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BOREHOLE STABILITY ANALYSIS

Research Report

TD93-18


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BOREHOLE STABILITY ANALYSIS

Research Report

Written by:

Wang Xiaojun Li Youngchi

Yu Jilin Guo Yong

Department of Modern Mechanics

University of Science and Technology of China

Hefei, Anhui, P.R. of China

July 30, 1993


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CONTENT

NOMENCLATURE

1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . .

2. THE PRINCIPAL PROBLEMS IN BOREHOLE STABILITY . . . . . .

2.1 Constitutive Models . . . . . . . . . . . . . . . .

2.2 Failure Criteria . . . - . . . . . . . . . . . . . .

2.3 Other Problems in Borehole Failure . . . . . . . . .

3. ANALYTICAL SOLUTIONS IN THE STUDY OF BOREHOLE STABILITY

3.1 Linear Elastic Models . . . . . . . . . . . . . . .

3.2 Elastic Model with Stress Dependent Modulus . . . .

3.3 Elastic-plastic Models and Rigid-plastic Model . . .

4. FINITE ELEMENT METHOD APPLIED TO ELASTIC-PLASTIC MODEL

IN THE BOREHOLE STABILITY . . . . . . . . - - . . . . .

4.1 Requirements of Numerical Simulations . . . . . . .

4.2 Brief Description on Published Finite Element Method

of Elastoplastic Models . . . . . . . . . . . . . .

4.3 Numerical Method in BSTAB 1.0 . . . . . . . . . . .

5. MODEL DESCRIPTIONS AND CASE STUDIES . . . . . . . . . .

5.1 Coordinate System and Transposition of

In-situ Stress State . . . . . . . . . . . . . . . .

5.2 Failure Criteria . . . . . . . . . . . . . . . . . .

5.2.1 Failure Criteria for Elastic Models . . . . .

5.2.1.1 Tensile failure criterion . . . . . .

5.2.1.2 Compressive failure criteria . . . . .

5.2.2 Failure Criteria for Elastoplastic Model . . .

(I)

.

(1)

. (4)

. (4)

. (7)

(11)

(12)

(12)

(15)

(16)

(20)

(20

(21)

(27)

(33)

(33)

(34)

(34)

(35)

(35)

(38)


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5.3 Introduction

5.3.1 Linear

of the Models. . . . . . . . . . . . .

Elastic Models . . . . . . . . . . . .

5.3.1.1 Linear elastic model

with pore fluid flow . . . . .

5.3.1.2 Linear elastic model

with impermeable mudcake . . .

5.3.2 Stress-Dependent Modulus Elastic Model

5.3.3 Elastoplastic Model . . . . . . . . .

5.4 Introduction of the Finite Element

Numerical Simulation . . . . . . . . . . . .

5.4.1 Description of the Finite Element

Method Simulation . . . . . . . . . .

5.4.2 Boundary Condition . . . . . . . . . .

5.5 Case Studies. . . . . . . . . . . . . . . .

6. REFERENCE . .. . . . . . . . . . . . . . . . . . . . . . .

APPENDIX A. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

APPENDIX B. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . ..

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

(38)

(39)

(39)

(41)

(42)

(43)

(46)

(46)

(46)

(47)

(53)

(58)

(59)

(II)


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NOMENCLATURE

(Compressive stress assumed positive throughout)

Vertical principal in-situ total stress

Maximum and minimum horizontal principal in-situ

total stress

Initial pore pressure/far field formation pressure

In-situ total stresses to the borehole Cartesian

coordinate system

Total stresses to the borehole Cartesian coordinate

system

Total stresses to the borehole Cylindrical

system

Pore pressure

Well pressure

Maximum, intermediate and minimum total stresses

Mean pressure

Effective mean pressure

Effective stresses

Effective maximum principal deviator stress

Strains to the borehole Cartesian coordinate system

Deviator strain

Deviator stress

Shear stress intensity

(III)


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Shear strain intensity

Youngs modulus

Young's modulus for zero confining pressure

Poisson's ratio

Cohesive strength and friction angle of rocks

Ultimate plastic strain

Octahedral normal and shear stresses

Mohr-Coulomb criterion parameters

Drucker-Prager criterion parameters

Wu-Hudson criterion parameters

Hook-Brown criterion parameters

Biot's poroelastic parameter

Kronecker's note

(IV)


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1. INTRODUCTION

The failure of vertical, horizontal or inclined boreholes is

a great and continuing problem which results in substantial and

remarkable yearly expenditure for the petroleum industry. For years

the petroleum engineers have had to give close and careful

attention to drilling fluid programs, casing programs and operating

methods in drilling wells to keep the borehole wall from failure

and to minimize the costs in drilling and production procedure.

The cause of borehole failure comes from the redistribution or

concentration of stresses around the borehole. Before drilling the

formations are under a certain state of compressive in-situ stress,

which usually, with the exception of structurally complex area, can

stresses,

horizontal principal stress). When a well is drilled, the stresses

in the formation material around the borehole will be redistributed

because the support originally offered by the removed material is

replaced by the hydraulic pressure of the mud added into the

borehole. If the formation material around the borehole is not

strong enough, or if it is too much weakened by the interaction

with the drilling mud, the borehole failure (borehole instability)

will be initiated caused and developed.

On the whole stress induced borehole failure can be categor-

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ized into the following three classes:

1) Hole size reduction caused from the too much ductile yield

flow of the wall rock into the borehole. This is one kind of

compressive failure which will result in stuck pipe, hole closure

or even full loss of the open

2) Hole size enlargement

hole section.

caused from the brittle failing and

falling of the wall rock into the borehole. This is another kind of

compressive failure which will result in poor directional control,

poor cementing and fill on trips.

3) Unintentional hydraulic fracturing or formation breakdown

of the wall rock caused from excessive bore mud pressure. This is

one kind of tensile failure which will result in severe loss of

drilling fluid into formation (lost circulation), and hence lost

time as

To

well as increased

understand and

costs .

solve the borehole failure problems,

petroleum engineers have to study in detail the following subjects:

1) To find out the stress redistribution in rocks around the

borehole under various conditions such as different in-situ

stresses, different mud pressures, different borehole orientations

and so on. For this purpose, the most important thing is to have a

reasonable constitutive model for the formation materials no matter

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they use analytical or numerical methods.

2) To have a proper failure criterion for the formation

materials.

As a summation of our work in the past year we have completed

this report which consists of 5 sections altogether, including the

introduction (section 1) . Section 2 outlines the principal problems

in borehole stability, particularly the constitutive models and

failure criteria for the formations widely used in references.

Section 3 presents a brief review about the analytical solutions in

the study of borehole stability published so far. Section 4

discusses the application of elastoplastic finite element method to

the borehole stability analysis. Finally, section 5 gives a

detailed description of the models used in BSTAB 1.0, and provides

several indicative case examples.

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2. THE PRINCIPAL PROBLEMS IN BOREHOLE STABILITY

2.1 Constitutive Models

The mostly often used constitutive model for the formations is

the Homogeneous, Isotropic and Linear-Elastic model (HILE) . This is

mainly because with HILE constitutive model the

used to determine the stress field around the

simple and hence analytical solutions can be

many cases (C.Hsiao[2], C.H.Yew et.al. [3], B.

governing equations

borehole are fairly

easily obtained in

S.Aadnoy [4]), and

with analytical solutions it is easy to reveal the effects of

various factors on the stress distribution. Another advantage of

the HILE model is of much less material parameters with it. The

disadvantage of using HILE model is that it usually underpredicts

the hole stability (i.e., it is too conservative), especially when

it is used together with the category B or D criteria (see 2.2).

Besides, for better description of the real formation properties

some researchers used the Anisotropic Linear-Elastic Model

(B. S. Aadnoy[5]) .

With the knowledge that the use of HILE model underpredicts

hole stability and that the onset of plastic yield flow does not

indicate the complete failure of a borehole researchers begin to

try the use of Elastoplastic Model (EP) . Most of the works

(R. T. Ewy[6], C. A. M. Veeken et. al. [7], E.Detournay[8]) use perfect-

plasticity with Mohr-Coulomb (MC) yield criterion or Drucker-Prager

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(DP) yield criterion:

(Me) (1.1)

(2.3)

the octahedral shear stress and octahedral normal stress respect-

ively. Associated or nonassociated flow rule can be used. The

advantage of the former is the symmetry of the plastic stiffness

matrix, but it usually produces too much dilation. The later can

avoid the production of dilation but increase the calculation

difficulty. Some works incorporated hardening and softening

behaviors in the EP model (C.A.M.Veeken et.al. [9], N.Morita

et.al. [10]). The main problem with EP model is the difficulty for

the resolution because the governing equations are relatively

complex and the elastic-plastic interface is usually unknown in

advance. The second problem is the lack of proper failure criteria.

A simpler failure criterion is the ultimate threshold of

plastic strain (see Eq. (2.17)). However, it is somewhat

equivalent

arbitrary.

Rigid-Plastic constitutive model (RP) has also been developed

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in the bifurcation analysis for borehole failure (J.Vardoulakis

et.a. .[ll], J.Sulem et.al. [12]). The RP model neglects the elastic

strain and assumes a form of deformation theory with dilatancy:

(2.5)

where eij and Sij are the deviator strain and deviator stress respect-

shear stress intensity and shear strain intensity respectively. The

in (2.5) can be determined by only uniaxial tests for any formation

materials. The second advantage is that by the combination of RP

model and the requirement that the internal and external stress

tensor across a shear band boundary

bifurcation criterion (instability)

solution for the governing equations

should be in equilibrium a

can be obtained, and the

gives accurate prediction of

hollow cylinder failure. So it seems to be a promising model.

However, further study should be done about whether model (2.5) can

represent the formation behaviour in general stress state and

whether it can give good prediction of borehole failure in various

conditions.

A new kind of constitutive model is the Stress Dependent

Moduli Elastic model (SDM) (F.J.Santarelli et.al. [13], M.E,D,Fama

[14]). SDM model takes the form of HILE constitutive equations,

however, the elastic moduli are not constant but functions of

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stress state. A commonly used model is to assume the Youngs

modulus,

given by

(2.6)

2.2 Failure Criteria

Tensile failure criterion has the form


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minimum principal stress. The existing compressive failure

criteria, as summarized by M.R.McLean et.al. [1], can be classified

into the following 4 categories:

Category A: linear and with intermediate principal stress

effect;

Category B: linear and with no intermediate principal stress

effect;

Category C: nonlinear and with intermediate principal stress

effect;

Category D: nonlinear and with no intermediate principal

stress effect.

Some of the most commonly used criteria are listed as follows

(M.R.McLean et.al[l], [15], J.R.Marsden et.al. [16]):

Drucker-Prager criterion (D-P criterion, category A) ,

r Ott

Mohr-Coulomb criterion (M-C criterion, category B) ,

(2.8)

(2.9)

Wu-Hudson criterion (W-H criterion, category C)

(2.10)


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( 2 . 1 1 )

After the triaxial failure

pressures are completed one

any equations of (2.8)-(2.11

experiments with different confining

can fit the experimental points with

) in corresponding planes, and obtain

the respective material parameters in them.

According to M.R.McLean et.al. [1] the following conclusions

are widely accepted:

(1) Failure criteria with no consideration of the influence of

the intermediate principal stress (Category B or D) are usually

conservative in the prediction of the borehole stability, particu-

larly when they are used in association with HILE models. Although

the true triaxial experiments show the intermediate principal

stress effects, the failure criteria incorporating the intermediate

principal stress (Category A or C) trend to overpredict the

formation strength and the borehole stability. A possible better

and (2.10) as

where the value of n is around 1. However, we would like to give a

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conjectural idea about this problem: The reason for the predictive

deviation of Eqs. (2.8) and (2.10) might be because they were

If failure criteria

triaxial experiments

with 02 effects were obtained by the true

the results might be fairly better.

(2) In most cases the linear failure criteria are adequate for

application. However, for very weak formations or for confining

pressures greater than 14MPa, the nonlinear failure criteria are

necessary.

The above failure criteria are all for the use in association

with elastic constitutive models. For plastic constitutive models

it seems to lack satisfactory failure criteria so far. A relatively

simple criterion is the ultimate plastic strain cri-

terion (B.G.D.Smart et.al. [17]):

(2.15)

As pointed out above, however, the threshold parameter

difficult to determine and somewhat arbitrary. To seek proper

failure criteria for the plastic constitutive models is an

important research work which should be done.

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2.3 Other Problems in Borehole Failure

The following problems are also important in the borehole

stability:

(1) Wellbore stresses produced by moisture adsorption. C.H.Yew

et.al. [18] gave a comprehensive study for the problem by assuming

the following deformation laws from moisture adsorption:

(2.16)

&v- (2.17)

are the moisture induced strains along the horizon-

anisotropic ratio, K1 and K2 the expansion coefficients, W the water

content in the formations. They show that the moisture-adsorption

process is governed by a diffusion equation and that the governing

equations for the moisture-induced stresses around the hole are

similar to those used in thermoelasticity.

(2) Wellbore stresses produced by temperature difference

between the hole and formations. For this problem K.Hojka et.al.

[19] and A.L.Siu et.al. [20] gave preliminary analysis, which we

would like to leave out here.

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3. ANALYTICAL

STABILITY

SOLUTIONS IN THE STUDY OF BOREHOLE

A number of analytical solutions

literature. Most of them can be put into

linear elastic models, elastic models with

and elastic-plastic models.

are available in the

three categories, namely

stress dependent modulus

3.1 Linear Elastic Models

The simplest model for borehole stability study is the linear

elastic and isotropic model. Normally a plane stress condition is

assumed and proper failure criterion is adopted. The equations for

stress around a vertical or inclined borehole can be found in

W.B.Bradley[21] where the effect of borehole angle and borehole

direction on the stability of wellbores was discussed. An extended

von Mises yield criterion is applied for predicting wellbore

collapse while the maximum effective stress criterion is applied

for predicting bore fracture. B.S.Aadnoy and M.E.Chenevert [4]

studied the influence of different in-situ stress states (normal or

tectonic) and different criteria (a modified von Mises criterion or

the Mohr-Coulomb criterion) on the stability of borehole with

various inclination.

A three-dimensional linear elastic analysis of a deviated well

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was presented by C. H. Yew and Y. LI [3]. Only tensile fracture is

discussed using the strain-energy density criterion. It was found

that the off-plane shear-stress components have significant effects

on the wellbore breakdown pressure.

An anisotropic stress model was used by B.S.Aadnoy[5] to take

the directional properties of real rocks into account. In this

model, rock property was described by a transversely isotropic

linear constitutive relation. Directional shear and directional

tensile strengths were also considered. The results showed that the

anisotropy in the tensile strength and shear strength is more

important than the anisotropy in elastic properties. Assuming

isotropic rock properties instead of the real anisotropic elastic

constants introduced only a small error in the failure-pressure

prediction except for highly anisotropic rocks.

Linear elastic theory predicts that failure always occurs at

the borehole wall. So one only need to check the stress distribu-

tion at the wall if only permissible borehole pressure is of

interest. This makes numerical calculation very easy and fast.

Although linear elastic models for borehole stability analysis have

been well established, however, the results are sensitive to the

failure criterion used, as discussed by many authors (M.R.McLean

and M.A.Addis[15]; R.T.Ewy[6]). Hence the reliability of these

models is limited.

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Most of rock media in petroleum operations contain pore fluid.

The presence of pore fluid in porous or fissured rock masses can

substantially alter the stress distribution around a borehole. This

phenomena was first investigated by Paslay and J.B.Cheatham [22]

thirty years ago. A fracture analysis for a borehole in a non-

hydrostatic stress field with one of. the principal stresses

parallel to the wellbore axis was given by B.Haimson and C.Fairhu-

rst[23]. It was assumed that the fluid flow through the porous

elastic rock obeys Darcys law. In their analysis only an

additional stress field arising from the radial fluid flow was

considered and deformation and diffusion processes were uncoupled.

Due to the linearity of the problem, the model can be easily

extended to more complicated situations. A theoretical model of

horizontal-wellbore failure has been developed by C.Hsiao[2] based

on maximum-normal-stress theory (for tensile fracture) and Drucker-

Prager failure theory (for compressive failure) . Both normal in-

situ stress condition and tectonic in-situ stress condition were

considered. It was demonstrated that the flow induced stresses is

a significant part to the total stress distribution around the hole

and the permissible borehole operating-pressure range is signifi-

cantly affected by the in-situ stresses, borehole orientation and

rock properties.

A few analytical models of wellbore stability in permeable

rock formation which account for coupled hydraulic-mechanical

processes have been developed. E.Detournay and A.H-D.Cheng[24]

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presented a model for vertical wellbores under non-hydrostatic in-

situ stresses. The Biot theory of poroelasticity was employed. The

Laplace transform technique and an approximate numerical inversion

technique were used to deal with the transient progress assuming a

borehole was instantaneously drilled. An asymptotic solution for

small time was also given. Their analysis suggested that shear

failure could be initiated at a small distance inside the rock,

rather than at the borehole wall. More recently, C.Y.Yew and Gefei

Liu[25] presented a method for estimating the permissible borehole

pressure of a

employed and a

criterion for

deviated wellbore. The same poroelastic theory was

steady state solution was given. The Drucker-Prager

shear collapse and the maximum effective tensile

stress criterion for fracture were used for failure analysis. The

results showed that the plastic failure of borehole is very

sensitive to the pore pressure.

3.2 Elastic Model with Stress Dependent Modulus

Porous or elastic rocks often have elastic moduli which are

not constant but increase with increasing minor principal stress.

To illustrate this behaviour, F.J.Santarelli et al.[26] developed

an elastic model with stress dependent modulus in which the Youngs

modulus varies with the minor principal stress. The results showed

that the tangential stress near the borehole wall are much lower

than those predicted by linear elastic theory and the maximum

tangential stress occurs some distance inside the surrounding rock.

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This is in agreement with experimental observations. Unfortunate-

ly, due to the nonlinearity introduced in the model, analytical

solutions are only available for axisymmetric cases with a few

the primitive of its invert must be

easily expressible. A complete closed form solution for a power

et al. [27]. Its use is restricted to deep wellbore problems. Two

other forms, and were treated by

E.T.Brown et al. [28]. In these cases numerical integration is

3.3 Elastic-plastic Models and Rigid-plastic Model

Normally numerical simulation is required if a plastic

constitutive relation is used. In the simplest cases, e.g. when a

wellbore is in a hydrostatic stress field and the rock is assumed

to be elastic-perfectly plastic,however, closed form solutions for

the stress and strain distribution exist. One of the early works

contributing to this problem is the study by H.M.Westergaard[29].

During the last decade, more analytical results were published

which incorporated either

post-yield behaviors.

A three-dimensional

the permeability of the rock or different

axisymmetric elastic-perfectly plastic

analysis with the Mohr-Coulomb criterion was presented

et al. [30]. The associated flow rule of the theory of

--16--

by R.Risnes

plasticity,


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commonly used for metals, was adopted. However, experiments showed

that the dilatancy of rocks is far less than that predicted by the

associated flow rule with the Mohr-Coulomb criterion and rocks are

rarely able to maintain the strength after shear failure occurs.

To account for this behaviour, E.M.Airey[31] presented a solution

for a roadway in a hydrostatic in-situ stress field, assuming the

rock is elastic-plastic with strain softening. E.Hock and

E.T.Brown[32] suggest an empirical peak strength criterion for rock

masses that the initial strength of the rock mass is described by

(3.1)

while the residual strength of the broken rock mass is described by

m, s, mr, s, are material constants. Using this

(3.2)

of the intact rock,

criterion, E.T.Brown

et al.[33] presented a closed form solution for an elastic-brittle-

plastic material behaviour model and a step-wise calculation

procedure for an elastic-strain softening plastic model. A

solution of the stress and strain distribution near a deep tunnel

in a hydrostatic in-situ stress field for a rock obeying the Mohr-

Coulomb yield criterion with variable dilatancy was obtained by

E.Detournay[3 4].

Besides these results, it is worth to pay special attention to

the following two models published recently. One is by E.Detournay

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and C.Fairhurst[8] and the other is by I.Vardoulakis et al. [11].

Detournay and Fairhurst[8] presented a two-dimensional

elastoplastic analysis of a long, cylindrical cavity under non-

hydrostatic in-situ stress. It was assumed that the axis of the

cavity is parallel to one of the far-field principal stresses and

the rock is assumed to be linear elastic, perfectly plastic. Non-

associated flow rule was used and the yield function and the

potential were described by linear Mohr-Coulomb functions. A

solution for the later stage, when the cavity is completely

surrounded by a yield zone, was obtained by self-similar analysis.

Explicit formulation of the stress field in elastic and plastic

regions and of the displacement field in the elastic field is

provided while the displacement field in the plastic region can be

obtained numerically by the method of characteristics. The model

is, however, subject to several restrictions. Only a limited range

of deviation from hydrostatic loading can be considered. Since no

explicit solution for the early stage exists, an apriori assump-

tion was introduced that at the end of the early stage the elastic-

plastic interface coincides with the prediction of this solution.

Nevertheless, this is the only model which can deal with non-

hydrostatic in-situ stress. It seems possible to apply these

results approximately to the stability analysis of inclined

wellbores if neglect the influence of the off-plane shear stress,

as assumed in most two-dimensional elastic models.

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Vardoulakis et al[ll] proposed a bifurcation analysis for

borehole stability. A wellbore in an axisymmetric far-field stress

under plane strain state was analyzed. The rock is assumed to be

a rigid-plastic pressure sensitive material with dilatancy. The

failure of the rock is considered as shear band formation and

predicted by the situation when a non-trivial and an axisymmetric

solution exists. The application of the model to real cases showed

that this bifurcation analysis is in good agreement with experimen-

tal and field observations. The model, although somewhat complex,

has the advantage of only requiring uniaxial test data for defining

the constitutive behaviour. However, it is not sure whether this

model can be extended to the case of non-hydrostatic in-situ stress

field.

In comparison with methods of numerical simulation such as

finite element method and finite difference method, analytical

models can give a clear picture of the parameter. dependence of the

problem and require less calculation time and memory storage.

Besides the above three types of models, an attempt was made

by J.Sulem and 1.Vardoulakis[12] who tried to analysis the scale

effect found in many laboratory tests on failure of rock masses.

A Cosserat continuum model was used and coupled by a bifurcation

analysis. However, the scale effect obtained was rather small.

This is not unexpected since the grain size was used as the

intrinsic length of the model which is too small in comparison with

the characteristic length of the problem.

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4.

4.1

FINITE ELEMENT METHOD APPLIED TO ELASTOPLASTIC

MODEL IN THE BOREHOLE STABILITY

Requirements of Numerical Simulations

The analytical solutions described in section 3 are reasonable

and useful in the study of wellbore stability but the assumptions

imposed on the solutions often do not hold both in field applica-

tions and hollow cylinder tests. For example, the mechanical

behaviour of rocks tested in uniaxial or triaxial compression show

that the material response to loading is far from simple

neither linear elastic nor elastic-perfectly plastic. The

stress field we met in engineering usually is anisotropic

a horizontal plane and hence the axial symmetric condition

one but

in-situ

even in

can not

be applied even for a vertical wellbore. In fact, many factors may

influence the wellbore instability such as wellbore axis relative

to the in-situ stress field, material strength and behaviour,

viscous and thermal effects, wellbore pressure and mud cake etc. In

principle, an ideal numerical model could take account, both

qualitatively and quantitatively, of all the imposed boundary

conditions and the relevant material behaviour. Nevertheless, some

numerical analyses would be extremely complex and consequently some

assumptions are still imposed on the numerical simulations such as

stress-strain relationships, other material properties, failure

criterion, and so on. The accuracy of the numerical model is

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limited usually by the model choice and the reliability of the

input parameters, for instance, the in-situ stresses and the

material parameters which may be extremely uncertain in some cases.

It is no doubt that the numerical simulations will give much better

and reasonable results than those of analytical solutions, provided

the input parameters can be determined in an appropriate way.

Therefore a number of petroleum engineers are devoted to developing

the finite element or finite difference method. It should be

pointed out that both the hollow cylinder tests in the laboratory

and the field drilling observations are very important to assess

the correctness and the accuracy of the numerical results. For this

reason, in the study of wellbore stability three aspects, namely

numerical simulations, experimental research and the field

applications, should be noticed simultaneously.

4.2 Brief Description on Published Finite Element Method of

Elastoplastic Models

The finite element method has now been recognized as a general

method of wide applicability to engineering and physical science

problems. As a result, the method has gained wide acceptance by

civil engineers for designing and studying the wellbore stability

in the petroleum industry.

The main element required in a wellbore stability model is the

constitutive model. Even for permeable, isotropic, homogeneous rock

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formations the materials

response to loading are far from.

simple linear elastic. Typical

curves for sandstones tested at

confining pressures are shown in

Figure 4.1. It can be seen that

before the peak strength is

reached the material deforms with

strain hardening and upon reach-

ing the peak strength the sample

loses strength and undergoes a

portion of strain softening. In

Figure 4.1 Triaxial Test Data

for a Carboniferous Sandstone

order to evaluate the potential for wellbore stability a realistic

constitutive model must be used to compute the stress distributions

around the wellbore.

A lot of nonlinear constitutive equations were developed in

the past to

K.E.Gray[lO]

are composed

fit the stress-strain

proposed a equation in

curves of rocks. N.Morita and

which they assumed the strains

of four parts, that is an initial nonlinear part, an

elastic part, a plastic part, and the volume change of the rock

matrix due to fluid pressure. Each part was carefully examined and

expressed by mathematically consistent equation and then applied to

a finite element simulator. Two dimensional parabolic isoparametric

element with eight nodes was used to study the vertical fracture

initiation during drilling. The breakdown

--22--

pressures obtained in the


30/68

simulations provide an upper bound of actual pressure. It is

meaningful to estimate the tensile failure of brittle rocks but

since the compressive failure occurring in the form of spalling or

hole closure was not considered in their work the applications to

the borehole instability analysis are obviously limited.

C.A.M.Veeken et al[9] presented an elastoplastic finite

element approach incorporating strain hardening and softening for

the prediction of stability of vertical and horizontal borehole. A

non-associated flow rule with zero dilatancy is used in their

numerical simulations. The hollow cylinder collapse experiments on

sandstone samples were performed and both the axisymmetric and

plane strain finite element simulations were carried out. The

purpose of their work is limited to verify the tests of hollow

cylinder collapse and consequently only boundary loading is

considered. The distinctive feature of the paper is in the

successful numerical simulation of localised failure patterns,

which is in good agreement with experimental observations.

The plane strain analysis mentioned above is confined to in-

neglected. Such neglect is appropriate to hollow cylinder tests but

is improper to the real wellbore. Therefore, the 3D effects (i.e.

effects) on the plane strain wellbore failure model must be

estimated in advance and it was completed by R.T.Ewy[6] with a

finite element program. The rocks was modeled as elasto-perfectly

--23--


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plastic, with zero volume change during plastic flow. It can be

shown theoretically, and was verified with numerical models, that

the size of plastic zone is independent of the amount of dilation

occurring in the yield material. The amount of dilation does affect

the size of hole closure following yielding. Excavation unloading

simulating the hole drilling is achieved by reducing the support

pressure PW

in the hole from its pre-existing values (in-situ

stresses) . Both the hydrostatic and non-hydrostatic cases are

studied and it is concluded that if a 2D yield criterion adequately

describes the rocks, then for special, limited cases a 2D stress

analysis may be sufficient, but for most cases other than hydro-

static the 3D stress analysis is required. Although the constitut-

ive models they used are fairly simple, their method is still valid

as one incorporates more realistic and complex material behaviour

models into stability predictions. These findings also provide

guidance for the important development of fully 3D numerical models

to analyze the stability of deviated and extended-reach wells.

Recently the sand production near the wellbore unsolidated

sand are studied by an elastoplastic finite element formulation.

The model presented by A.F.Polillo[35] is completely based on the

analysis of elastic/plastic deformations. The area around a

wellbore under plastic deformations is assumed to be a relative

movement area of sand grains and therefore instability occurs. The

stability is considered to be elastic behaviour and the plastic

deformation is assumed to indicate failure. The model can provide

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information on whether a wellbore will exhibit sand production

problem.

As we make a summary of the elasto-plastic models in the

finite element study of wellbore stability we have to mention the

work of M.R.McLean[36]. The conception that the plane strain should

be subdivided into simple plane strain and complete plane strain is

proposed by McLean in his thesis. For simple plane strain the

displacement parallel to well axis (z) is restricted to constant

and hence only in-plane strains are nonzero. Such

restriction is suitable only for the well where the axial direction

of excavation is parallel to a principal in-situ stress direction,

namely for vertical or horizontal wellbores. However, the condition

of constant axial displacement is an unreasonable restriction on

plane strain, or in other words,

Actually, the plane strain

requirements are met when displacements are independent of the

axial coordinate, without any further restrictions. Thus, a more

popular definition named as complete plane strain is introduced in

which the axial displacement is independent of

z but a function of x and y. Using the general

ment relationships we will know that all the

the axial coordinate

3D strain-displace-

strains are nonzero

be used to study the directional or deviated wellbore instability.

McLean developed the plane strain finite element method suggested

by D.R.J.Owen and E.Hinton[37) with the aim of simulating the

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stress distributions around an inclined borehole. Both the

associated and non-associated flow rule are considered and

different boundary conditions (boundary loading and excavation

unloading) are included in his work. His research can be considered

as a contribution to the further work of borehole stability

analysis.

Most of the papers up to now on the elasto-plastic model of

finite element analysis are limited to the study of

tests and only a few of them are applied to the

Theoretically, the elasto-plastic model should be

hollow cylinder

field borehole.

much reasonable

and reliable than the linear elastic model and usually it will

predict a more reasonable and economic mud pressure that will be a

great benefit for drilling and completing wells. The reasons why

the elasto-plastic model was not often applied to field analysis

can be summarized as: (1) The input material parameters defined in

the elasto-plastic relations are hard to be obtained, especially

for deep formations. Although we can get the material constants

from the rock sample tests the behaviour of formation rock are

often quite different from that of small rock samples. (2) Another

problem in the application of elasto-plastic model is due to the

fact that the failed

stability of wellbore

plastic deformation,

researchers suggested

material still has strength and hence the

is always depending on the chosen allowable

which is often arbitrary in nature. Many

to

the wellbore instability

use an ultimate strain as a threshold in

analysis. If the ultimate strain is not

--26--


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exceeded the hole is stable. It sounds a good idea to determine the

collapse of an opening hole but the value of this ultimate strain

is hard to be obtained. Field application of borehole stability

prediction, therefore, is influenced by input data uncertainty.

Since the absence of laboratory and field validation, the current

numerical prediction are mainly of qualitative use. In order to

improve the quality of a sophisticated theoretical model, field

observations, laboratory experiments and numerical modelling should

be combined as far as possible and then a realistic and consistent

approach to wellbore instability prediction may be obtained.

4.3 Numerical Method in BSTAB 1.0

The finite element method used in BSTAB 1.0 are based on the

work presented by Owen and Hinton. Generally speaking, the yield

function F of elasto-plastic material can be written in a form,

(4.1)

denotes the stress tensor,

connected with the plastic work WP.

The stress-strain relations during plastic loading usually are

expressed in an incremental form,

--27--


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(4.2)

the elasto-plastic matrix De p

in the above equation can be

derived from plasticity theory,

where I is the identity matrix, D is the linear elastic matrix, a

is a vector determined by derivative of the yield function F with

which can be written in polar coordinates

and

(4.4)

(4.5)

the yield criterion.

It is convenient to rewrite the yield function in terms of

stress invariants for numerical computations. The main advantage of

this formulation is that it permits the computing code of the yield

function and the flow rule in a general form. For example, Mohr-

Coulomb yield criterion can be written in the form,


36/68

(4.6)

the rocks respectively, J, is the first stress invariant, J2 the

second deviatoric stress invariant,

stress invariant J3 and J2 as,

(4.7)

similarly, the Drucker-Prager yield criterion can be expressed as,

(4.8)

where a,

Therefore; the flow vector a can be conveniently expressed in a

form that is suitable for numerical computations,

(4.9)

Applying the virtual work principle to the equilibrium equations,

w e have,

where N are shape functions, the gradient matrix B are defined

by derivatives of

vector of boundary traction,

computational domain, T is the part of the boundary on which the

--29--


37/68

boundary traction are prescribed. It is well know that in the

wellbore the initial body force do not participate as equivalent

forces in the equilibrium equations and thus equation (4.10)

becomes,

(4.13)

The displacement increment vector

be given by the initial stiffness method,

(4.14)

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Several restrictions are imposed on the constitutive relations

in BSTAB 1.0. (1) The material behaviour is assumed to be elasto-

perfectly plastic and thus both the strain hardening and strain

softening are omitted. (2) Only associated flow rule is considered

and so that the dilation of plastic strain may be bigger than that

observed in experiments. The main advantage of this assumption is

that it will ensure a

by the frontal solver

The condition of

isoparametric element

symmetric stiffness matrix which is required

listed in Owen and Hintons book.

plane strain is imposed on the quadrilateral

with eight nodes. The computational domain is

subdivided into 400 elements with the external boundary set at 11

times radius from the center of the hole. Figure 4.2 shows the

inner part of computational meshes. Displacements of the nodes at

the external boundary are cons-

trained and 2*2 Gauss integrating

rule is used.

In order to simulate the

inclined wellbore stability with

the plane strain element the

effects of the antiplane shear

must be ignored.

It is our experience that the

the common in-situ stresses areFigure 4.2 Inner Part ofElement Mesh

--31--


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Therefore, a small error may be accompanied by BSTAB 1.0 and a more

accurate computational code can be expected in the new version.

At the beginning to run the program, the in-situ stresses

should be placed on each Gauss point of the elements. Boundary

loading which can simulate the hollow cylinder tests is not

considered. Excavation unloading is simulated by applying a

sequence of negative pressure increments at the inner boundary. At

first, large pressure increments are chosen to simulate the elastic

unloading. Then the magnitude of the pressure increment is selected

carefully so that each increment would induce a small increase of

the size of the plastic zone.

Both the Mohr-Coulomb criterion and the Drucker-Prager

criterion are inserted in BSTAB 1.0. The stress distributions and

the plastic zone around the borehole will be simulated. It is

necessary to assign an ultimate strain as a threshold to study the

borehole stability. If the ultimate strain is not exceeded the hole

is stable.

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5. MODEL DESCRIPTIONS AND CASE STUDIES

Generally, the two main elements in a borehole stability

models are the constitutive behavior model to determine the

redistribution of the stress state around the borehole and the

failure criterion to decide whether the borehole is stable. The

models used in our software are two linear-elastic analysis models

and two finite element method analysis models using stress-

dependent modulus elastic model and elastoplastic model.

5.1 Coordinate System and Transposition of In-situ Stress State

The coordinate system in our analysis is set up as shown in

Figure 5.1. The z-axis is parallel to the borehole axis and the x-

axis lies in a horizontal plane.

the azimuthal angle.

Before determining the stress distributions around an inclined

borehole, it is necessary to transpose the in-situ stress tensor

relative to our borehole coordinate system. The transposed stress

state is given as

--33--


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(5.ld)

(5.le)

(5.lf)

Figure 5.1 Coordinate System and Transposition of In-situStresses

5.2 Failure Criteria

5.2.1 Failure Criteria for Elastic Models

For both the linear elastic models and the stress-dependent

modulus elastic model, the peak-strength criteria are used to

predict the rock failure. At the point where the stress state

exceed the formation strength (either in tension or compression)

failure is considered to have initiated.

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5.2.1.1 Tensile failure criterion

The criterion for tensile failure initiation is simply

determined by whether the minimum effective stress is less than the

tensile strength of the formation. Thus, failure occurs when

(5.2)

5.2.1.2 Compressive failure criteria

In the compressive failure analysis, two most commonly used

criteria, namely the Mohr-Coulomb and the Drucker-Prager, are

incorporated in our program.

(1) Mohr-Coulomb criterion

The Mohr-Coulomb criterion can be expressed in terms of

principal effective stresses as

(5.3)

where kp, Co. are Mohr-Coulomb strength parameters, which can be

derived from two material constants, the cohesive strength c and

--35--


43/68

(5.4a)

(5.4b)

(2) Drucker-Prager criterion

The Drucker-Prager criterion is expressed in terms of

octahedral stresses as

three choices to determine the Drucker-PragerThere are

parameters. These choices have come about through comparing the

Drucker-Prager criterion with the Mohr-Coulomb criterion. The

projection of the Mohr-Coulomb criterion and one of the Drucker-

--36--


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Mohr-Coulomb

Outer circle

criterion are given by:

(5.7)

--37--


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Middle circle

Inner circle

(5.8)

(5.9)

5.2.2 Failure Criteria for Elastoplastic Model

For the elastoplastic model, the failure of a borehole is con-

sidered to occur when the maximum equivalent plastic strain of the.

whole region exceeds the ultimate plastic strain, that is

(5.10)

the

ultimate plastic strain.

5.3 Introduction of the Models

To determine the distribution of the stresses around a bore-

hole, certain governing equations must be incorporated. These

equations include equilibrium equations, compatibility equations

and constitutive relations. For linear elastic models, exact

analytic solutions can be obtained based on these equations. But

for stress dependent modulus elastic model and elastoplastic model,

--38--


46/68

it is difficult to derive analytic solutions, especially under

tectonic in-situ stresses or in an inclined borehole, thus, the

numerical simulations with the finite element method are used to

determine the stress distributions.

5.3.1 Linear Elastic Models

For the brittle, homogeneous, isotropic and porous rocks, the

analytic linear elastic models can be used to predict

stability. There are many versions of linear elastic

divers considerations for different situations and

the borehole

models, with

some special

treatments. In our program, we use two linear elastic models

described respectively in C. Hsiao[2] and M.R. McLean & M.A.Addis

[15].

5.3.1.1 Linear elastic model with pore fluid flow

Assuming that

obeys Darcys law,

situ stresses, the

The stresses

the fluid flow through the pore is steady and

the stress redistribution is induced by the in-

well pressure and the pore fluid flow.

around a borehole, induced by the in-situ

stresses, as a function of radial distance away from the wellbore

can be found in W.B.Bradley [21] and can also be seen in Appendix

A at the end of this paper. The total stress components at the

borehole wall are given as (C.Hsiao[2])

--39--


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(5.lla)

(5.llb)

(5.llC)

(5.lld)

= 0 (5.lle)

(5.llf)

The stresses at the borehole wall due to the steady pore fluid

flow and the well pressure are given as following (the detailed

description can be seen in Appendix B):

(5.12a)

= well pressure;

= initial / far field pore pressure;

= Poissons ratio;

(5.12b)

(5.12c)

bulk modulus of the solid skeleton

bulk modulus of the interport material

--40--


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is Biots poroelastic parameter.

Therefore, the total stress components at the borehole wall

are the sum of Eq. (5.11) and Eq. (5.12).

5.3.1.2 Linear elastic model with impermeable

For a borehole with a perfect mudcake,

mudcake

the model given by

M.R.McLean and M.A.Addis[15] describes the total stress components

and pore pressure at the wall as:

1 -v

pore fluid pressure at the borehole wall,

(5.13a)

(5.13b)

(5.13C)

(5.13d)

(5.13e)

(5.13f)

(5.14)

--41--


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5.3.2 Stress-Dependent Modulus Elastic Model

For many porous or elastic rocks, the elastic moduli deter-

mined from triaxial testing are not constant but increase with

increasing minor principal stress (F.J.Santarelli[26]), thus, it is

better to use the stress-dependent modulus elastic model to predict

the borehole stability. E.T.Brown etc[28].

The stress-dependent modulus elastic model is an empirical

constitutive behavior model in which the variation of the secant

elastic modulus with the minor principal stress is obtained by

triaxial testing (see Figure 4.1). The relations between the moduli

and the minor effective principal stresses can be written as (M.A.

McLean and M.A. Addis[l])

and

(5.15)

E=EO (5-.16)

There are many expressions for the function The two

most commonly used relationships are power laws

(5.17)

and exponential laws

--42--


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where EO ,

and Dl, D2 and m are material constants derived from the triaxial

testing data.

The Poissons ratio v , though slightly varied with the minor

principal stress, can be considered a constant because the

variation of the Poissons ratio has only a little effect on the

stress distribution around a borehole.

Thereafter, the relationship of the strain components to the

stress components takes Hooks law:

where

5.3.3 Elastoplastic Model

For the elastoplastic

a linear elastic, perfect

i,j=l,2,3 (5.19)

(5.20)

model, the rock is assumed to behave as

plastic isotropic material, which is

characterized by a cohesive-frictional yield strength and dilatant

behaviour during yielding.

--43--


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In the elastic state,

ensures that the constitutive equation can be expressed exclusively

in terms of the plane components of the stress and strain tensors.

The Cartesian components of the incremental elastic strain are.

given by Hook's law:

(5.21a)

(5.21b)

(5.21C)

can be expressed as a

function of the normal effective stress increments in the x-y plane

by

where E, G and v are

modulus and Poissons

(5.22)

respectively elastic Young's modulus, shear

ratio of the rocks.

The yield function F is described by linear Mohr-Coulomb

function, which is independent of the intermediate principal

(5.23)

or is described by Drucker-Prager function in terms of the

octahedral stresses

--44--


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(5.24)

In the plastic state, the total strain can be decomposed into

two parts, the elastic strain and the plastic strain.

(5.25)

An associated flow rule is used in conjunction with Eq (5.23) or

(5.24), so that the incremental plastic strains can be obtained by

the following equations,

(5.26)

The constitutive relations can be written as follows,

(5.27a)

(5.27b)

(5.27c)

(5.27d)

--45--


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(5.27e)

(5.27f)

5.4 Introduction of the Finite Element Numerical Simulation

The computational method used in our numerical simulations is

quoted from the book written by D.R.J.Owen and E.Hinton[36].

5.4.1 Description of the Finite Element Method Simulation

In our analysis, a real borehole is simulated by a thick

cylinder" with its outer boundary subjected to the far-field in-

situ stresses and the inner surface to well pressure.

The elements we use are

with eight nodes. The typical

external boundary set at

hole.

5.4.2 Boundary Condition

For stress dependent

is a force boundary, that

11

quadrilateral isoparametric elements

mesh is shown in Figure 4.2, with the

times radius from the center of the

modulus elastic model, the outer boundary

is, the stress states of the nodes at

external boundary are given as far field in-situ stresses..

--46--

the

The


54/68

loading path we use in our program is named as excavation unload-

ing, in which the problem domain is subjected to the in-situ

stresses and the boundary of the excavation is then unloaded

suddenly to the mud pressure. The boundary loads are unloaded at

only one step.

For elastoplastic model, the displacements of the nodes at the

external boundary are constrained, thus , it is a displacement

boundary. Only excavation unloading is simulated by applying a

sequence of negative pressure increments at the inner boundary.

Detailed discussion can be found in section 4.3.

5.5 Case Studies

The main purpose of this part is to highlight the difference

in the predicted values of safe mud weight with different consti-

tutive models and different compressive failure criteria for

elastic models. Two examples are given. Example 1 is to study the

relation between the safe mud weight and the inclined angle and

Example 2 is to show the relation of the safe mud weight to the.

azimuthal angle for horizontal wellbores.

The rock formation in Example 1 is a sandstone in the Cyrus

reservoir in the UK Continental Shelf [M.R. McLean and M.A. Addis

(15)]. A Mohr-Coulomb criterion is fitted to the laboratory

strength data , giving a cohesion, C, of 6.OMpa (860psi) and an

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The mean value of Poissons ratio

is 0.2. This formation is normally pressured (equivalent to

10.2 kPa/m, or 0.45 psi/ft) and at a depth of about 2,600 mTVD. The

vertical stress at this depth is assumed to be the weight of

overburden (taken at 22.6 KPa/m, or l psi/ft) . Both horizontal

stresses in the reservoir are assumed to be equal to 17.O KPa/m

(0.75 psi/ft) . A further assumption is that the tensile strength of

the rock is negligible. For stress dependent modulus elastic model

and elastoplastic model, the Young's modulus at

be 17.41 GPa.

The in-situ stresses in Example

total in-situ stress is 6,000 psi and

2 are tectonic. The vertical

the two horizontal principal

total stresses are 5,000 psi and 4,000 psi. The formation pressure is

2,000 psi. The cohesive strength,

the rock are 790 psi and 21.60

, respectively. The Poisson's ratio

is equal to 0.30 and the Youngs modulus with zero confining

pressure is assumed to be 500,000 psi.

Figures 5.3-5.7 show the results of the examples. Figures 5.3-

5.5 show the safe mud weight range varied with the inclined angle

of Example 1 and Figure 5.6-5.7 are for Example 2.

Figure 5.3, in which the linear elastic model

cake [M.R. McLean and M.A. Addis(15)]

values of mud weight causing compressive

--48--

is used,

collapse

with perfect mud

shows that the

are dependent on


56/68

the collapse failure criteria used.

Figure 5.4 shows the results of the linear elastic model given

by C.Hsiao(2), using only the Mohr-Coulomb criterion. With a

perfect mud cake (impermeable,

shown in Figure 5.3,

weight range shrinks.

Figure 5.5 gives the results with the stress dependent modulus

(SDM) elastic model and the elastoplastic (EP) model. The power

relation [Eq. (5.17)] is used for SDM model and the two parameters

are D1=O.08 and m=O.403. The two lower mud weight bounds obtained

by SDM model and EP model are almost similar because the ultimate

plastic strain

But the safe mud weight range is wider than those predicted by

linear elastic models, especially for horizontal wellbore.

Figure 5.6 shows the relations of the well pressure causing

the borehole instability with the azimuthal angle (Example 2),

using the impermeable linear elastic model and Mohr-Coulomb failure

criterion. The two models of M.R. McLean and M.A. Addis(l5) and of

C. Hsiao(2) (impermeable) give the same results. But

pressures exist.

with permeable

no safe well

Figure 5.7 gives the results of Example 2 with SDM model and

--49--


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EP model. As in Example 1, the lower well pressure bound given by

EP model with

SDM model. the lower bound given by EP model

is much l ow e r than that by SDM model.

Figure 5.8 shows a stress distribution of a vertical borehole

in Example 1 using SDM model and linear elastic model, in which the

well pressure is 26.52 MPa. Figure 5.9 gives a stress distribution.

of a vertical borehole in Example 2 using EP model, in which the

well pressure is 500 psi. Figure 5.10 shows the plastic zone under

this condition.

--50--


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3

INCLINED ANGLE (DEG)

Figure 5.5 Range of Safe MudWeight Using SDM and EP Modelsand Mohr-Coulomb Criterion

1

AZIMUTHAL ANGLE (DEG)

Figure 5.6 Range of Safe WellPressure Using McLean and Ad-diss Model

--51--


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moo

AZIMUTHAL ANGLE (DEG)

Figure 5.7 Range of Safe WellPressure Using SDM and EP

Models

Figure 5.9 Stress Distribution

Figure 5.8 Stress Distribution

Figure 5.10 Plastic Zone

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1.

2.

3.

4.

5.

6.

7.

8.

9.

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Nobuo Morita & K. E. Gray, A Constitutive Equation for Nonlinear

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M.E. Duncan Fama,

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C.H. Yew, M.E. Chenevert, C.L. Wang & S.O. Osisanya, Wellbore

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23. B. Haimson & C. Fairhurst, Initiation and Extension of Hydraulic

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24. E. Detournay & A.H-D. Cheng, Poroelastic Response of a Borehole

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Sci. & Geomech. Abstr. Vol.25, No.3, pp.171-182, 1988

25. C.H. Yew & Gefei Liu, "Pore Fluid and Wellbore Stabilities,

SPE 22381, (1992)

26. F.J. Santarelli, E.R. Brown & V. Maury, Analysis of Borehole

Stresses Using Pressure-Dependent, Linear Elasticity",

Technical Note, 1986

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27.

28.

29.

30.

31.

32.

F.J. Santarelli, Pau & France, "Performance of Wellbores in Rock

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Axisymmetric Boreholes, Rock Mechanics and Rock Engineering

22, 189-203 (1989)

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J. Boston Sot. of Civ. Engrs., 27 (1940), 1-5.

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36. M.R. Mclean, Analysis of Wellbore Stability, A Thesis

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APPENDIX A

The stresses around a borehole as a function of radial

distance away from the wellbore center are:

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(B-1)

where

i i

K(bulk modulus of the solid skeleton)K=(bulk modulus of the interpose material)

Equations of equilibrium are,

-o

From Eqs. (B-1), (B-2) and (B-3), we can obtained,

--59--

(B-2)

(B-3)


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(B-4),

Plane strain assumption is used in our analysis. In the polar

coordinate, Eq. (B-4) can be written as:

where

u is the radial displacement component.

Integrate eq. (B-5), it yields,

(B-5)

(B-6)

Hence the stress components are

Rewrite Eq. (B-7)

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From the assumption of plane strain

(B-8)

(B-9)

where

Finally, the stresses on the borehole wall can be expressed

as,

(B-1O)

borehore stability

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