Jiang et al 2023a - Revision history

Rimni at 13:34, 26 January 2024

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Rimni: /* 2.2 Physical and mechanical parameters of sandstone and its joint plane */

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‎2.2 Physical and mechanical parameters of sandstone and its joint plane

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← Older revision		Revision as of 13:34, 26 January 2024
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−	In [[#img-13\|Figure 13]], the stability evolutions during a complete rising-drawdown process are illustrated for the following parameters: <math display="inline"> \theta = t0^\circ</math>, <math display="inline"> t = 3</math> m, <math display="inline"> v = 3</math> m/d and <math display="inline"> k=1\times 10^{-5} </math> m/s. It can be observed that when the RWL remains stable at either 145 m or 175 m, a higher slope height <math display="inline"> h </math> generally leads to a smaller value of FoS. As the slope height <math display="inline"> h </math> decreases, the increase in FoS during the rising stage and the decrease in FoS during the drawdown stage become more noticeable. The difference in FoS between <math display="inline"> h = 70</math> m and <math display="inline"> h = 90</math> m is slightly amplified during the rising stage, while it is slightly reduced during the drawdown stage.	+	In [[#img-13\|Figure 13]], the stability evolutions during a complete rising-drawdown process are illustrated for the following parameters: <math display="inline"> \theta = 50^\circ</math>, <math display="inline"> t = 3</math> m, <math display="inline"> v = 3</math> m/d and <math display="inline"> k=1\times 10^{-5} </math> m/s. It can be observed that when the RWL remains stable at either 145 m or 175 m, a higher slope height <math display="inline"> h </math> generally leads to a smaller value of FoS. As the slope height <math display="inline"> h </math> decreases, the increase in FoS during the rising stage and the decrease in FoS during the drawdown stage become more noticeable. The difference in FoS between <math display="inline"> h = 70</math> m and <math display="inline"> h = 90</math> m is slightly amplified during the rising stage, while it is slightly reduced during the drawdown stage.

	<div id='img-13'></div>		<div id='img-13'></div>

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 |-style=text-align: center;"
 |  style="text-align: left;"|The top zone
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 |-style="text-align:center"
 ! style="text-align: left;vertical-align:center;"| Zone !! Heavy <br> <math display="inline">\gamma</math> (kN·m<sup>-3</sup>)!! Elastic-modulus <br> <math display="inline">E</math> (GPa) !! Poisson ratio <br> <math>\mu </math> !! Cohesion <br> <math>C</math> (MPa) !! Friction angle <br> <math display="inline">\varphi \,(\circ )</math> !! Tensile strength <br> <math display="inline">\sigma_t</math> (MPa)
-|-
+|-style=text-align: center;"
 |  style="text-align: left;"|The top zone
 |  style=text-align: center;"|26.0
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 |  style="text-align: left;"|The RWL fluctuation affected zone
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-|  style="border: 1pt solid black;text-align: center;"|25
+|  style="text-align: center;"|25
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 |  style="text-align: left;"|The long term inundation zone
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 Based on the investigation of different factors on stability in Section 5.2, we can gain insight into the conditions that lead to better stability in anti-dip layered rock slopes. However, it is important to note that the stability can vary significantly during a complete rising and drawdown process of RWL, which requires careful control and prevention measures. In other words, the greater the stability variation, the more attention should be given to the impact of RWL fluctuations on the stability. Therefore, it is necessary to assess the relative influence of different factors on stability variation.
-The amplitude of variation in FoS, which measures the difference between the maximum and minimum values of the FoS during a complete RWL rising and drawdown process, is employed to indicate the level of stability variation. Range analysis is performed to rank the impact of different factors. Likewise, the pseudo-horizontal method is utilized for range analysis, considering that only two levels are selected for the permeability coefficient <math display="inline"> k </math>. This results in 16 combinations of different levels for the thickness <math display="inline"> t </math>, dip angle <math display="inline"> \theta </math>, fluctuation rate <math display="inline"> v </math>, permeability coefficient <math display="inline">  k</math>, and slope height <math display="inline"> h </math>, as shown in Table 6. Furthermore, the outcomes of the amplitude of variation in FoS are presented in [[#tab-6|Table 6]].
+The amplitude of variation in FoS, which measures the difference between the maximum and minimum values of the FoS during a complete RWL rising and drawdown process, is employed to indicate the level of stability variation. Range analysis is performed to rank the impact of different factors. Likewise, the pseudo-horizontal method is utilized for range analysis, considering that only two levels are selected for the permeability coefficient <math display="inline"> k </math>. This results in 16 combinations of different levels for the thickness <math display="inline"> t </math>, dip angle <math display="inline"> \theta </math>, fluctuation rate <math display="inline"> v </math>, permeability coefficient <math display="inline">  k</math>, and slope height <math display="inline"> h </math>, as shown in [[#tab-6|Table 6]]. Furthermore, the outcomes of the amplitude of variation in FoS are presented in [[#tab-6|Table 6]].
 <div class="center" style="font-size: 75%;">'''Table 6'''. Amplitude of variation in FoS resulted by pseudo-horizontal orthogonal experiment</div>

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 ===3.2 Seepage analysis of the conceptual model during a complete rising-drawdown process===
-In the numerical model used for the coupled analysis of the seepage-stress field, the fundamental unit is a triangular element with a side length of 1 m. Since the numerical models have three distinct heights, as indicated in [[#tab-3|Table 3]], the number of elements varies from 19,180 to 23,450. The left and bottom boundaries are treated as fixed boundaries. Throughout the entire model, the initial groundwater level is assumed to be 145m, and it will change according to RWL fluctuations. On the right boundary, a variable water level is applied to simulate the rise and drawdown of the RWL. The entire process of rising and drawdown is divided into five stages. These stages include: a stable RWL at 145 m, RWL rising from 145 m to 175 m, a stable RWL at 175 m, RWL falling from 175 m to 145 m, and finally, a stable RWL at 145 m. To reduce computation costs, the first and final stages each last for 10 days, while the third stage lasts for 30 days. The durations of the second and fourth stages are determined by the RWL fluctuation rate <math display="inline">v  </math>, as shown in Table 3. For instance, if <math display="inline">v = 2</math> m/d, the second and fourth stages will each last for 15 days.
+In the numerical model used for the coupled analysis of the seepage-stress field, the fundamental unit is a triangular element with a side length of 1 m. Since the numerical models have three distinct heights, as indicated in [[#tab-3|Table 3]], the number of elements varies from 19,180 to 23,450. The left and bottom boundaries are treated as fixed boundaries. Throughout the entire model, the initial groundwater level is assumed to be 145m, and it will change according to RWL fluctuations. On the right boundary, a variable water level is applied to simulate the rise and drawdown of the RWL. The entire process of rising and drawdown is divided into five stages. These stages include: a stable RWL at 145 m, RWL rising from 145 m to 175 m, a stable RWL at 175 m, RWL falling from 175 m to 145 m, and finally, a stable RWL at 145 m. To reduce computation costs, the first and final stages each last for 10 days, while the third stage lasts for 30 days. The durations of the second and fourth stages are determined by the RWL fluctuation rate <math display="inline">v  </math>, as shown in [[#tab-3|Table 3]]. For instance, if <math display="inline">v = 2</math> m/d, the second and fourth stages will each last for 15 days.
 The coupled analysis of the seepage-stress field is conducted using GDEM software, and the calculation is terminated once that a default setting criteria that the non-balance ratio is less than <math display="inline">1\times 10^{-5}</math> is satisfied. To facilitate the analysis, eight critical moments are selected: the moment when RWL is stable at 145 m, the moments when RWL rises to 155 m, 165 m, and 175 m, the moment when RWL is stable at 175 m, and the moments when RWL falls to 165 m, 155 m, and 145 m. The groundwater levels and the fracture water pressures at these critical moments are recorded for a detailed analysis and subsequent comparison. For example, in the case of <math display="inline">t = 2</math> m, <math display="inline">\theta = 50^{\circ}</math>, <math display="inline">v = 2</math> m/d, <math display="inline">h = 70</math> m, and <math display="inline">k = 1\times 10^{-4}</math> m/s, the groundwater levels and the fracture water pressures are shown in [[#img-2|Figure 2]], in which the unit of water pressures is kPa.

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-Last, the influence of the RWL fluctuation rate <math display="inline">v  </math> is evaluated using models with constant values of <math display="inline">\theta = 50^{\circ}</math>, <math display="inline">t = 2</math> m, <math display="inline">h = 70</math> m, and <math display="inline">k = 1\times 10^{-4}</math> m/s as examples. Figure 7 illustrates the groundwater levels under three different <math display="inline">v</math> settings. It is evident that a higher RWL fluctuation rate <math display="inline">v  </math> leads to a more pronounced lagging effect. This observation aligns with the existing conclusions regarding the impact of the RWL fluctuation rate <math display="inline">v  </math> on the groundwater level in slopes located in TRGA [32].
+Last, the influence of the RWL fluctuation rate <math display="inline">v  </math> is evaluated using models with constant values of <math display="inline">\theta = 50^{\circ}</math>, <math display="inline">t = 2</math> m, <math display="inline">h = 70</math> m, and <math display="inline">k = 1\times 10^{-4}</math> m/s as examples. [[#img-7|Figure 7]] illustrates the groundwater levels under three different <math display="inline">v</math> settings. It is evident that a higher RWL fluctuation rate <math display="inline">v  </math> leads to a more pronounced lagging effect. This observation aligns with the existing conclusions regarding the impact of the RWL fluctuation rate <math display="inline">v  </math> on the groundwater level in slopes located in TRGA [32].
 <div id='img-7'></div>
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 Numerous studies have been conducted on the assessment method of anti-dip rock slope stability. In this study, the numerical analysis based on the strength reduction method is considered a feasible choice for evaluating stability. This is because all the numerical models for the coupled analysis of the seepage-stress field have been established in Section 3, and it has previously been employed by Azarafza et al. [51] for assessing the stability of a complex secondary toppling failure case. However, there are two issues that may arise when implementing the strength reduction method in the GDEM numerical models. The first issue is the unfavorable computation cost due to the coupled analysis of the seepage-stress field. The second issue is the absence of a reliable criterion for determining the FoS in discrete element method modeling [52]. Therefore, this study utilizes the cantilever limit equilibrium method proposed by Aydan et al. [14] to evaluate the stability of the anti-dip layered rock slope.
-This cantilever limit equilibrium method [14] considers the toppling failure of the rock stratum as a result of the tensile stress induced by bending, using the calculation model as illustrated in Figure 8. The model begins by assuming a basal failure plane for the anti-dip layered rock slope, usually perpendicular to the strata and represented by the red line in [[#img-8|Figure 8]](a). The rock layers above the basal failure plane are numbered from bottom to top, and their equilibrium conditions are examined by treating them as individual cantilever beams.
+This cantilever limit equilibrium method [14] considers the toppling failure of the rock stratum as a result of the tensile stress induced by bending, using the calculation model as illustrated in [[#img-8|Figure 8]]. The model begins by assuming a basal failure plane for the anti-dip layered rock slope, usually perpendicular to the strata and represented by the red line in [[#img-8|Figure 8]](a). The rock layers above the basal failure plane are numbered from bottom to top, and their equilibrium conditions are examined by treating them as individual cantilever beams.
 <div id='img-8'></div>

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 ==6. Conclusions==
-To evaluate the impact of various factors on the stability of anti-dip rock slopes under fluctuating water levels in the TRGA, a conceptual model is constructed. The model focuses on sandstone as the strata and considers variations in strata thickness <math display="inline"> t </math>, strata dip angle ''θ'', permeability coefficient ''k'', RWL fluctuation rate ''v'', and slope height ''h''. Seepage field and groundwater lines are obtained through seepage-stress field coupled analysis using GDEM software. The FoS is determined using the improved cantilever beam limit equilibrium method. The variations of the groundwater line and the FoS during the entire process of RWL rising and drawdown are investigated. Range analysis is employed to rank the impact of different factors. Based on the findings, the following conclusions can be drawn:
+To evaluate the impact of various factors on the stability of anti-dip rock slopes under fluctuating water levels in the TRGA, a conceptual model is constructed. The model focuses on sandstone as the strata and considers variations in strata thickness <math display="inline"> t </math>, strata dip angle <math display="inline"> \theta </math>, permeability coefficient <math display="inline">  k</math>, RWL fluctuation rate <math display="inline"> v </math>, and slope height <math display="inline"> h </math>. Seepage field and groundwater lines are obtained through seepage-stress field coupled analysis using GDEM software. The FoS is determined using the improved cantilever beam limit equilibrium method. The variations of the groundwater line and the FoS during the entire process of RWL rising and drawdown are investigated. Range analysis is employed to rank the impact of different factors. Based on the findings, the following conclusions can be drawn:
-(1) The changes in groundwater levels in anti-dip layered rock slope exhibit a clear lag behind RWL fluctuations. During the rising stage, the groundwater level inside the model is lower than RWL, while during the drawdown stage, it is higher. Moreover, the gap between the groundwater level and RWL widens as we move from the slope surface to deeper regions. Range analysis confirms that the permeability coefficient ''k'' has the most significant impact on the lagging effect, followed by the fluctuation rate ''v'', the thickness <math display="inline"> t </math>, and the dip angle ''θ''.
+(1) The changes in groundwater levels in anti-dip layered rock slope exhibit a clear lag behind RWL fluctuations. During the rising stage, the groundwater level inside the model is lower than RWL, while during the drawdown stage, it is higher. Moreover, the gap between the groundwater level and RWL widens as we move from the slope surface to deeper regions. Range analysis confirms that the permeability coefficient <math display="inline">  k</math> has the most significant impact on the lagging effect, followed by the fluctuation rate <math display="inline"> v </math>, the thickness <math display="inline"> t </math>, and the dip angle <math display="inline"> \theta </math>.
 (2) The stability of anti-dip layered rock slopes varies throughout the entire RWL rising and drawdown process. Stability is improved during the rising stage but deteriorates during the drawdown stage. The FoS reaches its highest value when the RWL reaches 175 m and its lowest value when the RWL reaches 145 m. When the RWL remains stable at 175 m for a longer period, the FoS gradually decreases from its maximum value to a certain level. Conversely, when the RWL remains stable at 145 m for a longer period, the FoS increases from its minimum value to a certain level. Importantly, the FoS is higher when the RWL is stable at 145 m compared to when it is stable at 175 m.
-(3) Different factors influence the stability of anti-dip layered rock slopes and its variation during an entire RWL rising and drawdown process. A smaller permeability coefficient ''k'' and a greater fluctuation rate ''v'' lead to a more significant enhancement and damage of stability during the RWL rising and drawdown stages, respectively. A larger strata thickness <math display="inline"> t </math>, a smaller dip angle ''θ'', and a smaller slope height ''h'' generally contribute to improved stability. As the strata thickness <math display="inline"> t </math> increases, the dip angle ''θ'' increases, and the slope height ''h'' decreases, the enhancement of stability during the rising stage and the damage of stability during the drawdown stage become more noticeable. Range analysis shows that the permeability coefficient ''k'' has the greatest impact on the stability variation, followed by the fluctuation rate ''v'', the thickness <math display="inline"> t </math>, the dip angle ''θ'', and the slope height ''h''.
+(3) Different factors influence the stability of anti-dip layered rock slopes and its variation during an entire RWL rising and drawdown process. A smaller permeability coefficient <math display="inline">  k</math> and a greater fluctuation rate <math display="inline"> v </math> lead to a more significant enhancement and damage of stability during the RWL rising and drawdown stages, respectively. A larger strata thickness <math display="inline"> t </math>, a smaller dip angle <math display="inline"> \theta </math>, and a smaller slope height <math display="inline"> h </math> generally contribute to improved stability. As the strata thickness <math display="inline"> t </math> increases, the dip angle <math display="inline"> \theta </math> increases, and the slope height <math display="inline"> h </math> decreases, the enhancement of stability during the rising stage and the damage of stability during the drawdown stage become more noticeable. Range analysis shows that the permeability coefficient <math display="inline">  k</math> has the greatest impact on the stability variation, followed by the fluctuation rate <math display="inline"> v </math>, the thickness <math display="inline"> t </math>, the dip angle <math display="inline"> \theta </math>, and the slope height <math display="inline"> h </math>.
-−
-<math display="inline"> t </math> !!   style="text-align: center;"|<math display="inline"> \theta </math> !!   style="text-align: center;"|<math display="inline"> v </math> !!   style="text-align: center;"|<math display="inline">  k</math> !!   style="text-align: center;"|<math display="inline"> h </math>
-−
 ==Acknowledgments==

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-|  style="text-align: center;"|<math display="inline"> K_1 </math>
+|  style="text-align: center;"|<math display="inline"> k_1 </math>
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-|  style="text-align: center;"|<math display="inline"> K_2 </math>
+|  style="text-align: center;"|<math display="inline"> k_2 </math>
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 |  style="text-align: center;"|0.00672
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-|  style="text-align: center;"|<math display="inline"> K_3 </math>
+|  style="text-align: center;"|<math display="inline"> k_3 </math>
 |  style="text-align: center;"|0.00469
 |  style="text-align: center;"|0.00465