Coupling a void growth model with an elastoplastic material model for a simultaneous prediction of the borehole plastic zone and critical collapse pressure

The elastoplastic shear stability of highly inclined boreholes is explored in this study. First, the stress concentrations in the elastic and plastic zones around a highly inclined borehole are derived. Unlike other previously published elastoplastic wellbore stability solutions, this work considered the contributions of the anti-plane shear stresses. Furthermore, a new method for estimating the collapse pressure and extent of the plastic zone around the borehole is presented. In this approach an elastoplastic material model is coupled with a void-growth model to simultaneously predict the collapse pressure and plastic zone. The predicted collapse mud weights from the proposed approach compared very well with observed field data. The pure shear strength of the formation is a key material property that determines the contributions of the anti-plane shear stress in the stability of the borehole. If the pure shear strength is disregarded, as is the case with other previously published solutions, the predicted collapse pressure will be overly optimistic, underestimating the collapse pressure, if the pure shear strength of the rock is not negligible compared to the loading conditions. After conducting several sensitivity studies, the elastic brittle-plastic model performs better in predicting the shear stability of boreholes than the elastic perfectly-plastic model.


Introduction
Once the peak strength of rock surrounding a borehole is reached during drilling or fracturing operation, the rock yields but does not fall into the wellbore.Instead, an annulus of weakened rock, generally called a yielded or plastic zone, is formed around the borehole, and the peak stress is redistributed away from the surface of the borehole.Galin (1946) presented a solution for estimating the plastic zone around a circular borehole under non-uniform loading, assuming Tresca failure criterion.His solution showed that the elastoplastic interface is elliptic in shape.His solution pertains to cases where failure as occurred all around the hole and the problem is statically determinate.Cherepanov (1963) solved the Galin problem with a plane stress assumption.Detournay (1986), using plane strain approximation as Galin, and Mohr-Coulomb failure criterion found that the interface appears to be a compressed ellipse.Detournay and Fairhurst (1987) further indicated that the orientation of the major axis of the elliptical plastic zone aligns with the major compression direction, and that its shape is primarily dependent on the friction angle and obliquity.In their formulations, the continuity of all stresses was required, which is not in general necessary.Central to the model development are the hypotheses that there is no elastic unloading in the plastic zone and the problem remains statically determinate from the formation of the first plastic zone.Similarly, Huang et Wang and Dusseault (1991a), Wang and Dusseault (1991b), Detournay and John (1988), and Westergaard (1940) have also developed models simulating the yield zones around the wellbore.But none is applicable to highly inclined boreholes.Tokar (1990) extended Galin's problem by using Coulomb's failure criterion in the yielded zone and assumed both continuous and discontinuous stress distributions at the elastoplastic interface.He noted that discontinuous circumferential stresses at the elastoplastic interface greatly influence the shape of the plastic region.With the increase in the stress difference across the interface, the plastic zone shrinks.Detournay and Fairhust (1987) listed a few restrictions on the parameters defining the elastoplastic problem and nature of the expected solution.A key assumption is that the problem is two-dimensional, and that a plane strain restriction can be imposed on the elastic zone.While in the plastic zone, the constitutive relations can only be expressed as plane components if the out-of-plane stress is the intermediate principal stress.Detournay (1986) and Detournay and Fairhurst (1987) provided a sufficient condition on which this specific restriction holds.The statical determinacy and no elastic unloading assumptions suggests that the stress field in the plastic zone is axisymmetric.Lu et al. (2010) considered the impact of including the axial in-situ stress into the elastoplastic formulation on the failure distribution in the plastic zone.In their work they showed that the plastic zone will significantly depend on the axial stress if it exceeds a critical threshold value, but insignificant below this threshold.The failure zone was observed to increase with higher values of the axial stress.Ewy (1991) also observed that accounting for the axial stress can have significant implications on the predicted wellbore stability results.The use of all stress components in a 2D failure criterion was pointed out as a means in which the 3D effects can be included in wellbore stability analysis.Florence and Schwer (1978) and Risnes et al. (1982) have considered special cases in which the axial stress can be incorporated into the stability analysis.
To determine the critical collapse mud weights in the elastoplastic stability analysis of boreholes, a closure model or framework is needed alongside the elastoplastic material model.This is because there are two unknown variables and one equation.So, the problem is indeterminate.Hawkes and McLellan (1996) proposed the Normalized Yielded Zone Area (NYZA) method for estimating the critical collapse pressure, and several elastoplastic wellbore stability studies (Salehi et al. 2010, Huang et al. 2018, and Behnam et al. 2019) have been based on this approach.The NYZA is the area of the plastic zone around the borehole divided by the initial area of the borehole.In their method, the wellbore is stable when NYZA is less than 1.But when NYZA is greater than 1 the plastic region around the borehole will collapse.With this approach, the yielding of the borehole is not made on a single point, which the conventional elastic shear failure framework is based.One limitation of this method is that the criterion for failure, NYZA = 1, is not based on any scientific formulation.In fact, many field observations showed that failures have occurred at NYZA greater than 1.
In this work, a new framework for simultaneously estimating the plastic zone extent around the borehole and critical collapse pressure.In this framework, the elastoplastic material model is coupled with a voidgrowth model, and the two variablescollapse pressure and plastic radius are determined simultaneously by selecting any orientation around the borehole.The orientation of maximum compression is preferably selected in this work (Fig. 1).Hence, the normalized yielded zone radius (NYZR) used in this work refers to the radius of the plastic zone in the direction of maximum compression divided by the initial radius of the borehole.This is related to the NYZA through this equation NYZR=√(NYZA+1), assuming the plastic zone is circular.
Continuum-based void-growth models are well-established approach for investigating failure by void growth in ductile materials.Extensive works have been done in this area, as compiled by Tvergaard (1989), Gologamu et al. (2001), and Benzerga and Leblond (2010).Voids are nucleated by either decohesion of the particle-matric interface or by particle fracture, and subsequently grows due to the plastic straining of the neighboring material.These voids are nucleated in the skeleton of the porous rocks and are totally different from the initial porosity of the rocks.McClintock (1968) and Rice and Tracey (1969) presented the first sets of micromechanical studies on the growth of a single void in an elastoplastic solid.However, the most widely known void-growth model was developed by Gurson (1977a) based on Bishop and Hill (1951) averaging techniques.He developed a macroscopic yield criterion for a porous ductile material, assuming the voids to be spherical or cylindrical.Other void growth models have since been developed after Gurson, but they are all improvements on his model (Gologanu et al. 1997 The primary goal of this paper is to introduce to the scientific community a new approach for simultaneously estimating the plastic extent around a wellbore, with any orientation, and the critical collapse pressure.In this work, an elastoplastic material model (elastic perfectly plastic or elastic brittleplastic) is coupled with a Gurson void-growth model to determine the critical collapse pressure and the extent of the plastic zone around the borehole.

Governing assumptions and equations
In this section, the stress concentrations in the elastic and plastic regions around the borehole are first derived, based on some simplifying assumptions.

Governing assumptions
The model development is based on the following assumptions: a.The near-wellbore region is isotropic and homogeneous.b.No yield was generated before passive loading.c.The deformation is small.d.The wellbore geometry is circular.e.The near-wellbore region behaves as an elastoplastic material body, while at far-field the rock is linearly elastic.The formation matrix is homogeneous and isotropic.f.The principal in-situ stresses do not align with the borehole axes.g.Inside the cavity acts a fluid pressure,   , and there is no fluid transfer between the cavity and the host material body (the rock formation).Hence, there is no change in the pore pressure around the wellbore.
h.The size of the hole is very small compared to the size of the linearly elastic material body; in fact, it is assumed infinite.i. Compressive stress is assumed to be positive while tensile stress is taken to be negative in this study.

Material modeling
Laboratory and field studies have demonstrated that the behavior of rocks abutting excavated boreholes are inelastic, especially cracked ductile shale formations.Thus, the use of linear elastic modeling to describe the stability of these near-well regions may not be accurate.And very pertinent works to this are the studies done by Hoek and Brown (1997) and Gonzalez-Cao et al (2013), who suggested that the use of a strainsoftening model best represents the mechanical response of a typical rock mass (Fig. 1).Wang et al. (2011) presented a procedure for analyzing the elastoplastic stability of circular boreholes using the strainsoftening model.The procedure follows the approach for analyzing the stability of a rock with a brittleplastic response.Pourhosseini and Shabanimashcool (2014) and Zhang et al. ( 2012) also contributed to the application of the strain-softening model for investigating the elastoplastic stability of circular boreholes.This material modeling approach assumes a gradual reduction of the rock strength, from the peak to a residual value.
Furthermore, there are other two classifications of the stress-strain curves of a rock specimen, which constitute the limits of the strain-softening response.The elastic-perfectly plastic response assumes the rock strength remains constant after reaching the peak value, while the elastic-brittle-plastic behavior suggests that the rock strength decreases sharply, from the peak to the residual values.In this work, the stability of the wellbore will be explored using these two extreme assumptions and comparing the results with stability predictions from linear elastic models, based on Mohr-Coulomb, Mogi-Coulomb, modified Lade, and Drucker-Prager shear failure criteria.In the modeling framework, the near-well region, (II), is assumed to be plastically deformed while the farregion from the wellbore, (I), has a linear elastic response (Fig. 2).The elastic properties of the region (I) include an elastic modulus of  0 , Poisson ratio  0 , cohesive strength  0 , unconfined compressive strength  0 , and friction angle  0 .The yielding of the elastic region is assumed to obey Mohr-Coulomb failure criterion, defined as  In the plastic region, around the near-wellbore (II), (Fig. 2b), the rock failure satisfies the modified Mohr-Coulomb criterion if the elastic-brittle plastic response is assumed.In this region, the cohesion of the material is altered.On the other hand, with an elastic-perfectly plastic response, Eq. (1) satisfies the yield function.

Stress state around a circular wellbore
Consider a wellbore inclined with respect to the cartesian coordinate system  ′ ,  ′ ,  ′ as shown in Fig. 3 below.For generality, the in-situ maximum horizontal stress is oriented in a direction that is  0 from  ′ −axis.
The in-situ stresses can be transformed to local coordinate system , ,  through the tensor transformation relation   =       , where   is the cosines matrix of the coordinate axes of the reference frame with respect to the local axis to the wellbore,   are the in-situ stresses, and   are the transformed stresses.The z-axis of the local system coincides with the wellbore axis.The directions of the transformed stress components depend on the wellbore inclination angle (from vertical), , azimuthal angle referenced from the orientation of maximum horizontal stress, and the wellbore position from the x-axis, .

Elastic region.
It should be noted that the 3D stress state in both the elastic and plastic regions around the borehole are decomposed into four subproblems.In this section of the paper, only the stress profiles in the elastic region are considered.As derived in Appendix A, the stress distribution in the elastic region are:

Plastic Region
In the plastic region, the stress distributions are determined by substituting the yield function in Eq. (4) into the stress equilibrium equation.With the 3D problem divided into smaller subproblems, the in-plane and anti-plane stress distributions in the plastic region will be derived, using the appropriate stress equilibrium equation components.In the model development, it is assumed that the plastic zone surrounds the borehole, and it results from a monotonic loading.Furthermore, there is no elastic unloading in the plastically deformed region.The stresses in the plastic region, as derived in Appendix B, are Eq.( 16) assumes the Poisson ratio of the near wellbore region is not damaged by plastic deformation.

Shear failure at the wellbore wall
In a vertical well, shear failure occurs in the direction of the minimum horizontal in-situ stress.But in a deviated well, the orientation of shear failures significantly depends on the state of stress around the wellbore, which is a function of well angles (inclination, azimuth) and in-situ stresses.
The effective principal stresses at the wellbore wall are Where σ 1 ′() is the maximum effective principal stress and σ 3 ′() the least effective principal stress at the wellbore wall.The derived equations for the principal stresses result from the non-zero value of the antiplane shear stress,   () , at the wellbore wall.These equations reduce to the well-known principal stress equations when   () = 0. Hence, the shear failure and collapse of the borehole is assumed to occur, for an elastic-brittle plastic behavior of the rock, when

Failure of the plastic zone
Based on the foregoing elastoplastic analysis of the borehole, the stability of the borehole strictly depends on the stability of the plastic zone.In the previous section of this paper, the yielding of the elastic zone is assumed to obey Mohr-Coulomb failure criterion, and the corresponding stresses in the plastic zone are thus derived based on this assumption.However, it is further assumed in this work that the plastic zone fails when the void created by the plastic straining in the rock matrix reaches a critical value.The most widely known void growth model by Gurson (1977a) is used in this work.His yield condition is combined with Eq. ( 55) to form a system of equations, which will be solved simultaneously for plastic radius and critical collapse pressure.This micromechanical model assumed the voids are spherical or cylindrical, which are reasonable assumption, except at the brink of final failure, where extensive void coalescence has occurred.
For a given ductile material body containing a volume fraction  of voids, created by plastic straining, Gurson (1977a) and Gurson (1977b) proposed an approximate yield condition of the form Φ(  ′ ,  0 , ) = 0, where   ′ is the average macroscopic Cauchy effective stress tensor,  0 is the uniaxial compressive strength, and  is the void volume fraction.Assuming the nucleation of random spherical voids in the plastic zone, with no unloading, the approximate yield function for the plastic zone with stress-induced voids is:  (1977), Gurson (1977a), Tvergaard (1982), and Tvergaard and Needleman (1984).It should be noted that the assumption that the spherical voids are random thus implies that the macroscopic response is isotropic.
The von Mises stress is defined as, while the effective mean stress is The modeling framework explored in this paper rests with the scope of the Terzaghi's effective stress concept, which allows the poro-elastoplastic problem to be treated as an equivalent dry problem.Hence, it suggests that this framework is valid for an incompressible matrix, with an associated plastic flow rule.
It can be deduced from the material modeling approaches, used in this work, that the porous rock has little ductility.Hence, the use of 0.15 critical volume fraction may yield overly optimistic collapse pressure in some cases.From the works of Tvergaard and Needleman (1984) and Koplik and Needleman (1987).and other experimental studies on void nucleation, the choice of a critical volume fraction of 0.015 for the onset of failure of the plastic region, around the borehole, is reasonable.Nevertheless, conducting a laboratory experiment should provide a more reasonable and accurate value of the critical volume fraction at the onset of the failure of the plastic zone.

Parametric studies
Several sensitivity studies are conducted in this work to investigate the impact of the rock mechanical properties, wellbore pressure, wellbore diameter, and well inclination and azimuthal angles on the variation of the plastic zone around the borehole and the mechanical stabilities of the borehole.Results from the elastoplastic models are compared with observed critical collapse mud weights from highly inclined wells.

Model validation
In this section of the paper, the shear and tensile stability predictions from the elastoplastic models are compared with linear elastic-based models, using published field and laboratory data.
The linear elastic-based shear failure criteria can be categorized into two main groups, based on how they can be fitted to the triaxial test data (Colmenares and Zoback, 2002;Benz and Schwab, 2008) and accurately predict the critical collapse mud weight.The study by Mclean and Addis (1990) concluded that some failure criteria can predict unrealistic results for some stress states while performing very well in other stress conditions.And as studied by Yi et al 2005, any failure criterion that can fit well to the polyaxial test data will perform very well in predicting the shear failure of the rock, thus providing a reliable collapse mud weight.
The shear failure criteria, which specify the stress conditions at failure, can also be classified into two categories based on the linearity or nonlinearity of the governing equations and consideration for the effect of intermediate principal stress in the governing equations.From the investigation conducted by Morita and Nagano (2016), using a failure criterion with a linear stress-strain requires adjusting the unconfined compressive strength of the rock to match the failure observed on image logs.  1 provides a summary of the five main failure theories, which are under consideration.

Test cases
The performance of both the elastoplastic-based models and linear elastic-based shear failure criteria are compared with observed critical collapse pressures in several field cases.Assuming a minor breakout width of less than 60 0 to be the onset of shear failure is sometimes practiced in field analyses (Zoback, 2010).But without caliper logs, the onset of shear failure can be inferred from drilling events.In the quest of comparing our analytical predictions with the field-reported mud weights, this criterion was considered.
Tullich field, North Sea.Shear failure was observed along the horizontal section of the well through a sandstone reservoir with inter-bedded claystone.The stress regime in this area was reported to be a normal faulting and the orientation of the maximum horizontal stress is between 40 0 and 60 0 (Russel et al. 2006) Table 2. below provides the summary of the rock properties and stress data for the reservoir zone of interest in this field.
Comparing the predictions from the elastoplastic models and the linear elastic failure criteria with the observed collapse mud weight of 10.94 ppg, shows the accuracy of the elastic brittle-plastic model (Table 3).In this field case, a normalized yielded zone area (NYZA) of 1 and the proposed framework gave close results to the observed collapse mud weight (Fig. 5 and Fig. 6).It should be noted that a critical volume fraction of 0.015 was used.Similarly, the Mogi-Coulomb, Drucker-Prager (circumscribed), and modified Lade (expressed by Ewy) linear elastic-based shear failure criteria also predicted closer values to the observed mud weight (Fig. 12).On the other hand, the elastic perfectly-plastic model underpredicted the observed mud weight, with an error of about 16%, which is significantly lower than the observed mud weight (Fig. 8).
Although the elastic-brittle plastic material model was coupled with the void-growth model in Fig. (7), the incomplete principal stress equations were used for the computation.No significant distinction between the results predicted with the complete and incomplete principal stress equations was observed in this case, because the rock pure shear strength is very low in this case.On the other hand, when the elastic-perfectly plastic material model is coupled with the Gurson model, there is a clear distinction between the computed results with the full and reduced principal stress equations (Fig. 8 and Fig. 9).This difference can be traced to the peak rock strength value, which is part of the formulation of the elastic-perfectly material model.11) are plotted to demonstrate that it is very important to choose wisely the critical volume fraction, if there is no experimental data on that rock sample.In this case, the rock material within the zone of interest is not ductile, and the choice of a default critical volume fraction value of 0.15 will not be appropriate, considering the predicted mud weights.
From these results, it is worth noting that the commonly used shear failure criteria predict the onset of failure or yielding of the borehole.In essence, the shear failure criteria will perform well in predicting the onset of yielding for both brittle and ductile rocks.For brittle rocks, the yielding and ultimate collapse pressure are almost the same, as yielding is not pronounced.But for ductile formations, the yielding is pronounced, and the ultimate collapse pressure can be significantly different from the yielding pressure.where,   is the octahedral shear stress and the mean normal stress.Where,        Northwest Shelf, Australia.The two wells of interest, Well A-1 and Well B-4 are in different stress regimes.Field A is located in a normal-strike slip faulting regime and the well was drilled in the direction of the minimum horizontal stress to reduce the risk of wellbore instability.On the other hand, field B is located in a normal faulting regime.Well B-4 experienced severe instabilities due to insufficient mud weight at about 9.6ppg while Well-A-1 experienced minor instabilities at a mud weight of 10.44ppg.Table 4 presents the rock properties and stress states at the depths of failure in the two wells (Chen et al. 2002).The predicted critical collapse mudweights for Well B-4 for the different models are shown in  observed critical mud weight, except the Drucker-Prager (circumscribed) failure criterion (Table 5 and Fig. 23).Similarly in the case of Well A-1, the results showed that the proposed framework, using the elastic brittleplastic material and a critical volume fraction of 0.015, and the NYZA approach with elastic brittle-plastic material model yielded close results to the observed field value of the critical collapse mud weight (Table 8, Fig. 25, and Fig. 26).The predictions of the critical collapse pressure and extent of the plastic zone are not close to the observed field data when the reduced forms of the principal stresses are used in this example case as well (Fig. 27).
And when the elastic perfectly-plastic material model was coupled to the Gurson model, using the reduced forms of the principal stresses, there was no solution (Fig. 29).24) is a cartoon representation of the predicted plastic zone around Well B-4, based on the proposed framework.Unlike in the case of Well 9/23a-T2, plastic deformation is observed to form all around wells A-1 and B-4 (Fig. 24 and Fig. 31).However, the extent of the plastic deformation around Well B-4 is small compared to Well A-1, which has similar stress loadings, except different rock properties.This thus indicates that the extent of plastic deformation strongly depends on the rock properties.But in this scenario (Well A-1), when the critical volume fraction of 0.15 is used, there are no possible solutions when the elastic-brittle plastic and elastic-perfectly plastic material models were coupled with Gurson void-growth model (Fig. 32 and Fig. 33).It thus indicates that it is very important to note that the use of 0.15 will not yield possible solutions in all cases, especially for relatively brittle rocks.

Discussions
From the test cases above, it is evident that the proposed framework, which couples the void-growth model with the material failure model of the rock, can predict realistic critical collapse mud-weights, provided the correct critical volume fraction is chosen.It is recommended to conduct a laboratory experiment to determine the critical volume fraction for each rock sample.But if there is no experiment, this study shows that the choice of a critical volume fraction of 0.015 and 0.15, for a slightly ductile and ductile formation respectively, will yield a realistic critical collapse pressure.
Furthermore, the test cases show that the critical volume fraction, rock strength, and in-situ stresses are few of the key parameters that greatly affect the extent of the plastic zone and the critical collapse pressure.Also, the elastoplastic material failure model chosen is critical.The results show that the elastic-brittle plastic model performs better than the elastic-perfectly plastic model, especially when the complete or full principal stresses are used.However, for weak rocks, with relatively low pure shear strengths, the use of

Material Model
Gurson either the incomplete or complete principal stresses in the modeling framework will yield relatively similar values.But if the formation has a high pure shear strength the complete principal stress should be used.Generally, the collapse mud weights, predicted with the elastic brittle-plastic model, are often closer to the observed mud weights than the elastic perfectly-plastic model.The elastic perfectly-plastic model predicts a lower collapse mud weight than the elastic brittle-plastic model, for a given critical volume fraction.But it is worth noting that the difference between the predictions from these two elastoplastic models reduces with decreasing rock strength.
In addition, the critical collapse mud weight predicted by the proposed framework compare very well with observed field data.However, though the NYZA = 1 approach may yield in some cases results close to the observed data, the predicted plastic zone extent around the borehole can be significantly different from that predicted by the proposed framework.More so, it is important to note that all the linear elastic-based shear failure criteria can be categorized into two main groups, based on how they can be fitted to the triaxial test data (Colmenares and Zoback, 2002;Benz and Schwab, 2008) and predict the onset of shear failure of the borehole.From the different cases explored in this work, it is evident that these models should be used with caution.Nevertheless, they provide the upper-bound for collapse mud weight estimation.However, the Drucker-Prager (circumscribed) criterion, known to underestimate the onset of shear failure, because it exaggerates the influence of the intermediate principal stress, yielded close values to the observed collapse mud weight in the field, as the elastic brittle-plastic model.

Sensitivity Studies
To highlight the impacts of the rock strength, wellbore diameter, critical volume fraction, and in-situ stresses on the extent of the plastic zone and collapse pressure of a borehole, the following sensitivity studies are conducted.

Influence of rock strength
The influence of rock strength on the shear stability of slightly and highly inclined boreholes are considered in this subsection.Using the data from Well 9/23a-T2, the uniaxial compressive of the formation in the elastic zone was varied from 2000 psi to 3000 psi, and the corresponding compressive strength and pure shear strength in the plastic zone were varied from 1400 psi to 2100 psi and 79.38 psi to 119.07 psi respectively.
Figs. (34, 35, and 36) show that with increase in rock strength, the shear stability of the borehole increases, as expected.In this sensitivity analysis, the elastic-brittle plastic material model was coupled with the Gurson void-growth model.The results also indicated that the well inclination and azimuthal angles have some impact on the extent of the failure zone.With increasing well inclination angle the extent of the plastic zone and collapse pressure increases, for a given set of rock mechanical properties.

Influence of in-situ horizontal stress difference
It is well-known in petroleum geomechanics that in wellbore stability analysis the shear stability of a borehole decreases as the difference between the maximum and minimum in-situ horizontal stresses increases.Data from Well 9/23a-T2 are used in this sensitivity study, with the minimum horizontal stress kept constant.The maximum in-situ horizontal stress was varied from 4687.5 psi to 6562.5 psi, at an interval of 937.5 psi.The overburden stress was set to 6000 psi.In this scenario as well, the critical collapse pressures were predicted with the complete equations of the principal stresses and an elastic brittle-plastic material model.

Effect of wellbore radius
Small boreholes have been observed from geomechanical analysis to be more stable than bigger boreholes.The essence of this sensitivity analysis is to see the impact of wellbore size, combined with varying well angles, on the extent of the plastic zone and the shear stability of the borehole.The data from Well 9/23a-T2 is used and wellbore radii values of 0.2 m, 0.4 m. and 0.6 m are considered for this analysis.
As the wellbore size increase, the shear stability of the borehole reduces with increasing well inclination angle (Figs.40, 41, and 42).This agrees with the generally observed behavior in borehole failure tests and numerical simulations (Gao and Ren, 2022;Meier et al. 2013).The plastic extent around the borehole also increases as the borehole size increases.

Impact of the critical volume fraction
It was briefly mentioned earlier in this paper that the choice of the critical volume fraction,  * , has significant impact on the extent of the plastic zone at the point of failure.The data from Well 9/23a-T2, and critical volume fractions of 0.015 and 0.09 are used for this sensitivity analysis.Figures (43), (44), and (45) shows that the extent of plastic zone increases as the critical volume fraction values increases.On the other hand, the critical collapse pressure reduces with increasing critical volume fraction.These results are indicative of the fact that the extent of plastic zone around the borehole increases with the ductility of the formation.And with increasing well inclination, the extent of the plastic zone around the borehole increases with increasing critical volume fraction.

Application
to mud performance design.

Summary
A new elastoplastic-based framework for estimating the critical collapse pressure and plastic zone around a borehole is presented in this work.In this framework an elastoplastic material model can be coupled with a void-growth model to simultaneously estimate the critical collapse pressure and plastic zone around the borehole.In this paper, an elastoplastic Mohr-Coulomb material model and Gurson void-growth model were coupled to introduce this new framework, although different elastoplastic material model and voidgrowth model can be coupled together.But caution should be taken in ensuring the yield surfaces of the material and void-growth model are not too far apart, to prevent non-convergence in the numerical computation.The proposed framework was validated with data from three different fields.The collapse mud weights predicted with the proposed framework compared well with observed critical mud weights in these wells.The proposed framework performed better than the normalized yielded zone area (NYZA) approach, though the difference can be negligible in some cases.
More so, the performances of the elastic brittle-plastic and elastic perfectly-plastic models for predicting the collapse pressure of a borehole were explored in this work.The stress distributions in the elastic and plastic zones around a highly inclined borehole are derived.Unlike other previously published elastoplastic wellbore stability solutions, this work considered the contributions of the anti-plane shear stresses.The pure shear strength of the formation is a key material property that determines the contributions of the anti-plane shear stress in the stability of the borehole.If the pure shear strength is disregarded, as is the case with other previously published solutions, the predicted collapse pressure will be overly optimistic, underestimating the collapse pressure.After conducting several sensitivity studies, the elastic brittle-plastic model performs better in predicting the shear stability of boreholes than the elastic perfectly-plastic model.
Furthermore, the linear elastic-based shear failure models should be used with caution.They provide mud weights corresponding to the onset of failure.For ductile or moderately ductile formations, their predictions will be overly conservative.However, the Drucker-Prager (circumscribed) criterion yielded close values to the observed collapse mud weight in the field, as the elastic brittle-plastic model.
In addition, this study also affirms that the shear stability of the borehole increases with rock strength.The use of the complete principal stress equations, which include the anti-plane shear stresses, is advised for stability analysis of highly inclined boreholes in formations with high strength.Disregarding the pure shear strength of the formation will result in underestimating the critical collapse pressure of a borehole.The results also indicated that the well inclination and its orientation have some impacts on the magnitude of the collapse pressure and the extent of the plastic zone around the borehole.
More so, with increasing horizontal stress difference, there is a decrease in the shear stability of the borehole.The shear stability of the borehole further reduces with increasing well inclination.
Also, this study also showed that the borehole shear stability reduces with increasing borehole size, and the extent of the plastic zone increases with borehole size.
Finally, the study demonstrated that a ductile formation, under the same loading conditions as a brittle formation, is more stable in shear.This was demonstrated through the critical volume fraction parameter.
A ductile formation has higher critical volume fraction than a brittle formation, hence, with increasing critical volume fraction, the critical collapse pressure reduces.

Declaration of Competing interests
The author declares that there are no conflicts of interest.These fictitious principal stresses now act on both the elastic and plastic region.To determine the stress concentration equations in the elastic region due to the actions of these stresses, the following boundary conditions hold: The compatibility equation by Lekhnitskii (1968), in cylindrical coordinates, is given as In the plastic region, the stress distributions are determined by substituting the yield function in Eq. ( 4) into the stress equilibrium equation.With the 3D problem divided into smaller subproblems, the in-plane and anti-plane stress distributions in the plastic region will be derived, using the appropriate stress equilibrium equation components.In the model development, it is assumed that the plastic zone surrounds the borehole, and it results from a monotonic loading.Furthermore, there is no elastic unloading in the plastically deformed region.

Fig 1 .
Fig 1.The stress-strain result of an experiment conducted on a rock sample showing an elastic-strain softening response (Li et al. 2015).

1 Fig 2 .
Fig 2. The stress-strain result of an experiment conducted on a rock sample showing both and elasticperfectly plastic and elastic-brittle plastic responses.

Fig 3 .
Fig 3. A representation of an inclined wellbore in an isotropic formation where   ,   ,  ℎ are the in-situ stresses.

Fig. ( 10 )
Fig.(10) and Fig.(11) are plotted to demonstrate that it is very important to choose wisely the critical volume fraction, if there is no experimental data on that rock sample.In this case, the rock material within the zone of interest is not ductile, and the choice of a default critical volume fraction value of 0.15 will not be appropriate, considering the predicted mud weights.

4 tan 2
(9−7 sin ) 1−sin   0 is the cohesive strength of the rock Nonlinear model Yes • Sometimes underpredicts or overpredicts the polyaxial strength of rock with relatively large margin.• Stress state exceeding the failure point does not depart from the failure surface with significant distance.Drucker-Prager An extension of the Von-Mises theory: √ 2 =  0 +  0  1 ; where for inscribed failure envelope the material parameters are  0 • Underestimates the critical collapse mud weight • Exaggerates the influence of the intermediate stress effect.

Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .
Fig. 5.The critical collapse pressure predicted from the elastic-brittle plastic and elastic-perfectly plastic models using the (a) full or complete principal stresses and (b) reduced principal stresses.The standard approach is to assume failure occurring at NYZA = 1 or NYZR = 1.414.These plots are based on the data for Well 9/23a-T2.(a) (b)

Fig. 10 .Fig. 11 .Fig. ( 13 )
Fig. 10.The proposed framework for the simultaneous prediction of the critical collapse pressure and the yielded/damaged zone around the wellbore using the elastic-brittle plastic material model and the Gurson void-growth model.In these plots, the complete or full principal stresses are used.Plots (a) and (c) depict the failure zone at the orientation of the least compression, while plot (b) shows the failed zone and the corresponding critical collapse pressure at the orientation of maximum compression.These plots are based on the data for Well 9/23a-T2 and a critical volume fraction of 0.15.

Fig. 12 .
Fig. 12.Using data from Well 9/23a-T2, the estimated collapse pressure and orientation is predicted using Mohr-Coulomb, Mogi-Coulomb, Modified Lade, Drucker Prager (inscribed), and Drucker Prager (circumscribed) failure criteria.In these plots, the complete or full principal stresses are used.

Fig. 13 .
Fig. 13.A representation of the failed zone around the wellbore (Well 9/23a-T2) predicted by the proposed framework.

Fig. 14 .Fig. 15 .Fig. 16 .Fig. 17 .Fig. 18 .Fig. 19 .Fig. 20 .
Fig. 14.The critical collapse pressure predicted from the elastic-brittle plastic and elastic-perfectly plastic models using the (a) full or complete principal stresses and (b) reduced principal stresses.The standard approach is to assume failure occurring at NYZA = 1 or NYZR = 1.414.These plots are based on the data for Well B-4.

Fig. 21 .Fig. 22 .
Fig. 21.The proposed framework for the simultaneous prediction of the critical collapse pressure and the yielded/damaged zone around the wellbore using the elastic-perfectly plastic material model and the Gurson void-growth model.In these plots, the complete or full principal stresses are used.Plots (a) and (c) depict the failure zone at the orientation of the least compression, while plot (b) shows the failed zone and the corresponding critical collapse pressure at the orientation of maximum compression.These plots are based on the data for Well B-4 and a critical volume fraction of 0.15.

Fig. 23 .
Fig. 23.Using data from Well B-4, the estimated collapse pressure and orientation is predicted using Mohr-Coulomb, Mogi-Coulomb, Modified Lade, Drucker Prager (inscribed), and Drucker Prager (circumscribed) failure criteria.In these plots, the complete or full principal stresses are used.

Fig. 24 .
Fig. 24.A representation of the failed zone around Well B-4 predicted by the proposed framework.

Fig. 25 .Fig. 26 .Fig. 27 .Fig. 29 .Fig. 28 .
Fig. 25.The critical collapse pressure predicted from the elastic-brittle plastic and elastic-perfectly plastic models using the (a) full or complete principal stresses and (b) reduced principal stresses.The standard approach is to assume failure occurring at NYZA = 1 or NYZR = 1.414.These plots are based on the data for Well A-1.

Fig. 30 .
Fig. 30.Using data from Well A-1, the estimated collapse pressure and orientation is predicted using Mohr-Coulomb, Mogi-Coulomb, Modified Lade, Drucker Prager (inscribed), and Drucker Prager (circumscribed) failure criteria.In these plots, the complete or full principal stresses are used.

Fig. 31 .
Fig. 31.A representation of the failed zone around Well A-1 predicted by the proposed framework.

Fig. 32 .Fig. 33 .
Fig. 32.In this scenario, there is no possible solution coupling the elastic-brittle plastic material model with the Gurson void-growth model.In these plots, the complete or full principal stresses are used.Plots (a) and (c) are at the orientation of minimum compression, while (b) is at the orientation of maximum compression.These plots are based on the data for Well A-1 and a critical volume fraction of 0.15.

Fig. ( 33 )
Fig. (33) shows that the failure envelope of the material shrinks with increasing critical volume fraction.The failure envelope of the Gurson void-growth model reduces to the von-Mises failure criterion when the critical volume fraction is zero.If the stress state in the material is not in the plastic region for both failure envelopesvoid growth and Mohr-Coulomb, there will not be a solution.As the critical volume fraction increases, the solution space increases, then it shrinks as void-growth failure envelope shrinks.Coupling the Gurson void-growth model with Mohr-Coulomb elastoplastic model the available solution space is limited, although it may increase with increasing critical volume fraction.This is one limitation of coupling the Mohr-Coulomb elastoplastic material model with the Gurson void-growth model, which is a form of von-Mises failure criterion.Despite this limitation, the proposed framework of coupling a void-growth model with an elastoplastic material model is sound.To improve on this present work, a Mogi-Coulomb elastoplastic material model may be coupled with the Gurson void-growth model for a future paper.

Fig. 33 .
Fig. 33.The failure envelopes of Mohr-Coulomb material model and Gurson void-growth model on the π-plane.

From Figs. ( 37 , 38 , and 39 )
it can be confirmed that the shear stability of a borehole reduces with increasing horizontal stress difference.And with increasing well inclination the borehole stability reduces.There are cases in this sensitivity study where there are no possible solutions when coupling the Mohr-Coulomb elastoplastic material model and the Gurson void-growth model (Figs.37a, 37d, 37g, 37j).The reason behind this has been discussed earlier, and the reader is advised to refer to the comments on Fig.(33).

Fig. A. 1 .
Fig. A.1.Rotating the coordinate by  0 , so only the fictitious principal stresses act in the plane for subproblem-1.

Table 1 :
A summary of the four commonly used rock linear elastic-based shear failure criteria

Table 2 :
Stress state and rock mechanical properties for Well 9/23a-T2

Table 4 :
Stress state and rock mechanical properties for wells A-1 and B-4

Table 6
The results showed that coupling the elastic brittle-plastic material model with the Gurson model and using a critical volume fraction of 0.015 yielded close results to the observed field value of the critical collapse mud weight.But if the reduced forms of the principal stresses are used the predicted (15)Figures(15)to (18).critical mud weights are less than the observed field data.The predictions from the elastic perfectly-plastic material model, with both the NYZA and the proposed framework yielded lower critical collapse mud weights.Furthermore, the predictions from the linear elastic-based shear failure criteria are not close to the

Table 5 :
Predicted mud weights for well B-4 using linear elastic shear failure criteria

Table 6 :
Predicted mud weights for well B-4 using elastoplastic material models

Table 7 :
Predicted mud weights for well A-1 using linear elastic shear failure criteria

Table 8 :
Predicted mud weights for well A-1 using elastoplastic material models