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International Journal of Greenhouse Gas Control

journal homepage: www.elsevier.com/locate/ijggc

Effect of thermal stress on wellbore integrity during CO2 injection

Pratanu Roy⁎, Joseph P. Morris, Stuart D.C. Walsh, Jaisree Iyer, Susan CarrollLawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94551, USA

A R T I C L E I N F O

Keywords:Wellbore integrityCO2 injectionThermal stressStress intensity factorCarbon captureCarbon storageCement sheath failure

A B S T R A C T

Wellbore integrity is a critical component of long-term carbon storage. Depleted reservoirs that are potential CO2

storage sites, typically contain several wells. Due to years of operations and abandonment, these wells can havecracks in the cement, cement-casing interface, and/or cement-formation interface. During CO2 injection, changesin temperature may result in stress variations that can further damage the well, threatening its integrity. Thetemperature difference between the cold injected CO2 and warm reservoir, and different thermal properties forthe wellbore casing, cement, and the lithology, will stress the near wellbore environment, potentially extendingpre-existing defects creating leakage pathways from the storage reservoir to the overlying strata. We haveconducted a systematic numerical study to explore the role of CO2 injection temperature, downhole effective in-situ horizontal stress, and the thermo-mechanical properties by coupling a linear elastic stress model with heatconduction. We consider conditions in non-perforated casing above the injection zone where conductive heatingis dominant. The injection temperatures considered covers current industrial practice as well as sub-zero tem-peratures. The latter represents direct injection following ship transport, without pre-heating.

In this study, we consider the connection between damage risk and the temperature difference between theinjected CO2 and the formation and downhole effective in-situ horizontal stress. The study found that the ne-gative impacts of thermal stress in the wellbore environment are mitigated by the presence of effective in-situhorizontal stresses. Stresses normal to the well have the potential to reduce the tensile stress and stress intensityfactor. In the absence of sufficient effective in-situ horizontal stress, thermal stress may cause the fracture topropagate due to high stress concentration near the fracture edges. In general, formations with large effective in-situ horizontal stress can prevent leakage paths from growing even when large temperature difference existsbetween the formation and the injected CO2. Our simulations suggest that CO2 can be injected at sub-zerotemperatures, suitable for ship transport, when the downhole effective in-situ horizontal stress is greater than 10or 12MPa, depending on the location of the pre-existing cracks. For onshore transportation, injection of liquidCO2 results in minimal damage, provided there is ample in-situ horizontal stress.

1. Introduction

Carbon Capture and Storage (CCS) is a key mitigation strategy tohelp meet international greenhouse-gas reduction targets (Haszeldine,2009; Herzog and Golomb, 2004). An effective method of long-termstorage of CO2 is to inject it into underground saline or depleted oil andgas reservoirs through wells. However, without proper care, injectedCO2 may find leakage pathways out of the reservoir, affecting thesurrounding environment and ultimately re-emerging into the atmo-sphere. Preventing leakages from wells is crucial to establish long-termstorage of CO2 (Viswanathan et al., 2008; Carroll et al., 2014).

Maintaining long-term well integrity is challenging due to thecomplex geochemical and geomechanical interactions between sub-surface fluids, wells, and reservoir materials, combined with the

elevated temperatures and pressure found at depth (Zhang and Bachu,2011; Carey, 2013; Carroll et al., 2016). Well barrier materials, thecasing and cement, provide mechanical stability and hydraulic sealingthat prevent fluid migration. Damage to well barrier materials can re-sult in a loss of well integrity, allowing leaks to occur.

Thermal loading, especially repeated thermal loading, of wells is apotential source of failure in well barrier materials. In particular, CO2

injection wells are often subjected to large temperature variationsarising from differences between the injected fluid temperature andthat of the surrounding rock. The CO2 temperature may vary between15–25 °C for onshore pipeline transport, and 6–7 °C for offshore pipelinetransport (Lindeberg, 2011). Ship transportation offers an alternativeway of transporting CO2 to an offshore storage, especially from sourcesin the Nordic region. On the ship, CO2 is stored in a liquid state at a very

https://doi.org/10.1016/j.ijggc.2018.07.012Received 6 October 2017; Received in revised form 22 June 2018; Accepted 7 July 2018

⁎ Corresponding author.E-mail address: roy23@llnl.gov (P. Roy).

International Journal of Greenhouse Gas Control 77 (2018) 14–26

1750-5836/ © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

T

low temperature and high pressure (−50 °C and 0.7MPa) (Aursandet al., 2017)1. Before the injection, the CO2 is usually heated to atemperature above sea temperature (∼4.5 °C). These CO2 tempera-tures, typical for different transportation options, are much lower thanthe formation temperature. The temperatures of CO2 storage reservoirsin United States, Canada, and Algeria at different depths (1–3.2 km) canvary between 50-200 °C (Karamalidis et al., 2012). Consequently, in-jection of much colder CO2 causes the temperature of the near-wellboreregion to fall. During breaks in injection due to shut-downs or wellservicing, heat from the host rock will cause the wellbore temperatureto rise again. These perturbations in temperature induce stress in thewellbore materials, which in turn reduce the critical pressure needed toinitiate fractures (Luo and Bryant, 2011).

Experimental studies have attempted to replicate thermal effects byapplying mechanical loading, such as metal rings (Boukhelifa et al.,2004), hydraulic presses (Teodoriu et al., 2013) and pressure-cyclingcells (Shadravan et al., 2015). Radial cracking perpendicular to theapplied stress and disking failures with increased number of cycles wereobserved. One obvious limitation of these studies is the lack of a cor-relation between the mechanical loading in the experiment and thecorresponding temperature perturbation required to create the ob-served damage. Furthermore, the thermal properties of the materialsand the interfaces, which may play an important role in variation ofthermal stresses, are not considered.

Instead of mimicking the thermal effects by mechanical loading,several researchers used thermal platforms to study the effects ofthermal fluctuations in a more controlled manner. Carpenter et al.(1992) used a thermal shock apparatus to evaluate the cement bondstrength at elevated temperature and pressure. Their thermal shocktesting showed that sudden temperature reductions in casing can de-crease the cement bond strength by as much as 69%. Recent experi-ments by Albawi et al. (2014) and De Andrade et al. (2014) studiedthermally induced stresses using a thermal platform. They appliedcyclic thermal loading on a downsized wellbore sample to study howannular cement integrity varies as a function of time and temperature.Their results showed that the pre-existing cracks in the samples canextend during thermal cycling operations. Experiments conducted byTodorovic et al. (2016) on similar downscaled well section revealedthat extreme cooling can create radial fractures upon freezing the porewater. It was also found that the thermal and mechanical properties ofcement defects are crucial for accurately predicting the effect ofthermal cycling (Torsæter et al., 2017). For example, damage in-troduced by the defects in cement can be minimized by using a lowerstiffness and/or high thermal conductivity cement. Gomez et al. (2017)used an experimental wellbore mock up to investigate the effect ofconfining stresses on microannuli between the cement sheath and steelcasing. Using two microannuli joints with initial apertures of 136 μmand 19 μm, they found that the hydraulic aperture decreases mono-tonically and non-linearly with the increase in confining stress.

In addition to the experimental work, the role of thermal stressesduring CO2 injection has been studied numerically. Thiercelin et al.(1998) used a linear elastic model to assess the mechanical response ofset cement in a cased wellbore subjected to thermal perturbations. Theyconcluded that cement failure and debonding can be avoided bychoosing appropriate thermo-elastic properties of wellbore materials,which play a significant role in determining the stress state. Luo andBryant (2011) described a simplified model for heat transfer in thewellbore, showing that thermal stress is high for commercial scale in-jection and hence should not be ignored. Their analysis also showedthat due to the difference in mechanical properties, the thermo-elasticstress in a shale caprock can be twice as high as that in sandstone

storage aquifer. Preisig and Prévost (2011) used a coupled multi-phasemodel to study the thermo-poromechanical effects of CO2 injection at InSalah, Algeria. Their study revealed that the temperature differencebetween the injected CO2 and the reservoir can reduce the compressivestresses and even create tensile stresses due to the inability of the rockto contract freely, leading to creation or re-opening of fractures. Usingthe geomechanical properties and in-situ stresses of Krechaba field (InSalah, Algeria), Gor and Prévost (2013) showed that when CO2 is in-jected at 90 °C (the reservoir temperature), the stresses remain com-pressive preventing wellbore failure. However, when the injectiontemperature is 40-50 °C, the stresses inside the caprock near the injec-tion well may become tensile and even overcome the tensile strengthleading to failure of the caprock within 10 years. Asamoto et al. (2013)performed a numerical case study on the possibility of cement crackingand microannuli creation at the well material interfaces by using aclassical thermo-mechanical finite element model. They found that therisk of compressive failure and tensile cracking inside the cement issmall in the absence of large casing eccentricity or initial defect insidethe cement. Multistage finite element model was used by Nygaard et al.(2014) and Weideman (2014) to evaluate the wellbore leakage risk inthe Wabamun area. They reported that the near-wellbore stresses canbe reduced by thermal cooling thereby posing a higher risk of de-bonding and tensile failure, particularly for cement with high elasticmodulus and Poisson's ratio. Lavrov and Torsæter (2016) conductedfinite element simulations to study the effect of casing temperaturechange for different combinations of thermal expansion coefficients ofwellbore materials. They observed large increase in tensile radial stressduring cooling for cases where the casing thermal expansion coefficientis higher than those of cement and rock. They also showed that cementwith low elastic modulus (i.e. soft cement) and high tensile strengthupon hardening can be beneficial for preventing thermal failure.However, these two desirable properties do not always coincide. Recentwork by Lavrov (2017) emphasized that to get the full advantage of softor flexible cement in CO2 injection wells it should always be set againststiff rocks with high elastic modulus. Cement sheath failure analysis forvarying thermo-mechanical properties of casing-cement-formation wellsection was performed by De Andrade and Sangesland (2016). Althoughthey identified the elastic moduli of cement and formation as the keyphysical properties for the failure mechanism, they concluded thatthermal loading and unloading are relatively less sensitive to the elasticmodulus and thermal properties such as thermal expansion coefficientcan be important for these scenarios. Aursand et al. (2017) used acoupled two-phase flow model and radial heat transfer model to si-mulate CO2 injection cycles that may occur during a ship-based Northsea storage scenario. Based on their results, they concluded that the lossof well integrity can be minimized by carefully choosing the injectionparameters such as injection temperature, injection rate, and theduration of planned shut-down. Roy et al. (2017) studied the effect ofcooling rates, and showed that the stresses generated by thermalloading or unloading depend on both temperature difference of thematerial from its initial undeformed state and the spatial gradient oftemperature.

Thermal stress induced damage in cement sheath can exhibit dif-ferent types of failure. Radial cracking is likely to occur in the absenceof confining stress due to the thermal expansion of wellbore materialsduring the heating stage (Roy et al., 2016). Conversely, the thermalcontraction of the materials during the cooling stage may result in in-terfacial debonding (Roy et al., 2016; Lavrov and Torsæter, 2016). Bothof these failures correspond to tensile failure mode of the material.Shear failure may also occur if the compressive stresses are high enoughduring the heating stage (De Andrade and Sangesland, 2016). Typically,thermal stress induced failure is evaluated by comparing the stressesinside a material with its strength. This method works well for studieswhere it is assumed that no pre-existing cracks are present in the ma-terials. However, for pre-existing cracks the theoretical stress field atthe crack tip becomes infinite as the stress field around the crack is

1 Optimal conditions for ship transportations are achieved by pressurizingCO2 close to the triple point at−52 °C and 0.65MPa, producing a liquid densityof 1100 kg/m3 (Aspelund et al., 2006).

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inversely proportional to the square root of the distance from the cracktip (σ∝ r−1/2). Consequently, the above failure criterion will not workfor analyzing the stress state around pre-existing cracks. In this paper,we avoid this problem associated with this mathematical singularity inthe stress field by using stress intensity factor which characterizes themagnitude of the stresses near the crack tip region. The failure criterionbased on stress intensity factor compares its value with the fracturetoughness of the material. Although experimental studies (Bachu andBennion, 2009) suggested that the pre-existing mechanical defects suchas radial cracks in the cement or annular spaces at the cement-casinginterface can significantly increase the effective well permeability, veryfew studies have tried to address this issue during thermal loading orunloading. Our work is an attempt to reduce this gap in our under-standing of the impacts of thermal and far-field stresses on well in-tegrity in presence of pre-existing cracks. Our primary interest lies onthe conditions under which a pre-existing flaw propagates or widens inresponse to cooling.

Wellbore damage caused by thermal stress depends on a number ofkey factors including injection and formation temperature; state ofstress of the formation; and thermal and mechanical properties of thewell barrier materials. In this work, we have systematically investigatedthe role of cement properties, in-situ stress state, and injection tem-perature on thermally induced expansion and contraction of a wellsection. This understanding can aid in designing safe operations tominimize the risk of wellbore damage during the injection of CO2. Westudy the parameters that influence the stress distribution duringthermal loading or unloading operations, how likely the thermally in-duced stresses will propagate an existing fracture, and how to minimizethese fracture activations. Our analysis shows that over a critical valueof effective in-situ horizontal stress it is possible to safely inject CO2 at avery low temperature without further damaging the wellbore.

2. Numerical modeling

In this section we describe how we model the evolution of ther-moelastic stresses and heat transfer around the wellbore when CO2 isinjected. We consider conditions in non-perforated casing above theinjection zone. We neglect any convective heat transfer process asconduction is the dominant mode of heat transfer in this zone. We as-sume that the temperature of CO2 has reached a quasi-steady statecompared to the heat transfer process within the wellbore materials.Furthermore, we assume that there is no change in the CO2 temperaturealong the wellbore length and specify a constant injection temperatureat the wellbore wall. Previous studies by Luo and Bryant (2011) de-monstrated that below a critical depth the CO2 temperature gradientapproaches the geothermal gradient, essentially producing a constantΔT= To− Tinj, where To and Tinj are the formation temperature andinjection temperature respectively. This further justifies our assumptionof using a constant injection temperature Tinj for a fixed formationtemperature To. Under these assumptions, the thermoelastic deforma-tion in wellbore materials can be modeled by coupling two physicalprocesses: (i) equilibrium of stress and (ii) heat conduction. The equili-brium of stress implies that for a stationary body the sum of all forcesacting on it is zero. If the body force is neglected, the equations forstress equilibrium can be expressed in the following form2:

∂∂

=σx

0j

ij

(1)

= +σ σ σM Tij ij ij (2)

where σij denotes the components of the stress tensor, which consists ofthe material stress (σ M

ij ) and the thermal stress (σTij ). σ M

ij and σTij re-

present the change from the initial stress state due to external loadingand temperature change, respectively. For linear elastic deformations,the material stress can be expressed in terms of the strain tensor (εij) byusing Hooke's law for an isotropic material (Landau and Lifshitz, 1986;Fjar et al., 2008):

=+ −

++

σ Eνν ν

ε δ Eν

ε(1 )(1 2 ) (1 )

Mij vol ij ij (3)

where εvol is the volumetric strain or relative change in material vo-lume, which is identical to the trace of the strain tensor, i.e.εvol= εii= ε11+ ε22+ ε33. E is the modulus of elasticity, ν is thePoisson's ratio and δij is the Kronecker delta. Similarly, thermal stresscan be expressed in terms of thermal strain resulting from the change intemperature in the following way:

=−

−σ Eαν

T T δ1 2

( )T Tij 0 ij (4)

where αT is the linear coefficient of thermal expansion, and T0 is thetemperature of the material at undeformed condition. Heat conductionthrough the materials is governed by the Fourier law, which can beexpressed as2:

⎜ ⎟∂∂

= ∂∂

⎛⎝

∂∂

⎞⎠

ρC Tt x

k Txp

j j (5)

where ρ, Cp, and k are the density, specific heat capacity and thermalconductivity of the material, respectively.

The governing equations of linear elastic stress model coupled withconduction heat transfer model were numerically solved using theLLNL-GEOS code, a massively parallel, multi-physics, multi-scale si-mulator (Settgast et al., 2012, 2017). A finite element method was usedto solve the linear elastic stress equations (Eqs. (1–4)) whereas the heatconduction equation (Eq. (5)) was solved by a finite volume method.The solvers are able to handle different thermal and material propertiesin different regions. This is essential for accurate estimation of thethermal stress in the casing, cement, and rock regions.

We use a finite element framework to model the solid deformationand fractures in our model. At the location of a fracture, we duplicatethe topological entities associated with the finite element mesh such asthe finite element nodes, faces, and edges. Aperture is defined as thedifference in the displacement values between two initially spatiallycoincident faces on the fracture. At the nodal locations, this is just adifference in the values of displacement on the nodes on either side ofthe fracture. At any intermediate location on the face that do not cor-respond with the nodes, this is a difference in the value of the inter-polated displacement field on either side of the fracture. Initial value ofthe aperture is assumed to be zero.

To evaluate whether the pre-existing fracture would further openup, we calculate the stress intensity factor (SIF) in our simulations. Thestress intensity factor (SIF) calculations follow the Virtual Crack ClosureTechnique (see Raju (1987), Krueger (2004)). The idea behind theapproach is that the energy required to extend the crack surface by asmall length from a to a+ Δ is the same as the energy required to closethe crack tip from a+ Δ to a. The fracture energy release rate is givenby the following expression:

= − °A

G f w12

( )closure t (6)

where A is the area of the newly generated fracture surface, fclosure isthe closure force at the crack tip, wt is the crack opening at the nodetrailing the crack tip, and (°) denotes the entrywise product (Hadamardproduct). G is a vector with components GI, GII, GIII where the directionsI, II, III refers to the directions of crack opening, crack sliding and cracktearing, respectively. The stress intensity factor for each mode can beestimated using the corresponding component of G. For example, the

2 The tensor notation and Einstein summation rule have been used here.Einstein summation rule states that repeated indices are implicitly summedover i.e. ⎛

⎝⎞⎠

= + +∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂( )( )( )k k k kxj

Txj x

Tx x

Tx x

Tx1 1 2 2 3 3

.

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stress intensity factor for mode 1 failure is calculated by using thefollowing formula:

=− ν

SIF EG1I

I2 (7)

A detailed discussion on the method is provided in Settgast et al.(2017). The sliding contact and crack propagation techniques used inGEOS can be found in Annavarapu et al. (2015) and Annavarapu et al.(2016).

2.1. Wellbore model

In this paper, the geometry of the near-wellbore region is modeled asthree concentric rings of casing, cement, and rock (Fig. 1). The wellboremodel has an inner radius of 0.1m, steel thickness of 0.02m, cementthickness of 0.04m, and an outer radius of 2.0m. The total length of thewellbore section is 10m. The model dimensions, thermoelastic and othermaterial properties used in the simulations are listed in Table 1. A gridresolution of 75×36×80 is used in radial, azimuthal and longitudinaldirections, respectively. The grid independence of simulations conductedat this resolution is demonstrated in Appendix A.

The temperature at the inner surface (wellbore wall) was either heldconstant or increased/decreased as a constant rate. A zero heat flux (i.e.insulated) boundary condition is applied at the outer radius. For thisstudy, the initial temperature of the well and host rock was set to 70 °Cat the beginning of each simulation. Then, the temperature at the innerradius of the casing was increased or decreased instantaneously or at aspecific heating/cooling rate, depending on the nature of the numericaltest. The following assumptions are made:

(i) The materials can expand along all axes.(ii) The casing/cement and cement/rock interfaces are perfectly

bonded with no sliding or detachment unless the pre-existingfracture is at the interface.

(iii) Curing does not alter the internal stress state of cement.(iv) The pore pressure (Pp) is equal to the pressure at the inner well-

bore wall or the fluid pressure. The effective stress (σeff) is relatedto the total stress (σ) and pore pressure (Pp) by the following re-lationship: = −σ σ δ P( ) peff ij ij ij , where δij is the Kronecker delta.Hereafter, we drop the subscript from σeff and denote the effectivestress by σ only.

(v) The horizontal effective stresses (σro) are assumed to be isotropic.Unless otherwise specified σro=0.

(vi) A nominal effective vertical stress of =σ 5v MPa was used, whichcorresponds to a depth of 500m. The sensitivity of stress intensityfactors and aperture values due to the variation in vertical stresswas found negligible between 0 to 1000m depth.

(vii) A zero effective horizontal in-situ stress is considered as a re-ference case. Although in reality in-situ horizontal stresses are

Fig. 1. Wellbore model composed of casing, cement and rock. (left) simulated domain, (right) closeup of near wellbore region.

Table 1Dimensions and thermoelastic properties employed in the simulations (valuestaken from (De Andrade et al., 2014; Thiercelin et al., 1998; Aursand et al.,2017; Wang et al., 2012; Sellevold and Bjøntegaard, Ø., 2006; ThyssenKruppMaterials International, 2015)).

Properties Materials

Steel Cement Rock

Density (kg/m3) 8000 2300 2500Thermal expansion coeff. (×106 K−1) 12.0 6.0–20.0 10.0Thermal conductivity (W/(m K)) 50 0.5–2.0 2.1Specific heat capacity (J/kg/K) 450 1600 2000Young's modulus (GPa) 200 10–40 40Poisson's ratio 0.26 0.2 0.2Radius (m) 0.1–0.12 0.12–0.16 0.16–2.0Length (m) 10.0 10.0 10.0

Fig. 2. A three-dimensional view of the crack showing its length and azimuthalextent and its position relative to the sliced region of the casing and cement atthe wellbore mid-length.

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always present, we have chosen this case as a reference to showthe importance of in-situ horizontal stresses.

(viii) The initial stress state of wellbore is stress-free in all directions,i.e. the effective stresses are zero (σr= σθ= σz=0).

(ix) The model does not account for fracture propagation; it onlypredicts the stress state assuming the fracture remains static.

We consider a single pre-existing concentric fracture in the cementregion with a finite length and azimuthal extent (Fig. 2). The fracturewas located at a radius of 0.15m from the center of the wellbore. Unlessotherwise specified, the vertical extent of the fracture was fixed at0.625m, which is 6.25% of the total length of the computational do-main and the injection temperature was taken as 15 °C. It was assumedthat the fracture is filled with water, and thermophysical properties ofwater were used in the heat conduction model for the fracture region.Fig. 3 presents the radial stress distribution around a fracture with anazimuthal extent of 16.6% for zero effective confinement stress and aninjection temperature of 15 °C. We use the convention that compressivestresses are negative and tensile stresses positive. Initially, the systemhas zero stress before the drop in temperature. Once the casing is cooleddown by applying the injection temperature, the system experiencestensile stresses due to contraction of the casing. The maximum tensilestress is concentrated on the fracture edges near the crack tips. As de-monstrated in Fig. 3, the radial stress (σr) peaks inside the cement atvalues of 2 times higher than the rest of the cement. Near the cement-casing interface, the stresses are about 1.5 times higher. Also, strongvariation of σr along the cement periphery occurs with the increase inazimuthal extent of the fracture (not shown here). We use the stressintensity factor with mode I or tensile failure (SIF-I) to quantify thestress state near the crack tip. If the SIF-I exceeds the fracture toughnessof the material, failure is likely to occur and the fracture is likely togrow (Kuna, 2013). The tangential stress will dictate the stress intensityfactor for mode II failure (SIF-II). From our previous studies we haveobserved that the tangential or azimuthal stress can be significant forheating problems. In our study, the dominant stress is in the radialdirection, which is also perpendicular to the direction of the crackplane. For this reason, our analysis is based on the tensile failure mode(SIF-I) and shear failure mode is neglected as SIF-II values are very low.

3. Results

Key parameters that affect the stress distribution and maximumaperture can be broadly divided into two categories: (i) downhole

conditions and (ii) thermal and mechanical properties. The injectiontemperature, and the effective in-situ horizontal stress are the primarydownhole conditions that influence the stress state near the wellbore.The thermo-mechanical properties include the thermal conductivity,thermal expansion coefficient and elastic modulus of cement. We willstudy the effect of parameters from each of these categories. In addi-tion, we will study the influence of azimuthal extent of a single pre-existing fracture. These controlling parameters are systematicallyvaried to reveal what conditions cause the SIF-I to exceed the fracturetoughness and how the maximum apertures vary for a given tempera-ture range. During the variation of one controlling parameter, otherparameters are kept constant. The baseline or reference case, whichdefines these constant parameter values, is highlighted by a solidmaroon line in all figures, and tabulated in Table 2. We have consideredzero effective horizontal in-situ stress as a baseline or reference case.However, it should be noted that in reality, in-situ horizontal stressesare always present around the wellbore. The baseline case was used as areference point for analyzing the effects of in-situ horizontal stressesduring thermal loading or unloading operations.

Our study spans an injection temperature range between −15 °C to20 °C. We chose this range to capture the most realistic operating condi-tions and to explore the possibility of injection at sub-zero temperature forship transportation. The lower limit of the temperature range was chosenconsidering the fact that the freezing point of brine is approximately−21 °C. According to the industry practice, the injection temperatureusually varies between 4 °C and 30 °C to keep CO2 in a liquid state,3 and

Fig. 3. An example of radial stress distributionaround a single pre-existing fracture due to thecooling of the casing: (a) sliced view of thewellbore, (b) closeup view around the fracture.The cross-sectional snapshots are taken after30min of CO2 injection at 15 °C. The initialtemperature of the surrounding formation is70 °C. The black arc segment denotes the frac-ture cross section which is located inside thecement. The azimuthal extent of the fracture is16.6%. The axes units are in meters. The col-orbar denotes the stresses in Pa.

Table 2Parameter specifications for baseline case.

Parameter Value

Injection temperature 15 °CFormation temperature 70 °CEffective in-situ horizontal stress 0MPaFracture length 0.625mFracture azimuthal extent 16.66%, 0.156mCement elastic modulus 22 GPaCement thermal conductivity 1.0W/(m K)Cement thermal expansion coefficient 14.6× 10−6 K−1

3 The critical temperature of CO2 is 31.1 °C, and the critical pressure is7.3MPa.

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avoid excessive costs of heating. Injecting liquid CO2 is energetically moreefficient than its supercritical gaseous counterpart because higher densityof liquid CO2 enables it to flow down the wellbore easily due to gravityand requires much less compression energy (Vilarrasa et al., 2013;Vilarrasa and Rutqvist, 2017). Our numerical experiments consider bothinstantaneous cooling of the wellbore with a constant injection tempera-ture and thermal cycling operations at a constant cooling/heating rate. Inthe following sections we investigate the feasibility of injecting liquid CO2

at low temperatures without initiating fracture instability inside cement orformation.

3.1. Effect of injection temperature and effective in-situ horizontal stress

When the temperature of the inner periphery of the casing is sud-denly subjected to an injection temperature, the SIF and aperture aremaximum due to the initial thermal shock. As heat is transferredthrough the wellbore materials, the system goes into thermal equili-brium resulting in a gradual drop of SIF and aperture values with time.Fig. 4(a,b) shows how the maximum apertures and stress intensityfactors (SIF-I) of a pre-existing fracture vary with time for injectiontemperatures spanning from −15 °C to 20 °C. Once the injection starts,the casing is cooled almost instantaneously due to its high thermalconductivity. The thermal contraction of casing during the cooling isresisted by the surrounding cement and rock. Consequently, high stressintensity is experienced around the fracture inside the cement. Duringthe initial stage of the injection, the thermal shock results in a steep risein maximum aperture and stress intensity factors (SIF-I). After the in-itial thermal perturbation, the heat is transferred through wellborematerials and the rate of heat exchange between the cold CO2 and thewarm reservoir decreases. As a result, we observe a gradual decrease inaperture and SIF-I values over 20 days. For the studied temperaturerange, the maximum aperture values varied between 90 μm and160 μm. The SIF-I values varied between 2.8 and 5.2MPam

12 , well

above the fracture toughness of cement, which is about 1MPam12 (Zhu

and Xu, 2007; Kosior-Kazberuk, 2016).If the formation temperature is fixed, the thermal stress is directly

proportional to the injection temperature, resulting in a parallel shift ofthe aperture and SIF-I curves with decreasing injection temperatures, asobserved in Fig. 4(a and b). Fig. 4c presents the maximum SIF-I over thetime as a function of injection temperature. For the reasons statedabove, the maximum SIF-I varies linearly with injection temperature.The SIF-I curves are all above 1MPam

12 in the absence of any effective

in-situ horizontal stress, and the fracture in the cement will almostcertainly grow even when the injection temperature equals 20 °C(ΔT=50 °C). Moreover, the colder the injection temperature, thegreater the probability of fracture growth and the larger the fractureapertures.

Effective in-situ horizontal stress acting on the well has the potentialto prevent fracture growth in the wellbore materials. To study its effect,we varied the effective in-situ horizontal stress (σro) from 0MPa to10MPa at injection temperatures of 15 °C and −15 °C (Fig. 5). For in-jection temperature of 15 °C, the SIF-I was as high as 3MPam

12 when no

effective in-situ horizontal stress was applied. Increasing the effectivein-situ horizontal stress to 5MPa reduced the SIF-I to 1MPam

12 , which

is close to the fracture toughness of cement. Further increase in effec-tive in-situ horizontal stress (7.5MPa and 10MPa) resulted in zero SIF-Ivalues, indicating a very low probability of fracture growth. Fig. 5bshows a similar trends of effective in-situ horizontal stress for an in-jection temperature of −15 °C. In this case, a minimum of 10MPa ef-fective in-situ horizontal stress is required to ensure zero fracturegrowth suggesting that fracture growth would be quite likely at thesevery low injection temperatures. In these conditions, a non-zero effec-tive in-situ horizontal stress is essential to avoid the activation of pre-existing fractures.

Fig. 4. Effect of injection temperature on (a) maximum aperture and (b) maximum SIF-I. (c) Maximum SIF-I as a function of injection temperature.

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3.2. Effect of thermal cycling

In the previous section, we have demonstrated that without effec-tive in-situ horizontal stress even a small thermal perturbation can in-duce sufficient stress to activate the defects in the wellbore. Thesenumerical experiments were conducted assuming a constant injectiontemperature over time. However, during the shut-down or well servi-cing period, the well temperature will rise to the formation tempera-ture. To understand the thermal loading and unloading situations, wesimulated several thermal cycling tests for an injection temperature of−5 °C, assuming a formation temperature of 70 °C. As shown in Fig. 6,these thermal cycling tests consist of 2 days of injection at constantinjection temperature, followed by a rise of well temperature to theformation temperature within 2 days (heating rate ∼1.56 °C/h), fol-lowed by 2 days of constant formation temperature, followed by a fallof temperature to the injection temperature in 2 days (cooling rate∼1.56 °C/h), and so on. Note that, the injection temperature is appliedinstantaneously at the beginning of the cycling test, whereas the in-jection temperature is ramped down at a constant cooling rate duringthe second half of the cycle. Several things can be inferred from Fig. 6.First, in absence of effective in-situ horizontal stress, SIF-I values aremost likely to exceed the critical fracture toughness during the con-tinuous injection of CO2. Once the injection is stopped, the well tem-perature starts to rise and the stress intensity factor falls below thecritical fracture toughness or becomes zero within a short period.Second, the peak SIF-I values are lower for gradual cooling than theinstantaneous cooling as the thermal equilibrium relaxes the stressduring gradual cooling. It implies that the instantaneous cooling orthermal shock during the injection poses a greater risk of fracture thangradually cooling the well at a constant rate. This observation is con-sistent with our previous study on thermal stress using a smaller

wellbore model (Roy et al., 2016). Third, an increase in effective in-situhorizontal stress reduces the risk of fracture, which conforms with ourprevious discussion on the role of effective in-situ horizontal stress.

3.3. Effect of thermo-mechanical properties, fracture azimuthal extent andradial position

The thermal stresses in wellbore materials depend on the thermo-mechanical properties of the material, the azimuthal extent and radialposition of the fracture in addition to injection temperature and effec-tive in-situ horizontal stress. In this section, we consider how theaperture and stress states are affected by the variation of thermalconductivity, thermal expansion coefficient and modulus of elasticity ofcement. We also study how the variation of azimuthal extent and radialposition of the fracture affects the stress intensity factors around thefracture. We choose 15 °C and −15 °C as the representative injectiontemperatures. The former is usually used in low pressure CO2 injectionwells and the latter one represents an extreme condition that maycorrespond to the injection temperature during ship transportation.

Thermal conductivity is one of the controlling parameters thatdictate how much heat is transferred through the wellbore materials fora given thermal disturbance. For our parameter study, thermal con-ductivity of cement was varied between 0.29–1.8W/(m K). The thermalconductivity of the interstitial fluid was assumed to be 0.6W/(m K),equal to that of water. Fig. 7 shows the variation of maximum apertureand SIF-I with time for the above mentioned range of thermal con-ductivities and an injection temperature of 15 °C. At early times in thesimulation, larger apertures (∼100 μm) were observed for cases withhigh thermal conductivity. However, a high thermal conductivity alsomeans higher rate of heat transfer allowing the system to approach thethermal equilibrium more quickly. As a result, the aperture decreases ata faster rate. The maximum aperture varied between 97 μm to 102 μm,which is a narrow range compared to the range of thermal con-ductivities considered. The SIF-I values varied between 2.8 to3.4 MPam1/2 indicating that SIF-I is also weakly dependent upon thethermal conductivity of cement.

The maximum aperture and SIF-I exhibit similar monotonic varia-tions with time for all thermo-mechanical properties of cement. In theinterest of space, these results are placed in Appendix B and will not bediscussed here in detail. For each parameter, we can find the maximumvalue of SIF-I over time during the whole injection period and plot themas a function of the corresponding parameter. The rest of this sectiondiscusses these effects showing how the maximum SIF-I over the in-jection period varies as a function of thermo-mechanical properties. Aswas seen in Fig. 7, Fig. 8a shows that the cement thermal conductivitydoes not significantly influence the maximum SIF-I.

The coefficient of thermal expansion (αT) quantifies how much amaterial will contract or expand with the change in temperature. For

Fig. 5. Effect of effective horizontal stress on SIF-I for (a) injection temperature=15 °C and (b) injection temperature=−15 °C.

Fig. 6. Influence of effective in-situ horizontal stress on thermal cycling forinjection temperature=−5 °C.

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rock and cement, the thermal expansion coefficient is determined usingthe weighted average of the constituent values. As thermal expansioncoefficient of quartz is much higher than the other common rock-forming minerals, the weighted value of thermal expansion coefficientfor rock is a strong function of silica content (Zoback, 2010). Similarly,thermal expansion coefficient for the Portland cement strongly dependson the saturated cement paste content and the moisture content. In thisstudy, the cement thermal expansion coefficient was varied between6.0×10−6 and 2.0× 10−5 K−1. The lower bound represents thethermal expansion coefficient of regular cement after setting whereasthe upper bound represents the thermal expansion coefficient of a ce-ment with high moisture content before setting (Sellevold andBjøntegaard, Ø., 2006). We observe that the maximum SIF-I is far moresensitive to the choice of cement thermal expansion coefficient thancement thermal conductivity (Figs. 8b and B.13). A variation of about2MPam

12 in maximum SIF-I was observed for an injection temperature

of 15 °C. For an injection temperature of -15 °C, this variation is evenlarger (> 3MPam1/2). For both injection temperatures, the maximum

SIF-I varies linearly as a function of thermal expansion coefficient,which is consistent with Eq. (4). In fact, it is this predictable lineardependence that guarantees that its influence on SIF-I will be sig-nificant.

As the thermal stress is directly proportional to the elastic modulusof the material, the stresses are affected by the change in elastic mod-ulus. The variation of maximum SIF-I with time for different elasticmoduli exhibits similar trends as injection temperature and effectivestress. However, the maximum aperture shows an opposite trend i.e. itincreases with decrease in cement elastic modulus (see Figs. 8c andB.14). Although increasing the elastic modulus will increase thethermal stress, it will also result in smaller strains for the same appliedstress. As the elastic modulus appears in both Eqs. (3) and (4), the in-teractions among the total stress, cement modulus and displacement arecompeting, which may result in the observed trends shown here.

A large reduction in SIF-I can be achieved by reducing the cementelastic modulus, as shown in Fig. 8c. For injection temperature of 15 °C,and −15 °C the maximum SIF-I was reduced by more than 50% (from

Fig. 7. Effect of cement thermal conductivity on (a) maximum aperture and (b) maximum SIF-I.

Fig. 8. Maximum SIF-I as a function of (a) ce-ment thermal conductivity, (b) cement thermalexpansion coefficient, (c) cement elastic mod-ulus, and (d) azimuthal extent of fracture. Thevariation of maximum aperture has beenshown in (c) and (d). The solid lines representan injection temperature of 15 °C and the da-shed lines represent an injection temperatureof −15 °C.

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4.5 to 2.0 MPam1/2 and from 7.5 to 3.0 MPam1/2, respectively). A si-milar conclusion was achieved by Thiercelin et al. (1998) who studiedthe cement mechanical response and showed that thermal stress gen-eration can be minimized by using a lower Young's modulus cement. Toreduce the risk of interfacial debonding and cement failure, severalresearchers proposed using soft or flexible cement formulations withlow or varying cement elastic moduli (Boukhelifa et al., 2004; DeBruijnet al., 2009; Lavrov and Torsæter, 2016; Lavrov, 2017). By comparingsoft cement (very low elastic modulus of 1 GPa) with regular cement(elastic modulus of 10 GPa), Aursand et al. (2017) showed that radialstress and axial stress can be reduced by 5 and 2 times, respectively.Our observed dependence of maximum SIF-I on cement modulus con-firms these findings indicating that the low cement elastic modulus canminimize the risk of tensile failure and debonding provided that suffi-cient compressive in-situ horizontal stress is present. However, aspointed out by Lavrov (2017), this result may also be an artifact ofplacing a soft cement adjacent to a stiff rock with relatively high elasticmodulus. Although choosing a soft cement in conjunction with stiff rockcan be beneficial for preventing the cement sheath failure, the aperturesof pre-existing fractures can increase when cement elastic modulus isdecreased.

The stress state around the fracture will also depend on the azi-muthal extent of the fracture. To study this dependence, the azimuthalextent of the pre-existing fracture was varied between 8.3–50% of thetotal circumference, which corresponds to 0.078–0.471m. The simu-lations indicated that maximum aperture varied between 50 μm to170 μm and the SIF-I values varied between 2.0 to 3.6 MPam1/2 as wevaried the fracture dimensions (see Fig. B.15). Unlike injection tem-perature and thermo-mechanical properties, the aperture and SIF-Ivalues showed a non-linear and non-monotonic behavior with the azi-muthal extent (Fig. 8d). The aperture and SIF-I first increases with theazimuthal extent and then saturates after a certain threshold of azi-muthal extent is crossed. In our study, SIF-I saturates afterθextent=33.3% is reached. Here, θextent=33.3% corresponds to afracture width of 0.314m, which is approximately half of the length ofthe fracture. In practical situations, the length of the pre-existing cracks,that are hydraulically connected, will be much longer than the crackwidth. We also performed case studies with varying fracture lengthwhile keeping the fracture width and other parameters constant. Thevariation of fracture length did not result in any significant change inthe stress state implying that fracture length values considered for thisstudy are saturated.

In all these simulations the radial position of the fracture was fixed atr=0.15m, which is located inside the cement. But cooling inducedtensile failure may occur in other locations, especially at the cement-casing or cement-rock interfaces as the interfacial strength is oftensmaller than the material strength. To assess this situation we havevaried the radial position of the pre-existing fracture between the ce-ment-casing interface (r=0.12m) and the cement-rock interface(r=0.16m). Fig. 9 presents the maximum stress intensity factor (SIF-I)and aperture as a function of radial distance. SIF-I of the fracture locatedat the cement-casing interface is almost twice as high as the SIF-I offractures at other radial locations. The maximum aperture, on the otherhand, is highest at the cement-rock interface and varies between 30 μmto 50 μm. The SIF-I values for fractures located inside the cement exhibitsmall variations with radial position of the fracture (∼2.5MPam1/2).The fracture at the cement-rock interface produces slightly lower SIF-Ithan that of the fractures inside the cement. This implies that micro-annuli present at the cement-casing interface can be subjected to muchhigher stress concentration than micro-cracks present in other parts ofthe cement. The stress intensity factor depends not only on the tensilestress but also on the elastic moduli of the materials (Eq. (7)). As steelhas a much higher elastic modulus than cement, the SIF-I value is highestat the cement-casing interface. Thus, during cooling operations, anymicro-annulus or crack at the cement-casing interface is more prone tofailure than pre-existing cracks inside the cement.

4. Implications to CO2 storage

Our parametric analysis suggests that in absence of any effective in-situ horizontal stress, temperature perturbations during the injectionwill induce sufficient thermal stress and cause any pre-existing fractureto grow. A non-zero effective in-situ horizontal stress is required toprevent the growth of fractures. We can use these results to generate amap that suggests the downhole and operational conditions required tominimize the growth of damage zones within the wells. The drivingmechanism for fracture growth under thermal perturbations is not theabsolute value of the injection temperature, rather the difference be-tween the formation temperature and injection temperature(ΔT= To− Tinj). Armed with some understanding of the formationtemperature and downhole effective in-situ horizontal stress, we candefine a minimum injection temperature for safe operations (equivalentto finding an upper bound for ΔT). Fig. 10 maps three regions showingwhere fracture growth inside cement is likely (red), unlikely (green)and near the boundary between these two regions (yellow) by plottingthe critical effective in-situ horizontal stress as a function of injectiontemperature (or ΔT). The upper and lower solid lines bounding theyellow region represents SIF-I= 0 and SIF-I= 1MPam1/2. The greenregion above the upper solid line denotes low risk failure zones whereno activation of the existing fracture is expected under the conditionsconsidered in this study. The red region below the lower solid line re-presents high risk failure zones where SIF-I value exceeds the fracturetoughness of cement (∼1MPam1/2). For any combination of effectivein-situ horizontal stress and injection temperature in this region, thereis a high probability of growing the pre-existing fracture. The yellowzone represents an intermediate risk failure zone where depending onthe fracture toughness of cement the pre-existing fracture may or maynot grow for a given injection temperature and formation temperature.

These results do not apply to a well without any damage or pre-existing fractures. However, wells that have been abandoned for dec-ades are very likely to have damage and for those wells such a map canprovide an understanding of the risk of increasing damage. In this re-spect, the risk map shown in Fig. 10 has some deeper implications forCO2 injection. First, a lower ΔT or higher injection temperature resultsin lower risk. However, lower ΔT also means that the transported CO2

should be heated as close to the formation temperature as possible toavoid any kind of failure risk, which may be very expensive and in somecases impossible. Secondly, the map emphasizes the importance ofcorrectly estimating the stress state at downhole conditions. The mapshows that it is possible to safely inject CO2 even at −15 °C(ΔT=85 °C) if the effective in-situ horizontal stress is more than10MPa. For safe injection of CO2 at 5-15 °C, the required minimumeffective in-situ horizontal stress reduces to 6-8 MPa. This can drama-tically reduce the heating cost of CO2, especially for ship transportation

Fig. 9. Variation of maximum SIF-I and aperture as a function of radial positionof the crack.

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where the CO2 is stored at a very low temperature in shipping con-tainers (∼−50 °C), and for offshore pipe transport where the CO2 isstored at a 6-7 °C. For onshore transport it implies that the CO2 can beinjected below the supercritical temperature, in a liquid state with aminimum effective in-situ horizontal stress of 5MPa. Apart from low-ering the heating cost, liquid CO2 has the added advantage of makingthe injection energetically more efficient. To summarize, a properchoice of the stress state and formation temperature may enable lowerinjection temperatures while avoiding wellbore damage resulting in alower operating cost.

The risk map shown by the solid lines in Fig. 10 assumes that thecrack is located inside the cement sheath. As shown in Fig. 9, placingthe pre-existing crack at the cement-casing interface can dramaticallyincrease the SIF-I value. As a result, the yellow zone will be movedupwards and this shift is denoted by the dashed lines in Fig. 10. In-terestingly, when the injection temperature is close to the formationtemperature (ΔT≤ 30 °C), the shifting in risk zones are negligible. ForΔT=40–85 °C, the intermediate risk zones move upwards by about20–30%. These risk zones are also highly dependent on the azimuthalextent of the fracture. For example, it was observed from our simula-tions (not shown here) that for a pre-existing crack located at the ce-ment-casing interface, decreasing the fracture azimuthal extent from16.6% to 8.3% moves the dashed lines back to the original position ofthe solid lines in Fig. 10, which represents a pre-existing crack insidethe cement with an azimuthal extent of 16.6%. Similarly, incerasing theazimuthal extent of the fracture up to 30% will shift the intermediaterisk zone upward. Further increase in azimuthal extent has negligibleeffect on the SIF-I and the risk zones remain unaltered.

Changing the cement's thermo-physical properties shifts the riskzones in accordance with the trends shown in the previous section(Section 3). For example, increasing the cement's thermal conductivitywill result in a slight upward shift of the yellow zone, meaning a smallincrease in the area of high risk failure zone. However, high thermalconductivities will help the system to approach the thermal equilibriumquickly by increasing the heat transfer along the wellbore materials andthereby closing down any pre-existing fractures. This observation isconsistent with the previous findings by Lavrov et al. (2015) andTorsæter et al. (2017) who experimentally showed that a high thermal

conductivity is beneficial for preventing tensile cracks in cement.Thermal expansion coefficient has a larger effect on the overall stressstate around the fracture. A relatively small increase in thermal ex-pansion coefficient is expected to result in a large increase in the pro-pensity for fracture propagation. For example, increasing the thermalexpansion coefficient from 6×10−6 K−1 to 2.0× 10−5 K−1 can shiftthe yellow zone upwards by ∼20% for injection temperatures rangingbetween 15 °C and −15 °C, requiring higher effective in-situ horizontalstress to limit the damage. This suggests that a low cement thermalexpansion coefficient may greatly reduce the risk of failure. The effectsof cement elastic modulus on failure zones is more complex. As dis-cussed in the previous section, the competing interactions among theelastic modulus, total stress and displacement can result in an increasein aperture of pre-existing fractures with decreasing cement elasticmodulus. These factors should be considered when selecting a soft ce-ment to act as barrier material.

The maximum apertures for all cases studied here are in the order of100 μm. These aperture values decrease with time and will have a re-sidual aperture once the system reaches the thermal equilibrium. This isbecause even after reaching the thermal equilibrium, the final tem-perature of the materials around the fracture will be higher than thetemperature of the undeformed state. These small fractures are likely tobe sealed due to geochemical interactions between the brine and ce-ment (Iyer et al., 2017; Carroll et al., 2017). However, it should benoted that, in absence of sufficient effective in-situ horizontal stress, thefracture may propagate and fracture apertures may increase beyondhundreds of micrometers for which even chemical healing will not besufficient to prevent the fracture growth.

Acknowledgement

This work was performed under the auspices of the U.S. Departmentof Energy by Lawrence Livermore National Laboratory under theContract DE-AC52-07NA27344. We gratefully acknowledge the dis-cussions with Dr. Chandrasekhar Annavarapu at different stages of codedevelopment, simulations and interpretations of the results. We alsoacknowledge Dr. Pengcheng Fu for his assistance in setting up the si-mulations during the initial stage of this work.

Appendix A. Grid independence study

A grid independence study was conducted to ensure that the grid size resolution used for the parametric studies produced consistent results. Themesh size was varied in the radial (Nr), azimuthal (Nθ) and longitudinal (Nz) directions. To explore mesh dependence on Nθ and Nz, a wellbore with asingle fracture (fracture length= 0.375m) was subjected to an injection temperature of 15 °C at t=0 s. The temperature at the wellbore wall wasgradually increased to the formation temperature (10 °C) in 10 days, and then the formation temperature was maintained through the rest of thesimulation. The blue dashed line denotes the variation of temperature with time in Fig. A.11. The black lines present the variation of maximumaperture with time for different grid sizes. Due to the initial thermal shock during the injection, the aperture jumps to a finite value and reaches amaximum after which it gradually decreases as the thermal equilibrium is reached. Through repeated simulations, it was determined that Nθ=36

Fig. 10. Different risk zones for a pre-existing crack based onvariation of effective in-situ horizontal stress and injection tem-perature or temperature difference (ΔT= To− Tinj) with referenceto the baseline case model scenario. The ‘Low Risk Failure Zone’(green shaded area) denotes operating conditions where SIF-I re-mains zero. The ‘High Risk Failure Zone’ (red shaded area) de-notes the operating conditions with high likelihood of fracturegrowth as SIF-I remains greater than 1MPam1/2. The‘Intermediate Risk Failure Zone’ demarcates the region between

0MPam12 <SIF-I< 1MPam1/2. The dashed lines represent the

shifting of the risk zones when the pre-existing crack is located atthe cement-casing interface.

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and Nz=80 were sufficient to achieve grid independence.The grid dependence for radial direction was tested on a case where a constant injection temperature of 15 °C was applied for 5 days. Simulations

were performed at grid sizes (Nr) of 40, 75, and 150. Fig. A.12 presents the effect of grid size in radial direction on the stress intensity factor (SIF-I).From these results, we can conclude that Nr=75 is sufficient to produce grid indenpdent results.

Appendix B. Effect of cement thermo-mechanical properties and fracture azimuthal extent

Fig. A.11. Grid independence study for (a) longitudinal direction and (b) azimuthal direction.

Fig. A.12. Grid independence study for radial direction.

Fig. B.13. Effect of cement thermal expansion coefficient on (a) maximum aperture and (b) maximum SIF-I.

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