Assessment of shearing phenomenon and porosity of porous media using microfocus computed tomography

∂−∂ ∂ =ω …………...………….…………..(3.21) 22 23 cby w z v yz −= ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂−∂ ∂ =ω ……………………….…………..(3.22) After calculating all the strain components, the local dilatancy angles (θ) can be calculated as (Tatsuoka, 1987): ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − +−= − 31 311 2 2 sin εε εεψ l ………………………..………………………. (3.23) Where ε1 and ε3 are the major and minor principle stresses calculated from the strain tensor εij. 32 Tatsuoka (1987) suggested Equation (3.23) for axisymmetric triaxial compression based on comparison of laboratory measurements of plane strain and axisymmetric triaxial compression experiments. (dεv / dε1) • 3D Visualization After obtaining the coordinates of all the particles, they were used to generate 3D renderings of the particles. Figures 3.12 through 3.15 show multiple views of a cluster of eight particles at four different compression stages. And Figures 3.16 through 3.18 show multiple views on individual beads at the four compression stages. It is also possible to create 3D animations showing the movement of the particles throughout the experiment. 33 (a) x-z view (front view) (b) x-y view (top view) Figure 3.12. 3D rendering of a cluster of particles before compression ( zε = 0%) 34 (a) x-z view (front view) (b) x-y view (top view) Figure 3.13. 3D rendering of a cluster of particles at zε = -7.8% 35 (a) x-z view (front view) (b) x-y view (top view) Figure 3.14. 3D rendering of a cluster of particles at zε = -13.7% 36 (b) x-y view (top view) Figure 3.15. 3D rendering of a cluster of particles at (a) x-z view (front view) zε = -23.5% 37 (a) x-z view (front view) (b) y-z view (side view) Figure 3.16. Sequence of movement of Bead 2 at the four compression stages (b) y-z view (side view) Figure 3.17. Sequence of movement of Bead 2 at the four compression stages = -13.7% εa εa= 0% εa= -7.8% εa= -23.5% (a) x-z view (front view) 38 (b) y-z view (side view) Figure 3.18. Sequence of movement of Bead 8 at the four compression stages (a) x-z view (front view) 39 CHAPTER FOUR RESULTS AND DISCUSSION .1 RESULTS .1.1 Translation and Rotation The translation and rotation es were calculated as described in Chapter 3. The data was pre tions. A statistical analysis as performed, and the data was then fitted to the closest probability density distribution. Then intervals were calculated. The Frequency distributions for the translation for the first stage of the experiment are presented in Figure 4.1, and for the other stages, the figures are presented in Appendix A. Since the scanning was not performed on uniform strain intervals, the translation values were normalized with respect to the value of the displacement at the top of the specimen at every stage of the experiment, and are presented as percentages of that value. It should be noted that the translation values are not presented in cumulative form, i.e. the presented values represent the translation that took place between the scan under consideration, and the previous scan. This way of presentation was selected to make it easier to note the changes in the translation rate that takes place at different stages throughout the experiment. The values for the horizontal translations (X and Y directions) were taken as absolute values. 4 4 for four hundred particl sented in the form of frequency distribu w the “90% level of confidence” • Translation 40 Figure 4.1: Normalized Translation Histograms at εz = -7.8 % (Normalization Value = 9 mm) 0 5 10 15 20 0 50 100 Normalized Translation (%) , X Direction Fr eq ue nc y (P ar tic le s) 0 5 10 15 20 0 50 100 Normalized Translation (%), Y Direction Fr eq ue nc y (P ar tic le s) 0 10 20 30 0 20 40 60 80 Normalized Translation (%), Z Direction Fr eq ue nc y (P ar tic le s) 41 Table 4.1 summarizes the results of the translation values throughout the experiment, and the pe of the probability distribution that best fits the data. It also presents the lower limit (L) of the 90% confidence interval (CI) where 5% of the values are less than or equal to this value, and the upper limit (U) of the 90% co the values are less than or equal this value. The 90% CI is the interval lying between L and U, where 90% of the sample lies. ty nfidence interval where 95% of to Table 4.1. Summary of normalized translation data and distribution fitting Actual Data Statistical Fit Global St. (%) St. (%) εz (%) Direction Mean (%) Dev. Distribution Mean (%) Dev. L (%) U (%) 90% CI X 5.32 4.673 Log-Normal 4.11 2.72 0 8.72 8.72 Y 6.47 6.44 Log-Normal 5.47 3.38 0.89 11.71 10.82-7.8 Z 11.25 4.50 Log-Normal 10.68 3.42 5.14 16.4 11.26 X 7.93 6.24 Beta 8.12 6.06 0.99 20.16 19.1 Y 6.5 6.66 Weibull 4.61 3.60 0 11.35 11.35-13.7 Z 20.97 8.59 Log-Normal 20.21 8.03 7.01 33.43 26.42 X Beta 8.23 73 21.25 20.529.88 7.77 6.50 0. Y 8.84 7.75 Beta 5.51 0.61 19.01 18.4 7.46 -23.5 Z 24.27 8.96 Log-Normal 26.138 14.08 43.43 29.359.27 • tation As mentioned earlier, using spherical coordinat o n s o ed, represented by Ro es, tw rotatio angle were c nsider φ and θ . The fre y f ot th o ent are illustrated in Figures 4.2 and e the ion da pre d m w the total amount of rotation that the particles undergo, the Figures show the sum of the absolute values of the rotations taking place throughout the experiment. A summary of the results quenc distributions or the r ation rough ut the experim 4.3.Th rotat ta are sente in a cu ulative form. To sho along with the statistical data and distribution fitting for the rotations is presented in Tables 4.2 and 4.3. 42 Figure 4.2. φ Angle rotation throughout the experiment Table 4.2. Statistical summary of rotation angleφ Actual Data Statistical Fit εz (%) Dev ( Distribution Me(degree.) (degree) (degree.) (degree.) 0% CI (degree) Mean (degree) Std. degree) an St. Dev. L U 9 -7.8 -0.074 4.62 9.51 4.74 Normal -0.13 2.93 -4.89 -13.7 -0.93 8.79 Normal -0.76 4.67 -8.40 6.90 15.3 -23.5 -2.34 11.46 Normal -2.00 11.46 -17.10 13.09 30.19 0 10 20 30 40 0 10 20 30 40 Fr eq ue nc y (P ar tic le s) φ Absolute Cumulative Rotation (degree) φ Angle Rotation (degree) (a) εz = -7.8 % 10 5 0 5 10 0 20 60 40 Fr eq ue nc y (P ar tic le s) φ Angle Rotation (degree) (b) εz = -13.7% 10 5 0 5 10 0 20 40 60 Fr eq ue nc y (P ar tic le s) φ Angle Rotation (degree) (c) εz = -23.5 % 20 10 0 10 20 0 10 30 20 Fr eq ue nc y (P ar tic le s) 43 44 θ Angle Rotation (degree) (a) εz = -7.8 % θ Angle Rotation (degree) (b) εz = -13.7 % θ Angle Rotation (degree) (c) εz = -23.5 % θ Angle Cumulative Rotation (degree) 0 20 40 60 0 10 20 30 Fr eq ue nc y (P ar tic le s) 40 20 0 20 40 0 10 20 30 40 Fr eq ue nc y (P ar tic le s) 30 40 60 80 20 10 0 10 20 30 0 20F re qu en 30 cy (P ar tic le s) 20 10 0 10 20 30 0 20 80 40 60 Fr eq ue nc y (P ar tic le s) Figure 4.3: θ angle rotation throughout the experiment Table 4.3: Statistical summary of rotation angleθ Actual Data Statistical Fit εz (%) Mean (degree.) Std. Dev (degree) Distribution Mean (degree) St. Dev. (degree) L (degree) U (degree) 90% CI (degree) -7.8 0.51 7.63 Normal 0.75 4.91 -7.22 8.73 15.95 -13.7 1 al 0.69 .16 11.92 Norm 7.60 -11.8 13.2 25 -23.5 al 1.46 16.26 1.66 20.85 Norm -25.3 28.2 53.5 4.1.2 Local The local strains are calculated using the me h r (Equations 3.14 through 3.22). The results for the first stage of the experime pre in Strains thod described in t e previous Chapte nt are sented 45 cy di tions he r or th owing ges are presented in Appendix B. Cumulative values of the axial and radial strains (εz, εx and εy), shear strains (εxy, εxz, and εyz), and rotation strains(ωxy, ωxz, and ωyz) at the different stages on the nted in each Figure. Then a summary of the results along with the best fit frequen Figure 4.4. Local strains histograms at εz = -7.8% Figures 4.4 and 4.5 in the form of frequen stribu , and t esults f e foll sta experiment are prese cy distribution and the confidence intervals are shown in Tables 4.4, 4.5 and 4.6. εz 0.4 0.2 0 0.2 0.4 0 20 40 60 80 Fr eq ue nc y (P ar tic le s) εx 0.4 0.2 0 0.2 0.4 20 40 0 Fr eq ue nc y (P ar tic le s) εy 0.4 0.2 0 0.2 0.4 20 0 40 Fr e nc y (P a tic le s) qu e r εxy 0.4 0.2 0 0.2 0.4 0 20 40 60 Fr eq ue nc y (P ar tic le s) εxz 0.4 0.2 0 0.2 0.4 0 20 40 60 Fr eq ue nc y (P ar tic le s) ε 0.4 0.2 0 0.2 0.4 0 20 40 60 Fr eq ue nc y (P ar tic le s) yz Figure 4.5. Rotation strains histograms at εz = -7.8% ωyz0.6 0.4 0.2 0.2 0.4 0.6 0 20 40 60 80 0 Fr eq ue nc y (P ar tic le s) yzω 0.4 0.2 0 0.2 0.4 20 40 60 80 0 Fr eq ue nc y (P ar tic le s) ω xzωxz 0.4 0.2 0.2 0.4 0 20 40 60 0 Fr eq ue nc y (P ar tic le s) ω xyωxy 46 Table 4.4: Summary of axial and radial strains data and distribution fitting Actual Data Statistical Fit Global εz (%) Component Mean St. Dev. Distributi on Mean St. Dev. L U 90% CI εx 0.026 0.15 Log-Norm 0.065 0.10 -0.045 0.262 0.307al εy Log- 0.359 0.3960.014 0.15 Normal 0.089 0.15 -0.037 -7.8 εz -0.014 0.08 Normal -0.013 0.04 -0.080 0.055 0.135 εx 0.041 0.16 LogNormal 0.056 0.12 -0.115 0.275 0.390 - εy 0.035 0.15 Log-Normal 0.053 0.10 -0.089 0.223 0.312 -13.7 εz -0.016 0.14 Normal 0.012 5 0.09 0.205 0.06 -0.11 εx 0.085 0 0 -0.240 0.611 .32 Log-Normal 0.133 .26 0.851 εy 0.070 0.36 Log-Normal 0.140 0.27 -0.244 0.621 0.865 -23.5 0.002 0.39 -0.049 0.19 -0.358 0.259 0.617εz Logistic ble 4.5 ary of sh s data and distribution fitting Ta : Summ ear strain Actual Data Statistical Fit Global εz (%) Com nt D Distribution pone Mean St. ev. Mean St. Dev. L U 90% CI εxy 0.006 0.14 Logistic 0.003 0.07 -0.104 0.111 0.215 ε -0.013 0.13 xz Logistic -0.006 0.06 -0.101 0.088 0.189-7.8 -0.007 0.12 Logistic εyz 0.000 0.06 -0.100 0.100 0.200 εxy 0.010 0.16 Logistic 0.01 0.09 -0.133 0.150 0.283 εxz -0.002 0.15 Logistic -0.008 0.09 -0.152 0.136 0.288-13.7 εyz -0.001 0.14 Logistic 0.007 0.09 -0.143 0.156 0.299 εxy 0.036 0.35 Logistic 0.031 0.27 -0.409 0.472 0.881 εxz 0.002 0.36 Logistic -0.015 0.25 -0.429 0.398 0.827-23.5 εyz Logistic 0.003 9 0.435 0.8640.020 0.38 0.27 -0.42 47 T le 4.6: ar ota s d u i Actual Data Statistical Fit ab Summ y of r tion strain ata and distrib tion fitt ng Global εz (%) Component Mean St. Dev. Distribution Mean St. Dev. L U 90% CI ωxy -0.015 0.13 Logistic -0.006 0.07 -0.121 0.109 0.230 ωxz -0.001 0.13 Logistic -0.006 0.05 -0.094 0.083 0.177-7.8 ωyz -0.003 0.13 Logistic 0.004 0.06 -0.092 0.100 0.192 ωxy -0.022 0.15 Logistic -0.020 0.11 -0.193 0.152 0.345 ωxz -0.023 0.15 Logistic -0.025 0.10 -0.173 0.123 0.296-13.7 ωyz 0.015 0.15 Logistic 0.012 0.12 -0.177 0.202 0.379 ωxy -0.007 0.34 Logistic -0.038 0.23 -0.419 0.342 0.761 ωxz -0.018 0.38 Logistic -0.054 0.30 -0.540 0.433 0.973-23.5 ω Logistic 0.004 1 0.438 0.869yz 0.046 0.40 0.27 -0.43 4.1.3 Dilatancy Angles The loc latanc s lcu ng n . s cal dila ngle gram d nt the m a the results as well a stat t ation are shown in Table 4.7. .2 DISCUSSIO OF R TS 4.2.1 Translation The translation values in the lateral direction (x and y) and axial direction (z) were studied with the aid of Figures 4.1, A1, and A2 (Appendix A) and Table 4.1. The lateral translation during the first stage of the experiment (ε = -7.8%) looks similar in the x and y directions, a mean value of 5.3% of the displacement at top of the specimen was obtained in the x direction, and a value of 6.5% was obtained in the y direction (Figure 4.1). It is also realized that the shapes of the histograms in x and y direction look similar. This is due to the axisymmetric conditions of the experiment. al di y angle are ca lated usi Equatio 3.23. Figure 4 6 show the lo tancy a histo s at the iffere stages of experi ent. A statistic l summary of s the istical fi inform 4 N ESUL z 48 Figure 4.6: Local dilatancy angle histograms 100 80 60 40 20 0 20 40 60 80 100 0 10 20 30 40 50 60 70 80 Fr eq ue nc y (P ar tic le s) lψ (Degrees) 100 80 60 40 20 0 20 40 60 80 100 0 10 20 30 40 50 60 70 80 Fr eq ue nc y (P ar tic le s) lψ (Degrees) (a) εz = -7.8% (c) εz = -23.5% (b) εz = -13.7% lψ (Degrees) 100 80 60 40 20 0 20 40 60 80 100 0 10 20 30 50 40 60 70 80 Fr s) eq ue nc y (P ar tic le 49 Table 4.7: S Actual Data ummary of dilatancy angle data and distribution fitting Statistical Fit εz (%) Mean (degree) U (degree) 90% CI (degree) St. Dev. (degree) Distribution Mean (degree) St. Dev. (degree) L (degree) -7.8 10.19 -41.9 70.8 112.7 31.9 Logistic 14.47 34.73 -13.7 22.85 30.8 -21.7 75.5 97.2 7 Logistic 26.87 29.93 -23.5 21.68 26.5 -12.6 65.3 77.9 3 Logistic 26.34 23.97 On the other hand, looking at the values of translation in the z direction, the mean value as 11.25% of the displacement at the top en. It can be realized from the low value f standard deviation, and the sha gram for the z direction translation, the homogeneity of the values of the axial distance here most of the values lie between 8% and 12%. The values of translation in the lateral and axial directions were relatively low, that can be explained that during the early stages of the e particles exhibit small re ge t li s f im Du e se tage peri lo -13 n inc in th of anslation in the lateral direction was observed. The mean values for the translation were 7.93%, and 6.5 e translation in the lateral directio w of the specim o pe of the histo traveled, w experiment, th arran ment due o the app cation of stre s on top o the spec en. ring th cond s of the ex ment (G bal εz = .5%) a rease e rate tr % in the x and y directions, respectively (Figure A1). The mean value for the translation in the axial direction also increased to be 20.98%, which is about double the amount obtained in the first stage of the experiment. This increase in the translation values in the lateral and axial direction could be caused by the collapse of the large voids due to the continuous compression, causing higher translation value. Moreover, particles from the upper levels of the specimen migrate into the lower levels by displacing the underlying particles axially and laterally. In the last stage of the experiment, (global εz = -23.5%), th n continues to increase, where the mean value for the translation in the x direction is 9.9%, and 8.84% in the y direction (Figure A2). It can be seen that the mean values in the x and y are still similar. Also, the similarity of the histograms of the translation in the x and y direction 50 can still be clearly seen where they both take the shape of a “Beta distribution”. As mentioned earlier, this is an expected result due the axisymmetric conditions of the experiment. The translation in the axial direction also increases, to have a mean value of 24.27% and the values in the 90% level of confidence reach up to 43%. This stage of the experiment is best described as the critical state, where the shear resistance of the specimen is very small and greater strains can be gene As mentioned earlier, a vertical (φ) and a horizontal (θ) component of the rotation were considered. From Figure 4.2 and Table 4.2, the vertical rotation histograms during all the stages of the experiment take the shape of normal distributions. During the first stage of the experiment, 90% of the vertical rotation values lie between -4.9 and 4.6 degrees. While in the second stage they lie between -8.4 and 6.9 degrees. In the final stage of the experiment 90% of the values lie between -17.1 and 13.1 degrees. Taking the absolute values of all the rotations, the cumulative vertical rotation values reach up to 30 degrees. On the other hand, the horizontal rotation had higher values, where 90% of the values of the θ angle rotation in the first stage of the experiment -7.22 and 8.73 degrees (Figure 4.3). During the second stage, the values lie betwee rated with relatively small stresses. 4.2.2 Rotation range between n -11.8 and 13.2 degrees. On the final stage of the experiment, the horizontal rotation values range between -25.3 and 28.2 degrees. Like the vertical rotation, the horizontal rotation histograms took the shape of normal distributions. Taking the absolute values of all the rotations, the cumulative horizontal rotation values reach up to 60 degrees. 51 4.2.3 Local Strains Studying the local strains distributions in Figures 4.4, 4.5 and Appendix B (Figures B1 to B4), a considerable similarity can be noticed between the lateral strains (εx and εy) throughout the experiment. They always had a Log-Normal distribution that tends to have more positive values then negative. The positive sign here indicates dilation, or expansion, and that’s what is expected to happen during the compression of the specimen, where it expands laterally. During the last stage of the experiment, it is noted that there is some difference in the shape of the lateral strains histograms. This happens because the bulging at failure in the middle portion of the specimen (where the CT scans are taken) is not perfectly symmetric around the z axis. The specimen might expand laterally in one direction more than the other, but there is an overall similarity between the lateral stain values, due to the axisymmetry. On the other hand, the axial strain (ε ) distributions take a normal distribution shape in the first and second stages of the experiment, the distributions always have a negative mean that indicates compression, and this is the expected result when axially compressing a specimen. On the final stage, the negative values dominate, resulting in a Logistic distribution where most of the values lie in the negative (compression) area. This result indicates the higher values in axial strains obtained at failure. All the local shear strains histograms took the shape of a Logistic distribution. The positive or negative signals for the shear strains only indicate the direction of shearing, and are not related to the expansion or compression. In the first stage of the experiment 90% of the values of the horizontal shear strains (ε ) range between -0.104 and 0.111, while in the second last stage of the experiment, where 90% of the z xy stage they lie between -0.133 and 0.150. An as seen in the axial and lateral stains as well as the translation, the greatest increase is noted in the 52 data is strains values ranges between –0.193 xy xz yz xy xz yz 4.2.4 Dilatancy Angles The dilatancy angles in the first level of compression took a wide range of values, between -41.9 and 70.8 degrees (Figure 4.6 and Table 4.7), where the negative sign indicates contraction whereas positive sign indicates dilation. In this stage of the test some contraction took place in the specimen, and then dilation started (Figure 3.7), it can be realized that the dilatancy angles at this level had more positive values than negative. As the dilation of the between -0.401 and 0.472. The other shear strains (εxz and εyz ) distributions also shows a similarity that is caused by the axisymmetric conditions. Both of them have 90% of the values between -0.1 and ~0.1 in the first stage of the experiment. In the second stage, the values range between -0.152 and 0.136 for εxz , and -0.143 and 0.156 for εyz. As expected, the larger increase takes place in the last stage of the experiment, where the values of εxz range between -0.429 and 0.398, and the values of εxz lie between -0.429 and 0.435. The rotation strain histograms also took the shape of logistic distributions (Table 4.6). The similarity of the rotation strain distributions throughout the specimen, specially the rotation strains that have a vertical component (ωxz , ωyz). During the first compression stage, (global εz = 7.8%), the 90% confidence interval lies between -0.121 and 0.109 for ωxy, -0.094 and 0.083 for ωxz, and -0.092 and 0.100 for ωyz (Figure 4.5). The values of the rotation strains increased during the second compression stage, where 90% of the rotation and 0.152 for ω , -0.173 and 0.123 for ω , and -0.177 and 0.202 for ω (Figure B3). Like all the other strain components, the largest increase in the rotation strains took place in the last stage of the experiment. ω had a 90% confidence interval between -0.419 and 0.342. ω and ω had 90% confidence intervals of -0.540 to 0.433, and -0.431 to 0.438, respectively (Figure B4). 53 specimen continues at the later levels of the experiment, the fraction of positive dilatancy angles gradually increases, where at the final stage of the test, positive values dominate the dilatancy angle distribution, and a smaller range of dilatancy angle values were obtained (-12.6 to 65 degrees). 54 CHAPTER FIVE SPATIAL POROSITY DISTRIBUTION OF ROCK CORES STUDIED BY µCT .1 INTRODUCTION In this part of the thesis, CT technology was used to study the porosity distribution of synthetic rock cores. Several rock co ain size distributions, and prepared u umber to the bulk density was conducted, and then the correlation was used to obtain the porosity distribution for the cores. They were compared to determine the effect of grain size distribution and consolidation pressure on the local porosity distribution of the cores. 2 The main rtz powder. The cores were prepared by thoroughly mixing the s as a cementing agent. Table 5.1 shows a list of the cores along with their grain size distribution, and the compaction pressure used to prepare them. The cores have a cylindrical shape, with an average radius of 28 mm and an average height of 29 mm (Figure 5.1). 5 res having different gr nder different consolidation pressures were scanned. A density calibration to correlate the CT n 5.2 EXPERIMENTAL WORK 5.2.1 Specimen Description Twelve synthetic rock cores were used in this study. The cores were prepared by mixing Quartz (SiO ) with four classes (Powder, 0.126-0.149 mm, 0.149-0.177 mm, and 0.177-0.210 mm) and consolidating (Compacting) them under 3000, 4000 or 5000 psi pressures. constituent of the cores is the Qua particles and adding liquid glas 55 Table 5.1: Rock cores list Consolidation Grain Size [%] Core Name Pressure Quartz 0.126 - 0.149 0.149 - 0.177 0.177 - 0.210 [psi] flour mm mm mm 3k_100 3000 100 0 0 0 4k_100 4000 100 0 0 0 5k_100 5000 100 0 0 0 3k_80 3000 80 6 6 8 4k_80 4000 6 8 80 6 5k_80 5000 80 6 8 6 3k_60 3000 60 12 12 16 4k_60 4000 60 12 12 16 5k_60 5000 60 12 12 16 3k_40 3000 40 18 18 24 4k_40 4000 40 18 18 24 5k_40 5000 40 18 18 24 Figure 5.1: A Sample Core 56 5.2.2 CT Scanning The CT scans were performed using the Washington State University CT laboratory in ugust, 2003, using a system similar to the MSFC CT system (Figure 5.2), with a different x-ray source. The x-ray was generated using an X-tek 225 kV microfocus x-ray source (Figure 5.3). It has a 5 A µm beam with m focal spot size x-ray source with microfocus option that can produce an intense x-ray inimum penumbra. It took approximately 6 minutes to scan each specimen at keV and 0.158 mA current. The distances from the x-ray source to the specim energy level of 142 en and from the specimen to the detector panel were 244.5 and 416 mm, respectively, producing about 700 slices per core. Each slice is about 620 x 620 pixels yielding a spatial resolution of about 47.83 µm/pixel. Figure 5.4 shows example CT renderings of one of the scanned cores. Figure 5.2. The X-ray CT System of Washington State University (WAX-CT) 57 Figure 5.4. Example wedge view and axial CT sections of a core Figure 5.3. The X-tek 225 kV Microfocus x-ray source 58 5.3 POROSITY CALCULATIONS 5.3.1 Density Calibration Density calibration was performed using the ASTM guidelines (ASTM E1935, 1997). Three materials d Acrylic. The ensities (ρ) of the materials were calculated through weight and volume measurements, and the ass attenuation coefficients (MCA) were obtained from the NIST website (Hubbell and Seltzer, 997). Then the linear attenuation (Ca) was calculated as the product of the density and the mass ed at the same energy and geometric ass attenuation coefficients, densities and the average CT numbers for the The correlation in Figure 5.5 was then used to calculate the bulk density of the rock cores. The mass coefficient of attenuation (MCA) of quartz was calculated at 142 keV as 0.1447 (cm2/g), and then from the CTN data using the correlation in Figure 5.5 the bulk density can be calculated from Equation 5.1 after rearranging the terms. Then the porosity (n) can be calculated using in Equation 2.5. Table 5.2: The Density Calibration Data. Material Attenuation (µCA) (cm2/g ) (g/cm 3) Attenuation (Ca) (cm-1) CTN were used to perform the calibration: Aluminum, Graphite, an d m 1 attenuation (Equation 5.1). The scanning was perform settings as the cores. The m calibration materials are shown in Table 5.2, and the density calibration curve is shown in Figure 5.5. CA (cm-1) = ρ (gm/cm3) * MCA (cm2/gm) ………………………… (5.1) Mass Density (ρ) Linear Aluminum 0.1434 2.66 0.3814 7840 Graphite 0.1374 1.76 0.2421 4028 Acrylic 0.1425 1.2 0.1710 2046 59 uter prog m was written using the Interactive Data Language (IDL) to calculate the local porosity distribution for each core. The porosity is calculated for small volumes of five voxels, yielding about 15 million n values for each core. Histograms of the local porosity distribution for each core are prepared in order to be able to compare the local porosity distribution for all the cores. And to demonstrate the effect of u compactions pressures on the local porosity distribution. 0.50 Ca = 4E-05CT R N + 0. 2 = 1 0.00 0.10 0.20 0.40 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 CTN L in ea r A tte nu at io n (c m- 1 ) phite Acrylic luminum 0963 A 0.30 C a Gra Figure 5.5. The density calibration curve After subtracting the edges, the 3D data array obtained by stacking the cross sectional slices for each core has the size of approximately (350x350x550) voxels. A comp ra sing different grain size distributions and 60 5.4 RESULTS AND DISCUSSION 5.4.1 Effect of Grain Size Distribution The local porosity histograms for the cores, grouped by the compaction pressure, are shown in Figures 5.6 through 5.8. For each compaction pressure, four cores having different grain size distributions are analyzed, where the percentage of the quartz powder is reduced each me, and different grain size distributions are used. Figure 5.6. Porosity Histograms for 3000 psi Cores ti 0 1 2 3 4 8 0 0.1 0.2 0.3 0.4 0.5 0.6 Porosity N or m al iz ed eq ue nc y 9 3k_100 3k_80 3k_40 3k_60 5 6 7 F r (% ) 61 Figure 5.8. Porosity Histograms for 5000 psi Cores Figure 5.7. Porosity Histograms for 4000 psi Cores 62 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 Porosity N or m al iz ed F re qu en cy (% ) 4k_100 4k_80 4k_60 4k_40 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 Porosit N or m al iz ed F re qu en cy (% ) 5k_100 5k_40 5k_60 5k_80 y As seen from the histograms, increasing the non uniformity of particles sorting reduces the average porosity of the cores, and this is a well know fact, since providing different sizes of grains helps in filling the small and large voids ith the different sizes of particles, resulting in better packing of the particles. To clearly illus re 5.9(a) shows a poorly graded packing, while Figure 5.9(b) shows a well graded packing. It can be seen that having different particle sizes helps in filling most of the voids, resulting in a lower porosity value. . Figure 5.9. (a) A poorly graded packing. (b) A well graded packing eviation) with increasing the grain size distribution, where a wider porosity histogram indicates lager variety of void sizes within the core. If the core has one grain size, it has nearly same pore ze, so, if we look into a small volume (five voxels in our case) we will find that it will have early the same percentage of pores and solids, therefore, we can say that it is a homogeneous powder cores. Figure 5.10a shows three ra t all the areas w trate this concept Figu (a) (b) Furthermore, the shapes of the histograms tend to be wider (i.e. higher standard d a si n medium, and that results in the “narrow” histograms for the 100% ndom areas selected in a homogeneous medium. It can be seen tha 63 contain nearly the same proportions of solid and void areas. On the other hand, looking at a core having (a) (b) Figure 5.10. (a) Selection of a random area in a poorly graded specimen (b) selection of a a variety of grain sizes, it will have different pore sizes, resulting from the different sizes of the particles. Therefore, considering a small volume (5 voxels) in different positions, will yield in a wide variety of void to solid ratios (Figure 5.10b), resulting in a high standard deviation value and wider porosity histograms. A summary of the means and standard deviations of the local porosity histograms is shown in Table 5.3. random area in a well graded specimen. Table 5.3: Summary of the statistical parameters for the porosity distributions Core Mean Std. Dev. Variance Skewness Kurtosis 3k_100 0.331951 0.0243841 0.0005946 -0.677971 9.15775 4k_100 0.324608 0.0307659 0.0009465 -0.742309 12.1766 5k_100 0.340829 0.0218543 0.0004776 0.387745 38.6906 3k_80 0.324644 0.031345 0.0009825 -1.16835 13.1452 4k_80 0.276654 0.0419643 0.001761 -0.67183 5.63001 5k_100 0.303162 0.0370358 0.0013717 -0.805755 5.93213 3k_60 0.237291 0.0385848 0.0014888 -1.47545 12.0596 4k_60 0.257841 0.0506909 0.0025696 -0.630058 5.4647 5k_60 0.220352 0.0413722 0.0017117 -1.286 10.4692 3k_40 06179 0.046165 0.0021312 -0.997361 7.97292 0.2 4k_40 0.20234 0.0628318 0.0039478 -0.630799 6.53747 5k_40 0.199239 0.0426337 0.0018176 -1.38796 10.8378 64 5.4.2 Effe res are into s. Th each e the ain size distribu ut ea was at a different comp essur porosity histogra the fo s are shown in Figure ough 5 Figure 5.11. Porosity Histograms for 100% Quartz Powder Cores ct of Compaction Pressure The co grouped four group e cores in group hav same gr tion, b ch core prepared action pr e. The ms for ur group s 5.11 thr .14. 0 1 2 3 4 7 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6Porosity N or m al iz q (% ) 3k_100 4k_100 8 5k_100 5 6 ed F re ue nc y 65 01 2 3 4 5 6 7 N or m al iz ed F re qu en cy (% ) . 3k_80 4k_80 5k_80 0 0.1 0.2 0.3 0.4 0.5 0.6 Porosity Figure 5.12. Porosity Histograms for 80% Quartz Powder Cores Figure 5.13. Porosity Histogram for 60% Quartz Powder Cores 0 1 2 3 4 5 6 7 8 0 0.6 N or m al iz ed F re qu en cy (% ) 4k_60 5k_60 3k_60 0.1 0.2 0.3 0.4 0.5 Porosity s 66 from histogram the particles particles` sh are more suscep cores having a 0 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 Porosity N or m al iz ed F re qu en cy (% ) 4k_40 5k_40 3k_40 Figure 5.14. Porosity Histograms for 40% Quartz Powder Cores It is expected that increasing the compaction pressure, should decrease the porosity of the cores. This was not the case for the cores under investigation. Increasing the compaction pressure 3000 psi to 4000 psi resulted in decreasing the porosity as expected. But on the other hand, increasing the compaction pressure to 5000 psi did not reduce the porosity. The porosity s for the 5000 psi cores were located and different location each time with respect to the 3000 and 4000 psi cores histograms (Figures 5.11 – 5.14). This may be due to the crushing of that occurred at the very high pressure that resulted in the irregularity of the apes, resulting in the unexpected values for the porosity. Moreover, large particles tible to crushing than smaller ones, and it can be seen in the 67 higher percentage of large particles (40% and 6 Quartz powder) experience an irregularity of results 5.4.3 Vertical Profiles of Porosity 0% that can be justified by the crushing of the large (0.177 - 0.210 mm) particles at lower compaction pressures of 4000 or even 3000 psi. It can also be realized that the porosity values for the cores having the same grain size distribution, and different consolidation pressures are very close to each other, because the lowest consolidation pressure used (3000 psi) is already a very high pressure, and it might be enough to achieve the highest possible compaction for the particles, and not much more compaction can be achieved by increasing the consolidation pressure. As a result of the closeness of the porosity values for these cores, any imperfections in the preparation process or the measurement could result in misleading trends in the results. This could be another reason for the irregularity of the results for those cores. To demonstrate the variation of porosity along the height of the cores, the average porosity values for each slice were calculated for every core, and plotted against the distance from the bottom of the core (Figure 15.15). All the profiles followed the same trend, where the porosity increases near the ends, and decreases near the middle of the cores, this could be due the boundary conditions of the cores where the edges doesn’t get as compacted as the middle part of the cores. 68 0.15 0.2 0.25 0.35 0 5 10 15 20 25 30 Distance from bottom (mm) Po 5k_100 4k_100 4k_80 4k_60 3k_60 5k_60 4k_40 3k_40 5k_40 0.3 0.4 ro si ty 3k_100 3k_80 5k_80 Figure 15.15. Vertical profiles of the porosity distributions of the rock cores. 69 CHAPTER SIX CONCLUSIONS AND RECOMMENDATIONS .1 CONCLUSIONS This thesis has two objectives. The first is to use micro-focus computed tomography (µCT) study the shearing of granular materials, and the second is to characterize the variation of the cal porosity distributions of synthetic rock cores with changing the grain size distribution and onsolidation pressure using µCT. To achieve the first objective a triaxial specimen was scanned t different strain levels to track the translation and rotation of the particles within the specimen. he obtained values for translation and rotation were used to calculate the local strain istribution within the specimen, and the distributions were compared to study the behavior of the particles at different stages. To ach tive, twelve different rock cores were scanned. The cor e different grain ze distributions. Density calibration was performed to correlate the CT numbers to the bulk Local porosity distributions of the cores were obtained using this orr particles in the triaxial specimen were obtained at a high degree of precision, for all stages of the compression. This data was used to calculate the local strain distribution of the specimen throughout the test. The calculated local strains were increasing throughout the experiment, especially in the last stage, were a large increase in the local strains values was noticed. It was also realized the local strains in the radial directions (ε and ε ) showed a similarity throughout 6 to lo c a T d ieve the second objec es were prepared at different compaction pressures, and hav si density and porosity. c elation. They were porosity compared to find the effect of compaction pressure and grain size distribution. Using the 3D volumetric data obtained by CT, the values of translation and rotation of the x y 70 the test, where all the histograms took the shape of a Log-Normal distribution, with a greater number of positive (extension) values then negative (contraction) values. This is due to the axisymmetric conditions of the test. All the local shear strain histograms took the shape of a logistic distribution; their values in the first two stages of the experiment lied in the narrow range of ± 10% to 15%. On the last stage of compression a large increase in the shear strains values in in ly graded grains, higher porosity values and homogeneous pore size r creased to reach up to ± 40%. A similar trend was noticed in the rotation strain values, where all the histograms had the shape of a logistic distribution. During the first and second stages of compression the rotation strain values had values within ± 20%. They increased to reach about ±50% during the final stage. A wide range of values was noticed for the dilatancy angles. At the beginning of the test, the values where nearly evenly distributed between positive (dilatancy) and negative (contraction). At the later stages, more positive values where obtained due to the dilative nature of the specimen, until the last stage of the experiment where the positive values where dominant. On the other hand, µCT showed an excellent ability to track the changes in the local porosity distribution, and the homogeneity of pore sizes of the of the synthetic rock cores. It was found that when using well graded grains less porosity values and inhomogeneous pore sizes were obtained, while when us g poor s were obtained, this was obtained from the mean and standard deviations of the porosity distributions of the cores. No clear trend was obtained when increasing the consolidation pressure. This could be due to the crushing of the particles at the ve y high consolidation pressures, and some imperfections in the preparation process of the rock cores, which resulted in inconsistent results for the porosity distribution. 71 6.2 RECOMMENDATIONS • The data presented in the first part of this thesis could be compared with the values obtained from different theoretical models. This could help in improving modeling of the constitutive behavior of granular materials. • In this study, plastic beads where used to model the behavior of granular materials. The possibility of using other materials, like sand, should be investigated. This requires the improvement of the CT scanner resolution, and the development of computer programs to identify the particles and track their translation and rotation. • In order to be able to relate the local values of strains and dilatancy angles to the global values, the whole specimen should be scanned. This will require a larger scanning times, and data sizes. It will also require very high computer capabilities and storage space. So, in order to achieve this goal all these requirements should be considered. in the preparation process, to reduce the • Further studies should be conducted on the effect of increasing the consolidation pressure on the porosity distribution of rock cores. A wider range of consolidation pressures should be considered, with special care taken amount of error caused my imperfections in the preparation of the cores. 72 REFERENCES M.C. and Wong, R.C.K. (2001): Some Applications of Computerized Tomography (CT) ental Geotechnics, Proc. 15th International Conference in Soil Mechanics and chnical Engineering , Istanbul, Turkey, 2001, Vol. 1, pp. 337-342 i, K.A., Batiste, S.N., Swanson, R.A., Sture, S., Costes, N.C., Lankton, Alfaro Scan in Experim Geote Alshibl M.R., (2000a): Quantifying Void Ratio Variation in Sand Using Computed Tomography, Geotechnical Measurements: Lab and Field, Proceedings of the sessions of Geo-Denver 2000, Denver, Colorado, August 5-8, 2000. Alshibli, K.A., Sture, S., Costes, N.C., Frank, M.L., Lankton, M.R., Batiste, S.N., Swanson, R.A., (2000b). Assessment of Localized Deformations in Sand Using X-Ray Computed Tomography, Journal of Geotechnical Testing, 23(3):274-299. Alshibli, K.A., (2004): Unpublished Work mersham Health, (2004): Encyclopaedia of Medical Imaging by NICER (Medycyclopedia) online version, Anandarajah, A., (2004): Sliding and Rolling Constitutive Theory for Granular Materials, Journal of Engineering Mechanics, Vol. 130, No. 6, June, 2004. B A Bansal, A., Islam, M. R., (1991): State-of-the-Art Review of Nondestructive Testing with Computer-Assisted Tomography, SPE 22127, International Arctic Technology Conference, SPE, Anchorage, Alaska, May 29-31, 1991. Bossi, R.H., Friddel, K.D., Lowery, A.R., (1990): Computed Tomography. Non-Destructive Testing of Fiber-Reinforced Plastic Composites, New York, Elsevier Applied Science. arrington Medical Imaging LLC Website, 2004, Cromwell, V., Kortum, D.J., Bradely, D.J., (1984): The Use of a Medical Computer Tomography (CT) System To Observe Multiphase Flow in Porous Media, SPE 13098, 59th Annual Technical Conference and Exhibition, SPE, Houston, Texas, September 16-19, 1984 Desrues, J., Chambon, R., Mokni, M., Maz Bands in Triaxial Sand Specimens Studie erolle, F., (1996): Void Ratio Evolution Inside Shear d by Computed Tomography, Geotechnique 46, No. re During Shear", ASCE, ujii, M., Uyama, K., (2003): Recent Advances on X-ray CT, , Proceedings of the International Workshop on X-ray CT for Geomaterials, Kumamoto, Japan, 6-7 November 2003. 3, 529-546. rost, J.D., and Jang, D.J., (2000): Evolution of Sand MicrostructuF Journal of Geotechnical and Geoenvironmental Engineering, Vol. 126, No.2, pp. 116-130. F 73 Honarpour, M.M., Cromwell, V., Hatton, D., Satchwell, R., Reservoir Rock (1985): Descriptions Using Computed Tomography (CT), SPE 14272, 60th Technical Conference and Exhibition, and Mass Energy-Absorption Coefficients”, NIST web site SPE, Las Vegas, Nevada, September, 22-25, 1985. Hubbell, J.H., and Seltzer, S.M., (1996): Tables of X-Ray Mass Attenuation Coefficients ( . Kanatani (1979): A micropolar continuum theory for the flow of granular K materials, International Journal of Engineering Science, Vol. 17, No. 4 (1979), pp. 419-432 K mographic imagery: Application to the Geosciences, Computer & Geosciences, 27:381-400. K he American Society for Nondestructive Testing (ASNT). K uhn, M. R., Bagi, K., (2002): Particle Rotations in Granular Materials. 15th ASCE Engineering M ournal of Physics Education 36, 442-451. ukunoki, T., Otani, J., Obara, Y., Kaneko, K., (2003): Artifacts of X-ray CT Data in the 3. da, M., Iwashita, K., (edt.), (1999): Introduction to Mechanics of Granular Materials, Balkima, da, M., Konishi, J., and Nemat-Nasser, S. (1982): Experimental micromechanical evaluation of O ita, K., Kakiuchi, T., (1997): Importance of Particle Rotation in Mechanics of Granular Materials, Powders & Grains 97, Behringer & Jenkins (edt.), Balkema, pp. 207-210, O hod for Characterization of Failure in Soils, Soils and Foundations, Vol. 40, No. 2, 111-118. etcham, A.R., Carlson, D.W., (2001): Acquisition, and optimization and interpretation of X-ray computed to ropas-Hughes C., Neel, S., (2000): Basics of Computed Tomography. Website of T TH Royal institute of technology (2001): Computed Tomography, November 2001. K Mechanics Conference, June 2 – 5, 2002, Columbia University, New York, NY. ichael, G., (2001): X-ray Computed Tomography, J M Analysis of Geomaterial Properties, Proceedings of the International Workshop on X-ray CT for Geomaterials, Kumamoto, Japan, 6-7 November 200 O 1999. O strength of granular materials: Effects of particle rolling. Mech. of Materials, 1(4), 269–283. da, M., Iwash (1997). tani, J., Mukunoki, T., Obara, Y., (2000): Application of X-Ray CT Met 74 Rowe, P.W., (1962): The Stress Dilatancy Relation for Static Equilibrium of an Assembly of heats, M.J., (2000): Advanced Data Processing for Volumetric Computed Tomography Data, A trumas, N., Antonyshyn, O., Yaffe, M. J., Mawdsley, G., Cooper, P., (1995): Computed g of the Canadian Society of Plastic Surgeons, Saskatoon, Saskatchewan, June 1995. drs), “Applications of X-ray Computed Flow Visualization. 62nd Annual Technical Conference and Exhibition of the Society of W Yoshikawa, H., (2004): X-ray Computed Tomography. Website of Inouye Labs, Japan. Particles in Contact. Proceedings of the Royal Society. Vol. 269, Series A, pp 500-527. S Masters Thesis, University of California Davis. Steude, J.S., and F. Hopkins. (1994): Industrial X-ray computed tomography applied to soil research. Tomography of soil-water-root processes. SSSA Spec. Publ. 36. SSSA, Madison, WI. S Tomography Artifacts: An Experimental Investigation of Causative Factors, 49th Annual Meetin Tatsuoka, F. (1987): Discussion: The Strength and Dilatancy of Sands. Geotechnique, 37(2), 219-226. Van Geet, M., Lagrou, D., and Swennen, R. (2003): Porosity Measurements of Sedimentary Rocks by Means of Microfocus X-ray Computed Tomography (µCT),” from Mees, F. Swennen, R., Van Geet, M. & Jacobs, P. (e Tomography in the Geosciences”, Geological Society of London, Special Publication, Vol. 215, pp. 51-60. Vardoulakis, I., Sulem, J., (1995), Bifurcation Analysis in Geomechanics, Blackie Academic & Professional, Bishopbriggs, Glasgow, 1995. Wang, L.B., Frost, J.D., and Lai, J.S., (1999): Non-Invasive Measurement of Permanent Strain Field Resulting from Rutting in Asphalt Concrete, Transportation Research Record, No. 1687, pp. 85-94. Washington State University (2004). X-Ray Computed Tomography Facility WAX-CT ( Wevers, M., De Meester, P., Swennen, R., (2000): Microfocus X-ray Computer Tomography in Materials Research, Proceeding of the 15th World Conference on Nondesructive Testing, Roma, Italy, 15-21 October 2000. Withjack, E. M., (1987): Computed Tomography for Rock-Property Determination and Fluid- Petroleum Engineers, Dallas, TX, September, 1987. ood, D.M. (1990): Soil Behavior and Critical State Soil Mechanics, Cambridge, Cambridge University Press. 75 APPENDIX A: TRANSLATION AND ROTATION HISTOGRAMS Figure A1: Normalized Translation Histograms at εz = -13.7 % (Normalization Value = 6.6 mm) 0 10 20 30 60 40 eq ue nc y (P 0 20Fr Normalized Translation (%), X Direction ar tic le s) 0 10 20 30 0 20 60 40 Normalized Translation (%), Y Direction Fr ar tic le s) eq ue nc y (P 0 10 20 3 40 50 0 20 40 0 Normalized Translation (%), Z Direction Fr eq ue nc y (P ar tic le s) 76 Figure A2: Normalized Translation Histograms at εz = -23.5 % (Normalization Value= 11.4 mm) 0 10 2 0 0 20 40 0 3 Normalized Translation (%), Y Direction Fr eq ue nc y (P ar tic le s) 0 10 20 30 40 50 0 10 20 30 40 Normalized Translation (%), Z Direction Fr eq ue nc y (P ar tic le s) 0 10 2 0 0 20 40 0 3 Normalized Translation (%), X Direction Fr eq ue nc y (P ar tic le s) 77 APPENDIX B: LOCAL STRAIN HISTOGRAMS 78 Figure B1: Local strains histograms at εz = -13.7% εxz 0.4 0.2 0.2 0.4 0 20 40 Fr eq ue nc y (P ar tic le s) εx0.4 0.2 0 0.2 0.4 0 30 40 20 10 Fr eq ue nc y (P ar tic le s) εy0.4 0.2 0 0.2 0.4 0 10 20 30 40 Fr eq ue nc y (P ar tic le s) εz0.4 0.2 0 0.2 0.4 0 20 60 0 40 Fr eq ue nc y (P ar tic le s) εyz 0.4 0.2 0.2 0.4 40 0 0 20 Fr eq ue nc y (P ar tic le s) 0.4 0.2 0 0.2 0 0.4 40 20 Fr at ic le s) eq ue nc y (P εxy Figure B2: Local strains histograms at εz = -23.5% εxy 0.5 0 0.5 0 15 20 10 5 Fr eq ue nc y (P ar tic le s) εx0.5 0 0.5 0 5 10 15 20 Fr eq ue nc y (P ar tic le s) εy0.5 0 0.5 0 5 10 15 20 Fr eq ue nc y (P ar tic le s) εz0.5 0 0.5 0 10 20 30 Fr eq ue nc y (P ar tic le s) εyz 0.5 0 0.5 0 5 10 15 20 25 Fr eq ue nc y (P ar tic le s) εxz 0.5 0 0.5 0 5 10 15 20 25 Fr eq ue nc y (P ar tic le s) γxz 79 Figure B3: Rotation strains ωxz 0.4 0.2 0 0 10 20 30 40 Fr eq ue nc y (P ar tic le s) ω 0.4 0.2 0 0 20 40 60 Fr eq ue nc y (P ar tic le s) ω 0.6 0.4 0.2 0 0 10 20 30 40 Fr eq ue nc y (P ar tic le s) ωωyz ωxy 80histograms at ez = -13.7% 0.2 0.4xy 0.2 0.4xz 0.2 0.4 0.6yz ωx z1 0.5 0 0.5 1 0 10 20 Fr eq ue nc y (P ar tic le s) ωxz 1 0.5 0 0.5 1 0 10 20 Fr eq ue nc y (P ar tic le s) ωy Figure B4: Rotation strains histograms at εz = -23.5% zωyz 1 0.5 0 0.5 1 0 10 20 30 Fr eq ue nc y (P ar tic le s) ωx yωxy 81 VITA Bashar Alramahi was born on September 7th, 1979, in Beirut, Lebanon. He studied high school at Al-Ittihad School in Amman, Jordan, and graduated in 1997. He received his bachelor’s degree in civil engineering from Birzeit University, Ra allah, Palestine, in August, 2002. He came to the United States in January, 2003 to pursue a master’s degree in geotechnical engineering at Louisiana State University, Baton Rouge, Louisiana. It is anticipated that he will fulfill the requirements for the master’s degree in civil engineering in August, 2004. m 82