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Re-Examination of the Wellbore Mechanics Basis for Interpreting Leak-off Tests – Impacts on Stress State and Pore Pressure Determinations


The Hubbert and Willis (1957) analysis of wellbore stress (and subsequent modifications to it), based on Kirsch's (1898) analytical model, still informs the interpretation of leak-off test information, underpinning estimations of far-field stress and pore pressure. Given the importance of these estimates – in relation to efficiency of well construction, to safety issues associated with loss of well control, and to impact on subsequent well stimulation – we have re-examined the wellbore mechanics model that is the basis for these activities. The Kirsch model is said to be the solution to the mechanical problem of determining the stress distribution associated with a hole that is located within a plate. Finite-element solutions to that idealised problem differ from the Kirsch expressions by as much as 10-15% at critical locations on the wall of the “opening”. A deep analysis reveals that the Airy stress function concept, which underlines Kirsch's approach, is not mechanically correct, and that the stratagems employed by Kirsch (and those who followed) introduce impossible contradictions that render the method invalid. However, the issues with the analytical “solution” are only a diversion. The real issue is that the classical mechanical problem – ie, loading a plate that has a pre-existing hole – is wrongly conceived. The reality is that a hole is created into a pre-loaded domain. When this problem is simulated, the outcomes (in terms of tangential stress at the wall of the hole) are smaller than in the pre-hole case by factors ranging from 2 to 50. These differences occur in fully-elastic conditions where no material failure is allowed. The elastic strain energy of the “excavation” case is only about ~60% of the energy contained in the pre-hole (or analytical) problem. If failure modes are introduced into the numerical solutions, the stress/energy differences become even greater. A superficial conclusion is that these results do not matter, since the classical estimates of far-field stress (which are over-estimated) or pressure are re-used within the same model framework to make operational decisions. But, they do matter if we seek to understand the phenomena taking place, and they matter a great deal if we wish to employ rock mechanics knowledge to understand and then predict rock failures and their consequences – especially for hydraulic fracturing activities.