Application of quantitative risk assessment in wellbore stability analysis

Journal of Petroleum Science and Engineering 135 (2015) 185–200

Contents lists available at ScienceDirect

Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol

Application of quantitative risk assessment in wellbore stability analysis Raoof Gholami a,n, Minou Rabiei b, Vamegh Rasouli b, Bernt Aadnoy c, Nikoo Fakhari d a

Department of Petroleum Engineering, Curtin University, Malaysia Department of Petroleum Engineering, University of North Dakota, USA c Department of Petroleum Engineering, University of Stavanger, Norway d Department of Chemistry, Shahrood University, Iran b

art ic l e i nf o

a b s t r a c t

Article history: Received 11 December 2014 Received in revised form 11 August 2015 Accepted 10 September 2015 Available online 14 September 2015

Elastic and strength parameters, together with pore pressure and in-situ stresses are key parameters required to be known for determination of safe mud weight window (MWW) in vertical wellbores. Estimation of these parameters, however, is subjected to wide uncertainties mainly due to lack of adequate calibration information including lab and field test data. While there are literatures on the applications of probabilistic and risk analysis on wellbore stability evaluation, limited numbers of publications report on the impact of the chosen failure criteria in estimation of safe MWW under uncertain condition. In this study, data corresponding to a wellbore located in south part of Iran was analyzed using quantitative risk assessment to consider the effect of uncertainty on estimation of safe MWW using different failure criteria. The results indicated that Mogi–Coulomb and Hoek–Brown are more robust against the uncertainty of input parameters and mud weight used for this wellbore could have slightly been increased to reduce the shear failure of the borehole wall. The uncertainty in the input data might also be very critical for casing design when only a simple margin together with pore and fracture pressures are used to select the grade of the casing against burst or collapse loads. It was also noted based on sensitivity analysis that the maximum horizontal stress is the most effective parameter in estimation of MWW. This emphasizes the importance of a reliable estimation of insitu stresses for safe drilling. The results presented here are based on a single case study, and further studies are still required to get any ultimate conclusion. & 2015 Elsevier B.V. All rights reserved.

Keywords: Failure criteria Quantitative risk assessment Maximum horizontal stress Sensitivity analysis

1. Introduction In drilling operations, a proper mud weight needs to be used in order to avoid wellbore instability. The input data used to estimate a safe Mud Weigh Windows (MWW) for drilling practice includes rock elastic and strength properties as well as pore pressure and in-situ stresses (Aadnoy and Looyeh, 2010). In practice, the input parameters are determined using log based analysis and calibrated against core and field data (Rasouli et al., 2011). However, the input data are subject to uncertainty due to limited number of calibration points acquired because of technical or financial constraints. According to Aadnoy (2011), the results of wellbore stability analysis may be uncertain due to lack of calibration data and poor interpretation of in-situ stresses. Therefore, MWW obtained from deterministic analysis is subject to a high degree of uncertainty which needs to be quantified before making any n

Corresponding author. Fax: þ 60 85443837. E-mail address: [email protected] (R. Gholami).

http://dx.doi.org/10.1016/j.petrol.2015.09.013 0920-4105/& 2015 Elsevier B.V. All rights reserved.

recommendations. This has raised the need in developing probabilistic methods for prediction of MWW and performing wellbore stability analysis. A statistical approach based on Quantitative Risk Analysis (QRA) has been presented in the last decades to provide a means to assess uncertainty associate with the input datasets used for determination or prediction of many petroleum related parameters (Moos et al., 2003). A large number of literature have reported the use of probabilistic analysis in petroleum related applications ranging from drilling exploratory prospects (Cowan, 1969), optimum casing setting depth selection (Turley, 1976), directional drilling (Thorogood et al., 1991), wireline operations (Sam et al., 1994), special remedial operations (Cunha, 1987), and prediction of pore pressure and fracture gradient (Liang, 2002). Limited work is however available on applications of risk analysis methods in drilling. Morita (1995) was the first one who published the results of a study on the effects of uncertain parameters on the wellbore stability analysis. He used a derivative based uncertainty assessment for probabilistic analysis of mud loss and break-out prediction. Dumans (1995) applied Monte-Carlo

186

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

simulation and Fuzzy sets methodologies to estimate the uncertainty of wellbore collapse and tensile failures. Later, Ottesen et al. (1999) presented his model for assessment of uncertainties in break-out pressure prediction using operationally tolerable limits. Liang (2002) considered tensile failure as the upper limit and pore pressure as the lower limit of mud weight to present a complete scheme of risk analysis in wellbore stability studies. Moos et al. (2003) presented an uncertainty analysis for borehole stability and did a sensitivity analysis for determination of key input parameters. Later, Sheng et al. (2006) used Monte-Carlo simulation and numerical model to predict the MWW. Luis et al. (2008) compared deterministic and probabilistic analysis for safe drilling. Aadnoy, (2011) did a wellbore stability analysis by quantifying the uncertainties in mud weight prediction. Mostafavi et al. (2011) presented an approach for wellbore stability analysis using analytical models. Udegbunam et al. (in press) indicated the importance of risk analysis in wellbore stability studies and used Monte-Carlo simulation to determine the risk involved in estimation of pore pressure, strength and in-situ stress parameters. However, there are few studies on the application of different failure criteria in determination of MWW under uncertain conditions. For instance, Al-Ajmi and Al-Harthy (2010) did a study on the applications of Mogi–Coulomb and Mohr–Coulomb failure criteria in determination of mud weight collapse pressure in vertical and deviated boreholes. However, they did not estimate the input parameters required for the analysis and rather used some predetermined values. In this paper, the QRA is applied to consider the uncertainty of input parameters in determination of MWW when different failure criteria are used.

2. Quantitative risk assessment Quantitative Risk Assessment (QRA) is one of the most commonly used probabilistic analysis approaches introduced by Ottesen et al. (1999) for oil and gas drilling applications. In the QRA technique, errors involved in input parameters is firstly evaluated and quantified by selecting a suitable distribution function. This is followed by considering an appropriate constitutive model to relate input parameters to desire output. Once the constitutive model is identified, thresholds between failure and success are specified according to Limit State Function (LSF) and a response surface is built using iterations. This response surface is applied to obtain a likelihood of success (LS) by quantifying uncertainty involved in estimation of input and output parameters using probabilistic distribution functions (Aadnoy and Looyeh, 2010). The latter step can be done using an interactive numerical simulation method such as Monte-Carlo technique. Monte-Carlo simulation has been replaced by traditional deterministic methods in petroleum industry to quantify the uncertainty included in any input datasets. According to Murttha (1997), Monte-Carlo simulation is a statistical analysis yielding the probability and relationship of key parameters. It has been used recently for hydrocarbon production forecast (Murttha, 1997), well control (Arlid et al., 2009), well time and cost estimation (Adams et al., 2010) and underbalance well planning (Undebunam et al., 2013).

Generally speaking, as the mud weight decreases, probability of breakout incident increases. On the other hand, high mud weights increase the risk of lost circulation and fracturing the formation. In both of break-out and induced fracture cases, distributions function are fitted to input parameters such that 99% of values lie between maximum and minimum of the curves fitted to them. Once uncertainty of input parameters is specified, response surfaces for wellbore break-out and fracture pressures can be defined. These response surfaces are quadratic polynomial functions of input parameters and their unknown coefficients can be determined by linear regression analysis. These theoretical values are calculated for various combinations of input parameters by taking samples from their distributions. After determination of the response surfaces, Monte-Carlo simulation can be efficiently used to establish uncertainty analysis for wellbore stability to see the possibility of success and failure under given indeterminate condition (Aadnoy and Looyeh, 2010).

3. Key input parameters for wellbore stability analysis The input parameters considered as the key variables for quantitative risk assessment of borehole stability are elastic parameters (Young's Modulus and Poisson's ratio), Uniaxial Compressive Strength (UCS), pore pressure (Pp), principal in-situ stresses (i.e. sv, sH, sh), borehole inclination, its azimuth and geometry as well as mechanical properties of bedding plane (Aadnoy and Looyeh, 2010; Zhang, 2013; Han and Meng, 2014). By estimation of these parameters and utilizing a constitutive model, a relationship can then be established through different failure criteria for determination of safe MWW using QRA. The principles and correlation used for estimation of these input parameters which also used in this study have been presented in the literature (e.g. Maleki et al., 2014).

4. Constitutive models 4.1. Limit state and probability failure functions The wellbore stability analysis is a combination of conventional analytical models calibrated against the operational thresholds obtained from in-situ tests including facture tests. These thresholds can be used to determine the possibility of failure and success together with generating a limit state function (LSF) formulated as below:

fL (X ) = fC (X ) − f (X )

(1)

where f is basic failure function obtained from a deterministic analysis, fC is critical failure function and fL is LSF function value of the same input parameters. The parameter X is stochastic vector representing key input parameters involved in stability analysis (Ottesen et al., 1999). It should be noticed that depending on which failure criterion is used, the input parameters may be different. According to Ottesen et al. (1999), the critical failure occurs when:

2.1. QRA applied to wellbore stability analysis

fL (X ) ≤ 0

To maintain wellbore stability, the mud weight used to drill the well should be between break-out and induced fracture pressures limits. Wellbore collapse pressure which is also known as breakout pressure is the mud pressure required to avoid wellbore failure in shear mode which is induced due to excessive tangential stresses around the wellbore wall exceeding the rock strength.

The value of LSF is not usually known for drilling operations as there is no direct equation available to estimate it. However, Monte-Carlo approach can be applied for point-by-point evaluation of this state function with different random values. Using Monte-Carlo approach and defining a probability failure function according to the key input parameter (X) as below:

(2)

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

Fig. 1. Geological Stratigraphy of the field (taken from Bosold et al. (2005)).

187

188

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

Pf (X ) =

∫f (X)≤ f (X) P(X )dX

(3)

C

the possibility of failure at any drilling operation can then be

Well_A

Scale : 1 : 2500

DEPTH (3624.38M - 4125.16M)

DB : Case Study_Dr (2)

Gamma GR 0. 120. 100. DEPTH (M)

calculated. This possibility is defined by a symmetrical bell-curve shape probability distribution function which is divided into two parts indicating the probability of shear failure in the left and tensile failure in the right.

P-wave

3/20/2015 10:42

S-wave

DTCO (us/f)

Density and Porosity

DTSM (us/ft) 40. 300.

NPHI (V/V) 40. 0.45

-0.15 RHOB (G.CM3)

1.5

Fig. 2. Petrophysical logs of Well A used for this study.

3.2

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

189

6. Estimation of input parameters and their distribution functions

4.2. Rock failure criteria To relate the input parameters to mud weight determination based on a constitutive model (i.e., Kirsch's model was considered for the purpose of this study), rock failure criteria are used. Mohr– Coulomb criterion is a conventional criteria used for wellbore instability and risk assessment analysis. However, it has a linear equation and neglects the effect of intermediate stress in rock strength prediction. There are other three famous failure criteria known as Hoek–Brown, Modified Lade and Mogi–Coulomb which have been successfully used for wellbore stability analysis (Maleki et al., 2014; Ewy, 1999). The input parameters required for defining a probability distribution function for each of these three criteria are as below:

PMC (X ) = P (UCS, PP , σH , σh, σv, φ)

(4)

PHB(X ) = P (UCS, PP , σH , σh, σv, m)

(5)

PML(X ) = P (PP , σH , σh, σv, φ, C )

(6)

In the above equations, PP is pore pressure, sH and sh are respectively, maximum and minimum horizontal stresses, sV is vertical stress, ϕ is friction angle, C is cohesion and m is Hoek–Brown constant parameter defined according to rock type. It should be noticed that the input parameters for Mohr–Coulomb and Mogi–Coulomb criteria are similar as listed in Eq. (4). In this paper, these four failure criteria were used for wellbore stability analysis and their results were compared for evaluation of their application when uncertainty is included in input data. The principle and equations of these failure criteria developed for wellbore stability analysis can be found in the literature (e.g., AlAjmi and Zimmerman, 2006; Gholami et al., 2014; Zhang and Radha, 2010; Ewy, 1999) and their applications will be discussed in this paper only. In the next section, data from a well drilled in a carbonate field in South of Iran is used for the above purpose. It would be a vertical and onshore well which is referred to as Well A.

5. Study area and available data This oilfield is located in state of Kuzestan, South of Iran, close to the petroliferous area of Dezful Embayment. The middle and upper Cretaceous sediments of this Embayment form one of the richest petroleum systems in the Middle East, with presence of the Gurpi, Khazdumi and Gadvan source rocks and Lurestan, Asmari, Khuzestan, Fahliyan, and Khami/Bangestan reservoirs. The field is an oriented symmetrical fold with the dimension of approximately 24  10 km2. The Fahliyan carbonate formation is the reservoir in this field which has an exposure in the state of Fars. The current oilfield seems to be a carbonate complex with sequence of shale and bioclastic limestone (James and Wynd, 1965). Fig. 1 shows the geological stratigraphy of the field. The conventional well logs used in this study to determine the MWW are shown in Fig. 2. These include gamma ray, compressional and shear sonic slowness, density and porosity logs together with caliper log. A number of plugs were cut out of the cores acquired from different intervals and used for mechanical testing in the lab. The results of these tests were used later to calibrate the mechanical parameters estimated based on developed correlations.

To establish the QRA for wellbore stability analysis, key input parameters are required to be estimated and quantified by fitting a distribution function to their variations. These distribution functions are used later for Monte-Carlo simulation and assessing the changes in the variation of output parameter (i.e. mud weight). This quantification is important since log data, which are used as the sources of information for prediction of input parameters, are subjected to various uncertainties due to common mechanical or electrical malfunctioning. In addition, the estimated properties from log data need to be calibrated against field or core data which are exposed to their own uncertainties during measurements. The effect of variability in geological setting of the filed may also increase the level of uncertainty when required calibration points are not adequate. The QRA will allow determining the degree of accuracy in estimation of the safe MWW when all of these uncertainties included in the database. 6.1. Uncertainty in log data It is generally known that there are uncertainties in logs data due to electrical or mechanical deficiencies during acquisition and human errors during the operation. There will therefore be two possible scenarios which increase the sources of uncertainty in the log data based analysis: (1) malfunactionaing of device and human error during acquisition and (2) complicated variation and significant changes in properties of rocks which may cause difficulty in fitting a reasonably well representative distribution function to the logs. This difficulty in finding a suitable distribution function will reduce the accuracy of estimation made by Monte-Carlo as the entire variation of input parameters will not be included in the risk analysis. There are four logs including P- and S-wave transit times together with density and porosity logs which were used for the purpose of this study. These logs together with the best distribution functions fitted to them are shown in Fig. 3. Looking at the distribution functions, it is seen that none of them are able to predict the entire variation of the log data. In fact, the best distributions functions found for porosity and density logs do not seem to be very efficient and this will increase the uncertainty of simulation as Monte-Carlo uses these distributions functions for ultimate MWW determination. Three distribution functions, including Beta-General, InverseGaussian and Logistic, were used for describing the possible changes in the logs data. Beta-General was the best function selected for P- transit time log as it could capture as much as 99.9% of data variation. This distribution specifies a beta distribution with a defined minimum and maximum value using shape parameters α1, α2 which are always greater than zero. As it is shown in Fig. 3 (top left), α1, α2, and minimum and maximum values for Beta-General distribution function selected for P-wave are respectively 2.09, 7.52, 46.88 and 107.68. On the other hand, Inverse-Gaussian function which was fitted to porosity and S-wave transit time is described by a mean (μ) and shape parameter (λ). It should be noticed that both of these two parameters must be greater than zero as otherwise the function may not be stable. The values of μ and λ for the functions fitted to porosity log were respectively 0.13 and 0.17 while those of shear wave log were 43.59 and 356.02. As it can be seen, this function has a poor performance in covering the changes in the variation of porosity log. This will be a significant amount of uncertainty which will even be increased when this log is used for prediction of any input parameters for MWW determination.

190

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

InvGauss(43.593, 356.029) Shift=+71.576

BetaGeneral(2.0912, 7.5217, 46.883, 107.688) 4.0

7

Mean = 60.113

Mean = 115.17 3.5

6

Mean = 60.111

Mean = 115.17 3.0

Fit

2.5

2.0

Input

3

Probability (%) Values x 10^-2

Fit

4

Input

Probability (%) Values x 10^-2

5

1.5

2 1.0

1

0.5

0

0.0

50

40

60

80

70

90

100

60

110

80

P-wave Transit Time (Us/ft) 0.0% 47.00

100

140

120

160

180

200

S-wave Transit Time (Us/ft)

99.9%

0.0%

0.1%

99.8%

0.2% > 180.0

84.0

90.00

Logistic(2.48102, 0.10897)

InvGauss(0.13438, 0.17547) Shift=-0.025936

3.5

10

Mean = 2.4460

Mean = 0.10844

9

3.0

8

Mean = 2.4810

Mean = 0.10844

1.5

Fit

6

5

Input

2.0

Probability (%)

Fit

7

Input

Probability (%)

2.5

4

3

1.0

2 0.5

1 0.0

0 1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

Density (g/cc) < 0.0% 1.420

95.0%

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Porosity (V/V) 5.0% > 2.802

3.1% 0.0000

95.8%

1.1%

>

0.5500

Fig. 3. Distribution functions fitted to P-wave transit time (top left), S-wave transit time (top right), Density (RHOB) (bottom left) and porosity (PHIT) (bottom right) logs.

There was another function known as logistic which was used for describing the changes in the values of density log. This distribution function specifies a logistic distribution with two parameters of alpha and beta where beta parameter must have a positive value. As is seen in Fig. 3, the logistic function is only able to cover 95% of density log variation which increases the uncertainty of estimating any other parameter from density log. From the above discussion, it can be concluded that the sources of uncertainty would be very high in estimation of geomechanical parameters since these four logs should be used. This uncertainty

would be even higher for any estimation when neutron porosity and density logs are being used. 6.2. Elastic parameters Determination of dynamic elastic parameters is conventionally done through the use of sonic wire line log where Young's modulus (E), and Poisson's ratio (υ) are estimated by P- and S-wave velocities together with bulk density of subsurface layers (Fjaer et al., 2008; Maleki et al., 2014). However, these dynamic

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

parameters are overestimated values and need to be converted to static properties (Zoback, 2007). There are many correlations developed for different fields and formations which can be used for such conversion. 6.2.1. Young's modulus Young's Modulus of subsurface layers was estimated using dynamic elastic equation (Fjaer et al., 2008) and then converted to their static values using correlation proposed by Wang (2001). The

191

result obtained from the above correlation was calibrated against the corresponding lab test results. There are many uncertainties involved in calculations of static Young’s modulus. As explained before, logs data used for estimation of dynamic Young's modulus suffer from uncertainties due to malfunctioning of device and complexity of data recorded. Apart from this, the correlations used for dynamic to static conversion are field specific and in many cases extracted with inadequate representative data. The lab data which are used to calibrate the

BetaGeneral(4.3051, 3.0440, 1.1094, 30.772)

Logistic(0.309437, 0.010341)

9

35

Mean = 18.511

8

Mean = 0.31038 30

Mean = 18.486

7

Mean = 0.30944

Fit

20

Input

4

Probability (%)

Fit 5

Input

Probability (%) Values x 10^-2

25

6

15

3 10

2 5

1

0

0 0

5

10

15

20

25

30

0.05

0.00

35

0.10

0.15

0.1% 4.50

99.9%

0.25

0.20

0.30

0.35

0.40

0.45

0.50

Poisson's ratio

Young's Modulus (GPa) 0.0% 30.50

<

0.1%

BetaGeneral(4.3051, 3.0440, 6.8285, 128.45)

99.9% 0.24000.4250

0.0% >

BetaGeneral(1.2683, 3.8894, 20.828, 53.141)

2.0

9

Mean = 78.174

1.8

Mean = 28.793

8 1.6

Mean = 78.072

Mean = 28.774

7

0.8

Fit

6

5

Input

1.0

Probability (%) Values x 10^-2

Fit

1.2

Input

Probability (%) Values x 10^-2

1.4

4

3

0.6

0.4

2

0.2

1

0.0

0 0

20

40

60

80

100

120

140

15

20

UCS (MPa) 0.1% 20.0

25

30

35

40

45

50

Friction Angle (Degree)

99.9% 128.0

0.7% 21.00

99.3%

Fig. 4. Distribution functions fitted and used for quantifying the variation of elastic and strength parameters.

0.0% 49.50

55

192

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

BetaGeneral(3.5907, 10.404, 36.547, 47.535)

BetaGeneral(1.4985, 0.88472, 36.641, 50.095)

0.40

0.25

Mean = 39.3656

Mean = 45.108

0.35

0.20

Mean = 39.3661

Mean = 45.101

Fit

0.15

Input

0.20

Probability (%)

Fit

0.25

Input

Probability (%)

0.30

0.10

0.15

0.10

0.05 0.05

0.00

0.00 36

37

38

39

41

40

42

43

36

38

40

42

Pore Pressure (MPa) 99.1%

> 42.700

36.500

5.0% 38.65

Weibull(5.6386, 132.40) Shift=+12.513

48

50

52

90.0%

5.0% 49.79

Weibull(5.4021, 50.568) Shift=+31.574

1.8

4.5

Mean = 134.70

1.6

Mean = 78.127

4.0

Mean = 134.92

1.4

Mean = 78.210

3.5

1.2

0.8

0.6

Fit 2.5

Input

1.0

Probability (%) Values x 10^-2

Fit

3.0

Input

Probability (%) Values x 10^-2

46

44

Sv (MPa)

2.0

1.5

0.4

1.0

0.2

0.5

0.0 40

60

80

0.0

100

120

140

160

180

200

40

50

SHmax (MPa) < 0.4% 62.0

99.2%

60

80

70

90

100

110

Shmin (MPa)

> 192.0

< 0.2% 48.00

99.2%

0.6% > 100.00

Fig. 5. Distribution functions fitted and used for quantifying the variation pore pressure and effective in-situ stresses.

log based extracted properties are usually limited to few samples which may not be well representative of the entire log section. In addition, it should be noticed that the lab tests themselves are subjected to uncertainties associated with sampling and preparation methods, test procedure and equipment errors. These will result in inherent uncertainties with the estimated values for the static Young’s modulus. While based on the published literature, Normal, Uniform, Triangle and Lognormal distribution functions have been recommended as the best functions for geomechanical analysis (Moos et al., 2003), Beta-General distribution function was found

to be more appropriate for describing up to 99.7% of data variation corresponding to Young's Modulus. Fig. 4 (top left) shows the distribution function for Young's modulus. This distribution function was introduced earlier by parameters α1, α2, minimum value and maximum value which were obtained to be 4.30, 3.04, 1.10 and 30.77, respectively. 6.2.2. Poisson' ratio Dynamic Poisson's ratio was estimated using dynamic elastic equation (Fjaer et al., 2008) but unlike Young's modulus it is not necessary to convert dynamic Poisson's ratio to its equivalent

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

193

Mud Weight (Mohr-Coulomb) 1

Well_A

Scale : 1 : 2500

0.9 Gamma GR 0. 150. 0. DEPTH (M)

Mud Weight (Mohr-Coulomb)

Caliper

MW (G/CM3)

0.8

HDMX (in) 3. 6.

Kick

30. BS (in)

6.

30.

0.7

Loss Break-down

Probability (%)

Break-out

0.6 0.5 0.4 0.3 0.2 0.1

3700

0

1.30

1.35

1.40

1.45

1.50

1.55

2

Mud Weight (G/CC)

3800

3900

4000

4100

Fig. 6. Safe MWW predicted based on Mohr–Coulomb criterion (left) and the corresponding response surface (right).

static parameter (Cheng and Johnston, 1981; Simmons and Brace, 1968; Wang, 2001). Although the uncertainty due to converting dynamic to static parameters is not included in determination of Poisson's ratio, the risk of using uncertain log data will make the estimation of this parameter uncertain. The inaccurate core sample data used for calibration purposes will also pose another source of uncertainty in estimation of Poisson's ratio. Bearing this in mind, the best distribution functions found to cover the variation of Poisson's ratio was Logistic function shown in the top left of Fig. 4. Logistic function, as it was indicated earlier, can be defined by two parameters (α) and (β) which were obtained respectively as 0.30 and 0.01.

6.3. Uniaxial compressive strength Uniaxial compressive strength (UCS) is one the most important mechanical parameters required to be determined for wellbore stability analysis (e.g. Moos et al., 2003). This parameter is determined in the lab through performing mechanical tests on the core samples. However, core samples are very sparse to be used for continues assessment of the given intervals. Thus, empirical correlations are conventionally used to determine rock strength from wireline log data (e.g., Horsrud, 2001). These correlations must however be calibrated against the core samples taken from the same interval for accuracy measure. To estimate the UCS of the formations intersecting Well A,

194

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

Mud Weight (Hoek-Brown) 1

Well_A

Scale : 1 : 2500

DEPTH (3660.04M - 4125.16M)

0.9 Gamma GR 0. 150. 0. DEPTH (M)

Mud Weight (Hoek-Brown)

Caliper

MW (G/CM3)

HDMX (in) 3. 6.

Kick

0.8

30. BS (in)

6.

30.

0.7

Break-out

Probability (%)

Loss Break-down

0.6 0.5 0.4 0.3 0.2

3700

0.1 0

0.7

1

1.3

1.6

1.9

2.2

2.5

Mud Weight (G/CC)

3800

3900

4000

4100

Fig. 7. Safe MWW predicted based on Hoek–Brown criterion (left) and the corresponding response surface (right).

correlation proposed by Bradford et al., (1998) was used and the results were compared against the lab test data. UCS is one of the key parameters in wellbore stability analysis with the highest level of uncertainty in its prediction. There are in fact many approaches developed for estimation of UCS using different physical properties of logs but they can only be used in certain fields with similarities in geological settings. In the same time, the only approach for calibration of these correlations is core lab data results which may not be very representative due to stress relief problem and error of the device used for measuring the UCS. For the purpose of this study, Beta-General function was used to capture 99.8% variation of UCS. Fig. 4 (bottom left) shows this distribution function where parameters α1, α2 and minimum and maximum values were respectively obtained to be 4.30, 3.04, 6.82 and 128.45.

6.4. Friction angle Estimation of friction angle is not straightforward as there are limited numbers of relationships being proposed to determine this parameter (Zoback, 2007). This parameter may have a relationship with lithology, porosity, as well as volume of shale, confining pressure, and Young’s modulus (Zoback, 2007). However, there might be other physical or mechanical parameters linked to friction angle which have not yet been well understood. Here, to estimate the friction angle, the correlation proposed by Plumb (1994) was applied which has been successfully used in many studies (Maleki et al., 2014). In this correlation two logs of gamma ray and porosity are used for prediction of friction angle. These logs are subjected to uncertainties during their acquisition as discussed before, which in turn expose uncertainties in the values of friction angle being estimated up to 5% to 10% based on

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

195

Mud Weight (Modified Lade) 1

Well_A

Scale : 1 : 2500

DEPTH (3660.04M - 4125.16M) Gamma GR 0. 150. 0. DEPTH (M)

Mud Weight (Modified Lade)

0.9 Caliper

MW (G/CM3) Kick

0.8

HDMX (in) 3. 6.

30. BS (in)

6.

30.

0.7

Loss Break-down

Probability (%)

Break-out

0.6 0.5 0.4 0.3 0.2 0.1

3700

0

0.7

1

1.3

1.6

1.9

2.2

2.5

Mud Weight (G/CC)

3800

3900

4000

4100

Fig. 8. Safe MWW predicted based on Modified Lade criterion (left) and the corresponding response surface (right).

what has been experienced. This uncertainty is summed with 0.7% uncertainty of Beta-General distribution function which was fitted to the variation of friction angle estimated from Plumb correlation. This distribution function is shown at the bottom left side of Fig. 4. Looking at this Figure, the parameters included in Beta-General distribution function were obtained to be 1.26, 3.88, 20.82 and 53.14, respectively. 6.5. Pore pressure Pore pressure is a key parameter required to be determined during the drilling and production phases of hydrocarbon reservoirs. Any mistakes in accurate estimation of this parameter may cause well blowouts, kicks and partial or total loss of the well (Tingay et al., 2009).

Direct determination of pore pressure using formation tester is a very expensive and time consuming task (Chopra and Huffman, 2006). Besides, the data will be limited to few intervals along the borehole wall. There have therefore been many indirect methods proposed to predict the pore pressure gradient using physical properties of rocks obtained from logging tools (Eaton, 1975; Gutierrez et al., 2006; Zhang, 2011). Eaton equation is one of these correlations successfully used for pore pressure prediction (Maleki et al., 2014; Zhang, 2011), where overburden stress, hydrostatic pressure and Normal Compaction Trend (NCT) line are reuired for acurate calculations (Zhang, 2011). To estimate the pore pressure variation for the purpose of this study then, Eaton equation was used (Eaton, 1975). As a result, different NCT curves were fitted to the compression sonic slowness (DTCO) log to obtain the best match between the estimated pore

196

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

Mud Weight (Mogi-Coulomb) 1

Well_A

Scale : 1 : 2500

DEPTH (3660.04M - 4125.16M)

0.9 Gamma GR 0. 150. 0. DEPTH (M)

Mud Weight (Mogi-Coulomb)

Caliper

MW (G/CM3)

HDMX (in) 3. 6.

Kick

0.8

30. BS (in)

6.

30.

0.7

Break-out

Probability (%)

Loss Break-down

0.6 0.5 0.4 0.3 0.2

3700

0.1 0

0.7

1

1.3

1.6

1.9

2.2

2.5

Mud Weight (G/CC)

3800

3900

4000

4100

Fig. 9. Safe MWW predicted based on Mogi–Coulomb criterion (left) and the corresponding response surface (right).

pressure and available MDT pressure data, as calibration points. However, the MDT data are not available for the entire interval of the borehole and the measured values are limited to the reservoir interval only. Therefore, uncertainties expected in estimation of pore pressure might be due to acquirement of P-wave transit time log data and measurement of the MDT data as calibration points. Beta-General function was used at the end to capture 99.1% of pore pressure variation as shown in Fig. 5 (top left).

uncertainty in acquisition of density log is a source of error when it is used to estimate vertical stresses. Adding to this the uncertainties of estimating pore pressure, as explained in previous section, the estimated effective stress is subjected to uncertainties. The Beta-General distribution function was applied to present the variation of effective vertical stress as is shown at top left side of Fig. 5. The results of this figure indicate that this function covers 90% of data variation, hence 10% uncertainties is expected when vertical stress is used to determine MWW.

6.6. In-situ stresses 6.6.1. Vertical stress Estimation of vertical stress is usually done using the bulk density (RHOB) log. Effective vertical stress can then be estimated by subtracting the vertical stress from pore pressure. The

6.6.2. Horizontal stresses Horizontal stress determination is not an easy and straightforward task and there have been many discussions on the methodologies which can be used to determine these in-situ stresses in tectonically relax or active regions (Fjaer et al., 2008).

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

197

Fig. 10. Sensitivity analysis using Tornado graph representing the maximum horizontal stress and pore pressure parameters as the most and least effective parameters on the MWW estimation.

Poro-elastic method is one the successful approaches developed and successfully used in determination of horizontal stresses in tectonically active regions (Maleki et al., 2014). For the purpose of this study then, minimum and maximum horizontal stresses were estimated using poro-elastic equations. Young's modulus, Poisson's ratio, vertical stress and pore pressure are the parameters included in these equations. Biot coefficient (α) which modifies the effect of pore pressure with respect to the compressibility of the rock bulk and matrix components is commonly assumed to be 1 (Zoback, 2007) and the value of tectonic strains (εx, εy) were estimated through a trial and error practice till an acceptable match was observed between the estimated minimum horizontal stresses and LOT data. In addition to the uncertainties corresponding to acquisition of each individual log, the operational and data interpretation errors related to the LOT, which is used for calibration of minimum horizontal stress, will add more uncertainties to the estimation of horizontal stresses. The difficulty in direct calibration of maximum horizontal stress is a major source of uncertainty in estimation of this principal stress. The use of dipole shear sonic has been proposed to make a better estimation of maximum horizontal stress, however, the use of this tool is not commonly practiced in industry and there are some debates about the application of this method in estimation of stress anisotropy (Sinha et al., 2008). Weibull distribution function was used to observe the data variation associated with minimum and maximum horizontal stresses corresponding to Well A. This distribution generates a distribution with the shape parameter alpha and a scale parameter beta. The results are shown at the bottom of Fig. 5. Although, there might be 0.8% uncertainty in using this distribution function, the fitted function seem to be very sophisticated. The only problem is the fact that nobody can be sure about the accuracy of the values obtained for the maximum and minimum horizontal stresses in the first place.

7. Mud weight determination under uncertain condition The parameters required to determine the safe MWW, were estimated in the previous sections. Here, with the use of different failure criteria, it is attempted to assess which criteria will have more sensitivity to the uncertainty in input parameters in determination of safe MWW. The results of using Mohr–Coulomb, Hoek–Brown, modified Lade and Mogi–Coulomb criteria in determination of safe MWW are shown in Figs. 6–9 correspondingly. It should be noticed that the mud weight used to drill Well A was 1.3 gr/cm3. Comparing the predicted shear failures by different failure criteria and observations from calipers, it can be seen that Mohr–Coulomb underestimates the minimum mud weight to avoid breakouts while Mogi–Coulomb and Hoek–Brown criteria give better prediction with respect to real observations. Modified Lade criterion, however, yields results between these two criteria. The upper bound of the MWW, i.e. the break-down or fracture pressure curve (i.e. blue line), predicted by Mohr–Coulomb appears to be very close to the mud weight used during drilling in most intervals, which suggests the potential for mud loss. In the lack of image logs, checking on the drilling reports there was no indication of any mud loss during drilling of this well which invalidates the prediction of Mohr–Coulomb criteria. Finally, comparing the predicted and observed breakouts, Mogi–Coulomb criteria appears to yield better results than Hoek–Brown and modified lade. This can be associated to the fact that this criteria is a three dimensional model where the effect of the intermediate stress is considered in failure analysis (Al-Ajmi and Zimmerman, 2006). To compare the performance of the above criteria when input data are uncertain, the response surfaces of each failure criterion were calculated to determine the likelihood of preventing wellbore breakout and induced facture as a function of mud weight. These response surfaces are presented in the right side of Figs. 6–9 for each criterion.

198

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

Conductor Pipe 30 ” @ 30 m

24” Hole Surface Casing 18 5/8“ @ 250 m

133/8“DV Collar @ 700 m

171/2” Hole

Intermediate Casing 13 3/8“ @ 2200 m

121/4” Hole

Top of 7” Liner @ 3870 m

Production Casing (10 3/4” X 9 5/8”) @ 4020 m 81/2” Hole Liner 7” @ 4340 m

57/8” Hole

Well TD @ 4510 m Fig. 11. Schematic of Well A with size of borehole and casings at different intervals.

Halving the response surfaces, the vertical lines shown in these figures indicate the range of the mud weights which provides at least a 90% certainty in avoiding breakouts (on the left side) and lost circulations (on the right side) simultaneously. As it can be seen in Fig. 6, according to the Mohr–Coulomb criteria, the mud weight has to be between 1.42 gr/cm3 and 1.48 gr/cm3 for having a safe drilling operation without exposing the well to break-outs and mud losses. Looking at Figs. 7–9, the likelihood of safe drilling based on Hoek–Brown, modified Lade and Mogi–Coulomb is almost the same indicating that to have a safe drilling with more than 90% certainty, the mud weight should be between 1.37 gr/cm3 and 1.8 gr/cm3. Comparing these results with the real mud weight used for drilling this well, the density of the mud needs a larger increase if Mohr–Coulomb criteria is employed comparing to other three criteria. This large increase in mud weight may result in excessive mud loss due to exceeding the loss circulation pressure gradient which was obtained from the inversion of minimum horizontal stress. However, according to the results presented, the mud

weight used for this borehole could have been slightly increased to reduce the shear failure and have a safer borehole.

8. Sensitivity analysis Sensitivity analysis has been widely employed in different fields of research to assess the influence of different parameters on the state of the system as well as gaining the insight into the model behavior (Saltelli et al., 2000). In particular, in wellbore stability analysis, sensitivity analysis is used to (i) measure how sensitive the probability of failure is to small changes in elastic parameters, UCS, pore pressure and in-situ stresses, (ii) recognize which one of these key parameters have more influence on the MWW, and (iii) explore the sensitivity to model assumptions and uncertainty in input variables. In this paper, one of the well-known types of sensitivity analysis known as tornado graph was used to determine the effective

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

parameters whose variation may have a significant impact on the MWW prediction. Tornado graph is a special type of Bar chart, where input parameters are listed vertically according to their impact on the system. Fig. 10 shows the results obtained from tornado graphs after performing sensitivity analysis on the effect of changes in variation of geomechanical parameters on MWW predicted. Depicted in Fig. 10, UCS, Young's Modulus and pore pressure are the least effective parameters in the analysis implying that uncertainty involved in estimation of these parameters may not have remarkable impact on the results of wellbore stability analysis for MWW determination. Changing the values of UCS by 10% can cause the mud weight to change around 5% which is more pronounced than that of the Young’s modulus. These changes will be less pronounced for the pore pressure when mud weight increases or decreases by only 2% as the pore pressure increases by 10%. Poisson’s ratio and friction angle were also part of this sensitivity analysis, but had a very negligible effect on the ultimate MWW determination. On the other hand, the predictions are extremely sensitive to SHmax and Sv indicating that these two parameters have a considerable impact on the MWW estimation and any error in their calculations may result in serious problems due to under or overestimation of mud weight. In fact, the top ranked parameter, SHmax is the one with the highest uncertainty. This was the parameter which could not be directly calibrated against the field or core data. According to Tornado graph, 10% changes in the magnitude of maximum horizontal stress may cause approximately 30% to 46% changes in the predicted mud weight which is a large value for the situation where any minor error may cause loss of borehole. This is aligned with the conclusion drawn by Aadnøy and Hansen (2005) which indicates the in-situ stresses as the main parameters with extreme impact on the MWW determination.

9. Future field development The wellbore used for the purpose of this study was drilled in 2011 where initial well plan and development was based on the seismic data and appraisal wells drilled in the field at the early stages. This wellbore was one of the first boreholes drilled in the field and was very vulnerable to uncertainty and design errors such as inaccurate design of casings due to incorrect estimation of pore and fracture pressures. The casing scheme designed and used for this well is shown in Fig. 11. According to the sensitivity analysis presented earlier, the pore pressure was not among the most sensitive parameters. However, common practice in the industry is to apply a margin for pore pressure and use it to determine the mud weight. This will induce too much risk in casing design due to over or underestimation of mud weight. As shown in Fig. 11, a liner was used across the production zone which avoided potential differential sticking due to wrong estimation of pore pressure. However, the significant uncertainty in estimation of pore pressure may result in choosing wrong casing grade which in turn leads into potential casing damage or loss of financial resources. The fracture pressure is determined using the inversion of minimum horizontal stress which indicated to be one of the parameters with reasonably large impact on MWW determination. While checking the integrity of casing shoes during formation integrity test is a good practice, this does not necessarily provide information about the effectiveness of the casing along its entire length. In addition the safe MWW obtained from the variation of pore and fracture pressure might be different when estimated values are subject to uncertainty as discussed in this study. As a result, the casing setting depths and scheme shown in Fig. 11 may be different than when deterministic approaches are used for

199

similar design purposes. Therefore, the best strategy would be to consider QRA even during casing design to quantify the risk which may be included in estimation of pore and fracture pressure since they are the influential information required for this practice.

10. Conclusions In this paper, quantitative risk assessment was used to evaluate the wellbore stability using different failure criteria when the input data are uncertain. The results indicated that safe drilling requires a MWW of between 1.42 gr/cm3 and 1.48 gr/cm3 when Mohr–Coulomb criterion is applied. This window would be between 1.37 gr/cm3 and 1.8 gr/cm3 if one uses Hoek–Brown, modified Lade and Mogi–Coulomb criteria for the analysis. Considering the real mud weight used for drilling this wellbore, it was concluded that the mud weight used for this borehole could have been slightly increased to have a more stable borehole. It was also shown that careful casing design using QRA need to be done in order to quantify the amount of risk which might be included in estimation of pore ad fracture pressures. In addition, the results of sensitivity analysis performed using tornado graph indicated that the maximum horizontal stress is the most influencing parameter in wellbore stability analysis. This was correlated to the difficulty in direct calibration of this parameter. The results obtained in this study may be generalized after performing similar studies on several wells.

References Aadnoy, S.B., 2011. Quality assurance of wellbore stability analysis. In: Proceedings of SPE/IADC Drilling Conference and Exhibition. Paper SPE/IADC 140205, 1–3 March 2011. Amsterdam, Netherlands. Aadnøy, B.S. and Hansen, A.K. 2005. Bounds on In-Situ Stress Magnitudes Improve Wellbore Stability Analyses. SPE Journal 10 (2): 115–120. http://dx.doi.org/10. 2118/87223-PA 93, pp. 51–74. Aadnoy, S.B., Looyeh, R., 2010. Petroleum Rock Mechanics: Drilling Operation and Well Design. Elsevier Publication, Amsterdam, p. 359. Adams, A.J., Gibson, C., Smith, R., 2010. Probabilistic well-time estimation revisited. In: Proceedings of Drilling and Completion Journal, SPE 119287, vol. 25, pp. 472–499. Al-Ajmi, A.M., Zimmerman, R.W., 2006. Stability analysis of vertical boreholes using the Mogi–Coulomb failure criterion. Int. J. Rock Mech. Min. Sci. 43, 1200–1211. Al-Ajmi, A.M., Al-Harthy, M.H., 2010. Probabilistic wellbore collapse analysis. J. Pet. Sci. Eng. 74 (3), 171–177. Arlid, O., Ford, E.P., Loberg, T., Baringbing, J.W.T., 2009. Kick Risk—A well specific approach to the quantification of well control risk. In: Proceedings of Asia Pacific Oil and Gas Conference and Exhibition, SPE 124034. Jakarta, Indonesia. Bosold, A., Schwarzhans, W., Julapour, A., Ashrafzadeh, A.R., Ehsani, S.M., 2005. The structural geology of the High Central Zagros revisited (Iran). Pet. Geosci. 11, 225–238. Bradford, I.D.R., Fuller, J., Thompson, P.J., Walsgrove, T.R., 1998. Benefits of assessing the solids production risk in a North Sea reservoir using elastoplastic modeling. In: Proceedings of SPE/ISRM Eurock ‘98 held in Trondheim, Norway, 8–10 July, 1998, pp. 261–269. Cheng, C.H., Johnston, D.H., 1981. Dynamic and static moduli. Geophys. Res. Lett. 8, 39–42. Chopra, S., Huffman, A., 2006. Velocity determination for pore pressure prediction. CSEG. Cowan, J.V., 1969. Risk Analysis as Applied to drilling and developing an exploratory Prospect. In: Proceedings of the 44th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, September–1 October, 1969. Paper SPE 258328, Denver, Co. Cunha, J.C., 1987. Risk Analysis Theory Applied to Oilwell Fishing Operations (M.Sc. dissertation). U. of Ouro Preto, Brazil (in Portuguese). Dumans, C.F.F., 1995. Quantification of the Effect of Uncertainties on the Reliability of Wellbore Stability Model Predictions (Ph.D. dissertation). University of Tulsa, Oklahoma. Eaton, B.A., 1975. The Equation for Geopressure Prediction from Well Logs Society of Petroleum Engineers of AIME. Paper SPE5544. Ewy, R., 1999. Wellbore-stability predictions by use of a modified Lade criterion. In: Proceeings of SPE Drill Completion, vol. 14, pp. 85–91. Fjaer, E., Holt, R.M., Hordrud, P., Raaen, A.M., Risnes, R., 2008. Petroleum Related Rock Mechanic (Development in Petroleum Sciences. Elsevier, Amsterdam. Gholami, R., Moradzadeh, A., Rasouli, V., Hanachi, J., 2014. Practical application of of

200

R. Gholami et al. / Journal of Petroleum Science and Engineering 135 (2015) 185–200

rock failure criteria in wellbore stability problem. J. Rock Mech. Geotech. Eng. 6, 13–25. Gutierrez, M.A., Braunsdore, N.R., Couzens, B.A., 2006. Calibration and ranking of pore pressure prediction models. The leading Edge, Dec 1516–1523. Han, Y., Meng, F., 2014. Selecting safe mud weight window for wellbore in shale while drilling using numerical simulation. In: Proceedings of IADC/SPE Drilling Exhibition Conference 167991. Horsrud, P., 2001. Estimating mechanical properties of shale from empirical correlations. SPE Drill. Complet. 16, 68–73. James, G.A., Wynd, J.G., 1965. Stratigraphic nomenclature of Iranian Oil Consortium Agreement Area. AAPG Bull. 49 (12), 2182–2245. Liang, G.J., 2002. Application of quantitative risk analysis to pore pressure and fracture gradient prediction. In: Proceedings of Annual Technical conference and Exhibition. Paper SPE 77354. San antonio, TX. Luis, A., M Vargas, J., Vaz, L., Gonçalvesc, C., 2008. Borehole stability analysis considering spatial variability and poroelastoplasticity. Int. J. Rock Mech. Min. Sci. 46 (1), 90–96. Maleki, Sh, Gholami, R., Rasouli, V., Moradzadeh, A., Ghvami, R., Sadeghzadeh, F., 2014. Comparison of different failure criteria in prediction of safe mud weigh window in drilling practice. Earth Sci. Rev. 136, 36–58. Moos, D., Peska, P., Finkbeiner, T., Zoback, M., 2003. Comprehensive wellbore stability analysis utilizing Quantitative Risk Assessment. J. Pet. Sci. Eng. 38 (3–4), 97–109. Morita, N., 1995. Uncertainty Analysis of Borehole Stability Problems. In: Proceedings of the SPE Annual Technical Conference and Exhibition, 22–25 October 1995, Dallas, Texas. Mostafavi, V., Aadnoy, B.S., Hareland, G., 2011. Model – Based Uncertainty Assessment of Wellbore Stability Analyses and Downhole Pressure Estimations. In: Proceedings of the 45th US rock Mechanics/Geomechanics Symposium, ARMA 11-127, San Francisco. Murttha, J.A., 1997. Mont Carlo Simulation: its status and future, SPE 37932, SPE Distinguished Author Series, JPT. April. Ottesen, R.H., Zheng, R.H., McCann, R.C., 1999. Borehole Stability Assessment Using Quantitative Risk Analysis. In: Proceedings of SPE/IADC Drilling Conference, 9– 11 March 1999. Paper SPE/IADC 52864. Amsterdam, Holland. Plumb, R.A., 1994. Influence of composition and texture on the failure properties of clastic rocks. In: Proceedings of Eurocks 94 Rock mechanics in Petroleum Engineering Conference. Delft, Netherlands, pp. 13–20. Rasouli, V., Zacharia, J., Elike, M., 2011. The influence of perturbed stresses near faults on drilling strategy: a case study in Blacktip field. J. Petrol. Sci. Eng. 76,

37–50. Saltelli, A., Chan, K., Scott, E., 2000. Sensitivity Analysis. John Wiley and Sons, Chichister. Sam, G., Alderson, A., Pousset, M., 1994. Accidental Risk Analysis of Wireline Operations. In: Proceedings of the Second International Conference in Health Safety & Environment in Oil & Gas Exploration & Production, 25–27 January, 1994. Paper SPE 27224. Jakarta, Indonesia. Sheng, Y., Reddish, D., Lu, Z., 2006. Assessment of Uncertainties in Wellbore Stability Analysis. In: Modern Trends in Geomechanics, Springer publications. Simmons, G., Brace, W.F., 1968. Comparison of static and dynamic measurements of compressibility of rocks. J. Geophys. Res. 70, 5649–5656. Sinha, B.K., Wang, J., Kisra, S., Li, J., Pistre, V., Bratton, T., Sanders, M., 2008. Estimation of formation stresses using borehole sonic data. In: Proceedings of SPWLA 49th Annual Logging Symposium, May 25–28, 2008. Thorogood, J.L., Hogg, T.W., Williamson, H.S., 1991. Application of risk analysis methods to subsurface well collisions. In: Proceedings of SPE Drilling Engineering, December 1991, pp. 299–304. Tingay, M.R.P., Hillis, R.R., Swarbrick, R.E., Morley, C.K., Damit, A.R., 2009. Origin of overpressure and pore-pressure prediction in the Baram province, Brunei. AAPG Bull. Turley, J.A., 1976. A Risk Analysis of Transition Zone Drilling. In: Proceedings of the 51st Annual Fall Meeting of the Society of Petroleum Engineers of AIME, 3–6 October 1976. Paper SPE 6022. New Orleans, LA. Undebunam, J.E., Arild, O., Fjelde, K.K., Lohne, H.P., 2013. Uncertainty Based approach for predicting the operating window in UBO design, In: Proceedings of the 75th EAGE Conference and Exhibition. SPE 164916. Jakarta, Indonesia. Udegbunam, J.E., Aadnøy, B.S., Fjelde, K.K., 2014. Uncertainty evaluation of wellbore stability model predictions. J. Pet. Sci. Eng. 124, 254–263. Wang, Z., 2001. Dynamic versus static elastic properties of reservoir rocks. In: Wang, Amos, Nur, Amos (Eds.), Seismic and Acoustic Velocities in Reservoir Rocks, pp. 531–539. Zhang, J., 2011. Pore pressure prediction fromwell logs: Methods, modifications, and new approaches. Earth Sci. Rev. 108, 50–63. Zhang, J., 2013. Borehole stability analysis accounting for anisotropies in drilling to weak bedding planes. Int. J. Rock Mech. Min. Sci. 60, 160–170. Zhang, L., Radha, K.C., 2010. Stability analysis of vertical boreholes using a threedimensional Hoek–Brown strength criterion. In: Proceedings of the Geo Florida, West Palm Beach, Florida. Zoback, M.D., 2007. Reservoir Geomechanic. Cambridge University Press, United Kingdom.