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Title of the paper:

Well Bore Stability Using the Mogi-Coulomb Failure Criterion and Elasto-Plastic

Constitutive Model.

Authors (name and last name): 1- Ali Mirzaghorbanali 2- Mahmoud Afshar

Authors (E-mail): 1- alimirzaghorbanali@yahoo.com 2- mafshar@put.ac.ir

Authors (institutional address) 1- Petroleum University of Technology, Tehran, Iran 2-

Assistant professor of Petroleum University of Technology, Tehran, Iran.

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Abstract

Saving time and money are the results of the stable bore hole design. During drilling there

are two main instability problems, namely, bore hole collapse and fracture. The consequences

of these drilling problems are severe, even the most simple bore hole collapse or break down

can lead to the loss of millions of dollars in equipment and valuable natural resources.

The main aspect of the well bore stability analysis is to mitigate these drilling problems.

This is typically investigated by a constitutive model to estimate stress around the well bore,

coupled with failure criterion.

The most common approach for stability analysis is a linear elastic and isotropic

constitutive model in conjunction with linear failure criteria like Mohr-Coulomb. The Mohr-

Coulomb failure criterion only involves the maximum and minimum principle stresses and

therefore assumes that the intermediate principle stress has no influence on rock strength. In

addition, it is believed that the fluid barrier, and a part of the bore hole wall, behave

plastically which provides higher fracturing pressure than conventional elastic theory.

In this paper, a model for the mud weight window determination, using Mogi-Coulomb

failure criterion and the elasto plastic model is developed. This is based on the hypothesis

that, indeed elastic constitutive model does not fit with the reality of the well bore wall

behavior and intermediate principle stress plays an important role on rock strength. This

hypothesis is verified and is used in this paper for the South Pars gas field (phases 6, 7, and 8)

in the Persian Gulf. This model leads to easily computed expression for the critical mud

pressure required to maintain well bore stability.

Key words: Wellbore stability, Mud weight window, Mogi-Coulomb failure criteria,

Elasto-Plastic theory.

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1- Introduction

Nowadays, deliberate access to geological formations bearing petroleum through drilled

wells is relatively difficult because most shallow formations are close to end of their

economic life. Drilling operations have to get access to deeper reservoirs through deviated

bore hole, meanwhile more harsh conditions lead to substantial increase in failure potential.

The consequences of failure are severe: even the most simple bore hole collapse or break

down can result in the loss of millions of dollars in equipment and valuable natural recourses.

It was 1980's that geomechanical studies were extensively applied to confront with well

bore instabilities. It was understood through performed geomechanical surveys that well bore

stability problems might be alleviated or often eliminated by pertinent determination of mud

weight window. Thus Kirsch's equation, which was developed one hundered years ago

coupled with Mohr-Coulomb failure mechanism were utilized to compute the safe mud

density (Fjaer et al. (1992)).

Fleming et al. (1990) performed field study on mechanical borehole stability. They claimed

that a program based on plastic analysis could be more proper and compatible with the

ductile lithologies. Similar results were reported by Russell et al. (2003), and Winterfeld et al.

(2005).

Aadnoy et al. (2007) imitated wellbore by hollow concrete cores and through performing

experiments, they found that the linear elastic model underestimates four to eight times the

fracture pressure measured on hollow concrete cores. They concluded that the fluid barrier

and a part of bore hole wall, behaved plastically.

Fersheed K et al. (2007), and Mclean and Addis (1990) studied the sustainable development

of geomechanics technology to reduce well construction costs. They suggested that it was

better and reasonable to develop bore hole stability analysis based on elasto-plastic theory,

especially when shale formations were drilled because yielded rocks in the vicinity of the

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hole might transfer a portion of applied stresses to rock deeper inside the formation.

Furthermore, they investigated the role of failure criteria in safe mud weight window and

suggested that traditional Mohr-Coulomb failure criterion, which ignores the influence of the

intermediate principle stress, would lead to a conservative and thin mud weight window in

contrast with three dimensional failure criteria. Their conclusions are in the same direction

with Collins (2002), and Simangunsong et al. (2006).

Al. Ajmi, and Zimmerman introduced the fully polyxial Mogi-Coulomb failure criterion

(2004), and proposed a new 3D analytical model (2006) to approximate the mud weight

needed to avoid failure for the vertical wells based on Mogi-Coulomb failure mechanism

coupled with elastic theory. Their study shows the significant role of intermediate principle

stress in rock strength, where by using three dimensional Mogi-Coulomb failure criterion

greater mud weight windows than Mohr-Coulomb failure mechanism have been obtained.

Albeit in their research plastic zone in the vicinity of the well bore has not been considered.

In this paper the plasticity of the near bore hole region and the mid principle geotechnical

stress and their effects in the mud window are investigated. In the next sections model

development and verification will be presented.

1-1- Regional geology

South Pars field is located on the Qatar-Fars Arch, one of the major structural elements

of the Central Persian Gulf Area. Persian Gulf is part of what is referred to in the Plate

Tectonic literatures as the "Arabian Plate" and/or the "Middle East Sedimentary Basin"

which is approximately 3000km in length and 2000km in width. It is bounded from the

north by Turkey Bitlis Suture, from the west by the Red Sea, from the east by Zagros

Mountain (Zagros Thrust), and from the south by the Gulf of Oman and Owen Fracture

Zone in the Arabian.

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2- Model development

Well bore stability evaluation is divided in to the shear and tensile failure mechanisms,

each part determines one of the boundaries of the mud window.

2-1- Shear failure

In this section a new shear failure model based on the elasto plastic theory and Mogi-

Coulomb failure mechanism is developed.

2-1-1- Intermediate border line of the mud window

In the elasto plastic theory, the largest stress concentration occurs at the onset of the elastic

zone where plastic shield terminates (Fjaer et al. (1992). the stresses at the plastic elastic

interface in the vertical wells are given by Equations (1) to (3) (Mirzaghorbanali, 2009).

√ln (1)

3√

ln (2)

2 (3)

To avoid plastic elastic interface collapse, there are three cases of three principle stresses

need to be investigated, as follows:

(1)

(2)

(3) (Mirzaghorbanali, 2009).

By utilizing the effective stress concept and in term of the first and the second stress

invariants, Mogi-Coulomb law can be represented as in Equation (8).

(4)

(5)

2 cos (6)

sin (7)

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3 . 2 ْ (8)

To simplify mathematical operations, parameters A, B, M, (K, K', K''), (F, F', F''), H, L, and

G are defined as in Equations (9) to (16).

A 3 (9)

B 2 (10)

√ln (11)

2 ˚ , 2 2 ˚ , 2 ˚ (12)

2 3 2 , 2

3 2 , 2 3

2 (13)

4 3 4 12 (14)

2 ˚ (15)

(16)

By substituting the principle stresses into Equations (4) and (5), the first and the second

stress invariants are obtained by Equations (17) and (18).

2 (17)

2 (18)

Consider the first scenario of bore hole collapse, where the intermediate principle stress is

the tangential stress and the well pressure is equal to the minimum allowable mud pressure to

avoid plastic elastic interface collapse, introducing the Equations (17) and (18) into Equation

(8), Equation (19) is obtained.

2 3 2 (19)

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Solving Equation (19) for , two roots will be obtained. To prevent bore hole collapse,

the smaller root has to be considered as the intermediate limit of the mud window as shown

in Equation (20).

(20)

Similarly the intermediate limit of the mud window for other cases are calculated and

presented in Table (1)

Table (1) the intermediate limit of the mud window in different cases for vertical wells

Intermediate limit of the mud window Cases Case number

1

2

3

2-1-2- Lower limit of the mud window

When the interface between elastic and plastic zones fails, plastic shield coupled with the

well pressure resist against the well bore collapse. After interface failure, rest of the

formation behaves elastically with a radius equal to the actual well radius plus plastic zone. In

this situation radial stresses are exerted on the elastic zone instead of the plastic zone to avoid

well bore collapse. Therefore to keep stability, plastic shield should be subjected to the same

radial stresses with elastic zone as shown in Figure (1).

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Figure.1 Plastic shield under tangential, radial, and well pressure

The equilibrium equation for the arrangement shown in Figure (1) is as follows:

2 2 2 2 (21)

Substitution for tangential stress into Equation (21), and by integrating over plastic zone

thickness (Mirzaghorbanali, 2009), Equation (22) is obtained.

2√

ln 2 2 2 (22)

By rearrangement of Equation (22), Equation (23) is obtained.

√ln (23)

The required radial stress for vertical wells is obtained from elastic theory and Mogi-

Coulomb failure criterion (Al.Ajmi and Zemmerman, 2006). By substituting radial stress into

the Equation (23), minimum allowable mud pressure for vertical wells is obtained as shown

in Table (2).

Table (2) Minimum allowable mud pressure to avoid well bore collapse for vertical wells

Minimum allowable mud pressure Cases Case number

3 2 12√

ln 1

12 2 � 3 2√

ln 2

3 2 12√

ln 3

Tangential stresses ( ) 

Well pressure ( ) 

Radial stresses ( ) 

Plastic zone

Plastic elastic interface

Elastic zone 

Well radius (a)

Plastic thickness (t)  

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Deduction of the plastic term ( √

ln ) from the conventional elastic equations of the

Table (2), would increase the formation durability against well bore break out. The advantage

of this model is described in section (3).

2-2- Tensile failure

When the effective stress at the well bore exceeds the tensile strength of the formation,

tensile failure, and as a result, induced fracture is imminent as shown in Equation (24).

˚′ (24)

The upper limit of the mud window to avoid well bore fracture is determined directly from

Equation (24).

Aadnoy and Belayneh (2004) used elasto plastic constitutive model to obtain the effective

tangential stress around the vertical wells, and by introducing this term into the Equation

(24), the upper limit of the mud weight window based on elasto plastic model is obtained as

shown in Equation (25).

3 ,′

,′ 1.1547 ˚ (25)

The plastic zone thickness is determined by leak off test, and eventually fracturing pressure

for entire of the well trajectory is calculated by Equation (25), this equation shows that

similar to the shear failure, the role of the plastic zone in vicinity of the well bore is like a

pressure shield.

2-3- Proposed mud weight window

In the new model developed in this study, unlike the other approaches, three different limits

are proposed. As in previous models the upper and lower limits are fracture pressure and

collapse pressure, respectively. The intermediate limit acts as a border line, if the well

pressure becomes greater than this limit, drilling without joints will be gained. Otherwise, as

the well pressure becomes less than this limit, joints and discontinuities will form around the

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well bore. Consequently, to reach stable drilling it is necessary to keep the well pressure

between the upper and lower limits as shown in Table (3).

Table (3) Mud weight window for different cases for vertical wells

Maximum allowable mud

pressure

Minimum allowable mud pressure (without discontinuities) Minimum allowable

mud pressure

Cases Number

1.1547 '-√

ln

1

1.1547 '-√

ln

2

1.1547 '-√

ln

3

X', Y', Z', and, W' for different scenarios are determined as shown in Table (4).

Table (4) X', Y', Z', and, W' for different scenarios

W' X', Y', and Z' Case number

W'=3 , , ˚ X'= 3 2 12 1

W'=3 , , ˚ Y'= 12 2 � 3 2 2

W'=3 , , ˚ Z'= 3 2 12 3

3- Model verification and discussion of results

To assess the maximum mud window limit of the proposed model, the results obtained

from leak off tests, and accidental geomechanical instabilities are compared with the fracture

gradient obtained from the proposed model, along with the conventional approach results in

Figure (2) for South Pars gas field (phases 6, 7, and 8) located in the Persian Gulf.

The new developed model follows the fracture gradient results obtained from leak off tests

and accidental geomechanical instabilities more closely than the conventional approaches.

The variance of the new proposed model from the real fracture gradient is 0.001422, while

for the conventional approach, this value is 0.032539.

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As there is no suitable data to assess the minimum mud window limit of the proposed

model in South Pars gas field, one of the available field data in literature (Al.Ajmi and

Zemmerman, 2006) which is for Wanaea field located in Northwest Shelf of Australia is

incorporated to check the minimum mud window limit of the proposed model.

As shown in Figure (3), the results obtained by the new developed model are 80% closer to

the real breaking out gradient, in comparison to the conventional approach.

Figure.2 Comparison between the maximum limits of the mud weight windows on South Pars

gas field (phases 6, 7, and 8) in Persian Gulf

1420

1620

1820

2020

2220

2420

1,6 1,7 1,8 1,9Depth (m)

Well pressure (Gradient)

Conventional approach

New developed 

method

Real fracture gradient

1950

1970

1990

2010

2030

2050

2070

2090

2110

2130

5000 6000 7000 8000 9000 10000

Depth (m)

Minimu allowable mud density (KPa)

Real gradient

New developed model

Conventional model

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Figure.3 Comparison between the minimum limits of the mud weight windows on Wanaea

field, Northwest Shelf of Australia

Regarding the new developed model verification and its superior accuracy in comparison to

the conventional approaches, the main advantage of this model is discussed in the following.

One of the main challenges in the drilling operation is the safe mud weight determination in

the deviated wells. The anticipated mud weight windows for the deviated wells are severely

narrow and sometime no safe mud window is obtained based on the conventional approaches

for the highly deviated wells. The new proposed model predicts a wide mud weight window

and can be worked out for higher deviations than the conventional approaches as shown in

Figure (4) for South Pars gas field (phases 6, 7, and 8) in Persian Gulf in depth equal to 2500

(m).

Figure.4 Comparison between the mud weight windows in the highly deviated wells for

South Pars gas field (phases 6, 7, and 8) in Persian Gulf

As shown in Figure (4), the new proposed model results in the safe mud weight window for

the deviated wells up to the 58°, but the conventional approach is restricted for the deviated

wells up to 38°. This would prevent challenges faced by drilling operators for wells deviated

more than the maximum mud weight allowed by the conventional approach.

0,8

1

1,2

1,4

1,6

1,8

2

0 20 40 60 80

Mud density gradient

Well deviation (degree)

Minimum limit of the new model

Maximum limit of the new model

Minimum limit of the conventional 

approach

Maximum limit of the conventional 

approach

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4- Conclusion

In this paper a new model, based on the Mogi-Coulomb failure criterion and the elasto

plastic constitutive model for the mud weight window is introduced and its accuracy is

verified against field data. The new developed model results in an enlarged mud weight

window in comparison to the conventional approach which enables well drilling operators to

drill safely on more deviated wells. The accuracy and the precision of the proposed model are

higher than the conventional approach.

Acknowledgement

The financial support of the National Iranian Drilling Company for this research is highly

appreciated.

Nomenclature

σ Stress (KPa)

r Radius (Inch)

Maximum horizontal stress (Kpa)

Minimum horizontal stress (Kpa)

Yield point (Kpa)

Vertical stress (Kpa)

˚ Pore pressure (Kpa)

Poisson's ratio

Maximum principle stress (Kpa)

Intermediate principle stress (Kpa)

Minimum principle stresses (Kpa)

C Cohesion (Kpa)

Internal friction angle (Degree)

Well pressure (Kpa)

′ Effective tangential stress (Kpa)

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Tensile strength (Kpa)

,′ Effective Minimum horizontal stress (Kpa)

,′ Effective maximum horizontal stress (Kpa)

Plastic thickness (Inch)

Well radius (Inch)

Radial stress (Kpa)

Tangential stress (Kpa)

Axial stress (Kpa)

Plastic elastic interface radius (Inch)

First stress invariant (Kpa)

Second stress invariant (Kpa)

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