How much does slip on a reactivated fault plane constrain the stress tensor?

Abstract

Given a fault plane and its slip vector, the stress tensor which caused the displacement is sought. Two constraints are considered: first, a geometrical constraint that the shear stress applied to the fault plane is parallel to the slip (Wallace, 1951; Bott, 1959); second, a frictional constraint that the shear to normal stress ratio equals tanϕ 0 (Coulomb, 1776). In a first step, the stress tensors that satisfy the geometrical constraint are sought. For tensors belonging to the vectorial space of solutions, shear and normal stress magnitudes become a function of the orientation of the principal stresses, , , , and are mapped, extending a study by McKenzie (1969). In a second step, it is investigated which of these tensors also satisfy the frictional constraint. Within this more restricted vectorial space, there is a relationship between the magnitudes, δ = (σ 1 − σ 2 )/(σ 1 − σ 3 ) and s = (σ 1 − σ 3 )/σ 1 , and the orientations, , , , of the principal stresses. Both the range of s and the spatial distribution of , , are more restricted than when the geometrical constraint alone is considered. As when the geometrical constraint is solely considered (McKenzie, 1969), the orientations of the principal stresses, , , , may lie significantly away from and up to right angle to the P, B, T axes. However, this can happen only in two cases: (1) either the effective stress difference s has reached a high value, which is unlikely to happen if enough preexisting fractures are available to release the stress, or (2) σ 2 becomes close to either σ 1 or σ 3 and therefore barely distinguishable from it; in that case the delocalization of the orientations of the principal stresses is best described by a tendency for to exchange role with either or . When the stress difference remains small and σ 2 reasonably away from σ 1 and σ 3 , , , approach positions that we define as the P f , B, T f axes and that are obtained from the P, B, T axes by a rotation of angle ϕ 0 /2 around B and toward the slip vector. This explains why the P, B, T axes gives reasonable estimates of the orientations of the principal stresses (Scheidegger, 1964) despite objections (McKenzie, 1969). However, whenever the fault plane can be distinguished from the auxiliary plane, P f , B, T f should give a better estimate (Raleigh et al., 1972). In an area where many fault planes are available and a uniform tensor is assumed, the scatter in the plane orientations contains information about both the relative position of σ 2 , represented by δ, and the relative stress difference s: the higher s or the closer δ to either 0 or 1, the more scatter. This information could then be extracted by inverse methods. Because a friction law would constrain these inverse method more tightly, it may show the necessity of nonuniform tensor to explain scattered fault plane.