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Naifei Liu, Liang Cui, Yan Wang, "Analytical Assessment of Internal Stress in Cemented Paste Backfill", Advances in Materials Science and Engineering, vol. 2020, Article ID 6666548, 13 pages, 2020. https://doi.org/10.1155/2020/6666548
Analytical Assessment of Internal Stress in Cemented Paste Backfill
Abstract
To analytically describe the internal stress in a fill mass made of granular manmade material (cemented paste backfill, CPB), a new 3D effective stress model is developed. The developed model integrates Bishop effective stress principle, water retention relationship, and arching effect. All model parameters are determined from measurable experimental data. The uncertainties of the model parameters are examined by sensitivity analysis. A series of model application is conducted to investigate the effects of field conditions on the internal stress in CPB. The obtained results show that the proposed model is able to capture the influence of operation time, stope geometry, and rock/CPB interface properties on the effective stress in CPB. Hence, the developed model can be used as a useful tool for the optimal design of CPB structure.
1. Introduction
As a relatively new backfilling method, cemented paste backfill (CPB, a mixture of dewatered tailings, hydraulic binder, and water) technology has gradually become a standard practice around the world [1, 2]. Due to the rapid strength acquisition [3], CPB can provide reliable ground support to the adjacent stopes and thus enhance mining cycles and productivity. In addition, since the tailings are used as the major ingredient for the CPB preparation and then pumped into minedout underground space (called stope), CPB technology can be considered as an effective and attractive alternative to the conventional tailings disposal on the surface (i.e., tailings impoundments or dams) [4] and thus enhances the associated mechanical (strength and stability) [5] and microstructural (total porosity and pore size distribution) properties [6] and minimizes the associated environmental issues (e.g., the acid mine drainage, acid/sulphate attack, heavy metal concentration, and so on) [7–9]. Consequently, the CPB technology has been widely adopted in underground mining and started to replace the conventional backfilling methods (i.e., hydraulic fill and cemented rockfill) [10, 11].
However, the binder costs may reach up to 80% of the total backfill operating costs [12,13]. Therefore, optimal design of CPB becomes one of the major tasks for backfill engineers and researchers. The prerequisite for the optimal CPB design is to reliably assess the internal stress in CPB. Previous laboratory and field studies [14] have found that the rock/CPB interface resistance (i.e., the interface friction stress and adhesion) can dramatically reduce vertical stress acting on the stope floor (i.e., the arching effect). Moreover, the water drainage through the barricade (i.e., a retaining structure for fresh CPB constructed near the stope entrance) can cause the dissipation of porewater pressure (PWP) [15] and thus affect the effective stress in CPB mass. Therefore, to accurately and reliably assess and predict the internal stress in CPB, the hydraulic and mechanical processes under the influence of rock/CPB interface interaction must be fully considered.
To study the variation of internal stress and the associated influencing factors, extensive laboratory experiments and insitu monitoring programs have been conducted. For example, previous laboratory studies [16–20] have found that curing stress, water drainage, and structural filling properties (e.g., filling rate and sequences, filling time, filling interval time, and filling surface angle) affect the evolution of material properties (e.g., coefficient of permeability, shear strength parameters, elastic modulus, and so on) and the internal stress in CPB. Moreover, field monitoring programs have been carried out to assess the evolution of PWP [21] and total stress [22] in stopes with various mixture recipes and field conditions. The field measurements provide direct evidence for the arching effect in stopes. In addition, since the analytical models can provide a simple closedform solution to assess the total stress in CPB, the analytical method has been extensively used in the optimal design of CPB structure. Correspondingly, a series of analytical methods has been proposed through limit equilibrium analysis [23, 24]. However, the skeletal deformation and its strength development are controlled by the effective stress rather than the total stress [25]. Therefore, it is necessary to investigate the changes of effective stress in CPB structure. However, the available analytical methods focus on the evaluation of total stress rather than the effective stress in CPB. Therefore, This study aims to (1) develop a 3D effective stress analytical model to evaluate the internal stress in CPB, which fully considers the effects of hydraulic and mechanical processes in CPB under the influence of rock/CPB interaction and (2) investigate the changes of internal stress of CPB mass with various filling operation time, stope geometry, and rock/CPB interface properties.
2. Formulation of a 3D Effective Stress Analytical Model
To develop a 3D effective stress analytical model for the characterization of internal stress in CPB, the prerequisite is to determine the mechanisms responsible for the stress change. Then, the adopted assumptions for the derivation of the effective stress model should be identified. Afterwards, the limit equilibrium analysis can be carried out to derive the mathematical model. A schematic diagram (see Figure 1) of insitu CPB mass in underground stope is adopted to elucidate the mechanisms of stress changes in CPB.
As shown in Figure 1, the selfweight load increases with the backfilling operation after placement into stope. Meanwhile, the PWP progressively builds up with the filling process. However, the dissipation of PWP induced by barricade drainage accompanies the filling process. Correspondingly, negative PWP appears in the top area of CPB mass and changes the magnitude and spatial distribution of effective stress in CPB. In addition, the rock/CPB interface interaction can weaken the influence of selfweight stress on the vertical stress in fill mass [26]. Consequently, the resultant vertical stress can be apparently lower than the selfweight stress for a given elevation in CPB. As aforementioned, the backfilling operation, rock/CPB interface interaction, and barricade drainage control the changes of effective stress in fill mass. Considering the realistic underground mining conditions where variable rocks may exist as neighbour rock, the hard rock (igneous or metamorphic rock) is considered in this study.
2.1. Definition of Effective Stress in CPB
Due to the barricade drainage, PWP dissipation takes place in CPB and thus causes the change of the saturation state. As a result, an unsaturated zone gradually appears in the top regime of fill mass and is separated from the saturated zone by the local water table. The variation of PWP affects the magnitude and spatial distribution of effective stress in CPB. To analytically describe the change of effective stress, Bishop effective stress σ’ [27] is adopted:where σ, , and P_{a}, respectively, represent total stress, PWP, and pore air pressure and χ is the parameter of Bishop effective stress (χ = 1 for saturated CPB and χ = 0 for dry CPB). It has been found that Bishop’s parameter χ is strongly dependent on the degree of saturation S (i.e., χ = χ (S)) [28]. The latter can be determined by matric suction ( − P_{a}) via the water retention curve. Therefore, Bishop’s parameter χ can be related to matric suction as well. In this study, the empirical model proposed by Khalili and Khabbaz [29] is adopted to characterize the relationship between χ and matric suction ( − P_{a}):where P_{e} represents the air entry value and R is a fitting parameter. Based on the experimental studies on 14 soils including glacial till, silts, sandy clay, and clays, R = −0.55 is obtained [29]. In terms of particle size, the tailings can range from fine sand down to claysized particles, which is very close to the investigated soils in the study conducted by Khalili and Khabbaz [29]. Therefore, R = −0.55 is adopted in the present study. Moreover, based on the previous study on the water retention curves (WRCs) of CPB [30], the air entry value P_{e} = −200 kPa is used in this study. In addition, it should be noted that equation (2) is only valid when the matric suction is greater than the air entry value; otherwise, the value of χ is equal to 1 [31], namely, the CPB remains fully saturated.
The pore air pressure is assumed to be zerogauge pressure against atmospheric pressure (i.e., P_{a} = 0). Therefore, by substituting equation (2) and P_{a} = 0 into equation (1), the effective stress can be rewritten as follows:
To incorporate the effect of hydraulic process into the effective stress, the PWP in CPB (including both saturated and unsaturated zones) should be determined. For this purpose, the location (i.e., H_{wt}) of local water table should be identified first. Then, the positive PWP in the saturated zone can be considered as hydrostatic pressure, and the negative PWP above the local water table can be determined by the spatial distribution of saturation degree and water retention relationship (i.e., the WRC model).
It should be noted that the location of local water table changes with time t due to the water drainage through barricade. Hence, the thickness of the unsaturated zone H_{wt} can be expressed as follows:where is the movement velocity of local water table, which is related to the outflow rate of pore water through barricade. In this study, a linear relationship between and is assumed as follows:where L_{A} and L_{B}, respectively, represent the length and width of the stope, la and lb refer to the barricade dimensions, and is a constant scaling parameter which is related to ratio of coefficient of permeability of CPB and barricade ( = K_{CPB0}/K_{B} with K_{CPB0} and K_{B} as the initial coefficient of permeability of CPB and barricade, respectively). The detailed information about the determination of K_{CPB0} and K_{B} will be presented in the Section of Determination of Model Parameters. The average porewater flow velocity through barricade can be calculated by Darcy’s law:where K_{b} denotes the coefficient of permeability of barricade, W_{b} is the thickness of barricade, and H refers to the backfilling height and can be determined by filling rate and filling strategy:where H_{ri} is the total filling height after ith filling sequence, t_{ri} is the rest time between ith filling and (i+1)th filling sequences, and n is the total filling sequences adopted by the backfilling operation. As indicated in equation (7), the filling rate and filling sequences are fully considered in the present study. Substituting equations (5) and (6) into equation (4) and applying the initial condition (H_{wt} = 0 when t = 0) yield the following:
Then, the effective saturation degree S_{e} (S_{e} = (θ − θ_{r})/(θ_{s} − θ_{r}) with θ_{s} as saturated water content, θ as current water content, and θ_{r} as residual water content) is assumed to be linearly distributed in the unsaturated zone (i.e., S_{e} = h/H_{wt} with h as the thickness of CPB relative to the top surface). Hence, the negative PWP in the unsaturated zone can be determined by the water retention curve (WRC). Previous studies [30, 32] on CPB have proved that the van Genuchten model [33] is able to accurately predict the water retention capacity of CPB materials.where α and m denote the parameters of the van Genuchten model. The detailed information about the determination of parameters of the WRC model will be provided in the Section of Determination of Model Parameters.
Therefore, the negative PWP can be derived by substituting the effective saturation degree (i.e., S_{e} = h/H_{wt}) into the WRC model (i.e., equation (9)):
Hence, the spatial distribution of effective stress in CPB mass can be expressed as follows:withwhere refers to the unit weight of pore water.
As previously discussed, the arching effect causes the discrepancy between vertical stress and the selfweight stress. Therefore, the total stress in equation (3) cannot be represented by the overburden stress. The detailed discussion about the determination of total stress in CPB is presented in the following section.
2.2. Total Stress in CPB under Arching Effect
To evaluate the total stress in CPB under arching effect, the force analysis of CPB structure is first conducted through limit equilibrium analysis. Then, the limit equilibrium analysis is used to derive the analytical model for the total stress in CPB. To perform the force analysis in CPB mass, a representative thinlayer CPB is selected from the fill mass. The relevant forces acting on the thin layer are plotted in Figure 2. It should be noted, since the constant material properties are assumed in this study, the spatial variation of material properties due to the different filling and curing conditions is neglected. Consequently, the selection of representative thinlayer CPB is dependent only on the position of local water table. Therefore, one representative layer is selected above water table to determine the total stress in the partially saturated zone, while the other representative layer is selected below the local water to derive the total stress in the fully saturated zone.
As shown in Figure 2, the forces acting on the representative thin layer can be classified into two categories including body force and surface forces. Specifically, the former refers to the selfweight load (i.e., G_{CPB}) due to the gravitational effect, and the latter consists of interface resistance force (i.e., interface shear force F_{s} and interface adhesion force F_{c}) and the vertical forces acting on the top (i.e., ) and bottom (i.e., + ) surfaces of the CPB layer. In addition, the interface shear force (i.e., F_{s}) should be considered as a type of reaction force induced by the horizontal force (i.e., F_{h}) on the rock/CPB interface. Since the representative CPB layer is in equilibrium under the action of body force and surface forces, the equilibrium analysis approach can be utilized to derive the total vertical stress. Then, the horizontal stress can be calculated through reaction coefficient.
2.3. Total Stress in Unsaturated Zone
Based on the force analysis on the thin layer of CPB, the body force G_{CPB} in the unsaturated zone (i.e., h < H_{wt}) can be expressed as follows:where γ_{CPBu} is the unit weight of unsaturated CPB, L_{A} and L_{B}, respectively, denote the length and width of the stope, and dh is the thickness of the representative thin layer.
In addition, the representative element is also subjected to the surface forces including the vertical stress on the top and bottom surface and the interface resistance force (i.e., the interface shear force and adhesion). The former (i.e., the vertical force F_{V}) can be calculated through the vertical stress σ_{V}:
For the interface resistance force, the interface friction component (i.e., F_{As} and F_{Bs}) can be determined by the horizontal stress (i.e., σ_{hAu} and σ_{hBu}) and interface friction angle ɸ:
The horizontal stress can be calculated by the product of vertical stress and the reaction coefficient K.
Substituting equation (17) into equations (15) and (16), the interface shear force can be rewritten as follows:
The interface adhesion force (i.e., F_{Ac} and F_{Bc}) can be calculated as follows:where c represents the rock/CPB interface adhesion.
To derive the analytical solution for the total vertical stress in the unsaturated zone of CPB mass, the force equilibrium equation in the vertical direction is utilized:
Then, substituting equations (13), (14), (18)–(21) into equation (22), the total vertical stress above the local water table (h < H_{wt}) can be solved by applying the initial condition (i.e., σ_{vu} = 0 when h = 0):
2.4. Total Stress in Fully Saturated Zone
Below the water table (i.e., h ≥ H_{wt}), similar force analysis can be conducted on a representative thin layer of CPB. The body force (i.e., CPB gravity G_{CPB}) in the fully saturated zone can be expressed as follows:where γ_{CPBs} is the unit weight of CPB in the fully saturated state.
In addition, the surface forces acting on the saturated thin layer can be determined in a similar way adopted in the unsaturated CPB layer. The vertical force F_{V} can be expressed as follows:with σ_{vs} as the vertical stress acting on the saturated thinlayer surface.
The rock/CPB interface resistance force (i.e., interface friction and adhesion forces) can be defined as follows:
Below the local water table (i.e., h ≥ H_{wt}), the force equilibrium equation in the vertical direction can be obtained by substituting equations (24)–(29) into equation (22). To derive the vertical stress in the saturated zone, the boundary condition on the top surface of local water table (i.e., h = H_{wt}) is needed. The corresponding vertical stress can be expressed as follows:
Then, applying the initial condition (i.e., equation (30)), the total vertical stress in the saturated zone (i.e., h > H_{wt}) can be derived through integrating equation:
It should be noted that interface shear strength parameters (adhesion c and friction angle ɸ) adopted in equations (24) and (33) refer to the total stress shear strength parameters which can be measured through direct shear tests on rock/CPB samples. Since the Bishop effective stress (equation (11)) is adopted to define the effective stress in unsaturated (i.e., χ < 1) and fully saturated (i.e., χ = 1) zones, the effective shear strength parameters are not required in the present study.
2.5. 3D Effective Stress Model of CPB
The effective stress rather than total stress controls the mechanical behaviour of CPB. Therefore, to characterize the variation of internal effective stress, a 3D effective stress model for CPB materials is needed, which can be derived by substituting equations (23) and (31) to equation (11)) based on the definition of Bishop effective stress.with
As indicated in equation (32), there exist a number of model parameters (including reaction coefficient K, WRC parameters α and m, interface friction angle ɸ, adhesion c, and the coefficient of permeability K_{b}). The detailed discussion about the model parameters is provided in the Section of Determination of Model Parameters.
3. Determination of Model Parameters
To implement the developed 3D effective stress model (i.e., equation (32)), the associated model parameters are required. With respect to the components of the backfilling system, the associated model parameters can be divided into three groups including (1) CPB material properties, (2) rock/CPB interface properties, and (3) properties related to barricade. The relevant discussion about determination of each group of model parameters is provided in the following subsections.
3.1. Rock/CPB Interface Properties
As aforementioned, the vertical stress is mainly affected by the rock/CPB interface shear force and adhesion. Correspondingly, the rock/CPB interface properties including interface friction angle ɸ and adhesion c are required. In this regard, extensive experimental studies have been carried out [14, 34, 35]. The measured rock/CPB interface friction angle is listed in Table 1. The obtained results indicate that the interface properties show no evident variation for each group of experimental data, which proves that the assumption of constant interface properties adopted in this study is reasonable. Hence, the average value of interface friction angle and adhesion from the collected data in Table 1 is used in this study, namely, ɸ = 31.5° and c = 14.2 kPa.
3.2. CPB Material Properties
Apart from the rock/CPB interface properties, the total horizontal stress σ_{h} is needed to calculate the interface friction force. The horizontal stress can be defined by the total vertical stress and reaction coefficient K (i.e., σ_{h} = K). Therefore, the reaction coefficient is required to assess the horizontal stress. During and after backfilling operation, the rock wall movement is expected to be small. For this reason, the reaction coefficient at rest K_{r} (i.e., K_{r} = 1−sinφ with φ as the internal friction angle of CPB) is adopted in this study. Moreover, previous studies [14, 34] showed that the internal friction angle of CPB φ is almost equal to the rock/CPB interface friction angle ɸ (i.e., φ = ɸ). As discussed previously, the ɸ = 31.5° is adopted. Thus, the internal friction angle of CPB can be obtained, namely, φ = 31.5°. Compared with the measured data of φ reported in the literature (see Table 2), φ = 31.5° is reasonable for CPB materials.
To determine the water retention relationship, the WRC model parameters (i.e., α and m) are required. The parameter α is related to the inverse of air entry value P_{e} (i.e., α = −/P_{e} with as the unit weight of pore water). As aforementioned, the air entry value P_{e} = −200 kPa is adopted in this study. Hence, the parameter α = 0.049 m^{−1} can be obtained. Compared with the reported values of α (i.e., 0.002 m^{−1} to 0.065 m^{−1}) in Table 3, α = 0.049 m^{−1} is a reasonable value and employed in this study. In addition, the dimensionless parameter is related to the width of the pore size distribution of solid particles [33]. Based on previous studies on WRCs of CPB (e.g., 30, 39, and 40), the parameter m is in the range from 0.25 to 0.62 (see Table 3). Therefore, the average value m = 0.44 is used in this study.
3.3. Material Property of Barricade
The CPB barricade can be constructed by various materials (e.g., porous brick, concrete, timber frame, shotcrete, and so on). However, to avoid excessive PWP acting on the barricade structure, the permeable concrete barricade bricks are commonly adopted to construct the barricade in practice [41, 42]. The measured data of coefficient of permeability of the concrete brick (i.e., K_{b}) are normally in the range of 0.03 cm/s to 0.31 cm/s (see Table 4). Therefore, the average value of K_{b} = 0.17 cm/s is employed in the present study.
As shown in equation (5), the constant scaling parameter is required for the definition of movement velocity of local water table . In this study, the constant scaling parameter is defined in terms of the ratio of coefficient of permeability of CPB and barricade (i.e., = K_{CPB0}/K_{b}). In this study, the initial coefficient of permeability is assumed to be equal to the counterpart of tailings. The measured data of coefficient of permeability of tailings are listed in Table 5, and the average value of coefficient of permeability of fresh CPB K_{CPB0} = 3.4 × 10^{−4 }cm/s is obtained. As discussed previously, the average value of K_{b} = 0.17 cm/s is adopted in this study. Therefore, the scaling parameter = 2 × 10^{−3} is used in this study.
4. Sensitivity Analyses
The material parameters adopted in the developed model may change from mine to mine. Therefore, it is necessary to conduct the sensitivity analysis to analyze the uncertainties induced by the variation of model parameters. In this study, the effect of interface friction angle ɸ (i.e., the rock/CPB interface property) and the coefficient of permeability of barricade K_{b} (i.e., the property of barricade) were investigated. To clearly demonstrate the effect of investigated parameters on the internal stress in CPB, a control stope is selected as a reference. The dimensions of stope and barricade and filling strategies and rate adopted in the control stope are listed Table 6. The model parameters listed in Table 7 are employed for the control stope. The monitoring point is located at the stope floor.

 
Symbol “” indicates dimensionless unit. 
For the model implementation, the backfilling rate will be used to calculate the present filling height, H, based on equation (7). It should be noted that multiple filling sequences may be adopted in the stopes. Correspondingly, the rest time (t_{ri}) is considered at the end of each filling sequence, and filling height will be kept constant during each rest time t_{ri}. Hence, the piecewise function (equation (7)) can be used to capture the backfilling strategy used in practice. After the backfilling height is obtained, the evolution of local water table, H_{wt}, will be quantitatively evaluated by equation (8). As indicated in equation (8), the local water table will change its spatial position as time elapses, which is used to capture the water loss due to the water drainage. When the filling height, H, and local water table position, H_{wt}, are determined, the corresponding total stress, pwp, and effective stress can be calculated by equation (32). Therefore, the proposed model is able to analytically describe the spatiotemporal evolution of internal stresses in CPB. Then, the proposed model was implemented to perform the sensitivity analysis and model application.
4.1. Interface Friction Angle
The interface friction angle ɸ can directly contribute to the development of interface shear stress between CPB mass and surrounding rock and thus affect the internal stress in CPB. In this study, three different interface friction angles including 25°, 31.5°, and 40° were chosen. The development of internal stress in CPB with different interface friction angles is plotted in Figure 3. From this figure, it can be found that the internal stress is sensitive to the change of interface friction angle and thus to the interaction between CPB and rock walls. With the increase in interface friction angle, a decrease in effective stress was observed. This is because larger interface friction angle can cause higher interface shear stress and thus, to a larger extent, reduce the internal stress in CPB (i.e., strengthen the arching effect).
4.2. Coefficient of Permeability of Barricade
Water drainage through barricade structure causes the variation of PWP and thus affects the effective stress in CPB. Correspondingly, coefficient of permeability K_{b} plays a crucial role in the process of barricade drainage. Therefore, it is necessary to investigate the effect of K_{b} on the variation of internal stress in CPB. In this study, a range of coefficient of permeability including 0.2 cm/s, 0.17 cm/s, and 0.14 cm/s were selected. As shown in Figure 4, the effective stress demonstrates an increasing trend with the increase in K_{b}. This is because higher value of K_{b} can enhance the barricade drainage and thus reduce the PWP to a larger extent. Consequently, a higher effective stress was observed in CPB with a larger value of K_{b}.
4.3. Model Application
Due to the irregularities of ore bodies, various mining methods, and backfilling strategies, the resultant backfilling conditions may differ from mine to mine [38]. Hence, the developed model was used to address the practical problems including the effects of operation time, stope geometry, and rock/CPB interface properties on the internal stress in CPB. To clearly demonstrate the effects of factors investigated in the following subsections, a control stope is chosen as a reference. All investigations are conducted through some specific adjustments to the control stope. The model parameters listed in Table 7 are employed for the control stope. The dimensions of stope and barricade and filling strategies and rate adopted in the control stope are listed Table 8. The monitoring point is located at the stope floor.

4.4. Effect of Operation Time
Due to staged placement of CPB into stope and drainage through barricade, the internal stress (effective stress, total stress, and PWP) can demonstrate strongly timedependent characteristics [2, 47]. As indicated in equation (32), the developed model incorporates the evolution of PWP (including both positive and negative PWP) and total stress (i.e., different unit weight of CPB in saturated and unsaturated states) into the prediction of effective stress in CPB. Hence, the variation of effective stress with time can be described by the developed model. In this study, the control stope is chosen to assess the change of internal stress in CPB with time. The evolution of internal stress in CPB is plotted in Figure 5. From this figure, it can be observed that (1) the effective stress gradually increases during the filling stage and postfilling stage, which is mainly attributed to the increase in total vertical stress with the fresh CPB poured into stope and to the water drainage through barricade and (2) during the rest period (from 1 day to 2 days), the effective stress shows an increasing trend although no fresh CPB is placed into stope during this stage. This is because of the porewater loss by barricade drainage. As shown in Figure 5, the PWP decreases from 86 kPa to 64 kPa during the rest period, which contributes to the enhancement of effective stress in CPB. Therefore, the staged filling operation with a specified rest time is favorable to the improvement of stability of CPB structure. The obtained results indicate that the developed model is able to describe the change of internal stress in CPB with operation time.
Moreover, due to the water drainage through barricade, negative PWP will be generated in CPB. Therefore, it is necessary to investigate the spatial distribution of PWP in CPB with operation time. The comparison of PWP versus stope height for different operation time is plotted in Figure 6. From this figure, it can be observed that during the filling stage (from 0 to 4 days), the PWP shows an increase for a given stope elevation. As discussed previously, this is due to the continuous placement of fresh CPB into stope. Moreover, the location of water table also increases with time during the filling stage. However, the PWP shows an opposite trend during the postfilling stage. Specifically, PWP becomes more negative for a given elevation, and the water table decreases with time. Therefore, the obtained results indicate that filling operation and water drainage through barricade can significantly affect the distribution of PWP in CPB. The developed model is able to characterize the spatial evolution of PWP in CPB.
4.5. Effect of Stope Geometry
As aforementioned, the stope geometry may differ from one stope to another due to the irregularities of ore body and various stoping methods. To investigate the effect of stope geometry, three different stope heights including 15m, 30m, and 45m were selected. In this case, the values of model parameters are same to those adopted in control stope. Figure 7 presents the development of internal stress for different stope heights. The results show, as expected, that stope height (i.e., the backfilling heights) has a significant effect on the variation of internal stress in CPB. Specifically, a higher level of effective stress is obtained in the CPB with a higher stope height (see Figure 7(a)) after the filling completion. This is due to the larger total stress developed in CPB with higher filling height (see Figure 7(b)). However, the contribution of filling height to effective stress becomes progressively smaller with the increase in stope height. For example, at the end of filling operation in these three stopes, the effective stress for 30m and 45m cases, respectively, increases by 40% (163 kPa) and 52% (178 kPa) with respect to the counterpart (117 kPa) obtained in stope with a height of 15m. This is partly because the arching effect can reduce the total stress to a larger extent for CPB with higher height and partly because higher PWP is observed after filling operation for higher filling height case (see Figure 7(c)). The increased PWP can reduce the development of effective stress and thus further contribute to the nonlinear variation of effective stress in CPB. The obtained results are consistent with the previous study [38] on the variation of internal stress in CPB.
(a)
(b)
(c)
4.6. Effect of Rock/CPB Interface Adhesion
Due to the rock/CPB interface resistance, the resultant total vertical stress in CPB is less than its selfweight stress, namely, the arching effect is attributed to the interface interaction. The rock/CPB interface interaction consists of two components including interface friction stress and adhesion. From a mathematical point of view, the contribution of interface friction angle and adhesion to the arching effect is same. Hence, only the interface adhesion c is selected to investigate the effect of rock/CPB interface interaction on the internal stress in CPB. For this purpose, a range of interface adhesion values: 10 kPa, 15 kPa, and 20 kPa were chosen in this study. The comparison of internal stress in CPB with different interface adhesion is presented in Figure 8. It can be observed that lower total stress is obtained in CPB with higher interface adhesion (see Figure 8(a)), namely, the arching effect is enhanced by the increased interface adhesion. Moreover, it should be noted that as indicated in equation (32), the interface adhesion only affects total stress, which means the resultant PWP is same for these three cases. Consequently, with the change of interface adhesion, the development of effective stress is dominated by the variation of total stress. Hence, the lower total stress can reduce the level of effective stress in CPB (Figure 8(b)). The obtained results show that the change of interface adhesion can significantly affect the internal stress in CPB. Therefore, the interface interaction should be incorporated into the optimal design of CPB structure.
(a)
(b)
5. Discussion
As complex reactive porous media, the behaviour and performance of CPB are strongly affected by arching effect [48], curing conditions (e.g., temperature of surrounding rock and water drainage) [49], mix recipe (cement content, tailings types, and watertocement ratio) [50], and backfilling operation (filling rate and sequences) [51]. The focus of the present study is on the effect of rock/CPB interface behaviour on the internal stresses. The analytical model (equation (32)) was developed based on limit equilibrium analysis and several assumptions (including constant material properties and uniformly distributed stresses in the horizontal direction). Therefore, it is necessary to identify the associated limitations of the developed model for its engineering application and future study.
First, constant material properties are assumed in the present study. Therefore, this model is not applicable for the prediction of internal stresses under the effects of (1) binder hydration on the improvement of material properties such as the CPB cohesion and interface adhesion and (2) volume change on the porositydependent material properties such as hydraulic conductivity and associated development of excess PWP in CPB. Second, to perform equilibrium analysis on the representative thinlayer CPB, uniformly distributed stresses (total stress, effective stress, and PWP) are assumed in the horizontal direction. However, due to differential settlement [52] and the rock/CTB interface interaction [38] in CPB, especially for the CPB in narrow stope, the nonuniform distribution of internal stresses may develop in CPB mass. Consequently, the proposed model (equation (32)) may overestimate the arching effect in stopes. Third, it is assumed that the change of PWP is attributed to the water drainage through the barricade. However, there exist several additional contributors (e.g., water consumption by binder hydration, water evaporation through top surface of CPB, and water exchange between CPB and fractured rock walls) to the variation of PWP in the field [51]. Consequently, the obtained results from this equation (10) may underestimate the change of PWP in CTB. The abovementioned aspects will require more work related to advanced mathematical modelling (especially the multiphysics modelling) for the reactive CPB.
6. Conclusions
Based the limit equilibrium analysis, a new 3D analytical solution was developed to predict the internal stresses (total stress, effective stress, and PWP) under the influence of rock/CPB interface interaction. In this model, the changes of the saturation state in CPB due to the water drainage through barricade and rock/CPB interface behaviour were taken into account. The model parameters were determined in terms of measurable model parameters. The uncertainties induced by model parameters were assessed by the sensitivity analysis. Moreover, the developed model was applied to address the practical problems including the effects of operation time, stope geometry, and rock/CPB interface properties on the internal stress in CPB. Based on the obtained results in the present study, the following conclusions were drawn:(1)A 3D effective stress model is developed in this study to assess the evolution of internal stress in CPB. The model fully considers the influence of rock/CPB interface interaction, backfilling conditions, and barricade drainage.(2)The variation of internal stress (effective stress, total stress, and PWP) in CPB can demonstrate strongly nonlinear and timedependent characteristics.(3)The effect of stope geometry on the total stress becomes progressively weak with the increase in filling height, which can further affect the enhancement of effective stress in CPB and thus its stability.(4)The rock/CPB interaction significantly affects the arching effect and thus the variation of internal stress in CPB. The obtained results show that the total stress and effective stress are sensitive to the change of interface adhesion. With the increase in interface adhesion, the arching can develop to a larger extent.(5)The reaction coefficient can affect the interface shear stress and thus the internal stress in CPB. The decreased reaction coefficient can reduce the contribution of interface shear stress to the arching effect.
The proposed analytical model can be used as a helpful tool to assess the effects of interaction between CPB mass and surrounding rock, backfilling conditions (e.g., filling strategies and stope geometry), and drainage conditions for both filling and postfilling stages. Hence, the developed model can be employed as an effective tool for the optimal design of CPB structure.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery grant, the Shaanxi Natural Science Foundation (grant no. 2019JQ756), the Special Scientific Research Project of Shaanxi Education Department (grant no. 19JK0452), and the China Postdoctoral Science Foundation (grant no. 2019M663648).
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