Measurements and Inverse Modeling for Far-Field State of Stress Estimation
Authors: Sankaran Sathish , Deeg Wolfgang , Nikolaou Michael , Economides Michael J. 
Published: December 2007

Abstract
The near-well stress concentration is affected by several factors, including the far-field stresses, the well deviation relative to the principal stresses, and the well completion configuration (e.g., open-hole or cased, cemented and perforated.) The stress concentration has been known to greatly affect the fracture initiation pressure and the near-well fracture geometry. Under adverse conditions this will cause problems during execution such as excessive fracturing pressures and premature screenouts caused by tortuosity. The latter also affects the post-treatment well performance with especially severe consequences in high-permeability formations.
For an open-hole completion, predictive models are available to evaluate both the fracture initiation pressure and the near-well fracture tortuosity, given the far-field stresses and all the angles that describe the well position and the fracture initiation point. The solution to this forward problem, i.e., the calculation of the fracture geometry and fracturing pressure when everything is known, can be used to avoid highly undesirable situations a priori. A typical conclusion would be that wells intersecting the preferred fracture plane are generally far less attractive candidates for hydraulic fracturing than vertical or horizontal wells aligned with this plane.
The inverse problem is more interesting and is posed as follows: Assuming that any pressure perturbation (fracture or not) or displacement can be detected around the well's periphery and that the pressure profile is measured, is it possible to calculate the far-field stresses and the fracture direction vis a vis the fracture plane? Although the measurement method is not described in this paper, we have determined that direct measurements of pressure perturbations and displacements at the wellbore are possible and can be used as suggested by this study.
A closed-form analytical model has been developed to analyze the symmetric state of stress within an arbitrary number of concentric cylinders. This solution can be used to calculate the induced stresses within these cylinders in direct contact with each other, with known contact pressures. The general case of an asymmetric stress distribution such as in inclined wells is modeled numerically with boundary element techniques. Further, the results for a fractured two-zone case (cement sheath and formation) are compared to the non-fractured case to understand the effect of the redistributed stress concentration on the well completion (e.g., casing or cement). A parametric study provides clues to determine the nature and choice of well architecture when a hydraulic fracture is considered.
An inverse problem is described and solved for the open-hole situation with the least available information, where the far-field stresses and the well departure angle (angle of departure on the horizontal plane from the principal horizontal stress direction) are unknown. The method illustrates how the displacement measurements at the wellbore can be used to back-calculate the far-field stresses and, thus, the true well departure angle. Some numerical issues are addressed, which also determine the number of wellbore measurements needed for the inverse problem solution. Finally, the effect of noisy measurements on the accuracy of the results is studied.