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10 <strong>ISRM</strong> (India) Journal STUDY OF WELLBORE STRESSES AND STABILITY BASED ON A HOLLOW CYLINDER MODEL P.A. Nawrocki, Z. Qi and D. Wang Department of Petroleum Engineering, The Petroleum Institute, P.O. Box 2533, Abu Dhabi, UAE ABSTRACT The linear elastic theory has been used in the parametric analysis of borehole stresses in the hollow cylinder model. The analysis has been conducted in terms of two major parameters that introduce different geometries and loading conditions and have significant influence on the critical wellbore pressures. Different outer diameters and different hole sizes have been considered and their impact on stresses and failure investigated, both for dry and saturated rock. The Mohr-Coulomb, the Drucker-Prager and the Modified Lade criterion have been used and the safe mud weight window has been defined in each case. It has been shown that pore pressure plays an important role in borehole stability and that the Mohr-Coulomb criterion is a safe but apparently conservative choice as it predicts higher well collapse pressure than the other two criteria. On the other hand, the Drucker-Prager criterion is a very optimistic criterion that predicts very lowest values of collapse pressure, but it significantly overpredicts the σ 2 strengthening effect. Providing predictions between the extremes of the other two criteria, the Modified Lade criterion is a moderate criterion that seems to account for the σ 2 strengthening effect in a reasonable way and give reasonable mud weight choices required for maintaining well stable. SUBJECT : Stress analysis, wellbore stability, modelling, numercial methods KEYWORDS : Numerical modelling, rock stress, stability analysis, rock failure, rock properties. 1. INTRODUCTION Hollow cylinder modeling is often used in wellbore stress and strain distribution estimation, borehole stability analysis and sand production prediction [1-4] . Hollow cylinders have the advantage that their configuration is similar to underground openings and that makes them an ideal tool to simulate wellbore situations. As for stress and strain distribution estimation and borehole stability analysis, a number of theories and approaches of hollow cylinder modeling have been proposed. Classic elastic theory [1] has been widely used to analyze the stress concentrations around open holes and borehole stability. It assumes the formations with isotropic properties and elastic behavior and can be used to provide approximation to the stress, strain and displacement distribution around underground openings. Elastic-plastic theory [4,5] has been proposed to analyze more complex rock behaviour and poroelastic theory [6,7] has been also used as it can account for the coupled interaction between rock matrix and saturating fluids. In addition, finite element simulations have been successfully conducted [8] to simulate the stress distributions and rock failure around wellbores under different loading conditions. The linear elastic theory has been used in this paper to conduct the analysis of stress, the displacement distribution and borehole stability. Firstly, different loading conditions have been applied to the hollow cylinder to analyze the influence of the internal and external pressure on the stress and displacement. Secondly, different outer diameters and hole sizes have been chosen to investigate their impact on stress and displacement distribution. Furthermore, three widely used failure criteria, the Mohr- Coulomb (considered the conventional “triaxial” criterion), the Drucker-Prager, and the Modified Lade criterion have been used to explore and compare the variation trends of critical wellbore pressures under the above two conditions. Both dry and saturated conditions have been considered and some practical recommendations for maintaining wellbore stability have been formulated. 2. ROCK FAILURE CRITERIA Failure criteria provide limits to wellbore stresses and knowledge of rock strength is essential for accurate rock failure analysis and wellbore instability prediction. They are derived from laboratory tests on rock samples and can be typically divided into those that depend on all three principal stresses, σ 1 , σ 2 , and σ 3 , and those that neglect the effect of the intermediate principal stress σ 2 on failure. The Mohr-Coulomb criterion belongs to the latter group and is thus applicable to conventional triaxial test data (σ 1 > σ 2 = σ 3 ). The two “triaxial” criteria, i.e. the Modified Lade and the Drucker-Prager criterion, consider the influence of the intermediate principal stress in polyaxial strength tests (σ 1 >σ 2 >σ 3 ). They are compared to Mohr- Coulomb in Fig. 1. Volume 1 No. 2 July 2012
Study of Wellbore Stresses and Stability based on a Hollow Cylinder Model 11 ...(6) Note that S o can be linked to C o and ϕ through S o = C o /2q 1/ 2 where q = [(μ 2 +1) 1/2 + μ] 2 = tan 2 (π/4 + ϕ/2]. Fig. 1 : Comparison of different rock failure criteria in the π-plane. Mohr proposed that when shear failure takes place across a plane, the normal stress σ n and the shear stress τ across that plane are related by |τ| = S o + μs n ...(1) where S o = cohesion and μ = the coefficient of internal friction of the material which is related to the angle of internal friction ϕ of that material by μ = tanϕ. Since the sign of τ only affects the sliding direction, only the magnitude of τ matters. Presented in terms of principal stresses, the Mohr-Coulomb criterion is: σ 1 = C o + σ 3 tan 2 β ...(2) where σ 1 = the major principal effective stress at failure, σ 3 = the least principal effective stress at failure, C o = the uniaxial compressive strength, and β gives the orientation of the failure plane and is related to internal friction angle ϕ as β = π/4 + ϕ/2. In some formulations tan 2 β is replaced by q, where q = [(μ 2 +1) 1/2 + μ] 2 . The Lade criterion [9] is a three-dimensional failure criterion. Originally proposed for cohessionless sands, the criterion was then adopted for analyzing rocks with finite values of cohesion and tensile strength [10] and such a formulation was later linked [11] with standard rock mechanics parameters such as ϕ and S o to obtain: where I 1 ’ and I 3 ’ are stress invariants ...(3) I 1 ’ = [(σ 1 + S – P p )+(σ 2 + S – P p ) +(σ 3 + S – P p ) ...(4) I 3 ’ = [(σ 1 + S – P p )(σ 2 + S – P p ) (σ 3 + S – P p ) ...(5) where S and η are material constants, and P p is the pore pressure. The parameter S is related to the cohesion of the rock, while η represents the internal friction. These parameters can be derived directly from the Mohr-Coulomb cohesion S o and internal friction angle ϕ by The typical behaviour of the Modified Lade criterion is shown in Figs 1 and 2. In principal stress space criterion [3] has the form of a convex, triangularly shaped cone. The parameter η determines the shape of the cross-section in the π-plane: increasing values of h correspond to the crosssectional shape changes from circular to triangular with smoothly rounded edges, Fig. 1. When plotted in σ 1 - σ 2 space, the criterion first predicts a strengthening effect with increasing intermediate principal stress s 2 followed by a slight reduction in strength once s 2 becomes ‘‘too high’’, Fig. 2. Thus, the Modified Lade criterion seems to provide a good alternative to the Mohr-Coulomb criterion. Fig. 2 : Behaviour of different failure criteria in σ 1 -σ 2 space for different values of confining stress σ 3 . The extended von Mises yield criterion, or the Drucker- Prager criterion, was originally developed for soil mechanics. The yield surface of that criterion in principal stress space is a right circular cone equally inclined to the principal-stress axes, Fig. 1. The intersection of the p-plane with this surface is a circle and the Drucker-Prager yield function has the form: 1/2 J 2 = k + α 1 J 1 ...(7) where J 1 = (σ 1 +σ 2 +σ 3 )/3 ...(8) J 2 = [(σ 1 -σ 2 )+(σ 2 -σ 3 ) +(σ 1 -σ 3 ) 2 ]/6 ...(9) The material parameters α 1 and k can be determined from the slope and the intercept of the failure envelope plotted 1/2 in the J 1 - J 2 space: α 1 is related to the internal friction of the material and k is related to its cohesion. In this way, the Drucker-Prager criterion can be compared to the Mohr- Coulomb criterion. The Drucker-Prager criterion can be further divided into an outer bound criterion (or Circumscribed Drucker-Prager) and an inner bound criterion (or Inscribed Drucker-Prager). These Volume 1 No. 2 July 2012
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