Multivariable Regression 3D Failure Criteria for In-Situ Rock

Wellbore stability problems increase with the exploration and development of oil and gas reservoirs. A new 3D non-linear failure criterion is proposed as a trigonometric function considering the intermediate principal stress (σ 2 ) on the triaxial compression test data. Mohr-Coulomb and Hoek-Brown are well-known failure criteria, but they do not consider the influence of (σ 2 ) on rock strength. This new criterion produces a concave surface on the principal stress space (σ 1 ,σ 2 ,σ 3 ) with the influence of intermediate principal stress. In this study, sensitivity analysis for the variable is also done to understand the significant influence of parameters on the accuracy of the proposed criterion. Further validation of this non-linear criterion on three principal stresses (σ 1 ,σ 2 ,σ 3 ) was done compared with linear regression and second-degree polynomial regression results. It has been observed that the new non-linear 3D criterion with five material parameters reveals a good fit compared to linear regression and second-de-gree polynomial regression, which have four and six material parameters, respectively. The new non-linear criterion was further validated by comparison with existing criteria like the Priest, Drucker-Prager, and Mogi-Coulomb. It has been observed that the new 3D non-linear criterion shows a more accurate result than these existing criteria as certain rock types exhibit coefficient of determination (DC) values near one, precisely 0.95 for inada granite, 0.94 for orikabe monzonite, and 0.91 for KTB amphibolite. In contrast, other rock types have DC values ranging from 0.7 to 0.9. The new 3D non-linear criterion also yields lower root means square error (RMSE) values than the Mogi-Coulomb criterion for seven rock types. Specifically, the RMSE values by the new criterion are as follows: KTB amphibolite - 40.03 MPa, Dunham dolomite - 15.16 MPa, Shirahama sandstone - 9.08 MPa, Manazuru andesite - 22.14 MPa, Inada granite - 35.47 MPa, and Coconino sandstone - 19.047 MPa. This new 3D criterion gave precise predictions of the failure of the formation under in-situ stresses and was further helpful for the simulation of the wellbore in the petroleum industry. The variable in the new 3D criterion should be calculated from triaxial compression test data for each formation rock before applying this criterion to the wellbore stability problem and the sand production problem.


Introduction
Better prediction of the failure of rock near the wellbore has been essential over the last three to four decades.New oil and gas fields develop on the uncertainty of subsurface reservoir pressure, tectonic effect, and temperature gradient.More recent technology is required in drilling and production operations for deeper reservoirs.The problem of wellbore stability increased at high-pressure, high-temperature reservoirs with an inclination of the well in directional drilling.The prominent wellbore instability is brack-out of rock by a combination of tensile and compressive force on the geomechanical stress near the wellbore results from formation, overburden, and hydrostatic pressure (Mahetaji et al., 2020).Lots of wells are missed due to the instability of rock fractures.Geomechanical modeling with maximum wellbore stability solves uncertainty about the drilling well.Geomechanical stress is vital in numerical modeling (Zhang, 2019).
To recognize a failure phenomenon, a compatible and specific criterion must be used (Bou-Hamdan, 2022).While some materials fail in shear, some may fail due to plastic deformation.The reasons behind the near-wellbore instability problems are tensile failure of formation, shear failure without plastic deformation, formation collapse or compressive failure, erosive-cohesive failure, and creep failure during drilling.(Aadnoy and Looyeh, 2019).
Various failure criteria are introduced in the literature for the failure of different-different materials.Mohr-Coulomb criterion and Hoek-Brown failure criterion are well-known failure criteria that consider maximum and minimum principal stress (σ 1 , σ 3 ) and neglect the intermediate principal stress (σ 2 ) (Culshaw andUlusay, ISRM Suggested Method 2007-2014).Mohr-Coulomb criterion is driven by the linear relation of mean and shear stress.Mohr-Coulomb failure criterion is mainly recommended when principal stresses σ 1 , σ 2 , and σ 3 are compressive and in a limited range of mean stress.The Hoek-Brown failure criterion is the empirically derived relation between two principal stress σ 1 and σ 3 (Hoek and Brown, 1997;Hoek et al., 2002).Hoek-Brown criterion is a non-linear, parabolic curve with the independence of the effect of intermediate principal stress.Experimental evidence shows that the intermediate principal stress influences the failure of rock material (Takahashi and Koide, 1989).Mohr-Coulomb and Hoek-Brown's failure criterion does not consider the effect of intermediate principal stress, so the rock failure prediction under the triaxial stress condition is unreliable.These lead to developing several three-dimensional Hoek-Brown criteria given by Pan and Hudson (1988); Priest (2005); Zhang and Zhu (2007); Zhang (2008); Jiang and Yang (2020).Singh et al. (2011) and Mahetaji et al., (2023) developed a modified non-linear Mohr-Coulomb criterion considering the effect of intermediate principal stress on strength behavior.
Furthermore, failure criteria are introduced after considering the effect of intermediate principal stress, such as the Drucker and Prager criterion (Drucker and Prager, 1952) and the Lade criterion (Kim and Lade, 1984).Drucker-Prager criterion was a curvilinear 3D generalized form of the Mohr-Coulomb criterion for soils.Drucker-Prager criterion is the relation between normal and octahedral shear stress through material constant.The Lade criterion (Kim and Lade, 1984) and Modified Lade criterion (Ewy, 1999) relate to the respective first and third invariants of the effective stress tensor at the failure.Modified lade criterion is further used for sand production to produce well and wellbore stability problems with the effect of intermediate stress (Yi et al., 2005;Yi et al., 2006).
The failure criterion from the triaxial experimental testing data in terms of the power law, considering three principal stresses, is given by Mogi (1971).Other rock triaxial testing data also validate failure criterion based on the experiment data by Haimson and Chang (2000), Chang and Haimson (2000), Takahashi andKoide (1989), andXiaodong andHaimson (2016).Priest (2005) proposed the 3D failure criterion based on the extended Hoek-Brown yield criterion combined with the Drucker-Prager criterion.Melkoumian et al., 2008 further developed the 3D Hoek-Brown yield criterion by considering the weighting of intermediate principal stress.Further research and triaxial rock testing are necessary before applying any of the 3D failure criteria.
The study aims to propose a 3D non-linear failure criterion from the actual triaxial compression data.The proposed criterion may apply to a subsurface rock formation using the triaxial compression test of that rock formation.After that, this failure criterion is helpful for the wellbore stability problem and the prediction of sand production in producing well.A new failure criterion is proposed by considering the effect on the strength of the rock.Proposed failure criteria produce a curve surface as a failure envelope on a 3D space of (σ 1 , σ 2 , σ 3 ) that is easy to derive and interpret.Available triaxial data in the literature from various researchers (Mogi, 1971;1974;2007;Takahashi and Koide, 1989;Chang and Haimson, 2000;and Xiaodong and Haimson, 2016) as shown in Table 1 are used for the non-linear regression analysis of the proposed 3D criterion on principal stress space (σ 1 , σ 2 , σ 3 ).
Table 1.Sources of Laboratory Test Data for Ten Rock Samples.Rectangular prism 1.9 cm × 1.9 cm × 3.8 cm

Theory
The failure criteria are visualized and expressed with principal stress space f(σ 1 ,σ 2 ,σ 3 ), stress invariant f(I 1 ,J 2 ,θ), f(J 1 ,J 2 ,J 3 ) or f(I 1 ,I 2 ,I 3 ), or a version of 3D Haigh-Westergaard coordinates stress space (ξ, ρ, θ) which is a direct physical interpretation geometrically (Ottosen, 1977).The Geometric representation of a stress state in Haigh-Westergaard and principal stresses space is given in Figure 1.Haigh-Westergaard coordinates are widely used invariant in (ξ, ρ, θ) where ξ is the projection on the unit of the hydrostatic axis, ρ is the radial distance from a failure point P and θ is the similarity angle range of 0 to π/3 from the rotation of the axis. (1) The ξ-ρ plane is also called the rendulic plane.Principal stress σ 1 ,σ 2 , and σ 3 in the terms of the (ξ, ρ, θ) are given by Brannon et al., 2009.
Further validation of the proposed 3D criterion is done by comparison of predicted results with actual triaxial data on ten rock types on 3D Haigh-Westergaard coordinates stress space (ξ, ρ, θ).In addition, the sensitivity of the parameter on the prediction error and comparison of the proposed criterion with the Priest criterion are discussed on the ten rock types.The priest criterion combines the three-dimensional Drucker-Prager criterion (Drucker and Prager, 1952) and the two-dimensional Hoek-Brown criterion.The proposed 3D nonlinear failure criterion considers the trigonometric relation between on principal stress space (σ 1 , σ 2 , σ 3 ).Based on that, predicted strength by the new 3D criterion might indicate a more accurate result compared to the existing failure criteria.
Angle θ is the stress angle, and cos3θ is the Load parameter.When θ equals zero in compressive meridian and for θ is equal to 60֩ in tensile meridian

New Failure Criterion
In the conventional Mohr-Coulomb and Hoek-Brown criterion, the effect of the intermediate stress is neglected on the rock failure prediction.The experimental result of the triaxial compression failure test indicates the impact of the intermediate principal (σ 2 ) on the rock strength.Rock strength increases with the increment in the value.Mogi's (1971) failure criterion is based on the true triaxial testing data regarding the power law.That failure criterion best fits the actual triaxial experimental data by obeying the power law between normal octahedral stress and octahedral shear stress τ oct .
The sine function, often denoted as sin(x), is a fundamental trigonometric function that relates an angle in a right triangle to the ratio of the length of the side opposite the angle to the length of the hypotenuse.However, the sine function can also be extended to real numbers beyond the scope of triangles.The plot of the sine function exhibits a smooth wave that oscillates for the value of the θ in the range of 0 to 2π.It has been observed from the experimental data that the non-linear relation between principal stresses is matched with the upper crest side oscillating sine function wave that is constructed in a curvilinear plotting range of 0 to π (Shvarts and van Helden, 2022).So, choosing this sine function for the regression analyses of principal stress ( σ 1 ≥ σ 2 ≥ σ 3 ) for actual triaxial compression test data with five material parameters gives the best fitting result compared to linear and second-degree polynomial regression.The new 3D non-linear criterion as trigonometric sine function between the principal stresses as proposed, where σ 1 ,σ 2 , and σ 3 are the principal stresses and a,b 1 ,b 2 ,d 1 ,and d 2 are material constants for the best fitting with the actual triaxial data point.
Relation of the a and UCS with adopting σ 2 = 0 and σ 3 = 0 as, The value of σ 1 must be positive for σ 1 ≥ σ 2 ≥σ 3 compression failure in equation ( 8).For a positive value of σ 1 , the value of sin(b 1 σ 2 + d 1 ) and the value of sin(b 2 σ 3 + d 3 ) must be positive, so the range of (b 1 σ 2 + d 1 ) and (b 2 σ 3 + d 3 ) must be between 0 to π.All the strength parameters are easy to derive by the triaxial compression test data for a minimum six-core sample.
Equation 8 is a proposed non-linear failure criterion directly derived from non-linear 3D regression analysis considering the effect of σ 2 on the rock strength.Equation 8 produces a concave surface on the principal stress space (σ 1 ,σ 2 ,σ 3 ).
Proposed non-linear failure equation in the Haigh-Westergaard coordinate is given by the function of (ξ,ρ,θ) as This equation is complex to solve with the arithmetic method but easily solved by the numerical method.Newton Raphson's method is recommended to solve this equation with a quick convergent rate.Newton Raphson's method has a limitation in that f'(ξ,ρ,θ) is not zero.If that f'(ξ,ρ,θ), then it must be required to modify the Newton-Rapson method.Modifying code for the Newton-Rapson method is done by making a bridge for that range at f'(ξ,ρ,θ) nearly equal to zero or using another method like the bisection method.9represent failure criterion in the coordinate of (σ 1 ,σ 2 ,σ 3 ) and (ξ, ρ, θ), respectively.Till date, no such equations are available in the literature to predict the rock failure as Equation 8 and Equation 9.
Usually, in-situ condition rock break-out done by the compression and rock fracture is due to the combination of compression and tension (Bui et al., 2023;Ghassemi, 2017;Lee et al., 2012).Experimental data on true triaxial compression tests are available in the literature review.Estimating the strength parameters a,b 1 ,b 2 ,d 1 ,and d 2 is done by two methods: the non-linear regression analysis on the triaxial compression test data result and the analytical solution of the new failure criterion by the triaxial compression test data.Non-linear three-dimensional regression analysis was done to find the material parameters (a,b 1 ,b 2 ,d 1 ,and d 2 ), as shown in Table 2.The strength parameter is dependent on the rock property.The failure equation for each rock type is shown in Table 2.This equation considers three principal stresses in relation to the function of sine

Linear Multivariable Regression
Linear multivariable regression analysis is conducted on all these tenrock types to find the linear criterion that considers all three principal stresses in the linear relation.Linear regression analyses give the linear relation for the KBT amphibolite as Where, p 1 = co-efferent of the σ 2 =0.529 The value p 1 , p 2 ,and c 1 is dependent on the formation rock property that is given as the weighting of the intermediate principal stress, minimum principal stress on the strength of the formation, and compressive strength of the formation, respectively.The value of the maximum principal stress is calculated based on the above equation as a combination of the value at the yield point.The yield point is the point where the first crack is initiated.The equation for all ten-rock types based on the linear regression with a variable is given in Table 2.

Second Degree Polynomial Regression
Polynomial regression analysis on these ten-rock types for the failure in true-triaxial compression test is done the finding appropriated relationships between these three principal stresses.Polynomial regression analysis on the data of KBT amphibolite's triaxial compressive test is given as, Where, q 1 = co-efferent of the σ 2 = 1.03, q 2 = co-efferent of the σ 3 = 8.56, q 3 =co-efferent of the σ 2 2 = 0.0013, q 4 = co-efferent of the σ 2 σ 3 = 0.0026, q 5 = co-efferent of the σ 3 2 = 0.0248, c 2 = intersection =187.25 MPa The value of q 1 , q 2 , q 3 , q 4 , q 5 , and c 2 is determined by the formation rock property, which is defined as the weighting of the σ 2 , σ 3 , σ 2 2 , σ 2 , σ 3 , σ 3 2 , and compressive strength of the formation, respectively.The maximum principal stress is calculated using the above equation as a combination of the σ 2 and σ 3 values at the first crack begin.Table 2 shows the equation for all ten rock types based on linear regression with variables.(2005; 2009) developed a three-dimensional failure criterion by combining the three-dimensional Drucker-Prager criterion (Drucker and Prager, 1952) and the two-dimensional Hoek-Brown criterion.Three-dimensional effective failure stress (σ 1 f ) is estimated by

Priest
Where, σ 3hb is the minimum two-dimensional Hoek-Brown effective stress at failure, and σ 1hb is the maximum two-dimensional Hoek-Brown effective stress at failure given by σ 1hb is calculated by Hoek-Brown criterion (1997) Where, σ 1hb calculated by Hoek-Brown criterion (1997), w is a weighting factor, m b is the Hoek-Brown constant value, and s is a fracture parameter of the rock mass.

Drucker-Prager Criterion
The Drucker-Prager failure criterion is a curvilinear 3D generalized form of the Mohr-Coulomb criterion for soils.Drucker-Prager criterion is the relation between octahedral stress and octahedral shear stress τ oct through material constant.
Drucker-Prager Criterion is given by the equation (Drucker and Prager, 1952 λ and k are material constants.Drucker-Prager Criterion in Haigh-Westergaard coordinates.(ξ, ρ, θ) is given by equation. Here, Material constant a and b is simply related to the angle of internal friction (Ø), and cohesion (c) is given by Mogi-coulomb criterion in Haigh-Westergaard coordinates (ξ, ρ, θ) is given as.

Result and Discussion
Failure Prediction and Validation on Principal Stress Space (σ 1 , σ 2 , σ 3 ) The root means square error (RMSE) function was used to find the misfits between the triaxial test data and predicted strength by the failure criterion.The standard deviation of the residuals defines RMSE.
Where n is the number of the test data point of a specific rock.Residuals or prediction errors are defined by (21) σ i calc is i th calculated value by the failure criterion, and σ i test is the i-th tested data.The best fitting RMSE value is the minimum, and if prediction is the best fit of test data, the mean of the residuals is always equal to zero.
Figure 2 indicates the prediction of rock strength by the proposed criterion on ten rock types with a triaxial compression test.Scatter data indicate triaxial compression test data, and solid lines represent the data prediction by a single equation for all rock types shown in Table 2.
In Figure 2, the black dotted line is for σ 1 = σ 2 > σ 3 or σ 1 > σ 2 = σ 3 ; color solid is the proposed failure criterion, and scatter data is the experimental triaxial compression data point.The single-color solid line is for the fixed value of the minimum principal stress.For the shirahama sandstone, the RMSE value is minimum among all the rock types,, so we will discuss the shirahama sandstone shown in Figure 2. Color indication for a solid line when the value of σ 3 equal to 5 MPa, 8 MPa, 15 MPa, 20 MPa, 30 MPa, and 40MPa is gradually blue, red, forest green, purple, maroon, and lime.All solid line is plotted by a single equation given in Table 2 Table 3 shows the RMSE values of σ 1 for shirahama sandstone is a minimum of nearly 9.08 MPa and for the KTB Amphibolite is a maximum of nearly 65.03 MPa for the best fitting triaxial data point.RMSE values of σ 1 are between 9.08 MPa to 65.03 MPa for the rest of all rock types shown in Table 3. Figure 3 shows the distribution of the error percentage for predicted σ 1 for all the rock types.The equation defines the prediction error, calc is the i th value of predicted σ 1 from the triaxial data point.σ 1i test is the tested value of σ 1 on the triaxial data.The predicted error distribution for all the rock types is as a histogram shown in Figure 3.The distribution of data points is excellent and distributed equally in the range of -10 % to 10%, and the maximum number of data points distributed is near zero percentage error.Like Gaussian distribution, this type of indicator with a maximum number of data having an error percentage around zero is preferable.Error distribution for the shirahama sandstone, yamaguchi marble, inada granite, orikabe monzonite, manazuru andesite, and coconino sandstone is distributed assuredly as a Gaussian distribution, so it suggests that the new failure criterion works better for that rock types.Error distribution for the remaining rock types is also good, with a maximum error percentage near zero, but the distribution shape is not like a bell curve, shown in Figure 3. Validation on Haigh-Westergaard Coordinates Stress Space (ξ, ρ, θ) The proposed failure criterion is validated with the help of Haigh-Westergaard coordinates stress space (ξ, ρ, θ).The non-linear failure criterion generates a curve in stress space (ξ, ρ, θ), so validation can be done with the coefficient of determination (DC).To validate the proposed non-linear failure criterion here, we compare it with experimental triaxial test data of ten rock types, as shown in Figure 4.
Equation 9 is the failure criterion in Haigh-Westergaard coordinates.(ξ, ρ, θ).Here ρ is a dependent variable, ξ is an independent variable, and θ depends on rock type.Figure 4 Where ρ i calc i-th calculated value data, ρ i test is i-th tested data on the triaxial test, ρ - i test is the mean value of tested data by the triaxial test.DC=1 means it is an ideal case where test data agree with the zero misfits.The desirable value of the DC must be near one.Desirable values of DC are for Inada granite (DC=0.95),Orikabe monzonite (DC=0.94),and KTB amphibolite (DC=0.91).DC for the rest of the rock types (dunham dolomite, yamaguchi marble, mizuho trachyte, solnhofen limestone, shirahama sandstone, manazuru andesite, coconino sandstone) in the range of 0.7 < DC < 0.9 shown in In Figure 5 (i), the RMSE decreases from 66.83 MPa to 14.36 MPa when it increases from 200 to 300 and then increases again to 19.70 MPa when a value is more significant than 320.Hence, the best-fit value for the shirahama sandstone a is 300 with the minimum RMSE value (14.36 MPa).Figures 5(ii   The prediction of principal stress (σ 1 ) is made with the help of the combination of another two principal stresses (σ 2 and σ 3 ) by the failure criterion discussed above.Maximum principal stress at the failure using the New nonlinear criterion given in Figure 2 and by Priest Criterion is given as shown in APPENDIX I. Maximum principals stress is not directly driven by the Drucker-Prager criterion and Mogi-Coulomb criterion.The equation for the Drucker-Prager and Mogi-Coulomb criterion is difficult to determine the maximum principal stress as a dual result of the quadratic equation of the principal stresses.The solution of the Drucker-Prager and Mogi-Coulomb criterion is done with the help of the Haigh-Westergaard coordinates (ξ, ρ, θ) and then converted into the principal stresses for the comparison of the all-failure criterion which was discussed above.APPENDIX II and APPENDIX III gradually show the Drucker-Prager and Mogi-Coulomb criterion result.Comparisons of the RMSE value with linear regression and second-degree polynomial regression and with existing failure criteria are given in Table 3 and Figure 6 for all rock types.RMSE in MPa for the linear regression, the Drucker-Prager criterion, and the Priest criterion are higher for all the rock types than the New non-linear criteria, Mogi-Coulomb criterion, and polynomial equation.RMSE by New non-linear criteria, second-degree polynomial, and Mogi-Coulomb criteria gave the lowest value among the ten-rock type.Seven rocks have lower RMSE by the New non-linear criterion than RMSE by the Mogi-Coulomb criterion, as shown in Figure 6.The minimum RMSE value by Mogi-Coulomb is 8.36 MPa for yamaguchi marble, and the maximum is 55.51 MPa for orikabe monzonite.The maximum RMSE by the New non-linear criterion is 50.04 for orikabe monzonite, and the minimum is 9.08 for shirahama sandstone.

Figure 1 .
Figure 1.Geometric Representation of a Stress State in the Haigh-Westergaard and Principal Stresses Space (Recreated after Lian et al., 2013). 2 k can be expressed in terms of cohesion intercept and internal friction angle(Yi et al., 2005)    and Zimmerman (2005) proposed a failure criterion by considering the Mogi criterion (1971) and linear Mohr-coulomb failure criterion called the Mogi-Coulomb criterion.The Mogi-Coulomb criterion is the linear relation between octahedral stress (τ oct ) and mean effective stress (σ m2 ).= + * 2 . The regression analysis estimates parameters a = 299.3,b 1 =0.00754, d 1 =0.9390, b 2 =0.02235, and d 2 =0.2695.It has been observed that the parameter is independent of the principal stresses.

Figure 2 .
Figure 2. Failure Prediction by Proposed Criterion in Space for Ten Rock Type a

Figure 3
Figure 3 Error Percentage Distribution as Histogram for Ten Rock Types.
compares the proposed 3D failure criterion with triaxial compression test data.Scatter data on the plot indicate the triaxial test data, where the solid red line indicates the failure criterion by the equation (9).Validation of the proposed 3D failure criterion is done with the help of DC with the reliability prediction of by the failure criterion.

Figure 4 .
Figure 4. Validation of The New 3D Failure Criterion for the Ten Rock Types ), 5(iii), 5(iv), and 5(v) show the influence of the parameter (b 1 , b 2 , d 1 , and d 2 ) gradually on RMSE (MPa) of σ 1 when remain parameters are static and given in Table2for the shirahama sandstone.Hence best-fit value for b 1 is 0.0055 with minimum RMSE (9.33 MPa); is d 1 0.8 minimum RMSE (11.63MPa); b 2 is 0.0225 minimum RMSE (14.36 MPa), and d 2 is 0.27 minimum RMSE (14.35MPa) exclusively.Best fitting value of parameters a, b 1 ,b 2 , d 1 , and d 2 with minimum RMSE given by optimization methods like gradient descent and stochastic gradient descent in machine learning and deep learning.

Figure 5 .
Figure 5. Sensitivity Analysis of Parameters a, b 1 ,b 2 , d 1 , and d 2 of New Non-linear Trigonometric Criterion for Failure Prediction in RMSE.

Figure 6 .
Figure 6.Comparisons of RMSE for by New 3D Non-linear Failure Criterion with Linear, Second-degree Polynomial, and Existing Criteria: Dracker-Prager, Mogi-Coulomb, and Simplified Priest Criterion.

Table 2 .
Regression Analysis for Ten Rock Types.

Table 3 .
Comparison of New 3D Non-linear Criterion with Linear, Polynomial Criterion, and Existing Failure Criterion for RMSE of σ1 (MPa)