Issue 45

Q.-C. Li et alii, Frattura ed Integrità Strutturale, 45 (2018) 86-99; DOI: 10.3221/IGF-ESIS.45.07 89 tot eff p P   = + (1) In order to accurately determine the effective stresses around the arbitrary borehole, four coordinate systems (Fig.2) are defined, that is, the in-situ stress coordinate system ( X S -Y S -Z S ), the geographic coordinate system ( X G -Y G - Z G ), the borehole coordinate system ( X B -Y B -Z B ) and the wellbore polar coordinate system ( r-θ-Z B ). The in-situ stresses ( σ H , σ h , and σ v ) can be transformed from the in-situ stress coordinate system to the borehole coordinate system with two transformation steps [30]. Firstly, the in-situ stresses are transformed from the in-situ stress coordinate system to the geographic coordinate system. Then the stresses in geographic coordinate system are transformed from the geographic coordinate system to the borehole coordinate system. The whole transformation process can be expressed by the following equation. x T = = xx y xz B B B yx yy yz T B B B B B S P S B zx zy zz B B B R R R R                        (2) where the tensor R S and R B are expressed as Eqn. (3) and Eqn. (4). - = - + - - S cos cos sin cos sin R cos sin sin sin cos sin sin sin cos cos cos sin cos sin cos sin sin sin sin cos cos sin cos cos                                                        (3) ( ) ( ) ( ) ( ) - ( ) = - ( ) ( ) 0 ( ) ( ) ( ) ( ) ( ) B cos Az cos Inc sin Az cos Inc sin Inc R sin Az cos Az cos Az sin Inc sin Az sin Inc cos Inc           (4) The stresses around borehole in the cylindrical coordinate system can be expressed as Eqn. (5) when the drilling fluid pressure is P m [31]. ( ) ( ) ( ) y y x = - = + -2 - 2 -4 2 - - = -2 - 2 +2 2 - =2 - r m p xx yy xx yy x B B B B B m p zz xx yy x z B B B B p yz z z B B P P cos sin P P cos sin P cos sin                             (5) The maximum and minimum effective principal stresses need to be determined when Mohr-Coulomb criterion is used for borehole stability analysis. However, determining the three effective stresses around borehole ( σ i , σ j and σ k ) in borehole cylindrical coordinate system is the basis, which can be calculated using Eqn. (6). ( ) ( ) ( ) ( ) 2 2 2 2 = = - =0.5 + +0.5 - +4 =0.5 + 0.5 - +4 i r m p j z z z k z z z P P                     − (6) The maximum and minimum effective principal stresses can be obtained by comparing these three effective stresses.