Peng S., Zhang J., Engineering Geology for Underground Rocks

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7.3 Wellbore stability – elastoplastic solution 175

When r = R

is the radius of the failed zone), the radial stress can be

obtained both from Eqs. 7.21 and 7.22. Using the continuity of the radial

stress for r = R

, the constant C

can be obtained.

)1(

pfc

Rrr













(7.23)

and,











1

)1(

pfc

(7.24)

Therefore, the stresses in the elastic zone are as follows:



























)1(

pfc

(7.25)

In the elastic region, when r = R

, the stresses also satisfy Mohr-

Coulomb’s failure criterion expressed in effective stress form; i.e.:

)(

precpe

pqp  

(7.26)

where

is the uniaxial compressive strength in the elastic rock zone,

)sin1()sin1(

 q

and

is the angle of internal friction of the rock.

Substituting Eq. 7.25 into Eq. 7.26, the radius of failed zone can be ob-

tained:

)1]}()1([)1({

))1)(1()1]()1([)1)(2(













pfcfm

pfpfcfc

qpqqp

pqqqpqq

VVV

(7.27)

Equation 7.27 shows that the entire rock block will remain purely elas-

tic, if

. Therefore, the mud pressure (p

) to keep wellbore from

plastic deformation can be solved from Eq. 7.27:

)1(

)1(2





(7.28)

176 7 Wellbore/borehole stability

7.4 Wellbore stability – fracture gradient determinations

Fracture gradient is one of important parameters for wellbore stability

modeling, mudweight design, and real-time application. Fracture gradient

is the upper bound of the mudweight, while shear failure gradient is the

lower bound of the mudweight. If the mudweight is lower than the rock

fracture gradient, then the wellbore will have tensile failure, which may

cause losses of drilling mud, or even lost circulation. There are several ap-

proaches to calculate fracture gradient. The following two methods are

commonly used in the drilling industry; i.e. the minimum stress method

and tensile failure method.

7.4.1 The minimum stress method for fracture gradient

The minimum stress method is based on that most formations are naturally

fractured media. Therefore, in naturally fractured formations with closed

fractures the drilling mud loss occurs, once the mudweight is higher than

the minimum stress. The reason is that at this case the minimum stress is

the closure pressure to make the fracture closed.

Hubbert and Willis (1957) presented the following equation to estimate

fracture gradient:

PPPPOBGFG 



)(

(7.29)

where FG is fracture gradient; OBG is the overburden stress gradient; PP

is the pore pressure gradient;

is the Poisson’s ratio.

Daines (1982) developed this method by introducing a superimposed

tectonic stress correction:

SPPPPOBGFG 



)(

(7.30)

where S

is the superimposed tectonic stress gradient. This value may be

evaluated from the first leak off test on drilling. It is regarded as being

constant for the rest of the well.

Matthews and Kelly (1967) introduced a variable of effective stress co-

efficient into fracture gradient prediction:

PPPPOBGKFG  )(

(7.31)

where K

is the effective stress coefficient, K

c ;

c is the mini-

mum effective stress;

c is the maximum effective stress.

7.4 Wellbore stability – fracture gradient determinations 177

In this method the values of K

were established on the basis of fracture

threshold values derived empirically in the field. The effective stress

coefficient is variable and dependent on depth (Mouchet and Mitchell,

1989). The application of this method in a new area requires local varia-

tions in K

to be determined in relation to depth. The K

can be obtained

from leak-off tests (refer to Chap. 3 at Sect. 3.3).

Compared to Eqs. 7.29 and 7.31, one obtains that Matthews and Kelly

method and Hubbert and Willis method are very similar. They are identi-

cal, when the Poisson’s ratio and the K

have the following relationship:



(7.32)

It should be noted that Eqs. 7.29 - 7.32 are based on that the overburden

stress is the maximum principal stress, which is the normal faulting stress

regime. In other stress regimes these equations need to be modified.

Figure 7.10 presents a case to predict fracture gradient and shear failure

gradient (Madge et al. 2006) using Drillworks software, in which both

lithology and formation depth effects are considered for fracture gradient

prediction. It can be seen that the limestone has very low fracture gradient

because of its low Poisson’s ratio.

7.4.2 The tensile failure for fracture gradient prediction

For intact rocks, only when the wellbore appears tensile failures, drilling

mud loss can occur. In this case, fracture gradient can be calculated from

Eq. 7.12, when the minimum tangential stress is equal to the tensile

strength by noticing that the tensile stress is negative. For a vertical well

fracture pressure can be expressed as follows:

3 TpF

TpHhp



(7.33)

where F

is the fracture pressure; p

is the pore pressure;

is the thermal

stress induced by the difference between the mud temperature and the

formation temperature, and T

is the tensile strength of the rock. In some

cases the temperature effect can be neglected (

= 0).

This fracture pressure is also called the fracture breakdown pressure. For

the naturally fractured formation (with pre-existing fractures) this method

will overestimate the fracture gradient.

178 7 Wellbore/borehole stability

Fig. 7.10. Predicted fracture gradient (FG), pore pressure gradient (PP) and shear

failure gradient (SFG). The drilling events, such as drilling mud losses, tight hole,

stuck etc. coincide to the predicted fracture gradient and shear failure gradient.

From left to right, the curves are mudweight (MW), PP, SFG, FG and OBG, re-

spectively (from Madge et al. 2006). To keep wellbore stability, the mud weight

should be in between SFG and FG in each hole sections.

7.5 Wellbore stability – the finite element poroelastic

solution

Naturally fractured formations are often treated ideally as double porosity

media, which can be solved by using the finite element method (Zhang and

Roegiers 2005), as described in Chap. 6. The first step in the solution of

the coupled fluid flow and solid deformation problem is to discretize the

problem domain. This can be done by replacing the problem domain with a

7.5 Wellbore stability – the finite element poroelastic solution 179

collection of nodes and elements, referred to as the finite element mesh.

The second step is to derive an integral formulation for the governing

equations. This integral formulation leads to a system of algebraic equa-

tions that can be solved for values of the field variable at each node in the

mesh (Zhang et al. 2003). This section introduces boundary conditions,

boundary forces, and several failure criteria for the finite element modeling

of wellbore stability. A number of numerical examples are also given.

7.5.1 Boundary stresses and forces determinations

In the finite element analysis, the common approach is to subtract the

constant far-field stresses and original pore pressures from the field

quantities. The boundary conditions are thus modified; i.e., on the outer

boundary surfaces of the finite domain, all the tractions and pore pressures

vanish. At the borehole wall equivalent stresses and pressures are given.

After solving this problem, the final solution can be obtained by adding

back the constant in-situ stresses and original pressures (Zhang and

Roegiers 2002).

The equivalent stresses and pressures at any node at the borehole wall

can be calculated by using the following equations (Zhang et al. 2003):







frfr

mama

yzxzzxy

xyzyyy

zxyxxx

mSlSnS

lSnSmS

nSmSlS

(7.34)

where l, m, and n are the direction cosines between the normal to the

inclined plane and the x-, y-, and z-axes, respectively.

The boundary nodal forces along the wellbore can be determined by

using the following equations:



xyz

(7.35)

where A is the elemental area shown in Fig. 7.11.

180 7 Wellbore/borehole stability

Fig. 7.11. Boundary nodal forces at the wellbore wall.

7.5.2 Stress transformation

When analyzing stress and pore pressure distributions in and around well-

bores, the polar coordinate system is generally adopted. For the general-

ized plane strain formulation, the stresses in polar coordinates are related

to the Cartesian coordinate stresses according to the following rules:













TWTWW

TTWTTVVW

TTWTVTVV

sincos

)sin(coscossin)(

cossin2cossin

cossin2sincos

xzyzz

yzxzrz

xyxyr

xyyx

xyyxr

(7.36)

where r and

are the radius and the angle with reference to the center of

the wellbore in the polar coordinate system, respectively.

The strains relate to the Cartesian coordinate strains as:

7.5 Wellbore stability – the finite element poroelastic solution 181













TJTJJ

TTJTTHHJ

TTJTHTHH

sincos

)sin(coscossin)(

cossin2cossin

cossin2sincos

xzyzz

yzxzrz

xyxyr

xyyx

xyyxr

(7.37)

7.5.3 Mud weight Considerations at the Wellbore

Permeable boundary

A suitable mud weight of the drilling fluid during drilling plays a very

important role in helping to protect the borehole wall from breakouts and

other failures. The mud weight pressure, p

, acts both as a radial stress and

as fluid pressure at the borehole wall when the wellbore formation is

permeable. Considering mud pressure effects, the stresses at the borehole

wall can be expressed as (Zhang 2002):











mfrfr

mmama

yzxzzxy

mxyxyzymyyy

zxmyxyxmxxx

ppp

mSlSnS

SSlnSSSm

nSSSmSSl

)()(

(7.38)

where S

, S

,and S

myx

are stresses induced by the mud pressure (p

) as

the radial stresses, which can be obtained by:



mmxy

mmy

mmx

lmpS

pmS

plS

(7.39)

182 7 Wellbore/borehole stability

Impermeable boundary

Mud cake can be formed during drilling which may make the borehole

wall impermeable. In this case, the flow rate at the borehole wall is zero

and the mud weight pressure, p

, acts only as a radial stress at the borehole

wall under such conditions (Zhang and Standifird 2007). Therefore, the

stresses at the borehole wall can be written as:











)()(

frfr

mama

yzxzzxy

mxyxyzymyyy

zxmyxyxmxxx

mSlSnS

SSlnSSSm

nSSSmSSl

(7.40)

7.5.4 Failure Criteria

Borehole walls may fail when the surrounding stress exceeds the tensile,

the compressive, or the shear strengths of the rock formation, whichever is

reached first. The effective stress concentration near the wellbore due to

the far-field stresses and the specific borehole orientation need to be

considered for failure initiation. It is commonly accepted that wellbore

failure is controlled by Terzaghi’s effective stress; i.e.

maijijij



(7.41)

where

is the effective stress tensor, p

is the matrix pore pressure and

is the Kronecker delta.

Various failure criteria, under tensile, Mohr-Coulomb and Drucker-

Prager, are considered and incorporated into the dual-porosity finite ele-

ment model, as individually described below.

Tensile Failure

There are two possible modes of tensile failure. One is due to radial frac-

turing at the borehole wall as the mud pressure in the wellbore is increased.

The other is the outburst failure (or spalling) due to an effective radial ten-

sile stress caused by a very rapid depressurization of the borehole.

7.5 Wellbore stability – the finite element poroelastic solution 183

Spalling failure

As discussed above, spalling (or splintering) will occur when the effective

radial tensile stress equals or is greater than the formation tensile strength;

that is:

marr



(7.42)

where

is the total radial stress;

is the effective radial tensile stress;

and T

is the tensile strength of the formation.

Usually, the formation tensile strength is very small, especially for frac-

tured formations. Therefore, it is usually considered to be zero. It should

be pointed out that, due to poroelastic effects, the tensile fracture does not

necessarily initiate at the borehole wall. In addition, tensile failure condi-

tions vary with time.

Fracturing failure

If mud pressure is too high, radial tensile failure will occur when the mini-

mum principal effective stress or effective circumferential stress is equal to

the formation tensile strength, as shown in Eq. 7.43. In this condition the

drilling induced tensile fractures will occur due to high mud weight, which

may induce hydraulic fracturing in the wellbore and cause drilling mud

losses.

(7.43)

where

is the minimum effective principal stresses (note that

is nega-

tive).

Shear Failure

Shear failure occurs when the shear strength of the formation is exceeded.

The Mohr-Coulomb failure criterion can be used to determine such a fail-

ure mechanism; i.e.:

tan

 c

(7.44)

where

is the angle of internal friction and c is the cohesion.

In the principal space, (

), the Mohr-Coulomb failure criterion

is given by:

184 7 Wellbore/borehole stability



(7.45)

where

are the maximum and minimum effective principal stresses,

respectively; and

)sin1()sin1(

 q

Collapse Failure

The effect of the intermediate principal stress is not considered in the fail-

ure criteria mentioned above. Laboratory data have shown that the inter-

mediate principal stress plays an important role in the failure of rocks and

can be conveniently represented by the Drucker-Prager yield condition;

that is:



(7.46)

where

and

are material constants; and, I

and J

are the stress invari-

ants, which can be expressed as:

zyx



(7.47)

222222

])()()[(

zxyzxyxzzyyx

WWWVVVVVV



(7.48)

7.6 Finite element application - Case studies

7.6.1 Wellbore in a strike slip faulting stress regime

For this case study the in-situ stresses and initial pore pressures are ob-

tained from Woodland (1990): S

= 29 MPa, S

= 20 MPa, S

= 25 MPa, and p

= 10 MPa. Two wellbore inclinations are considered:

= 0 and 90q. The wellbore radius is R = 0.1 m. Biot's effective stress co-

efficients of rock matrix and fractures are,

= 0.771 and

= 0.91. The

corresponding parameters (Table 7.1) for the dual-porosity poroelastic

model are similar as the case described in Chap. 6, Sect. 6.4.