1. 1 Tatiana Chichinina * Irina Obolentseva and Geser Dugarov (Trofimuk Institute of Petroleum Geology and Geophysics, Russian Academy of Sciences Siberian Branch ) ( Instituto Mexicano del Petróleo), ИНГГ СОРАН Effective-Medium Anisotropic models of Fractured Rocks of TI Symmetry: Analysis of Constraints and Limitations in Linear Slip model

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3. 3 Schoenberg (1980); Schoenberg & Sayers (1995) Linear Slip model Сompliance based models ε = S σ, where S = S 0 + S fr Effective Stress Field model Sayers & Kachanov, 1991; (4) Plate- stack model Independent components Equal components Components related to others by certain equation Hudson (1980) Penny-shaped crack model Bakulin A., Grechka V., Tsvankin I., 2000, Estimation of fracture parameters from reflection seismic data – Part I: HTI model due to a single fracture set : Geophysics, v. 65, p. 1788–1802. С = S -1

4. 4 Effective medium models Сompliance based σ = С ε ε = S σ Stiffness based S = S 0 + S fr С = С 0 + С fr  С = S -1 Differential Effective Medium methods Single effective inclusion methods (Eshelby, 1957), Nishizawa, 1982; Sheng, 1990; Hornby et al., 1994 Linear Slip model Schoenberg, 1980; Schoenberg and Sayers, 1995; Jakobsen et al. 2003 Agersborg et al, 2009 T-matrix method General Singular Approximation Shermergor, 1977 Bayuk,2010 Self-Consistent methods O’Connell & Budiansky (1974), Willis, 1977; Hoenig, 1979 Sayers & Kachanov, 1991; Schoenberg & Sayers 1995; Sayers, 2008 ; Kachanov, 1992; Kachanov et al., 2003; Grechka and Kachanov, 2006a, 2006b) Effective Stress Field (ESF) method Hu & McMechan, 2009 Kuster, Toksöz, 1974 Penny- shaped crack model Smoothing methods Hudson 1980, 1981,1994. Hudson et al., 1996 Mori & Tanaka, 1973; Benveniste, 1987; Zhao et al., 1989;

5. С 13 → V P 45 5 с 13 с 13 2 с 13 с 66 = с 33 (с 11 – 2с 66 ) - с 2 13 Hsu & Schoenberg, 1993 Constraint on С 13 for the Linear Slip model VTI-model stiffness tensor C (5)(5)

6. ∆T∆T ∆N∆N Fracture weakness - Normal weakness - Tangential weakness where and 4 independent parameters : ∆ N, ∆ T, λ and µ The Linear- Slip model stiffness tensor C (4) Hsu & Schoenberg, 1993 C = S -1 6 ( of the VTI symmetry)

7. 7 First step (#1): S 0 + S fr = S Schoenberg, M., 1980. Elastic wave behavior across linear slip interfaces, J. Acoust. Soc. Am., 68 S  C, where С = S - 1 Second step (#2) : Schoenberg M., Muir F., 1989, A calculus for finely layered anisotropic media: Geophysics, 54 Algorithm of the Linear-Slip model constructing Constructing compliance tensor S с 13 2 с 13 с 66 = с 13 = с 33 (с 11 – 2с 66 ) - с 2 13 s 13 = s 12 Constraint for the Linear-Slip model (4) S = c 13 Constraint on c 13 in in the Linear-Slip model

8. (2) S0S0 S 0 - Compliance tensor for isotropic host rock (no cracks) 8 Independent components Equal components -Components related by certain equations with others -- Equal components E -- Young’s modulus ν -- Poisson’s ratio E -- Young’s modulus ν -- Poisson’s ratio s 13 = s 12

9. 9 Isotropic rock Isotropic rock σεaσεa E = Young’s modulus Poisson’s ratio x3x3 x1x1 x2x2 ε r εaεa ν =ν = σ σ εaεaεaεa εrεr However in transversely isotropic rock, there are two Young’ s moduli: Е 1 and Е 3, and three Poisson’ s ratios: ν 31, ν 12 and ν 13. Two horizontal Poisson’ s ratios Vertical Poisson’ s ratio Poisson´s ratio is always greater than zero: ν > 0 (excluding auxetic materials)

10. 10 (4) ZNZN ZTZT Fracture compliances: - normal - tangential compliance S 12 = S 13 S 31 = S 21 E is Young’s modulus ν is Poisson’ s ratio of isotropic host rock (no fractures) S 0 + S fr = S Compliance tensor S for the Linear Slip model The first step (№1): Poisson´s ratio is always greater than zero: ν > 0 (excluding auxetic materials) S  C, where С = S -1 The second step (№2):

11. 11 S 21 S31S31 Two Young’ s moduli: Е 1 and Е 3 Three Poisson’ s ratios: ν 31, ν 12 and ν 13. Two horizontal Poisson’ s ratios Note that in the overall TI model, in contrast to the LS model, the two compliances S 12 and S 13 are not equal to each other. S 12 ≠ S 13 (5)(5) S 21 ≠ S 31 due to ν 12 ≠ ν 13 (4)(4) Linear Slip model Overall TI model Compliance tensor S of overall Transversely Isotropic (TI) model (VTI)

12. 12 σ σ εaεaεaεa εrεr Vertical (0°) plug (0°) Vertical (0°) plug (0°) ε r εaεa ν =ν = Vertical Poisson’s ratio Vertical Young’s modulus E =E = σεaσεa Vertical Young’s modulus E3 =E3 = σ 33 ε 33 ν 31 = Vertical Poisson’s ratio ε 31 ε 33 σ Horizontal plug (90°) Horizontal plug (90°) σ Two horizontal Poisson’s ratios One horizontal Young’s modulus x3x3 x1x1 x2x2 1 Vertical Poisson’ s ratio and 2 horizontal Poisson’ s ratios in Transversely Isotropic rock (VTI)

13. 13 x3x3 x1x1 x2x2 Horizontal Young’s modulus E 1 = σ 11 ε 11 ν 13 > ν 12 > 0 Horizontal plug (90°) Horizontal plug (90°) 13 ε 13 ε 11 ε12 ε12 ν 12 = and ε 11 ν13 = ν13 = Two horizontal Poisson’s ratios εrεr σ εaεaεaεa σ εrεr εrεr Two horizontal Poisson’ s ratios, ν 13 and ν 12, in Transversely Isotropic rock

14. 14 Two horizontal Poisson's ratios in transversely isotropic (TI) medium: One of them ( ν 13 ) will be always greater than another one ( ν 12 ): ν 13 > ν 12 Uniaxial compressionDeformation of the cylinder εrεr εaεa x1x1 x2x2 x3x3 ν 13 ν12ν12ν12ν12 BeforeAfter ν 13 > ν 12 Horizontal plug of TI-rock under uniaxial compression: Anisotropic expansion in the cross-section Poisson's ratios Yan F., Han D.-H., Yao Q., 2013, Physical constraints on c 13 and Thomsen parameter delta for VTI rocks. SEG Yan F., Han D.-H., Yao Q., 2015, Geophysical Prospecting. Article first published online: 27 JUL 2015

15. 15 x3x3 x2x2 x1x1 ν 13 > ν 12 ε12ε12ε12ε12 ε 13 εrεr ε 11 σ ε 12 ε 13 After x3x3 x1x1 x2x2 Before Deformation of the cross-section of the plug under uniaxial stress It was a circle Now it has turned into an oval. ε 13 > ε 12 Radial strains Two horizontal Poisson’s ratios

16. 16 Dry unsaturated samples of VTI-type layered rocks and layered materials such as graphite (experimental data from: Chenevert & Gatlin, 1965; Blakslee et al., 1970; Colak, 1998; Gercek, 2007; Ruiz-Pena, 1998; Gross et al., 2011; Sone & Zoback, 2013.) Yan F., Han D.-H., Yao Q., 2013, Physical constraints on c 13 and Thomsen parameter delta for VTI rocks. SEG, 2013 Yan F., Han D.-H., Yao Q., 2015, Geophysical Prospecting. Article first published online: 27 JUL 2015 13 > 12 ν 12 = ν 13 εrεr σ σ 12 13 Static measurements of Poisson's ratios 13 and 12 in rock samples

17. 13 > 12 Dynamic Poisson’s ratios, ν 12 versus ν 13, are estimated from ultrasonic velocity measurements in unsaturated shale samples (the data source from Thomsen, 1986; Johnston and Christensen, 1995). ν 12 = ν 13 17 Dynamic Poisson's ratios ν 12 and ν 13 estimated on rocks samples

18. С 13 2 С 13 С 66 С 11 С 33 – С 13 2 ν 13 = С33С33 (C 11 - 2С 66 ) С 11 С 33 – С 13 2 ν 12 = – С 13 2 18 С33С33 (C 11 - 2С 66 ) С 11 С 33 – С 13 2 ν 12 = – С 13 2 Dynamic horizontal Poisson's ratios ν 13 and ν 12 are estimated by the following formulas (from the velocity measurements): LS C 13 C 13 2 C 13 C 66 = C 33 (C 11 – 2C 66 ) - C 2 13 s 31 = s 21 LS The restrictions on the c 13 and s 31 for the Linear Slip (LS) model are linked with each other by the constraint equality: ν 13 = ν 1 2 = The equality ν 13 = ν 1 2 is not physically feasible.   Therefore the constraint on C 13 for the LS model is not physically feasible, too.

19. С 13_min = С 33 (C 11 - 2С 66 ) + С 66 2 – С 66 19 С 33 (C 11 - 2С 66 ) – С 13 С 11 С 33 – С 13 2 22 2 ν 12 ν 12 = 2 2 С 13 С 66 С 11 С 33 – С 13 2 ν 13 ν 13 = > > 2 С 13 С 66 С 33 (C 11 - 2С 66 ) – С 13 2 С 13 + 2 С 13 С 66 - С 33 (C 11 - 2С 66 ) > 0 2 С 13 > С 13_min Yan F., Han D.-H., Yao Q., 2013, SEG Yan F., Han D.-H., Yao Q., 2015. Geophysical Prospecting. Physical constraint on C13 Physical constraint on C13! The lower limit of the С 13 can be revealed from the inequality of the Poisson’s ratios

20. 20 S 21 S31S31 ν 13 > ν 12 | S 31 | > |S 21 | The Poisson ratio ν 13 is always greater than the Poisson ratio ν 12, therefore S 31 is always greater than S 21 in the compliance tensor of overall transversely isotropic media Whilst the equality for the Linear Slip model, S 31 = S 21, is not feasible, as well as the formula for the С 13_LS is not feasible, too. С 13 > С 13_LS, where С 13_LS ≡ С 13_min Yan F., Han D.-H., Yao Q., 2013, SEG Yan F., Han D.-H., Yao Q., 2015. Geophysical Prospecting.

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22. Colin Sayers justifies the Linear Slip model for shales by the following manner: “Because the regions between clay particles are expected to be more compliant than the particles themselves are, they can be treated as “horizontal fractures”. Therefore it is important to account for the additional compliances (and/or the fracture weaknesses) of such regions in any model of elastic wave propagation through shales. (Sayers, 1999, 2008, 2013).“ Sayers, C.M., 1999, Stress-dependent seismic anisotropy of shales: Geophysics, 64, 93–98. Sayers, C.M., 2008, The effect of low aspect ratio pores on the seismic anisotropy of shales: 78th Annual International Meeting, SEG, Expanded Abstracts, 2750–2754. Sayers, C.M., 2013, The effect of anisotropy on the Young’s moduli and Poisson’s ratios of shales: Geophysical Prospecting, 61, 416– 426. VTI Linear Slip model for shales z 22

23. The upper and the lower bounds of V P (θ) and V SV (θ) are calculated using the physical constraints for C 13 proposed by Yan, Han and Yao (2013): 23 θ is the wave incidence angle relatively the symmetry axis of VTI rock С 13_min < С 13 < С 13 _MAX. Phase velocities V P (θ), V SV (θ) and V SH (θ) Yan F., Han D.-H., Yao Q., 2013, Physical constraints on c13 and Thomsen parameter delta for VTI rocks: SEG Expanded Absracts С 13_min С 13 _MAX We used the elastic moduli c 33, c 11, c 44, and c 66, measured in shale cores according to the data from the paper of Sone & Zoback (2014). Haynesville-1 shale at the pressure P = 90 MPa

24. Shales It is the deviation ΔC 13, which is calculated as 18 % 21 % 19 % 14 % Estimations of the difference, ΔC 13, between the C 13 measured in shale rocks´ samples in comparison with the theoretically predicted C 13_LS in the Linear Slip model Results of our statistical analysis (of total 140 shale-rock samples) Such a column indicates a number of rock samples falling into the 10%-width interval of the ΔC 13 - deviation. ΔC 13, % Number of samples 24 N [We took the original data from Thomsen, 1986; Jakobsen, Johansen, 2000; Wang, 2002].

25. Results of statistical analysis of the deviation ΔC 13 measured in sandstones and carbonates, N – number of samples [We took the original data from Thomsen, 1986; Jakobsen, Johansen, 2000; Wang, 2002]. ΔC 13 is a relative error in estimating of C 13 by the Linear- Slip theory in comparison with the C 13 measured in real rock samples. These columns indicate a number of rock samples (N) falling into the [0; 10%]- interval of the error ΔC 13 For majority of the sand- stone and carbonate samples, a greater number of samples falls into the ΔC 13 - interval [0; 10%] than into another ΔC 13 - intervals. 10% The LS is better suited for sandstones and carbonates than for shales, because shales have the greatest number of samples fallen into the С13 - interval [0; 30%], which is three times wider than the interval [0; 10%] for the case of sandstones and carbonates. Significant portion of shale samples got the C 13 -error of more than 100%, whilst in the case of sandstones and carbonates it never occurred. 30% 25

26. 26 The LS is better suited for sandstones and carbonates than for shales. Shale rocks have greater anisotropy, and therefore the greater Δ N and Δ T. This implies the greater error in C13-estimating by the LS equation.  For a weak anisotropy, the weaknesses Δ N and Δ T are small, as it is displayed here for sandstones and carbonates; and therefore the Linear Slip (LS) model may work well, as if it were isotropic rock. (For isotropic rock the LS-equation is always satisfied.) Δ N  0, and Δ T  0

27. 27  For a weak anisotropy (Δ N  0, and Δ T  0) the Linear Slip model may work well.  Also in some special theoretical cases the Linear Slip model can successfully work, for example in such cases when: Δ N = Δ T, as well as Δ N = 0, and Δ T ≠ 0 (fluid-saturated cracks).  In these cases, the restriction on C 13 for the LS model turns into mathematical identity, i.e. it is automatically satisfied.  For a weak anisotropy (Δ N  0, and Δ T  0) the Linear Slip model may work well.  Also in some special theoretical cases the Linear Slip model can successfully work, for example in such cases when: Δ N = Δ T, as well as Δ N = 0, and Δ T ≠ 0 (fluid-saturated cracks).  In these cases, the restriction on C 13 for the LS model turns into mathematical identity, i.e. it is automatically satisfied.

28. Data from Sone & Zoback (2014) S H -wave S V -wave Velocities in Bossier / Haynesville shale, pressure P = 17.2 MPa δ = 2g(Δ N - Δ T ) / (1 - Δ N ) = 0 Thomsen´ parameter δ =0: ΔС 13 ≈ 0 In some special cases, the Linear Slip model can work successfully, for example in such a case when Δ N = Δ T 27 P-wave In this case, the restriction on C 13 for the LS model, turns into mathematical identity, i.e. it is automatically satisfied. С 13_LS = С 33 (C 11 - 2С 66 ) + С 66 2 – С 66, (where )

29. 29  This is an example of SATURATED SHALES (Δ N  0, and Δ T ≠ 0) The LS is better suited for saturated shales; that is the greatest number of samples fallen into the available ∆С13- interval [0; 10%]. 10% 30% In the interval [0.30 %], there is a tendency of ∆C 13 -reducing to 10 % for saturated shales. 30% In the interval [0.30%], there is a tendency of decrease of the relative error in the C13-estimating for saturated shales In the interval [0.30%], there is a tendency of increase of the relative error in the C13- estimating for all shales up to 30%.

30. 30 Application of the Linear Slip model is not recommended in estimating of the attributes, which include the C 13, such as: Thomsen’s parameter δ, the normal weakness Δ N, the normal compliance Z N, and the ratio Z N /Z T. Chichinina T., Obolentseva I., Gik L., Bobrov B., 2009, Attenuation anisotropy in the linear–slip model: Interpretation of physical modeling data: Geophysics, 74. Hsu, C.-J., and M. Schoenberg, 1993, Elastic waves through a simulated fractured medium: Geophysics, 58. Far M., 2011; Far M. et al, 2014 For providing these conclusions we analyzed the physical modeling data from: Plate- stack model Representative medium of compressed perspex plates

31. Two ways in δ estimating The input data from the paper of Hsu & Schoenberg, 1993 Plate- stack model C 13 from ultrasonic measurements of the velocity V P 45 C 13 from ultrasonic measurements of the velocity V P 45 V P 45 C 13 from the Linear Slip formula C 13 from the Linear Slip formula Error in estimating of Thomsen’s anisotropy parameter δ by the Linear-Slip formula Chichinina et al.,2015, SEG True 29% error Erroneous The error of 4% in the constant C 13 entails the 30%- error in estimating of the parameter δ. 31

32. The input data from the paper of Hsu & Schoenberg, 1993 Plate- stack model 18% error The error of 4% in the constant C 13 entails the 18%-error in estimating of the normal weakness ∆ N Estimating of the normal weakness ∆ N by the Linear-Slip formula True Erroneous Estimating of the normal weakness ∆ N from real data on C 13 32

33. Error in the estimating of C 13 by the Linear Slip theory Plate-stack model with inclusions (Far. 2011) cannot be identified by the Linear Slip model ∆C 13 ≤ 87 % (90/120 kHz) ∆C 13 ≤ 69 % (480 kHz) Model with inclusions (Far. 2011) 0 % ≤ ∆C 13 ≤ 15 % 87 % 69 % 33

34. 34 Physical modeling of Far et al. 2011 Far M., Figueiredo J.J.S., Stewart R.R., Castagna J.P., Han D-H., Dyaur N., 2014, Measurements of seismic anisotropy and fracture compliances in synthetic fractured media: Geophys. J. Int., 2014, v. 197. –No data on Vp_45 Far M., 2011, Seismic characterization of naturally fractured reservoirs: PhD Thesis, 2011, University of Houston. -- There are the data of Vp_45 Model 2. As above but “cracks” contain rubber pellet inclusions Model 1. Representative medium of compressed perspex plates

35. 35 Plate-stack model no inclusions Error in the estimating of C 13 by the Linear Slip theory (data from Far, 2011, no inclusions)

36. Error in the δ – estimating by the LS formula Thomsen’s anisotropy parameter δ. δ δ 90/120 kHz 480 kHz Data from Far (2011), Plate-stack model, no rubber inclusions 67%-error The error of 20% in the constant C 13 entails the 67%- error in estimating of the parameter δ. 100%-error The error of 15% in the constant C 13 entails the 100%- error in estimating of the parameter δ. 36

37.  The main conclusion is that the most popular effective-medium models of fractured media, such as the Linear Slip model of Schoenberg and Sayers and the Penny-Shaped Hudson´s model should be used with great caution.  Typically, in general, the formula for the C 13 of the Linear-Slip theory is not applicable for real rocks of TI-symmetry.  In any case, first of all you must see how great is the error ΔC 13 : The Linear-Slip model should be used with the caution in every attribute calculation, which is linked to the C 13 (For example, the velocity estimation at 45⁰, Thomsen’s parameter δ, the normal weakness Δ N, the normal compliance Z N, the ratio Z N /Z T, and the minimum horizontal stress σ h ). Conclusions 37

38. 38 Thank you! !Gracias! http://chichinina.rusiamexico.com/wp-content/uploads/2015/10/Chichinina_SEG2015.pptx

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40. ∆T∆T ∆N∆N Fracture weakness - normal - tangential weakness Z N и Z T -- Fracture compliance Bakulin A., Grechka V., Tsvankin I. Estimation of fracture parameters from reflection seismic data – Part I: HTI model due to a single fracture set : Geophysics, 2000, v. 65, p. 1788–1802. Z N / Z T  0 - saturated (oil / water), Z N / Z T  1 - dry fractures (gas) The normal compliance Z N

41. Two ways in ∆ N -estimating by Schoenberg C 13 from the Linear Slip formula С 13 → V P 45 C 13 from ultrasonic measurements of the velocity V P 45 The normal weakness ∆ N Hsu & Schoenberg, 1993

42. 1 -st way (LS): Δ N1 – with the use of the equation for C 13 of the Linear Slip (LS) model 2 -nd way of Δ N calculation Four ways in ∆ N – estimating Various C ij -combinations in formulas: 4 -th way of Δ N calculation 3- rd way of Δ N calculation 1 2 3 4 The normal weakness ∆ N By Tatiana Chichinina Chichinina et al.,2015, SEG

43. Normal weakness ∆ N estimated by 4 ways Input data from the paper of Hsu & Schoenberg, 1993 Plate- stack model Chichinina et al.,2015, SEG

44. 44 Input data from the paper of Hsu & Schoenberg, 1993 Normal compliance Z N estimated by 4 ways Chichinina et al.,2015, SEG

45. Four ways of Z N /Z T -estimating Z N / Z T  0 - saturated, Z N / Z T  1 - dry fractures (gas) Saturated (by honey), Chichinina et al.,2015, SEG

46. 46 Physical modeling of Far et al. 2011 With inclusions No inclusions Representative medium of compressed perspex plates. Besides of that “cracks” contain rubber pellet inclusions Representative medium of compressed perspex plates

47. 47 Plate-stack model with inclusions It cannot be identified by the Linear Slip model Error in the estimating of C 13 by the Linear Slip theory Chichinina et al.,2015, SEG

48. 48 The type of the velocity data, which we used Terrible scatter of the estimated Δ N –s in the model with Inclusions (90/120 kHz) The data from Far (2011), the model with inclusions, 90/120 kHz Chichinina et al.,2015, SEG

49. 49 The normal fracture weakness ∆ N estimated by 4 ways in plate-stack model, no inclusions 90/ 120 KHz 480 KHz ! Coincide Chichinina et al.,2015, SEG

50. 50 Normal compliance Z N estimated by 4 ways in plate-stack model, no inclusions, 480KHz ZTZT Z T  [ 10 -8 ; 3·10 -7 ] Z N  [ 10 -8 ; 5·10 -8 ] [m Pa -1 ] ZNZN Chichinina et al.,2015, SEG

51. 51 The ratio of fracture compliances Z N / Z T estimated by 4 ways in plate-stack model, no inclusions, 480KHz Chichinina et al.,2015, SEG

52. 52 Saturated Dry Z N /Z T Choi et al. 2014 Min-Kwang Choi, Antonio Bobet, and Laura J. Pyrak-Nolte, 2014, The effect of surface roughness and mixed-mode loading on the stiffness ratio κx∕κz for fractures, GEOPHYSICS, VOL. 79, NO. 5, P. D319–D331, Chichinina et al., 2015, SEG

53. 53 Verdon & Wustefeld, 2013 0.1 - 0.48  480 kHz 0 - 0.15  480 kHz Chichinina et al.,2015, SEG

54. 54 Compilation of normal, (Z N ) and shear (Z T ) fracture compliance data (Worthington, 2008). 10 -8 - 3·10 -7 1–5 ·10 -8 AGL Laboratory. Data from Far, 2011. Representative medium of compressed perspex plates, 480KHz Chichinina et al.,2015, SEG

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56. 56 Axial compressionDeformation of plug ε r εaεa ν =ν = Poisson ratio Укорочение ε a Deformation of extension in the cross-section ε r εrεr εrεr BeforeAfter Isotropic rock Isotropic rock Poisson´s ratio is always greater than zero: ν > 0 (excluding auxetic materials)

57. 57 Horizontal plug 57

58. 58 Uniaxial loading After Penny-Shape crack Disclose of the crack The gap between the layers increases due to the crack disclosure Uniaxial loading Before Uniaxial loading

59. 59 Physical constraints on C 13 for overall TI model С 13_min < С 13 < С 13 _MAX Yan F., Han D.-H., Yao Q., 2013, Physical constraints on c13 and Thomsen parameter delta for VTI rocks: SEG Expanded Absracts Yan F., Han D.-H., Yao Q., 2015. Geophysical Prospecting. С 13_min = С 33 (C 11 - 2С 66 ) + С 66 2 – С 66 The lower limit coincides with the expression for the C13 of the Linear Slip model: С 13_min = С 13 LS ν 13 > ν 12. The lower limit of C 13, that is C 13_min, follows from the inequality of Poisson´s ratios: ν 13 > ν 12. С33С33 (C 11 - 2С 66 ) С 13 _MAX = ν 12 > 0 ν 13 > 0. And the upper limit of C13, that is C 13_MAX, can be inferred from the physical property of these Poisson ´s ratios, which should be always greater than zero : ν 12 > 0 and ν 13 > 0.

60. 60 С 13 2(С 11 – С 66 ) ν 31 = (ν 31 = ν 32 ) This follows from the positive value of the vertical Poisson's ratio: ν 31 > 0 σ σ ε a ≡ ε 31 The limitation on the С 13 should be: С 13 > 0 due to ν 31 > 0, and therefore С 11 > С 66 Thus the most simple and evident limitation on С 13 should be С 13 > 0  Limitation on the constant С 13 : С 13 > 0 ν 31 Poisson's ratio Radial strain