Characterization of magnitudes and directions of principal stresses is vital for building a geomechanical model. The magnitude of the maximum horizontal stress is most challenging to estimate among three principal stresses. Meanwhile a well known Kirsch solution formulated in terms of displacements, allows for estimating the ratio of minimum to maximum horizontal stress in vertical wells when displacements of the borehole wall can be measured (e.g., by multi-arm caliper). This study evaluates the limits of applicability of such analysis.
Multi-arm caliper readings may be indicative of both inelastic (i.e., rock failure) and elastic (e.g., contraction) deformation. In case of inelastic deformation (e.g., breakouts or key seats) Kirsch solution is not applicable. Meanwhile within the limits of elasticity, the solution can be more useful. Drilling history or a BHI log may assist in identifying zones where no failure or yielding have occurred. Then, potentially, the solution allows for reconstructing two of the following parameters: i) ratio of the minimum to maximum principal stresses acting in the plane orthogonal to the wellbore axis, ii) static Young's modulus, and iii) static Poisson ratio. The suggested approach includes the following two steps. In the first step multi-arm caliper data are used to calculate the ratio of minimum to maximum displacement Ku and translate it to the ratio of stresses where v is the static Poisson ratio. In the second step the analytical solutions for non-Kirsch displacements can be used to estimate static shear modulus.
Practical application of the methodology highlighted the need for additional calibration of the log derived data. Firstly, the displacements derived from the multi-arm caliper need to be corrected in regard to the orientation of the caliper arms relative to the principal stress directions. Because of the irregularities in caliper signatures this can be challenging especially if the principal directions are not known. Secondly, careful analysis of the resistivity logs indicates that caliper measuremenst may be affected by the formation of a filter cake. Finally, the estimate of the elastic moduli relies on accurate calculations of displacements. The case study discussed in this paper, illustrates that the desirable accuracy may not be attainable because the actual borehole diameter may be uncertain (e.g., due to back reaming or a filter cake) or the accuracy of caliper readings may be inadequate. Meanwhile the estimation of the maximum stress relies on the displacement ratio Ku rather than the absolute value of the displacement. Hence, the estimate for the ratio is expected to be more accurate.
The suggested technique allows for easy estimation of the maximum horizontal stress. In some cases when borehole diameter is certain and caliper readings are accurate, the technique also provides the means for estimating static shear modulus. The estimate can be further used to decrease the uncertainty of the results obtained with more conventional and sophisticated approaches.