Subsurface stress assessment from cross-coupled borehole acoustic eigenmodes

We present an approach, based on cross-coupling of quadrupole and monopole borehole acoustic modes caused by anisotropy, to investigate the in situ stress state, a critical parameter for effective CO₂ sequestration and for determining subsurface injection bounds in general. We focus on in situ stress states where the vertical direction is a principal stress direction, and we aim at determining the minimum and maximum horizontal stresses. Because of non-linear elastic effects, three unequal principal stresses in an otherwise isotropic rock may create three orthogonal planes of symmetry, which characterize an orthorhombic elastic medium. Near a wellbore, where the stress field is perturbed, a stress sensitive material causes material axes and moduli to form a spatial distribution to which sonic logging is sensitive. We present a method for differentiating between stress induced and intrinsic anisotropy. Using finite element modelling, we demonstrate that in either case the quadrupole fundamental mode includes an axis-symmetric (monopole) component. We demonstrate that the acoustic amplitude at the borehole centre divided by the maximum acoustic amplitude at the wellbore periphery (dominated by the acoustic profile cos(2θ)) is an indicator of elastic anisotropy. We denote this ratio I_A and argue that I_A > 0 when the elastic anisotropy is of entirely intrinsic origin (meaning the elastic moduli are in-sensitive to stress), and further that I_A increases for decreasing frequency for such cases. We demonstrate that I_A attains negative values for increasing frequency in stress-sensitive formations where a cross-over (from negative to positive values) is attributed to the perturbed velocity/moduli/stress fields near the wellbore. In synthetic data, we show that the ratio I_A, in combination with the phase velocity dispersion, uniquely determines the state of stress in stress-sensitive formations. In stress in-sensitive formations, we argue that I_A at lower frequencies, that is at frequencies slightly above the cut-off frequency, is very sensitive to elastic anisotropy. We argue that in quadrupole eigenmodes, evidence of intrinsic anisotropy is present at low frequencies whereas stress induced anisotropy is better gauged at moderate to high frequencies. Finally, we discuss the practical implications of these findings.