Haimson and Fairhurst (1967) derived equations for the near-wellbore stresses (axial σyy, tangential σθθ, radial σrr...) around a pressurized hole in a porous, permeable "poroelastic" medium considering fluid infiltration from the borehole into the surrounding rock. Their expressions include Biot's constant, A which depends on the Biot poroelastic coefficient, αB.
Should at αB = 1.0 those expressions give the same answer as when using the standard near-wellbore stress equations for linearly elastic media (the Kirsch solutions modified later by Hubbert and Willis to incorporate pp, on pages 170 and 174 in Zoback, 2010)? I was of the impression that the answer will be "yes" but the math seems to prove me wrong.
Dear Andreas,
it will be helpful if you provide your math which does not add up.
Sergey
Apparently, the approximation Haimson and Fairhurst (1967) used (from Terzaghi) to modify the Hubbert and Willis (1957) expressions to account for fluid infiltration from the wellbore to the porous-permeable surrounding rock does not match the corresponding equations for linearly elastic media.
The discrepancy between the two models increases with (pw - pp) increase and is minimized to zero at pw = pp. This leads to the conclusion (which I should have known) that using αB = 1.0, despite yielding the same effective stress (σij = Sij - αΒδijpp), it will not yield the same wellbore stresses values (namely axial σyy and tangential σθθ). Fig. 7b on my recent JPSE paper (link below) shows that taking account of linear poroelasticity in expressions used to approximate the stress state on the perforation base (intersection of two cylinders of different diameter) yields different results compared to using the corresponding expressions for linear elasticity.
Article Analytical Orientation Criteria for Drilling and Completion-...
I attached three figures demonstrating that drawn using inputs (Moos, 2012) for a horizontal well drilled parallel to Shmin the Barnett Shale in Texas:
(1) Variation of σyy and σθθ with angle θ for pw = Shmin
(2) Variation of σyy and σθθ with pw at the top of the hole (θ = 0 deg)
(3) Variation of σyy and σθθ with pw at the top of the hole (θ = 90 deg)
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