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The role of rheology in extensional basin formation …...The role of rheology in extensional basin formation modelling a * M. Fernandez , , G. Ranalli b " Institute of Earth Sciences

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  • E L S E V I E R Tectonophysics 282 (1997) 129-145

    TECTONOPHYSICS

    The role of rheology in extensional basin formation modelling a *

    M . F e r n a n d e z , , G . R a n a l l i b

    " Institute of Earth Sciences (J. Almera), CS1C, Lluis Sold Sabarfs s/n, 08028 Barcelona, Spain b Department of Earth Sciences and Ottawa-Carleton Geoscience Centre, Carleton University, Ottawa, K1S 5B6, Canada

    Accepted 11 March 1997

    Abstract

    The rheology of the lithosphere determines its deformation under given initial and boundary conditions. This paper presents a critical discussion on how rheological properties are taken into account in extensional basin modelling. Since strength envelopes are often used in models, we review the uncertainties (in temperature and rheological parameters) and assumptions (in type of rheology and mode of deformation) involved in their construction. Models of extensional basins are classified into three groups: kinematic, kinematic with rheological constraints, and dynamic. Rheology enters kinematic models only implicitly, in the assumption of an isostatic compensation mechanism. We show that there is a critical level of necking that reconciles local isostasy with the finite strength of the lithosphere, which requires a flexural response. Kinematic models with rheological constraints make use of strength envelopes to assess the initial lateral variations of lithospheric strength and its evolution with time at the site of extension. Dynamic models are the only ones to explicitly introduce rheological constitutive equations (usually in plane strain or plane stress). They usually, however, require the presence of an initial perturbation (thickness variations, pre-existing faults, thermal inhomogeneities, rheological inhomogeneities). The mechanical boundary conditions (kinematic and dynamic) and the thermal boundary conditions (constant temperature or constant heat flux at the lower boundary of the lithosphere) may result in negative/positive feedbacks leading to cessation/acceleration of extension. We conclude that, while kinematic models (with rheological constraints if possible) are very successful in accounting for the observed characteristics of sedimentary basins, dynamic models are necessary to gain insight into the physical processes underlying basin formation and evolution.

    Keywords: deformation; isostasy; stress; strain; velocity structure

    1. In troduct ion

    Extensional (rifted) basins are formed by subsi- dence of the Earth 's surface as a consequence of large-scale l i thospheric stretching. This subsidence is produced by the replacement of crust by denser mantle rocks consequent upon thinning, and by the

    * Corresponding author. Tel.: +34 (3) 4900 552; fax: +34 (3) 4110 012; e-mail: [email protected]

    thermal cool ing of the l i thosphere and mantle. Two end member mechanisms have been proposed to generate extensional basins (e.g., Turcotte and Emer- man, 1983; Neugebauer, 1983; Keen, 1985; Morett i and Froidevaux, 1986; Bott, 1992): active rifting in which the ascent of the asthenosphere causes con- vective thinning, domal uplift and li thospheric ex- tension; and passive rifting where horizontal tectonic stresses produce l i thospheric thinning and passive mantle upwelling. Actually, most rifts show active

    0040-1951/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PH S0040- 195 1 (97)002 16-3

  • 130 M. Ferngmdez, G. Ranalli/Tectonophysics 282 (1997) 129-145

    and passive signatures and the two mechanisms are complementary (Khain, 1992; Wilson, 1993). The resulting basin geometry, rates of subsidence/uplift and sedimentation/erosion, depositional style, occur- rence of magmatism, etc., are surface expressions of processes that operate at crustal and mantle levels. These processes are directly related to the deforma- tional pattern of the lithosphere when subjected to deviatoric tensile stresses, and consequently to its rheological behaviour (Vilotte et al., 1993; Quinlan et al., 1993).

    Laboratory experiments show that rocks can de- form in a brittle or ductile manner depending on pressure and temperature conditions (e.g., Goetze and Evans, 1979; Carter and Tsenn, 1987; Kohl- stedt et al., 1995). In modelling, the mode of litho- spheric deformation can be prescribed either implic- itly, by imposing a deformation pattern which the lithosphere is assumed able to sustain (kinematic models) or explicitly, by specifying equations gov- erning the rheological behaviour (dynamic models). Kinematic models have been widely used in passive rifts to account for a large variety of observations such as rates of subsidence/uplift of the basement, differential stretching, crustal and lithospheric de- tachments, volcanism, etc. (e.g., McKenzie, 1978; Royden and Keen, 1980; Buck et al., 1988; McKen- zie and Bickle, 1988; Weissel and Karner, 1989; Cloetingh et al., 1993, 1994a,b, 1995a,b). Dynamic models, on the other hand, have been applied to pas- sive and active rifting mechanisms and also to litho- sphere/asthenosphere interaction (e.g., Braun and Beaumont, 1987; Lynch and Morgan, 1987; Dun- bar and Sawyer, 1989; Sonder and England, 1989; Bassi et al., 1993; Keen and Boutilier, 1995). The lithosphere can be treated as a single viscous layer with Newtonian or non-Newtonian rheology, or as a layered medium with an elasto-visco-plastic rheol- ogy according to the predominant type of rocks at each depth.

    Some models use a mixed approach where the mode of deformation of the lithosphere is imposed kinematically, while the deformation of the under- lying substratum is treated dynamically (e.g., Keen, 1985; Buck, 1986; Keen and Boutilier, 1995). Like- wise, rheological controls based on the depth vari- ation of brittle and ductile strength, and the effects of gravitational buoyancy forces arising from lateral

    thickness variations, have been used to externally constrain the mode of deformation imposed in kine- matic models (e.g., Kusznir and Park, 1987; Buck, 1991; Negredo et al., 1995).

    In the last few years, the origin and evolution of sedimentary basins has been the subject of a focused research effort by a Task Force of the In- ternational Lithosphere Program (Cloetingh et al., 1994a). As a result, the available database on natural basins, as well as the number of models of basin formation, have increased enormously (cf. Cloetingh et al., 1993, 1994b, 1995a,b). The purpose of this paper is to analyse critically the role of rheology in extensional basin modelling, and to discuss the consequences and limitations associated with the different rheological assumptions. We begin with a review of the strength envelope concept, its uncer- tainties and assumptions. Basin formation models are subdivided into three categories from the rheologi- cal viewpoint, namely, kinematic models, kinematic models with rheological constraints', and dynamic models, which are discussed in turn. Some problems (nature of the initial perturbation leading to basin development, and the effects of different boundary conditions) are treated in separate sections. Finally, we offer some general remarks on the role of rheol- ogy in basin models and discuss the relative merits of the different approaches.

    2. Rheology of the lithosphere and strength envelopes

    The rheology of the lithosphere is a function of its composition and structure, pressure, temperature, and state of stress. The concept of strength envelope (rheological profile), firstly developed by Goetze and Evans (1979), is well known (cf. Ranalli, 1995). On the basis of a generalization of experimental results, it is assumed that the deformation regime for any given rock can be subdivided into two domains: brit- tle or frictional, governed by the Coulomb-Navier shear failure criterion, and ductile, governed by the power-law creep equation. The brittle/ductile transi- tion is defined by the equality of frictional strength and ductile strength (for a given strain rate). Al- though undoubtedly an oversimplification of reality, strength envelopes have proven very useful in rhe- ological modelling of lithospheric processes, espe-

  • M. Ferngmdez, G. Ranalli/Tectonophysics 282 (1997) 129-145 131

    cially in providing rheological constraints for basin modelling (see discussion in the following sections).

    Assumptions and uncertainties associated with strength envelopes naturally affect the final model of basin formation. Reviews of the construction and applications of strength envelopes can be found else- where (e.g., Ranalli, 1995, 1997a); here, we focus on their uncertainties and their limits of applica- bility. The uncertainties can be divided into two groups. Operational uncertainties derive from imper- fect knowledge of composition and structure of the lithosphere, errors in the estimated temperature dis- tribution, scatter in experimentally determined rheo- logical parameters, lack of constraints on pore fluid pressure, and similar factors. Methodological uncer- tainties stem from the basic assumptions used in the construction of strength envelopes, which can be re- lated to the rheology (i.e., the assumption of simple brittle-over-ductile behaviour), or to the deformation regime (i.e., uniform strain and constant strain rate), and which are not necessarily realistic.

    The main operational uncertainties (assuming that the composition and structure of the lithosphere are reasonably well known) derive from temperature, rheological parameters, and pore fluid pressure. Not much - - except direct observation - - can be done about the last, and usually the hydrostatic assumption (pore pressure equal to the pressure of an overlying column of water at any depth) is adopted. Most of the uncertainty in the geotherm derives from the scatter of surface heat flow values (Chapman and Furlong, 1992). Uncertainties in the lower crust of 4-100°C are the norm rather than the exception (cf. also Lamontagne and Ranalli, 1996). This can result in peak-to-peak variations in estimated creep strength of about one order of magnitude, and consequent displacements of several kilometres of the estimated depth of the brittle/ductile transition (Fadaie and Ranalli, 1990).

    Rheological parameters in the brittle regime are usually assumed constant for all rock types. Pre-exist- ing faults are often taken to be cohesionless, with a coefficient of friction # = 0.75. The uncertainties introduced by these approximations are low com- pared to those generated by the lack of constraints on the pore fluid pressure. Rheological parameters in the ductile regime for different rock types, on the other hand, show considerable scatter (see e.g., Kirby

    and Kronenberg, 1987a,b; Ranalli, 1995). Table 1 is a compilation of representative rheological parameters for different lithospheric layers (upper crust, lower crust, upper mantle) used in basin formation mod- els. Although occasionally the individual values of creep parameters differ considerably, their applica- tion in the creep equation in any given case results in creep strengths that usually vary no more than 4-50% at any given depth. A notable exception is the softer upper crust, and to a lesser extent lower crust, result- ing from the parameters adopted by Burov and Di- ament (1995) and Cloetingh and Burov (1996). The harder lower crust in the models of Mareschal (1994) and Lamontagne and Ranalli (1996) is a consequence of a specific composition (mafic granulite) which may apply only to certain areas. The parameters listed in Table 1 can be compared to compilation of experi- mental results (e.g., Kirby and Kronenberg, 1987a,b; Ranalli, 1995, 1997a).

    Methodological uncertainties are potentially very significant. They arise from basic assumptions used in the construction of strength envelopes. Leaving aside the consideration that the brittle/ductile transi- tion in nature is transitional rather than sharp, rhe- ological assumptions used in strength envelopes are that the Coulomb-Navier frictional criterion applies at any depth where the material is not ductile, and that power-law creep is the only kind of behaviour in the ductile field. Both of these are only first-order approximations.

    The Coulomb-Navier frictional criterion (Byer- lee's law) is experimentally confirmed only up to pressures corresponding to mid-crustal depths (By- erlee, 1967; Jaeger and Cook, 1979). Its linear ex- trapolation to lower crustal and upper mantle condi- tions results in unrealistically high brittle strengths. The assumption of a linear Moho envelope (constant friction coefficient) is often not confirmed in practice, and there are indications that the coefficient of friction decreases with increasing pressure (Jaeger and Cook, 1979). Furthermore, other deformation mechanisms such as high-pressure fracture (Shimada, 1993) and plastic yielding (Ord and Hobbs, 1989), expected to take over from frictional failure as pressure increases and being only weakly dependent on pressure (cf. Ranalli, 1997b), are neglected in strength envelopes. Their effect would be to decrease the pressure depen- dence of strength as depth increases.

  • 132 M. Ferngmdez, G. Ranalli/Tectonophysics 282 (1997) 129-145

    Table 1 Ductile creep parameters A (pre-exponential factor), n (stress exponent), and E (activation energy) in power-law equation/: = Ac~"exp(-E/RT) used in basin modelling studies for upper crust (UC), lower crust (LC), and lithospheric upper mantle (UM); W denotes hydrated conditions

    Layer A (MPa -n s - 1 ) n E (kJ mol- I ) Ref.

    UC 2.5 x 10 8 3.0 138 [1] UC 1.3 x 10 -3 2.4 219 [4] UC 3.4 x 10 .6 2.8 185 [5, 13] UC 1.3 × 10 -7 2.9 149 [6] UC 1.0 × 10 .6 2.8 150 [7] UC 1.6 × 10 -9 3.0 123 [9] UC 1.3 × 10 -9 3.2 144 It0[ UC 5.8 × 10 -5 2.4 142 [11] UC 2.0 × 10 4 1.9 141 [12] UC 5.0 x 106 3.0 190 [14, 15] UC 2.5 × 10 - s 3.0 140 [16] UCW 1.0 × 10 -2 1.8 151 [2] UCW 2.0 × 10 -4 1.9 134 [3] UCW 2.9 x 10 -3 1.8 150 [5] UCW 3.3 x 10 .6 2.4 134 [6] UCW 3.1 × 10 7 3.1 135 [8] UCW 5.6 × 10 -5 2.4 160 [13] LC 3.2 x 10 .3 3.0 25l [1] LC 1.3 x 10 3 2.4 219 [3] LC 3.2 x 10 3 3.3 268 [4[ LC 3.3 × 10 4 3.2 238 [5, 10] LC 8.9 × 10 4 3.2 238 [6] LC 1.0 x 10 -3 3.0 230 ]7] LC 3.2 × 10 3 3.2 270 [7] LC 3.8 x 10 .2 3.1 243 [8] LC 2.0 × 10 -4 3.4 260 [9] LC 3.3 × 10 4 3.2 384 [11] LC 1.4 × 10 4 4.2 445 [12, 17] LC 2.3 x 10 6 3.9 235 [13] LC 1.3 2.4 212 [14, 15] LC 0.13 3.1 276 [14, 15] LC 3.2 × 10 -3 3.0 250 [16] LC 8.0 x 10 -3 3.1 243 [17] LCW 3.0 × 10 -2 3.2 239 [2] LCW 3.3 × 10 -4 3.2 238 [5] LCW 6.3 x 10 -3 2.8 271 [17] UM 1.0 x 103 3.0 523 [1, 16] UM 7.0 × 104 3.0 510 [3] UM 3.2 × 104 3.6 535 [4] UM 2.9 × 104 3.6 535 [5, 13] UM 1.0 × 103 3.0 500 [6] UM 3.2 x 104 3.5 535 [7] UM 1.9 x lO 3 3.0 420 [9] UM 1.4 × 105 3.5 535 [10] UM 7.0 × 104 3.0 530 [111 UM 4.3 × 102 3.0 527 [121 UM 7.0 × 104 3.0 520 [14, 15] UMW 1.9 x 105 4.5 498 [2] UMW 2.0 x 103 4.0 471 [4]

    Table I (continued)

    Layer A (MPa " s l) n E (ld m o l l ) Ref.

    UMW 1.4 x 104 3.4 445 [5] UMW 8.6 x 103 3.0 420 [8] UMW 4.0 × 102 4.5 498 [12]

    References. [1] Lynch and Morgan, 1987; [2] Braun and Beau- mont, 1989b; [3] Dunbar and Sawyer, 1989; [4] Fadaie and Ranalli, 1990; [5] Bassi, 1991; [6] Buck, 1991; [7] Ranalli, 1991; [8] Govers and Wortel, 1993; [9] Liu and Furlong, 1993; [10] Lowe and Ranalli, 1993; [11] Boutilier and Keen, 1994; [12] Mareschal, 1994; [13] Bassi, 1995; [14] Burov and Dia- ment, 1995; [15] Cloetingh and Burov, 1996; [16] Negredo et al., 1995; [17] Lamontagne and Ranalli, 1996. Notes. Upper crust is usually taken as quartz-rich or granitic ex- cept in [4] where it is assumed to be quartz-dioritic: lower crust varies between intermediate (quartz-dioritic) and basic composi- tion (the parameters in [12] and [17] apply to mafic granulites); upper mantle is ultrabasic (olivine-rich).

    In t h e d u c t i l e r e g i m e , at l e a s t fo r p e r i d o t i t i c r o c k s ,

    t h e r e is e v i d e n c e tha t a t r a n s i t i o n o c c u r s w i t h in-

    c r e a s i n g t e m p e r a t u r e b e t w e e n shear zone ductility, w h e r e f l o w is c o n c e n t r a t e d a l o n g d i s c r e t e s h e a r

    z o n e s , a n d bulk ductility, w h e r e f low is p e r v a s i v e ( D r u r y e t al. , 1991; V i s s e r s e t al. , 1991). T h e c o m -

    b i n e d e f f e c t s o f t h e s e t h e o l o g i c a l c o m p l i c a t i o n s are

    d e p i c t e d q u a l i t a t i v e l y in F ig . 1 fo r t h e s i m p l e c a s e

    o f a c r u s t o f u n i f o r m f e l s i c c o m p o s i t i o n a n d a l o w

    g e o t h e r m a l g r a d i e n t r e s u l t i n g in a b r i t t l e u p p e r m a n -

    t le. C r i t i c a l t e m p e r a t u r e s a r e a l s o s h o w n . A l t h o u g h

    it c a n n o t y e t b e q u a n t i f i e d , t h e r h e o l o g i c a l l a y e r i n g

    o f t h e l i t h o s p h e r e is l ike ly to be m o r e c o m p l e x than

    u s u a l l y a s s u m e d in s t r e n g t h e n v e l o p e s .

    A s s u m p t i o n s c o n c e r n i n g d e f o r m a t i o n s ty l e a re an

    i n t e g r a l pa r t o f s t r e n g t h e n v e l o p e s . T h e a s s u m p -

    t i o n o f u n i f o r m s t r a in i m p l i e s t ha t al l l a y e r s su f f e r

    t h e s a m e e x t e n s i o n a l d e f o r m a t i o n , a n d c o n s e q u e n t l y

    s t r i c t l y l i m i t t he a p p l i c a t i o n to p u r e s h e a r e x t e n s i o n

    w i t h t h e s t r e t c h i n g f a c t o r i n d e p e n d e n t o f d e p t h . T h i s

    is o f c o u r s e n o t n e c e s s a r i l y t rue. D e f o r m a t i o n in t h e

    l o w e r d u c t i l e p a r t o f t he c r u s t is v e r y h e t e r o g e n e o u s

    ( R u t t e r a n d B r o d i e , 1992), a n d s i m p l e s h e a r p l a y s

    an i m p o r t a n t ro l e in l i t h o s p h e r i c e x t e n s i o n ( B u c k ,

    1991). S o m e i m p l i c a t i o n s fo r s t r e n g t h e n v e l o p e s a re

    s h o w n in F ig . 2. T h e a p p a r e n t e x t e n s i o n a l s t r a in ra te

    ( A L / L ) / t , w h e r e t is t i m e , is n o t t he p h y s i c a l l y re l - e v a n t o b s e r v a b l e . T h e ac t ua l s t r a in ra te is t he s h e a r

    vJd, w h e r e v~ is t h e r e l a t i v e v e l o c i t y a n d d the t h i ck -

  • M. Ferngmdez, G. Ranalli/Tectonophysics 282 (1997) 129-145 133

    B . . . . . . . . . I . . . . . . . . . . . . . . . . .

    M L1

    L2

    L3

    L 4

    Z

    (3"1- - ~ 3

    300±50oc ..........

    iiiiiiiiiiiiiiiiiiiii 6 o oooc " ~ - 911 +50°C .......

    ................. 1250+100oc ...............

    3F BF ..

    ... -II

    D D

    BF • " -'II

    3~

    D

    D

    Fig. 1. Type rheological profiles (left), critical temperatures (where known), and variation of rheology with depth (right) for a uniform felsic crust and low geothermal gradient. Stan- dard strength envelope is shown by the full line; changes due to high-pressure failure and to shear ductility by dashed lines. Critical depths B: crustal brittle/ductile transition; Ll: mantle brittle/ductile transition; F1F2, ML2: ranges of high- pressure failure (qualitative) in crust and upper mantle; L3: mantle shear ductility/bulk ductility transition; M: Moho; L4: lithosphere/asthenosphere boundary. Variations of rheology with depth (left column: corresponding to standard strength envelope; right column: modified strength envelope): BF: brittle frictional; D: ductile (power-law); HP: high-pressure failure; DS: shear zone ductility.

    ness of the shear zone. The strength in this case is determined by the variations of rheological proper- ties along the zone separating the two lithospheric blocks, not along the vertical direction.

    The assumption of constant strain rate (upon

    which strength envelopes are built) has implications for the extensional velocity that will be considered when discussing the effects of boundary conditions.

    The above considerations are not meant to deny the usefulness of strength envelopes as a first-order tool to constrain models of basin formation, but rather as a plea not to overlook their limitations and to take into consideration more realistic rheologies and deformation styles.

    3. Kinematic models

    A common characteristic of kinematic models of lithospheric extension is that deformation is imposed by prescribing a velocity field which is linked, as an advective term, to the heat transport equation, and no constitutive equations are incorporated. Only vertical forces are considered, related to loading/unloading associated with infilling of basins, erosion of shoul- ders, and mass redistribution due to lithospheric stretching. In this section we will focus on the rela- tionship between isostatic mechanisms and rheology, which governs the response of the lithosphere to vertical forces.

    A very simple one-dimensional approach to ex- plain the subsidence observed in passive margins and sedimentary basins was proposed by McKenzie (1978). The model assumes local isostatic compen- sation, pure shear deformation and instantaneous lithospheric stretching followed by thermal relax- ation. Extensions of this model to two dimensions consider the effects of finite duration of rifting, lat- eral heat transport, and differential stretching (e.g., Royden and Keen, 1980; Jarvis and McKenzie, 1980; Cochran, 1983; Buck et al., 1988; White and McKen- zie, 1988). These models have permitted to bet- ter constrain the observed basin subsidence, crustal

    k L+ A L ~t I( L :,1

    . . . . . . . . - - - - - - , - - . - - . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . .

    " " " " .

    "

    ~ ° ° ° ° ° .

    Fig. 2. Extension of lithosphere in simple shear. Increase in width AL is accomplished by relative motion across the shear zone. See text for details.

  • 134 M. Ferngmdez, G. Ranalli/Tectonophysics 282 (1997) 129-145

    structure, and surface heat flow. They show that flank uplift is primarily produced by thermal effects and therefore it progresses during the syn-rift phase but tends to vanish during the post-rift phase (Buck et al., 1988). Changes in the crust-mantle ratio caused by differential stretching can explain post-rift strati- graphic onlaps (White and McKenzie, 1988) and prominent and permanent rift shoulders (Zeyen et al., 1996).

    Local isostasy assumes that the lithosphere is un- able to support vertical shear stresses. Therefore, any vertical force is compensated by lithosphere buoy- ancy. From a rheological viewpoint, this assump- tion implies that the lithosphere behaves as a solid with zero threshold shear stress under any vertical load, while at the same time possessing high lateral strength to prevent deformation caused by horizontal stress gradients (since the lithostatic pressure at a given depth above the compensation level varies in different columns). Thus, local isostasy appears at first sight difficult to reconcile with any self-consis- tent rheological model of the lithosphere.

    If, on the other hand, it is assumed that the litho- sphere retains a finite strength when loaded, then it responds to vertical loads by flexure, resulting in regional isostatic compensation. The simplest model is that of a thin elastic plate. The elastic behaviour of the lithosphere has been successfully proven for the oceanic lithosphere, where combined gravity and bathymetry analyses have shown that topographic features with a wavelength less than 100 km are compensated by flexural isostasy (McKenzie and Bowin, 1976).

    Flexural studies indicate that the effective elastic thickness (ire) of the oceanic lithosphere is correlated with the 450-600°C isotherm and therefore to its age (Watts, 1978; McNutt and Menard, 1982). How- ever, this empirical relationship is much more debat- able for continental lithosphere, since the effective elastic thickness depends on the crustal thickness, the lithospheric thickness through the temperature structure, and the interaction of these factors dur- ing lithospheric deformation. For a thermally young lithosphere, Te is dominated by quartz-feldspar rhe- ology, while for older lithospheres it is dominated by olivine rheology (Kusznir and Karner, 1985). Pure- shear flexural models have been used to account for the basement subsidence and stratigraphy at pas-

    sive margins and sedimentary basins (e.g., Watts and Ryan, 1976; Beaumont et al., 1982; Watts et al., 1982). Decreasing Te results in deeper and narrower basins; at the limit Te = 0 is equivalent to local isostasy.

    From the rheological viewpoint, several problems arise when modelling the lithosphere as a linear elastic plate. First, in a bent elastic plate, the calcu- lated stresses can be considerably higher than those deduced from lithospheric strength envelopes; sec- ond, the effective elastic thickness depends not only on age, but also on plate curvature and load; and third, the initial elastic response of the lithosphere to loading is followed by a delayed viscous compo- nent. Several attempts have been made to reconcile the flexural behaviour of the lithosphere with its ac- tual rheology, constraining the flexural stresses by the strength envelopes (e.g., Bodine et al., 1981; McAdoo et al., 1985; McNutt et al., 1988; Burov and Diament, 1992; Ranalli, 1994). The main re- sult is that the effective elastic thickness depends on rock rheology and curvature for a given structure, composition, and thermal regime.

    Whatever mechanism of isostasy is assumed, lo- cal or regional, pure-shear models do not account for the asymmetry and/or high uplift of the flanks observed in many basins. An alternative kinematic model of basin formation was proposed by Wernicke (1985), who assumed a detachment across the en- tire lithosphere to explain the uplift of the Colorado Plateau adjacent to the Basin and Range by local isostasy. Simple-shear deformation produces a lat- eral offset between mantle and crustal thinning and asymmetry in the resulting basin (see e.g., Buck et al., 1988, and Kusznir and Egan, 1989, tbr a quan- titative analysis). Depending on the depth, dip, and number of detachments, different styles of deforma- tion can be reproduced, such as simple shear (Buck et al., 1988), combined simple shear and pure shear (Kusznir et al., 1987), and cantilever (Kusznir and Ziegler, 1992). Including detachments and faults in kinematic models implies that some lithospheric lev- els can act as decoupling horizons. However, faults and detachments merely play a role of slip surfaces and no considerations on the stress necessary to produce this slip are taken into account.

    An important concept in the study of the kine- matics of lithosphere extension is the level of neck-

  • M. Ferngmdez, G. Ranalli/Tectonophysics 282 (1997) 129-145 135

    ¢, ,,

    (a) "'- -t--." (i) . . . . . ~ . . . . . .

    (c)

    h c

    hm

    Pc

    Pm

    h o

    hm/[3

    ha

    Fig. 3. (a) Upward, and (b) downward flexure of necked lithosphere. (c) Equilibrium level of necking, resulting in no flexure and therefore local isostasy (mass defects, denoted by minus signs, compensate mass excesses, denoted by plus sign). Thicknesses (h) and densities p of crust, mantle, and sediments denoted by subscripts c, m, and o, respectively;/3 is the stretching factor.

    ing, defined as the level which, in the absence of buoyancy forces, would not move vertically during extension (Braun and Beaumont, 1989a; Weissel and Karner, 1989; Kooi et al., 1992). Because of the changes in mass distribution related to extension, different necking levels result in different flexural re- sponses (see Fig. 3). Deep necking levels produce regionally supported high basin shoulders, while shallow necking levels produce downwarped basin flanks. Note, however, that if the depth of necking is such that no lithospheric loading or unloading results, no flexural isostatic deflection will occur, and the basin will be in local isostatic equilibrium (Fig. 3c). This 'neutral' level of necking is indepen- dent of the stretching factor, and is about 10 km deep for a lithosphere initially 100 km thick including a 33 km crust, if the basin is filled with sediments (the actual value depends on initial configuration and adopted densities). Therefore, local isostasy can fit the evidence in some sedimentary basins, without contradicting the fact that the lithosphere has finite strength.

    Kinematic necking models account for a variety of basin morphologies (e.g., Kooi et al., 1992; van der Beek et al., 1995; Spadini et al., 1995). Although

    in principle the necking level should coincide with the level of maximum lithospheric strength, a review of Mediterranean and intracratonic basins (Cloetingh et al., 1995a) shows depths varying from 4 to 35 km (see also next section). This is, in part, a consequence of large lateral variations in lithospheric strength, but it also reflects that the relationship between strength envelopes and kinematic level of necking is more complicated than previously thought.

    4. Mechanical controls on kinematic models: 'back-door rheology'

    Kinematic models are very powerful in account- ing for the main features of extensional sedimentary basins and their evolution through time. This capa- bility is due to the high variety of deformation modes that can be imposed by predefining the velocity field. However, in kinematic models there is no control over the compatibility between the imposed mode of deformation and the actual mechanical behaviour of rocks. Lithospheric rheology predicts that two com- peting effects arise during finite continental exten- sion: weakening produced by lithospheric thinning and strengthening produced by thermal relaxation

  • 136 M. Fern?mdez, G. Ranalli/Tectonophysics 282 (1997) 129-145

    (see e.g., England, 1983). According to the interplay of these effects, the locus of extension may migrate and the predefined velocity field will no longer be valid. A simple procedure to evaluate the progress of extension is to compare the total strength of a stretched lithospheric column with that correspond- ing to an undeformed lithosphere on the basis of some rheological model. These one-dimensional ap- proaches do not provide the actual deformation of the lithosphere, but they introduce rheological controls on the mode of deformation. We call them 'back- door rheological models'. One example is the anal- ysis of the correlation of necking level, mentioned in Section 3, with strength envelopes (Cloetingh et al., 1995a). The necking level usually corresponds to a strong layer in the lithosphere (upper-middle crust or uppermost mantle, according to geothermal gradient). When two well-defined strong layers are present, however, the necking level loses its geomet- ric meaning.

    Assuming a thin sheet viscous lithosphere, Eng- land (1983) showed that the force required to deform the lithosphere at a given strain rate is inversely proportional to the geothermal gradient and depends exponentially on the Moho temperature. In the early stages of rifting, this force decreases until strain ex- ceeds a value which depends on the P6clet number, the rheological parameters, and the initial crustal thickness. Once this critical value is reached, the strength increases very rapidly and limits further ex- tension. The effect of lithospheric strengthening is higher at low strain rates. As an example, England (1983) concludes that if the transition to oceanic lithosphere is produced by a stretching factor fl rang- ing from 3.25 to 6, the duration of rifting must be less than 10-20 Ma.

    The thin sheet approach assumes that the litho- sphere consists of a single viscous layer with olivine rheology, and that the thermal gradient within this layer is constant. This is a simplification of the me- chanical behaviour of the lithosphere that tends to overestimate lithospheric strengthening, since nei- ther the role of crustal rocks (both in the brittle and in the ductile fields), nor the changes in ther- mal gradient are considered (Sawyer, 1985). Kusznir and Park (1987) considered a layered rheology for the lithosphere and analyzed its response to an ap- plied tectonic tensile force by assuming conservation

    of the total horizontal force and uniform horizontal strain with depth. The model predicts high stretch- ing factors and localized extension for fast strain rates, while wide basins with low stretching fac- tors result for slow strain rates. Fast strain rates are only possible in initially hot lithosphere; cold litho- sphere requires an unrealistic high force to begin extension. Intermediate geotherms (heat flow ~ 6 0 - 70 mW m -2) result in pronounced zones of low stress and low ductile strength at the base of the middle and lower crust, which can act as detachment surfaces.

    A similar one-dimensional approach, using strength envelopes in conjunction with a constant strain rate and gravitational buoyancy forces, was proposed by Buck (1991). In this model, changes in total lithospheric strength during extension are as- sociated not only with changes in crustal thickness and temperature distribution, but also with lateral pressure differences which may drive flow in the lower crust. The results obtained by Buck (1991) dif- fer from those of Kusznir and Park (1987). Narrow rifts develop for initially cold lithosphere (heat flow

  • M. Ferngmdez, G. Ranalli/Tectonophysics 282 (1997) 129-145 137

    part of the basin becomes stronger than the adja- cent undeformed areas and hence any new extension event will shift the locus of deformation.

    5. Dynamic models

    The use of 'back-door rheology' puts important constraints on models of lithospheric extension, and yields some relations between the mode of defor- mation (narrow/wide rifts) and strain rate, initial geotherm, and crust/mantle thickness ratio. However, a complete account of lithospheric deformation re- quires the use of two- or three-dimensional dynamic models. By definition, a dynamic model involves constitutive equations which relate dynamic quanti- ties (stress) to kinematic quantities (strain or strain rate) through material parameters (Ranalli, 1995). There are several constitutive equations that describe rheological behaviour as a function of material, tem- perature, pressure, and stress conditions (elastic, vis- cous, plastic, viscoelastic, etc.). Accordingly, dy- namic models of lithospheric extension cover a wide range of lithospheric rheologies. For simplicity, and because most of the models are two-dimensional, we classify them into plane-strain and plane-stress models.

    5.1. Plane-strain models

    The hypothesis of plane strain assumes that flow is everywhere parallel to a plane containing any two axes and independent of the third axis. Consequently, one principal strain component is zero. Plane-strain models are usually applied to lithospheric cross- sections of elongated geological structures such as basins and rifts, orogenic belts, and subduction zones. Some early analyses of lithospheric extension adopted this approach, and assumed that the mantle behaves as a viscous fluid while the crustal deforma- tion is imposed kinematically (see e.g., Mareschal, 1983; Neugebauer, 1983; Keen, 1985, 1987; Buck, 1986). These studies incorporated progressively the effects of pressure, temperature, and strain rate on mantle viscosity, and allowed discussion of the ef- fects of mantle convection on topography, as well as of the role of active versus passive rifting mecha- nisms. The results are highly sensitive to the adopted viscosity, but show that in general mantle convection

    induces uplift of the rift flanks (e.g., Buck, 1986). A dynamic component of elevation is generated due to mantle flow, which has to be added to the isostatic component (Keen, 1987).

    Improvements in experimental rock mechanics and in computing facilities allowed for the incor- poration of more realistic theologies in plane-strain models (see e.g., Braun and Beaumont, 1987; Dun- bar and Sawyer, 1989; Lynch and Morgan, 1990; Bassi, 1991). In this second generation of models, the lithosphere is assumed to consist of three layers which correspond to the upper crust, lower crust and lithospheric mantle. Each of these layers behaves as an elastic/viscous/plastic medium according to chosen rheological equations (non-linear viscosity is adopted for ductile flow). The results of these mod- els have clarified the relations between the exten- sional deformation of the lithosphere, its rheology, and boundary conditions. The initiation of extension depends usually on the presence of a perturbation (strength heterogeneity). The nature of these hetero- geneities is discussed separately in Section 6.

    Apart from the initial perturbation, most plane- strain dynamic models focus on the relative effects of extension rate and theology. In principle, high strain rates produce narrow rifts, since the thermally induced lithospheric weakening concentrates defor- mation. The opposite, however (i.e., that low strain rates induce wide rifts), is not always true. The oc- currence of wide rifts depends largely on the adopted theology. In particular, wide rifts are produced at low strain rates only when combined with a soft and viscous lithosphere.

    Conversely, if the adopted rheology is plastic, deformation cannot migrate due to the reduction in strength contrasts within the lithosphere and to the mechanical instability associated with plasticity, which tends to concentrate deformation (Bassi et al., 1993; Bassi, 1995). Localization of deformation in the lithosphere can be produced by incorporating a power-law breakdown in the yield stress envelopes (Bassi, 1991), or by deep-seated weaknesses af- fecting the lithospheric mantle, e.g., asthenospheric upwelling (Lynch and Morgan, 1990; Christensen, 1992; Chery et al., 1992).

    Another common result of plane-strain models is that, in the absence of restoring forces, the Earth's surface subsides due to lithospheric necking, produc-

  • 138 M. Fernandez, G. Ranalli / Tectonophysics 282 (1997) 129-145

    ing large isostatic anomalies. Then, if the lithosphere retains a finite elastic rigidity, the consequent iso- static adjustment can result in different basin mor- phologies. In particular, when differential stretching occurs with larger deformation at deep levels, the isostatic adjustment leads to an upward elastically supported rebound and rift shoulders are formed (Lynch and Morgan, 1990; Chery et al., 1990, 1992; Bassi, 1991). These results agree qualitatively with the previously discussed kinematic necking.

    In general, results of plane-strain modelling show that strain rate is not the only factor determining the style of extension, but that rheology, and con- sequently initial geotherm and structure, also play a major role in the resulting mode of deformation. Therefore, initial heterogeneities causing lateral vari- ations in the mechanical properties of the lithosphere exert a strong influence on the deformation pattern. These aspects will be treated in more detail in the following section.

    5.2. Plane-stress models

    The hypothesis of plane stress assumes that ver- tical gradients of horizontal velocity are small com- pared with horizontal gradients, and that deviatoric stresses above and beneath the lithosphere vanish. This assumption, and the integration with depth of the differential equations for stress equilibrium, per- mit to treat the lithosphere as a thin sheet or plate (e.g., Bird and Piper, 1980; Vilotte et al., 1982; England and McKenzie, 1982, 1983). In this way, lithospheric deformation is studied in map view us- ing a quasi-three-dimensional model. In order to have a vertically averaged rheology, lithospheric strength envelopes are approximated by a single viscous (England and McKenzie, 1982, 1983) or visco-plastic (Bird, 1989) layer, which accounts for ductile flow and frictional sliding on fault surfaces. The plane-stress approach requires several assump- tions for its application: theological parameters are depth-independent, deformation is anelastic, topog- raphy is locally supported, and heat conduction is vertical. However, gravitationally induced stresses related to variations in crustal and lithospheric thick- ness can be taken into account. Moreover, plane- stress models are not restricted to a single layer and it is possible to define two layers in order to simulate

    the effects of detachments between crust and mantle (Bird, 1989).

    Plane-stress models have been widely used to simulate different tectonic settings such as conti- nental collision (e.g., England and McKenzie, 1982, 1983; England and Houseman, 1985; Vilotte et al., 1986), extensional basins (e.g., Houseman and Eng- land, 1986; Sonder and England, 1989; Bassi and Sabadini, 1994), and strike-slip regimes (e.g., Bird and Piper, 1980; England et al., 1985; Sonder et al., 1986). Of particular interest are the results obtained by Sonder and England (1989), which show that for low strain rates the locus of maximum strain rate mi- grates during extension from regions of high strain to regions of low strain.

    Clearly, a vertically averaged rheology implies a simplification of the actual mechanical behaviour of the lithosphere. However, as shown by Sonder and England (1986), a single power-law rheology is sufficient to describe the major features related to lithospheric deformation. The main advantage of the plane-stress approach lies in the ability to model those geological structures that show a three-dimen- sional geometry (arcuate orogens, finite structures, transpression, etc.). As a trade-off, details of the vertical variations of deformation are lost.

    6. The problem of the initial perturbation

    Dynamic models need initial perturbations (lat- eral heterogeneities) in order to concentrate defor- mation in a finite area. This is a necessary con- dition in plane-strain approaches which usually as- sume boundary conditions such as depth-indepen- dent strain rate or velocity, or a constant extensional force. In contrast, plane-stress approaches can con- centrate deformation without initial perturbations by applying specific boundary conditions such as in- denters, fixed/slipping boundaries, and the like. Both types of approach, as well as pure or simple shear kinematic models, can incorporate initial perturba- tions in order to modify the mode of deformation. Initial perturbations may be grouped in four cate- gories: thickness variations, pre-existing faults, ther- mal anomalies, and rheological inhomogeneities. All of them result in a lateral strength variation.

  • M. Ferngmdez, G. Ranalli/Tectonophysics 282 (1997) 129-145 139

    6.1. Thickness variations

    Crustal thickening produces lithospheric weaken- ing due to both the replacement of stronger mantle material by softer crustal rocks, and the tempera- ture increase produced by radiogenic heat sources. Therefore, thermally relaxed orogens and areas with relatively high crustal thickness but without promi- nent lithospheric mantle roots are favourable struc- tures to concentrate deformation (see e.g., Lynch and Morgan, 1990; Harry and Sawyer, 1992; Bassi et al., 1993). Several plane-strain dynamic models, based on perturbation theory, address the problem of the evolution and stability of the initial perturbation during extension. Different rheologies are used: for example, a strong plastic layer overlying a viscous layer (Fletcher and Hallet, 1983); two strong lay- ers separated by a weak layer (Zuber and Parmentier, 1986; Zuber et al., 1986); an elastic layer with a plas- tic weak zone overlying an inviscid layer (Lin and Parmentier, 1990). These models show that, in most cases, an initial perturbation grows during exten- sion. The rate of growth depends on the wavelength of the initial perturbation, and the resulting charac- teristic wavelength depends on the layer thickness. Two strong layers give two predominant wavelengths under both extension and compression (Froidevaux, 1986; Ricard and Froidevaux, 1986; Burov et al., 1993). However, Bassi and Bonnin (1988) found that a layered lithosphere behaves stably unless it is more dense than the asthenosphere. Similar results were obtained by Govers and Wortel (1993, 1995), who used a two-dimensional layered model and found that initial boudinage is amplified only for very high strain rates.

    6.2. Pre-existing faults

    Deep faults can also concentrate deformation since they are assumed to behave as slip surfaces with very low resistance to motion. Faults and de- tachments are by definition surfaces of discontinuity, and therefore cannot be incorporated in continuum mechanics models. To avoid this problem in nu- merical algorithms, faults are commonly treated as narrow channels with very low viscosity (e.g., Braun and Beaumont, 1989b; Boutilier and Keen, 1994), as slippery nodes (Melosh and Williams, 1989) or as

    plastic shear bands (e.g., Dunbar and Sawyer, 1989; Makel and Walters, 1993). Their position and orien- tation are usually pre-determined. The incorporation of faults strongly modifies the resulting pattern of deformation because of the associated mechanical and thermal effects. Kinematic models show that heat advection related to the relative displacement between the hanging- and foot-wall can produce a downward displacement of the brittle-ductile tran- sition and consequent fault growth (Willacy et al., 1996).

    6.3. Thermal inhomogeneities

    Temperature is one of the most important param- eters in determining the total lithospheric strength: the higher the temperature the weaker the litho- sphere. Thus, initial lateral variations in temperature concentrate the deformation around the warmer ar- eas, which under extension will enhance the ther- mal anomaly. Thermal inhomogeneities applied to plane-strain models result in narrow rifts which are characterized by a very fast evolution and a large de- parture from local isostatic equilibrium (e.g., Chery et al., 1990; Lynch and Morgan, 1990). Plane-stress approaches are also capable to simulate thermal per- turbations by varying the initial surface heat flow distribution, which induces changes in the average lithospheric viscosity (Vilotte et al., 1986; Bassi and Sabadini, 1994).

    6.4. Rheological inhomogeneities

    Lateral variations in rock properties related to changes in composition, grain size, and pore fluid pressure lead to rheological inhomogeneities and consequent strength variations. Lithological changes are particularly common in the crust where pressure and temperature conditions allow for different sta- ble mineral phases. Granite intrusions at mid-crustal levels result in localized weaknesses which, when combined with an offset crustal thickening, can pro- duce a simple-shear mode of extension (Harry and Sawyer, 1992). Lateral variations in rheology ap- plied to plane stress models can simulate cratonic and basin environments, corresponding to hard and weak areas, respectively, which may affect the ini- tiation and propagation of shear zones (Tommasi et

  • 140 M. Ferng~ndez, G. Ranalli/Tectonophysics 282 (1997) 129-145

    al., 1995). Lateral variations in pore fluid pressure may enhance these heterogeneities. If the forma- tion of a shear zone is accompanied by dynamic recrystallization with consequent reduction in grain size (cf. Ranalli, 1995 for details), and subsequently the small grain size is frozen into the rock due to a decrease in temperature, a new heating episode will reactivate the shear zone. Therefore, rheologi- cal heterogeneities may sometimes be inherited from previous tectonothermal events.

    7. Boundary conditions

    For discussion purposes, we consider separately mechanical and thermal boundary conditions. With respect to the former, we focus essentially on passive rifting (in active rifting, the drag of the asthenosphere on the lithosphere is prescribed, either kinematically or dynamically). With respect to the latter, we dis- cuss the differences between constant heat flux and constant temperature conditions at the lower bound- ary of the lithosphere (the condition of zero heat flux in the horizontal direction is commonly imposed on the side boundaries of the model).

    7.1. Mechanical boundary conditions

    When conditions are imposed on the vertical sides of the extending lithosphere (passive rifting), they can be either kinematic (constant strain rate or con- stant velocity) or dynamic (constant tectonic force). In both cases, these conditions are usually coupled with the assumption that the deformation (implicitly assumed to be pure shear) is homogeneous, that is, different layers at a given site undergo the same amount of extension. Where this is not the case, i.e., where the stretching factor is a function of depth, the model is usually one-dimensional and the question of how differential stretching is compensated in the horizontal direction is left unanswered.

    Constant-velocity and constant-strain rate bound- ary conditions were discussed by England (1983). Constant-velocity boundary conditions imply that the across-strike width of the extending basin at time t since the beginning of extension is L = Lo + rot, where L,, is the initial width andvo the exten- sion velocity. Consequently, the extensional strain rate k(t) = (1 /L ) (dL /d t ) is a decreasing function

    of time. On the other hand, constant-strain rate boundary conditions (see e.g., Jarvis and McKen- zie, 1980) require that the width varies with time as L = Loexp(~ot), where ~o is the strain rate. Con- sequently, the extension velocity v = (dL/dt) is an increasing function of time.

    Kinematic boundary conditions cannot, by defini- tion, lead to an instability, unless they are coupled with the assumption of a pre-existing lateral strength inhomogeneity of the type discussed in the pre- vious section. Also, if constant-velocity conditions apply, rheological constraints imposed by strength envelopes are themselves a function of time (be- cause of the constant strain rate assumption used in the estimation of envelopes, as discussed previous- ly). Therefore, relevant strength envelopes at a given stage of basin formation should take into account not only varying lithospheric structure and tempera- ture distribution, but also varying strain rate. If, on the other hand, constant-strain rate conditions ap- ply, they must do so only for a limited period of time, as they imply exponentially increasing opening velocity.

    Neither of the above kinematic conditions is uni- vocally related to dynamic boundary conditions, where tectonic force (or stress) is prescribed on the boundaries of the extending lithosphere. A condi- tion of constant tectonic force (equivalent to constant tectonic stress at infinity, that is, away from the ex- tending zone, where the lithospheric thickness does not change) leads to a two-dimensional stress (and therefore strain rate) distribution depending on the spatial and temporal variations of lithospheric struc- ture and temperature (see e.g., Bassi, 1995). Even in the one-dimensional case, stress is concentrated in those parts of the lithosphere which are most resistant to deformation (Kusznir, 1982).

    At the site of necking (either imposed a priori or resulting from initial conditions), the reduction in lithospheric cross-sectional area results in geo- metric stress amplification, independently of and in addition to any dynamic stress amplification due to yielding of the softer layers. Consequently, under constant tectonic force, the deviatoric tensile stress at the site of necking increases with time. This may lead to an instability (accelerating strain rate), if not counteracted by structural and/or thermal hardening processes. The interplay between stress concentra-

  • M. Fern~ndez, G. Ranalli/Tectonophysics 282 (1997) 129-145 14l

    tion, changes in lithospheric strength due to stretch- ing, and thermal relaxation depends on a variety of factors (structure and temperature of the lithosphere before extension, applied tectonic force and its time- dependence, magnitude of strength inhomogeneity). Only a fully coupled dynamic analysis can identify criteria for lithospheric instability.

    7.2. Thermal boundary conditions

    The boundary conditions to solve the heat trans- port equation (apart from fixed temperature at the surface and no lateral heat flow through the side boundaries) are either a fixed heat flow or a fixed temperature at the base of the lithosphere. Which conditions apply depends on the coupling between the lithosphere and the asthenosphere. If it is as- sumed that there are no heat sources such as small- scale convective cells in the asthenosphere, the heat flow trough the base of the lithosphere should be constant during extension. However, this requires that the temperature at the base of the lithosphere decreases with time as extension progresses. In con-

    trast, a constant basal temperature implies that the heat flow increases with time which, in turn, requires heat sources within the asthenosphere. A constant temperature boundary condition matches the defini- tion of a cooling lithospheric plate given by Parsons and Sclater (1977) and is, in fact, the most widely used in basin modelling. The temperature at the base of the lithosphere is taken to be the solidus of peri- dotite Tm, including hydration effects (for instance, 0.85 Tm; Pollack and Chapman, 1977). The tem- perature in the asthenosphere is considered to have a negligible gradient. This condition is particularly useful when the mechanical coupling between litho- sphere and asthenosphere is modelled (e.g., Buck, 1986; Keen and Boutilier, 1995).

    8. Concluding remarks

    Our critical review of the role of rheology in the modelling of lithospheric extension (basins and rifted margins), summarized in the diagram shown in Fig. 4, allows two important general statements to be made. The first is that kinematic and 'back-door rhe-

    Mode of Deformation 'a posteriori'

    Basin Formation Models

    f Kinematic ""

    No controls on actual rock-rheology Thermal and subsidence evolution

    , ~ of given basins

    constraints ~ , , ~ J of total strength ~ R h e ° l ° g y Partial controls on actual f ~

    rock-rheology j

    Dynamic

    Self-consistent with actual rock-rheology ,~ Fundamental processes j

    Fig. 4. Diagram showing the linking between the mode of deformation and the role of rheology in basin formation models. Consistency of models increases at the expense of versatility.

  • 142 M. Fernandez, G. Ranalli/Tectonophysics 282 (1997) 129-145

    ology' models are very successful in reproducing the subsidence~uplift and thermal histories of extensional basins. Given that these models are subject only to first-order rheological constraints, this success may appear surprising. Methodologically, it is due to the possibility of adjusting parameters freely, through the kinematically imposed deformation field. This is not to detract from their importance. The accurate prediction of sedimentary patterns and temperatures is not only interesting per se, but has also economic applications, for example in the estimation of the degree of thermal maturation of potential hydrocar- bon deposits. Furthermore, the fact that an imposed deformation field is compatible with observation im- plies that the mechanical properties of the lithosphere must be such as to allow that deformation field. From the predictive viewpoint, therefore, the present situ- ation in rheological modelling of sedimentary basins seems to be one of 'simplest is best', in the sense that kinematic conditions together with simplified rheological constraints produce excellent agreement with observation.

    On the other hand, kinematic and 'back-door rhe- ology' models cannot account, except in an ad hoc way, for physical processes such as the localiza- tion of extension, the weakening or strengthening of the lithosphere during extension (leading to whole lithospheric failure or to cessation of stretching), and for the interplay between tectonic forces and time- and space-dependent rheological properties. Although dynamic models explain consistently the mode of deformation, the high non-linearity of con- stitutive equations and the uncertainties on the actual structure, composition and rheology of the litho- sphere make the results very sensitive to the initial and boundary conditions. Consequently, a second general statement is that progress in the knowledge of the rheological properties of the lithosphere and further refinement of dynamic models are necessary conditions for new insights into the physics of litho- spheric deformation.

    In a sense, kinematic models (with or without rhe- ological constraints, and 'thermomechanical' in the sense that they yield velocity and temperature fields) and dynamic models have complementary roles. The former successfully simulate the formation and evo- lution of given basins. The latter give insight into the fundamental processes governing basin dynam-

    ics. Despite remarkable advances in both types of modelling, a lot of work remains to be done.

    Acknowledgements

    This paper was presented, in a preliminary form, at the Sixth Workshop on 'Origin of Sedimentary Basins' held at Sitges, Spain, in September 1995. The authors thank all the colleagues who, through discussion and criticism, have contributed to the evolution of their ideas. G.R. acknowledges sup- port from NSERC (Natural Sciences and Engineer- ing Research Council of Canada). M.E acknowl- edges support from the European Union 'Integrated Basin Studies' project (Nr. JOU2-CT92-0110). Fruit- ful suggestions and comments from E. Burov, S. Cloetingh, and G. Spadini have been incorporated during the review process.

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