, acting at depth z, may be written as crv = p g z

(4a)

and O"H

=fpgz

(4b)

Deviatoric or tectonic stress is the difference between the overburden (or lithostatic stress) and either the minimum horizontal stress (when f < 1.0) or the maximum horizontal stress (when f > 1.0). If f = 1.0 the stress field is lithostatic, f < 1.0 implies deviatoric tension exists, f > 1.0 implies the deviatoric stress is compressional. We might also call f the tectonic stress ratio. Of course, we are presently neglecting the out-of-plane component. The resulting crustal forces which I suggest also act along the fault (Fig. 1) are obtained by integrating the vertical and horizontal crustal stresses over the horizontal projection of the fault, and the depth of the fault, respectively. The crustal forces per unit strike length along the fault will be ( F v and FH, Fig. 1) F v (vertical crustal force) = 1 / 2 p g ( Z 2 - Z 2) tan d~

(5a)

and F H (horizontal crustal force) = 1 / 2 pgf(Z22 - Z 2)

(5b)

where Z 2 and Z l define the appropriate depth interval along the fault. For the case of no sediment, the location of Z = 0 is at the top of the graben surface and Z 1 = 0 , Z 2 = h - s . If sediments fill the depression, Z = 0 is at the original ground elevation; and because I choose the sediments to be unable to support the normal force, Z 1 = s and Z 2 = h (this point is mentioned below

when sedimentation is considered). Taking the fault normal components of Eqs. (5a) and (5b), and summing, yields the normal force acting across the fault plane per unit strike length ( F N = Fv, N + FH,N, Fig. 1), which is F N = 1 / 2 p g ( Z 2 - z Z ) [ t a n ¢f * sin ¢h + f c o s of] (6a) In equilibrium this force balances the normal component of forces acting along the fault plane from the graben. The vertical component of the normal force per unit strike length acting on the graben from the bounding crustal blocks (FN.v, Fig. 1) will tend to support the graben and is given by FN, v = FN(sin of)

(6b)

Lastly, the frictional force or resistance to shear along the fault per unit strike length (Ff) is given by (Halliday and Resnick, 1963) Ff =/x t * FN

(7)

where /xf is the coefficient of fault friction, not the coefficient of rock friction which is rather large in laboratory experiments. Note that /zf is an average coefficient of friction along the entire fault surface, possibly both a temporal and spacial average. If shears transferred across fault surfaces can be equated to shears transferred across rock surfaces in laboratory experiments, then one might relate rock and fault friction by /3.f = / J , R ( F N - P ) / F N , where /~R and P are, respectively, the coefficient of rock friction and pore pressure in laboratory experiments. Byerlee (1990) proposes a thin film model for fault gouge which may explain the existence of high pore pressures along fault zones and therefore why many faults weakly transmit shear (i.e. why faults have a relatively low coefficient of friction). The vertical component of the frictional force per unit strike length, Ff,v, will oppose vertical motion, and is given by Ff,v = Ff(cos oh)

(8)

By substituting Eqs. (2) through (8) appropriately into Eq. (1) one can solve for f ( = ~rH/cr v) in terms of brittle-upper crustal thickness (h), graben base width (W1), subsidence (s), the angle of the fault from the vertical (oh), and the effec-

~'L Reiter / Tectonophysics 246 (1995) 71-95

tive coefficient of fault friction (J./,f). For example, in the model of subsidence without sedimentation and with constant crustal density,

-s)

f = { 1 / 2 ( W 2 + W3)(h

W3h

-

- (Z~ - Z?)sin ~b tan ~b (sin d~ + / , f c o s d~)} ×

{(Z22 - ZZ)cos ~b(sin~b + / x f c o s ~b)} -~

(9)

In Eq. (9) parameters W2 and W3 are dependent on W1 (Fig. 1). We will compare graben subsidence with and without sedimentation (e.g. models shown in Figs. 2a and 2c or Figs. 2b and 2d, respectively). For the case of sedimentation we must consider the sediment weight and adjust the depth coordinate as mentioned above. When upper- and lower-

(a) D N D R O P 1 ( N O S E D I M E N T A T I O N ) DIFFERENT CRUSTAL THICKNESS AND FAULT FRICTION 1.0

1.0 ",,. \ ~ Q8

Subsidence = 2 0 0 0 m e = 25 ° 10 km brittle 15 km crustal . . . . 20 km th ckness

~'~.

0.9

~O

~OOO

~,~. /~, ", "O;o~

0.6

\

0.5

( b ) D N D R O P 1 (NO S E D I M E N T A T I O N ) DIFFERENT FAULT DIPS AND FAULT FRICTION Subsidence = 2000m

\ 0 9

--

0.8|\

0q \

0.7

f

79

06

\'-

\ \

~

brittle crustal thickness

15 km

\\

.

\

\\

V"

\\,

,'oo

0.4 0.3

0.3

\\

0.2 0.1 0

\ \\

0 (h = 10krn) 10 20 30 0 (h = 15~m) 10 20 0 (h = 20kin) 10

~:~k

02 Ol 40 30 20

• 40 30

o

o

~\\ "~k\

..,o \

\

\\

00 ((p = I10°) 10

• • 40

\

210

\

310

o(~ =2s°l to

40

ao

4o

20

30

(d) D N D R O P 2 (WITH S E D I M E N T A T I O N ) DIFFERENT FAULT DIPS AND FAULT FRICTION .

0.8

0.8

f

~

=~--~ . - - - =:-~: 25

ii

0.7

Ff = 0.20 .....

0.6

0.6

f

laf = 0.10

f 0.5

~ ¢,= 10 °

-

e =25°

[ 40 30 20

I

0.5

0.4

0.1

40

(c) DNDROP 2 (WITH SEDIMENTATION) DIFFERENT CRUSTAL TH ICKNESS AND FAULT FRICTION 0.9

0.2

•

2o

0 (~ = 40 °) 10 W 1 (km)

09

0.3

\

W 1 (km)

1.0

0.7

\

0.4 DNDROP 2 Subsidence = 2000m ¢, = 25 ° 10 km I brittle 15 km crustal . . . . 20 km thickness

0.3 0.2

DNDROP 2 Subsidence = 2000m

~

~

15 km

brittle crustal thickness

0.1

I I I I I I (n = 10krn) 10 20 30 0 (h = 15kin) 10 20 0 (h = 20km) 10

W 1 (km)

I

I 40 30 20

I

I • 40 30

I

0 • • 40

I I (q~ = 10 °) 10

F

I 20

0 (~, = 25°) t 0

I

E 30

20

0 (~ = 40 °) 10

I

I • 40 30

I

• • 40

W 1 (kin)

Fig. 3. Plots of tectonic stress ratio f(o'H/O" v) versus trapezoidal base width (W1) for 2000 m of subsidence; different brittle crustal thickness, fault friction, and fault angle are considered. Models DNDROP1 and DNOROP2 (Figs. 2b and 2d) are represented.

M. Reiter / Tectonophysics 246 (1995) 71-95

80

crustal densities are the same, models with rapid subsidence (displacement of the brittle-ductile transition, e.g. Figs. 2a and 2c) will give the same results as m o d e l s with slower subsidence (brittle-ductile transition maintains constant depth, e.g. Figs. 2b and 2d). Similar models for (a)

uplift in extensional regimes are shown in Figs. 2e and 2f; we pursue these uplift models a bit later in the study. Figs. 3 and 4 allow one to consider the significance of various potentially important geologic parameters on graben subsidence in extensional(b) D N D R O P 2 (WITH S E D I M E N T A T I O N ) D I F F E R E N T U P P E R C R U S T A L T H I C K N E S S ,FAULT DIP, A N D FAULT FRICTION

D N D R O P 1 (NO S E D I M E N T A T I O N ) D I F F E R E N T U P P E R C R U S T A L T H I C K N E S S A N D FAULT DIP

1.0

1

.o]~

0.9

09

0.8

04-

~

~

I~ =

\ \

t

\

(~.10 ~ &

~,40 o

o,L

0.7

0,00

~.....~,,

~=

-'. X%X

XX

\\

.

o,X, oo

40 °

,,

06 f 0.5 0.4 ] U f

5°

II1~=

: 0.20

25o ~ = 40o ' | i |

V.o=,o

0.2

'"

o

DNDROP I Wl = 1000m I~ = 0 0 0

0.1

0

0

t0km I brittle ° ~ 15kin { crustal -- -- - 20kin thickness

(h = 10km) 4 • 0 (h = 15km) 4 0 h = 20km) Subs dence km

•

4

8

I 2

•

(C) D N D R O P 2 (WITH S E D I M E N T A T I O N ) D I F F E R E N T UPPER CRUSTALTHICKNESS AND FAULT FRICTION 1.C

I 4

I I t 6 8 10 Subsidence (kin)

~

I

\

°°b

0e

-

-

-..-.

I 12

114

61 1

(d) D N D R O P 2 (WITH S E D I M E N T A T I O N ) D I F F E R E N T U P P E R C R U S T A L T H I C K N E S S A N D FAULT F R I C T I O N 1 0 ~ ~

09

O~

\

DNDROP 2 \ W1=10,000 10krn ) brittle - ~ 15kin I crustal -- -- - 20kin thickness

0

•

•

~

XI

!

¢ = 40° I

\ .

- ~

0

0.3

~ "~ ~

\

16

,,

X,o

"-.

02 0(~

f 0.5

04 03 DNDROP 2 @ = 25 ° - 1000m

02

02

Wl 0.1

10kin - 15km -- -- - 20kin •

2

1 4

I brittle

crustal thickness I I I 6 8 10 Subsidence (km)

I 12

I 14

I 16

0L.. 0

DNDROP 2 e = 25° Wl = 40,000m I 2

I 4

~

10km

brittle

] I I 6 8 10 Subsidence (km)

I 12

114

I 16

Fig. 4. Plots of tectonic stress ratio f(o'H/O-v) versus subsidence for several different trapezoidal base widths; different brittle crustal thickness, graben base width, fault friction, and fault angle are considered. Models DNDP.OP1 and DNDROP2 (Figs. 2b and 2d) are represented.

81

M. Reiter / Tectonophysics 246 (1995) 71-95 (f) DNDROP 2 (WITH SEDIMENTATION) INCREASED LOWER CRUSTAL DENSITY. DIFFERENT UPPER CRUSTAL THICKNESS AND FAULT FRICTION

(e) DNDROP 2 (WITH SEDIMENTATION) INCREASED LOWER CRUSTAL DENSITY, DIFFERENT UPPER CRUSTAL THICKNESS, AND FAULT FRICTION 1.0

1.0

019

~"~ 0 9 [" ~ ' ~ .

\\

0.8

"',,,,

0.8 ~

0.4

°" o.,

0.3

0.3

°° 0

~

DNDROP 2 q'= 250 Wt = 1,000m

~

~

1'o 1'~

"

02 X

10kin I brittle - ~ 15km crustal -- -- • 20kin thickness

;

"'',,

\\ ",,,.

o;i°\

06 f 0.5

01

"',

0.7 ~

0.7

0.2

~= 0.00

0.1 114

i

16

DNDROP 2 ~ = 25° Wl = 40.000m

0,0

Subsidence (kin)

%%

\ 91

4

,,

10kin brittle - ~ 15km I crustal -- -- • 20kin I thickness ;

8

t'O 1'2 (kin)

1'4

116

Subsidence

Fig. 4 (continued). stress regimes. From these figures one may note: (1) for any crustal thickness, subsidence of a given amount requires more deviatoric tension (lower f ) as the graben base width (W1) increases (Figs. 3a-3d); (2) given a set of graben parameters, greater subsidence requires larger deviatoric tension (lower f , Figs. 4a-4d); (3) for a given subsidence, as fault friction (/zf) increases the deviatoric tension required for subsidence increases ( f must decrease, Figs. 3a and 3b); (4) crustal blocks with a thicker elastic crust require less deviatoric tension than those with a thinner elastic crust to subside equal amounts (Figs. 3a, 3b, and 4a); (5) frictionless graben subsidence does not seem to depend greatly on fault angle 05 but does depend on brittle crustal thickness (Fig. 4a); (6) if the fault angle 05 is 10° equilibrium becomes somewhat more sensitive to friction than if the fault angle is 25 ° or 40 ° (Figs. 3b, 3d, 4a, and 4b). Some of these results have been previously obtained using different models (Bott and Mithen, 1983; Barroll and Reiter, 1987). We consider briefly the physical reasons for the observations in the last paragraph in the order given. Narrow grabens subside more easily (or alternatively wider grabens require greater deviatoric tension to subside) because the weight to buoyancy force ratio is greater for narrow

grabens than for wide grabens. This occurs because the weight of material outboard of verticals at the corners on the base of the trapezoid makes up much more of the total weight when the graben is narrow than when the graben is wide. Greater subsidence requires larger deviatoric tension because of increased buoyancy forces. Subsidence with increased fault friction requires greater deviatoric tension to overcome the increased resistance to motion along the fault. Thicker upper crustal blocks subside more easily than thinner upper crustal blocks because again the weight to buoyancy ratio is greater for thicker blocks. The greater weight to buoyancy ratio for a thicker brittle crust is also more important than fault angle when faults are considered frictionless. Lastly, vertical movement is most sensitive to friction along steeply dipping faults (05 = 10 ° ) because the frictional resistance along steep faults is directed more effectively to resist vertical movement (see also Eq. 8). To consider possibilities for maximum graben subsidence one must attempt to estimate horizontal stresses in the brittle crust. From measurements of minimum horizontal stresses in basins of the United States, data show a depth gradient for minimum horizontal stress which is ~ 0.62 times the overburden gradient above 2.3 km depth, and

82

M. Reiter / Tectonophysics 246 (1995) 71-95

~ 0.76 times the overburden gradient from 2.3 km to 5.1 km depth (McGarr and Gay, 1978). Values of f = 0 . 5 are noted in granite for US basins and f values as low as 0.3 are noted for tests less than 2 km deep in South Africa (McG a r r and Gay, 1978). D e e p e r data are surely desirable to better characterize stress conditions in the brittle crust. However, with presently available data 1 suggest that values of f ~ 0.6 are as small as might typically be expected (lower values of f can also be considered in Figs. 3 and 4). With a minimum value of f ~ 0 . 6 , one may notice that subsidence is typically unlikely unless the coefficient of fault friction is ~ 0.1-0.3 (Figs. 3 a - 3 d and 4b-4d). One may also note that without sedimentation, subsidence of only a few kilometers (maximum ~ 5 km) may be expected (Fig. 4a, DNDROP1, no friction). To consider the subsidence of deep grabens we must therefore account for deposition as illustrated in Figs. 2c or 2d. The general observations made several paragraphs above for parameters affecting graben subsidence without sedimentation apply for graben subsidence with sedimentation (Figs. 3c, 3d, and 4 b 4d, DNDROP2). Some of the important model conditions for the grabens with deposition are: sediments outboard of A and B (Fig. 1, Fig. 2c and 2d) are supported by neighboring blocks; the sediments are easily compressible and therefore do not transfer crustal stresses which would support the graben and generate friction; the void is 100% filled with sediments; and the sediment density is 2650 k g / m 3. All of these conditions are arguable and each could vary to modify results. For example, including sediment weight outboard of A and B would make graben subsidence easier; sediment transfer of crustal stresses would help support the graben and make subsidence more difficult; if the void is filled at less than 100%, the graben weight decreases and subsidence is made more difficult; and variation of sediment density will have an effect on the weight of the graben and therefore the amount of subsidence. Using the models as presented in the last paragraph we may consider the conditions necessary for graben subsidence of more than a few kilometer. Figs. 4 b - 4 d (DNDROP2) illustrate the potential for large amounts of subsidence given

several different graben base widths (W1). As expected, considerably more subsidence occurs when graben valleys are filled with sediments. Thus there is a feedback mechanism where deep subsidence depends on the availability of sediments, and deposition depends on the availability of an initial graben. From Figs. 4b-4d, we note that optimum conditions for deepest subsidence are: a low coefficient of fault friction (~f ~ 0.0), a thick brittle crust ( ~ 20 km), a small base width, and a high deviatoric tension ( f ~ 0.6). Graben models with different fault dips generally show about the same characteristics (Fig. 4b). I have also analyzed subsidence models when the lower crust is denser than the upper crust (Figs. 4e and 4f, 2950 vs. 2750 kg/m3). The statements made above concerning subsidence for models with equal upper- and lower-crustal densities generally apply to the models with greater lower-crustal densities. The main difference is that somewhat less subsidence is possible for models with greater lower-crustal density because of the increased buoyancy force. With or without sedimentation, models with a denser lower crust show about 1, 2, or 3 km less subsidence than models with a constant crustal density, for brittle crustal thicknesses of 10, 15, and 20 kin, respectively (graben base width is 1000 m, and the fault angle 4) is 25°; compare Figs. 4c and 4e for models with sedimentation). If the graben base width is quite large, then the effect of a higher density lower crust on subsidence becomes more important. For example, if the graben base width is 40 km, then about 3, 4, and 5 km less subsidence are noted for models with a denser lower crust vs. constant density crustal models (brittle crust is 10, 15, and 20 km thick, respectively, fault angle d) is 25 °, models include sedimentation; compare Figs. 4d and 4f). The subsidence of wider grabens is more sensitive to a denser lower crust than is the subsidence of narrower grabens because of a proportionally increased buoyancy force for wider grabens.

3.2. Uplift in extensional stress regimes The models of uplift presented in this section are shown in Figs. 2e-2g. The analysis for uplift

M. Reiter / Tectonophysics 246 (1995) 71-95

is similar to the analysis for subsidence; however, Eq. (1) must be rearranged to account for the differences in direction of movement and orientation of the trapezoidal block. In equilibrium the buoyancy force must equal the weight of the trapezoid, plus the crustal forces of the bounding blocks (which act to restrain movement in this case), plus the vertical component of the friction

1.0

X

"~'..

\

X

0.8 \

0.7 f

force. Rapid uplift (as discussed earlier) may be approximated with the base of the trapezoid moving upward, allowing lower crustal material to fill the void and remain ductile during the time of uplift (Fig. 2e). Slow uplift may be approximated by assuming that the viscous material which moves under the trapezoid cools and becomes brittle (Fig. 2f). Unlike the subsidence models, the net

(a) UPLIFT 1 (RAPID UPLIFT) DIFFERENT BRITTLE CRUSTAL THICKNESS AND FAULT FRICTION

0.9

"~.

".

~, \ ~(3 ,~k~

1.0

UPLIFT 1 Uplift = 2000m ¢=25° ,, ~10km I brittle ~, •--15km crustal ",~ .~ 20 krn I thickness

0.7

0.5

0.4

0.4

0.3

0.3

\\

0.1

,

0 0

, \\

10

, 20

,

,

30 W 1 (kin)

,f-~, e = 10° \

0.9

\

08

\ V

o.~

J

\I ~ - 0.00 ,.-"." ~ ~ -~

,

40

,

50

60

,

0 0

\

X

~10km =2=°

0.9

}brittle

%% - - - - 1 5 k i n }crustal " " ~-~'~- - 20 km ) thickness

.0.5

,

, \,

0.8

o.,

, 20

X\

~=

,

,

30 W 1 (km)

40°'~ x

,\\

,. 60

,

40

50

U P L I F T 3 (RAPID UPLIFT AND EROSION) DIFFERENT BRITTLE CRUSTAL THICKNESS AND FAULT FRICTION

~

~

~

"

"'" . ~:,~o ""-......

~

""

...\\\

"

"-..

o.~

0.5

,,

0.2

,,o\

\

-'o\

"~O~

0.1 0

thickness

\\ \\. \\ \% "\\

10

1.0

Uplift: 2000m

.\ -. \ "'-. ~ ..~\~ .~\.. ..........

0.6

15 km .... tal

--

0.1

\,\\,

"~'~.

f \ ~=25 ° ~'

\k

(d)

~

UP~iPL=IF2T10britt,e N -- ~

0.2

"-

(c) UPLIFT 2 (SLOW UPLIFT) DIFFERENT BRITTLE CRUSTAL THICKNESS AND FAULT FRICTION 1.0 .. UPLIFT 2 \

0 C ~ X'

0.6

0.5

° \

\ ~ # f ~

0.8

f

~\\

(b) UPLIFT 1 (RAPID UPLIFT) DIFFERENT FAULT ANGLE AND FAULT FRICTION

09

0.6

0.2

83

I

,

10

,

,

20

"~

, , 30 W 1 (km)

\ X

x

I 40

'

02 X %XI~ 50

60

Uplift = 2000m ,=~5:

--

O.~o,,..

--

\

"eo& \ \\

10kin'brittle X \ 0.1 -- ~ 1 5 k m }crustal ~ \ .... 20km)thickness ~ \ 0 I I I I I ~1 ~. I 0 10 20 30 W 1 (kin)

I 40

I

I 50

I

60

Fig. 5. Plots of tectonic stress ratio f(~'H/O'~) versus trapezoidal base width (W1) for 2000 m of uplift; different brittle crustal thickness, fault friction, and fault angle are considered. Models UPLIFT1, UPLIFr2, and UPUFT3 (Figs. l e - l g ) represented.

84

M. Reiter / Tectonophysics 246 (1995) 71-95

buoyancy force minus weight is not the same in the two uplift cases because the buoyancy forces along the sloping faults below the horst base in the rapid uplift model are directed toward the blocks bounding the horst (compare Figs. 2a and 2e). The net buoyancy force minus weight is

somewhat greater for the slow uplift model (Fig. 2f). In addition, the crustal and frictional forces derived from the bounding blocks will be greater for the slow uplift model because these forces will act over a greater fault length than in the case of rapid uplift (Figs. 2e and 2f).

(b)

(a)UPLIFT 1 (RAPID UPLIFT) DIFFERENT ELASTIC CRUSTAL THICKNESS AND FAULT ANGLE

~~&,

1.o

0.9

\%

.~,~. kkx ~,\\ \\""

0.8

\~ ~ 0.7 / / /

1.0

\

TopUfP ~loFrTt11 km ~ = 0.00 - 10 km ) brittle

~x, %,,

%~%

----15km

~,

------

UPLIFT 3 (RAPID UPLIFT-WITH EROSION) DIFFERENT ELASTIC CRUSTAL TH ICKNESS AND FAULT FRICTION

0.9 \

", #f~-- 0.00

UPLIFT 3 T~P=C2f5horst:1 km

"~,

0.8

crustal

20 krn ' thickness

0.7 0.6

o., ,ollT o.,,

too,

-\~o°

f

too'43

40 °

0.5

40°

/-"\

-T'-..

',

0.4 0.3

00"3 .2

f

0.2 0.1

O.t

0

~

0(h = 10kin) 4 • • 0 (h = 15 km) 4 • 0 (h = 20 kin) 4

UPLIFT (kin)

(C)

0

0(h = 10 kin) 4 • • 0(h= 15km) 4 • 0(h = 20 kin) 4

• • 8

UPLIFT (km)

U P L I F T 3 (RAPID UPLIFT- WITH EROSION) DIFFERENT BRIFILE CRUSTAL THICKNESS DIFFERENT FAULT ANGLE AND FAULT FRICTION

t.°l~---

(d)

~,,,

0 9 L~\ ' I~%

%~\ ~',

II~

\~

0 8 1 - 1 \ 3~ ' I-I ~ I 1% ~-

I I I O

"%",,

% x ~ '

~ \

% ,% %'~ %~ %

%0%

:~%

0'7t- | I ~

0.9

10km brittle -- --15km crustal .... 20 kmlthickness

0.8

U P L I F T 3 (RAPID UPLIFT- WITH EROSION) DIFFERENT BRII-FLE CRUSTAL THICKNESS AND FAULT FRICTION

!/ooo

-

-

%% I I pf - 0.30 /

0.5

',';

o4-11~l I lll l I llll

~ ~\ / ~ =' ' |~ . = o.I( ~ , , . ~ - o . 2 o

'

°-%/I \l I

U WO.20

~ll t, ~ 4 0 °

o.21-=1111,=4 o, ~ l \ ~ Io~o~ll '~ = ' °

o lF ! II

~

to}l/

,

I

'\

,

\

0.4

~ ~ I % I% l

~

', ",4-, %t I

~=25 ° 10 km brittle 15 km crustal .... 20 km th ckness

0.6 f

} ;

UPLIFT 3 Top of horst = 40km

%%

017

~

I\l\l

%%

%

%~%

.'L\._

0.6~ - - ~

1.0

UPLIFT 3 Topofhorst=10km

o. o

I

'~', = ~ , ~

0.2

I

0(h = 10 km} 4 • • 0(h= 15km) 4 • 0 (h = 20 kin) 4 UPLIFT (km)

• • 8

%%

\

01 I

% %

~.% I% % I %% I. %

\l \

0.3

=4°°

,,,,,,,

• • 8

I

%

% I

I

I

I

00(h= 10 krn) 4 • • 0(h = 15 km) 4 • 0(h = 20 krn) 4 UPLIFT (kin)

I

I

I

I

Fig. 6. Plots of tectonic stress ratio f(o'H/O"v) versus subsidence; different horst width, brittle crustal depth, fault friction, and fault angle are considered. Models UPLIFT1, UPUFT2, and UPLIFT3 represented (Figs. le-lg).

85

M. Reiter / Tectonophysics 246 (1995) 71-95

(f) UPLIFT 2 (SLOW UPLIFT-INCREASEDLOWER CRUSTAL DENSITY) DIFFERENTBRITTLE CRUSTAL THICKNESS, DIFFERENTFAULT FRICTION

(e) UPLIFT 2 (SLOW UPLIFT) DIFFERENT BRI'I-I-LE CRUSTALTHICKNESS, DIFFERENT FAULT FRICTION 1.0

~.~

o

\

1.0

o.oo\\\

= . .7

",,, \\

f

o.oo • \\-,,, \ \ ",, \\',,

\'-,, "',,

•°

p~ = 0.30

f

\,

",,,

,,

tf = 0.30

~5

.4 ~

~

~.

.3

"it 0

.4

.3

UPLI FT 2 Top of horst = 1 km ~, = 25 ° 10 km I brittle -15 km crustal 20 km thickness

I

I

I

4

8

12

UPLIFT 2 Top of horst = l k m = 25 °

.2

.1

~

0

UPLIFT (km)

I 4

I 8

10 km I brittle 15 km crustal 20 km thickness I 12

UPLIFT (km)

Fig. 6 (continued).

We now examine the dependence of horst uplift in an extensional tectonic-stress regime on various parameters such as crustal thickness, horst base width (W1), coefficient of fault friction (/z f), horizontal to vertical stress ratio ( f ) , fault angle (~b), and slow or rapid uplift. From Figs. 5 and 6 we may observe that many of the relations noted for the subsidence models are also valid for the uplift models: (1) for any crustal thickness, uplift of a given amount requires more deviatoric tension (lower f ) as the horst-base width (W1) increases (Figs. 5a-5d); (2) given a horst width and a coefficient of fault friction (/./,f), greater uplift requires larger deviatoric tension (Figs. 6a-6d); (3) for a given uplift, deviatoric tension must increase as fault friction increases (Figs. 5a-5d); (4) for a given uplift and fault angle, somewhat less deviatoric tension is required as the brittle crust thickens (e.g. Figs. 5a, 5c and 6b); (5) for a given uplift and brittle crustal thickness, frictionless uplift requires less tectonic tension for shallow vs. steep faults (i.e. ~ = 40 ° vs. ~ = 10°); (6) the models with steep faults (~h = 10°) are more

sensitive to friction (Figs. 5b and 6c). The physical reasons for these observations are much like those reasons given for graben subsidence except when uplift becomes easier, the orientation of the trapezoid has allowed the buoyancy force to weight ratio to be preferentially enhanced; i.e. the buoyancy force becomes greater with respect to trapezoidal weight. Uplift also appears easier for the slow vs. rapid uplift models (Figs. 5a and 5c; Figs. 6a and 6e; Figs. 2e and 2f). This results because of the increase in buoyancy force minus weight for the slow uplift model, even though the stress acting from the bounding crustal blocks acts over a greater fault length in the slow uplift model (Fig. 2f). Several kilometers more uplift appear possible for slow vs. fast uplift if the top of the trapezoid is 1 km wide (Figs. 6a and 6e). The parameter dependencies discussed in the above paragraph for models without erosion are also generally true for the erosional uplift model as well (Fig. 2g). One may note that 2000 m of uplift is easier for the erosion model as the base width (W1) increases (compare Figs. 5a, 5c, and

86

M. Reiter / Tectonophysics 246 (1995) 71-95

5d). This occurs because of the increased amount of material eroded as the horst becomes wider. Somewhat more uplift is in general possible if erosion occurs; for example, compare Figs. 6a and 6b (fault angle 4~ = 25 ° and fault friction / z f = 0.00), which shows a few hundred meters more uplift is predicted for the erosion model when f --- 0.6. An increase in lower crustal density also has some effect on uplift (i.e. from 2750 to 2950 kg/m3). One approach toward appreciating the consequences of different crustal densities is to consider the model with a constant brittle-ductile transition, and assign an increased density to the material between the bottom of the original horst and the brittle-ductile transition (Fig. 2e). This straightforward approach will reduce the amount of uplift because the weight of the horst becomes greater. The most significant change in uplift is observed for narrow horsts; e.g. 1 km wide at the top, where the uplift is about 1 km less for the model with increased lower crustal density than for the model with constant crustal density (Figs. 6e and 6f). This difference in uplift becomes very small (less than 100 m) when the upper surface of the horst is 40 km wide, because the small amount of uplift under the wide horst does not permit as great a percentage weight increase as for the narrow horst (consider Figs. 6b and 6d). The difference in uplift due to different lower crustal densities is much less than the uplift difference due to movement of the brittle-ductile transition discussed a few paragraphs earlier (slow vs. fast movement, Figs. 6a and 6e). It is instructive to compare the results of the uplift and subsidence models in order to appreciate the possible relative movements in extensional tectonic regimes. Consider first possible movement for models without erosion or deposition. From Figs. 3a and 3b and 5a and 5b it appears that uplift of 2000 m is about as easy as subsidence of 2000 m (i.e. requires about the same deviatoric stress for equivalent trapezoids). Figs. 4a a n d 6a show that for equivalent trapezoids, and fault friction = 0.00, uplift can exceed subsidence by ~ 300-1000 m in the case of no erosion or deposition. In the uplift model the buoyancy force less trapezoidal weight is some-

what greater than is the weight less buoyancy force in the subsidence model. Next, consider the relative amounts of vertical movement likely to occur during uplift and subsidence when erosion and deposition take place. In general considerably more subsidence than uplift is possible, for example, if the base of the graben and the top of the horst are 1000 m wide, then > 16 km of subsidence appears possible but only 7-10 km of uplift (Figs. 4c, 6b and 6e, fault angle q5 is 25 °, brittle crustal thickness is 20 kin). The deviatoric tension needed for subsidence and uplift is shown for various fault angles, brittle crustal thicknesses, and fault friction coefficients in Figs. 4b and 6c. Both uplift and subsidence appear very sensitive to the coefficient of fault friction and the brittle crustal thickness, and uplift appears to be somewhat more sensitive to fault angle than subsidence (Figs. 4b and 6c). Also note that as the width of the graben base, or horst top, increases to 40 km, subsidence of ~ 14 km may still be possible if optimum conditions are present; however, uplift of only ~ 2 km is possible (Figs. 4d and 6d). The general reason for the much greater subsidence possible during deposition as opposed to uplift possible during erosion is that so much more material can fill the void created by trapezoidal subsidence than can be eroded during trapezoidal uplift.

3.3. Upward mouement in compressional stress regimes The last models to be discussed are shown in Figs. 7a-7c. These simple illustrations show uplift along high-angle faults in compressional environments. In the first model rapid uplift occurs (much greater than 100-200 m / m y ) and the brittleductile transition moves upward with the base of the trapezoid (Fig. 7a). The second model assumes uplift is slow (less than 100-200 m / m y ) so that the depth of the brittle-ductile transition remains constant (Fig. 7b). The third model assumes both rapid uplift and erosion, as shown in Fig. 7c. The second model will differ from the first because the fault lengths differ, as does the net buoyancy less weight. In equilibrium, the weight of the trapezoid plus the vertical compo-

M. Reiter / Tectonophysics 246 (1995) 71-95

nent of frictional resistance equals the buoyancy force plus the vertical component of the normal force from the bounding blocks.

(A) UPPUSH 1 (B-DT MOVES)

\_vv_,_

Pc

= 2750kg/m3

Brittleductile transition Pc= 2750kg/m3 (2950kg/m3)

(B) UPPUSH 2 (B-DT STATIONARY)

Brittleductile transition (C) UPPUSH 3 (B-DT MOVES)

~--

W2

-~

~[-1/4 W2 ~-

Eroded material

Fig. 7. Illustration of trapezoidal blocks undergoing upward movement in a compressional tectonic environment. (a) Uplift without erosion, trapezoidal base moves upward with brittleductile transition (B-DT; rapid geologic movement). Model is uPPusH1. (b) Uplift without erosion, trapezoidal base stays at constant depth with brittle-ductile transition (B-DT, slow geologic movement). Model is uePUSn2. (c) Uplift with erosion, trapezoidal base moves upward with brittle-ductile transition (B-DT; rapid geologic movement). Model is uPeusH3.

87

One may again observe the relations between upward movement (veetJsr0 and a number of geologic parameters (Figs. 8 and 9, f increases with compression). Relations which hold true for the three models are: (1) for a given crustal thickness, uplift of a given amount requires more compression (greater f ) as the horst-base width (W1) increases (Figs. 8a-8d); (2) given a horst width and a coefficient of fault friction (/zf), greater uplift requires greater compression (Figs. 9a-9d); (3) for a given uplift and base width compression must increase if fault friction increases (Figs. 8a-8d); (4) for a given uplift and fault angle, somewhat less compression is required as the elastic crust becomes thicker (Figs. 8a-8c and 9a); (5) for a given uplift and elastic crustal thickness, it requires less compression to uplift trapezoids with fault angles of 25° or 40 ° than with fault angles of 10 ° (Figs. 8b and 9a-9d). The physical reasons for these observations are in some instances a bit different from similar observations in the extensional environment. With increasing horst base width, increased compression becomes necessary for a given uplift so that the bounding brittle crustal blocks can effectively support the wider horst. Greater uplift requires more compression because of reduced buoyancy forces and reduced crustal support from bounding blocks (Fig. 7a). Increased fault friction requires more driving force from compression to overcome frictional resistance to motion. As the brittle crust thickens, uplift requires less tectonic stress because more support along increased fault lengths can be acquired from the bounding blocks; this is also the case as the fault angle becomes shallower. For two thousand meters of uplift, with a given crustal thickness and a given graben base width, model veetJsH3 requires slightly less tectonic stress than ueeusH2, which in turn requires slightly less tectonic stress than ueetJsH1 (Figs. 7a-7c and 8a-8d). For slow uplift (tJeetJsH2) the weight of the trapezoid is supported over a greater fault length by the bounding blocks; for the erosional model (uPeUSH3),the weight of the trapezoid is reduced. The relation between rapid vs. slow uplift (or a moving vs. stationary brittle-ductile transition, Figs. 7a and 7b) can be further appreciated by

88

M. Reiter / Tectonophysics 246 (1995) 71-95

comparing uplift calculations presented in Figs. 9a and 9e for a trapezoidal base width of 10 km. As mentioned above, the slow uplift model provides an increased fault length for adjacent blocks to transfer uplifting tectonic compression, so that

uplift is easier and more uplift is possible when the brittle-ductile transition remains at a given (deeper) depth. Note that uplift becomes unlikely in any case if the coefficient of fault friction is greater than 0.15-0.20 (see Figs. 9b and 9e). At

(a)UPPUSH 1 (RAPID UPPUSH) DIFFERENT BRITTLE CRUSTAL THICKNESS AND FAULT FRICTION

(b)UPPUSH 1 (RAPID UPPUSH) DIFFERENT FAULT ANGLES AND FAULT FRICTION 2.0

1.9

/

1.8

Z

~/

1.7

/

' 1.5

)"

"/

1.4

/

,,"#

~/

,"

/

1.3

,,@

1.1

~b~£~y../

.

/ /

-., \'.-~.," -"

/

~. , . "

.... I • 30 20

1.0 I I I 0 (h = 10km) 10 20 30 0 (h = 15kin) 10 20 0 (h = 20km)10 Wl (kin)

/

, sL 1.7t

/

I

I

o,/

1.2

10 km I brittle 15k . . . . . tal 20 krn hickness I • • 40 • 30 40

,

,,

W

/

',

/

,I"

/

/ 'S ~,Z

,

""

/ /

/

1,00/ (h = tOkm) 10 I

."

/

,.."

.~

_,''"

I 20I ~ 30 0 (h = 15krn) 10 20 0 (h = 20km)l 0 Wl (kin)

~,"

Uppush=2000m ~=25 o - - --10kin Ibrittle - " - 1 5 k m crustal -- -- 20 km I thickness •I 30 20

./~/

•I 40 30

• 40

-,

/ /

(e = 10°)

///

/

UPPUSH 1 Uppush

tt,

=2000°

~'

-- --15k

I I I 10 20 30 0 (e = 25°) 10 20 0 (~, = 40°) 10 W1 (kin)

I • 30 20

brittle ..... tal th ckness

I • 40 30

• • 40

(d) UPPUSH 3 (RAPID UPPUSH WITH EROSION) DIFFERENT BRITTLE CRUSTAL THICKNESS AND FAULT FRICTION 1.8

(b/

1.7

#1 I//

f14 /

/

/ ~b?/ --~-

1.1 / 1.0

~ / /

/ ~

1.6

f

oL /// - ' ~ / <~'/ I / 111-/

/ l

,3 UPPUSH 1 Uppushe==25 °2000m

(c) UPPUSH 2 (SLOW UPPUSH) DIFFERENT BRII-I-LE CRUSTAL THICKNESS AND FAULT FRICTION 1.9

/

1,4

..

/

1.2

-"

5°

/ /

~1.5

.-" ~.-"

/ /

o

10o/

,o

. ~"

/

/

1,7

/-"

/

/

1.8

/~,,"

/

' =10° I ,~=o,o

1.9

/

~/

,,

., / "

11 / " /

/

/ /

/

/

,

I

/

/ /

~/

.~/

-/"- ." ~ /

/'.2

/

."

~'/

/

/

.|

£~Q~b ~ Uppus =2000m I , ," ,=25 / ~£A"~ --10km )brittle 1.0 / ,,..~ ----15km crustal / 20 km I thickness I I I I I / 0.9 (h= lOkm)10 30 • • 20 • 0 (h = 15kin) 10 20 30 40 • 0 (h = 20km)10 20 30 40 Wl (krn) j.~

.~k/'/

Fig. 8. Plots of tectonic stress ratio f(O'u/O" v) versus trapezoidal base width ( W I ) for 2000 m of uplift; different brittle crustal thickness, fault friction, and fault angle considered. Models uePusH1, uppusrf2, and upeusH3 represented (Figs. 7a-7c).

M. Reiter / Tectonophysics 246 (1995) 71-95

large trapezoidal base widths of 40 km the uplifts calculated for slow and rapid uplift models are nearly the same (as given in Fig. 9d) because the difference in fault length and therefore adjacent crustal support is not very significant in proportion to the graben weight and buoyancy force.

The effect of increased lower crustal density on uplift might be investigated simply by increasing the density of the material between the base of the original trapezoid and the brittle-ductile transition (Fig. 7b). Figs. 9e and 9f show that uplift is only slightly more difficult for the case of

(a) U P P U S H 1 (RAPID U P P U S H ) D I F F E R E N T

(b) UPPUSH 3 (RAPID UPPUSH WITH EROSION) D I F F E R E N T BRI-r-rLE CRUSTAL THICKNESS, FAULT ANGLE AND FAULT FRICTION

BRI-I-FLE CRUSTAL T H I C K N E S S A N D FAULT ANGLE 19 Q ¢//¢ ~ / @ II~

2.0 1.9

1°°1 2;i/"s° 's°/,

40°

1.8-

/

//

,

I

1.6

II

f

1.4

II

/

II

I II

1.2 1.1

,

1.0"

b'(h = 10km) 4

•

•

0 (h = 15km) 4 8 0 (h = 20km)4 UPPUSH (km)

II

/

4h0 1.8

°

r/ I I I

10°/

, I

II

1.7 1.6

I /

/

1

1'/ .

•

•

• 8

• t2

i

/i

,

/~,~=o

¢=/ ~u t,~25°11 I 10°

i I //

,

,

tl' I

o=// 25 o, ,~=

f 1.5

I/I,,

,',,

I /I

1.3

/

1.2 /

1.1 /

/

[

I

/

I

/__1 /"/ // /

ii

,

//

t

II

/

~

II

/I / / ~,/ -" .," s'" - " "" .'~

,

/ UPPUSH 3 ~, = o.oo

1G ' 0 (h = 10kin) 4 8 • 0 (h = 15kin) 4 8 0 (h = 20km) 4 UPPUSH (km)

Wl = lO,O00m

-- - . . . .

• • 8

•

I

I I 1.0 I 0 (h = 10km) 4 • • 0 (h = 15km) 4 8 0 (h = 20kin) 4 UPPUSH (kin)

lOkm )brittle 15km }crustal on u~ } thickness • •

12

UPPUSH 3 Wl = lO,O00m 10 km I brittle -- -15 km crustal .... 20 km thickness I I • • 8

(d) UPPUSH 3 (RAPID UPPUSH WITH EROSION) D I F F E R E N T BRITI'LE CRUSTAL THICKNESS AND FAULTANGLE 1.9M II

I ltt '=25

o '

~

I

//-~,;o11°

~-°

I

/

,sHIb=°o I M-:25o; IIII

I'

1,7

I

//

I

;81111 1.511"// 14

t

/I

/

1.1

'.

/ /I /////I/// /i//40°

1.4

/

:;-...,

"

lift= u.Lu

~/

1.2

D I F F E R E N T BRITTLE CRUSTAL T H I C K N E S S A N D FAULT ANGLE 1.9

015

'~=

/~-e\~

." i 0.20 I /

/

T

ii

(c)UPPUSH 3 (RAPID UPPUSH WITH EROSION) 2.0

i

1.3

=

UPPUSH 1 ,f o.oo Wl = lO,O00m lOkm~ -- --15km cb;itt/teal -/-20 kr~ I thickness

I II

,,""-..~'0=2s °

1

J ~ , /(p=40o, ,

I

1.4

I I/ 11#'# ~

r" %/

/0.15

1.5

/ H

// l// 7

-/I.h= 0.20 I /

f

I/

I

#

I

,o

,"

,;" II

,

/

II

i,'; ." ,,,,,,

,I II

1.3

1.7

,,

"

// //

1.5

I

1.8 r/ / ;L=~4- 0 "o / / /

,,I I

,I

/I

,L--d? =25°'

1.9[ l J ~ / ~ ¢ ' = 2 5 ° / [I / " - 0 1 5 I

II

!

1.7

2.oj L / ~ = 1°° L I _ . 4 0 = l O O / ~ , t , ¢ = 10° ,o.o5, ~'-..~,--o.o5 I / /~ = o.os''-r

25/(/14S°'1

I I

89

:I , ,

I/

I

/"

,

II

13111

,/

12j

,

H

"

1 13-

;I , ,

0

I

','I

, /

I

ueeusH

/

li,"

ii I

"" I | 1.0 / 0 (h = 10km)

/

:i/''

,/

IlU

¢-o

, 25

.°=40

, I

I

,

II I I[? • I • • = 15km) 4 • 0 (h = 20kin) 4 UPPUSH (km)

3

w::°°;:°m --10km -.

-.

.

I

.

i brittle 15 km } crustal 20 km } thickness I

Fig. 9. Plots of tectonic stress ratio f(o'H/O"v) versus uplift, different trapezoidal base widths, brittle crustal thickness, fault angle, and fault friction considered. Model UPPUSH1, UPPUSH2, and UPPUSH3 shown (Figs. 7a-7c).

90

M. Reiter / Tectonophysics 246 (1995) 71-95

UPPUSH 2 (SLOW UPLIFT-GREATER LOWER CRUSTALDENSITY) DIFFERENT BRITTLE CRUSTALTHICKNESS,DIFFERENT FAULT FRICTION

(f)

(e)UPPUSH 2 (SLOW UPLIFT) DIFFERENT BRITTLE CRUSTALTHICKNESS, DIFFERENT FAULT FRICTION

2.0

(,,

,,,f = 0 . 2 0

2.0 ] ~ f UPPUSH 2 ¢ = 25 °

1.9

1.9

W l = 10,000m 1.8

1.8

10 km brittle 15 krn crustal 20 km th c k n e s s

-. . . .

1.7

1.6

1.6

f

f

1.5

1.5

14

1.4

1.3

1.3

¢=25 °

r

Wl

= t0,000m

10 km brittle 15 km crustal 20 km t h i c k n e s s

12

1.2

1.0

UPPUSH2

. . . .

1.7

1.1

= 0.20

/." I 2

d

11

I 4

I 6

I 8

I 10

1.0

U P P U S H (km)

/// / I 2

I I 4 6 UPPUSH (kin)

I 8

I 10

Fig. 9 (continued).

increased lower crustal density when the horst block width is 10 km. T o appreciate the possible maximum uplift of the trapezoids shown in Fig. 7 we n e e d to estimate the potential magnitude of compressive stress in the brittle crust. Measurements of crustal stress such that the maximum horizontal stress exceeds the vertical stress are presented for data taken in Canada by McGarr and Gay (1978). Tabulation of maximum horizontal-stress data to depths of 2150 m allows one to estimate a least mean squares depth gradient for the maximum horizontal stress, which is about 1.6 times the vertical stress depth gradient (r [correlation coefficient] = 0.8, n [number of data] = 17). S o m e of the data from this set show maximum horizontal stresses as great as 2.5 times overburden. With this limited and relatively shallow data it is difficult to predict a maximum value for f; I suggest a maximum value of f ~ 1.6-2.0 for compressional stress regimes; although it would certainly be desirable to obtain more data in areas of compressional tectonics. With a preliminary value for

f(max) of 1.6, one can make initial estimates for uplift along high angle faults in compressional settings; larger f values may also be considered in Figs. 8 and 9. One can notice that uplift is enhanced if erosion occurs (Figs. 9a and 9c). If fault friction is zero, the amount of additional uplift occurring during erosion will vary from about 500 m (fault angle & = 10 ° and brittle crustal thickness h = 10 km), to about 4 km (fault angle 4' = 40 ° and brittle crustal thickness h = 20 km; Figs. 9a and 9c). Fault friction, fault angle, and brittle crustal thickness are also important. If the coefficient of fault friction is changed from 0.00 to 0.05 (for fault angle ¢b = 10 ° and brittle crustal thickness h = 10 km) the maximum uplift for UPPUSH3 is reduced by a little more than a kilometer (Figs. 9b and 9c); if the coefficient of fault friction is changed from 0.00 to 0.15 (for fault angle 4' = 25 ° , brittle crustal thickness h = 20 km) the change in possible uplift is about 8.5 km (Figs. 9b and 9c). If all parameters are taken to maximize uplift when the horst base width W1 = 10,000 m; then ~ 7

M. Reiter / Tectonophysics 246 (1995) 71-95

km of uplift appears possible for UPPUSH1 (Fig. 9a) and ~ 10.5 km for UPPUSH3 (Fig. 9c). If W1 = 40,000 m, UPPUSH3 yields a maximum uplift of ~ 4.5 km (Fig. 9d). Potential uplifts in extensional and compressional tectonic regimes may be c o m p a r e d (e.g. models with erosion, UPLIFT3 VS. UPPUSH3). The present analysis suggests that for equivalent trapezoids uplift is sometimes easier, or more uplift is sometimes possible, in a compressional environment. From Figs. 6c and 9c one may notice that when the coefficient of fault friction is zero, about 0 to 6 km more uplift is possible for UPPUSH3 VS. UPLIFT3; the amount of uplift differential depends largely on the brittle crustal thickness and the fault angle. The difference in potential uplift between models with erosion in compression or extension becomes much less when brittle crustal thickness (h) = 10 km and the fault angle is 10 ° or 25 ° (Figs. 6b, 6c, and 9c). From Figs. 6c and 9b, it is observed that when the coefficient of fault friction is not zero, the differences in uplift between UPLIFT3 and UPPUSH3, as well as the amounts of uplift, are less. For equivalent conditions the effects of friction in a compressional environment are somewhat greater than in an extensional environment; this may be most easily noticed in Figs. 5c and 8c where higher fault friction during extension vs. compression results in similar uplift. For models without erosion or friction possible uplift in extensional regimes is comparable with possible uplift in compressional regimes (Figs. 6a and 9a). For no e r o s i o n - n o friction models potential uplifts can be hundreds of meters greater in compressional vs. extensional regimes if brittle crustal thickness is 20 km and fault angle is 40°; alternatively, about 0.7-2.4 km more uplift is possible in extensional regimes when the fault angle is 10 ° a n d / o r brittle crustal thickness is 10 km (uplift is more sensitive to fault angle in compression than extension; Figs. 6a and 9a). In models with erosion (UPLIFT3 and UPPUSH3) the uplift of trapezoids defined by steep faults (tb = 10 ° ) is more sensitive to fault friction than is the uplift of trapezoids defined by lower angle faults (th = 25 ° or 40 ° ; Figs. 6c, 9b and 9c). If the trapezoidal width is 40 krn at the base or 40 km at

91

the top (UPLIFT3 or UPPUSH3, Figs. 6d and 9d) one notices that slightly larger amounts of uplift are predicted possible in the compressional environment (fault angle is 25 ° , fault friction is zero). Fig. 8d shows that for t h e normal force proposed to act across the fault plane in the present study, the trapezoidal blocks shown in Fig. 7c can be in equilibrium even if f < 1.0 (fault friction equal zero). O f course, for uplift to occur, the blocks bounding the trapezoid of interest have to move inward. Uplift appears to be enhanced in both extension and compression when the brittleductile transition remains at a constant depth (Figs. 6e and 9e). It is again noted that these potential vertical movements will be sensitive to possible deviatoric stresses.

4. Geologic examples of the models I shall present a few geologic examples that are consistent with the models discussed above. The narrow grabens along Atlantic Canada (such as the grabens within the East Newfoundland Basin and the Abenaki Basin; see Wade et al., 1977) provide good examples for comparison to the present subsidence models. These grabens may have 10-12 km of sediment and are only about 25 km wide. The models presented above are compatible with such figures, and also predict that the base of the grabens should be less than 18 km wide if the brittle crust is 20 km thick (depending on the fault angle). The above models also seem to be compatible with wide basins of intermediate subsidence, such as the 8 km of deposition over 100 km width in the East Newfoundland Basin. For example, model DNDROP2 can predict 8 km of subsidence if the brittle crustal block is 20 km thick, the fault angle ~b is 40 °, the trapezoidal base width W1 is 66 km, the coefficient of friction /xf= 0, and the tectonic stress ratio f = 0.82. (I have also calculated 8 km of subsidence predicted for the same geometric p a r a m e t e r s if fault friction is 0.15 and the stress ratio f is 0.59.) However, basins 100 km wide are more than twice the width estimated above for structurally coherent basin blocks. The very wide and very deep basins, such as the Scotian Basin

92

M. Reiter / Tectonophysics 246 (1995) 71-95

along Atlantic Canada are somewhat more difficult to explain (about 12 km of sediments over about 100 km width), requiring a value of f = 0.65 for parameters just given above for the East Newfoundland Basin (p.f=0.00). For these wide basins the assumption of rigid upper crustal blocks is probably no longer valid. Therefore in such cases either the subsidence process is involving regions below the crust; or if the present model is applicable, the graben is preferentially subsiding along widely spaced faults even though the brittle crust may not be structurally coherent across the total graben width. Examples of uplifted symmetrical horst blocks are difficult to find; one example is the Jornada del Muerto in the Rio Grande rift, south-central New Mexico (New Mexico Geological Society, 1982). The structure is about 50 km wide, has boundaries which are rotated and faulted, and a central basin probably remanent of Laramide compression (Kelley, 1955). As elsewhere, the amount of uplift is difficult to separate out of the differential m o v e m e n t b e t w e e n neighboring grabens and horsts. Comparing the pre-Cambrian elevations at the center of the Jornada del Muerto and the Tularosa Basin to the east, suggests differential vertical movement of 1.0-1.5 km between the two areas (New Mexico Geological Society, 1982). This comparison implies uplift a good deal less than 1.0-1.5 km because the neighboring Tularosa Basin has 1.0-1.5 km of Cenozoic sediments deposited during subsidence. A relatively small amount of uplift (less than a kilometer) would be consistent with uplift models for wide horsts having a base of about 40 km, a coefficient of fault friction between 0.00 and 0.30, a n d / o r a brittle crustal thickness of less than 15 km (e.g. UPLWT3; Fig. 6d). As an example of uplift along high angle faults in a compressional environment we consider the interpretation of uplift in the central core of the Wind River Mountains by Mita et al. (1988). A trapezoidal block, about 12 km wide at the top, about 7 km wide at the bottom, and about 12 km high is pictured to have differentially risen about 1.5 km along boundary faults with angles 4) of ~ 1 0 ° and ~ 2 5 ° (or dip angles of ~ 8 0 ° and ~ 65 ° ). One may notice from Figs. 9 a - 9 c that the

models of uplifted blocks having small coefficients of fault friction a n d / o r large amounts of tectonic compression, are generally consistent with the geologic interpretation. Models in Figs. 9a and 9c show that 1 km to 1.5 km of uplift appear possible with brittle blocks having base width and thickness of 10 km, a fault angle 4) of 10° and a coefficient of fault friction equal to zero. Uplifts of 2 km to 3 km appear possible if the fault angle 4) = 25°, the horst base and brittle crustal thickness are both 10 km, and the coefficient of fault friction is zero. More uplift is possible as the crustal thickness and tectonic compression increase (Figs. 9a and 9c); however, uplift can be reduced with increases in fault friction (Fig. 9b). For model UPPUSH3 I have calculated 1.5 km uplift for the following conditions: a base width of 7 km, a fault angle ~b equal to 17 ° , a brittle crust 12 km thick, a coefficient of fault friction p.f= 0.10, and a horizontal to vertical stress ratio f = 1.75. With these considerations, uplifts for the models presented in the present analysis are generally consistent with the uplift of the central core of the Wind River Mountains.

5. Discussion

From the models presented above it is suggested that considerable topography, up to several tens of kilometers width, can develop by the movement of brittle crustal blocks along high angle faults in response to tectonic stresses. In many instances (i.e. reverse faulting) this may require that faults developed during previous extensional tectonism remain weak over geologic time. There is geologic evidence for reoccurring zones of mountain building (e.g. along the Rio Grande rift; Chapin, 1979) and there are reasonable theories for maintaining weak faults (e.g. thin-film high-pore pressure fault gouge; Byerlee, 1990). Although the development and transferal of crustal stresses is beyond the scope of this study, it may be important to note that many significant geologic features, such as major rift structures and individual mountain ranges, could develop from tectonic stresses and the resulting movements operating only in the crust.

M. Reiter / Tectonophysics 246 (1995) 71-95

It appears that normal and reverse movement along high-angle faults often require approximately equivalent conditions. For some models, in the absence of erosion and deposition, subsidence and uplift in extensional regimes, as well as uplift in compressional regimes, require about the same tectonic stress and fault friction. This occurs because the buoyancy force at the base of the brittle crust acts to reduce the weight of the graben or horst, making movement in the brittle crust more susceptible to horizontal tectonic forces instead of the force of gravity. Significant vertical movement of the brittle crustal block can then occur largely due to the redirection of the horizontal tectonic forces along the high-angle faults. In the case of narrow blocks being uplifted in a compressional regime, a constant depth brittle-ductile transition (slow geologic movement) promotes long boundary faults across which tectonic stresses act and enhance uplift. Slow geologic movement (a constant depth brittle-ductile transition) also increases uplift in an extensional environment because of increased buoyancy forces. The tectonic stress and fault friction required for m o v e m e n t along high angle faults is very nearly the same as that calculated necessary for m o v e m e n t along low angle faults where buoyancy is not considered as a driving mechanism (Reiter, in prep.). It is important that relatively low coefficients of fault friction have been estimated in several different studies. Bird and Kong (1994) recently estimated the "time averaged frictional coefficients of major faults as 0.17-0.25". Reiter et al. (1992) calculate the coefficient of fault friction necessary for half-graben rotation to occur as < 0.1-0.2. The coefficient of fault friction required for normal or reverse movement along low-angle faults is also estimated as < 0.2-0.3 (Reiter, in prep.). Bott and Mithen (1983) using very different models from the present study, also suggest that very low fault friction is necessary for subsidence. In the present study, the coefficient of fault friction necessary for m o v e m e n t along high angle faults ( < 0.1-0.3 in extensional regimes and < 0.1-0.2 in compressional regimes) agrees well with these other estimates. Low fault friction due to pore pressure was perhaps first suggested

93

by H u b b e r t and Rubey (1959) in order to explain thrust faulting. The models in this study predict enhanced vertical m o v e m e n t of brittle crustal blocks occurs with narrower and thicker crustal blocks, greater tectonic stress, and lower coefficients of fault friction. Uplift and subsidence, without erosion or sedimentation, are about equally difficult; however, deposition can cause much greater subsidence than erosion can cause uplift. The relative movement of grabens and horsts should depend therefore on the availability of sediments. From the models it appears that narrow structures of the order of 10 km width may experience considerable vertical movement; e.g. the very deep grabens offshore Atlantic Canada are about 25 km wide. Interestingly, the present models predict the possibility of broad subsidence of considerable amount (even though wide brittle blocks, > 30-40 km, are probably not structural coherent); whereas broad uplift is much less likely for the present models. Considerable uplift of about 10 km width appears possible in both extensional and compressional environments; however, with the effects of erosion and the possibility of greater tectonic compression than tectonic extension, it seems likely that greater uplift can occur in compressional regimes vs. extensional regimes. Very deep grabens require optimum conditions for subsidence, including deeply penetrating bounding faults. Rifts and deep grabens are typically associated with processes that raise crustal temperatures and should therefore reduce the depth to the brittle-ductile transition. It is interesting to consider the possibility of deep bounding graben faults in the context of an extending environment which may have a relatively shallow brittle-ductile transition. The possibility that bounding faults deepen with crustal cooling in late stages of rifting is inconsistent with the facts that most differential graben subsidence occurs during early rifting and the stresses at passive margins during lithospheric cooling are probably neutral or compressive (Bott, 1982; Scrutton, 1982; Zoback and Zoback, 1980; Barroll and Reiter, 1987). Perhaps during the early stages of rifting crustal temperatures remain cool enough to allow a rather deep brittle-ductile transition,

94

M. Reiter / Tectonophysics 246 (1995) 71-95

with the deep graben bounding faults that will promote a great amount of graben subsidence (Barroll and Reiter, 1987). It may also be possible that although earthquakes do not nucleate below the brittle-ductile transition, the bounding faults envisioned in the brittle crust project a zone of weakness below the brittle-ductile transition which defines the trapezoid in the lower crust where viscous creep occurs. Such a process may require pre-existing deep faults and a tectonic environment that is highly extensional.

Acknowledgements I thank William H a n e b e r g and Jiri Zidek for reading and commenting on an early version of the manuscript. Bruno Vendeville and an anonymous reviewer made many helpful comments to improve the paper. Lynne H e m e n w a y typed the manuscript, Michael Wooldridge and Kathryn Campbell drafted the illustrations.

References Bally, A.W., 1982. Musing over sedimentary basin evolution. Philos. Trans. R. Soc. London, Ser. A, 305: 325-338. Barrientos, S.E., Stein, R.S. and Ward, S.N., 1987. Comparison of the 1959 Hebgen Lake, M o n t a n a and the 1983 Borah Peak, Idaho, earthquakes from geodetic observations. Bull. Seismol. Soc. Am., 77: 784-808. Barroll, M.W. and Reiter, M., 1987. Influences of stress and temperature regimes on graben s u b s i d e n c e - - A note. Tectonophysics, 133: 157-163. Bieber, D.W., 1983. Gravimetric evidence for thrusting and hydrocarbon potential of the east flank of the Front Range, Colorado. In Rocky Mountain Foreland Basins and Uplifts, Rky. Mt. Assoc. Geols.: 245-255. Bird, P. and Kong, X., 1994. Computer simulation of California tectonics confirm very low strength of major faults. Geol. Soc. Am. Bull., 106: 159-174. Bott, M.H.P., 1976. Formation of sedimentary basin of graben type by extension of the continental crust. Tectonophysics, 36: 77-86. Bott, M.H.P., 1981. Crustal doming and the mechanics of continental rifting. Tectonophysics, 73: 1-8. Bon, M.H.P., 1982. Stress based tectonic mechanisms at passive continental margins. In: R.A. Scrutton (Editor), Dynamics of Passive Margins. Am. Geophysical Union, Washington, DC, pp. 147-153.

Bott, M.H.P. and Mithen, D.P., 1983. Mechanism of graben f o r m a t i o n - - T h e wedge subsidence hypothesis. Tectonophysics, 9 4 : 1 1 - 2 2 . Brown, L.D., Krumhansl, P.A., Chapin, C.E., Sanford, A.R., Cook, F.A., Kaufman, S., Oliver, J.E. and Schilt, F.S., 1979. C O C O R P seismic reflection studies of the Rio Grande rift. ln: R.E. Riecker (Editor), Rio Grande Rift: Tectonics and Magmatism. Am. Geophysical Union, Washington, DC, pp. 169-184. Burchfiel, B.C., Cowan, D.S. and Davis, G.A., 1992. Tectonic overview of the Cordilleran orogen in the western United States. In: B.C. Burchfiel, P.W. Lipman and M.L. Zoback (Editors), The Cordilleran Orogen: Conterminous US, Geol. Soc. Am., The Geology of North America, G - 3 : 407-479. Byerlee, J., 1990. Friction, overpressure and fault normal compression. Geophys. Res. Lett., 17: 2109-2112. Chamberlin, R.M. and Anderson, O.J., 1989. The Laramide Zuni uplift, southeastern Colorado Plateau: a microcosm of Eurasian-style indentation-extrusion tectonics? New Mexico Geological Society, Socorro, Guidebook 40: 81-89. Chapin, C.E., 1979. Evolution of the Rio Grande r i f t - - A summary. In: R.E. Riecker (Editor), Rio Grande Rift: Tectonics and Magmatism. Am. Geophysical Union, Washington, DC, pp. 1-5. Enos, P., 1991. Sedimentary parameters for computer modeling. In: E.K. Franseen, W.L. Watneg, C.G. St.C. Kendall and W. Ross (Editors), Sedimentary Modeling: Computer Simulations and Methods for Improving Parameter Definition. Kansas Geol. Sur., Lawrence, pp. 63-99. Gans, P.B., 1987. An open-system, two-layer crustal stretching model for the eastern Great Basin. Tectonics, 6: 1-12. Halliday, D. and Resnick, R., 1963. Physics. John Wiley and Sons, Inc., New York, pp. 1122. Hanks, T.C. and Rayleigh, C.B., 1980. The conference on magnitude of deviatoric stresses in the earth's crust and uppermost mantle. J. Geophys. Res., 85: 6083-6085. Itartse, H.E., 1991. Simultaneous hypocenter and velocity model estimation using direct and reflected phases from microearthquakes recorded within the central Rio Grande rift, New Mexico. Ph.D. thesis, New Mexico Institute of Mining and Technology, Socorro, 251 pp. Heiskanen, W.A. and Vening Meinesz, F.A., 1958. The Earth and Its Gravity Field. McGraw-Hill, New York, 470 pp. Hubbert, M.K. and Rubey, W.W., 1959. Role of fluid pressure in mechanics of overthrust faulting. Geol. Soc. Am. Bull., 70: 115-166. Jackson, J.A., 1987. Active normal faulting and crustal extension. In: Continental Extensional Tectonics. Geol. Soc. Spec. Publ. London, 28: 3-17. Jackson, J.A. and White, N.J., t989. Normal faulting in the upper continental crust: observations from regions of active extension. J. Struct. Geol., 11: 15-36. Johnson, G.R. and Olhoeft, G.R., 1984. Density of rocks and minerals. In: R.S. Carmichael (Editor), C R C Handbook of Physical Properties of Rocks Vol. Ill. C R C Press Inc., Boca Raton, pp. 1-38.

M. Reiter / Tectonophysics 246 (1995) 71-95 Kelley, S.A., Chapin, C.E. and Corrigan, J., 1992. Late Mesozoic to Cenozoic cooling histories of the flanks of the northern and central Rio Grande rift, Colorado and New Mexico. New Mexico Bur. Mines and Mineral Res., Socorro, Bull. 145: pp. 39. Kelley, V.C., 1955. Regional tectonics of south-central New Mexico. In: J.P. Fitzsimmons and H.H. Krusekopf, Jr. (Editors), South-Central New Mexico. New Mexico Geological Society, Socorro, Guidebook 6: 96-104. Lister, G.S. and Davis, G.A., 1989. The origin of metamorphic core complexes and detachment faults formed during Tertiary continental extension in the northern Colorado River region, USA. J. Struct. Geol., 11: 65-94. McGarr, A., 1988. On the state of lithospheric stress in the absence of applied tectonic forces. J. Geophys. Res., 93: 13,609-13,617. McGarr, A. and Gay, N.C., 1978. State of stress in the Earth's Crust. Ann. Rev. Earth Planet. Sci., 6: 405-436. Meissner, R. and Strehlau, J., 1982. Limits of stresses in continental crusts and their relation to the depthfrequency distribution of shallow earthquakes. Tectonics, 1: 73-89. Mita, G., Hull, J.M., Yonkee, W.A. and Protzman, G.M., 1988. Comparison of mesoscopic and microscopic deformational styles in the Idaho-Wyoming thrust belt and the Rocky Mountain foreland. In: C.J. Schmidt and W.J. Perry, Jr. (Editors), Interaction of the Rocky Mountain foreland and the Cordilleran thrust belt. Geol. Soc. Am. Mem. 171: 119-141. New Mexico Highway Geologic Map, 1982. New Mexico Geological Society in cooperation with New Mexico Bureau of Mines and Mineral Resources, Socorro, NM. Olsen, K.H., Keller, G.R. and Stewart, J.N., 1979. Crustal structure along the Rio Grande rift from seismic refraction profiles. In: R.E. Riecker (Editor), Rio Grande Rift: Tectonics and Magmatism. Am. Geophysical Union, Washington, DC, pp. 127-143. Percival, J.A. and Berry, M.J., 1987. The lower crust of the continents. In: K. Fuchs and C. Froidevaux (Editors), C o m p o s i t i o n , Structure and Dynamics of the Lithosphere-Asthenosphere System. Am. Geophysical Union, Washington, DC, pp. 33-59. Reiter, M., Barroll, M.W. and Cather, S.M., 1992. Rotational buoyancy tectonics and models of simple half graben formation. J. Geophys. Res., 97: 8917-8926. Reiter, M., Eggleston, R.E., Broadwell, B.R. and Minier, J., 1986. Estimates of terrestrial heat flow from deep petroleum tests along the Rio Grande rift in central and southern New Mexico. J. Geophys. Res., 91: 6225-6245.

95

Rosendahl, B.R., 1987. Architecture of continental rifts with special reference to East Africa. Annu. Rev. Earth Planet. Sci., 15: 445-503. Sanford, A.R., Jaksha, L. and Wieder, D., 1983. Seismicity of the Socorro area of the Rio Grande rift. In: C.E. Chapin (Editor), Socorro Region II. New Mexico Geological Society, Socorro, Guidebook 34: 127-131. Scholz, C.H., 1988. The brittle-plastic transition and the depth of seismic faulting. Geol. Rundsch., 77(1): 319-328. Scrutton, R.A., 1982. Passive continental margins: a review of observations and mechanisms. In: R.A. Scrutton (Editor), Dynamics of Passive Margins. Am. Geophysical Union, Washington, DC, pp. 17-29. Stein, R.S. and Barrientos, S.E., 1985. Planar high-angle faulting in the Basin and Range: geodetic analyses of the 1983 Borah Peak, Idaho, earthquake. J. Geophys. Res., 90: 11,355-11,366. Stewart, J.H., 1978. Basin-range structure in western North America: a review. In: R.B. Smith and G.P. Eaton (Editors), Cenozoic Tectonics and Regional Geophysics of the Western Cordillera. Geol. Soc. Am. Mem. 152: 1-31. Strehlau, J. and Meissner, R., 1987. Estimation of crustal viscosities and shear stresses from an extrapolation experimental steady state flow data. In: K. Fuchs and C. Froidevaux (Editors), Composition, Structure and Dynamics of the Lithosphere-Asthenosphere System. Am. Geophysical Union, Washington, DC, pp. 69-87. Toppozada, T.R. and Sanford, A.R., 1976. Crustal structure in central New Mexico interpreted from the GASBUGGY explosion. Bull. Seismol. Soc. Am., 66: 877-886. Turcott, D.L., 1987. Rheology of the oceanic and continental lithosphere. In: K. Fuchs and C. Froidevaux (Editors), Composition, Structure and Dynamics of the Lithosphere-Asthenosphere System. Am. Geophysical Union, Washington, DC, pp. 61-67. Turcott, D.L. and Schubert, G., 1982. Geodynamics, Applications of Continuum Physics to Geological Problems. John Wiley and Sons, Inc., New York, 450 pp. Vening Meinesz, F.A., 1950. Les grabens africains, resultat de compression ou de tension dan la cr6ute terrestre?. Bull. Inst. R. Colon. Belg., 21: 539-552. Wade, J.A., Grant, A.C., Sanford, B.V. and Barss, M. S, 1977. Basement structure, eastern Canada and adjacent areas. Geol. Sur. Canada., Map 1400A. Zoback, M.L. and Zoback, M., 1980. State of stress in the conterminous United States. J. Geophys. Res., 85: 61136156.