International Journal of Petroleum and Geoscience Engineering Volume 05, Issue 01, Pages 8-23, 2017 ISSN: 2289-4713 Stability Analysis of Vertical, Directional and Horizontal wellbores using the Three Dimensional Hoek Brown Criterion Mohammad Tabaeh Hayavi *, and Mohammad Abdideh Department of Petroleum Engineering, Omidiyeh Branch, Islamic Azad University, Omidiyeh, Iran. * Corresponding author. Tel.:+98 939 183 5971 E-mail address: m.hayavi.put@gmail.com A b s t r a c t Keywords: Wellbore stability, Mud pressure, Hoek Brown criterion, Analytical model, Constitutive model. Accepted: 15 Mar 2017 Wellbore stability is one of the crucial issues in oil and gas industries. The issues related to instability of wells, impose significantly unwanted costs on drilling operation. Hence, in many oil companies, wellbore stability analysis is one of the major activities in the well design stage. The objective of this paper is to present the 3D wellbore stability prediction models for vertical wellbores. The 3D Hoek Brown strength criterion developed by Zhang and Zhu in conjunction with linear poroelastic constitutive model is utilized to develop the models. Furtheremore, safe mud pressure window required to stabilize the directional and horizontal wellbores in different well trajectories and in-situ stress regimes during drilling operation were determinded. The analytical model is applied to real field case in order to verify the applicability of the developed models. The results indicate that the decreasing of the Biot s coefficient and increasing the UCS and Poisson s ratio, Young s modulus, bulk modulus and ratio of shear wave travel time to compressional wave travel time will increase the optimum mud pressure window. Also, in different in-situ stress regimes, the inclination and azimuth have a significant role in wellbore stability during drilling. Academic Research Online Publisher. All rights reserved. 1. Introduction When a well is drilled, the rock surrounding the borehole must take up the load previously supported by the rock hat has been removed. This results in the development of a stress concentration at the borehole wall. If the rock is not strong enough, the wall will fail [1]. The integrity of the wellbore plays an important role in petroleum operations. Hole failure problems cost the petroleum industry several billions of dollars each year. Prevention of wellbore failure requires a strong understanding of the interaction between formation strength, in-situ stresses, and drilling practices. As in-situ stress and rock strength cannot be easily controlled, adjusting the drilling practices is the usual way to inhibit wellbore failure [2,3]. During drilling, there are two types of mechanical borehole failure: compressive and tensile failures. Compressive failure occurs when the wellbore pressure is too low compared with the rock strength and the induced stresses. On the other hand, tensile failure occurs when the wellbore pressure is too high [4]. The main aspect of the wellbore stability analysis is to mitigate these drilling problems [5]. 8 P a g e
through altering the applied mud pressure and the orientation of the borehole with respect to the insitu stresses. In engineering practice, a linear poroelasticity stress model in combination with a rock strength criterion is commonly used to determine the minimum and maximum mud pressures required for ensuring wellbore stability. Therefore, a main aspect of wellbore stability analysis is the selection of an appropriate rock strength criterion [6,7,8,9]. Zhou [10] introduced a modified Wiebols and Cook [11] criterion and developed a computer program for the wellbore stability analysis. The results indicated the importance of the intermediate principal stress on the stability of wellbores. Ewy developed the Modified-Lade failure criterion and presented the advantages of this new criterion over Mohr-Coulomb and Drucker-Prager [12]. Colmenares and Zoback evaluated seven different rock failure criteria based on polyaxial test data, and they concluded that the Modified Lade and the Modified Wiebols and Cook fit best with polyaxial test data [13]. Aadnoy [14] developed an analytical solution to study the stability of inclined wellbores drilled into rock formations modeled as a transversely isotropic material. He showed that neglecting the anisotropic effects arising from the directional elastic properties can result in errors in the wellbore stability analysis. Al-Ajmi and Zimmerman [15] developed the Mogi Coulomb failure criterion, according to polyaxial failure data of the variety of rocks. They concluded that Mohr Coulomb failure criterion is conservative in estimating of collapse pressure during drilling and using Mogi Coulomb failure criterion can minimize the conservative nature of the mud pressure predictions. Hoek Brown failure criterion is another wellknown criterion successfully applied to a wide range of rocks for almost 30 years [16,17]. Zhang and Zhu [18] developed a 3D Hoek-Brown strength criterion for rocks. This criterion properly considers the effect of the intermediate principal stress. Also this criterion has the advantage over the other 3D strength criteria in that it uses the same input parameter as the most widely used Hoek Brown criterion. Zhang et al [19] compared minimum mud weight prediction of five common rock failure criteria. They recommended Mogi- Coulomb and 3D Hoek-Brown to be used for wellbore stability analysis. In this paper, the 3D Hoek-Brown strength criterion developed by Zhang and Zhu is used to analyze wellbore stability. Furthermore, the analytical models are applied to field data in order to verify the applicability of the developed models. 2. Stress Concentration around a Wellbore at Drilling Condition Drilling a borehole will alter the in situ principal stresses, the vertical stress and the maximum and minimum horizontal stresses, in a manner so as to maintain the rock mass in a state of equilibrium. This leads to a stress concentration around the wellbore [20]. The degree of stress concentration depends on the wellbore orientation, the magnitude and orientation of in-situ stresses, and the wellbore pressure [21]. When the elevated stress exceeds the rock strength, the rock will fail resulting in the development of wellbore failure [22]. If excessive, the cavings produced by the spalling of broken materials into the wellbore can cause drilling problems such as pack-off, over-pulls, stuck-pipe and poor cementing, to name a few [23]. 9 P a g e
To evaluate the stability of a wellbore, a constitutive model is required to compute the stresses around the borehole [24]. Although different constitutive models are available, the linear poroelasticity stress model is commonly used in industry practice [19]. The stress concentration around a well drilled in an isotropic, elastic medium under anisotropic in-situ stress condition (Maximum and minimum horizontal stresses are different) is described by the Kirsch equations. The general expressions for the stresses at the wellbore wall for a deviated well in the drilling situation are [25]: where i is wellbore inclination and is the azimuth angle due to the maximum horizontal stress ( ) direction (Degree) as illustrated in Fig. 1. Where and are the effective radial, tangential and axial stresses, respectively. P w is the well pressure, P p is the pore pressure, is the Biot s coefficient, is the Poisson s ratio, θ is the angular position around the wellbore circumference and measured clockwise from the azimuth of maximum horizontal stress. Fig. 1: Axes and inclination and direction angles of the inclined well [26]. Effective induced stresses created at the borehole wall for a vertical borehole ( ) can be obtained from equations. 1-3 in the following [25]: The shear stresses at the wellbore wall are denoted, and, while the in-situ stresses in (x, y, z) coordinate system, denoted,,,, and, and they are defined as [21]: 10 P a g e
According to Eqs. 13-15 the tangential and axial stresses are functions of the angle θ. This angle indicates the orientation of the stresses around the wellbore circumference, and varies from 0 to 360. Consequently, the tangential and axial stresses will vary sinusoidally. The tangential and radial stresses are functions of the well pressure, but the vertical stress is not. Therefore, any change in the mud pressure will only influence the tangential and radial stresses. Inspection of these equations reveals that in the vertical well, both tangential and axial stresses reach a maximum value at θ = 90, 270 and a minimum value at θ = 0, 180.Therefore, the shear failure known as breakouts is expected to happen at the point of maximum tangential stress where the rock is under maximum compression (at θ = 90 ). Tensile failure known as hydraulic or induced fracture, however, is expected to occur at the point where minimum tangential stress is applied to the rock (at θ = 0 ): an orientation 90 away from the location of shear failures around the wellbore (Fig. 2) [24]. The magnitudes of three effective principal stresses around the wellbore to analyze the initiation of induced fracture can be obtained as: Fig. 2: The location of breakout and tensile fractures on the borehole wall [27]. For shear failure or breakouts to occur, the magnitude of effective principal stresses around the wellbore are estimated as Based on linear elasticity, maximum stresses, occur in the wellbore wall (Fig. 3). Therefore, borehole instability is expected to initiate at the borehole wall [28]. Fig. 3: Stresses around a vertical borehole in a linear elastic formation [25]. 11 P a g e
3. Three-Dimensional Hoek-Brown Strength Criterion A great number of rock strength criteria have been proposed over the past decades. Of these different strength criteria, the Hoek-Brown strength criterion has been used most widely, because: (1) it has been developed specifically for rock materials and rock masses; (2) its input parameters can be determined from routine unconfined compression tests, mineralogical examination, and discontinuity characterization; and (3) it has been applied for over 20 years by practitioners in rock engineering, and has been applied successfully to a wide range of intact and fractured rock types [29]. For intact rock, the Hoek Brown strength criterion may be expressed in the following form [17]. which depends upon the rock type (texture and mineralogy) as tabulated in Table 1. For jointed rock masses, the Hoek Brown strength criterion can be expressed as follows [32]: where Table 1: Values of m i for different rocks [30, 31]. Class of rock Group Coarse Conglomerate (21±3) a Texture Medium Fine Very fine Sandstone Siltstone Claystone (17±4) (7±2) (4±2) Clastic Non- clastic Carbonate Evaporite Organic Breccia (19±5) Crystalline limestone (12±3) Sparitic limestone (10±2) Gypsum (8±2) Greywacke (18±3) Micritic Limestone (9±2) Anhydrite (12±2) Shale (6±2) Marl (7±2) Dolomite (9±3) Chalk (7±2) where is the Uniaxial Compressive Strength (UCS) of intact rocks, and are respectively the major and minor effective principal stresses, and mi is a material constant for the intact rock, The parameter mb is a reduced value of mi, which accounts for the strength reducing effects of the rock mass conditions defined by Geological Strength Index (GSI). Adjustments of s and a are also done according to the GSI and D values [33]. GSI was estimated from the chart of Marinos et al. [34] (Fig. 4). D is a 12 P a g e
factor which depends upon the degree of disturbance to which the rock mass has been subjected by blast damage and stress relaxation. Where is the intermediate effective principal stress. 4. Building the Vertical Wellbore Stability Prediction Models The modes of shear and tensile failures may be different depending on the order of magnitude of three effective principal stresses around the wellbore wall. These stresses are, and presented in equations. 13-15. Fig. 4: GSI chart for jointed rocks [34]. It varies from 0 for undisturbed in situ rock masses to1 for very disturbed rock masses [32]. As can be seen from above, the Hoek Brown strength criterion does not take account of the influence of the intermediate principal stress. Much evidence, however, has been accumulating to indicate that the intermediate principal stress does influence the rock strength in many instances [18,21,35,36]. Zhang and Zhu [18] proposed a 3D version of the original Hoek Brown strength criterion for rock mass (Eq. (5) with a=0.5): Where and are, respectively, the effective mean stress and the octahedral shear stress defined by: When mud pressure decreases, increases towards the compressive strength. Thus, the lower limit of the mud pressure, Pwb, is associated with borehole collapse, in which should be greater than. There are three permutations of the three principal stresses that need to be investigated in order to determine the minimum allowable mud pressure: (1) (2) (3). Substituting each of these scenarios in the 3D Hoek-Brown failure criterion presented in equation 21, and introducing equations 16 and 19-20, gives Solving this equation for Pwb will give four roots. The smallest root is the lower limit of the mud pressure in order to avoid breakouts (collapse pressure). The constants P4, P3, P2, P1, P0 are shown in Appendix. If the well pressure falls below Pwb, borehole collapse will take place. On the other hand, when Pw increases, decreases towards the tensile strength. Therefore, the upper limit of the mud pressure, Pwf, is associated with fracturing, where should be less than. Considering this constraint and the 13 P a g e
relative magnitude of the axial stress, there are three permutations of the three principal stresses that need to be investigated in order to determine the maximum allowable mud pressure: (1) Where (2) (3) Similarly, for each case, by introducing equation 16-18 into equation 21, gives Solving this equation for Pwf will give four roots. The smallest root is the upper limit of the mud pressure in order to avoid borehole fracturing (fracture pressure). The constants P4, P3, P2, P1, P0 are shown in Appendix. If the well pressure rises above the fracture initiation pressure, Pwf, tensile failure will take place. Reduction of mud pressure, corresponding to lower confining pressures, increases the potential for shear failure. On the other hand, increasing the mud pressure above a certain limit causes the tensile failure to happen. This discussion indicates that there is a optimum window for the mud weight to drill the wellbore in a stable condition. The lower limit for this window corresponds to shear failure (breakouts) with its upper limit being the fracture initiation pressure [20,37]. 5. Evaluation of Directional and Horizontal Wellbore Stability In an arbitrarily oriented wellbore, the radial stress, is one of the effective principal stresses. Other two effective principal stresses can be calculated by using the theory of combined stresses. Equations of these three effective principal stresses,,, and can be written as follows [38]: where and are the effective maximum and minimum effective principal stresses and is the effective intermediate principal stress. Regarding the fact that radial and tangential stresses are functions of wellbore pressure, Pw, the principal stresses are therefore also functions of well pressure. So an iterative loop should be applied to obtain minimum and maximum allowable mud pressure in oriented wellbores. In this study, a computer program is developed to obtain the safe mud pressure required to maintain wellbore stability. This program using several input parameters, including: pore pressure and in-situ stresses (vertical stress, maximum and minimum horizontal stresses), rock strength parameters (Tensile strength, Biot s coefficient, Poisson s ratio, material constants (m and s)), well inclination and azimuth. In drilling situation wellbore pressure increases from minimum horizontal stress until the condition for tensile failure satisfied. Furthermore, well pressure decreases from minimum horizontal stress to formation pore pressure until the shear failure occurs. These analyses have been done for different well inclination (i=0 to i=90 ) and azimuth ( =0 to =180 ) in several cases of in-situ stress regimes. 6. Results and Discussion Table 2 shows thee cases of different in-situ stress regimes and the input parameters for wellbore 14 P a g e
stability analysis. According to these data minimum and maximum bottomhole pressure that mud weight must be provided to prevent wellbore wall instability are determined. In Case 1, the formation is in the normal regime. Fig. 5 shows the 3-D plot of collapse and fracture pressures as function of inclination and wellbore azimuth for Case 1. The vertical axis are collapse and fracture pressures, and horizontal axes indicate wellbore inclination and azimuth. Fig. 3 show that a vertical wellbore has a less collapse pressure (Pwc) and higher fracture pressure (Pwf) (larger safe mud pressure window) than the horizontal borehole. So, the vertical borehole is more stable than horizontal and deviated boreholes in almost all directions. In addition, it is obvious that drilling parallel to the minimum horizontal stress direction ( =90 ) is the best trajectory to stabilize the borehole and associated with a larger safe mud pressure window than the maximum horizontal stress direction ( =0 ) in this case. However, in this particular case, deviated borehole with an inclination of 50 and azimuth of 93 have the largest safe mud pressure window. This means that instability problems can be minimized if the well would be deviated 50 from the vertical and parallel to the direction of the minimum in-situ stress. Furthermore, it shows that the collapse and fracture pressures are highly sensitive to the inclination in all direction or azimuth. In Case 2, the formation is in the strike-slip regime. Fig. 6 shows that, the horizontal boreholes are more stable (larger safe mud pressure window) than the vertical or all deviated boreholes in all directions. Moreover, wellbores drilled in the direction of maximum horizontal stress show the highest stability (largest safe mud pressure window) and the others bored along minimum horizontal stress are the least stable. Finally Case 3 indicates a formation in the reverse fault regime. Fig. 7 shows that a horizontal borehole which is drilled parallel to the maximum horizontal stress has a largest safe mud pressure window. Here also the risk of wellbore instability decreases with increasing the borehole inclination. Contrary to normal regime, in reverse and strikeslip regimes, the least variation of safe mud pressure window versus wellbore inclination are for the wellbores in the direction of minimum horizontal stress. 15 P a g e
Fig. 5: Collapse and fracture pressures as function of wellbore trajectories in normal stress regime (Case 1). Table 2: Different in-situ stress regimes used in this study. Case Depth Stress m b s (ft) regime 1 6500 Normal 1 0.85 0.75 0.442 3000 4.1 0.03 0.3 0.75 2 6500 strikeslip 0.9 1 0.85 0.442 3000 4.1 0.03 0.3 0.75 3 6500 Reverse 0.85 1.1 0.91 0.442 3000 4.1 0.03 0.3 0.75 16 P a g e
Fig. 6: Collapse and fracture pressures as function of wellbore trajectories in strike-slip stress regime (Case 2). Fig. 7: Collapse and fracture pressures as function of wellbore trajectories in reverse stress regime (Case 3). 17 P a g e
7. Field Case Study The developed analytical models will be applied to a well (called well A) drilled in Ahwaz oilfield (One of southern Iranian field in the Middle East) for investigation of stability analysis during drilling. This oil field, one of the most important Iranian super giant oil fields, was discovered in 1956 and now has more than 450 producing wells. This oil field has an anticline structure 72 km long and 6 km wide with NW-SE trending symmetrical anticlinal, located in central part of north Dezful region. Its main reservoir is the Asmari formation [39,40]. The big Asmari reservoir is complicated and heterogeneous in terms of reservoir rock features [41]. In-situ stresses and pore pressure profiles of Asmari formation are shown in Fig. 8. Fig. 9 shows the estimated log based geomechanical properties of Asmari formation. including UCS, Poisson s ratio, Young s modulus (E), bulk modulus (Kb), Biot s coefficient and ratio of shear wave travel time to compressional wave travel time (Dts/Dtc). For simplicity, the average values of mi for sandstone and limestone rocks are assumed to be 17 and 10, respectively. The value of D is considered 0.9 and the GSIs of rocks was estimated from the chart of Marinos et al. [34]. The most commonly observed order of magnitude of stresses around a wellbore in terms of shear failure is and in case of tensile failure [24]. Fig. 8: In-situ stresses and pore pressure profiles of Asmari formation. 18 P a g e
Considering this assumption and the real mud weight that had been used to drill Well A (i.e. 1.05 gr/cm3), the calculations were carried out to determine the potential for any shear failure (breakouts) or tensile failure (induced fracture). The results of such analysis and hole diameter log (caliper log) are shown in Fig. 10. It can be concluded that the minimum and maximum allowable mud pressures change as a function of depth and for well A are varied between the 21-29 MPa and 43-49 MPa, respectively. So, the optimum mud pressure window for this well is 29-43MPa. Fig. 9: Geomechanical properties of Asmari formation. 19 P a g e
pressure and increase in the maximum allowable mud pressure, while an increase in Biot s coefficient tends to increase the breakout pressure and decrease the fracture pressure. 8. Conclusions Fig. 10: Left: Determination of minimum and maximum allowable mud pressures for Well A using the 3D Hoek-Brown criterion; Right: Caliper log. From the caliper log shown in this figure, by applying the real mud weight that had been used to drill this well, several breakouts were happened at the interval of 2420-2530 m. It can be seen that a good agreement is reached between the results of caliper log and developed model in order to investigate the depths of borehole breakout. As Figs. 6 and 7 depict, optimum mud pressure window is a function of geomechnical properties. Therefore, increasing of UCS, Poisson s ratio, Young s modulus, bulk modulus and ratio of shear wave travel time to compressional wave travel time causes decrease in the minimum allowable mud In this paper the new models for prediction of vertical wellbore stability, based on the 3D Hoek- Brown failure criterion and the linear constitutive model were introduce. The result indicated that optimum mud pressure window increases with increasing the UCS, Young s modulus, bulk modulus, Poisson s ratio and ratio of shearwave travel time to compressional wave travel time and decreasing the Biot s coefficient. Also, a good agreement was achieved between the results of caliper log and developed model in order to investigate the depths of borehole breakout. Furthermore, this paper presents the sensitivity study of in-situ stress regime and well trajectory on collapse and fracture pressures (or safe mud pressure window). It was shown that the safe mud pressure window in drilling condition is highly affected by in-situ stress regimes and well trajectories. The other results of this study indicated that: 1- In the normal regime, a vertical boreholes has a larger safe mud pressure window than the horizontal borehole (Contrary to the reverse and strike-slip regimes). 2- Contrary to normal regime, in reverse and strikeslip regimes, drilling in the direction of minimum horizontal stress is less stable (smaller safe mud pressure window) than the other directions. 3- The collapse and fracture pressures of the boreholes drilled parallel to the minimum horizontal stress in the reverse and strike-slip regimes have the least sensitivity to the wellbore inclination (Contrary to normal regime). 20 P a g e
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[41] Abdideh, M., Fathabadi, M.R., Analysis of stress field and determination of safe mud window, J Petrol Explor Prod Technol, 15 (3): 105 110, 2013. Appendix (3) 2. Tensile failure: The constants A, B, C and D are calculated based on the effective principal stress states related to each case of shear and tensile failures occurred around the borehole. (1) 1. Shear failure: (2) (1) (3) (2) 23 P a g e