Viscoelastic Strains of Palaeozoic Shales under the Burger’s Model Description
Abstract
:1. Introduction
2. Characteristics of the Material
3. Laboratory Rheological Analyses of Paleozoic Rocks
3.1. Experimental Equipment
3.2. Initial Creep Analysis in Paleozoic Shales on 1D Models
4. Theoretical Basics of Rock Rheology in Triaxial Test Conditions for the Burger’s Model
- σm is axiator of the stress tensor σm = (σ1 + σ2 + σ3)/3 = σokt,
- εm is axiator of the strain tensor εm = (ε1 + ε2 + ε3)/3 = εokt,
- Sij is deviatoric stress tensor,
- eij is deviatoric strain tensor.
- and are the first and second derivative of the average stress, respectively,
- vol and vol are the first and second derivatives of the volumetric strain, respectively,
- ij and ij are the first and second derivatives of deviatoric stress tensor, respectively,
- ij and ij are the first and second derivatives of the deviatoric strain tensor, respectively.
4.1. Parameters of Burger’s Model for the Paleozoic Shales from the Baltic Basin
5. Verification of Burger’s Model for Creep Description in Paleozoic Shales in Conventional Triaxial Loading Conditions
6. Assessing the Significance of Volumetric Creep Description in Paleozoic Shales
7. Linear Creep Limit in Paleozoic Shales
8. Summary and Remarks on Creep in Shales with Regard to Loadings Exceeding Secondary Creep
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Deviatoric Creep | Volumetric Creep | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Sample | Sample | (σ1 − σ3) | GK | GM | R2 | KK | KM | R2 | ||||
Location | Number | [GPa] | [GPa] | [GPa/day] | [GPa/day] | [GPa] | [GPa] | [GPa/day] | [GPa/day] | |||
Pelplin | W260A | 0.23 | 473 | 25 | 8.0 | 292 | 0.89 | 420 | 40 | 38 | 48 | 0.87 |
0.46 | 237 | 19 | 7.6 | 167 | 0.98 | 315 | 35 | 31 | 94 | 0.90 | ||
0.69 | 81 | 16 | 3.0 | 63 | 0.99 | 221 | 39 | 16 | 65 | 0.89 | ||
W264A | 0.27 | 242 | 20 | 4.7 | 192 | 0.96 | 203 | 29 | 61 | 47 | 0.93 | |
0.55 | 166 | 20 | 3.0 | 63 | 0.99 | 413 | 46 | 38 | 70 | 0.90 | ||
B211A | 0.33 | 400 | 18 | 7.1 | 205 | 0.91 | 359 | 35 | 32 | 36 | 0.90 | |
0.65 | 133 | 18 | 2.2 | 86 | 0.99 | 231 | 62 | 57 | 68 | 0.91 | ||
M30B | 0.26 | 282 | 18 | 4.1 | 166 | 0.95 | 130 | 32 | 46 | 55 | 0.83 | |
0.52 | 118 | 15 | 2.7 | 60 | 0.98 | 281 | 35 | 6 | 67 | 0.87 | ||
W260B | 0.26 | 268 | 19 | 5.1 | 107 | 0.98 | 269 | 25 | 19 | 42 | 0.92 | |
0.52 | 128 | 17 | 6.1 | 145 | 0.98 | 174 | 37 | 7 | 59 | 0.89 | ||
Pasłęk | W278A | 0.28 | 372 | 20 | 7.4 | 224 | 0.90 | 450 | 38 | 67 | 33 | 0.90 |
0.56 | 162 | 17 | 4.7 | 191 | 0.97 | 178 | 28 | 18 | 66 | 0.92 | ||
0.84 | 15 | 13 | 1.1 | 20 | 0.99 | 331 | 60 | 56 | 81 | 0.95 | ||
W278B | 0.34 | 185 | 16 | 3.7 | 194 | 0.88 | 439 | 38 | 28 | 37 | 0.85 | |
0.69 | 20 | 12 | 0.6 | 43 | 0.98 | 485 | 39 | 19 | 59 | 0.82 | ||
B236A | 0.20 | 711 | 24 | 9.1 | 195 | 0.96 | 363 | 27 | 27 | 114 | 0.92 | |
0.41 | 257 | 21 | 5.3 | 138 | 0.90 | 61 | 51 | 16 | 117 | 0.88 | ||
W277B | 0.51 | 68 | 17 | 1.1 | 71 | 0.92 | 173 | 39 | 19 | 81 | 0.90 | |
Jantar | W305A | 0.25 | 244 | 18 | 1.8 | 76 | 0.96 | 6 | 24 | 10 | 61 | 0.78 |
0.49 | 123 | 14 | 1.3 | 32 | 0.98 | 226 | 34 | 10 | 45 | 0.89 | ||
W306A | 0.26 | 260 | 19 | 4.6 | 104 | 0.97 | 357 | 23 | 22 | 53 | 0.91 | |
0.52 | 110 | 14 | 3.3 | 80 | 0.99 | 182 | 37 | 22 | 73 | 0.94 | ||
0.78 | 42 | 10 | 1.3 | 31 | 0.99 | 175 | 48 | 19 | 59 | 0.95 | ||
B275B | 0.26 | 246 | 15 | 4.1 | 155 | 0.88 | 287 | 20 | 21 | 30 | 0.90 | |
0.52 | 98 | 12 | 1.6 | 34 | 0.94 | 150 | 31 | 7 | 43 | 0.90 | ||
B279D | 0.23 | 329 | 19 | 5.7 | 119 | 0.96 | 226 | 26 | 44 | 164 | 0.73 | |
0.46 | 145 | 17 | 3.0 | 67 | 0.98 | 464 | 45 | 77 | 50 | 0.81 | ||
M43BB | 0.23 | 285 | 20 | 2.8 | 110 | 0.95 | 90 | 26 | 22 | 50 | 0.83 | |
0.46 | 142 | 16 | 3.0 | 60 | 0.90 | 261 | 33 | 6 | 62 | 0.78 | ||
Sasino | M60B | 0.30 | 336 | 18 | 7.1 | 105 | 0.92 | 378 | 22 | 22 | 44 | 0.90 |
0.60 | 169 | 16 | 3.1 | 62 | 0.98 | 191 | 28 | 28 | 12 | 0.92 | ||
W320B | 0.21 | 314 | 18 | 1.9 | 185 | 0.92 | 106 | 22 | 44 | 39 | 0.82 | |
0.42 | 161 | 15 | 3.2 | 144 | 0.98 | 87 | 29 | 12 | 43 | 0.84 | ||
M62A | 0.24 | 319 | 22 | 3.7 | 257 | 0.88 | 116 | 24 | 30 | 73 | 0.64 | |
0.48 | 304 | 20 | 7.4 | 219 | 0.94 | 254 | 35 | 1 | 47 | 0.66 |
Sample | Sample | (σ1 − σ3) | B | n | R2 |
---|---|---|---|---|---|
Location | Number | [MPa−1] × 10−5 | [-] | ||
Pelplin | B-211-A | 0.65 | 1.6 | 0.043 | 0.93 |
M-30-B | 0.52 | 2.0 | 0.044 | 0.94 | |
W-260-A | 0.46 | 1.6 | 0.037 | 0.97 | |
W-260-B | 0.52 | 1.7 | 0.045 | 0.97 | |
W-264-A | 0.55 | 1.8 | 0.044 | 0.93 | |
Average | 1.7 | 0.043 | |||
Standard dev. | 0.11 | 4.4 × 10−5 | |||
Pasłęk | B-236-A | 0.41 | 1.7 | 0.023 | 0.94 |
W-277-B | 0.46 | 1.8 | 0.029 | 0.98 | |
W-278-A | 0.56 | 1.9 | 0.032 | 0.98 | |
W-278-B | 0.69 | 1.6 | 0.048 | 0.98 | |
Average | 1.8 | 0.033 | |||
Standard dev. | 0.06 | 3.4 × 10−4 | |||
Jantar | B-275-B | 0.52 | 2.6 | 0.040 | 0.99 |
B-279-D | 0.46 | 1.8 | 0.039 | 0.99 | |
M-43B-B | 0.46 | 2.0 | 0.036 | 0.97 | |
W-305-A | 0.49 | 2.0 | 0.048 | 0.76 | |
W-306-A | 0.52 | 2.0 | 0.046 | 0.93 | |
Average | 2.1 | 0.042 | |||
Standard dev. | 0.37 | 1.1 × 10−4 | |||
Sasino | M-60-B | 0.60 | 1.9 | 0.036 | 0.96 |
M-62-A | 0.48 | 1.8 | 0.020 | 0.91 | |
W-320-B | 0.42 | 2.4 | 0.027 | 0.82 | |
Average | 2.0 | 0.028 | |||
Standard dev. | 0.17 | 1.3 × 10−4 |
Shale | Ref. | B | n | QFP | Cb | Clay | σTCS | pc | Temp. |
---|---|---|---|---|---|---|---|---|---|
[MPa−1] × 10−5 | [-] | vol% | vol% | vol% | [MPa] | [MPa] | [oC] | ||
Barnett-1 (Ba1) | a | 2.0–2.6 | 0.012–0.021 | 48 | 2 | 50 | 210 | <30 | room |
Barnett-2 (Ba2) | a | 1.6–1.6 | 0.009–0.010 | 42 | 48 | 10 | 325 | <30 | room |
Hayneville-1 (Ha1) | a | 1.8–2.7 | 0.027–0.062 | 32 | 20 | 48 | 145 | <30 | room |
Hayneville-2 (Ha2) | a | 1.5–1.8 | 0.011–0.049 | 23 | 49 | 28 | 240 | <30 | room |
Eagle Ford-1 (Ea1) | a | 1.7–2.3 | 0.024–0.053 | 24 | 46 | 30 | 200 | <30 | room |
Eagle Ford-2 (Ea2) | a | 1.7–1.8 | 0.023–0.049 | 14 | 66 | 20 | 175 | <30 | room |
Pos_Dot | b | 2 | 0.05 | 14 | 42 | 43 | 175 | 20 | 20 |
Pelplin | c | 1.7 | 0.043 | 40 | 14 | 46 | 215 | 50 | 85 |
Paslek | c | 1.8 | 0.033 | 36 | 7 | 57 | 155 | 50 | 85 |
Jantar | c | 2.1 | 0.042 | 31 | 22 | 46 | 215 | 50 | 85 |
Sasino | c | 2 | 0.028 | 51 | 3 | 45 | 222 | 50 | 85 |
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Wilczynski, P.M.; Cieslik, J.; Domonik, A.; Lukaszewski, P. Viscoelastic Strains of Palaeozoic Shales under the Burger’s Model Description. Appl. Sci. 2023, 13, 10981. https://doi.org/10.3390/app131910981
Wilczynski PM, Cieslik J, Domonik A, Lukaszewski P. Viscoelastic Strains of Palaeozoic Shales under the Burger’s Model Description. Applied Sciences. 2023; 13(19):10981. https://doi.org/10.3390/app131910981
Chicago/Turabian StyleWilczynski, Przemyslaw Michal, Jerzy Cieslik, Andrzej Domonik, and Pawel Lukaszewski. 2023. "Viscoelastic Strains of Palaeozoic Shales under the Burger’s Model Description" Applied Sciences 13, no. 19: 10981. https://doi.org/10.3390/app131910981
APA StyleWilczynski, P. M., Cieslik, J., Domonik, A., & Lukaszewski, P. (2023). Viscoelastic Strains of Palaeozoic Shales under the Burger’s Model Description. Applied Sciences, 13(19), 10981. https://doi.org/10.3390/app131910981