1-1- 1. The title of my presentation is Constraints on C13 for shales. -2-2. Outline: C13 in shales and physical models. Different Poisson's ratios in VTI rock. Compliance tensor for VTI shale.
2-3- 3.It is a well-known fact that anisotropy in shale rocks can be caused by at least of two factors: the first is for the layered micro-structure in shale rocks. These are the horizontal clay micro-particles, or the horizontal plate-lets, which are parallel and they form "the layered model" at the micro-scale of rock, and the second is horizontal microcracks. Linear Slip model was developed by Mike Schoenberg. 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. -4-
3 4.The main topic of my presentation is on the C 13 -parameter-constraint in shale rocks. It is very important to get true C 13 parameter for a certain shale formation, for example, that is necessary in estimation of the minimum horizontal stress. The overall transversely isotropic model with the vertical axis of symmetry is a conventional model for shale rocks. There are five independent stiffness tensor components. For the C13-estimation, it is necessary to know the P-wave velocity at 45 degrees to the symmetry axis, but in practice these data is always absent for shales. In contrast to the overall VTI model for shales, in the Linear Slip model, the component C13 is linked to others by this equation. It is the restriction on the C13 established by Schoenberg. Following this formula for the C13, one can estimate C13 in shales, using only 3 velocity measurements in the shale rock. These are two P-wave velocity measurements at the main symmetry directions: 0 and 90 degrees relatively the vertical symmetry axis, and one shearwave-velocity measurement, that is the so-called SH-wave velocity at 90 degrees. However, this approach results in great errors in the C13-estimation as I demonstrated in my SEG paper last year. Recently, two important papers appeared on this topic. -5-5.Last year there was an interesting article by Yan, De-hua Han and Yao, Physical constraints on C 13 and δ for transversely isotropic hydrocarbon source rocks, published in Geophysical Prospecting.
4 The authors of the paper derived formulas for the lower and upper bonds for the C13-parameter. Their constraints on the C13 parameter are based on the following statements: First, Poisson s ratio should be always positive in rocks (that is obvious), and The second statement from their paper is not so obvious. They claimed the following inequality ν 13 > ν 12 for two Poisson s ratios in shale rocks. Based on the inequality between two Poisson s ratios, the authors derive the following formula for the lower bound of C 13. For brevity, I shall name it Yan s criterion by the name of the first author of the paper. -6-6. This year there was an article by Joel Sarout that was actually the answer to the paper of Yan, Han and Yao. The author finds out a counter example that is an opposite inequality between the two Poisson s ratios ν 13 < ν 12. The counter example is based on calculating of five anisotropic elastic constants {Cij} using Backus s averaging for a thin-layered medium (VTI) composed by two alternating layers. Finally, Sarout concludes that Yan constraint for the C 13 is not true. My presentation is about this contradiction and the ways to resolve it.
5-7- 7.I have found out that Yan s formula for the lower bound of C 13 coincides with the formula for the C 13 established by Schoenberg for the Linear Slip model. (I have explained it in details at SEG conference last year.) It helps me to resolve the problem of the two contradictory statements. (I have explained it in details at SEG conference last year.) -8-
6-9- 9.To resolve the problem, let me introduce a new parameter ΔC 13, which I defined as the relative difference between the real C 13 measured in shales and the theoretical C 13 proposed by Schoenberg for the Linear-Slip model. The ΔC 13 -parameter can be written also in the following way: because of the coincidence of the C13_min of Yan and the C 13 LS from the LS model. -10-10.I study the parameter ΔC 13 using published data on shales as well as physicalmodeling data. Let us begin with the results on shales. This figure shows the results for the data from the famous paper of Thomsen (1986). Shown at the vertical axis, N is the number of shale-rock samples fallen in the certain interval of C13- parameter, given in percent.
7 The negative ΔC13-parameter corresponds to the criterion of Yan, and the positive to the criterion of Sarout. The majority of the samples falls in the interval of negative ΔC13-parameter, so that the shales of Thomsen correspond to Yan s criterion. -11-11.By this table, I summarize what we have: there are three different constraints for the C13. The first is Schoenberg s restriction on the C13 for the Linear Slip model; note zero parameter ΔC13 for this case, and there are equal two Poisson s ratios, ν13=ν12, for the Linear Slip model that I have already reported at the SEG conference last year. Actually, the equality of these two Poisson s ratios required for the Linear Slip model is an exotic case in real data. That is almost never occurs in practice in real shale rocks... This equality takes place only in the theoretical Linear-Slip model devised by Mike Schoenberg, which was developed furthermore by Colin Sayers for the application to shales. In the last column of the table, I treat the physical meaning of these two Poisson s ratios in terms of compliance-tensor components.
8-12- 12.These are two Poisson s ratios estimated in real shale rocks, using the data from these three articles. This figure shows the line with the inscription Linear Slip (at which the two Poisson s ratios are equal: ν13=ν12); however as you can see, it is a rear case, because of only a few points fallen at the line. The line divides the variety of all considered shale samples into two groups; the first group is for the shales of Sarout (the upper region in the graph), and the lower region is for the shales of Yan. -13-13. For the Linear Slip model, the equality ΔC13=0 means that both Poisson s ratios are equal to each other, ν13=ν12, as shown by this line. The line with the inscription ΔC13=0 separates the variety of Poisson s ratios into two groups: the shales of Sarout ( С13>0) and the shales of Yan ( С13<0).
9-14- 14.For some shales, Yan s criteria works well as shown here. These are Thomsen s shales and the shales from the paper of Johnston and Christensen shown here. For other shales, the counter-example of Sarout suits better. -15-15.There are very different results for the shales from the paper of Wang. In contrast to the previous two examples, now only a smaller part of shales samples meets the criterion of Yan, corresponding to negative ΔC 13 -parameter. Whilst the majority of them exhibits positive ΔC 13 -parameter that is the criterion of Sarout.
10 Wang s shales represent very diverse collection. For example, there are Africa shales and several samples from the North-Sea and Gulf-Cost shales, which meet the criterion of Yan, however for the other shales, which are the majority of the samples considered, the criterion of Sarout fits better (it includes the brine-saturated Gulf-Coast shales, the Hard shales, the Siliceous shales and the Shaly coals). -16-16.Finally, according to the results of the analysis, we can conclude that the criterion of Yan and its counter-example of Sarout are both true for shales. For some shales, Yan s criteria works well as shown here. These are Thomsen s shales and the shales from the paper of Johnston and Christensen. However, for other shales, the counter-example of Sarout suits better as for the shales of Wang. -17-
11 17. In addition, I have also studied North-Sea mudrocks from the article of Morten Jakobsen and Johansen. The results of the analysis on ΔC13 parameter are shown here. The mudrocks exhibit similar anisotropic properties of the VTI-type as the shales of Wang due to horizontal alignment of the micro-grains. Interestingly, in the diagram for the mudrocks, there is a similar "two-maximum" trend in the ΔC13- distribution as that for the shales of Wang. Namely, one maximum is for negative ΔC13-parameter, and the other is for the positive (compare these two figures). Moreover, there is another obvious similarity in the predominance of the rock samples fallen into the interval of positive ΔC13- parameter, that meets the criterion of Sarout. -18- In addition, I studied carbonates and sandstones from the paper of Wang. We can see almost symmetric ΔC13-distribution (near zero) in both graphs. As is known, sandstones and carbonates have weak anisotropy. Following Wang, the average Thomsen s parameters epsilon and gamma are small for the sandstones: ε=0.07, γ=0.4. Moreover, for the carbonates, they are even smaller than for the sandstones: ε=0.03, γ=0.01. Whereas for the shales, the anisotropy parameters are 10 times greater: ε=0.21 and γ=0.20.
12-19- 19.So, sandstones and carbonates are almost isotropic, and they exhibit very small ΔC13-parameter. I can explain it in this way. Note that for isotropic rocks, the condition on C13 for the LS-model automatically works, and therefore ΔC13 is zero, here it is a very simple proof of this statement. Therefore, in weakly anisotropic rocks, the average ΔC13-parameter should be close to zero: ΔC13=0. -20-20.Results of the analysis of the data from physical modeling with plexiglass plate models are interesting, too. There are the results for the data from the paper of Hsu and Schoenberg, that demonstrate that they all are positive, and are identified by criterion of Sarout, that is to say the parameter ΔC13 is greater than zero.
13-21- 21.Now, let as consider the results, which I obtained from the data of the PhDthesis of Mehdi Far that is physical modeling data, with plexiglass plate model. In other his paper published in the Geophysical Journal International, there are no data for the incidence angle 45 degrees that is necessary for the C13-estimation, because Mehdi Far has taken away these data from the text after my revision of his article. -22-22.When I was reviewer of this paper, I have pointed out to a great difference in Thomsen parameter delta, which can be estimated by two ways. The first way in the delta-estimation was with the use of the C13 measured in the experiment. In addition, the second way was by using C13 calculated from Schoenberg s formula for the Linear Slip model. There was a great difference in the results for the anisotropy parameter delta by these two ways.
14 Therefore, I have written a letter to Mehdi Far about it, but he could not even explain why does the difference take place, and so he decided not to show the data at the angle 45 degrees. Unfortunately he took it away from the paper, and now all the original experimental data on the velocity measurements at the angle 45 degrees are absent in his published article in the Geophysical Journal International. --23-23.Using the original experiment data of Mehdi Far (published in his PhD thesis), I studied the ΔC13-parameter dependence on pressure. There is a critical pressure point P0, the threshold, at which the ΔC13-parameter turns to zero that is Shoenberg s restriction on the C13 parameters is satisfied at this pressure. As you can see for the initial pressures, which are smaller than the threshold pressure P0, the criterion of Yan takes place that is negative C13- parameter. Then, passing over the threshold pressure P0, for greater pressures, the C13-parameter turns into the positive that meets the counter-example of Sarout. I can give the following physical interpretation. At the threshold pressure P0, the contact areas between the cracks are subjected to the transition to a different qualitative state corresponding to the Linear Slip model. That is because of closure of the cracks. -24-
15 24.So that the contact areas between the cracks become forming thin layers, which are compliant and soft. There are the so-called welded contacts, for which the cracks can be represented as the layers of weakness following the Linear Slip model. At the threshold pressure P0, the displacement-discontinuity boundary condition of Schoenberg developed for the Linear Slip model works well. Actually, the cracks can be represented by infinitely thin soft layers embedded in the host rock. Therefore, Backus averaging for the thin-layered model with the two alternating layers can be used as for the counter example of Sarout. -25-25. Now let me summarize what we have at the moment. Sarout found the counterexample to Yan s criterion and therefore concluded that Yan s constraint for C13 was not true. However, I believe that these two are both right for certain shales under certain pressures.
16 For the first kind of shale rocks, which I named Yan s shales, I suppose "the cracks factor is responsible for the inherent inequality of two Poisson s ratios (ν13 > ν12). And for the shales of Sarout, I treat the cracks as thin compliant layers as in Schoenberg s Linear-Slip model; I name it the layer factor that is responsible for the opposite inequality of two Poisson s ratios (ν13 < ν12). Now, I would like to show you my pictures and models, which I have made for you, for the explanation of the physical meaning of these two Poisson s ratios. -26-26. - May I remember you that Poisson s ratio can be defined as transverse strain divided by the axial strain under uniaxial loading. So that the cylinder shrinks along its axis and simultaneously, it expands at the cross section. In TI rocks, there are 3 Poisson ratios, these are one vertical Poisson s ratio, and 2 horizontal Poisson s ratios. In addition, there are 2 Young s moduli. For measuring these 3 Poisson s ratios, it is necessary to have two cylinder rock samples: one is the vertical cylinder; its axis coincides with the vertical axis of TI rock, and the other is the horizontal cylinder. Given the vertical cylinder, loaded by the vertical uniaxial stress sigma33, we can measure the vertical Poisson's ratio and the vertical Young modulus. By uniaxial loading of the horizontal cylinder by the horizontal stress sigma 11, we can obtain two different horizontal Poisson's ratios and one horizontal Young modulus.
17-27- 27. Let us consider the transverse strains at the cross-section of the horizontal cylinder under horizontal uniaxial loading sigma_11. If you measure the transverse strain at the vertical direction x3 normal to the bedding plane, that is epsilon33, it gives you the horizontal Poisson's ratio ν13. You can measure the other strain at the horizontal direction x2 parallel to bedding plane that gives you the other Poisson's ratio ν12. In this presentation, we deal with these two horizontal Poisson s ratios. -28-28. This ellipse is for the deformation pattern corresponding to Yan s criterion. This is the cross section of the horizontal cylinder-sample of VTI rock under uniaxial loading by the horizontal compression stress σ11 at the x1-axis. So that the cylinder shrinks along the axis and simultaneously, it expands at the cross section x2x3. The expansion is anisotropic. You can see these two unequal transverse strains at two mutually perpendicular radial directions.
18 The strain epsilon_33 at the vertical x3-axis is larger than the strain epsilon_22 at the horizontal axis x2. Therefore, as you can see at this deformation pattern, the largest axis of the ellipse points the vertical x3-direction, and therefore the inequality of Poisson s ratios takes place that claimed in the paper of Yan and his coauthors. -29-29. I believe that a possible physical mechanism for Yan inequality of Poisson s ratios can be interpreted as the opening of the cracks, due to uniaxial stress. Here I drew these imaginary springs just for the illustration that the distance between two opposite crack sides has increased under the loading stress σ11. Therefore, it caused the resultant expansion greater at the vertical x3-direction than at the horizontal x2- direction. By this manner, I can explain why this Poisson s ratio ν13 is greater than that one ν12 that is the criterion of Yan. -30-
19 30. I painted another picture for the explanation of the same criterion of Yan, so maybe you would like it more. As you can see, this is a penny-shaped crack embedded into the host rock. The upper picture shows the crack before the loading, and the lower picture is for the same crack after the uniaxial loading. The crack shrinks at the uniaxial-loading direction, and at the same time, it expands at the vertical direction x3, and therefore the gap between the layers increases due to the crack disclosure. That means that total effective-medium deformation epsilon_33 becomes greater than the epsilon_22, and that is why the inequality between two Poisson s ratios takes place, claimed by Yan with coauthors. -31-31. Now let us consider the counter example of Sarout that is the opposite inequality between two Poisson s ratios: ν13<ν12. Now the largest axis of the resultant [эллиипс] ellipse-pattern is pointing the horizontal x2-direction, because of the horizontal strain epsilon_22 greater than the vertical strain epsilon_33. -32- -
20 32. For the counter-example of Sarout, I can explain the physical meaning of the inequality between two Poisson s ratios ν13<ν12 using the model composed by two alternating [Алтёнейтинг] layers as shown here. One layer is soft and compliant, and the other is hard and stiff as in the Linear Slip model for shales (for example as in the paper of Schoenberg & Muir). These compliant layers represent cracks treated as planes of weakness in the Linear-Slip model for shale rocks. This is the cross section view at the vertical plane x 2 x 3 of the VTI-rock sample in the form of cube after the uniaxial loading by the compression stress σ11. The compliant layers are stretched in the bedding plane under the uniaxial loading. Therefore, the horizontal strain ε22 becomes larger than the vertical ε33. My exaggerated sketch shows large elongation of all the soft layers at x2-direction. Now it becomes clear that the total strain ε 22 at the bedding plane will be greater than the strain ε 33 at the normal-to-bedding-plane direction. It leads to the inequality of two Poisson s ratios derived by Sarout (ν13 < ν12). Thus, both inequalities between two Poisson's ratios are true and can take place in shale rocks: ν13 > ν12, as well as ν13 < ν12. The crack factor can explain the physical meaning of Yan s criterion (ν13>ν12) that is the crack opening under uniaxial loading by stress σ11. In addition, the layers factor is resposible for the physical sense of the counter example of Sarout (ν13 < ν12) that is the compliant-layer stretching under the uniaxial loading -33-
21 33. Let us consider the compliance tensor for a VTI shale rock. Given the inequality of Poisson s ratios claimed by Yan with coauthors (the ν13 is greater than the ν12), one can infer that the compliance-tensor component S13 should be greater than the component S12. -34-34. For the counter example of Sarout, the component S31 should be less then S21. -35-35. The compliance tensor S for the linear slip model is the mathematical sum of the isotropic compliance tensor for the host rock, and the additional tensor for fractures, which contains excess [ээксес] fracture compliances Z N and Z T. Note that for the Linear-Slip model, there is the equality of two compliance-tensor components S31=S21, because it was inherited from the isotropic compliance tensor of the host
22 rock. However, this equality is not physically plausible for the transversely isotropic rock such as shale. -36-36. The stiffness tensor for the Linear Slip model is the mathematic inverse of the compliance tensor. The equality condition of two compliance components S13=S12 is transformed into the restriction for the C13-component in the Linear Slip model. I have introduced a new parameter ΔC defined as the relative 13 difference between the real C estimated from the real data in shale rock and this 13 theoretical C 13 established by Schoenberg for the Linear-Slip model. -37-37. In conclusion, I would like to remember that by the use of the new parameter ΔC 13, I have carried out the statistical analysis for the C13-component in different shales. The negative ΔC13-parameter corresponded to the criterion of Yan,
23 and the positive to the counter example of Sarout. The analysis proved that both considered inequalities between two Poisson's ratios could take place in shale rocks: due to their different microstructure (that is the crack factor and the layers factor) as well as different pressure conditions. In addition, I have found a correlation between the compliance tensor components S13 and S12 and these key factors and pressure conditions. -38- Literature 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. Chichinina, T.I., I.R. Obolentseva, and G.A. Dugarov, 2016a, Critical analysis of theoretical basics of displacement-discontinuity model for media with oriented fractures: Tehnologii seismorazvedki (Seismic Technologies), No.1, 31 43. Chichinina, T.I., I.R. Obolentseva, and G.A. Dugarov, 2016b, Estimating applicability of the displacement-discontinuity model using experimental data on measurements of elastic-wave velocities: Seismic Technologies ( Tehnologii seismorazvedki, in Russian), No.2, 48-56. Chichinina T.I., Obolentseva I.R., Dugarov G.A. Effective-medium anisotropic models of fractured rocks of TI symmetry: Analysis of constraints and limitations in
24 Linear Slip model // Expanded Abstracts of the 85th Ann. Int. Meeting SEG. 2015. P. 421 426. Far M. Seismic characterization of naturally fractured reservoirs: PhD Thesis. 2011. University of Houston. 208 pages. Far M., Figueiredo J.J.S., Stewart R.R., Castagna J.P., Han D-H., Dyaur N. Measurements of seismic anisotropy and fracture compliances in synthetic fractured media // Geophys. J. Int. 2014. V. 197. No. 3. P. 1845 1857. Hornby, B. E., L. M. Schwartz, and J. A. Hudson, 1994, Anisotropic effectivemedium modeling of the elastic properties of shales: Geophysics, 59, 1570 1583. Hsu C.-J., Schoenberg M. Elastic waves through a simulated fractured medium // Geophysics. 1993. V. 58. P. 964 977. Jakobsen M., Johansen T.A. Anisotropic approximations for mudrocks: A seismic laboratory study // Geophysics. 2000. V. 65. P. 1711 1725. Johnston J.E., Christensen N.I. Seismic anisotropy of shales // J. Geophys. Res. 1995. V. 100. N B4. P. 5991 6003. Sarout, J., 2016, Comment on Physical constraints on c13 and δ for transversely isotropic hydrocarbon source rocks by F. Yan, D.-H. Han and Q. Yao, Geophysical Prospecting 57, 393 411. Geophysical Prospecting (article first published online 2 February 2016). Sayers C.M. 1994. The elastic anisotropy of shales. Journal of Geophysical Research B 99, 767 774. Sayers, C.M., 1999, Stress-dependent seismic anisotropy of shales: Geophysics, 64, 93 98. Sayers C.M. Seismic anisotropy of shales // Geophys. Prosp. 2005. V. 53. P. 667-676. Sayers C.M. The effect of anisotropy on the Young s moduli and Poisson s ratios of shales // Geophys. Prosp. 2013. V. 61. P. 416 426. Sayers, C.M. and Dasgupta, S., 2015, Elastic anisotropy of the Middle Bakken Formation. Geophysics, 80, D23 D29. Schoenberg M. Elastic wave behavior across linear slip interfaces // J. Acoust. Soc. America. 1980. V. 68. 5. P. 1516 1521.
25 Schoenberg M. Reflection of elastic waves from periodically stratified media with interfacial slip // Geophys. Prosp. 1983. V. 31. P. 265 292. Schoenberg M., Muir F. A calculus for finely layered anisotropic media // Geophysics. 1989. V. 54. P. 581 589. Schoenberg M., Sayers C.M. Seismic anisotropy of fractured rock // Geophysics. 1995. V. 60. P. 204 211. Wang Z. Seismic anisotropy in sedimentary rocks, part 2: Laboratory data // Geophysics. 2002. V. 57. P. 727 735. Yan F., Han D.-H., Yao Q. Physical constraints on c13 and Thomsen parameter delta for VTI rocks // Expanded Abstracts of the 83rd Annual International Meeting. Houston, USA. SEG. 2013. P. 2889 2894. Yan F., Han D.-H., Yao Q. Physical constraints on c13 and δ for transversely isotropic hydrocarbon source rocks // Geophys. Prosp. 2015. V. 57, P. 393 411. Yan F., Han D.-H., Yao Q. Reply to Joel Sarout s comment on Physical constraints on c13 and δ for transversely isotropic hydrocarbon source rocks // Geophys. Prosp. July. 2016.