Determining the slope of the hydraulic sensitivity curves to erodibility for the selected rock mass parameters

Development of the eGSI erodibility index

The case studies of the eroded spillways documented by Pells (2016) produced RMR values >18; for these values, Equation 2.8 is used to determine GSI. GSI can also be determined using the lookup chart (GSI-Chart). The data points are classified according to the interpreted erosion categories presented in Table 2.3.
There exists a correlation between both GSI and the interpreted erosion categories; however, a stronger correlation is noted when the GSI-Chart is used (Pells, 2016a).
As such, Pells (2016) proposed an erodibility index eGSI derived from the GSI-Chart. Furthermore, use of the GSI-Chart is of considerable interest because:
1) The values determined from the lookup chart (Figure 2.7) are substantially easier to obtain than values obtained via calculation from RMR;
2) The lookup chart is not encumbered with the problematic RQD, as is the GSI-RMR. Indeed, Pells et al. (2017a) argued that at the time of its development, the RQD index was developed for a specific application and that this parameter is sometimes applied inconsistently in practice;
3) The lookup chart is also not dominated by the substance strength value of UCS, as occurs with GSI-RMR. In fact, Pells (2016) considered that the UCS of rock plays a very limited, if not negligible, role in the erodibility of fractured rock masses.
Factor F6, representing the discontinuity orientation adjustment, can be removed from RMR76 (Section 2.2.1.1), and a new erosion-discontinuity orientation adjustment factor (Edoa), which represents the vulnerability of a rock mass to erodibility, can be added (Pells 2016). Therefore, the proposed erodibility index (Equation 2.11) takes the form of the original RMR equation (Equation 2.7). eGSI = GSI + Edoa (2.11)
The Edoa factor can be determined from the curves presented in Figures 2.8 and 2.9. As reported by Pells (2016), the process of deriving these curves was inspired from the graphical presentation of Kirsten’s Js factor (Moore and Kirsten, 1988). Various pictograms were drawn for rock masses having two orthogonal joint sets at various orientations relative to the direction of flow and marked by various relative spacing (Figures 2.8 and 2.9). For the pictograms, the surface was defined primarily along joints observed at spillway sites. The surface formed in this manner creates a roughness and block shape that reflects joint structure. The Edoa values were derived purely by thought-experiment with the pictograms, assessing vulnerability to significant and ongoing erosion and considering the kinematics of block removal as well as the nature and direction of hydraulic loading, as intuited from field observation and the analysis of the numerous model tests. The process was also undertaken for non-orthogonal joint sets; however, the appraised values were not significantly different (Pells, 2016a).
The plotting of the force of flowing water versus the eGSI index is shown in Figure 2.10. The data points are classified according to the interpreted erosion classes presented in Table 2.3, and the erosion classes boundaries are contoured manually.
Pells (2016) stated that the inclusion of the factor Edoa, representing the erosion vulnerability due to rock block’s shape and orientation relative to the flow direction, increased the ‘‘spread’’ of the data and provided a subtle improvement in the correlation with erosion. Consequently, the eGSI index becomes more amenable for evaluating the hydraulic erodibility of rock (Pells 2016). It should be noted, however, that the GSI lookup chart remains semi-qualitative, and any subsequent evaluation can be greatly influenced by the judgment of the analyst in the field. Furthermore, it was not developed specifically to assess the hydraulic erodibility of rock; for example, it does not incorporate details related to joints opening that could play a determining role in the hydraulic erodibility process.

Rock mass erodibility index (RMEI)

The RMEI classification system was developed as alternative to represent the most important geological factors controlling the erosion mechanism. It is based on field observations of eroded case studies (Pells 2016). Pells (2016) conceived the RMEI classification system as the existing rock mass indices, including Kirsten’s index, did not represent the erosion actually observed during field investigations. The valuable addition to the RMEI classification system was the representation of the geological factors controlling the erosion mechanism, where the relative importance of each factor is based on the field observations of eroded spillway case studies. The structure of the RMEI classification system was inspired from that used in Fell et al. (2008); the subsequent classification system combines multiple factors (Pells, 2016a). For the RMEI system, the factors combining was applied to represent the likelihood factor (LF) concerning the detachment of rock blocks from the
spillway floor. The RMEI classification system is also based on the relative importance factor (RF). This factor places the greatest weight to those factors judged to be most important in controlling detachment and down-weights those judged least important (Pells, 2016a). The RMEI classification system is presented in Figure 2.11.
The value of RMEI is determined based on RF and LF as presented in Equation 2.12. The prefixes P1 to P5 are various sets of parameters that represent, respectively, the kinematically viable mechanism for detachment, the nature of the potentially eroding surface, the nature of the joints, the joints spacing, and the block shape.
To define the parameter of the nature of the joints (Figure 2.11), Pells (2016) grouped three rock mass characteristics (joints roughness, joints aperture, and the UCS of joints). On the other hand, Pells (2016) provided a suggested method for estimating the likelihood factor for this parameter as presented in Table 2.5. However, this table does not consider the UCS of joints.
The RMEI classification system for evaluating hydraulic erosion can also be considered as a method inspired from the engineering rock mass classification systems developed for assessing underground excavation stability and tunnel support design, such as the Q-system (Barton et al., 1974) similarly developed based on the field investigation. The Q-system, however, gives the most important rating to the Kb factor (rating ranges from 1–100), which is an indication of the rock block size, as compared with the Kd factor (rating ranges from 0.03–5.33) that represents the joints shear strength (Barton et al., 1974). In the RMEI classification system, rock block size is not included directly. However, joints spacing can provide an idea of rock block size given that greater spacing of joints generate a larger rock block volume than a tighter spacing of joints. Also, the joints shear strength is not included in the RMEI classification system; however, the nature of joints factor can be considered as synonymous given that this factor incorporates the natural condition of joints. In
contrast to the Q-system, the RMEI classification system considers the joints spacing factor as being less important (RF = 1) than the nature of joints factor weighted at RF = 2. This comparison demonstrates how the field evaluation is influenced highly by the judgment of the analyst.

Abstract

The most commonly used method for assessing the hydraulic erodibility of rock is Annandale’s method. This method is based on a correlation between the erosive force of flowing water and the capacity of rock resistance. This capacity is evaluated using Kirsten’s index, which was initially developed to evaluate the excavatability of earth materials. For rocky material, this index is determined according to certain geomechanical factors related to the intact rock and the rock mass, such as the compressive strength of intact rock, the rock block size, the discontinuity shear strength and the relative block structure. To quantify the relative block structure, Kirsten developed a mathematical expression that accounted for the shape and orientation of the blocks relative to the direction of flow. Kirsten’s initial concept for assessing relative block structure considers that the geological formation is mainly fractured by two joint sets forming an orthogonal fracture system. An
adjusted concept is proposed to determine the relative block structure when the fracture system is non-orthogonal where the angle between the planes of the two joint sets is greater or less than 90°. An analysis of the proposed relative block structure rating shows that considering a non-orthogonal fracture system has a significant effect on Kirsten’s index and, as a consequence, on the assessment of the hydraulic erodibility of rock.

Introduction

The assessment of the hydraulic erodibility of earth materials was studied initially for problems associated with the erosion of earth materials under bridges (Keaton, 2013). It has since been adopted for dams given that erosion phenomena can occur on downstream rocks during flood spill periods, as observed at the Tarbela Dam in Pakistan (Lowe et al., 1979) and the Kariba Dam in Zambia (Bollaert et al., 2012). Annandale’s method (Annandale 1995, 2006) is the most commonly used method for assessing the hydraulic erodibility of earth materials (Castillo and Carrillo, 2016; Hahn and Drain, 2010; Laugier et al., 2015; Mörén and Sjöberg, 2007; Pells et al., 2015; Rock, 2015). This method is based on a correlation between the erosive force of flowing water, namely the available hydraulic stream power, and the capacity of rock to resist the flow energy. This capacity is evaluated using Kirsten’s index (Kirsten, 1982, 1988), which was initially developed to evaluate the
excavatability of earth materials but has since been adopted to assess the hydraulic erodibility of earth materials. The interest of using Kirsten’s index was first mentioned at a symposium focused on rock mass classification systems (ASTM STP- 984, 1988), where it was argued that the processes of mechanical excavatability and hydraulic erodibility of earth materials could be considered as similar processes (Moore and Kirsten 1988). Since then, many researchers have analyzed the hydraulic erodibility of earth materials by using the excavatability index, where the “direction of excavation” of the original index has been replaced by the “direction of flow” (Annandale, 1995; Annandale and Kirsten, 1994; Dooge, 1993; Kirsten et al., 2000; Moore et al., 1994; Pitsiou, 1990; Van-Schalkwyk et al., 1994a). Hereinafter, the terms “direction of excavation” and “direction of flow” are considered as synonymous and the term corresponds to the direction of the acting force. For rock material, Kirsten’s index (N) is determined according to certain geomechanical factors related to the intact rock and the rock mass, such as the compressive strength of intact rock (Ms), the rock block size (Kb), the discontinuity shear strength (Kd) and the relative block structure (Js). Kirsten’s index can be calculated according to the following equation:

Relative block structure

Our modifications of Kirsten’s index focus on the relative block structure factor. This section describes the initial “relative block structure” concept of Kirsten, but we include a large review of the underlying concepts that were not included in the initial Kirsten paper. According to Kirsten (1982), the relative orientation of blocksand the spacing of joints affect the possibilities of both penetrating the ground surface and dislodging the individual blocks. Accordingly, Kirsten (1982) determined the effect of orientation and shape of blocks on the excavatability process by considering the kinematic possibility of penetration (Kp) and the kinematic possibility of dislodgment (Kd). The following first and second subsections describe Kp and Kd, respectively, while the third subsection describes the methodology followed by Kirsten to develop the relative block structure rating.

Kinematic possibility of penetration

To represent a rock block volume, at least three joint sets are required to be intersected (3D representation). In this work, a block is represented in 2D and consequently considered to be delineated by only two joint sets. Kp is directly related to the inclination of the joints bounding blocks. The respective dips of these two joint sets relative to the ground surface are labeled as  and , while S and S representtheir respective spacing (Kirsten, 1982) (Figure 3.1). As the reciprocal of the joint spacing provides the number of joints per unit length, defined as the joint frequency (),  can be given as 1/S and  can be given as 1/S. Accordingly, the dip weighted by the number of joints of the first joint set can be defined as .tan and as .tan for the second joint set. As the geological formation is assumed to be fractured by two intersected joint sets, the combined kinematic possibility of penetration is the arithmetic average of the relative dips weighted by the number of joints of joint sets (Eq. 3.2).

Kinematic possibility of dislodgement

Once there is penetration into the ground (Figure 3.2 depicts a bulldozer, which is moving from right to left), excavatability occurs according to the digging process of angle , followed by the riding process of angle  (Figure 3.2). The action of block dislodgement can be represented by a horizontal force behind the block while this block is free to move in a perpendicular direction to the ground surface (Kirsten, 1982). As a result, Kd shown in Figure 3.3 can be obtained by the vector product of the principal dislodging force and the principal degree of freedom. The vectors of the principal dislodging force and the principal degree of freedom can be decomposed into parallel coaxial components along the sides of the block (Kirsten, 1982). The coaxial components are identified as A, B, B′ and A′ in Figure 3.3.

Methodology

As already mentioned, the block dislodging action is controlled by Kp and Kd, while the  angle for non-orthogonal fracture systems could be larger or smaller than 90°. Kirsten’s Js equation (Eq. 3.9) could be used for this purpose. However, his initial relative block structure concept must be adjusted. As a modification of  angle in the equation for Kd subsequently modifies the equation for Kp, only Kd is adjusted.
This section describes the principle of the adjusted relative block structure concept that is used to develop a new set of equations for determining Kd. These new equations are then included as part of the equation for Js to propose a rating of Js for non-orthogonal fracture systems.

Principle of the adjusted concept

The RJS values, as well as angles  and  initially used by Kirsten to determine Js values, are presented in Table 3.1. Based on these unpublished data, a representation of two blocks is produced in this thesis (Figure 3.8). The planes of the joints associated with  and  are plotted in blue and red, respectively (Figure 3.8).
When the block is oriented in the direction of excavation, Kirsten considered  to be positive (e.g.  = 30°), while  is determined by adding an angle of 90° to  (e.g.  = 30°, thus  = 30°+90° = 120°). On the other hand, when the block is oriented against the direction of excavation, Kirsten considered  to be negative (e.g.  = –30°), while  is determined by again adding an angle of 90° to  (e.g.  = -30°, thus  = – 30°+90° = 60°). For these two orientations of block relative to direction of excavation, Kirsten always kept  = 90° between the planes of the joints associated to  and  to consider this as an orthogonal fracture system.

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Table des matières

LIST OF TABLES 
LIST OF FIGURES 
LIST OF APPENDICES
LIST OF SYMBOLS AND ABREVIATIONS
ACKNOWLEDGMENTS
CHAPTER 1 – INTRODUCTION 
1.1. General concern
1.2. Statement of the specific problem
1.3. Research objectives
1.4. Research methodology
1.5. Originality and contribution
1.6. Thesis outline
1.7. References
CHAPTER 2 – LITERATURE REVIEW
2.1. Existing methods based on Kirsten’s index
2.1.1. Kirsten’s index
2.1.1.1. Compressive strength of intact rock rating
2.1.1.2. Rock block size rating
2.1.1.3. Joints shear strength rating
2.1.1.4. Relative block structure rating
2.1.2. Critical observations on the Kirsten’s index
2.1.3. Comparative methods based on the Kirsten index
2.1.3.1. Moore et al.’s scour threshold
2.1.3.2. Van Schalkywk et al.’s scour thresholds
2.1.3.3. Annandale’s scour threshold
2.1.3.4. Kirsten et al.’s scour threshold
2.2. Pells’s methods
2.2.1. Geological strength index for erodibility (eGSI)
2.2.1.1. Determining GSI from RMR components
2.2.1.2. Determining GSI from the lookup chart
2.2.1.3. Development of the eGSI erodibility index
2.2.2. Rock mass erodibility index (RMEI)
2.3. References
CHAPTER 3 – DETERMINING RELATIVE BLOCK STRUCTURE RATING FOR ROCK ERODIBILITY EVALUATION IN THE CASE OF NONORTHOGONAL
JOINT SETS
3.1. Introduction
3.2. Relative block structure
3.2.1. Kinematic possibility of penetration
3.2.2. Kinematic possibility of dislodgement
3.2.3. Relative block structure rating
3.3. Methodology
3.3.1. Principle of the adjusted concept
3.3.2. Proposed Kd equation when the block is oriented in direction of flow
3.3.3. Proposed Kd equation when the block is oriented against direction of flow
3.3.4. Analysis of Kd behavior
3.3.5. Proposed equations for determining Js
3.4. Results and discussion
3.4.1. Determining Js when  is larger than 90°
3.4.2. Determining Js when  is less than 90°
3.4.3. Steps for determining the value of Js for non-orthogonal fracture systems
3.5. Impact of α angle
3.6. Conclusion
3.7. References
CHAPTER 4 – A METHOD TO DETERMINE THE RELEVANT GEOMECHANICAL PARAMETERS FOR EVALUATING THE HYDRAULIC ERODIBILITY OF ROCK 
4.1. Introduction
4.2. Description of the developed method
4.2.1 Step 1 – Establishing a dataset and an erosion-level scale
4.2.2 Step 2 – Selection of a geomechanical parameter
4.2.3 Step 3 – Classification of the selected geomechanical parameter
4.2.3.1 Classification of the UCS of rock
4.2.3.2 Classification of rock block size
4.2.3.3 Classification of joint shear strength
4.2.3.4 Classification of a block’s shape and orientation parameters
4.2.3.6 Classification of joint openings
4.2.3.7 Classification of NPES
4.2.4 Step 4 – Determining mean levels of erosion for given Pa categories
4.2.5 Step 5 – Evaluating all geomechanical parameter classes
4.2.6 Step 6 – Analysis of sensitivity curves to erodibility
4.2.7 Steps 7 and 8 – Analyze of all geomechanical parameters and the selection of the
relevant geomechanical parameters
4.3 Results and discussion
4.3.1 Effect of the UCS of rock on erodibility
4.3.2 Effect of rock block size on erodibility
4.3.3 Effect of joint shear strength on erodibility
4.3.4 Effect of a block’s shape and orientation on erodibility
4.3.5 Effect of joint opening on erodibility
4.3.6 Effect of NPES on erodibility
4.4 Validation of developed methodology
4.5 Conclusion
4.6 References
CHAPTER 5 – DEVELOPMENT OF A METHODOLOGY FOR DETERMINING THE RELATIVE IMPORTANCE OF ROCK MASS PARAMETERS THAT CONTROL THE HYDRAULIC ERODIBILITY OF ROCK 
5.1. Introduction
5.2. Background of the comparative methods
5.2.1 Background of comparative methods based on Kirsten’s index
5.2.2 Background of the Pells’s methods
5.3. Analysis of comparative methods
5.3.1 Comparing all methods
5.3.2 Comparing the Van Schalkwyk and Pells methods
5.3.3 Comparing the Pells methods
5.3.3.1 Comparisons based on erosion classes
5.3.3.2 Comparisons based on hydraulic stream power class
5.4. Description of the method
5.4.1 Step 1 – Selecting a rock mass parameter
5.4.2 Step 2 – Selecting a hydraulic stream power
5.4.3 Steps 3 and 4 – Determining erosion level based on the selected Pa
5.4.4 Steps 5 and 6 – Determining the slope of the hydraulic sensitivity curves to erodibility for the selected rock mass parameters
5.5 Results and discussion
5.5.1 Determining the hydraulic sensitivity curves to erodibility
5.5.2 Determining the relative importance of the selected parameters
5.5.3 Comparison with field observations
5.6 Conclusion
5.7 References
CHAPTER 6 – CONCLUSIONS 
6.1 Determining the relative block structure rating for evaluating rock erodibility in the case of non-orthogonal joint sets
6.2 A method for determining the relevant geomechanical parameters when evaluatingthe hydraulic erodibility of rock
6.3 A method to determine the relative importance of rock mass parameters that control the hydraulic erodibility of rock
6.4 Perspectives for future research
APPENDICES

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