An evaluation of viscous deformation of chalk on wellbore stability

Wellbore instability has become an increasing concern for horizontal wells, especially with the move towards using drilling innovations that the entire lateral sections are remained without casing during the lifetime of the reservoir and after plug and abandonment operations. We aim to understand the long-term effect of viscous deformation on wellbore stability for horizontal wells through advanced laboratory experiments and viscoplastic simulations. Two-stage single lateral hole tests were conducted on chalk samples, where a horizontal wellbore was drilled laterally at the center of a cylindrical specimen and loaded under triaxial condition with a constant stress ratio. The chalk samples were imaged at two stages in the creep phase using X-ray computed tomography CT(scanning) to precisely observe the breakout development of the damaged zones formed around the borehole. The results from the experiments and CT analysis confirmed model predictions. Results presented in this work clearly demonstrate that viscous deformation can trigger failure modes and induce breakout development around the wellbore in the creep phase.


Introduction
Wellbore stability deals with physically maintaining the shape of the hole during drilling operation and preventing solid particle influx and hole collapse during production.It is therefore an important issue in geothermal, gas and oil industry as a response to the increasing of exploration in complex areas during drilling and production especially where horizontal and extended reach wells are drilled with completely open hole lateral sections.This concern becomes more critical in recent drilling innovations such as high pressure jet drilling, underbalanced drilling techniques, re-entry horizontal wells and multiple laterals from a single vertical or horizontal well (Medetbekova et al., 2021(Medetbekova et al., , 2020a;;Mohamad-Hussein and Heiland, 2018;Parra et al., 2003).Wellbore instabilities take several forms (Paić et al., 2007).The hole size can reduce when plastic zone squeeze into the hole, and hole enlargement can occur by caving shales or hard rock spalling (Cheatham Jr., 1984).To maintain wellbore stability, it is essential to drill with a drilling fluid of appropriate density, mud weight, to control the induced wellbore stress (Aslannezhad et al., 2016;DARVISHPOUR et al., 2019).Excessive wellbore pressure beyond the upper limit of the mud weight window, also called the fracture gradient (Zhang, 2019), can cause tensile failure in the formation with lost circulation as results of inadvertent fracture growth.Wellbore pressure below the lower limit of the mud weight window can also cause shear failure in the wellbore wall and may result in a hole collapse.
Several methods have been proposed to investigate wellbore stability.McLellan (1996) categorizes three approaches for prediction methods of wellbore instability: empirical, deterministic and probabilistic methods.Empirical models are based on experience, observation and laboratory tests, while deterministic models rely on analytical solutions and numerical methods.The simplest deterministic method commonly used in wellbore stability is to assess when stress at one or more locations on the wellbore wall exceeds the rock strength (Gholami et al., 2014;Rafieepour et al., 2020).This method assumes a linear elastic model and completely disregards about stress states beyond the rock strength, near the hole away from the wellbore wall and the feasibility of rock detachment.
Analytical and numerical models are widely used to gauge well-bore instability by considering the plastic deformation for estimating the extent of the broken zone around the hole (Hsiao, 1988;Lv et al., 2019;Masoudian et al., 2021;Singh et al., 2019;Veeken et al., 1989), even with flow (Paslay and Cheatham, 1963;Risnes et al., 1982) and thermal and chemical effects (Aslannezhad et al., 2020;Gholilou et al., 2017;Gomar et al., 2015;Ma et al., 2016;Ma and Chen, 2015;Zhang et al., 2006;Zhou and Ghassemi, 2009), during drilling process and pressure depletion.Wellbore stability is also investigated in horizontal and inclined wells (Ma et al., 2015;Manshad et al., 2014;Wang and Sterling, 2007).Although viscous deformation is well known and commonly used in compaction analysis, less attention has been paid to the effect of viscous plasticity for wellbore stability and a good knowledge of the viscous effect in wellbore stability is still lacking.Recently, Hu et al. (2022) proposed an analytical approach for the fast estimation of time-dependent wellbore stability during drilling.They have considered a viscoelasto-plastic solution for the creep deformations around wellbore during drilling in methane hydrate-bearing sediments.They have looked at the deformations during a short period of time shortly after drilling and show that displacements increase slowly at different rate as time passing by.Not only during drilling, the creep effect can be critical for uncased sections of the well especially during the long-term production life and abandonment of hydrocarbon, geothermal and energy storage fields where the reservoir is experiencing more deformations due to depletions and creep.Despite some research on wellbore stability, this effect has been less studied in the literature.
This study aims to evaluate the effect of viscous deformation on wellbore stability in horizontal wells during the creep phase.For this purpose, a test called Single Lateral Hole test (SLH) was conducted to investigate the stability of open holes.With CT-Scan imaging at two stages of the SLH experiments, the breakout development caused by the creep phase were quantified.The effect of the flow on breakout zone was also highlighted.A constitutive model describing the rate dependent behavior of chalk material with two yield surfaces, composed of shear and pore collapse yield surfaces, was used for the numerical simulation.
This article begins by introducing the governing equations of constitutive model in Section 2. Single lateral hole test is described in Section 3 together with visualization and analysis of the experimental results.Section 4 presents the simulation results and compares them with laboratory observations.We discuss the impact of viscous deformation on wellbore failure in Section 5. Finally, concluding remarks are given in Section 6.

Theory
The constitutive model employed in the simulation section is based on a model recently developed by Hajiabadi and Nick (2020).The model simulates dilation and compaction with two active yield surfaces for shear failure and pore collapse.The two yield surfaces are briefly reviewed as follows:

Pore collapse yield surface
The Modified Cam Clay model represents the compaction behavior as a yield function for pore collapse deformation according to where p and q are the mean and deviatoric stresses, respectively.The parameter M defines the axis ratio of the ellipse and A is the attraction value (Fig. 1).The hardening behavior of the elasto-plastic materials for standard isotropic pore-collapse strength p c is replaced by a ratedependent value with εvp vol denoting volumetric visco-plastic strain rate.εref is a reference volumetric plastic strain rate, typically chosen as 0.1%/hr at laboratory conditions.Therefore, the yield surface can expand/shrink as p cc varies due to the volumetric plastic strain rate (hardening/softening).The exponent b is the creep material parameter determining the ratedependency (Amour et al., 2020).εgeo is quite a small intrinsic strain rate so called geological rate to avoid zero p cc at zero volumetric plastic strain rate and creep stops below geological rate.
In equation ( 2), hardening is also controlled by the standard isotropic pore-collapse strength p c which depends on the volumetric plastic strain ε vp vol as where χ c is the pore collapse hardening coefficient.Assuming the associated flow rule, the potential surface is the same as the yield surface for pore collapse.

Shear failure yield surface
The linear Drucker-Prager model describes the shear-failure yield surface as where β is friction angle and d denotes cohesion (Fig. 1).Hardening and softening in the case of shear failure are functions of the equivalent plastic strain ε pl eq , which is incrementally defined as where e pl ij is the deviatoric plastic strain.Below the equivalent plastic strain at peak strength ε pk eq , i.e. ε pl eq < ε pk eq , cohesion is assumed to be constant d 0 .Then it declines linearly to the residual cohesion d res , Assuming the non-associate flow rule, the same function as yield surface is used for the plastic potential, but the friction angle β is replaced by the dilatancy angle Ψ : The dilatancy angle Ψ is assumed constant here.

Single lateral hole test
The chalk material used in this study is from Gorm field's Ekofisk formation, well N31, in Danish part of the North Sea at reference depth of 2280 m.Detailed rock material properties of the North sea chalk can be found in the literature (Amour et al., 2021;Amour and Nick, 2021;Al Assaad et al., 2022).The specimen was end trimmed to length of about 20.4 cm and diameter of 10 cm for a test program called Single Lateral Hole Test (SLH) for investigating the stability of horizontal open holes.The porosity of specimen was 35% and the dry bulk density was 1.78 g/cm 3 .A lateral well boring was drilled into the sample in the middle, 20.5 mm in diameter and 57.6 mm in length perpendicular to the axis of the core piece.In order to avoid rubber membrane penetrating into the wellbore during the application of confining stress in triaxial cell, a plug of similar chalk material was shaped to fit the open end of the hole, as shown in Fig. 2. The plug has a hole inside to permit the passing of metallic tubing.To prevent leakage, the metallic tube and the rim of the plug were sealed with the Rencast epoxy mixture.A set of strain gauges were installed on the specimen to monitor local strains at selected points.The specimen enclosed with a rubber membrane was then installed in a Hoek cell and mounted in the load frame and two LVDT sensors were aligned to record axial deformations.Similar experiments are explained in detail by (Medetbekova et al., 2022).
The test program comprised two stages.In the first stage, the specimen was loaded under drainage condition with fixed confining to axial stress ratio of 0.4 until achieving 29 MPa axial stress.The axial load was applied at the strain rate of 0.001 per hour while the confining stress was adjusted to 0.4 of the axial stress.The loading phase lasted 6 h, and then followed by a creep period of 118 h while the axial and confining stresses were kept constant.The specimen was dismounted for CT scanning afterwards.
In the second stage of the experiment, the same specimen was loaded up to the previous level of stress under a similar stress ratio but at the axial strain rate of 0.0005 per hour.The loading phase lasted 20 h to provide previous level of loading stress corresponding to 29 MPa axial stress.Then, the specimen remained at constant stresses for 492 h, before the axial loading ram was fixed to the current position, i.e. relaxation, and the valve that provided the confining pressure was closed.Water was pumped from both top and bottom for 1 h at four constant rates with a maximum of 180 cm 3 /h, which was maintained for 3 h' time.The specimen was dismounted again for CT scanning at the end.
At the end of each stage, the specimen was dismounted for CT scanning.Since only the well boring segment of the core was of interest, only the middle part of the specimen was imaged for viewing "the region of interest" (Fig. 3).CT images demonstrated that the chalk pieces, broken off the chalk core body and accumulated in the well boring, are mainly from the sidewalls of the hole, not the ceiling and floor.
Comparing the damage zone between the CT images after the two stages of the test indicates significant breakout development in the second part.Fig. 4a shows the temporal evolution of strains to understand this observation.The average value of axial strain from two LVDTs and circumferential strains from two strain gauges are plotted in this figure.
During the loading in the first stage, the specimen experienced axial strain and circumferential strain both in compression.With creep phase starting at the end of loading phase, the axial strain continues with extra compressive strain.Meanwhile, the circumferential strains change sign and extends to the level of much higher than the absolute value experienced in the loading.This indicates the significant breakout development happening during creep phases.A similar behavior in strain curves was observed in the second stage.After achieving the previous level of stresses, the specimen deformed in the creep phase in the second stage.
As time passes by in the creep phase, strains continue to increase.
Approximately after 19 h passing in the creep phase, corresponding to t = 170 h in Fig. 4 (a), a sudden increase in the deformation rate was detected, which can also account for the breakout development in the specimen during the creep phase.
In the relaxation and pumping phases, the axial deformation was prevented by fixing the axial loading rim.Within 1 h of pumping into the specimen, 1.5 MPa pore pressure was established at two ends of the specimen with 0.16 MPa/cm pressure gradient between the two ends of the specimen and the hole.Assuming that fluid flows into the hole from the side wall of the hole, the average velocity of the fluid around the hole equals approximately 10 cm/h.Pumping continued at a constant rate for approximately three extra hours.The role of fluid flow was mainly restricted in breaking loose chalk pieces off the sample into the hole after failure.Therefore, a minor effect on breakout development is   expected during such period with observing negligible increase in the circumferential strain in this period.This highlights the long-term effect of creep phase on breakout development.

Results
In general, a 3D analysis is required for the simulation of proposed SLH test.Instead, for reducing computational costs, a 2D plane strain is also acceptable in the loading condition that the ratio of confining to axial stress reach minimum/plateau in uniaxial compaction test starting from initial hydrostatic stresses with zero lateral deformations.This ratio was set to 0.4 from a uniaxial compaction test, carried out on a similar intact specimen.The proposed constitutive model (Hajiabadi et al., 2020) was implemented in ABAQUS by means of UEL subroutine for a Cosserat continuum and has been tested for the simulation of SLH test.Cosserat continuum is an approach to introduce size effect into classical continuum to overcome convergence difficulties for material exhibiting softening behavior and localization (Khoei, 2005).Very few works has used Cosserat continuum for the stability analysis of wellbores (Papamichos, 2010;Papanastasiou, 2000;Papanastasiou and Vardoulakis, 1992), but without considering viscous effect.Due to symmetry, a quarter of the model was simulated.Fig. 5 shows the discretization, boundary, and loading conditions.The simulation was performed under stress controlled loading condition, although the experiment was carried out under strain controlled loading condition.This is due to the condition that was set for stress ratio in the experiment.In a simulation with the strain controlled loading condition, stresses on boundaries are unknown variables to be determined within the simulation.Therefore, an iterative procedure is required to impose the constant stress ratio condition between horizontal and vertical stresses.Alternatively, we used stress-controlled loading with history values obtained for axial and confining stresses recorded in the experiment for loading and unloading phases.The evolution with time of the stresses applied to the boundaries are plotted in Fig. 4b-c.
Simulation results are plotted in Fig. 4a for axial and horizontal strains.The axial strain in simulation results represents the vertical displacement at top left of the specimen with respect to the initial length of the specimen, i.e.H = 102.25 mm, that is compared with experimental values obtained from the average of two LVDTs and loading ram.Note that a 3D simulation is required to obtain true associated values for comparison with circumferential strains measured from strain gauges in the experiment.We used a 2D simulation for less computational cost and plotted the circumferential strain-gauge values with the horizontal strain in simulation at the right bottom of the specimen.Breakout zones indicated by equivalent plastic strain at the end of both stages of the simulation are compared with CT images shown in Fig. 6.
Good agreement was observed between the experiment and simulation results in the first stage (Fig. 4a).Comparing the CT-Scan#1 at the end of the first stage with simulation results reveals that the shape of the breakout zone predicted by the simulation agrees with the shape seen in the experiment (Fig. 6a-b).Fig. 7 shows the evolution of stresses with time and the stress path for two selected points near (A) and far (B) from the hole.As can be seen in Fig. 7a, stresses start to reduce right after creep period starts at t1 = 6.3 h around the hole while loading stresses are kept constant.Time passing in creep phase causes softening to happen in shear failure around the hole.That means breakouts are developing during the creep phase as Fig. 8 illustrating that the equivalent plastic strain develops between t1 and t2.Breakout  simulation, loading stresses approach to zero while axial strain reaches to a residual value (Fig. 4c).It should be emphasized that here we only focused on viscous behavior occurring because of the yielding cap and ignore the viscous deformation at the shear failure.Although at the intersection of the yield surfaces, the effect of viscous deformation is considered.The zone where both yield surfaces are active highlighted in yellow in Fig. 8c.This zone is considerable compared to the zone where only shear failure is meet.
After imaging the dismounted sample at the end of the first stage, a similar setup was reinstalled for continuation of the experiment.By reloading the specimen in the second stage, the axial strain approached the value experienced at the end of the creep phase.The reason for the discrepancy in strain curves between the simulation and experiment at early stage of reloading is that the strain measurements were set to measure strains from zero in the experiment while the strains of the simulation are the continuation of residual values.After 19 h in the creep phase, a sudden increase in the deformation rate in experimental results is ascribed to a local damage development, which is difficult to model/predict in the simulations.In order to deal with this difficulty, we modified two material parameters in the second stage of the simulation to achieve a better agreement with the experimental results (see Table 1).The results for two simulations with and without the changes in material parameters are plotted in Fig. 4a.The simulation results are presented until pumping and relaxation phase.As discussed in the previous section, it is worth mentioning that pumping and relaxation phase has little effect on the breakout development.The role of the flow is restricted to detaching the damaged zone during pumping phase where experienced low stresses due to the softening are close to the tensile strength around the hole.Comparing the breakout zone in Fig. 6c-d between CT-Scan#2 and the simulation results shows that the model could not predict the entire damaged area.The reason can be because of the local damage that happened after 19 h in the creep phase.Table 1 Material properties of the chalk specimen.

Discussion
We demonstrated the effect of viscous deformation on breakout development around the wellbore in SLH tests.This results can be compared with those of (Medetbekova et al., 2020b) for a similar test, but under lower stress condition on a chalk sample of a similar reservoir.The temporal evolution of stresses and axial strain in their work are shown in Fig. 9.The CT images around the hole at the end of both stages are compared in Fig. 10.A few small cracks on the wall of the hole observed in Fig. 10 at the end of the first stage of the experiment indicates that the failure around the hole has just stared to occur around the hole without developing enough to form a big breakout zone.At the end of the second stage, the failure zone does not grow into the bigger size.These observations imply that avoiding rock failure in perforation design plays the main role to prevent sand production due to visco-plastic deformation.Papanastasiou (2000) considered a similar role for fluid flow concluding no sanding is expected to occur because of the fluid flow providing that the rock failure conditions are avoided.Therefore, the failure in the form of instability around the wellbores mainly takes place due to the high stress concentration especially during the drilling and depletion.Once the shear/tensile strength of the rock is altered and reduced by the local damage around the wellbore (whether it results specifically from stress concentration during drilling operation or expands due to visco-plastic deformation in reservoir lifetime), the breakouts can be detached by the induced fluid pressure gradient.
To have a better assessment over the effect of viscous deformation, another simulation has been carried out with ignoring the viscous deformations.The results are compared in Fig. 11 where equivalent plastic strains ε pl eq are plotted for selected times.To remove viscous deformations, pore collapse yield surface is ignored in the new simulation.Comparing the equivalent plastic strain at the end of first loading phase, i.e., time t1 (6.3hr), between the simulation with and without viscous deformations, negligible differences are observed.This implies that viscous deformation has little impact during drilling.Although drilling process is not simulated here, similar behavior is expected during drilling process because viscous deformations are negligible due to quick process of drilling.Therefore, stress concentration is the main reason for determining the failure in this phase.With creep phase starting right after the first loading phase, the equivalent plastic strain starts increasing in the simulation with viscous deformation while there is no change in the simulation without viscous deformation.However, the viscous deformation does not extend the size of the failure area.It only causes equivalent plastic strain to increase significantly in the failure area where previously created in the loading phase.This indicates that breakout zone during the creep phase can expand in the failure zone which is created mainly by stress concentration and shear failure.Quit similar observations with less intensified equivalent plastic strain can be seen during unloading phase for both simulation case.Considering the results of the simulation with viscous deformation at the lower side wall of the hole, it indicates that equivalent plastic strain has intensified 5 times during the creep phase, i.e., ε pl eq increased from 0.045 to 0.225 between t2 and t3, while it has increased 1.3 times during unloading phase, i.e., ε pl eq increased from 0.225 to 0.292 between t2 and t3.This shows the key roles of creep phase in extending the breakout zone which is even higher than unloading phase.Reloading to previous stress in both simulation cases, ε pl eq increased.In the second creep phase similar behavior as the first creep phase is observed.
Another observation deals with the amount of the ε pl eq which is significantly higher if viscous deformation is considered.This validates the role of the viscous deformation in sand production in open hole section in the long term.In the reservoir, deformation is commonly taking place at very low rate.Therefore, viscous effect becomes important in sand production analysis.In this regard, the estimation of in situ stresses (Baouche et al., 2021;Ganguli and Sen, 2020) around the uncased wellbores becomes more critical where stress conditions are close to the shear failure of soft rock material.
The breakout can extend around damaged holes because of the viscous effect that in general lead to progressive deterioration of perforation and may reduce or even stop production.This finding helps optimize the choke management strategies in uncased wellbores.Choke management strategies, or bean-up operations, refer to process of gradually increasing the rate or drawdown toward achieving a stabilized rate (Andrews et al., 2017;Karantinos et al., 2017;Tawancy and   Alhems, 2016).Frequent and harsh shutdowns followed by rapid bean-ups can cause severe damage in uncased sections.Considering the viscous effect, the maximum allowable drawdown that a formation or completion can withstand without sanding and instability problems may decreases after a long time.Similar problems exist even if production stops during that time.It is, therefore, essential to consider the long-term viscous effects for designing the bean-up operation and wellbore stability.

Conclusion
In this study, we provided a general framework to explore the effect of viscous deformation in wellbore stability in horizontal wells during the creep phase through the experiment and numerical simulation.A rate dependent constitutive model was employed to simulate both dilation and compaction due to shear failure and pore collapse deformation.The localization of deformation and failure around the hole was considered based on the Cosserat continuum.The results show that viscous deformation can trigger mechanical instabilities and failure modes with breakout development around the wellbore in creep phase.The role of fluid flow is mainly restricted to breaking loose pieces of the sample into the hole after failure.It is therefore essential to avoid rock failure in the perforation design for sand control.Otherwise, the failure can extend around damaged holes because of the fluid flow and viscous effect that in general lead to progressive deteoriation of preformation and may reduce or even stop production.This can affect the choke management strategies for uncased wells to optimize bean-up operation.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.eq developed around the wellbore with and without viscous deformation at selected times.The amount of ε pl eq is much higher when viscous deformation is considered.The two bottom plots represent the equivalent plastic strain at different time along the horizontal edge at the bottom of the 2D cross section.
Imaging Center at The Technical University of Denmark is gratefully acknowledged for providing access to X-ray tomography equipment.

Fig. 3 .
Fig. 3. 3D Visualization with different slices of the segmented chalk samples at the end of two stages of SLH test.

Fig. 4 .
Fig. 4. SLH test comprising two stages.Each stage includes loading, creep and unloading phase while the second stage also have relaxation and pumping phase.At the end of each stage, the specimen dismounted for CT scanning.The time evolution of strains/stresses are plotted in (a) during the whole time; (b) during loading phase in the first stage (c) during unloading phase in the first stage followed by reloading in the second stage.

Fig. 6 .
Fig. 6.Comparison of break out zone between CT images and simulation results at the end of both stages in the experiment.
zone designated by shear failure zone, compaction and transient zone from compaction to shear failure are shown in Fig. 8.As shown in Fig. 8, the volumetric plastic strain develops in front of the breakout zone as time passes in the creep phase.During the unloading in the first stage of the

Fig. 7 .
Fig. 7. Evolution of the stresses in time (Top) and stress paths in p-q space (Bottom) at points A and B located near and far from the hole.

Fig. 8 .
Fig. 8.The evolution of (a) equivalent plastic strain ε pl eq , (b) volumetric plastic strain ε vp eq and (c) active yield surf ace during the 2D SLH simulation.

a
Values in parenthesis are material parameters applied for the second stage of the simulation to reflect property changes during the first stage.M.R.Hajiabadi et al.

Fig. 9 .
Fig. 9.The time evolution of measured stresses and axial strain for the SLH test carried out by Medetbekova et al. (2020b).

Fig. 10 .
Fig. 10.The breakout zone reported by Medetbekova et al. (2020b) in the SLH test under low stress condition at the end of both stages.

Fig. 11 .
Fig. 11.Comparison of equivalent plastic strain ε pl eq developed around the wellbore with and without viscous deformation at selected times.The amount of ε pl