# Probabilistic Approach in Wellbore Stability Analysis during Drilling

Authors: Mahmood R. Al-Khayari; Adel M. Al-Ajmi; Yahya Al-Wahaibi
Journal: Journal of Petroleum Engineering
 (2016)

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## Abstract

In oil industry, wellbore instability is the most costly problem that a well drilling operation may encounter. One reason for wellbore failure can be related to ignoring rock mechanics effects. A solution to overcome this problem is to adoptin situ stresses in conjunction with a failure criterion to end up with a deterministic model that calculates collapse pressure. However, the uncertainty in input parameters can make this model misleading and useless. In this paper, a new probabilistic wellbore stability model is presented to predict the critical drilling fluid pressure before the onset of a wellbore collapse. The model runs Monte Carlo simulation to capture the effects of uncertainty inin situ stresses, drilling trajectories, and rock properties. The developed model was applied to differentin situ stress regimes: normal faulting, strike slip, and reverse faulting. Sensitivity analysis was applied to all carried out simulations and found that well trajectories have the biggest impact factor in wellbore instability followed by rock properties. The developed model improves risk management of wellbore stability. It helps petroleum engineers and field planners to make right decisions during drilling and fields’ development.


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## Body

## 1. Introduction

Drilling is one of the most important and costly operations in oil industry due to the fact that if the wellbore stability cannot be accomplished, drilling will lead to higher than necessary costs. So, maintaining a stable wellbore is one of the main challenges encountered during drilling, whether a well is being drilled with overbalanced or underbalanced techniques [1].The possible mechanical wellbore failures (rock failures) can be grouped in two main types: tensile failure and shear failure. The tensile failure represents a hydraulic fracture within the formation which normally leads to a loss in system circulation during drilling, whereas the shear failure can be simply described as the collapse of the borehole wall which may result in stuck pipe and, in severe cases, a loss of the open hole section [2]. The upper limit of the mud weight is the maximum mud weight before the onset of tensile failure (fracturing). It can be obtained by conducting a leak off test. On the other hand, the minimum allowable mud pressure is the critical pressure before a wellbore collapse.The most common well failure during drilling is borehole collapse. The method of analyzing this case involves the use of wellbore stability modeling consisting of a constitutive model coupled with a failure criterion. The constitutive model describes the deformation properties of the rock and the failure criterion determines the limits of the deformation [3].This paper presents a model that accounts for uncertainties of rock properties andin situ stresses as well as wellbore trajectories in order to estimate the critical mud pressure during drilling. Unlike conventional stability models that deal with static values, this probabilistic model uses Monte Carlo simulation method in order to study the impact of uncertainties on wellbore instability [4]. It was developed by utilizing a linear elastic and isotropic constitutive model in conjunction with the Mogi-Coulomb rock failure law.Al-Ajmi and Al-Harthy [4] have developed a probabilistic borehole model to study the influence of uncertainty of input variables in vertical borehole stability. This paper discusses a probabilistic model to predict the collapse pressure for nonvertical wellbores (horizontal and deviated wells) as well as vertical wellbores during drilling. That will be done by applying Monte Carlo simulation to the stability model developed by Al-Ajmi and Zimmerman [5, 6].For nonvertical wellbores, there is no closed-form analytical solution to calculate collapse pressure in order to prevent borehole instability. To solve this problem, an excel program was developed to conduct stability analysis in vertical and nonvertical boreholes and perform risk analysis in wellbore instability.


### 1.1. Stresses around Boreholes

A constitutive model is a set of equations needed to determine the stress state of a borehole and describe its behavior. Different models have been developed and widely studied; however, linear elastic analysis is the most common approach due to its ease of application [7] and its requirements of fewer input parameters compared with the rest of complicated models.The stress components in cylindrical coordinates can be represented by(1)σr=σx′cos2⁡θ+σy′sin2⁡ϕ+2σx′y′sin⁡θcos⁡θ,σθ=σx′sin2⁡θ+σy′cos2⁡ϕ-2σx′y′sin⁡θcos⁡θ,σz=σz′,σrθ=σy′-σx′sin⁡θcos⁡θ+σx′y′cos2⁡θ,-sin2⁡θ,σrz=σx′z′cos⁡θ+σy′z′sin⁡θ,σθz=σy′z′cos⁡θ-σx′z′sin⁡θ,where σr, σθ, and σz are called radial, tangential, and axial stress components in a cylindrical coordinate system, respectively, while the virgin formation stresses in (x,y,z) coordinate system before excavation can be expressed as (2)σxo=σHcos2⁡α+σhsin2⁡αcos2⁡i+σvsin2⁡sin2⁡i,σyo=σHsin2⁡α+σhcos2⁡α,σzo=σHcos2⁡α+σhsin⁡αsin2⁡i+σvcos2⁡i,σxyo=0.5σh-σHsin⁡2αcos⁡i,σyzo=0.5σh-σHsin⁡2αsin⁡i,σxzo=0.5σHcos2⁡α+σhsin2⁡α-σvsin⁡2i.Equation (2) represents the virgin formation stresses which are denoted by the superscript “o” on the stresses. The virginin situ stresses will be altered by excavation. In cylindrical coordinates, the complete stress solution around an arbitrary oriented wellbore can be represented by the following:(3)σr=σxo+σyo21-a2r2+σxo+σyo21+3a4r4-4a2r2cos⁡2θ+σxyo1+3a4r4-4a2r2sin⁡2θ+Pwa2r2,σθ=σxo+σyo21+a2r2-σxo-σyo21+3a4r4cos⁡2θ-σxyo1+3a4r4-4a2r2sin⁡2θ-Pwa2r2,σz=σzo-v2σxo-σoya2r2cos⁡2θ+4σxyoa2r2sin⁡2θ,σrθ=-σzo-σyo21-3a4r4+2a2r2sin⁡2θ+σxyo1-3a4r4+2a2r2cos⁡2θ,σθz=-σxzosin⁡θ+σyzocos⁡θ1+a2r2,σrz=σxzocos⁡θ+σyzosin⁡θ1-a2r2,where “a” is the radius of the wellbore, Pw is the internal wellbore pressure, and v is Poisson’s ratio. The angle θ is measured clockwise from x-axis, as shown in Figure 1.Figure 1
Stress transformation system for deviated borehole.In a linear elastic material, the largest stress concentration occurs at the borehole wall. Therefore, borehole failure is expected to initiate there [8–10]. For wellbore instability analysis, consequently, stresses at the borehole wall are the ones that should be compared against a failure criterion.Estimating the stresses at borehole wall in the case of deviated wellbore is done by settingr=a in (2); the results will be as follows: (4)σr=Pw,σθ=σxo+σy0-2σxo-σyocos⁡2θ-4σxyosin⁡2θ-Pw,σz=σzo-v2σxo-σyocos⁡2θ+4σxyosin⁡2θ,σθz=2-σxzosin⁡θ+σyzocos⁡θ,σrθ=0,σrz=0.For vertical wellbore (Figure 2), the stresses at borehole wall are determined by setting the inclination angle i=0 in (2) and orienting the horizontal axes so that the direction θ=0 is parallel to σH (i.e., α=0). The stresses become (5)σr=Pw,σθ=σH+σh-2σH-σhcos⁡2θ-Pw,σz=σv-2vσH-σhcos⁡2θ,σθz=0,σrθ=0,σrz=0.The stresses around a horizontal wellbore wall after excavation (Figure 3) can be represented by setting i equal to π/2 and plugging the resulted equation into (2) which gives (6)σr=Pw,σθ=σv+σHsin2⁡α+σhcos2⁡α-σv-σHsin2⁡α-σhcos2⁡αcos⁡2θ-Pw,σz=σHcos2⁡α+σhsin2⁡α-2vσv-σHsin2⁡α-σhcos2⁡αcos⁡2θ,σθz=σh-σHsin⁡2αcos⁡θ,σrθ=0,σrz=0.In this case, where the wellbore axis lies along the maximum horizontal principal stress (i.e., α = 0), the stress at borehole wall is described as follows: (7)σr=Pw,σθ=σv+σh-2σv-σhcos⁡2θ-Pw,σz=σH-2vσv-σhcos⁡2θ,σθz=0,σrθ=0,σrz=0.Figure 2
Stress transformation system for vertical borehole [12].Figure 3
Stress transformation system for a vertical borehole.


### 1.2. Rock Failure Criteria (Mogi-Coulomb Criterion)

The second important element in wellbore stability is rock failure criteria. These criteria are sets of functions in stress or strain spaces that describe or predict the condition at which rock fails under external load. In literature, many of failure criteria have been developed by scientists who suggested different approaches trying to simulate actual rock failures. In this paper, the concentration will be on Mogi-Coulomb criterion.In 2005, Al-Ajmi and Zimmerman presented a linear form of Mogi criterion [11]. Al Ajmi found that the plotted polyaxial data of some rocks using Mogi failure envelope in τoct,σm,2 space can be fit to a linear model as follows: (8)τoct=a+bσm,2,where σm,2 is the effective normal stress defined by(9)σm,2=σ1+σ32;τoct is the octahedral shear stress defined by(10)τoct=13σ1-σ22+σ2-σ32+σ3-σ12.Anda and b are the intersection of the line on τoct-axis and its inclination, respectively (Figure 4). Al Ajmi [12] found and verified experimentally that failure under polyaxial stresses can be predicted from triaxial test data using the linear Mogi criterion. Accordingly, he developed a new true-triaxial failure criterion called Mogi-Coulomb criterion. For triaxial stress states (σ2=σ3) the linear Mogi criterion is exactly equivalent to Mohr-Coulomb criterion [13].Figure 4
Mogi-Coulomb failure envelope [12].Combining (8) and (9) and (7) and rearranging the equation, the failure function, F, will be (11)F=a+bσ1+σ32-13σ1-σ22+σ2-σ32+σ3-σ12.Failure occurs if F≤0.The Mogi-Coulomb parameters (a,b) can be related to Coulomb failure parameters (c,ϕ) as follows:(12)a=223ccos⁡ϕ,b=223sin⁡ϕ,where c is the shear strength or cohesion of the material and ϕ is the internal friction angle [13].


## 1.1. Stresses around Boreholes

A constitutive model is a set of equations needed to determine the stress state of a borehole and describe its behavior. Different models have been developed and widely studied; however, linear elastic analysis is the most common approach due to its ease of application [7] and its requirements of fewer input parameters compared with the rest of complicated models.The stress components in cylindrical coordinates can be represented by(1)σr=σx′cos2⁡θ+σy′sin2⁡ϕ+2σx′y′sin⁡θcos⁡θ,σθ=σx′sin2⁡θ+σy′cos2⁡ϕ-2σx′y′sin⁡θcos⁡θ,σz=σz′,σrθ=σy′-σx′sin⁡θcos⁡θ+σx′y′cos2⁡θ,-sin2⁡θ,σrz=σx′z′cos⁡θ+σy′z′sin⁡θ,σθz=σy′z′cos⁡θ-σx′z′sin⁡θ,where σr, σθ, and σz are called radial, tangential, and axial stress components in a cylindrical coordinate system, respectively, while the virgin formation stresses in (x,y,z) coordinate system before excavation can be expressed as (2)σxo=σHcos2⁡α+σhsin2⁡αcos2⁡i+σvsin2⁡sin2⁡i,σyo=σHsin2⁡α+σhcos2⁡α,σzo=σHcos2⁡α+σhsin⁡αsin2⁡i+σvcos2⁡i,σxyo=0.5σh-σHsin⁡2αcos⁡i,σyzo=0.5σh-σHsin⁡2αsin⁡i,σxzo=0.5σHcos2⁡α+σhsin2⁡α-σvsin⁡2i.Equation (2) represents the virgin formation stresses which are denoted by the superscript “o” on the stresses. The virginin situ stresses will be altered by excavation. In cylindrical coordinates, the complete stress solution around an arbitrary oriented wellbore can be represented by the following:(3)σr=σxo+σyo21-a2r2+σxo+σyo21+3a4r4-4a2r2cos⁡2θ+σxyo1+3a4r4-4a2r2sin⁡2θ+Pwa2r2,σθ=σxo+σyo21+a2r2-σxo-σyo21+3a4r4cos⁡2θ-σxyo1+3a4r4-4a2r2sin⁡2θ-Pwa2r2,σz=σzo-v2σxo-σoya2r2cos⁡2θ+4σxyoa2r2sin⁡2θ,σrθ=-σzo-σyo21-3a4r4+2a2r2sin⁡2θ+σxyo1-3a4r4+2a2r2cos⁡2θ,σθz=-σxzosin⁡θ+σyzocos⁡θ1+a2r2,σrz=σxzocos⁡θ+σyzosin⁡θ1-a2r2,where “a” is the radius of the wellbore, Pw is the internal wellbore pressure, and v is Poisson’s ratio. The angle θ is measured clockwise from x-axis, as shown in Figure 1.Figure 1
Stress transformation system for deviated borehole.In a linear elastic material, the largest stress concentration occurs at the borehole wall. Therefore, borehole failure is expected to initiate there [8–10]. For wellbore instability analysis, consequently, stresses at the borehole wall are the ones that should be compared against a failure criterion.Estimating the stresses at borehole wall in the case of deviated wellbore is done by settingr=a in (2); the results will be as follows: (4)σr=Pw,σθ=σxo+σy0-2σxo-σyocos⁡2θ-4σxyosin⁡2θ-Pw,σz=σzo-v2σxo-σyocos⁡2θ+4σxyosin⁡2θ,σθz=2-σxzosin⁡θ+σyzocos⁡θ,σrθ=0,σrz=0.For vertical wellbore (Figure 2), the stresses at borehole wall are determined by setting the inclination angle i=0 in (2) and orienting the horizontal axes so that the direction θ=0 is parallel to σH (i.e., α=0). The stresses become (5)σr=Pw,σθ=σH+σh-2σH-σhcos⁡2θ-Pw,σz=σv-2vσH-σhcos⁡2θ,σθz=0,σrθ=0,σrz=0.The stresses around a horizontal wellbore wall after excavation (Figure 3) can be represented by setting i equal to π/2 and plugging the resulted equation into (2) which gives (6)σr=Pw,σθ=σv+σHsin2⁡α+σhcos2⁡α-σv-σHsin2⁡α-σhcos2⁡αcos⁡2θ-Pw,σz=σHcos2⁡α+σhsin2⁡α-2vσv-σHsin2⁡α-σhcos2⁡αcos⁡2θ,σθz=σh-σHsin⁡2αcos⁡θ,σrθ=0,σrz=0.In this case, where the wellbore axis lies along the maximum horizontal principal stress (i.e., α = 0), the stress at borehole wall is described as follows: (7)σr=Pw,σθ=σv+σh-2σv-σhcos⁡2θ-Pw,σz=σH-2vσv-σhcos⁡2θ,σθz=0,σrθ=0,σrz=0.Figure 2
Stress transformation system for vertical borehole [12].Figure 3
Stress transformation system for a vertical borehole.


## 1.2. Rock Failure Criteria (Mogi-Coulomb Criterion)

The second important element in wellbore stability is rock failure criteria. These criteria are sets of functions in stress or strain spaces that describe or predict the condition at which rock fails under external load. In literature, many of failure criteria have been developed by scientists who suggested different approaches trying to simulate actual rock failures. In this paper, the concentration will be on Mogi-Coulomb criterion.In 2005, Al-Ajmi and Zimmerman presented a linear form of Mogi criterion [11]. Al Ajmi found that the plotted polyaxial data of some rocks using Mogi failure envelope in τoct,σm,2 space can be fit to a linear model as follows: (8)τoct=a+bσm,2,where σm,2 is the effective normal stress defined by(9)σm,2=σ1+σ32;τoct is the octahedral shear stress defined by(10)τoct=13σ1-σ22+σ2-σ32+σ3-σ12.Anda and b are the intersection of the line on τoct-axis and its inclination, respectively (Figure 4). Al Ajmi [12] found and verified experimentally that failure under polyaxial stresses can be predicted from triaxial test data using the linear Mogi criterion. Accordingly, he developed a new true-triaxial failure criterion called Mogi-Coulomb criterion. For triaxial stress states (σ2=σ3) the linear Mogi criterion is exactly equivalent to Mohr-Coulomb criterion [13].Figure 4
Mogi-Coulomb failure envelope [12].Combining (8) and (9) and (7) and rearranging the equation, the failure function, F, will be (11)F=a+bσ1+σ32-13σ1-σ22+σ2-σ32+σ3-σ12.Failure occurs if F≤0.The Mogi-Coulomb parameters (a,b) can be related to Coulomb failure parameters (c,ϕ) as follows:(12)a=223ccos⁡ϕ,b=223sin⁡ϕ,where c is the shear strength or cohesion of the material and ϕ is the internal friction angle [13].


## 2. Modeling Wellbore Stability

### 2.1. Vertical Wellbore Failure

To ensure mechanical wellbore stability in drilling vertical boreholes, the lower limit of the mud pressure (Pw) must be considered to be the collapse pressure. To find the collapse pressure Kirsch solution is applied to find stresses around a borehole: radial (σr), axial (σz), and tangential (σθ). This solution then must be coupled with a failure criterion (in this research the Mogi-Coulomb criterion will be used) to end up with collapsed pressure model.Using Kirsch solution, the stresses at the borehole wall can be defined by the following:(13)σr=Pw,σθ=A-Pw,σz=B.A and B are constants and can be defined by (14)A=3σH-σh,B=σv+2vσH-σh,where σv is verticalin situ stress; σH and σh are maximum and minimum horizontalin situ stresses, respectively. And v is a material constant called Poisson’s ratio.A closed-form solution for collapsed pressure can be achieved by applying (13) and (14) to the Mogi-Coulomb criterion. This solution can be used as a borehole failure model. However, in this model, there are three different permutations of the three principle stresses (σr, σz, and σθ) which need to be studied in order to determine the collapse pressure. These permutations are as follows: (1)  σz≥σθ≥σr, (2)  σθ≥σz≥σr, and (3)  σθ≥σr≥σz (Figure 5).Figure 5
In situ stress regimes: (a) normal fault, (b) reverse fault, and (c) strike slip [12].
(a) (b) (c)Equations (15) illustrate closed-form solutions to calculate collapse pressures in vertical wellbore forin situ stress regimes, normal faulting (σz>σθ>σr), reverse faulting (σθ>σz>σr), and strike slip (σθ>σr>σz), respectively: (15)Pwb1=16-2b′23A+2b′k-H+12K2+b′AK,Pwb2=12A-1612a′+b′A-2P02-3A-2B2,Pwb3=16-2b′23A+2b′G-H+12G2+b′AG.In these equations, failure occurs if Pw (internal wellbore pressure) ≤ Pwb (the lower limit of the mud pressure), where (16)H=A24b′2-3+B2-AB4b′2-12,(17)K=a′+b′B-2P0,(18)G=K+b′A.It has been proven in literature that the most common stress distribution is the second case whenσθ≥σz≥σr, and it is called reverse faulting stress regime (e.g., [4–6, 8–10]).


### 2.2. Probabilistic Approach Risk Analysis in Vertical Wellbore Stability

In 2010, Al-Harthi and Al-Ajmi applied risk analysis to wellbore stability. They utilized the Mogi-Coulomb stability model (see (15), (19), and (20)) to conduct a sensitivity analysis in vertical wellbores. They carried out Monte Carlo simulation using @Risk™ program. They also conducted a sensitivity analysis using Mohr-Coulomb stability model. The comparison between the two analyses showed that the Mohr-Coulomb criterion tends to overestimate minimum overbalance mud pressure that is required to keep borehole stable during drilling (as illustrated in Figure 6). They also concluded that risk analysis is a good tool to estimate the critical mud pressure of drilling fluid with existence of uncertainties in input parameters. The sensitivity analysis of the Mogi-Coulomb criterion (which is taking in account the effect of the intermediate principal stress) showed that the maximum horizontal stress (σH) is the most critical factor in vertical wellbore stability analysis under anyin situ stress regime (as illustrated in Figures 7 and 8).Figure 6
Overbalance pressure in the strike slip regime. The Mogi-Coulomb criterion on the left side of the graph and the Mohr-Coulomb criterion on the right side [4].Figure 7
Sensitivity analysis in the strike slip regime (tornado chart for Mogi-Coulomb) [4].Figure 8
Sensitivity analysis in the strike slip regime (tornado chart for Mohr-Coulomb) [4].


### 2.3. Nonvertical Wellbore Failure

In order to determine the critical mud pressure (the minimum mud pressure required to prevent borehole collapse) in deviated boreholes, the stresses at the borehole wall should be calculated using (4) and then compared with Mogi-Coulomb failure criterion. However, the resulting tangential and axial stresses are not principal stresses as the shear stress, σθz, may not be equal to zero. As a result, the tangential and the axial stresses should be determined using the following: (19)σ1=12σx+σy+τxy2+14σx-σy2,σ2=12σx+σy-τxy2+14σx-σy2,where σ1 and σ2 are tangential and axial stresses, respectively, σx and σy are normal stresses in x and y directions, respectively, and τxy is the shear stress in x-y direction.Introducing the resulting principal stresses to the Mogi-Coulomb failure criterion, a 4th-degree equation needs to be solved in order to find the collapse pressure. This means there is no closed-form analytical solution that can be derived for collapse pressure in deviated boreholes utilizing Mogi-Coulomb failure criterion similar to vertical boreholes or if Mohr-Coulomb failure criterion had been utilized [12].


## 2.1. Vertical Wellbore Failure

To ensure mechanical wellbore stability in drilling vertical boreholes, the lower limit of the mud pressure (Pw) must be considered to be the collapse pressure. To find the collapse pressure Kirsch solution is applied to find stresses around a borehole: radial (σr), axial (σz), and tangential (σθ). This solution then must be coupled with a failure criterion (in this research the Mogi-Coulomb criterion will be used) to end up with collapsed pressure model.Using Kirsch solution, the stresses at the borehole wall can be defined by the following:(13)σr=Pw,σθ=A-Pw,σz=B.A and B are constants and can be defined by (14)A=3σH-σh,B=σv+2vσH-σh,where σv is verticalin situ stress; σH and σh are maximum and minimum horizontalin situ stresses, respectively. And v is a material constant called Poisson’s ratio.A closed-form solution for collapsed pressure can be achieved by applying (13) and (14) to the Mogi-Coulomb criterion. This solution can be used as a borehole failure model. However, in this model, there are three different permutations of the three principle stresses (σr, σz, and σθ) which need to be studied in order to determine the collapse pressure. These permutations are as follows: (1)  σz≥σθ≥σr, (2)  σθ≥σz≥σr, and (3)  σθ≥σr≥σz (Figure 5).Figure 5
In situ stress regimes: (a) normal fault, (b) reverse fault, and (c) strike slip [12].
(a) (b) (c)Equations (15) illustrate closed-form solutions to calculate collapse pressures in vertical wellbore forin situ stress regimes, normal faulting (σz>σθ>σr), reverse faulting (σθ>σz>σr), and strike slip (σθ>σr>σz), respectively: (15)Pwb1=16-2b′23A+2b′k-H+12K2+b′AK,Pwb2=12A-1612a′+b′A-2P02-3A-2B2,Pwb3=16-2b′23A+2b′G-H+12G2+b′AG.In these equations, failure occurs if Pw (internal wellbore pressure) ≤ Pwb (the lower limit of the mud pressure), where (16)H=A24b′2-3+B2-AB4b′2-12,(17)K=a′+b′B-2P0,(18)G=K+b′A.It has been proven in literature that the most common stress distribution is the second case whenσθ≥σz≥σr, and it is called reverse faulting stress regime (e.g., [4–6, 8–10]).


## 2.2. Probabilistic Approach Risk Analysis in Vertical Wellbore Stability

In 2010, Al-Harthi and Al-Ajmi applied risk analysis to wellbore stability. They utilized the Mogi-Coulomb stability model (see (15), (19), and (20)) to conduct a sensitivity analysis in vertical wellbores. They carried out Monte Carlo simulation using @Risk™ program. They also conducted a sensitivity analysis using Mohr-Coulomb stability model. The comparison between the two analyses showed that the Mohr-Coulomb criterion tends to overestimate minimum overbalance mud pressure that is required to keep borehole stable during drilling (as illustrated in Figure 6). They also concluded that risk analysis is a good tool to estimate the critical mud pressure of drilling fluid with existence of uncertainties in input parameters. The sensitivity analysis of the Mogi-Coulomb criterion (which is taking in account the effect of the intermediate principal stress) showed that the maximum horizontal stress (σH) is the most critical factor in vertical wellbore stability analysis under anyin situ stress regime (as illustrated in Figures 7 and 8).Figure 6
Overbalance pressure in the strike slip regime. The Mogi-Coulomb criterion on the left side of the graph and the Mohr-Coulomb criterion on the right side [4].Figure 7
Sensitivity analysis in the strike slip regime (tornado chart for Mogi-Coulomb) [4].Figure 8
Sensitivity analysis in the strike slip regime (tornado chart for Mohr-Coulomb) [4].


## 2.3. Nonvertical Wellbore Failure

In order to determine the critical mud pressure (the minimum mud pressure required to prevent borehole collapse) in deviated boreholes, the stresses at the borehole wall should be calculated using (4) and then compared with Mogi-Coulomb failure criterion. However, the resulting tangential and axial stresses are not principal stresses as the shear stress, σθz, may not be equal to zero. As a result, the tangential and the axial stresses should be determined using the following: (19)σ1=12σx+σy+τxy2+14σx-σy2,σ2=12σx+σy-τxy2+14σx-σy2,where σ1 and σ2 are tangential and axial stresses, respectively, σx and σy are normal stresses in x and y directions, respectively, and τxy is the shear stress in x-y direction.Introducing the resulting principal stresses to the Mogi-Coulomb failure criterion, a 4th-degree equation needs to be solved in order to find the collapse pressure. This means there is no closed-form analytical solution that can be derived for collapse pressure in deviated boreholes utilizing Mogi-Coulomb failure criterion similar to vertical boreholes or if Mohr-Coulomb failure criterion had been utilized [12].


## 3. Probabilistic Numerical Models for Nonvertical Wellbores

The previous section illustrates that there is no closed-form solution to model deviated wellbore stability. However, Al-Ajmi [12] has applied iterative loops in a computer program to calculate the collapse pressure for nonvertical boreholes. MathCAD program was used to perform these iterated loops to find the critical mud pressure (see the Appendix).This model was utilized to build a stability model in the Excel program. In the developed model, equations of stresses and Mogi-Coulomb failure criterion have been converted from MathCAD language to the language of the Excel program. Also, the macro tool (which is using Visual Basic Programming Language) in the Excel has been utilized to do the iterated loops to calculate the collapse pressure. Figure9 illustrates the interface of the developed model.Figure 9
Interface of the deterministic instability model in Excel for nonvertical boreholes.The input parameters of the developed model are depth (h), vertical stress gradient (σvg), maximum horizontal stress gradient (σHg), minimum horizontal stress gradient (σhg), pore pressure gradient (P0g), cohesion (c), friction angle (ϕ), and Poisson’s ratio (v).The purpose of this model is to calculate the collapse pressure for a given set of input parameters that are constant (having static values) for 100 different well trajectory scenarios. In other words, the model will find out the collapse pressure for 10 values of inclination angle (i) ranging from zero to 90 degrees multiplied by 10 values of azimuth angle (k) ranging from zero to 90 degrees too.To ensure the validity of this model, it was run for a certain case (inputs are listed in Table1) and compared against Al-Ajmi’s MathCAD model. The output result was absolutely identical as illustrated in Figures 10 and 11.Table 1
Vertical wellbore failure for different stress regimes.
Input Value  σ v g (psi/ft) 1.1 σ H g (psi/ft) 1 σ h g (psi/ft) 0.9 P 0 (psi/ft) 0.46 Cohesion (psi) 870 Friction angle (deg.) 31.3 Poisson’s ratio 0.33Figure 10
Resulting collapse pressure from the developed Excel model.Figure 11
Resulting collapse pressure from the MathCAD model.The model is consistent with Al-Ajmi’s MathCAD model in calculating collapse pressure for static input parameters; however, the target is to end up with a probabilistic model in order to conduct instability risk analysis for nonvertical (deviated) wellbores. As a result, the @Risk program was utilized with the new Excel based instability model to give a probabilistic model (see Figure12).Figure 12
Interface of the probabilistic instability model in Excel for deviated boreholes.


### 3.1. An Overview of @Risk Program

@Risk is an add-in program to Microsoft Excel. It is using Monte Carlo simulation that allows user to test variety of scenarios in a given model and determine the probability distribution of each outcome. When combined with a well-built model, it can give you a good feel for how likely a given outcome is for a modeled plan or business venture. In this model, the sixth version of @Risk program is used to run Monte Carlo simulation. It allows users to make decisions using the sensitivity risk analysis. In this program, parameters can be set fixed or can be given a desired range and distribution. Due to the limitation ofin situ stresses and rock properties data, uniform distribution was used mostly in data simulations.


## 3.1. An Overview of @Risk Program

@Risk is an add-in program to Microsoft Excel. It is using Monte Carlo simulation that allows user to test variety of scenarios in a given model and determine the probability distribution of each outcome. When combined with a well-built model, it can give you a good feel for how likely a given outcome is for a modeled plan or business venture. In this model, the sixth version of @Risk program is used to run Monte Carlo simulation. It allows users to make decisions using the sensitivity risk analysis. In this program, parameters can be set fixed or can be given a desired range and distribution. Due to the limitation ofin situ stresses and rock properties data, uniform distribution was used mostly in data simulations.


## 4. Probabilistic Collapse Pressure Analysis

The developed probabilistic wellbore collapse pressure model was run to calculate the critical mud pressure for different cases. The model utilizes the @Risk program, and (as has been mentioned in the previous chapter) the model conducted Monte Carlo simulation for each case to capture the effect of uncertainty in input variables (in situ stresses and rock properties) in collapse mud pressure.In situ stresses and rock properties data of a sandstone formation provided in literature are used in this analysis. Uniform distribution was applied due to limitation of data. Then, for each case, a sensitivity analysis was carried out to figure out which parameter has more impact on collapse pressures.


### 4.1. Collapse Pressure Risk Analysis for Normal Faulting Stress Regime

In this case, Monte Carlo simulation was run to calculate the collapse mud pressure at normal faultingin situ stress regime. The borehole was assumed to be drilled in a sandstone formation with a depth of 3048 meters. The used field stresses condition and rock properties are described in Tables 2 and 3, respectively.Table 2
Normal faultingin situ stress regime [4].
Field stresses (kPa/m) Minimum Maximum  σ v g 21.94 22.62 σ H g 19.34 20.36 σ h g 18.76 19.34 P 0 10.11 10.52Table 3
Rock properties for a sandstone formation [4].
Rock properties Minimum Maximum Cohesion (kPa) 4500 5500 Friction angle (deg.) 31 33 Poisson’s ratio 0.2 0.3In the first simulation, all input values were given ranges of values with uniform distributions including well trajectories (inclination anglei and azimuth angle k). To calculate the collapse pressures, the developed probabilistic model was run applying Monte Carlo simulation with 10000 iterations using @Risk program [14].The histogram in Figure13 illustrates the resulting values of possible overbalance pressures (collapse mud pressure) and their occurrence frequencies within the given ranges of input parameters. The result shows that the minimum possible value of collapse pressure is 36,780 kPa, whereas the maximum possible value is 44,790 kPa. Table 4 summarizes the statistics of the simulation.Table 4
Normal faulting Monte Carlo simulation statistics (changing all input parameters).
Statistics P w (kPa)  Minimum 36,780 Maximum 44,792 Mean 40598 Standard deviation 1,687 90% confidence interval 38,310–43,250Figure 13
Overbalance pressure in normal faulting (changing all input parameters).To study the relations between input parameters and the resulting collapse pressures, scattered plots of collapse pressure versus input parameters were constructed (Figures14–16). The scattered plots of Pw versus input parameters show that Pw increases with deviation angle (i) in a relation similar to (S) shape. However, there are a lot of noises associated with this relation due to the effect of other parameters. For the rest of the parameters, no well-defined relations can be figured out from plots, even though the overall trends of plotted values may give indications of general relations between Pw and input parameters.Figure 14
Collapse pressure (kPa) versus deviation angle (degree).Figure 15
Collapse pressure (kPa) versus azimuth angle (degree).Figure 16
Collapse pressure (kPa) versus Poisson’s ratio.Another type of simulation was conducted while setting all input parameters constant except one in order to study the effect of uncertainty of unfixed parameter on the critical mud pressure. Then the simulation was repeated several times and in each time a different parameter was set variably. For the first case, deviation angle (i) was the only parameter that is given a range (from 0 to 90 degrees), whereas the rest of parameters were kept constant at the mid values of the given ranges. The occurrence frequencies of possible collapse pressures are illustrated in Figure 17. And Table 5 summarizes the statistics of the simulation. In order to study the relation between deviation angle (i) and collapse pressure Pw, a scattered plot has been constructed. The analysis of the resulting collapse pressure shows that deviation angle (i) has an (S) shape relation with the critical mud pressure (see Figure 18). ). Figures 19 and 20 illustrate the relation of collapse pressure versus (k) and collapse pressure versus (v), respectively.Table 5
Normal faulting Monte Carlo simulation statistics (constant input parameters excepti).
Statistics P w (kPa)  Minimum 38,360 Maximum 40,866 Mean 39,480 Standard deviation 928.71 90% confidence interval 38.363–40.852 thousandFigure 17
Overbalance pressure in normal faulting (constant input parameters excepti).Figure 18
Collapse pressure (kPa) versus deviation angle (degree).Figure 19
Collapse pressure (kPa) versus azimuth angle (degree).Figure 20
Collapse pressure (kPa) versus Poisson’s ratio (v).


### 4.2. Sensitivity Analysis

To study the effects of uncertainty of input parameters on collapse pressures, sensitivity analysis was carried out. The constructed tornado diagram (see Figure21) shows that deviation and azimuth angles have the most impact on the drilling fluid pressures. Friction angle and cohesion come next. This means well planner in such situation should pay more attention to well trajectories and rock properties.Figure 21
Sensitivity analysis in normal fault stress regime (tornado diagram).Similar simulations and sensitivity have been conducted for strike slip and reverse faulting stress regimes. Figures22 and 23 illustrate the result of sensitivity analysis of input parameters on both cases, respectively.Figure 22
Sensitivity analysis in strike slip stress regime (tornado diagram).Figure 23
Sensitivity analysis in strike slip stress regime (tornado diagram).


### 4.3. Collapse Pressure Risk Analysis in Terms ofIn Situ Stresses, Rock Properties, and Well Trajectories

In this case, the target was to study the effects of uncertainty ofin situ stresses, rock properties, and well trajectories as groups in borehole stability applying the developed instability model based on Mogi-Coulomb criterion.First, the model was run keeping rock properties and well trajectories at static values, whilein situ stresses were given ranges of values (see Figure 24). After that, the simulation was run, while rock properties were given ranges of values, and the rest were fixed, again keeping only well trajectories variables.Figure 24
Overbalance pressure (with variablein situ stresses).Sensitivity analysis was carried out to find out which of the three groups (in situ stresses, rock properties, or well trajectories) has more effects on collapse pressure under existence of uncertainty. The tornado chart in Figure 25 shows that well trajectories group has the most impact on collapse pressure. Rock properties group come next and thenin situ stresses group.Figure 25
Sensitivity analysis (tornado diagram).


## 4.1. Collapse Pressure Risk Analysis for Normal Faulting Stress Regime

In this case, Monte Carlo simulation was run to calculate the collapse mud pressure at normal faultingin situ stress regime. The borehole was assumed to be drilled in a sandstone formation with a depth of 3048 meters. The used field stresses condition and rock properties are described in Tables 2 and 3, respectively.Table 2
Normal faultingin situ stress regime [4].
Field stresses (kPa/m) Minimum Maximum  σ v g 21.94 22.62 σ H g 19.34 20.36 σ h g 18.76 19.34 P 0 10.11 10.52Table 3
Rock properties for a sandstone formation [4].
Rock properties Minimum Maximum Cohesion (kPa) 4500 5500 Friction angle (deg.) 31 33 Poisson’s ratio 0.2 0.3In the first simulation, all input values were given ranges of values with uniform distributions including well trajectories (inclination anglei and azimuth angle k). To calculate the collapse pressures, the developed probabilistic model was run applying Monte Carlo simulation with 10000 iterations using @Risk program [14].The histogram in Figure13 illustrates the resulting values of possible overbalance pressures (collapse mud pressure) and their occurrence frequencies within the given ranges of input parameters. The result shows that the minimum possible value of collapse pressure is 36,780 kPa, whereas the maximum possible value is 44,790 kPa. Table 4 summarizes the statistics of the simulation.Table 4
Normal faulting Monte Carlo simulation statistics (changing all input parameters).
Statistics P w (kPa)  Minimum 36,780 Maximum 44,792 Mean 40598 Standard deviation 1,687 90% confidence interval 38,310–43,250Figure 13
Overbalance pressure in normal faulting (changing all input parameters).To study the relations between input parameters and the resulting collapse pressures, scattered plots of collapse pressure versus input parameters were constructed (Figures14–16). The scattered plots of Pw versus input parameters show that Pw increases with deviation angle (i) in a relation similar to (S) shape. However, there are a lot of noises associated with this relation due to the effect of other parameters. For the rest of the parameters, no well-defined relations can be figured out from plots, even though the overall trends of plotted values may give indications of general relations between Pw and input parameters.Figure 14
Collapse pressure (kPa) versus deviation angle (degree).Figure 15
Collapse pressure (kPa) versus azimuth angle (degree).Figure 16
Collapse pressure (kPa) versus Poisson’s ratio.Another type of simulation was conducted while setting all input parameters constant except one in order to study the effect of uncertainty of unfixed parameter on the critical mud pressure. Then the simulation was repeated several times and in each time a different parameter was set variably. For the first case, deviation angle (i) was the only parameter that is given a range (from 0 to 90 degrees), whereas the rest of parameters were kept constant at the mid values of the given ranges. The occurrence frequencies of possible collapse pressures are illustrated in Figure 17. And Table 5 summarizes the statistics of the simulation. In order to study the relation between deviation angle (i) and collapse pressure Pw, a scattered plot has been constructed. The analysis of the resulting collapse pressure shows that deviation angle (i) has an (S) shape relation with the critical mud pressure (see Figure 18). ). Figures 19 and 20 illustrate the relation of collapse pressure versus (k) and collapse pressure versus (v), respectively.Table 5
Normal faulting Monte Carlo simulation statistics (constant input parameters excepti).
Statistics P w (kPa)  Minimum 38,360 Maximum 40,866 Mean 39,480 Standard deviation 928.71 90% confidence interval 38.363–40.852 thousandFigure 17
Overbalance pressure in normal faulting (constant input parameters excepti).Figure 18
Collapse pressure (kPa) versus deviation angle (degree).Figure 19
Collapse pressure (kPa) versus azimuth angle (degree).Figure 20
Collapse pressure (kPa) versus Poisson’s ratio (v).


## 4.2. Sensitivity Analysis

To study the effects of uncertainty of input parameters on collapse pressures, sensitivity analysis was carried out. The constructed tornado diagram (see Figure21) shows that deviation and azimuth angles have the most impact on the drilling fluid pressures. Friction angle and cohesion come next. This means well planner in such situation should pay more attention to well trajectories and rock properties.Figure 21
Sensitivity analysis in normal fault stress regime (tornado diagram).Similar simulations and sensitivity have been conducted for strike slip and reverse faulting stress regimes. Figures22 and 23 illustrate the result of sensitivity analysis of input parameters on both cases, respectively.Figure 22
Sensitivity analysis in strike slip stress regime (tornado diagram).Figure 23
Sensitivity analysis in strike slip stress regime (tornado diagram).


## 4.3. Collapse Pressure Risk Analysis in Terms ofIn Situ Stresses, Rock Properties, and Well Trajectories

In this case, the target was to study the effects of uncertainty ofin situ stresses, rock properties, and well trajectories as groups in borehole stability applying the developed instability model based on Mogi-Coulomb criterion.First, the model was run keeping rock properties and well trajectories at static values, whilein situ stresses were given ranges of values (see Figure 24). After that, the simulation was run, while rock properties were given ranges of values, and the rest were fixed, again keeping only well trajectories variables.Figure 24
Overbalance pressure (with variablein situ stresses).Sensitivity analysis was carried out to find out which of the three groups (in situ stresses, rock properties, or well trajectories) has more effects on collapse pressure under existence of uncertainty. The tornado chart in Figure 25 shows that well trajectories group has the most impact on collapse pressure. Rock properties group come next and thenin situ stresses group.Figure 25
Sensitivity analysis (tornado diagram).


## 5. Conclusions

(1)
A new probabilistic model has been developed for wellbore stability analysis. (2)
The developed model has been applied in normal faulting stress regime and in strike slip stress regime and reverse faulting stress regimes which are the most commonly encountered stress regimes in the field. The results indicated that the biggest impact factor in wellbore stability is the well trajectories followed by the rock strength parameters. (3)
Rock strength parameters are not precisely estimated in many field cases, while they have a significant impact in wellbore stability analysis. (4)
The critical mud pressure values in strike slip stress regime are higher than those estimated in normal faulting and reverse faulting stress regimes. That means the instability for a wellbore is high under strike slip stress regime compared to normal faulting and reverse faulting stress regimes. (5)
The influence of the formation pressure, maximum horizontal stress, and vertical stress on critical mud pressure determination is the same under normal fault and strike slip stress regimes. (6)
The increase in deviation angle in normal faulting will result in more stability. However, in strike slip regime, the higher the angle of drilling direction the less stability. (7)
Normal distribution gives narrower ranges of collapse pressure than uniform distribution (higher minimum values and lower maximum values).



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