International Journal of Petroleum and Geoscience … Tabaeh Hayavi, and Mohammad Abdideh / International Journal of Petroleum and Geoscience Engineering (IJPGE) 5 (1): 8-23, 2017

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International Journal of Petroleum and Geoscience Engineering

Volume 05, Issue 01, Pages 8-23, 2017 ISSN: 2289-4713

Stability Analysis of Vertical, Directional and Horizontal wellbores using

the Three Dimensional Hoek–Brown Criterion

Mohammad Tabaeh Hayavi *, and Mohammad Abdideh

Department of Petroleum Engineering, Omidiyeh Branch, Islamic Azad University, Omidiyeh, Iran. * Corresponding author. Tel.:+98 939 183 5971

E-mail address: [email protected]

A b s t r a c t

Keywords:

Wellbore stability,

Mud pressure,

Hoek–Brown criterion,

Analytical model,

Constitutive model.

Wellbore stability is one of the crucial issues in oil and gas industries. The issues related to

instability of wells, impose significantly unwanted costs on drilling operation. Hence, in

many oil companies, wellbore stability analysis is one of the major activities in the well

design stage. The objective of this paper is to present the 3D wellbore stability prediction

models for vertical wellbores. The 3D Hoek–Brown strength criterion developed by Zhang

and Zhu in conjunction with linear poroelastic constitutive model is utilized to develop the

models. Furtheremore, safe mud pressure window required to stabilize the directional and

horizontal wellbores in different well trajectories and in-situ stress regimes during drilling

operation were determinded. The analytical model is applied to real field case in order to

verify the applicability of the developed models.

The results indicate that the decreasing of the Biot’s coefficient and increasing the UCS and

Poisson’s ratio, Young’s modulus, bulk modulus and ratio of shear wave travel time to

compressional wave travel time will increase the optimum mud pressure window. Also, in

different in-situ stress regimes, the inclination and azimuth have a significant role in

wellbore stability during drilling.

Accepted: 15 Mar 2017 © Academic Research Online Publisher. All rights reserved.

1. Introduction

When a well is drilled, the rock surrounding the

borehole must take up the load previously

supported by the rock hat has been removed. This

results in the development of a stress concentration

at the borehole wall. If the rock is not strong

enough, the wall will fail [1].

The integrity of the wellbore plays an important

role in petroleum operations. Hole failure problems

cost the petroleum industry several billions of

dollars each year. Prevention of wellbore failure

requires a strong understanding of the interaction

between formation strength, in-situ stresses, and

drilling practices. As in-situ stress and rock

strength cannot be easily controlled, adjusting the

drilling practices is the usual way to inhibit

wellbore failure [2,3].

During drilling, there are two types of mechanical

borehole failure: compressive and tensile failures.

Compressive failure occurs when the wellbore

pressure is too low compared with the rock strength

and the induced stresses. On the other hand, tensile

failure occurs when the wellbore pressure is too

high [4]. The main aspect of the wellbore stability

analysis is to mitigate these drilling problems [5].

Mohammad Tabaeh Hayavi, and Mohammad Abdideh / International Journal of Petroleum and Geoscience Engineering (IJPGE) 5 (1): 8-23, 2017

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through altering the applied mud pressure and the

orientation of the borehole with respect to the in-

situ stresses. In engineering practice, a linear

poroelasticity stress model in combination with a

rock strength criterion is commonly used to

determine the minimum and maximum mud

pressures required for ensuring wellbore stability.

Therefore, a main aspect of wellbore stability

analysis is the selection of an appropriate rock

strength criterion [6,7,8,9].

Zhou [10] introduced a modified Wiebols and

Cook [11] criterion and developed a computer

program for the wellbore stability analysis. The

results indicated the importance of the intermediate

principal stress on the stability of wellbores. Ewy

developed the Modified-Lade failure criterion and

presented the advantages of this new criterion over

Mohr-Coulomb and Drucker-Prager [12].

Colmenares and Zoback evaluated seven different

rock failure criteria based on polyaxial test data,

and they concluded that the Modified Lade and the

Modified Wiebols and Cook fit best with polyaxial

test data [13].

Aadnoy [14] developed an analytical solution to

study the stability of inclined wellbores drilled into

rock formations modeled as a transversely isotropic

material. He showed that neglecting the anisotropic

effects arising from the directional elastic

properties can result in errors in the wellbore

stability analysis. Al-Ajmi and Zimmerman [15]

developed the Mogi–Coulomb failure criterion,

according to polyaxial failure data of the variety of

rocks. They concluded that Mohr–Coulomb failure

criterion is conservative in estimating of collapse

pressure during drilling and using Mogi–Coulomb

failure criterion can minimize the conservative

nature of the mud pressure predictions.

Hoek–Brown failure criterion is another well-

known criterion successfully applied to a wide

range of rocks for almost 30 years [16,17]. Zhang

and Zhu [18] developed a 3D Hoek-Brown strength

criterion for rocks. This criterion properly

considers the effect of the intermediate principal

stress. Also this criterion has the advantage over

the other 3D strength criteria in that it uses the

same input parameter as the most widely used

Hoek–Brown criterion. Zhang et al [19] compared

minimum mud weight prediction of five common

rock failure criteria. They recommended Mogi-

Coulomb and 3D Hoek-Brown to be used for

wellbore stability analysis.

In this paper, the 3D Hoek-Brown strength criterion

developed by Zhang and Zhu is used to analyze

wellbore stability. Furthermore, the analytical

models are applied to field data in order to verify

the applicability of the developed models.

2. Stress Concentration around a Wellbore at

Drilling Condition

Drilling a borehole will alter the in situ principal

stresses, the vertical stress and the maximum and

minimum horizontal stresses, in a manner so as to

maintain the rock mass in a state of equilibrium.

This leads to a stress concentration around the

wellbore [20].

The degree of stress concentration depends on the

wellbore orientation, the magnitude and orientation

of in-situ stresses, and the wellbore pressure [21].

When the elevated stress exceeds the rock strength,

the rock will fail resulting in the development of

wellbore failure [22]. If excessive, the cavings

produced by the spalling of broken materials into

the wellbore can cause drilling problems such as

pack-off, over-pulls, stuck-pipe and poor

cementing, to name a few [23].


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To evaluate the stability of a wellbore, a

constitutive model is required to compute the

stresses around the borehole [24]. Although

different constitutive models are available, the

linear poroelasticity stress model is commonly used

in industry practice [19].

The stress concentration around a well drilled in an

isotropic, elastic medium under anisotropic in-situ

stress condition (Maximum and minimum

horizontal stresses are different) is described by the

Kirsch equations. The general expressions for the

stresses at the wellbore wall for a deviated well in

the drilling situation are [25]:

Where and are the effective radial,

tangential and axial stresses, respectively. Pw is the

well pressure, Pp is the pore pressure, is the

Biot’s coefficient, is the Poisson’s ratio, θ is the

angular position around the wellbore circumference

and measured clockwise from the azimuth of

maximum horizontal stress.

The shear stresses at the wellbore wall are denoted

, and , while the in-situ stresses in (x, y,

z) coordinate system, denoted , , , ,

and , and they are defined as [21]:

where i is wellbore inclination and is the azimuth

angle due to the maximum horizontal stress ( )

direction (Degree) as illustrated in Fig. 1.

Fig. 1: Axes and inclination and direction angles of

the inclined well [26].

Effective induced stresses created at the borehole

wall for a vertical borehole ( ) can be

obtained from equations. 1-3 in the following [25]:


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According to Eqs. 13-15 the tangential and axial

stresses are functions of the angle θ. This angle

indicates the orientation of the stresses around the

wellbore circumference, and varies from 0° to 360°.

Consequently, the tangential and axial stresses will

vary sinusoidally. The tangential and radial stresses

are functions of the well pressure, but the vertical

stress is not.

Therefore, any change in the mud pressure will

only influence the tangential and radial stresses.

Inspection of these equations reveals that in the

vertical well, both tangential and axial stresses

reach a maximum value at θ = 90°, 270° and a

minimum value at θ = 0°, 180°.Therefore, the shear

failure known as breakouts is expected to happen at

the point of maximum tangential stress where the

rock is under maximum compression (at θ = 90°).

Tensile failure known as hydraulic or induced

fracture, however, is expected to occur at the point

where minimum tangential stress is applied to the

rock (at θ = 0°): an orientation 90° away from the

location of shear failures around the wellbore (Fig.

2) [24].

The magnitudes of three effective principal stresses

around the wellbore to analyze the initiation of

induced fracture can be obtained as:

Fig. 2: The location of breakout and tensile

fractures on the borehole wall [27].

For shear failure or breakouts to occur, the

magnitude of effective principal stresses around the

wellbore are estimated as

Based on linear elasticity, maximum stresses, occur

in the wellbore wall (Fig. 3). Therefore, borehole

instability is expected to initiate at the borehole

wall [28].

Fig. 3: Stresses around a vertical borehole in a

linear elastic formation [25].


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3. Three-Dimensional Hoek-Brown Strength

Criterion

A great number of rock strength criteria have been

proposed over the past decades. Of these different

strength criteria, the Hoek-Brown strength criterion

has been used most widely, because: (1) it has been

developed specifically for rock materials and rock

masses; (2) its input parameters can be determined

from routine unconfined compression tests,

mineralogical examination, and discontinuity

characterization; and (3) it has been applied for

over 20 years by practitioners in rock engineering,

and has been applied successfully to a wide range

of intact and fractured rock types [29]. For intact

rock, the Hoek–Brown strength criterion may be

expressed in the following form [17].

where is the Uniaxial Compressive Strength

(UCS) of intact rocks, and are respectively

the major and minor effective principal stresses,

and mi is a material constant for the intact rock,

which depends upon the rock type (texture and

mineralogy) as tabulated in Table 1.

For jointed rock masses, the Hoek–Brown strength

criterion can be expressed as follows [32]:

where

Table 1: Values of mi for different rocks [30, 31].

The parameter mb is a reduced value of mi, which

accounts for the strength reducing effects of the

rock mass conditions defined by Geological

Strength Index (GSI).

Adjustments of ‘s’ and ‘a’ are also done according

to the GSI and D values [33]. GSI was estimated

from the chart of Marinos et al. [34] (Fig. 4). D is a

Class of rock Group

Coarse

Texture

Medium

Fine

Very fine

Clastic

Conglomerate

(21±3)a

Sandstone

(17±4)

Siltstone

(7±2)

Claystone

(4±2)

Breccia

(19±5)

Greywacke

(18±3)

Shale

(6±2)

Marl

(7±2)

Non- clastic

Carbonate

Crystalline

limestone

(12±3)

Sparitic

limestone

(10±2)

Micritic

Limestone

(9±2)

Dolomite

(9±3)

Evaporite Gypsum

(8±2)

Anhydrite

(12±2)

Organic

Chalk

(7±2)


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factor which depends upon the degree of

disturbance to which the rock mass has been

subjected by blast damage and stress relaxation.

Fig. 4: GSI chart for jointed rocks [34].

It varies from 0 for undisturbed in situ rock masses

to1 for very disturbed rock masses [32]. As can be

seen from above, the Hoek–Brown strength

criterion does not take account of the influence of

the intermediate principal stress. Much evidence,

however, has been accumulating to indicate that the

intermediate principal stress does influence the

rock strength in many instances [18,21,35,36].

Zhang and Zhu [18] proposed a 3D version of the

original Hoek–Brown strength criterion for rock

mass (Eq. (5) with a=0.5):

Where and are, respectively, the

effective mean stress and the octahedral shear

stress defined by:

Where is the intermediate effective principal

stress.

4. Building the Vertical Wellbore Stability

Prediction Models

The modes of shear and tensile failures may be

different depending on the order of magnitude of

three effective principal stresses around the

wellbore wall. These stresses are , and

presented in equations. 13-15.

When mud pressure decreases, increases

towards the compressive strength. Thus, the lower

limit of the mud pressure, Pwb, is associated with

borehole collapse, in which should be greater

than . There are three permutations of the three

principal stresses that need to be investigated in

order to determine the minimum allowable mud

pressure: (1) (2)

(3) .

Substituting each of these scenarios in the 3D

Hoek-Brown failure criterion presented in equation

21, and introducing equations 16 and 19-20, gives

Solving this equation for Pwb will give four roots.

The smallest root is the lower limit of the mud

pressure in order to avoid breakouts (collapse

pressure). The constants P4, P3, P2, P1, P0 are

shown in Appendix. If the well pressure falls below

Pwb, borehole collapse will take place.

On the other hand, when Pw increases,

decreases towards the tensile strength. Therefore,

the upper limit of the mud pressure, Pwf, is

associated with fracturing, where should be less

than . Considering this constraint and the


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relative magnitude of the axial stress, there are

three permutations of the three principal stresses

that need to be investigated in order to determine

the maximum allowable mud pressure: (1)

(2) (3)

Similarly, for each case, by

introducing equation 16-18 into equation 21, gives

Solving this equation for Pwf will give four roots.

The smallest root is the upper limit of the mud

pressure in order to avoid borehole fracturing

(fracture pressure). The constants P4, P3, P2, P1,

P0 are shown in Appendix. If the well pressure

rises above the fracture initiation pressure, Pwf,

tensile failure will take place.

Reduction of mud pressure, corresponding to lower

confining pressures, increases the potential for

shear failure. On the other hand, increasing the mud

pressure above a certain limit causes the tensile

failure to happen. This discussion indicates that

there is a optimum window for the mud weight to

drill the wellbore in a stable condition. The lower

limit for this window corresponds to shear failure

(breakouts) with its upper limit being the fracture

initiation pressure [20,37].

5. Evaluation of Directional and Horizontal

Wellbore Stability

In an arbitrarily oriented wellbore, the radial stress,

is one of the effective principal stresses. Other

two effective principal stresses can be calculated by

using the theory of combined stresses. Equations of

these three effective principal stresses, , , and

can be written as follows [38]:

Where

where and are the effective maximum and

minimum effective principal stresses and is the

effective intermediate principal stress.

Regarding the fact that radial and tangential

stresses are functions of wellbore pressure, Pw, the

principal stresses are therefore also functions of

well pressure. So an iterative loop should be

applied to obtain minimum and maximum

allowable mud pressure in oriented wellbores.

In this study, a computer program is developed to

obtain the safe mud pressure required to maintain

wellbore stability. This program using several input

parameters, including: pore pressure and in-situ

stresses (vertical stress, maximum and minimum

horizontal stresses), rock strength parameters

(Tensile strength, Biot’s coefficient, Poisson’s

ratio, material constants (m and s)), well inclination

and azimuth.

In drilling situation wellbore pressure increases

from minimum horizontal stress until the condition

for tensile failure satisfied. Furthermore, well

pressure decreases from minimum horizontal stress

to formation pore pressure until the shear failure

occurs. These analyses have been done for different

well inclination (i=0° to i=90°) and azimuth ( =0°

to =180°) in several cases of in-situ stress

regimes.

6. Results and Discussion

Table 2 shows thee cases of different in-situ stress

regimes and the input parameters for wellbore


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stability analysis. According to these data

minimum and maximum bottomhole pressure that

mud weight must be provided to prevent wellbore

wall instability are determined.

In Case 1, the formation is in the normal regime.

Fig. 5 shows the 3-D plot of collapse and fracture

pressures as function of inclination and wellbore

azimuth for Case 1. The vertical axis are collapse

and fracture pressures, and horizontal axes indicate

wellbore inclination and azimuth.

Fig. 3 show that a vertical wellbore has a less

collapse pressure (Pwc) and higher fracture

pressure (Pwf) (larger safe mud pressure window)

than the horizontal borehole. So, the vertical

borehole is more stable than horizontal and

deviated boreholes in almost all directions.

In addition, it is obvious that drilling parallel to the

minimum horizontal stress direction ( =90°) is the

best trajectory to stabilize the borehole and

associated with a larger safe mud pressure window

than the maximum horizontal stress direction

( =0°) in this case. However, in this particular

case, deviated borehole with an inclination of 50°

and

azimuth of 93° have the largest safe mud pressure

window. This means that instability problems can

be minimized if the well would be deviated 50°

from the vertical and parallel to the direction of the

minimum in-situ stress. Furthermore, it shows that

the collapse and fracture pressures are highly

sensitive to the inclination in all direction or

azimuth.

In Case 2, the formation is in the strike-slip regime.

Fig. 6 shows that, the horizontal boreholes are more

stable (larger safe mud pressure window) than the

vertical or all deviated boreholes in all directions.

Moreover, wellbores drilled in the direction of

maximum horizontal stress show the highest

stability (largest safe mud pressure window) and

the others bored along minimum horizontal stress

are the least stable.

Finally Case 3 indicates a formation in the reverse

fault regime. Fig. 7 shows that a horizontal

borehole which is drilled parallel to the maximum

horizontal stress has a largest safe mud pressure

window. Here also the risk of wellbore instability

decreases with increasing the borehole inclination.

Contrary to normal regime, in reverse and strike-

slip regimes, the least variation of safe mud

pressure window versus wellbore inclination are for

the wellbores in the direction of minimum

horizontal stress.


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Fig. 5: Collapse and fracture pressures as function of wellbore trajectories in normal stress regime (Case 1).

Table 2: Different in-situ stress regimes used in this study.

Case Depth

(ft)

Stress

regime

mb s

1 6500 Normal 1 0.85 0.75 0.442 3000 4.1 0.03 0.3 0.75

2 6500 strike-

slip

0.9 1 0.85 0.442 3000 4.1 0.03 0.3 0.75

3 6500 Reverse 0.85 1.1 0.91 0.442 3000 4.1 0.03 0.3 0.75


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Fig. 6: Collapse and fracture pressures as function of wellbore trajectories in strike-slip stress regime (Case 2).

Fig. 7: Collapse and fracture pressures as function of wellbore trajectories in reverse stress regime (Case 3).


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7. Field Case Study

The developed analytical models will be applied to

a well (called well A) drilled in Ahwaz oilfield

(One of southern Iranian field in the Middle East)

for investigation of stability analysis during

drilling. This oil field, one of the most important

Iranian super giant oil fields, was discovered in

1956 and now has more than 450 producing wells.

This oil field has an anticline structure 72 km long

and 6 km wide with NW-SE trending symmetrical

anticlinal, located in central part of north Dezful

region. Its main reservoir is the Asmari formation

[39,40]. The big Asmari reservoir is complicated

and heterogeneous in terms of reservoir rock

features [41]. In-situ stresses and pore pressure

profiles of Asmari formation are shown in Fig. 8.

Fig. 8: In-situ stresses and pore pressure profiles of

Asmari formation.

Fig. 9 shows the estimated log based

geomechanical properties of Asmari formation.

including UCS, Poisson’s ratio, Young’s modulus

(E), bulk modulus (Kb), Biot’s coefficient and ratio

of shear wave travel time to compressional wave

travel time (Dts/Dtc). For simplicity, the average

values of mi for sandstone and limestone rocks are

assumed to be 17 and 10, respectively. The value of

D is considered 0.9 and the GSIs of rocks was

estimated from the chart of Marinos et al. [34]. The

most commonly observed order of magnitude of

stresses around a wellbore in terms of shear failure

is and in case of

tensile failure [24].


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Considering this assumption and the real mud

weight that had been used to drill Well A (i.e. 1.05

gr/cm3), the calculations were carried out to

determine the potential for any shear failure

(breakouts) or tensile failure (induced fracture).

The results of such analysis and hole diameter log

(caliper log) are shown in Fig. 10.

It can be concluded that the minimum and

maximum allowable mud pressures change as a

function of depth and for well A are varied between

the 21-29 MPa and 43-49 MPa, respectively. So,

the optimum mud pressure window for this well is

29-43MPa.

Fig. 9: Geomechanical properties of Asmari formation.


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Fig. 10: Left: Determination of minimum and

maximum allowable mud pressures for Well A

using the 3D Hoek-Brown criterion; Right: Caliper

log.

From the caliper log shown in this figure, by

applying the real mud weight that had been used to

drill this well, several breakouts were happened at

the interval of 2420-2530 m. It can be seen that a

good agreement is reached between the results of

caliper log and developed model in order to

investigate the depths of borehole breakout.

As Figs. 6 and 7 depict, optimum mud pressure

window is a function of geomechnical properties.

Therefore, increasing of UCS, Poisson’s ratio,

Young’s modulus, bulk modulus and ratio of shear

wave travel time to compressional wave travel time

causes decrease in the minimum allowable mud

pressure and increase in the maximum allowable

mud pressure, while an increase in Biot’s

coefficient tends to increase the breakout pressure

and decrease the fracture pressure.

8. Conclusions

In this paper the new models for prediction of

vertical wellbore stability, based on the 3D Hoek-

Brown failure criterion and the linear constitutive

model were introduce. The result indicated that

optimum mud pressure window increases with

increasing the UCS, Young’s modulus, bulk

modulus, Poisson’s ratio and ratio of shearwave

travel time to compressional wave travel time and

decreasing the Biot’s coefficient. Also, a good

agreement was achieved between the results of

caliper log and developed model in order to

investigate the depths of borehole breakout.

Furthermore, this paper presents the sensitivity

study of in-situ stress regime and well trajectory on

collapse and fracture pressures (or safe mud

pressure window). It was shown that the safe mud

pressure window in drilling condition is highly

affected by in-situ stress regimes and well

trajectories. The other results of this study indicated

that:

1- In the normal regime, a vertical boreholes has a

larger safe mud pressure window than the

horizontal borehole (Contrary to the reverse and

strike-slip regimes).

2- Contrary to normal regime, in reverse and strike-

slip regimes, drilling in the direction of minimum

horizontal stress is less stable (smaller safe mud

pressure window) than the other directions.

3- The collapse and fracture pressures of the

boreholes drilled parallel to the minimum

horizontal stress in the reverse and strike-slip

regimes have the least sensitivity to the wellbore

inclination (Contrary to normal regime).


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4-In all stress regimes the safe mud pressure

window is highly sensitive to wellbore inclination

in the direction of maximum horizontal stress.

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Appendix

The constants A, B, C and D are calculated based

on the effective principal stress states related to

each case of shear and tensile failures occurred

around the borehole.

1. Shear failure:

(1)

(2)

(3)

2. Tensile failure:

(1)

(2)

(3)

Documents

International Journal of Petroleum and Geoscience … Tabaeh Hayavi, and Mohammad Abdideh / International Journal of Petroleum and Geoscience Engineering (IJPGE) 5 (1): 8-23, 2017