Geodynamics Brittle deformation and faulting Lecture 11.6 - Predicting fault orientations Lecturer: David Whipp david.whipp@helsinki.fi Geodynamics www.helsinki.fi/yliopisto 1
Goals of this lecture Introduce Anderson s theory and how it can be applied to predict fault orientations 2
Faulting Beartooth Plateau, Wyoming, USA 3
Faulting Beartooth Plateau, Wyoming, USA 4
Anderson s theory Anderson formulated the Mohr-Coulomb criterion in terms of principal stresses and lithostatic pressure (σyy = ρgy) a 2 Anderson assumed σyy = σ3 for a reverse fault, σyy = σ1 for a normal fault and σyy = σ2 = (σ1 + σ3)/2 for a strike-slip fault 1 Normal fault 3 b 2 3 Reverse fault 1 c 2 1 Strike slip fault 3 Fig. 5.8, Stüwe, 2007 5
Anderson s theory c = 0; fs = 0.85 Fig. 5.7, Stüwe, 2007 In terms of differential stress, Anderson s theory is Reverse: Normal: Strike-slip: d = 1 3 = 2(c + f s( yy w gy) p f 2 s +1 f s d = 1 3 = 2(c f s( yy w gy) p f 2 s +1+f s d = 1 3 = 2(c + f s( yy w gy) p f 2 s +1 6
Predicting fault orientation Now we ll look at how to apply Anderson s theory to predict the dip angle β of normal and reverse faults Assume principal stresses yy = gy xx = gy + xx where Δσxx is the tectonic deviatoric stress, which is positive for a reverse fault and negative for a normal fault 7
Predicting fault orientation Fig. 8.8, Turcotte and Schubert, 2014 We first need to relate σxx and σyy to σn and σs in order to apply Amonton s law by using the equation for the normal and shear stresses in a coordinate system rotated by angle θ with respect to the principal stresses (see Lecture set 3) n = 1 2 ( xx + yy )+ 1 2 ( xx yy) cos 2 s = 1 2 ( xx yy)sin2 Note that here, θ is with respect to vertical, θ = π/2 - β 8
Predicting fault orientation Fig. 8.8, Turcotte and Schubert, 2014 If we plug in the values for σxx and σyy, we find xx n = gy + (1 + cos 2 ) 2 Inserting the values above into the form of Amonton s law that includes pore fluid pressure, τ = fs(σn - pw), yields ± xx 2 s = xx 2 sin 2 = f s apple gy p w + sin 2 xx 2 (1 + cos 2 ) Note that the upper sign is for reverse faults (Δσxx > 0) and the lower for normal faults (Δσxx < 0) 9
Predicting fault orientation Fig. 8.8, Turcotte and Schubert, 2014 The previous expression can be rearranged to solve for Δσxx 2f s ( gy p w ) xx = ± sin 2 f s (1 + cos 2 ) If we assume that faulting will occur with the minimum tectonic stress, then Δσxx should be minimized By setting dδσxx/dθ = 0, we find tan 2 = 1 or tan 2 = ± 1 f s f s where the upper sign again applies to reverse faults and the 10 lower to normal faults
Predicting fault orientation Dip angle Figs. 8.9, 8.10, Turcotte and Schubert, 2014 Tectonic stress = 2700 kg m 3 p w = w gy w = 1000 kg m 3 y =5km Finally, the two equations from the previous slide can be combined to yield the tectonic stresses corresponding to angle θ or β xx = ±f s( gy p w ) (1 + f 2 s ) 1/2 f s where the upper sign again corresponds to reverse faults and the lower to normal faults 11
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