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 tanϕ0 (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 = (σ1 − σ2)/(σ1 − σ3) and s = (σ1 − σ3)/σ1, 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) σ2 becomes close to either σ1 or σ3 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 σ2 reasonably away from σ1 and σ3, , , approach positions that we define as the Pf, B, Tf axes and that are obtained from the P, B, T axes by a rotation of angle ϕ0/2 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, Pf, B, Tf 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 σ2, 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.