Title of the paper: Well Bore Stability Using the Mogi-Coulomb Failure Criterion and Elasto-Plastic Constitutive Model. Authors (name and last name): 1- Ali Mirzaghorbanali 2- Mahmoud Afshar Authors (E-mail): 1- [email protected] 2- [email protected] Authors (institutional address) 1- Petroleum University of Technology, Tehran, Iran 2Assistant professor of Petroleum University of Technology, Tehran, Iran. 1 coupled with failure criterion. During drilling there are two main instability problems. This model leads to easily computed expression for the critical mud pressure required to maintain well bore stability. it is believed that the fluid barrier. The MohrCoulomb failure criterion only involves the maximum and minimum principle stresses and therefore assumes that the intermediate principle stress has no influence on rock strength. This is based on the hypothesis that. using Mogi-Coulomb failure criterion and the elasto plastic model is developed. In this paper. behave plastically which provides higher fracturing pressure than conventional elastic theory. indeed elastic constitutive model does not fit with the reality of the well bore wall behavior and intermediate principle stress plays an important role on rock strength. In addition. Key words: Wellbore stability. This is typically investigated by a constitutive model to estimate stress around the well bore. even the most simple bore hole collapse or break down can lead to the loss of millions of dollars in equipment and valuable natural resources. This hypothesis is verified and is used in this paper for the South Pars gas field (phases 6. Mud weight window. and 8) in the Persian Gulf. 2 .Abstract Saving time and money are the results of the stable bore hole design. and a part of the bore hole wall. The consequences of these drilling problems are severe. namely. The main aspect of the well bore stability analysis is to mitigate these drilling problems. 7. Elasto-Plastic theory. The most common approach for stability analysis is a linear elastic and isotropic constitutive model in conjunction with linear failure criteria like Mohr-Coulomb. Mogi-Coulomb failure criteria. a model for the mud weight window determination. bore hole collapse and fracture. deliberate access to geological formations bearing petroleum through drilled wells is relatively difficult because most shallow formations are close to end of their economic life. (2007). Drilling operations have to get access to deeper reservoirs through deviated bore hole. Similar results were reported by Russell et al. (2005). (1992)). They concluded that the fluid barrier and a part of bore hole wall. They claimed that a program based on plastic analysis could be more proper and compatible with the ductile lithologies. especially when shale formations were drilled because yielded rocks in the vicinity of the 3 . They suggested that it was better and reasonable to develop bore hole stability analysis based on elasto-plastic theory. Thus Kirsch's equation.1. and Mclean and Addis (1990) studied the sustainable development of geomechanics technology to reduce well construction costs. It was 1980's that geomechanical studies were extensively applied to confront with well bore instabilities. (1990) performed field study on mechanical borehole stability. they found that the linear elastic model underestimates four to eight times the fracture pressure measured on hollow concrete cores. Fleming et al. which was developed one hundered years ago coupled with Mohr-Coulomb failure mechanism were utilized to compute the safe mud density (Fjaer et al. The consequences of failure are severe: even the most simple bore hole collapse or break down can result in the loss of millions of dollars in equipment and valuable natural recourses. (2003). Aadnoy et al. behaved plastically. meanwhile more harsh conditions lead to substantial increase in failure potential. (2007) imitated wellbore by hollow concrete cores and through performing experiments. It was understood through performed geomechanical surveys that well bore stability problems might be alleviated or often eliminated by pertinent determination of mud weight window. Fersheed K et al.Introduction Nowadays. and Winterfeld et al. It is bounded from the north by Turkey Bitlis Suture. (2006). they investigated the role of failure criteria in safe mud weight window and suggested that traditional Mohr-Coulomb failure criterion. 4 . Persian Gulf is part of what is referred to in the Plate Tectonic literatures as the "Arabian Plate" and/or the "Middle East Sedimentary Basin" which is approximately 3000km in length and 2000km in width. Furthermore. would lead to a conservative and thin mud weight window in contrast with three dimensional failure criteria. Ajmi. which ignores the influence of the intermediate principle stress. Al. and from the south by the Gulf of Oman and Owen Fracture Zone in the Arabian. from the east by Zagros Mountain (Zagros Thrust). from the west by the Red Sea. and Zimmerman introduced the fully polyxial Mogi-Coulomb failure criterion (2004). and proposed a new 3D analytical model (2006) to approximate the mud weight needed to avoid failure for the vertical wells based on Mogi-Coulomb failure mechanism coupled with elastic theory. and Simangunsong et al. where by using three dimensional Mogi-Coulomb failure criterion greater mud weight windows than Mohr-Coulomb failure mechanism have been obtained. In this paper the plasticity of the near bore hole region and the mid principle geotechnical stress and their effects in the mud window are investigated. Their conclusions are in the same direction with Collins (2002). Albeit in their research plastic zone in the vicinity of the well bore has not been considered. 1-1. Their study shows the significant role of intermediate principle stress in rock strength.Regional geology South Pars field is located on the Qatar-Fars Arch. In the next sections model development and verification will be presented. one of the major structural elements of the Central Persian Gulf Area.hole might transfer a portion of applied stresses to rock deeper inside the formation. the stresses at the plastic elastic interface in the vertical wells are given by Equations (1) to (3) (Mirzaghorbanali. 2-1. (1992). each part determines one of the boundaries of the mud window. (4) (5) 2 cos (6) sin (7) 5 . the largest stress concentration occurs at the onset of the elastic zone where plastic shield terminates (Fjaer et al.Model development Well bore stability evaluation is divided in to the shear and tensile failure mechanisms. as follows: (1) (2) (3) (Mirzaghorbanali.Shear failure In this section a new shear failure model based on the elasto plastic theory and MogiCoulomb failure mechanism is developed. 2009). √ 3 ln (1) √ ln (2) 2 (3) To avoid plastic elastic interface collapse. By utilizing the effective stress concept and in term of the first and the second stress invariants. 2009).2.Intermediate border line of the mud window In the elasto plastic theory. there are three cases of three principle stresses need to be investigated. Mogi-Coulomb law can be represented as in Equation (8). 2-1-1. B. Equation (19) is obtained. K'. 2 (17) 2 (18) Consider the first scenario of bore hole collapse. parameters A. where the intermediate principle stress is the tangential stress and the well pressure is equal to the minimum allowable mud pressure to avoid plastic elastic interface collapse. (K. (F. H. 2 . 2 2 . K''). 2 2 ˚ (12) 2 3 (13) 4 3 2 4 12 (14) (15) ˚ (16) By substituting the principle stresses into Equations (4) and (5). F''). L.. A 3 (9) B 2 √ (10) ln (11) 2 ˚ . 2 2 3 3 2 ˚ . F'. 2 3 (19) 2 6 . the first and the second stress invariants are obtained by Equations (17) and (18). 3 2 (8) ْ To simplify mathematical operations. M. and G are defined as in Equations (9) to (16). introducing the Equations (17) and (18) into Equation (8). the smaller root has to be considered as the intermediate limit of the mud window as shown in Equation (20).Solving Equation (19) for .Lower limit of the mud window When the interface between elastic and plastic zones fails. plastic shield should be subjected to the same radial stresses with elastic zone as shown in Figure (1). 7 . (20) Similarly the intermediate limit of the mud window for other cases are calculated and presented in Table (1) Table (1) the intermediate limit of the mud window in different cases for vertical wells Case number Cases Intermediate limit of the mud window 1 2 3 2-1-2. plastic shield coupled with the well pressure resist against the well bore collapse. two roots will be obtained. rest of the formation behaves elastically with a radius equal to the actual well radius plus plastic zone. To prevent bore hole collapse. Therefore to keep stability. After interface failure. In this situation radial stresses are exerted on the elastic zone instead of the plastic zone to avoid well bore collapse. 2006). radial. Table (2) Minimum allowable mud pressure to avoid well bore collapse for vertical wells Case number Cases Minimum allowable mud pressure 1 3 2 12 3 3 2 12 3 2 2 12 ln √ 2 √ √ ln ln 8 . Tangential stresses ( ) Plastic zone Plastic thickness (t) Well radius (a) Well pressure ( Radial stresses ( ) ) Elastic zone Plastic elastic interface Figure. 2 √ ln 2 2 2 (22) By rearrangement of Equation (22). and by integrating over plastic zone thickness (Mirzaghorbanali. Equation (23) is obtained. √ ln (23) The required radial stress for vertical wells is obtained from elastic theory and Mogi- Coulomb failure criterion (Al. Equation (22) is obtained. By substituting radial stress into the Equation (23).Ajmi and Zemmerman. 2009).1 Plastic shield under tangential. minimum allowable mud pressure for vertical wells is obtained as shown in Table (2). and well pressure The equilibrium equation for the arrangement shown in Figure (1) is as follows: 2 2 2 2 (21) Substitution for tangential stress into Equation (21). tensile failure.Proposed mud weight window In the new model developed in this study.Deduction of the plastic term ( √ ln ) from the conventional elastic equations of the Table (2).1547 ˚ (25) The plastic zone thickness is determined by leak off test. Otherwise.Tensile failure When the effective stress at the well bore exceeds the tensile strength of the formation. Aadnoy and Belayneh (2004) used elasto plastic constitutive model to obtain the effective tangential stress around the vertical wells. The advantage of this model is described in section (3). drilling without joints will be gained. three different limits are proposed. The intermediate limit acts as a border line. 1. as the well pressure becomes less than this limit. induced fracture is imminent as shown in Equation (24). and by introducing this term into the Equation (24). joints and discontinuities will form around the 9 . the role of the plastic zone in vicinity of the well bore is like a pressure shield. ′ . the upper limit of the mud weight window based on elasto plastic model is obtained as shown in Equation (25). and eventually fracturing pressure for entire of the well trajectory is calculated by Equation (25). this equation shows that similar to the shear failure. would increase the formation durability against well bore break out. ′ ˚ (24) The upper limit of the mud window to avoid well bore fracture is determined directly from Equation (24). respectively. and as a result. if the well pressure becomes greater than this limit. 2-3. unlike the other approaches. 3 ′ . As in previous models the upper and lower limits are fracture pressure and collapse pressure. 2-2. well bore. along with the conventional approach results in Figure (2) for South Pars gas field (phases 6. Table (4) X'.1547 3 '- ln 1. ˚ 3. ˚ W'=3 . and. . W' for different scenarios are determined as shown in Table (4). 7. and accidental geomechanical instabilities are compared with the fracture gradient obtained from the proposed model. while for the conventional approach. The new developed model follows the fracture gradient results obtained from leak off tests and accidental geomechanical instabilities more closely than the conventional approaches. this value is 0. and Z' X'= 2 Y'= 3 Z'= 3 2 12 3 W' 12 2 2 3 12 2 W'=3 .1547 √ √ √ X'. Z'. ˚ W'=3 .001422. Y'. The variance of the new proposed model from the real fracture gradient is 0. and 8) located in the Persian Gulf. 10 . . Y'. and. . Z'.032539. Consequently. Table (3) Mud weight window for different cases for vertical wells Number Cases Minimum allowable Minimum allowable mud pressure (without discontinuities) Maximum allowable mud mud pressure pressure 1 '- ln 1. the results obtained from leak off tests. W' for different scenarios Case number 1 X'.1547 2 '- ln 1.Model verification and discussion of results To assess the maximum mud window limit of the proposed model. Y'. to reach stable drilling it is necessary to keep the well pressure between the upper and lower limits as shown in Table (3). 7. one of the available field data in literature (Al. 2006) which is for Wanaea field located in Northwest Shelf of Australia is incorporated to check the minimum mud window limit of the proposed model. and 8) in Persian Gulf 5000 6000 7000 8000 9000 10000 Minimu allowable mud density (KPa) 1950 1970 Real gradient 1990 New developed 2030 model 2050 Conventional model Depth (m) 2010 2070 2090 2110 2130 11 .2 Comparison between the maximum limits of the mud weight windows on South Pars gas field (phases 6. As shown in Figure (3).As there is no suitable data to assess the minimum mud window limit of the proposed model in South Pars gas field. the results obtained by the new developed model are 80% closer to the real breaking out gradient.7 2020 1. in comparison to the conventional approach.9 Depth (m) 1420 Conventional approach 1620 New 1820 developed method 1.6 1.8 Well pressure (Gradient) Real fracture gradient 2220 2420 Figure.Ajmi and Zemmerman. 1. 6 1.8 1. but the conventional approach is restricted for the deviated wells up to 38°. and 8) in Persian Gulf As shown in Figure (4). Northwest Shelf of Australia Regarding the new developed model verification and its superior accuracy in comparison to the conventional approaches.2 1 0. The new proposed model predicts a wide mud weight window and can be worked out for higher deviations than the conventional approaches as shown in Figure (4) for South Pars gas field (phases 6. the new proposed model results in the safe mud weight window for the deviated wells up to the 58°. 7. One of the main challenges in the drilling operation is the safe mud weight determination in the deviated wells. The anticipated mud weight windows for the deviated wells are severely narrow and sometime no safe mud window is obtained based on the conventional approaches for the highly deviated wells. 12 . 2 Minimum limit of the new model Minimum limit of the conventional approach Maximum limit of the conventional approach Mud density gradient Maximum limit of the new model 1.4 Comparison between the mud weight windows in the highly deviated wells for South Pars gas field (phases 6. This would prevent challenges faced by drilling operators for wells deviated more than the maximum mud weight allowed by the conventional approach. 7. the main advantage of this model is discussed in the following.Figure.3 Comparison between the minimum limits of the mud weight windows on Wanaea field.4 1.8 0 20 40 Well deviation (degree) 60 80 Figure. and 8) in Persian Gulf in depth equal to 2500 (m). The new developed model results in an enlarged mud weight window in comparison to the conventional approach which enables well drilling operators to drill safely on more deviated wells. based on the Mogi-Coulomb failure criterion and the elasto plastic constitutive model for the mud weight window is introduced and its accuracy is verified against field data. Nomenclature σ Stress (KPa) r Radius (Inch) Maximum horizontal stress (Kpa) Minimum horizontal stress (Kpa) Yield point (Kpa) Vertical stress (Kpa) ˚ Pore pressure (Kpa) Poisson's ratio Maximum principle stress (Kpa) Intermediate principle stress (Kpa) Minimum principle stresses (Kpa) C Cohesion (Kpa) Internal friction angle (Degree) Well pressure (Kpa) ′ Effective tangential stress (Kpa) 13 . Acknowledgement The financial support of the National Iranian Drilling Company for this research is highly appreciated.4.Conclusion In this paper a new model. The accuracy and the precision of the proposed model are higher than the conventional approach. (2004). Aadnoy. Bernt S. International Journal of Rock Mechanics and Mining Sciences . 179‐192. SPE Annual Technical Conference and Exhibition (pp. W.Tensile strength (Kpa) ′ ′ . C. Aadnoy. K. 1200‐1211. 14 . Journal of Petroleum Science & Engineering .M. Effective maximum horizontal stress (Kpa) Plastic thickness (Inch) Well radius (Inch) Radial stress (Kpa) Tangential stress (Kpa) Axial stress (Kpa) Plastic elastic interface radius (Inch) First stress invariant (Kpa) Second stress invariant (Kpa) References Adel. Elasto‐plastic fracturing model for wellbore stability using non‐penetrating fluids. B. Z.Babajan.von Winterfeld and S. (2006). (2005). (2007). Al‐Ajmi. E. M. M. Al‐Ajmi. Stability analysis of vertical borehole using the Mogi‐Coulomb failure criterion . R. Effective Minimum horizontal stress (Kpa) . a. International Petroleum Technology Conference (pp. 431‐439. R. A. Geomechanics Analysis of Crestal Region of an Omani Gas Field. Bernt S. International Journal of Rock Mechanics and Mining Sciences . 1‐12). California: SPE 110355. 1‐4). Doha: IPTC 10231. Elastoplastic Fracture Model Improves Predictions in Deviated Wells. A. Relation between the Mogi and the Coulomb failure criteria. (2004). Tehran. Fleming. Mirzahgorbanali. 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