Fracture-based evaluation of fi required to form a graben
Normal faults in the upper to middle levels of continental crust develop by brittle, or semi-brittle failure, and result from a lateral reduction in stress. Of course, the lateral extension model proposed by McKenzie will result in a reduction of stress, but, as we shall see, only quite small lateral extension is required to cause the initiation and development of normal faults with linear sections.
Consider the section of Figure 4.9d which is shown, simplified, in Figure 4.10. The reduction in lateral stress required to initiate faulting near the surface will be small, but will increase in magnitude at progressively deeper levels in the crust. The exact value of stress required will be controlled by the physical constants of the rock mass (which include the cohesive strength, the coefficient of friction and the fluid pressure within the rock mass (see Price and Cosgrove, 1990)). Let us assume that the rock mass is dry, for in such conditions the required reduction in magnitude of the horizontal stress is maximised and so will favour the
Figure 4.9 The B strain model and thinning of continental lithosphere and subsequent migration isotherms (after McKenzie, 1978).
McKenzie model. Let us further make the reasonable assumption that failure at depth takes place when the ratio of the maximum to least principal stresses is 4:1, and estimate the stresses required to cause failure at a depth of 20 km. At such a depth in the crust, the total vertical stress will be approximately 5.0 kb. Hence, to give rise to normal faulting (at a ratio of maximum to minimum principal stress of 4:1), the least horizontal principal stress must be 1.25 kb. As we have seen in Chapter 2, in conditions when the horizontal plane is unstrained, the initial ambient horizontal stress will be about 1/3 of the vertical stress (i.e. 1.7 kb). Hence, the lateral extension required is that which will reduce the lateral stress by approximately 0.45 kb. Let us now estimate the strain required to cause such a reduction in horizontal stress. If we take the average Young's modulus of strong rocks in the crust to be 105 bar, then the extensional strain (e) required to cause a lateral stress reduction of 450 bar is given by 450/100000=0.0045. That is, the line AA' (length 100 km) in Figure 4.10 need only be extended by 0.45 km to incur fault initiation (i.e. B=1.0045).
We now further assume that two normal faults form facing each other (Figure 4.10), and at the surface the faults are 100 km apart and both dip inward at 65°. Hence, at a depth of 20 km the fault planes will be 77 km apart. To bring about a 6.0 km vertical movement on the normal fault, the lateral extension needs to be 7 km (B=1.07). Boundary faults to the graben, with this amount of downthrow, would cut and displace the strong layer of the continent, thereby giving rise to rifting which may potentially split a continent. Even if we have overestimated the value of Young's modulus by a factor of 2.0, this would only lead to B=1.14. Real grabens develop over long periods, so may incur a significant proportion of ductile strain. The simple B model which has been applied to such 'natural' graben development, significantly overestimates the amount
Figure 4.10 Section through an idealised, hypothetical graben, indicating the extension required to generate downthrow of 6 km in the graben.
of stretching required to produce the structural effects caused by instantaneous brittle failure. To apply this concept to the instantaneous thinning of lithosphere by lateral stretching for B=20 to 50 is simply not valid.
The problem of lithospheric thinning and the quantity of melt that can be produced as a result of this thinning was addressed by McKenzie and Bickle (1988). Their argument, to which the reader is directed, is presented in a long and thorough paper which cannot be adequately summarised here. However, briefly, the authors point out that calculations of the volume and composition of melt induced by thinning of the lithosphere requires knowledge of the variations of melt fractions (X) with pressure P and temperature T. Such a study requires a detailed analysis of experimental data. (They note in passing that the simple model presented by McKenzie, 1978, which has been outlined in earlier paragraphs, omitted such analysis.) The approach used by these authors was empirical and quantitative; and consisted of three steps.
(1) The authors obtained analytical expressions for variations of the solidus Ts and the liquidus Tl temperatures. These expressions were based on experimental data derived from various sources.
(2) They then derived the melt fraction (X) as a function of P and T and
(3) established the melt composition as a function ofX and P.
It was further assumed that the thickness of melt was generated by instantaneous, adiabatic decompression of asthenospheric mantle.
McKenzie and Bickle then expressed the relationship between melt thickness and temperature for different values of B over an extended range of 2 to 50, as shown in Figure 4.11a. White (1992) later modified these findings by using what he considered to be the more correct value of 400 J (kg K-1) rather than 250 J (kg K-1) for the entropy change on melting. Accordingly, he considers that the relationship between melt thickness, temperature and B is better represented by Figure 4.11b.
We note that two physical parameters enter into the calculations conducted by White and by McKenzie and Bickle; they are the (1) the thickness of the oceanic crust and (2) the entropy change on melting. The range and mean thickness of oceanic crust are well established and are cited in both papers by these authors. The difference between entropy values of 400 and 250 J (kg K-1) is not trivial. It will be inferred from Figure 4.11a and Figure 4.11b that the White version requires significantly smaller values of B=20, rather than that of B=50, proposed by McKenzie and Bickle to generate oceanic crust. Thus, even if one accepts
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