Xu Yng,Xingho Shi,b,*,Yingfeng Meng,Xioyong Xie

a State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation,Southwest Petroleum University,Chengdu,610500,China

b Key Laboratory Deep Underground Science and Engineering,(Sichuan University),Ministry of Education,Chengdu,610065,China

c Yibin University,Yibin,644000,China

ABSTRACT Borehole instability was frequently encountered during shale gas drilling.Most conventional models are not applicable to layered formation's wellbore stability analysis on account of anisotropic strength characteristic.In this study,an empirical equation for predicting anisotropic strength was implemented in the Mogi-Coulomb criterion to describe variations of cohesive strength and friction angle of shale formations.A collapse pressure model and its appropriate solution method for layered shale formations were proposed.The impact of different strength criteria and rock anisotropy type on rock strength and collapse pressure was investigated.The analysis indicated that the predicted strength of our modified criterion was usually higher than the weak plane failure criteria.The collapse pressure calculated by the modified Mogi-Coulomb criterion was lower than the weakplane failure criteria.Furthermore,it was more consistent with real mud weight.Additionally,the anisotropy type of rock notably influences wellbore stability.More significant anisotropy coefficients correspond to higher strengths,which results in smaller collapse pressure values.Improper anisotropy coefficients can over- or underpredict the collapse pressure.Reasonable estimates of collapse pressure of anisotropic rocks can be made through the modified Mogi-Coulomb criterion using limited experimental data and the anisotropy rock type.

Keywords:Shale Wellbore stability Collapse pressure Anisotropy

1.Introduction

Although shale gas has got rapid development in recent decades,wellbore instability remains a severe problem of shale gas horizontal drilling [1,2].Borehole instability can lead to sticking,packing,loss of circulation,etc.It additionally causes significant challenges to cementing and multistage fracturing.

The failure criterion plays a significant impact on shale gas wellbore stability analysis.Several criteria for wellbore stability have been discussed in the literature [3].It is found that the Mohr-Coulomb criterion is one of the most widely used strength criteria for predicting collapse pressure [4].Because its parameters carry physical meaning and can be determined through relatively simple and inexpensive tests,however,many researchers have found that the safe mud density may be overestimated by the Mohr-Coulomb criterion,which does not consider the intermediate principal stress when performing wellbore stability analysis [2,5].Consequently,several researchers recommended using other criteria,such as the Drucker-Prager criterion,for wellbore stability analysis [6,7].However,some researchers indicated that the Drucker-Prager criterion overemphasizes its effect,leading to an underestimated minimum mud pressure [2,8].Meanwhile,the Mogi-Coulomb (MG-C)criterion was proven to estimate the true-triaxial failure data reasonably well.It was thus recommended for wellbore stability analysis of isotropic formations [4].

Shale formations exhibit direction-dependent strength characteristics owing to the influence of the weak plane [9,10].Most conventional wellbore stability analyses treated the rock as linear-elastic,which may lead to incorrect results if someone carries out wellbore instability analysis of layered rocks [11].Therefore,various strength criteria and wellbore stability models considering anisotropic strength were proposed [12-14].The results of these studies showed that neglecting the anisotropic effects can result in an inaccurate assessment of wellbore stability.

Several criteria for anisotropic rocks provide fairly accurate simulations of experimental data.The most widely used strength criterion for shale formation is the single plane of weakness criterion [15].Most approaches for anisotropic rocks depend on a wide range of core experiments.Jaeger [12]proposed the single plane of weakness criterion,which has a sliding mode and a non-sliding mode.Jaeger's criterion assumes that the strength at orientation angleβ= 0° is equal to that atβ= 90°.However,core experiments show that anisotropic strength changes with orientation angle [16].The maximum failure strength of anisotropic rocks usually occurs at orientation angle 0° or 90°,and its minimum failure strength is usually tested atβbetween 30° and 45°[13,17].The single weakness plane criterion can be used for predicting anisotropic strength owing to discontinuity plane presence [14].However,it does not adequately describe anisotropic strength because of the presence of bedding or foliation.

The purpose of this paper is to modify the conventional MG-C criterion for layered shale formations.The predicted anisotropic strength was compared with that of Jaeger's criterion.A collapse pressure gradient model for layered shale formations was proposed.A numerical method and its solution flowchart were additionally developed.The effect of different strength criteria and rock anisotropy type on collapse pressure was analyzed.

2.Anisotropic strength criteria

The strength criteria for layered rocks can be divided into the discontinuous model and continuous model.The discontinuous model effectively predicts the strength behavior of anisotropic rock due to the discontinuity plane presence.The continuous models are more suitable for inherently anisotropic rocks.

2.1.Single plane of weakness

Jaeger's criterion [12]is a simple approach for predicting anisotropic strength.However,its predicted strength does not continuously vary with orientation angleβ.Jaeger's criterion can be expressed as[12,15],

When a shear failure does not occur along the plane of weakness,rock failure will occur across the rock body.The single plane of weakness model handled rock as an isotropic body when shear failure was caused by the weak plane.

2.2.Modified MG-C criterion for anisotropic rocks

The intermediate principal stressis not taken into consideration in conventional linear M-C criterion and the single plane of weakness theory.Various studies indicated that this criterion underestimates the rock strength; consequently,the predicted collapse pressure mud weight is too conservative [2,4,18].Therefore,several researchers have proposed various true-triaxial failure criteria.The MG-C criterion was proven to estimate the true-triaxial failure data reasonably well.It was thus recommended for wellbore stability analysis of isotropic formations [19].The MG-C criterion was used to estimate shear failure for anisotropic rocks,

whereηβ(i.e.,cβorφβin Eq.(4))is the predicted anisotropic strength for shale at angleβ;anddenote predicted anisotropic strength at angleβ= 0° andβ= 90°,respectively;represents the predicted minimum value of;mandnare fitted parameters that describe layered rock's anisotropy type.It should be noted thatβ′ of the cohesion is not necessarily equivalent to that of the friction angle.Moreover,smaller values ofmandn(usually lower than three)indicate U-type anisotropy (cleavage and schistosity),while larger values ofmandnindicate shoulder-type anisotropy (bedding plane)[11,13].Authors can refer to Shi et al.[11]for calculating the anisotropic strength by using Eq.(5).Besides,η0°,ηmin,andη90°should be determined through experiment.

2.3.Evaluation of anisotropic strength criteria

In order to evaluate the predicted strength values of the modified MG-C criterion for anisotropic rocks,we compare it with the experimental data collected from the literature.R2and AAREP are used to evaluate the capabilities [11,20],

whereNis the total number of collected experimental data;is the experimental results of anisotropic strength;indicates the calculated values of anisotropic compressive strength;denotes the mean value of the experimental data.

Fig.1 shows the cohesion and friction angle for Longmaxi shale with orientation angleβ.The data points represent experimental results tested by Jia et al.[21].The fitted parametersmandnwere listed in Table 1.The minimum shear strength values were obtained at orientation angles of 30° and 60°,respectively.The cohesion and the friction angle of the weak plane were 13.36 MPa and 30.97°,respectively.

Fig.1.Variations in cohesion and friction angle with Longmaxi shale orientations (tested by Jia et al.[21]).

Table 1 Fitting parameters for Longmaxi shale (tested by Jia et al.[21]).

Fig.2.Experimental and predicted shale strength (tested by Jia et al.[21]).

Fig.2 shows the experimental and predicted anisotropic strength for Longmaxi shale.The solid lines represent predictions obtained from Jaeger's criterion,while dotted lines represent predictions calculated by the modified MG-C criterion.The single plane of weakness criterion involves sliding and non-sliding failure modes.It is assumed that the corner of the shape represents the transition between these two modes.The anisotropy curve for sliding mode is almost U-shaped.Jaeger's criterion assumed a non-sliding shear failure of rock body,leading to a constant strength at low or high value of angleβ.The continuous model(modified MG-C)handles the anisotropic rocks as continuous media;thereby,the anisotropic strength varies relatively smoothly.

The relationship between the compressive strength and orientation angle determined from core tests is discontinuous.Therefore,it is hard to know whether the curve exists a sharp corner or not.For inherently anisotropic rocks,researchers tend to expected that anisotropic strength varies continuously with the orientation angle.It can be seen from the figure that the minimum strengths for the single plane of weakness theory are always evident atβ= 30° throughout the confining pressure range.For the modified MG-C criterion,it is apparent that the minimum of the curve shifts to larger orientation angles when the confining pressure increases.Several researchers observed and reported similar experimental tests [13,22].

TheR2and AAREP values for the single plane of weakness criterion are 0.898 and 5.36%,respectively.TheR2and AAREP values of the modified MG-C criterion for anisotropic rocks are 0.982 and 2.79%.The continuous criterion performs slightly better than the single plane of weakness criterion.

3.Wellbore stability model

3.1.Stress distribution

Previous studies showed that the effect of anisotropic elasticity on stress distribution around a wellbore is relatively small,especially for a low degree of anisotropy [6,23].Therefore,most of the stress analysis for shale wellbore stability does not take into account anisotropic elasticity.Fig.3 shows the relationship between different coordinates:rectangular coordinate (x,y,z),in-situ stress coordinate (xs,ys,zs),and borehole coordinate (xb,yb,zb),whereαsis the azimuth of minimum horizontal principal stress,βsis the deviation of in-situ vertical stress,αbis the borehole azimuth,andβbis the borehole deviation.The effective stress around the shale wellbore can be obtained by stress transformation between different coordinates [2],

whereσr,σθandσzare the effective stresses in the cylindrical coordinate;,andare the shear stresses in the cylindrical coordinate;δis the filtrate coefficient;φis the porosity;pmis the downhole pressure;αis the Boit's coefficient;ppis the formation pore pressure;θis the circumferential angle;νis the Poisson's ratio;,andare the axial stresses in the rectangular coordinate;,,andare the shear stresses in the rectangular coordinate;K1is the percolation coefficient,

The principal stresses around a deviated well in the cylindrical coordinate system are [2,15],

3.2.Calculation of anisotropic strength

To calculate the anisotropic strength at the borehole wall,one needs to determine the distribution of orientation angleβin Eq.(5).It can be calculated as [15],

where

wherenis the normal vector of weak-plane in the global coordinate;Nis the direction vector of maximum principal stress;αwis the dip direction of the weak plane;βwis the dip angle of the weak plane;i,j,andkare the direction vectors;γis the angle betweenz-axis and axis of the major principal stress.When the orientation angleβis obtained,the anisotropic strengths in Eqs.(4)and (5)can be calculated.

3.3.Solution method of collapse pressure

When we get the stress,anisotropic strength,and strength criteria described in the previous sections,the collapse pressure can be calculated.If Jaeger's criterion is used,one can definefspas follows,

Fig.3.Relationship between different coordinate systems.(a)In-situ stress coordinate.(b)Borehole coordinate.

Shear failure happens whenfsp≤ 0.Therefore,the minimum drilling mud weight can be predicted by using Eq.(16).

The solution method for collapse pressure mud weight is proposed,as shown in Fig.4.It tells how to calculate the collapse pressure mud weights for a given condition.A flowchart for the singe plane of weakness theory is not shown because it can be readily obtained elsewhere [10].

4.Shale wellbore stability analyses and discussions

The parameters for shale wellbore stability analysis used in this section are listed in Table 2.These parameters were extracted from Wang [24]and Lan et al.[5].

4.1.Strength distribution at the borehole wall

Fig.5 shows the variation of orientation angleβwith circumferential angleθ.In order to increase the hydraulic efficiency,the horizontal well is designed to be parallel to the direction of in-situ minimum principal stress (N45°E).Orientation angleβis in the range of 0.12° and 61.27°,meaning wellbore failure may primarily result from the sliding mode in which the plane of weakness predominates.

Fig.6 shows the variation of compressive strengths at the wellbore with the circumferential angle using different anisotropic criteria.The compressive strength predicted by the modified M-C criterion is also provided.The Jaeger's criterion handles the rock as an isotropic material when failure of rock body occurs.As previously stated,the sharp corner,therefore,implies the transition point of sliding and non-sliding failure modes.We noted that the strengths predicted by Jaeger's criterion are usually lower than the modified MG-C criterion,except in certain exceptional circumstances.The strength obtained from the modified MG-C criterion for anisotropic rocks varies gradually with circumferential angleθ.The modified MG-C criterion takes the intermediate principal stress into account,which has a strong effect on shale failure strength [8].It is found that strengths determined by the modified MG-C criterion are 13.7-20.8 MPa higher than the M-C criterion,which means M-C criterion may underestimate the rock strength for general stress situations [19].

Fig.4.Solution flowchart for the determination of collapse pressure mud weight.

4.2.Collapse pressure distribution

Fig.7 shows the distribution of collapse pressure gradient under different failure criteria.The collapse pressure gradient determined by the modified M-C criterion is usually higher than the other two criteria.The modified M-C criterion calculates a higher collapse pressuregradient than Jaeger's criterion for most of the circumstances.However,the prediction of the modified M-C criterion is a little smaller than the Jaeger's criterion under other circumstances (0-0.32 g/cm3smaller in this case).In addition,the collapse pressure gradient calculated by the modified MG-C criterion is 0.31-0.48 g/cm3lower than the result predicted by the M-C criterion.

Table 2 Parameters used in the example.

Fig.5.Variation of orientation angle with circumferential angle.

Fig.6.Variation of compressive strength at the wellbore with circumferential angle.

Two horizontal well J1-HF and J1-3HF were drilled in the Longmaxi shale gas reservoirs in Jiaoshiba area,Southeastern Sichuan Basin.The orientations of the two horizontal wells are aligned with the direction of minimum principal in-situ stress.In Fig.7,we can find that the collapse pressure mud weights predicted by Jaeger's criterion (Eq.(16)),modified M-C,and MG-C (Eq.(17))criteria are 1.82,1.80,and 1.48 g/cm3,respectively.

Fig.7.Hemisphere plot of collapse pressure gradient with well path.(a)Single plane of weakness.(b)Modified M-C criterion.(c)Modified MG-C criterion.

The J1-HF field case indicated that no wellbore instability problems occurred when the oil-based mud weight was in the range of 1.45-1.55 g/cm3[5].The average hole enlargement rate of J1-HF was 1.05%.The water-based drilling mud used in the second open drilling of J1-3HF was 1.44 g/cm3,and serious collapse accidents occurred.The average hole enlargement rate of the second open drilling of J1-3HF was 17.95% [5].In the third section of J1-3HF,the oil-based drilling mud was 1.50-1.65 g/cm3.The average hole enlargement rate was-1.29% [5].Therefore,the collapse pressure gradient calculated by the modified MG-C criterion matches better with drilling practice than modified M-C and the single plane of weakness theory.

4.3.Effect of anisotropy type on collapse pressure

Fig.8.Strength variation with the circumferential angle for different cohesion anisotropy coefficients.

In the application of the modified MG-C criterion to anisotropic rocks,anisotropy coefficientsmandnin Eq.(5)must be determined.If sufficient compressive test data can be obtained,parametersmandncan be fitted with these data.As previously stated,these parameters describe the anisotropy rock type.In this section,we analyze how the anisotropy rock type influences the strength and collapse pressure gradient.We employ the anisotropy coefficient of cohesion as an example.For case (a),m=n= 2,and the anisotropy type is almost a Ushape.For case (b),m=n= 4,and for case (c),m=n= 6; the anisotropy curve is almost a shoulder shape.

Fig.8 shows the variation of anisotropic strength predicted by the modified MG-C criterion for anisotropic rocks with a circumferential angle.It is clear from this figure that larger values ofmandncorrespond to higher compressive strengths.

Fig.9 depicts the hemisphere plot of the collapse pressure gradient with the well path for different cohesion anisotropy coefficients.It is evident that the anisotropy rock type has a significant impact on the collapse pressure gradient.For a U-type case (i.e.,case (a)),the predicted collapse pressure gradient is much greater.For the shoulder-type anisotropy cases (i.e.,cases (b)and (c)),the difference in collapse pressure gradient between case (b)and case (c)is relatively small.If impropermandnvalues are used,an over- or under-predicted collapse pressure gradient may be obtained.The collapse pressure gradients of the horizontal well (αb= 45°,βb= 90°)for the three cases are 1.52,1.48,and 1.47 g/cm3,respectively.

In some cases,it is impossible to get the anisotropic strength for the whole range of orientation angleβon account of a sample shortage or economic reasons,the propermandnvalues should be chosen based on the anisotropy type.In such a situation,it is assumed that the minor shear strength (anisotropic cohesion and friction angle)is at orientation angle 30°,and core tests atβ= 0°,30°,and 90° should be conducted[11].Fig.10 shows the hemisphere plot of the collapse pressure gradient with well path using these limited data.It is evident that the predicted collapse pressures are always lower than in the above analysis(see Fig.9).For example,the collapse pressure gradient of the case (c)in Fig.10(c)is 0-0.18 g/cm3lower than that in Fig.9(c).This is because the friction angle atβ= 30° is larger than the minimum value of(=60°).The collapsepressuregradientofthehorizontalwell(αb=45°,βb=90°)for cases (a),(b),and(c)are1.49,1.48,and1.48 g/cm3,respectively.If the angle corresponding to the minimum strength is not at orientationβ= 30°,an under-predicted pressure collapse may be obtained.It should also be noted that the under-prediction is relatively small in the example cases.Therefore,reasonable estimates of the pressure collapse can be made through the modified MG-C criterion using limited test data and the anisotropy rock type.

5.Conclusions

Fig.9.Hemisphere plot of the collapse pressure gradient with well path for different cohesion anisotropy coefficients.(a)m = n = 2.(b)m = n = 4.(c)m = n = 6.

The modified MG-C criterion was proposed to determine the anisotropic strength of shale formations.The criterion handles inherently anisotropic shale formations as continuous media,and the gradual variation of the anisotropic strength withβis considered.The commonly used stress distribution model around the borehole and our modified anisotropic strength model were utilized to develop the collapse pressure calculation model for layered shale formations.The solution chart for the layered shale collapse pressure model was also presented.

Fig.10.Comparison of the collapse pressure gradient with well path based on only three strength measurements.(a)m = n = 2,(b)m = n = 4,(c)m = n = 6.

The anisotropic strength determined by the modified MG-C criterion was compared with the single weak plane strength criterion and the modified linear M-C criterion.The results indicated that the single weak plane strength criterion usually predicts lower anisotropic strengths than the modified M-C and MG-C criteria,except in exceptional circumstances.The modified MG-C criterion takes the intermediate principal stress into consideration,and its predicted strength was much higher.

The influence of anisotropy type on wellbore stability was presented.The analysis showed that the modified M-C criterion and the single weak plane strength criterion are conservative and over-predict collapse pressure.The collapse pressure gradient predicted by the modified MG-C criterion was lower than the other two criteria.Furthermore,it was more consistent with real mud weight.A shouldertype anisotropy (higher anisotropy coefficient)corresponded to a higher strength,which resulted in smaller values of collapse pressure.

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (No.51774248)and Sichuan Province in China Key Science and Technology Foundation (No.2019YFH0166).

Appendix A.Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.petlm.2019.11.002.