Harmonising Rock Engineering and the Environment – Qian & Zhou (eds) ? 2012 Taylor & Francis Group, London, ISBN 978-0-415-80444-8
Parametric study of formation stability u
sing a hollow cylinder model
P.A. Nawrocki, Z. Qi & D. Wang
Department of Petroleum Engineering, The Petroleum Institute, Abu Dhabi, UAE
ABSTRACT: The linear elastic theory has been used in the parametric analysis of borehole stresses in the hollow cylinder setting. 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 pressures. 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. Different outer diameters and hole sizes have been considered and their impact on stresses and failure investigated, both for dry and saturated rock. It has been shown that pore pressure plays an important role in borehole stability and that the Mohr-Coulomb criterion is apparently conservative. The Drucker-Prager criterion is non-conservative and overpredicts the σ2 strengthening effect. The Modified Lade criterion is a moderate one, between the extremes of the other two criteria. Subject: Keywords: Modelling and numerical methods numerical modelling, rock failure, rock properties, rock stress, stability analysis tensile strength (Lade, 1984) and such a formulation was later linked with ? and So (Ewy, 1998), to obtain:
1 INTRODUCTION Hollow Cylinder (HC) modelling has been a popular method for borehole stability analysis as geometry of HCs makes them an ideal tool to simulate wellbore situations. In this paper, the linear elastic theory has been used in the parametric analysis of borehole stresses in the HC setting. Different hole sizes and loading conditions have been considered and the influence of the internal and external pressure on stability has been analyzed. The three popular failure criteria, the conventional “triaxial” Mohr Coulomb (MC), the Drucker-Prager (DP) and the Modified Lade (ML) criterion have been used to explore the variation trends of critical wellbore pressures for both dry and saturated conditions.
where I1 and I3 are stress invariants
Pp is the pore pressure and S and η are material constants that can be derived directly from So and ?:
2 ROCK FAILURE CRITERIA Failure criteria can be divided into those that depend on all three principal stresses and those that neglect the effect of σ2 on failure. The MC criterion belongs to the latter group and is thus applicable to conventional triaxial test data (σ1 > σ2 = σ3 ). According to this criterion, the normal stress σn and the shear stress τ across the failure plane are related by
Note that So can be linked to Co and ? as So = Co /2q1/2 where q = [(?2 + 1)1/2 + ?]2 = tan2 (π/4 + ?/2). In principal stress space (2) has the form of a convex, triangularly shaped cone. The extended von Mises yield criterion, or DP 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. The intersection of the π -plane with this surface is a circle, and the DP yield function has the form:
where So is cohesion and ? is the coefficient of internal friction that is related to the angle of internal friction ? of the material by ? = tg ?. In terms of principal stresses, the MC criterion is σ1 = Co + σ3 tan2 ξ , where Co is the uniaxial compressive strength, and ξ gives the orientation of the failure plane, ξ = π /4 + ?/2. The two “triaxial” criteria, i.e. the ML and the DP criterion, consider the influence of σ2 in polyaxial strength tests (σ1 > σ2 > σ3 ). The Lade criterion (Lade & Duncan 1975) was originally proposed for cohessionless sands. It was then adopted for analyzing rocks with finite values of cohesion and
where J1 = (σ1 + σ2 + σ3 )/3, J2 = [(σ1 ? σ2 ) + (σ2 ? σ3 ) + (σ1 ? σ3 )2 ]/6, and the parameters α1 and k can be determined from the slope and the intercept of the failure envelope plotted 1/2 in the J1 and J2 space: α1 is related to ?, and k is related to Co (Colmenares & Zoback 2002). Thus the DP criterion can be compared to the MC criterion. It can be further divided into an outer bound (circumscribed) criterion and an inner bound (inscribed) criterion. The inner DP circle only touches the inside of the MC criterion and the outer DP circle coincides with the outer apices of the MC hexagon.
Figure 2. Effect of Pi on hollow cylinder stresses. Figure 1. Comparison of failure criteria in σ1 -σ2 space.
In Fig. 1 the behavior of both versions of DP is shown in comparison with other criteria for Co = 64.1 MPa and ? = 0.63. It can be seen in Fig. 1 that for fixed values of σ2 and σ3 , the inscribed DP predicts failure at lower stresses than the circumscribed DP, which behaves significantly different than the other criteria. The ML first predicts strengthening effect with increasing σ2 followed by a slight reduction in strength once σ2 becomes “too high”. Thus, the ML criterion seems to provide a good alternative to the MC criterion. Note that it was indicated before (Colmenares & Zoback 2002) that when trying to find the best criterion to fit the test data on different rocks, the MC failure criterion always yielded comparable misfits. Furthermore, the ML polyaxial criterion gave very similar fits of the data and the DP criterion did not accurately indicate the value of σ1 at failure and had the highest misfits. 3 ANALYSIS HC with an inner radius Ri and an outer radius Ro is considered. The external pressure is Po , the internal pressure Pi and the axial pressure is F. Utilizing linear elastic theory, the HC stresses, radial (σr ), hoop (σθ ), and axial (σz ), have been calculated from Hoskins (1969). The effective stresses are given as the difference between the stress and the pore pressure term αB ? Pp , where αB is Biot’s constant. When Pi and Po change, the HC stresses also change. From failure point of view, stresses at the wellbore wall are important as this is where failure is expected. Stresses corresponding to the situation when Po = 80 MPa and Pi varies from 0 to 144 MPa are shown in Fig. 2. Low Pi values favor shear failure; tensile failure at the wellbore wall can be expected for high hole (well) pressures. In fact, the shear stress defined as the difference between HC stresses is increasing when Pi is decreasing, so shear failure can occur for certain critical Pi = Pi coll . In field applications such situations have to be avoided to prevent failure and well pressure has to be maintained above that level. Note that Pi cannot get too high either as this may trigger tensile failure, i.e. fracturing. The difference between these two extreme pressures defines the “operational mud weight window” as during drilling pressure is inserted by the drilling mud. On the other hand, when Pi = const and Po changes, sq at Ri can become negative for low external pressures, Fig. 3, so the tensile failure may occur there first. When Po increases, sq can become positive and shear stress will be largest at the inner
Figure 3. Effect of Po on hollow cylinder stresses. Table 1. Depth (m) 2134 Rock properties and in situ stress data. σh (MPa/m) 0.0139 σv (MPa/m) 0.0235 Pp (MPa/m) 0.0113 So (MPa) 11.2 ? (degree) 32.7
wall so shear failure will also occur at Ri first, irrespective of the axial stress σz which is not going to change the failure pattern. Therefore, failure is always at the inner wall. Failure criteria mentioned above have been used in assessment of critical pressures, fracturing and collapse, that define safe mud weight window. To assess the influence of Pp on stability, analysis has been conducted with and without Pp . The set of data from one local oilfield has been used. It is shown in Table 1 where σh is the in-situ horizontal stress and σv is the vertical stress, and the analysis has been performed in terms of two parameters α and β defined as:
3.1 Maximum shear stress The stresses σr , σθ , and σz , at the inner wall have been calculated, σ1 , σ2 , and σ3 identified, and the maximum shear stress obtained as τmax = (σ1 ? σ3 )/2. Fig. 4 shows the variation of τmax with α. As α increases, τmax decreases, sharply at first and then remains almost constant. Thus, τmax is very sensitive to wall thickness if the HC wall is thin. Then a small increase in
Figure 4. Max shear stress at Ri as function of α and β.
Figure 6. β-collapse, dry case.
Figure 5. β-collapse, pore pressure included.
Figure 7. The effect of pore pressure on β-fracturing.
wall thickness can result in significant τmax reduction. However, τmax in thick-walled HC is much less sensitive to wall thickness. This can provide guidelines for scaling the laboratory HC stress data into the real wellbore situation where α is infinite. Fig. 4 also shows the impact of β on τmax . β = 0 means that Pi is zero and there is maximum pressure contrast between the HC boundaries. For β = 1.0 Pi and Po are equal and there is no pressure contrast. As β decreases from 1.0 to 0, τmax will increase accordingly, so the larger the difference between Pi and Po , the greater the τmax . Therefore, if Po is fixed (such as in-situ horizontal stress) and the internal well pressure is decreasing (such as mud weight-related drilling pressure), τmax at Ri will continuously increase until shear failure can occur. 3.2 Minimum well pressures Fig. 5 shows the critical stress ratio β for hole collapse. Pore pressure has been taken into account and effective stresses used. MC predicts the highest critical β and the DP criterion predicts the lowest. That is mainly because the MC criterion is a two-dimensional criterion that considers only σ1 and σ3 . Therefore, the strengthening effect of the intermediate principal stress is ignored and borehole strength is underestimated. Similarly, the DP criterion is significantly non-conservative. That is mainly due to its overestimation of the intermediate principal stress strengthening effect. The ML criterion is a moderate one, which lies between the above two extreme criteria as it seems to properly account for the influence of σ2 on rock strength. These results are consistent with the results obtained by Zhang et al. (2010). The results of similar analysis but without pore pressure (dry rock) are shown in Fig. 6.The three failure criteria qualitatively give similar results as the previous scenario. Again, the MC criterion is the most conservative one and the DP criterion
is the most non-conservative. The influence of radius ratio α on the critical β is significant. It is known from above that the maximum shear stress τmax will decrease as α increases. Therefore, the critical β needed for collapse will be reduced as the radius ratio α increases. 3.3 Maximum well pressures Critical well fracturing pressures have been calculated for both dry and saturated rock. Fig. 7 shows the variation of critical β for fracturing happening with change of radius ratio α. It can be seen that β-fracturing is increasing along with the increasing radius ratio for the two cases: with and without pore pressure. This indicates that the inner wall can sustain much greater internal pressures when the HC thickness increases. In addition, the gap between the two curves is indicating the effect of pore pressure on the critical β for fracturing. If pore pressure is ignored, β-fracturing will be much greater than that with pore pressure. In such case, the inner wall stability will be overestimated. 3.4 Safe mud weight windows – pore pressure effects Converting the critical β to the critical internal pressures Pi and then converting to critical mud weight, we can get the safe mud weight window for different HC geometries as shown in Fig. 8 and 9. Fig. 8 shows the safe mud weight window when pore pressures are taken into account and Fig. 9 the safe mud weight window when pore pressure is ignored. The differences are quite obvious. If pore pressure is ignored the mud weight window is much wider than if the pore pressure is considered. Pore pressure will definitely weaken the rock strength by increasing minimum mud weight and decreasing
Figure 8. The required mud weight with pore pressure.
Figure 9. The required mud weight without pore pressure.
maximum mud weight required for maintaining hole stability. For example, if the ML criterion is applied to calculate the minimum mud weight and α = 5, the safe mud weight is between 9.9 and 15.4 lb/gal when the effective stresses are considered. The safe mud weight is approximately between 1.9 and 24.3 lb/gal when the pore pressures are ignored. Again, the differences among the three failure criteria are quite obvious. The MC criterion is the apparently conservative one and the DP criterion is apparently non-conservative, and, accounting for the intermediate principal stress strengthening effect in a reasonable way, the ML criterion is a moderate one. These results also confirm the impact of HC size on the critical mud weight. As the HC thickness increases, the mud weight window is getting wider, so it is easier to avoid instability problems. Note that the minimum and maximum mud weights change only slightly and stay approximately constant when α is greater than a certain value, such as 3 or 4. This phenomenon can be used to predict the wellbore stability problems by virtue of HC testing. 4 CONCLUSIONS Parametric analysis of HC stresses and stability has been conducted in terms of two major parameters, namely α and β, that have significant influence on the critical internal pressures, as
the stress state in the HC very much depends on geometry and loading conditions. Different values of these parameters give rise to different stress distributions in the HC and, subsequently, affect the critical internal pressures. The critical values of pressure ratio β can be obtained using rock failure criteria as outlined above, thus the safe mud weight window can be defined. Several conclusions can be drawn through this work. Firstly, the HC model is confirmed as an effective tool to study wellbore stresses and stability. Its geometry and loading adaptability makes it ideal for reproducing stress states around wellbores and simulating a much wider variety of loading conditions than other available tests. Secondly, pore pressure plays an important role in borehole stability analysis and mud weight design. It can significantly weaken the rock narrowing down the safe mud weight window to a great degree. When pore pressure effects and effective stresses are considered, the safe mud weight window reduces significantly when compared to the dry rock case. Hence, neglecting pore pressure (or inaccurate pore pressure estimation) will mislead the safe mud weight choices and bring about wellbore instability problems. Finally, the MC criterion is apparently conservative. It is a two-dimensional failure criterion that does not account for the intermediate principal stress strengthening effect. This analysis has also re-confirmed that the DP criterion is apparently non-conservative and overpredicts the σ2 strengthening effect. The ML criterion is a moderate one, between the extremes of the other two criteria. Thus, it can provide reasonable mud weight predictions. ACKNOWLEGDEMENTS Support from the Petroleum Institute and local operating companies for the work covered in this paper is gratefully acknowledged.
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