NUMERICAL STUDY ON WELLBORE STABILITY ANALYSIS

NUMERICAL STUDY ON WELLBORE STABILITY ANALYSIS CONSINDERING VOLUMETRIC FAILURE

Michel N. Zahna, Lúcia Carvalho Coelhoa, Luiz Landaua and José Luis Drummond Alvesa

aLaboratório de Métodos Computacionais em Engenharia, Universidade Federal do Rio de Janeiro, Caixa Postal 68.552, CEP 21941-972, [email protected]

Keywords: Wellbore stability, Plasticity, Cap Models.

Abstract. When a wellbore is drilled, the equilibrium in-situ stress is changed. In order to support the stress relief induced by the drilling and to prevent hydrocarbon influx into the cavity, the borehole is filled with a fluid. These operations create new stresses configurations. A main point in wellbore projects is the definition of the drilling fluid density to keep the wellbore stable. The upper bound to the fluid density is the collapse stress that is the limit to shearing. The lower bound is the fracture stress that limits the tensile failure. The fluid densities between these limits is named safe mud weight window. Conventional wellbore stability analysis usually considers the effects of shear or tensile failure. Recent physical models pointed the possibility of compactive volumetric failure around boreholes. Other numerical studies showed that this type of failure may be attained by stress level compatible with reservoirs in under production in high porosity rocks. This paper applies numerical modeling by finite element method with ANSYS® software to study stability of oil wells. It simulates the compactive behavior of rocks using a cap plasticity model. The cap model data was collected from literature. Parametric studies were conducted in horizontal wellbores to evaluate the conditions of volumetric failure to occur. The study pointed the tangential stress concentration as a critical condition for volumetric failure around boreholes.

Mecánica Computacional Vol XXIX, págs. 8837-8858 (artículo completo)Eduardo Dvorkin, Marcela Goldschmit, Mario Storti (Eds.)

Buenos Aires, Argentina, 15-18 Noviembre 2010

Copyright © 2010 Asociación Argentina de Mecánica Computacional http://www.amcaonline.org.ar

1 INTRODUCTION

Hydrocarbon exploitation in marine coast of Brazil requires new technologies for its economic feasibility. Pre-Salt reserves are located in Santos Basin at 290 km from the coast. The water depth is about 1900 to 2400 meters and the reservoirs are located at 5000 meters underground, below a salt layer that may reach 2000 meters. The exploitation of these reserves require unconventional well technologies to make it feasible from an economic point of view.

The use of directional and horizontal wells has been increased in the last years, following the advances in the techniques on drilling and completing these wells. Horizontal wells present advantages in relation to vertical wells, as it has a bigger area exposed to flow. In offshore environments it is an additional advantage, since locating the platform is critical due to sea conditions.

These unconventional environments and well geometries led to the problem of wellbore stability analysis and sand prediction. This kind of problem has a big impact on wellbore costs.

Prediction models for wellbore stability analysis are based on rock mechanics models. The first model based on rock mechanics for this kind of problem was proposed by Bradley (1979). Several models based on continuous mechanics were developed since then. The assumptions of these models vary from simple elastic models to more elaborated elastic-plastic models. Morita (2004) proposed an analytical procedure based on elasticity to evaluate the stress state around the borehole. The stress level is compared to a compression or tensile failure criteria to evaluate stability. The most used failure criteria are Drucker-Prager, Mohr-Coulomb, modified Lade criterion (Ewy, 1999) or Hoek and Brown (Zhang and Zhu, 2007). Tensile criterion usually consists in the comparison of the minimum effective stress to the tensile strength of the rock.

These models are considered very conservative, since attaining the limit stress in a point around the borehole does not imply in instability. Numerical methods based on plasticity theory, such as finite difference or finite elements methods, present the advantage of showing the extent of the damaged region, leading to a better indicator of instability. These models usually consider two kinds of failure around the wellbore: shear failure and tensile failure. They became more complex, as they incorporated other physics associated to instability: poroelastic models (Detournay and Cheng, 1993), thermoporoelastic (Wang and Desseault, 2003), chemo-thermoporoelastic (Yu et al, 2001) and its versions associated to plasticity theory. Advances in elastic-plastic models to improve stability indicators were developed, such as bifurcation theory and localization of deformation along shear bands. The rock microstructure has been incorporated through the Cosserat continuum model (Papanastasiou and Vardoulakis, 1988, 1989) or gradient elastoplasticity (Zervos et al., 2001), which uses terms of strain gradient as state variable.

The models described above depend on a realistic constitutive model that must be able to reproduce the several failure modes around boreholes. Experimental studies show that rocks under high confining pressures can reach the hydrostatic stress strength. Once this limit is attained, severe compaction of porous media occur, leading to porosity and permeability reduction. This work names this kind of failure as volumetric failure.

Coelho et al. (2005) presented a numerical study on a wellbore drilled in a reservoir from Brazilian coast and showed that volumetric failure can be reached under those reservoir conditions. These results are compatible with experimental studies. Haimson (2007) identified a breakout pattern for porous rocks, distinct of those identified for rocks of lower porosity. Porous rocks presented a slit mode failure, surrounded by a highly compacted material,

M. ZAHN, L. CARVALHO COELHO, L. LANDAU, J.L. DRUMMOND ALVES8838


suggesting an “empty” compaction band. This work presents a study on the conditions for volumetric failure occurrence around

boreholes. It uses a constitutive model that is able to predict shear failure, tensile failure and compaction failure around boreholes. The data used in these analyses were taken from literature on porous rocks. Parameters that impact on wellbore stability were evaluated.

The numerical modeling of wellbore stability analysis is made considering small strain and small displacement finite element method and classical associated plasticity theory. The software used was ANSYS® and the mesh is based on linear triangles.

The well configuration studied was a horizontal wellbore under plane strain conditions. To take advantage of the problem symmetry, the mesh consisted of ¼ of the geometry. The rock is treated as an isotropic and homogeneous continuum medium. The constitutive model used was the cap model as implemented by ANSYS®.

2 FAILURE MODES AROUND BOREHOLES

Wellbore stability during drilling consists on evaluating the drilling fluid weight to mantain the borehole wall integrity. It means that pressure on the boreface must be less than formation fracture pressure and more than the collapse pressure to avoid fluid losses or borehole breakouts. These phenomena are associated to tensile failure (figure 1a) or shear failure (figure 1b), respectively.

Haimson and Song (1998) described two distinct breakout patterns in Berea sandstone under drilling in true triaxial laboratory conditions. For lower porosity (17%), the breakout showed dog-ear geometry (figure 1b). For a higher porosity (22.5%), slit-mode breakout (figure 1c) was obtained during the drilling. Haimson and Lee (2004) observed that these distinct modes of failure are dependent not only on porosity, but also on sandstone composition. Haimson´s work associated these fracture-like breakouts with empty compaction bands. This failure mode is not clear under field conditions.

Triaxial tests on rocks under higher confining pressures are described by Baud et al. (2004, 2006). They describe the volumetric behavior of rocks based in the stress-strain curve as shown in figure 2 for Tavel limestone. In this figure, compression is positive. At lower confining stress, the volumetric strain is dilative until point C’ of mean stress- volumetric strain curve (figure 2b). At a confining stress of 100 MPa, the rock first experiences compaction and then dilates. Under higher confining pressures the rock shows a compaction behavior. These characteristic points are presented in figure 3 in the differential stress-mean stress space for Indiana limestone. It can be seen the contour of a cap that limits the strength to rock hydrostatic compression. This graphic delimits different volumetric strain behavior: at lower confining pressures, dilation occurs. This region is followed by a brittle-ductile transition, and then the compaction cap is attained as confining pressure increases. Undergoing further compression, the cap expands in the stress space, until the material is so compacted that it returns to a dilation behavior. The micromechanisms associated to these failure modes are presented in figure 4. In the dilation regime, rock fails by the formation of a shear band; in the brittle-ductile transition, there is a mixed mode of deformation between dilatation and compaction. The cap delimits distinct mechanisms: at higher deviator stresses, the so-called shear enhanced compaction occurs, by the development of a compaction band. Close to the first stress invariant axis, the dominant deformation mechanism is pore collapse, which corresponds to a more diffuse deformation pattern.

Mecánica Computacional Vol XXIX, págs. 8837-8858 (2010) 8839


(a) (b) (c)

Figure 1: Failure modes around wellbores (a) tensile failure (hydraulic fracturing); (b) dog-ear shape breakout;

(c) fracture-like breakout. The most common elastic-plastic constitutive models used in wellbore stability analysis are

shear failure models such as Drucker-Prager and Mohr-Coulomb, associated to some tensile failure criteria. These models are representative of lower porosity rock behavior. The compaction mechanisms can be predicted by cap models (Fjaer et al., 2008).

Diffe

rent

ial s

tress

[MP

a]

0

50

100

150

200

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0 1 2 3 4 5 6 7

10203050100150200240

Axial strain [%]

(a)

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tress

[MP

a]

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150

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0 1 2 3 4 5 6 7

10203050100150200240

Axial strain [%]

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tress

[MP

a]

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0 1 2 3 4 5 6 7

10203050100150200240

Axial strain [%]

(a)

0

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-0.5 0 0.5 1 1.5 2 2.5 3

hydrostat10203050100150200240

Volum etric strain [%]

C'

C*

C*'

P*

Mea

n st

ress

[MPa

]

(b)

0

50

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250

300

350

400

-0.5 0 0.5 1 1.5 2 2.5 3

hydrostat10203050100150200240


C'

C*

C*'

P*

Mea

n st

ress

[MPa

]

0

50

100

150

200

250

300

350

400

-0.5 0 0.5 1 1.5 2 2.5 3

hydrostat10203050100150200240


C'

C*

C*'

P*

Mea

n st

ress

[MPa

]

(b)

Figure 2: Triaxial tests for Tavel limestone (a) differential stress-axial strain; (b) Mean stress-volumetric strain

(from Vajdova et al, 2004)

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350

C'peakC*C*'

M ean stress [MPa]

br ittl e fractu re

dilatan cy

dilatant cataclasti c flo w

shear-enhan ced compactio n

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eren

tial s

tress

[MP

a]

0

50

100

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200

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0 50 100 150 200 250 300 350

C'peakC*C*'

M ean stress [MPa]

br ittl e fractu re

dilatan cy



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eren

tial s

tress

[MP

a]

0

50

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0 50 100 150 200 250 300 350

C'peakC*C*'

M ean stress [MPa]

br ittl e fractu re

dilatan cy



Diff

eren

tial s

tress

[MP

a]

Figure 3: Triaxial tests for Indiana limestone (a) differential stress-axial strain; (b) Mean stress-volumetric strain

(from Vajdova et al, 2004)



Figure 4: Micromechanisms associated to distinct failure modes

(from: Baud et al, 2008)

3 STRESS STATE AROUND BOREHOLES

An underground rock mass is in equilibrium condition under compressive in-situ stress state, which can be decomposed in relation to a Cartesian coordinates system as vertical stress, parallel to the depth direction and two horizontal stresses: a major horizontal stress ( )HS and a minor horizontal stress ( )hS . Changes in this stresses are introduced by the drilling and production operations.

The governing poroelastic equations for a homogeneous isotropic media, considering the absence of body forces and fluid sources, are given by (Wang, 2000):

ijijijkkij pG δαεδλεσ Δ−+=Δ 2 (1)

( )kkMp αες −=Δ (2)

0, =jijσ (3)

0, =+ iiqdtdς (4)

ii pq ,κ−= (5)



Where tensile stresses are positive, ijσΔ is the total stress change tensor, ijε , the solid

strain tensor, Δp is the pore pressure differential, ς is the variation of fluid content, kkε is the volumetric strain, λ and G are the Lamé´s parameter for the porous rock, α is the Biot’s effective stress coefficient, M, the Biot modulus and ijδ is Kronecker delta.

The parameters that govern balance laws are:

μ

κ k= (6)

κ is the mobility coefficient, k is the permeability coefficient and μ is the fluid viscosity. Applying the constitutive relations to the equilibrium condition and expressing the

deformation term as displacement derivative, the governing equation in terms of displacements is given by:

ii

iki

k FxpuG

zw

yzv

xxuG

−∂∂

=∇+⎟⎟

⎠

⎞⎜⎜

⎝

⎛

∂

∂+

∂∂∂

+∂∂

∂⎟⎠⎞

⎜⎝⎛−

αν

22

222

21 (7)

Under steady state conditions, the the equation governing the pore pressure field uncouples (Detournay and Cheng, 1993):

αεMp −=Δ (8)

As the well is drilled and material is removed, a stress relief occurs. If the cavity is not fulfilled with fluid, the equilibrium is attained by tangential stress concentration. The drilling fluid introduces a radial pressure against the borehole wall. This pressure acts as a support to the boreface and relieves the generated tangential stress.

The stress state around the borehole may vary according to its radius and inclination angle. This variation depends on many factors as the wellbore direction related to in-situ stresses, the in-situ stresses magnitude, the rheology of the rocks and wellbore geometry.

As one moves from the borehole into the formation, the stresses tend to reach the in-situ stresses. According to Rocha and Azevedo (2009), the usual stress configuration around the well is: tangential stress is the major principal stress; axial stress is the intermediate stress although in lower depths it may become the major stress and radial stress is the minor principal stress.

4 ELASTIC PLASTIC MODEL

Figure 5: Cap model plasticity yield surface; tensile stress is positive (from Release 11.0 Documentation for

ANSYS)



The analyses were made using ANSYS® cap model (figure 5). The formulation of this

model is described in Theory Reference for Ansys and Ansys Workbench from Ansys documentation and is briefly summarized here.

It consists on a shear failure surface associated to an elliptical cap for compressive volumetric failure and a tensile cap, described by the following equation:

),(),(),,(),(

),,,,(),,(01

2010012

20032100

σσσβσσσ

IYIYKIYJKJJIYKY

stc−ΨΓ==

(9)

where : Γ : Lode’s angle function,

cY : compressive cap function

tY : tensile cap function

sY : shear failure surface

Figure 6: Shear yield function (from Release 11.0 Documentation for ANSYS)

The shear failure surface (figure 6) is described by:

1)(

001 1),( IAeIY IS ασσ β −−= (10)

2

00

0110001

),()(1),,(

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −−−=

σσ

KYR

KIIKHKIY

scc (11)

2

0

1101

),0()(1),( ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛−=

σσ

stt

YR

IIHIY (12)

In the above expressions, 1I is the first stress invariant, oσ is a constant associated to cohesion, β,A and α are constants obtained from experiments, cR and tR is the ratio between the main axis in 1I direction and main axis in 2J direction of the compressive cap and tensile cap, respectively. The parameter oK is the value of 1I at intersection point between the shear failure surface and the elliptical cap. Function H is the Heaviside function,



which assumes the values:

⎩⎨⎧

><

=o

oKIKI

H1

1 para ,1 para ,0

(13)

The parameters 0K and 0X are associated by:

),( 0000 σKYRXK sc+= (14)

The function cY is an elliptical function associated to Heaviside function, as show figure 7(a).

(a)

(b)

Figure 7: (a) Elliptical compressive cap; (b) Elliptical tensile cap (from Release 11.0 Documentation for ANSYS)

The tensile cap (figure 7(b) is similar to the compressive cap, but it is fixed in the stress

space and depends only on the first stress invariant and the cohesive term:

2

0

1101

),0()(1),(

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−=

σσ

syt

tYR

IIHIY (15)

The influence of the third deviator stress invariant is introduced by the function of Lode’s angle β and the triaxial compressive strength:

))31(131(21),( βββ sensen −

Ψ++=ΨΓ (16)

Lode’s angle is given by:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−= −

2/32

3132

2

3331),(

J

JsenJJβ (17)

These relations are shown in figure 8:

Figure 8: Failure surface in deviatoric plane (from Release 11.0 Documentation for ANSYS)



This work assumes a perfect elastic-plastic shear surface and a isotropic strain hardening

compressive cap, defined by:

}1{ )))(((1

0021 −= −−− XiXXiXDDcpv

cceWε (18)

The parameter iX represents actual position of cap parameter X, cW1 is the maximum

allowed plastic strain, cD1 and cD2 are experimental adjustment parameters for the hardening function.

5 MODELING PROCEDURE

Two loading steps were considered: the first corresponds to drilling phase and the second to steady state production phase.

Stability during drilling was modeled considering no fluid flow between the wellbore and the rock. The values of far-field effective stresses were applied to the boundaries of the model, as well as a pressure on the cavity, which is equivalent to the differential fluid pressure between the fluid in the cavity and reservoir pressure.

To simulate stability during production, the drawdown pressure was applied on the cavity face and poroelastic effects were taken into account, induced by fluid flow from the reservoir to the wellbore. In this work, it was assumed that one-phase fluid saturates porous media. The model assumes that at the outer boundary the pore pressure remains unaltered. This drainage radius may not be realistic, but it was assumed reasonable for the purpose of evaluating the stresses in the vicinity of the wellbore. It must be taken from a qualitative point of view.

To simulate these effects with ANSYS, the approach was to explore the analogy between poroelasticity and thermoelasticity.

5.1 Thermoelasticity and poroelasticity analogy

Fjaer et al. (2008) showed the analogy between poroelastic stresses, described by equation 1 (rewritten below) and thermoelastic stresses (equation 19):

ijijijkkfrij pG δαεδελσ Δ++= 2

ijTijijkkij TKG δαεδλεσ Δ++= 32 (19)

The term that couples the thermal/pore pressure field to the stress analysis is the parameter Tα (thermal stress expansion coefficient) or α (Biot´s effective stress coefficient),

respectively. Manipulating this term, poroelastic stress may be obtained by a thermoelastic analysis by making:

KT 3αα = (20)

where 1/K is the rock compressibility



5.2 Procedure validation

To check the procedure, the hollow poroelastic cylinder was modeled. The geometric model consists on a cylinder with a 2 meters internal radius and 10 meters external radius, with an internal pressure of 15 MPa. The mesh takes advantage of symmetry and represents ¼ of the geometry. It has 4753 nodes and 4608 linear isoparametric elements. Figure 9 shows de mesh. A poroelastic analysis was done by analogy with thermoelastic analysis. The elastic parameters used were an elastic modulus of 1200 MPa and Poisson ratio of 0.2. The thermal analysis was coupled to the structural analysis through the compressibility of the rock analogous to the thermal expansion coefficient following equation 20. The values adopted for Biot effective stress coefficient was 1 and 0005.0=tα .

Pore pressure field obtained through this analysis is presented in figure 10. It is compared to the analytical solution described by Wang (2000), and is shown in figure 11. It presents a good agreement between both analysis. The differences in the stresses in the vicinities of the wellbore may be attributed to the fact that in finite element analysis they are evaluated at the integration point rather then the nodal point at the boreface.

1

X

Y

Z

SEP 28 201016:22:34

ELEMENTS

MAT NUM

Figure 9: Hollow cylinder mesh

1

MN

MX

X

Y

Z

0

1.6673.333

56.667

8.33310

11.66713.333

15

SEP 28 201016:27:59

NODAL SOLUTION

STEP=1SUB =1TIME=1BFETEMP (AVG)RSYS=0DMX =.036837SMX =15

Figure 10:Pore pressure field



Stresses around a borehole during production

-20

-15

-10

-5

0

5

10

0 2 4 6 8 10 12

raio

tens

ão

radial stresstangential stressnumerical 1st principal stressnumerical 3rd principal stress

Figure 11: Comparison between analytical and numerical solution for stress states in the poroelastic hollow

cylinder (compression is positive)

6 PARAMETRIC STUDY OF A HORIZONTAL WELLBORE

6.1 Basic model

The basic model used for comparison is a open-hole horizontal wellbore, 8.5” diameter, drilled in a carbonate from Campos Basin. The data was obtained from Coelho et al (2006). It is presented in table 1.

Horizontal wellbore geometry takes advantage of the symmetry of the problem. The wellbore is represented by ¼ of the circle and the external boundary is 50 meters from the wellbore axis. The numerical model consists in a mesh with 1671 nodes and 3088 elements (figure 12). The in-situ stress state is represented by the introduction of a distributed vertical loading on the top of external boundary, which value is equivalent to the vertical in-situ stress and a lateral loading on the external boundary which value is the value of in-situ horizontal stress. The in-situ stress in the axial direction is simulated by setting an initial stress state equivalent to its value The pressure on the wellbore wall is represented by a pressure on the bore face.

1

X

Y

Z

OCT 1 201013:58:41

ELEMENTS

MAT NUM

Figure 12: Finite element mesh

It is assumed that the horizontal in-situ stresses are isotropic. Loading condition used in

this study assumes two load steps. The first one is the drilling phase, when the far-field in-situ stresses are applied to the boundaries as described above. A pressure at the borehole



equivalent to the difference between the fluid in the formation and the fluid in the cavity is set. The second load step assumes that the fluid flows into the borehole and poroelastic effects

are then simulated with the thermoelastic analogy procedure described in 5.2. The differential pore pressure stress at the boreface is considered by applying a tensile stress at this face equivalent to this differential.

Table 2 presents the reservoir data used in this work.

Parameter Value Elastic modulus (MPa) 1200.

Poisson ratio 0.2 oσ (MPa) 4.92 α 0.589707

Xo (MPa) -108. Rc 2.2 Rt 1.0 W 0.05 β 1.0 A 0.0

cD1 0.012 cD2 0.0

Table 1: Mechanical properties for carbonate rock for Campos basin.

Vertical effective in-situ stress

(MPa)

Horizontal effective in-situ

stress (MPa)

Static pressure in reservoir (MPa)

32.0 8.0 32 Table 2: Reservoir data.

To understand the effects of volumetric failure around the wellbore, analysis considering only shear failure in the compressive domain and analysis with the cap model were made. An evaluation of the stresses conditions on volumetric failure around the wellbore was done by analyses with parametric changes for different in-situ horizontal stress vs vertical stress ratio and different values of pressure on the borehole wall. Finally, a poroelastic-plastic analysis to simulate production phase under steady-state flow conditions is presented.

6.2 Shear failure analysis

Shear failure analysis was conducted by using the same failure surface setting 10000 =X MPa, such that cap would not be attained under the loading conditions used in this

work. Figure 13 shows the results in terms of principal stresses and plastic strain intensity for drilling phase and figure 14 presents these results for production phase. Failure occurred in tensile domain at the top of the wellbore during drilling, and the failed region increases during production. Production produces a tensile region at the top of the well, and a high magnitude concentration of tangential stress in X symmetry axis, but no failure is seen in this region.



1

MN

MX

X

Y

Z

-13.12

-11.231-9.342

-7.453-5.564

-3.675-1.786

.1025331.992

3.881

SEP 8 201013:55:08

NODAL SOLUTION

STEP=1SUB =1TIME=1S1 (AVG)DMX =1.191SMN =-13.12SMX =3.881

(a) Radial stress

1

MN

MX

X

Y

Z

-81.957

-72.981-64.005

-55.028-46.052

-37.076-28.1

-19.124-10.147

-1.171

SEP 8 201013:58:36

NODAL SOLUTION

STEP=1SUB =1TIME=1S3 (AVG)DMX =1.191SMN =-81.957SMX =-1.171

(b) Tangential stress

1

MN

MX

X

Y

Z

-34.367

-30.423-26.479

-22.535-18.591

-14.647-10.703

-6.759-2.815

1.128

OCT 2 201021:39:34

NODAL SOLUTION

STEP=1SUB =1TIME=1NLHPRE (AVG)RSYS=0DMX =1.191SMN =-34.367SMX =1.128

(c) Mean stress

1

MN

MX

X

Y

Z

0.120E-04

.239E-04.359E-04

.478E-04.598E-04

.717E-04.837E-04

.957E-04.108E-03

SEP 8 201014:07:48

NODAL SOLUTION

STEP=1SUB =1TIME=1EPPLINT (AVG)DMX =1.191SMX =.108E-03

(d) Plastic strain intensity

Figure 13: Results for shear failure analysis for drilling phase

1

MN

MX

X

Y

Z

-52.394

-45.723-39.053

-32.382-25.712

-19.041-12.37

-5.7.970618

7.641

OCT 2 201020:29:55

NODAL SOLUTION


(a) Radial stress

1

MN

MX

X

Y

Z

-177.799

-157.441-137.082

-116.724-96.365

-76.007-55.648

-35.29-14.931

5.427

OCT 2 201020:31:14

NODAL SOLUTION


(b) Tangential stress

1

MN

MX

X

Y

Z

-110.454

-97.557-84.66

-71.763-58.866

-45.969-33.072

-20.174-7.277

5.62

OCT 2 201020:44:30

NODAL SOLUTION


(c) Mean stress

1

MN

MX

X

Y

Z

0

.004129.008258

.012388.016517

.020646.024775

.028905.033034

.037163

OCT 2 201020:28:49

NODAL SOLUTION

STEP=1SUB =1TIME=1EPPLINT (AVG)DMX =1.511SMX =.037163

(d) Plastic strain intensity

Figure 14: Results for steady-state production phase



6.3 In-situ stress ratio variation

The impact of the in-situ stress ratio was evaluated by changing the horizontal in-situ stress as presented in table 3.

In-situ vertical

stress (MPa)

In-situ horizontal stress (MPa)

In-situ stress ratio

(MPa) 32 6,4 0,2 32 9 0,3 32 12,8 0,4 32 16 0,5

Table 3 In-situ stress changes.

A summary of the results are presented in the graphic of figure 15. The contours of principal sresses are presented in figure 16. Mean stress and plastic strain

intensity are presented in figure 17. The higher the horizontal stress, the higher is the confinement around the borehole. Since

the cap delimits the confining pressure, it is intuitive to believe that this condition would be more critical to volumetric failure. However, the results show that the critical condition is the higher difference between the in-situ stresses. As the radial stress is prescribed, the effect of the far-field confinement is to relax the tangential stress and reduce plastic strain. It suggests that the failure occurs in the domain described previously as “shear enhanced compaction”.

Maximum value of tangential stress vs far-field horizontal stress

-82

-81

-80

-79

-78

-77

-76

-75

-740 5 10 15 20

Far-field horizontal stress (MPa)

Tang

entia

l str

ess

(MP

a)

Figura 15: Tangential stress x far-filed horizontal stress



Radial Stress (MPa) Tangential Stress (MPa) 1

MN

MX

X

Y

Z

-13.15

-10.857-8.564

-6.271-3.978

-1.685.60833

2.9015.194

7.487

SEP 9 201010:42:42

NODAL SOLUTION


(a) 4,6=hσ MPa

1

MN

MX

X

Y

Z

-80.799

-71.632-62.465

-53.298-44.131

-34.964-25.797

-16.63-7.463

1.704

SEP 9 201010:44:03

NODAL SOLUTION


(b) 4,6=hσ MPa

1

MN

MX

X

Y

Z

-13.12

-11.222-9.323

-7.424-5.525

-3.626-1.728

.1712082.07

3.969

SEP 8 201015:22:25

NODAL SOLUTION


(c) 0,9=hσ MPa

1

MN

MX

X

Y

Z

-79.349

-70.658-61.967

-53.276-44.585

-35.893-27.202

-18.511-9.82

-1.129

SEP 8 201015:23:25

NODAL SOLUTION


(d) 0,9=hσ MPa

1

X

Y

Z

-13.316

-12.004-10.691

-9.378-8.065

-6.753-5.44

-4.127-2.814

-1.502

SEP 9 201010:54:33

NODAL SOLUTION


(d) 8,12=hσ MPa

1

X

Y

Z

-77.045

-69.219-61.394

-53.569-45.744

-37.919-30.093

-22.268-14.443

-6.618

SEP 9 201010:55:31

NODAL SOLUTION


(e) 8,12=hσ MPa

1

X

Y

Z

-13.244

-11.999-10.754

-9.509-8.264

-7.019-5.774

-4.529-3.284

-2.039

SEP 9 201011:04:29

NODAL SOLUTION


(f) 16=hσ MPa

1

X

Y

Z

-74.947

-68.35-61.754

-55.158-48.562

-41.965-35.369

-28.773-22.177

-15.58

SEP 9 201011:05:28

NODAL SOLUTION


(g) 16=hσ MPa

Figura 16: Principal stresses for in-situ horizontal stress variation



Mean stress (MPa) Plastic strain intensity 1

MN

MX

X

Y

Z

-32.948-28.895

-24.842-20.788

-16.735-12.682

-8.629-4.576

-.5228393.53

SEP 9 201010:45:48

NODAL SOLUTION


(a) 4,6=hσ MPa

1

X

Y

Z

0.181E-03

.361E-03.542E-03

.723E-03.903E-03

.001084.001264

.001445.001626

SEP 9 201010:47:52

NODAL SOLUTION


(b) 4,6=hσ MPa

1

MN

MX

X

Y

Z

-32.61

-28.876-25.143

-21.41-17.677

-13.943-10.21

-6.477-2.744

.989773

SEP 8 201015:24:33

NODAL SOLUTION

STEP=1SUB =1TIME=1NLHPRE (AVG)RSYS=0DMX =1.191SMN =-32.61SMX =.989773

(c) 0,9=hσ MPa

1

MN

MXX

Y

Z

0

.129E-03.258E-03

.386E-03.515E-03

.644E-03.773E-03

.901E-03.00103

.001159

SEP 8 201015:27:37

NODAL SOLUTION


(d) 0,9=hσ MPa

1

X

Y

Z

-31.973

-28.791-25.609

-22.426-19.244

-16.062-12.879

-9.697-6.515

-3.332

SEP 9 201010:57:21

NODAL SOLUTION

STEP=1SUB =1TIME=1NLHPRE (AVG)RSYS=0DMX =1.168SMN =-31.973SMX =-3.332

(e) 8,12=hσ MPa

1

X

Y

Z

0

.708E-04.142E-03

.212E-03.283E-03

.354E-03.425E-03

.496E-03.567E-03

.637E-03

SEP 9 201010:58:51

NODAL SOLUTION


(f) 8,12=hσ MPa

1

X

Y

Z

-31.255

-28.57-25.885

-23.2-20.514

-17.829-15.144

-12.459-9.774

-7.089

SEP 9 201011:06:32

NODAL SOLUTION

STEP=1SUB =1TIME=1NLHPRE (AVG)RSYS=0DMX =1.165SMN =-31.255SMX =-7.089

(g) 16=hσ MPa

1

X

Y

Z

0

.332E-04.665E-04

.997E-04.133E-03

.166E-03.199E-03

.233E-03.266E-03

.299E-03

SEP 9 201011:08:00

NODAL SOLUTION


(h) 16=hσ MPa

Figura 17: Mean stress and plastic strain intensity for in-situ horizontal stress variation



6.4 Effect of borehole pressure

The effect of borehole pressure was evaluated by applying different values of pressure on the cavity: -2, 0, 2 e 4 MPa. Figure 18 shows the graphic major principal stress vs pressure on the boreface. Figures 19 and 20 present the contours of principal stresses, mean stresses and plastic strain intensity.

It can be seen that the reduction on the cavity pressure concentrates tangential stress. This increases the volumetric plastic strain at the borehole wall. As before, the concentration on the tangential stresses is associated to the failure around the well. So, the higher the difference between the principal stresses is the most critical condition, where it seems to occur shear enhanced compaction.

Maximum value of tangential stress (MPa)

-82.5

-82

-81.5

-81

-80.5

-80

-79.5

-79

-78.5

-78

-77.5-3 -2 -1 0 1 2 3 4 5

Borehole pressure (MPa)

Tang

entia

l str

ess

(MPa

)

Figure 18: Tangential stress vs cavity pressure

6.5 Production phase

A numerical study on production phase was conducted assuming that the pore pressure on the outer boundaries is equal to the static pressure. At the borehole wall it was assumed a drawdown value of 10 MPa differential pressure.

The changes in pore fluid pressure were evaluated through the poroelastic stresses calculated by analogy with thermoelastic stresses. A pore pressure field was evaluated by thermal steady state analysis. This field was read by the thermostructural analysis and coupled through a compressibility parameter analogous to the thermal expansion coefficient, as described by the equation 20.

In this analysis, the pattern case used in previous models showed instability because both tensile failure and compaction failure. The model did not converge. It was necessary to major the tensile strength by setting the tensile cap parameter R as 2.0. The maximum allowed plastic compressive strain was also enlarged to 0.5.

Figure 21 presents the results for a steady-state analysis considering poroelastic effects, simulating production. It can be seen that production induces radial tensile stresses at the top of the wellbore and compressive stress concentration in the direction normal to the higher in-situ principal stress. The compressive stress concentration at this region associated to tensile failure at the top lead to a large damaged region by volumetric failure.



Radial Stress (MPa) Tangential Stress (MPa) 1

MN

MX

X

Y

Z

-12.187

-10.521-8.855

-7.189-5.523

-3.857-2.191

-.5243781.142

2.808

SEP 8 201015:41:07

NODAL SOLUTION


(a) p=-2,0 MPa

1

MN

MX

X

Y

Z

-81.938

-72.814-63.691

-54.567-45.443

-36.319-27.195

-18.071-8.947

.176476

SEP 8 201015:42:24

NODAL SOLUTION

STEP=1SUB =1TIME=1S3 (AVG)DMX =1.191SMN =-81.938SMX =.176476

(b) p=-2,0 MPa

1

MN

MX

X

Y

Z

-12.687

-11.017-9.347

-7.677-6.006

-4.336-2.666

-.995547.674704

2.345

SEP 8 201015:34:47

NODAL SOLUTION


(c) p=0 MPa

1

MN

MX

X

Y

Z

-80.708

-71.692-62.675

-53.658-44.641

-35.624-26.607

-17.59-8.573

.444038

SEP 8 201015:35:56

NODAL SOLUTION

STEP=1SUB =1TIME=1S3 (AVG)DMX =1.191SMN =-80.708SMX =.444038

(d) p=0 MPa

1

MN

MX

X

Y

Z

-13.12-11.222

-9.323-7.424

-5.525-3.626

-1.728.171208

2.073.969

SEP 8 201015:22:25

NODAL SOLUTION


(e) p=2 MPa

1

MN

MX

X

Y

Z

-79.349

-70.658-61.967

-53.276-44.585

-35.893-27.202

-18.511-9.82

-1.129

SEP 8 201015:23:25

NODAL SOLUTION


(f) p=2MPa

1

MN

MX

X

Y

Z

-13.812

-11.689-9.566

-7.443-5.32

-3.197-1.074

1.0493.172

5.295

SEP 8 201015:52:59

NODAL SOLUTION


(g) p=4 MPa

1

MN

MX

X

Y

Z

-77.89

-69.588-61.285

-52.982-44.679

-36.376-28.074

-19.771-11.468

-3.165

SEP 8 201015:54:34

NODAL SOLUTION


(h) p=4 MPa

Figure 19: Principal stresses for pressure changes in the borehole



Mean stress (MPa) Plastic strain intensity 1

MN

MX

X

Y

Z

-32.032

-28.352-24.671

-20.991-17.31

-13.63-9.949

-6.269-2.588

1.092

SEP 8 201015:43:40

NODAL SOLUTION


(a) p=-2,0 MPa

1

MN

MXX

Y

Z

0

.249E-03.498E-03

.748E-03.997E-03

.001246.001495

.001744.001993

.002243

SEP 8 201015:40:17

NODAL SOLUTION


(b) p=-2,0 MPa

1

MN

MX

X

Y

Z

-32.332

-28.616-24.9

-21.185-17.469

-13.753-10.037

-6.321-2.605

1.11

SEP 8 201015:36:42

NODAL SOLUTION


(c) p=0 MPa

1

MN

MXX

Y

Z

0

.179E-03.357E-03

.536E-03.715E-03

.894E-03.001072

.001251.00143

.001608

SEP 8 201015:33:19

NODAL SOLUTION


(d) p=0 MPa

1

MN

MX

X

Y

Z

-32.61

-28.876-25.143

-21.41-17.677

-13.943-10.21

-6.477-2.744

.989773

SEP 8 201015:24:33

NODAL SOLUTION


(e) p=2 MPa

1

MN

MXX

Y

Z

0

.129E-03.258E-03

.386E-03.515E-03

.644E-03.773E-03

.901E-03.00103

.001159

SEP 8 201015:27:37

NODAL SOLUTION


(f) p=2MPa

1

MN

MX

X

Y

Z

-32.864

-29.148-25.432

-21.716-18

-14.284-10.568

-6.852-3.136

.580204

SEP 8 201015:55:33

NODAL SOLUTION


(g) p=4 MPa

1

MN

MXX

Y

Z

0

.928E-04.186E-03

.278E-03.371E-03

.464E-03.557E-03

.650E-03.742E-03

.835E-03

SEP 8 201015:51:40

NODAL SOLUTION


(h) p=4 MPa

Figure 20: Mean stress and plastic strain intensity pressure changes in the borehole



1

MX

X

Y

Z

-14.479

-12.022-9.564

-7.106-4.648

-2.19.26763

2.7255.183

7.641

OCT 4 201020:17:35

NODAL SOLUTION


(c) Radial stress

1

MN

MX

X

Y

Z

-86.512

-76.296-66.081

-55.865-45.65

-35.434-25.219

-15.003-4.788

5.427

OCT 4 201020:15:58

NODAL SOLUTION


(d) Tangential stress

1

MN

MX

X

Y

Z

-31.409

-27.793-24.178

-20.563-16.948

-13.333-9.717

-6.102-2.487

1.128

SEP 30 201013:01:03

NODAL SOLUTION


(e) Mean stress

1

MN

MX

X

Y

Z

0

.004129.008258

.012388.016517

.020646.024775

.028905.033034

.037163

OCT 4 201020:19:53

NODAL SOLUTION


(f) Plastic strain intensity Figure 21: Results for steady-state production phase

7 CONCLUSIONS

The modeling procedure used was able to capture the volumetric compressive behavior of the rock around the borehole. Unlike shear models, as Drucker-Prager or Mohr-Coulomb, which have its parameters well defined and available data for correlation in literature. Cap models present different formulations. To use it, it is necessary to work directly on experimental data, once different formulations lead to different parameters. This is the major drawback of this model.

The analyses presented herein pointed that the tangential stress concentration is critical condition for volumetric failure in horizontal wellbores. During drilling, this tangential stress concentration is higher under high differential in-situ stresses and lower pressure at the boreface. This suggests that the dominant mechanism is shear enhanced compaction. During production, both differential pore-pressure at the bore face and poroelastic effects increase tangential stress, increasing the plastic zone.

Other wellbore configurations and geometries, like directional wellbores or anisotropic stress states will be evaluated to a better understanding or volumetric failure. The role of rock anisotropy should be also investigated.

Elastic-plastic cap plasticity models are able to define a damaged zone. The size of plastic zone that induce borehole instability is a matter of research. In the field, this mechanism is not completely understood. It is known that plastic compaction disaggregates the material, causing debonding and grain breakage. Whether this material will be carried by fluid flow or it will produce a compacted region that acts as a barrier to flux should be investigated by more refined coupled models. Localization models, multiscale models or FEM-DEM coupling would be helpful in understanding the role of compaction in wellbore instability.



AKNOWLEDGEMENTS

The authors are thankful to Fundação de Amparo à Pesquisa do Rio de Janeiro and Agência Nacional do Petróleo for their partial support to this research.

REFERENCES

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NUMERICAL STUDY ON WELLBORE STABILITY ANALYSIS