1. INTRODUCTION

The petroleum industry drills inclined and/or horizontal wellbores for many purposes. The wellbores are completed using casing and cement to maintain wellbore integrity. Moreover, these completed wellbores often experience severe conditions that threaten the integrity of the wellbore: eccentric casing; large changes in wellbore pressure; large changes in wellbore temperature and so on. These conditions often intensify near wellbore stress states, which, in turn, influence wellbore stability, sand production, zonal isolation, and hydraulic fracturing. Most oil companies spend significant amount of money to prevent the loss of wellbore integrity. Therefore, proper modeling of the stress states around wellbores is an important step in evaluating the effectiveness of a drilling/completion strategy.

For modeling stress states around wellbores, this work mainly focuses on two objectives: (1) develop comprehensive analytical and numerical models for an inclined cased wellbore; (2) apply the developed models 1 A commercial finite element analysis software program made by Dassault Systèmes S.A.

to study causes of the cement sheath failure. The following section introduces some background information relevant to the objectives.

1.1. Developing Models for an Inclined Cased Wellbore

Generally, several processes are required to model the state of stress of an inclined cased wellbore. Three main processes are considered as shown in Fig. 1: Phase 2, 3 and 4.

In this figure, Phase 1 shows the natural state of rock formation under in situ stresses before drilling.

Phase 2 illustrates the process of drilling an inclined wellbore. Drilling itself causes several complicated processes. However, these can be simplified into two basic processes: plain excavation and wellbore pressure of the drilling mud. Bradley calculated the stress distribution around an inclined open wellbore through this simplification along with the assumption of linear elasticity of the formation, and applied them to wellbore failure [1]. However, this elastic model had crucial limitations. It ignored the coupling between the rock matrix and the fluid contained inside. Moreover, the

ARMA 10-142

Mechanical Behavior of Concentric Casing, Cement, and Formation Using Analytical and Numerical Methods

Jo, H. BJ Services Company, Houston, Texas, USA

Gray, K.E. The University of Texas at Austin, Austin, Texas, USA

Copyright 2010 ARMA, American Rock Mechanics Association

This paper was prepared for presentation at the 44th US Rock Mechanics Symposium and 5th U.S.-Canada Rock Mechanics Symposium, held in Salt Lake City, UT June 27–30, 2010.

This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review of the paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented.

ABSTRACT: The primary goal of this research is to develop comprehensive analytical and numerical models for stress distributions around an inclined, cased wellbore by considering all wellbore processes, including amendments to models of other works. Most previous research has focused on individual wellbore processes rather than a comprehensive treatment which considers all wellbore processes simultaneously: a more realistic approach developed in this paper. To achieve this goal, this work utilizes an elastic approach by coupling a poroelastic, undrained condition and a steady state condition for stresses induced by wellbore temperature variations. The superposition principle is used to develop a comprehensive model, which is then applied to cement sheath failure. ABAQUS1 is utilized for numerical solutions to verify the comprehensive analytical model. These comprehensive models show analogous stress distribution results to those of previous models at each individual wellbore process when using the plane strain condition. However, ABAQUS model results show stress differences because the general plane strain model for the analytical solution and full 3D model for numerical solution are used. While there are differences between analytical and numerical solutions as noted, the comprehensive analytical model is a good alternate to costly numerical software programs, and it provides an improvement to plane strain models.

elastic model was not a 3D stress model because it used a plane strain condition. To consider the coupling effect of an inclined open wellbore, Cui et al. used Biot's fully coupled poroelastic theory and Detournay et al.'s poroelastic plane strain model. They developed a poroelastic stress distribution for an inclined open wellbore and applied it for wellbore failure [2, 3, 4, 5, 6, 7]. Nevertheless, this poroelastic model utilized the plane strain condition, and its solution was expressed in the Laplace-transform domain. To apply an inverse Laplace transform for the solution, the poroelastic model used a numerical inverse Laplace transform algorithm [8], which requires long convergent times. Interesting results of the poroelastic model can be summarized as follows: the elastic model is more conservative than the poroelastic model, and the results of the poroelastic model approach the elastic model as time approaches infinity [9].

Phase 3 depicts the well completion process involving the placement of casing and cement. The mechanism of how casing and/or cementing affects the stress distribution of an inclined cased wellbore is complicated and mostly unknown. The results of a few previous studies are applicable to general cases [10]. Chenevert et al. [11] made a semi-analytical model for the stress distribution induced by cement shrinkage and compared its results with experimental data by Cooke et al. [12]. Prohaska et al. developed a more intensive and reliable model than the previous work [13]. Recently, Zhou et al. collected the previous data and made a comprehensive model [14]. However, these analytical models are not widely applicable since they are based on limited experimental results. Thus, no generally applicable analytical models for stress distribution induced by

casing and/or cementing are available. To compensate for limitations of these analytical models, Gray et al. developed a reliable numerical model by using ABAQUS [10]. However, this present work ignores the effect of the cement shrinkage and assumes that the cement is a hydraulic column before it is set.

Phase 4 represents tectonic stress variation and/or wellbore temperature variation after a well is completed. There exist few analytical and/or numerical models that address those factors listed in Phase 4. Li developed a simple analytical model, which considered a well with only casing [15], for the stress distribution induced by tectonic stress variation after well completion. Atkinson et al. developed a more extended analytical model which considered a well with both casing and cement. However, the researchers mistook tectonic stress variations as the in situ stresses [16]. In addition, Carter et al. developed a numerical model [17] and recently, Morita et al. performed experiments to develop a more realistic analytical model [18]. For the stress distribution induced by temperature variation, Timoshenko et al. introduced a general solution [19]. Moreover, Ghosn derived the solution for the coupling between rock matrix and temperature [20] and Shahri tried to solve the stress distribution around an inclined cased wellbore under high temperature and high pressure using ANSYS2 [21].

Although separate models treating each phase in Fig. 1 have been developed, the comprehensive and combined effects of all the phases on the stress distribution around an inclined cased wellbore have not been intensively considered. Moreover, some previous work applied the 2 A commercial finite element analysis software program made by ANSYS, Inc.

Fig. 1. Processes of drilling and well completion, and related previous studies

incorrect boundary conditions, as mentioned above, for an inclined cased wellbore and, consequently, developed erroneous models. In order to address these errors, the present work develops an analytical and numerical model that considers the comprehensive and combined effects of all three phases. In addition, most previous analytical models used the plane strain condition to transform a 3D problem, which is non-solvable analytically, into a 2D problem, which is solvable analytically. Even though the plane strain condition (i.e. axial strains along a wellbore are assumed to be zero) can be applicable to most wellbores, non-zero axial strains occur along real wellbores. To compensate for this limitation, this research uses the generalized plane strain condition developed by Duncan et al. [22] and Zhenye et al. [23].

While only numerical results were available previously, this research develops an analytical model for the stress distribution induced by wellbore temperature variation. Although the analytical model assumes a steady state condition, it is economically beneficial because it does not require costly numerical programs.

1.2. Cement Failure Failure theories are constantly being developed. They can be categorized into two main areas: tensile failure and shear failure. Specifically, several criteria for shear failure have been proposed, while tensile failure has a simpler mechanism. Among them, Mohr-Coulomb criterion and Drucker-Prager criterion [24] are most commonly used. However, the former underestimates material strength by ignoring the effect of the intermediate stress, whereas the latter overestimates material strength [25]. To resolve this problem, this research uses Mogi-Coulomb criterion [25].

Handin [26] and Goodwin et al. [27] performed experiments to investigate cement failure under high pressure and/or high temperature environments. Moreover, Thiercelin et al. [28] and Bosma et al. [29] utilized numerical models to investigate cement failure and to design a cement sheath under high pressure and/or high temperature environments. These results were based on experiments or numerical programs which require significant time and expense; alternative models are needed to save time and expense. This research applies the developed analytical model for the cement failure under high pressure and/or high temperature.

2. DEVELOPING MODELS

2.1. Fully Coupled Poroelasticity This research distinguishes between partially coupled and fully coupled poroelasticity. Partially coupled

poroelasticity ignores the interaction between solid (solid stress) and fluid (pore pressure). However, fully coupled poroelasticity considers that interaction. This concept was first formulated by Biot [30] and developed by several others. Since rock is a porous medium, a fully coupled poroelastic approach is required. Nevertheless, the complexity of poroelastic solutions caused by the coupling between solid and fluid makes elastic approaches preferable.

This research applied the theory of fully coupled poroelasticity to an inclined cased wellbore which is a combined system of elastic and porous materials. In addition, it showed that a poroelastic, inclined cased wellbore system under undrained conditions became an elastic one with undrained properties instead of normal elastic properties, if it is assumed that the wellbore is perfectly cased and cemented and does not allow any leaking fluid from the rock formation. The complete derivation is shown in the dissertation by Jo [31].

2.2. Governing Equations Several governing equations were required to analyze an inclined cased wellbore system since the system had solid-fluid coupling and solid-temperature coupling. These governing equations and couplings are schematically represented in Fig. 2. Horizontal arrows represent coupling directions. Upper boxes describe governing equations for each coupling. Lower boxes show the governing equations for each area which have no direct relation with the couplings. If these equations are solved simultaneously under certain boundary conditions for an inclined cased wellbore system, the stress distribution of the system can be obtained and the mechanical behavior of casing, cement and rock can be analyzed and predicted. These analyses and predictions are the desired goal of this work.

However, there were technical problems in analytically solving these equations under general conditions. Numerical techniques or programs such as ABAQUS, DIANA3, ANSYS, I-DEAS4, etc. may be used to solve them. Thus, some special but reasonable boundary conditions were used in order to solve the governing equations simultaneously and analytically. These boundary conditions decouple the set of equations and make them analytically solvable.

To attain the goal, the undrained poroelastic condition and steady state wellbore temperature variation were introduced. The undrained poroelastic condition decouples solid-fluid coupling as stated above. Also, the steady state wellbore temperature variation decouples solid-temperature coupling. For this reason, if a steady

3 A commercial finite element analysis software program made by TNO DIANA 4 A commercial integrated design and engineering analysis software made by Siemens PLM Software

state is assumed, all time variation dependent terms disappear. Thus, only one term remains in the generalized Fourier's law [20]. This equation can be generally solved in terms of only spatial variables; for example r, θ and z. The solution can be substituted into the Duhamel-Neuman relation, which represents the solid-temperature coupling. The relation is reduced to an elastic equation which is solvable with ease. Therefore, the system becomes decoupled and analytically solvable.

3. STRESS DISTRIBUTION AROUND AN INCLINED CASED WELLBORE

Based on the results of the previous section, this research used elastic approaches to analyze an inclined cased wellbore. However, this elastic model is distinct from previous models [16, 22] because this research developed an analytical, elastic model including generalized plane strain, steady state temperature effect on the stress distribution, and correction for the erroneous modeling procedure of previous research [16].

To calculate the stress distribution induced after drilling, casing and cementing, some assumptions are required as follows.

3.1. Assumptions The following assumptions are required for developing a comprehensive model:

(i) The coordinate system used in this research is a local borehole coordinate system. Fig. 3 and 4 show the details.

(ii) For simplicity, the directions of in situ principal stresses coincide with the global coordinates.

(iii) The casing, cement and borehole are concentric circles.

(iv) The casing, cement and borehole are assumed to be perfectly bonded to each other at each interface. The perfectly bonding mathematically means that the continuity of stresses and displacements is satisfied at each interface.

(v) Casing and cement were homogeneous, isotropic, linearly elastic materials and rock was a homogeneous, isotropic, linearly poroelastic material.

(vi) The rock formation had an infinite boundary and it was a porous medium which was 100% saturated with one phase pseudo fluid.

(vii) Casing and cement were assumed to be nonporous. (viii) Drilling was assumed to happen instantaneously.

Since enough time (3 ~ 5 hours) had passed between drilling and well completion, poroelastic effects of pore pressure induced by drilling on the stress distribution could be ignored [3, 9].

(ix) The effects of invasion and cement shrinkage were negligible. Since reasonable analytical models for the cement shrinkage were not available, this research assumed that cement slurry was a hydraulic column before it was set.

(x) The poroelastic undrained condition was applied after well completion.

(xi) Generalized plane strain condition was used, which has many versions [23]. However, this research used the constant axial strain-generalized plane strain condition [23].

Fig. 2. Governing equations for an inclined cased wellbore system

Fig. 3. Global coordinate system and in situ principal stresses

(xii) The stresses induced by wellbore temperature variation were assumed to be steady state, i.e. not a function of time.

(xiii) The coupling between temperature and fluid contained inside of rock was ignored because perfectly bonded casing and cement at each interface sealed the wellbore, in other words, there was no fluid behavior.

(xiv) No history of previous phases affected any phases. This assumption becomes reasonable when a system has small displacements (or deformations) induced by each phase. Since the system of an inclined cased wellbore can have small deformations by each wellbore process, this assumption can be applicable to this system. This assumption is required because the superposition principle is valid under it.

In this paper, the sign convention for stress is positive in compression following the convention used in rock mechanics.

3.2. Methodology This research investigated the process of drilling and well completion and found that most previous models improperly analyzed the stress distribution for an inclined wellbore. For example, while previous work imposed in situ stresses after well completion, this research imposed in situ stresses before drilling. Since real rock formations are under in situ stresses before drilling, this model simulates the system more realistically by considering the effects of drilling in the presence of in situ stresses. Fig. 1 shows the process of drilling and well completion.

Fig. 4. In situ principal stresses and wellbore coordinate system [(x, y, z) = (xw, yw, zw)]

At Phase 1, the rock formation is under in situ principal stresses (or in situ stresses). It was assumed that the magnitude and direction of in situ stresses were known. Although methods of measuring in situ stresses have been reported, accurate measurement of the in situ stresses remains a challenging area in geoscience.

At Phase 2, a generalized plane strain condition is used. All detailed procedures may be found in the dissertation of Jo [31].

At Phase 3, the influence of casing and cementing, such as cement shrinkage on the stress distribution, should have been considered. However, this research ignored these contributions because they were considered second order effects compared to primary factors, such as changes in wellbore temperature and pressure; most results are applicable only under special conditions; and reliable models were not available [14]. The study on the influence of cement shrinkage is a challenging geomechanical research area [10].

At Phase 4, the stress distribution caused by tectonic stress variation was derived by modifying Li's model [15]. Details of the model are explained in Jo’s dissertation as well as the stress distribution induced by steady state temperature variation [31].

Through phases 1 to 4, the stress distribution induced by each factor was obtained. To combine each stress distribution caused by each factor, it was assumed that the system was linear elastic and the history of all phases was independent of each other. That is, the principle of linear superposition for the stress was applied. Considering that most geological materials exhibit non-linear behavior, this superposition principle may not always be appropriate. However, the superposition is useful and reasonable in a practical respect. The

superposition can be simply mathematically represented as follows.

disturbedinitialinduced (1)

Therefore, σinitial for phase 2 is in situ stress and Δσdisturbed for phase 2 is stress disturbed by drilling. σinduced for phase 2 is second order tensor summation according to Eq. (1). For phase 3, σinitial becomes σinduced for phase 2 and Δσdisturbed is stress disturbed by casing and cementing. By applying this sequential process, the stress distribution of an inclined cased wellbore at subsequent phases can be obtained.

4. APPLICATIONS (CEMENT FAILURE)

The comprehensive analytical model developed through the above processes is applicable to many wellbore phenomena. For illustration purposes, cement failure is considered here. While many factors are involved, this example illustrates the model’s utility.

Generally speaking, there are two kinds of failures: tensile failure and shear failure. For cement, tensile failure occurs when a stress on the cement exceeds the tensile strength of the cement. Similarly, shear failure happens when a stress surpasses the shear strength of the cement. However, since cement behaves differently under tension and compression, separate failure criteria are needed for each type of failure.

4.1. Tensile Failure The tensile failure is referred to as Mode I fracture (opening fracture). In addition, the tensile strength of cement is about ten times less than the compressive strength [32].

A tensile failure criterion can be described as follows:

fop TP (2)

where σp is the least principal stress, Po is the pore pressure in the cement and Tf is the tensile strength of cement [1, 33].

It is assumed that that Po=0 because the cement is assumed to be non-porous. Thus,

0 fp T (3)

If a location in the cement area satisfies Eq. (3), tensile failure occurs at that location.

Using this tensile failure criterion, this research investigates the influence of Young's modulus of the cement, the temperature of the wellbore and wellbore pressure. Both analytical and numerical models (ABAQUS models) are used to obtain the results.

4.2. Shear Failure The shear failure has been studied by many researchers and has many criteria. Among them, this work uses the Mogi-Coulomb criterion, which was experimentally verified for describing rock failure [25].

While the Mohr-Coulomb criterion consists of the shear stress (τ) and the normal stress (σn), and Drucker-Prager criterion uses the octahedral shear stress (τoct) and the octahedral normal stress (σoct), the Mogi-Coulomb criterion was developed in terms of the octahedral shear stress (τoct) and mean stress (σm,2) calculated as follows.

2132

322

213

1 oct (4)

2

312,

m (5)

In addition, Fig. 5 depicts the Mogi-Coulomb failure envelope. The envelope can be expressed mathematically as follows.

2,moct ba (6)

where a and b are Mogi strength parameters. Generally, they are obtained experimentally and given for failure analysis.

This research investigates the influence of the same factors as those of the tensile failure case on the cement shear failure.

4.3. Assumption The following assumptions are required for the failure criteria to be applied to the system properly:

(i) Cement is assumed to be homogeneous, linear, isotropic, and elastic. Although this assumption is simple and has some limitations when applied to real situations, it is practical and appropriate for many cases.

(ii) The failure of cement experiences no plastic behavior.

Fig. 5. Mogi-Coulomb failure envelope [25]

(iii) A linear failure envelope is used for Mogi-Coulomb failure. Higher order envelopes are not used because of their complexity.

(iv) Only failure in the cement is assumed to occur in order to ignore the coupling among failures of casing, cement and rock formation.

(v) Each factor affecting cement failure acts independently of all of the others.

4.4. Methodology Based on these assumptions, the following steps are applied to investigate the tensile and shear failure of cement:

(i) Calculate stresses in the cement in the area of interest. The stresses were calculated by using the developed analytical and ABAQUS models.

(ii) Calculate the least principal stress at every location in the cement area.

(iii) Investigate whether tensile failure in the cement has occurred by checking whether or not Eq. (3) is satisfied.

(iv) Calculate τoct and σm,2 in the cement area of interest. (v) Plot τoct versus σm,2. (vi) Draw a linear failure envelope for shear failure. (vii) Investigate whether shear failure of the cement has

occurred. (viii) Perform steps (i) through step (vii) to study the

influence of each of the following parameters: (a) Young's modulus of the cement

(b) Wellbore temperature variation

(c) Wellbore pressure

The tensile failure was investigated using steps (ii) to step (iii). The shear failure was studied using steps (iv) to step (vii).

5. RESULTS

Table 1 summarizes the input data. For the analytical analyses, MATLAB 5 is used for the analytical model whereas ABAQUS is utilized for the numerical analyses.

5.1. The Effect of Young’s modulus of the Cement on Cement Failure

For Young's modulus of the cement, this research considers the coupling between the tensile and compressive strength of the cement and Young's modulus of the cement. Thus, this study uses Lacy's empirical correlations between the unconfined compressive strength (UCS) and static Young's modulus as follows [34]:

5 A commercial software for technical computing developed by The MathWorksTM

)(485.22787.0)( 2 ksiEECUCS sso (7)

where Es is static Young's modulus (106 psi).

Even if Lacy's correlation was largely based on weaker sands, it is useful to this study. The tensile strength of cement (Tf) is assumed to one tenth of UCS [32]. In addition, Mogi strength parameters can be expressed as follows [25]:

13

22

q

Ca o (8)

1

1

3

22

q

qb (9)

where

sin1

sin1

q and is the angle of internal

friction of cement, which is normally assumed to be 30o.

For tensile failure, Fig. 6 describes the minimum value of σp+Tf in the cement for a given Young's modulus. Hereafter, large solid dots represent analytical results and small hollow dots represent numerical results. The location where tensile failure initially occurs is not considered. Only the value related to the initial tensile failure is considered. This diagram shows that the minimum value of σp+Tf declines and changes from a positive to a negative value as Young's modulus increases. This change in sign indicates that an initial tensile failure occurs in the cement. In other words, both the analytical and ABAQUS calculated results show that the chance of the cement tensile failure increases as the Young's modulus increases. In addition, σp+Tf values for the analytical solution showed a linear relationship with Young's modulus of the cement, while σp+Tf for the ABAQUS solution showed a quadratic relationship. The specific reason for this result is not understood. However, the difference between the analytical and ABAQUS models may cause this phenomenon: generalized plane strain model versus full 3D model.

Fig. 6. The effect of Young's modulus of cement on tensile cement failure

For shear failure, Mogi-Coulomb shear failure criterion is utilized [25]. In other words, if both analytical and numerical results are plotted in (σm,2, τoct) space and the plotted points lie on or over the given failure envelope, then this indicates that shear cement failure occurs at the interface between casing and cement. Fig. 7 shows the influence of Young's modulus on shear failure at the interface between casing and cement. Since the tensile and compressive strength of the cement and Young's modulus of the cement are coupled, the failure envelope changes whenever Young's modulus of the cement changes. Both analytical and numerical results show that as Young's modulus increases, the results move away from the corresponding failure envelopes. Therefore, as Young's modulus increases, the chance of shear failure decreases, while the chance of the tensile failure increases. These interesting phenomena may not be general because the calculations are specific to Lacy's empirical correlations. This observation can be

explained through following equations: Eq. (10) and (11). The slope of Tf/E2 is smaller than the slope of a/E2.

20657.0

13

22s

o Eq

Ca

(10)

202787.0

10 so

f EC

T (11)

However, even though the relationship between tensile and shear failure and Young’s modulus of cement depends on the correlations between the modulus and the strength of the cement, the results of this work can provide some good guidelines for the selection of cement to avoid the cement failure after the cement is set. In addition, the analytical results corresponding to each Young's modulus of cement are closer to the failure envelope than the numerical results. This means that the analytical results reach the shear failure faster than the numerical results.

Table 1. Input data for cement failure

Casing inner radius (R1) 0.1778 m (7 in) Casing outer radius (R2) 0.1905 m (7.5 in) Wellbore radius (R3) 0.2159 m (8.5 in) Young's modulus of casing (E1) 206.8 GPa (30 MMpsi) Young's modulus of cement (E2) 13.8 GPa (2 MMpsi) Young's modulus of rock (E3) 10.3 GPa (1.5 MMpsi) Poisson's modulus of casing (ν1) 0.3 Poisson's modulus of cement (ν2) 0.2 Poisson's modulus of rock (ν3) 0.2 Depth 1828.8 m (6000 ft) Wellbore pressure (Pw) 34.5 MPa (5000 psi) Azimuth 0o Inclination 0 o Reservoir pressure 19 MPa (2760 psi, or 0.46 psi/ft) Max. horizontal in situ stress (σH) 31 MPa (4500 psi , or 0.75 psi/ft) Min. horizontal in situ stress (σh) 26.9 MPa (3900 psi , or 0.65 psi/ft) Vertical in situ stress (σv) 41.4 MPa (6000 psi , or 1 psi/ft) Changed Max. horizontal in situ stress (σH2) 36.2 MPa (5250 psi ) Changed Min. horizontal in situ stress (σh2) 31.4 MPa (4550 psi) Changed Vertical in situ stress (σv2) 48.3 MPa (7000 psi) Thermal conductivity of casing (K1) 52W/mK (0.100 Btu(IT) inch/sec/ft2/ oF) Thermal conductivity of cement (K2) 0.9W/mK (0.00173 Btu(IT) inch/sec/ft2/ oF) Thermal conductivity of rock (K3) 1.6W/mK (0.00308 Btu(IT)inch/sec/ft2/ oF) Linear thermal expansion coefficient of casing (αT1) 0.000012 1/ K (0.00000667 1/ oR) Linear thermal expansion coefficient of cement (αT2) 0.0000144 1/ K (0.000008 1/ oR) Linear thermal expansion coefficient of rock (αT3) 0.0000125 1/ K (0.00000694 1/ oR) Temperature at wellbore (T∞,1) 90 oC (194 oF) Temperature at reservoir (T∞,4) 75 oC (167 oF) Mogi strength parameter (a) 9.8 MPa (1421.5 psi) Mogi strength parameter (b) 0.47 Cement density 1889.6 kg/m3 (15.77 ppg) Unconfined Compressive Strength (UCS) of cement 41.6 MPa (6030.8 psi) Tensile strength of cement (Tf ) 4.16 MPa (603.08 psi) Interesting area for cement failure R2

Fig. 8. The effect of wellbore temperature on tensile cement failure

5.2. The Effect of the Wellbore Temperature Variation on Cement Failure

Fig. 8 compares analytical tensile results caused by wellbore temperature variation with those of numerical models. As wellbore temperature increases, the possibility of tensile cement failure decreases. The interesting part is that analytical results show a linear relationship between wellbore temperature and the minimum value of σp+Tf, but numerical results show a logarithmic relationship. Even though further research has not yet been performed, this discrepancy may be

caused by the difference of the two models in dealing with stresses caused by temperature variation.

Fig. 9 shows the effects of wellbore temperature on shear cement failure. The chance of shear cement failure decreases as wellbore temperature increases for both analytical and numerical results. However, the numerical results have a smaller slope than the analytical results and show different values from those of the analytical results. Similar to the previous subsection, the analytical results for tensile and shear failure reach failure faster than the numerical results.

5.3. The Effect of the Wellbore Pressure on Cement Failure

Fig. 10 describes the effects of wellbore pressure on tensile cement failure. Contrary to the previous subsections, the analytical and numerical results show very similar behavior. Also, as the wellbore pressure increases, the chance of tensile cement failure increases. Moreover, the minimum of σp+Tf has a linear relationship with wellbore pressure. This linearity may be the result of attributing elastic properties to the cement.

Fig. 11 characterizes the aftermath of wellbore pressure on shear failure. Like the tensile failure, the analytical and numerical results show analogous results except for Pw = 0. Similar to the previous subsections, the analytical results reach failure faster than the numerical results. Similar to the tensile failure, the shear failure shows the linearity with respect to wellbore pressure.

Fig. 7. The effect of Young's modulus of cement on shear cement failure

Fig. 10. The effect of wellbore pressure on tensile cement failure

6. CONCLUSION AND FUTURE WORK

6.1. Analytical Model This research shows results in accord with the results of the previous research [16, 17] when taking the plane strain condition. Fig. 12 shows the comparison among them. The bottom of the figure shows input conditions for the comparison. is the angle from the direction of σxx. The values of σθθ at each interface for the each research are compared.

For the comprehensive models, even if further research on the effects of other wellbore conditions on cement

failure is required, the analytical model shows similar results to the numerical model and can be a good tool to quickly obtain the stress distribution without needing numerical simulators. Moreover, this analytical model can be extended to multiple, concentric, inclined cased wellbores thanks to the simple superposing principles. In addition, this analytical model can be summarized as having following characteristics:

(i) Uniqueness (a) Fully coupled poroelasticity is used.

(b) Solutions using simple superposition principles are used. This superposition extends an inclined cased wellbore model to multiple, concentric, inclined cased wellbores.

(c) Generalized plane strain is utilized.

(d) First analytical model considering all drilling and well completion processes for an inclined cased wellbore, including tectonic stress variation and temperature variation is developed.

(ii) Limitation

(a) Plasticity is not considered.

(b) Solution is not full 3D but pseudo 3D (generalized plane strain condition).

Fig. 9. The effect of wellbore temperature on shear cement failure

(c) The coupling between temperature and fluid contained in rock is ignored.

Fig. 12. Comparison of the results of the previous research and this study 6.2. Cement Failure First, as Young's modulus increases, the chance of shear failure decreases while the chance of tensile failure increases. These interesting phenomena are caused by the difference of slopes of a/E2 and Tf/E2. The

difference of slopes among them is due to the coupling among the tensile and compressive strength of the cement, and Young's modulus of the cement.

Second, the analytical and numerical models induced by wellbore temperature variation show distinct results for tensile and shear failure. The disagreement might be caused by the different modeling technique. That is, the analytical models are based on the generalized plane strain which is pseudo 3D, whereas the numerical models use a finite box model which is full 3D.

Third, the effects of wellbore pressure show very similar behavior between analytical and numerical results. Moreover, both analytical and numerical models show almost identical linear relationships between the tensile and shear failure and wellbore pressure, which is caused by the use of the linear elastic approximation.

Consequently, the following comprehensive conclusions can be made. First, the analytical models are more conservative than the numerical models, observed by the result that in every case, the analytical results always reach failure faster than the numerical results. That is, the analytical models may be more preferable to the numerical models in order to safely design cements for an inclined cased wellbore system. In addition, analytical models are faster and require less expense than numerical models. Second, this research can introduce a comprehensive map to determine which factor and to what degree will significantly affect the

Fig. 11. The effect of wellbore pressure on shear cement failure

tensile and shear cement failure, and provide the proper guidelines for the selection of cements.

6.3. Future Research Although these analytical and numerical models give beneficial and helpful results and insights into understanding mechanical behavior of concentric casing, cement, and rock formation systems, these models are based on elastic behavior. However, real casing, cement and rock behave elastically and plastically. That is, they are non-linear materials. Thus, further studies considering the plasticity of materials are needed. Also, the behavior of some materials depends on a time history: visco-elastic or visco-plastic materials. This time-dependent behavior should be considered, too.

In addition, this research ignores the influence of well completion, such as the effects of cement shrinkage on cement failure, because reasonable and generally applicable experimental data are lacking. However, the initial state of stress of the cement at set may be critical in predicting cement integrity. Therefore, further studies on well completion processes are needed.

Finally, these analytical models ignore the coupling between temperature and fluid contained in rock because of its complexity. Developing models that consider the coupling is needed.

ACKNOWLEDGEMENT

Ametek/Chandler Engineering, BJ Services, Chevron, ExxonMobil and Schlumberger are acknowledged for sponsoring the Life-Of-Well Rock, Fluid, and Stress Systems Research Program which funded this research.

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