An earth pressure coefficient based on the geomechanical and geometric parameters of backfill in a mine stope
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Abstract
The backfilling of underground stopes in mines has become common practice in the mining industry as it increases the stability of mine excavations and reduces environmental problems. The backfilling process involves several technical aspects that must be assessed to ensure that the backfilling objectives are achieved. The horizontal stress exerted by backfill on excavation walls must be estimated accurately to avoid the failure of mine pillars. The analytical Marston method is often used for evaluating this stress as it takes into consideration an earth pressure coefficient (K) that is dependent on the geotechnical properties of the backfill. This paper demonstrates that the accuracy of the Marston method for predicting horizontal earth pressure would be improved by using K values that are determined as a function of backfill geotechnical parameters and also excavation geometry. Both components vary inevitably from one backfilled site to another. A methodology is developed to adjust the K value to be a function of these parameters for its application within the Marston analytical method thereby providing a more accurate prediction of earth pressure. Based on various geometric and geomechanical parameters, two graphs are developed to evaluate proper earth pressure values. These graphs can be used for estimating more accurate K values that are based on the characteristics of the stope and the backfill material.
Keywords
Backfill Underground stopes Earth pressure Analytical method Geomechanical parametersNotations
 \( \upphi^{\prime } \)
friction angle of the backfill (degrees)
 \( \upphi^{\prime } \)_{res}
residual friction angle of the backfill (degrees)
 B
stope width (m)
 H
height of the backfill (m)
 H/B
height/width ratio of the backfill
 c
cohesion of the backfill (N/m^{2})
 c_{res}
residual cohesion of the backfill (N/m^{2})
 K
earth pressure coefficient
 K_{a}
active earth pressure coefficient (thrust)
 K_{p}
passive earth pressure coefficient
 K_{0}
atrest earth pressure coefficient
 V
vertical force at depth Z (N)
 W
backfill weight (N)
 m, s
constants in the Hoek and Brown failure criterion for the rock mass
 C_{0}
uniaxial compressive strength in the intact rock
 Z
depth below the top of the backfill (m)
 Z/H
normalized height at depth Z
 γ
unit weight of the backfill (MN/m^{3})
 δ
friction angle along the interface between the backfill and stope wall
 E
Young’s modulus of the rock or backfill (MPa)
 ν
Poisson’s ratio of the rock mass or backfill
 σ_{hz} = sigma xx
Horizontal stress exerted by the backfill on the wall at depth Z (Pa)
 σ_{vz} = sigma yy
Vertical stress at depth Z (Pa)
 τ
shear stress along the stope wall at depth Z (Pa)
 σ_{H}
in situ horizontal stress in the rock mass
 σ_{V}
in situ vertical stress in the rock mass
Introduction
Mine backfilling is a technique that has been used for decades in Canada and across the globe [1, 2]. This method has several advantages such as stabilizing the drifts and stopes of a mine and increasing worker safety. Backfilling serves to fill excavations in a mine and allows ore to be extracted more completely because the underground excavations have better support. Environmental benefits result from the use of tailings as backfill material: problems related to the disposal of surface tailings can be reduced as substantial quantities of potentially polluting tailings can be returned underground. This backfill material generally consists of a mixture of tailings, water and cement.
The backfilling process has several technical aspects that interact with one another. This includes the hydrogeology, mineralogy and geochemistry of both the host rock and backfill material as well as the mechanical behavior of these two media. Mechanical interactions occur at the interface between the backfill and the host rock mass. The mechanical properties of these two media are very different from each other. For example, the backfill is of lower density and rigidity than the host rock. These contrasting properties must be accounted for in a thorough analysis of the mechanical interactions between the two media. The freshly placed backfill tends to compress under its own weight and generates shear stress at the backfill–rock interface due to friction. A certain amount of weight is transferred laterally from the backfill to the rock mass. This transfer is associated with arching, which occurs mostly in narrow stopes and where a lower density material is placed between denser materials [3, 4]. The stresses exerted on the mine pillars by the backfill must be determined to avoid failure of the pillars. This requires accurate estimates of the geomechanical parameters of both the backfill and the rock mass.
The arching effect has been demonstrated by several authors through field measurements or analytical and numerical models. Important studies on this topic include that of Li and Aubertin [3], which presents the results of largescale numerical investigations focused on stope geometry and backfill properties. They conclude that the greatest effect on stress distribution within backfill results from the cohesion (c) and the friction angle (\( \upphi^{\prime } \)) of the backfill. However, the values of the earth pressure coefficient (K) used in these studies were obtained from the geotechnical engineering literature, particularly from analyses involving retaining walls, that applied values such as the Rankin, Coulomb and atrest earth pressure coefficients [5]. These analytical methods were based on simplifying assumptions that excluded the possibly important variations in K resulting from changes in the geomechanical parameters (e.g., E and c) and excavation geometry, thereby affecting the accuracy of the stress distribution models.
Ting et al. [6] studied the stress distribution along an inclined stope focusing on several geomechanical and geometrical parameters. They demonstrated that the inclination of a stope is another critical factor for predicting stress distribution in backfill. However, they did not consider that K in backfill will vary inevitably with variations in stope geometry.
It is necessary to develop more accurate values of K that account for the variability of geometrical and geomechanical parameters. A method is proposed for modifying the value of K depending on the geometry of the excavation and the geomechanical parameters of the backfill. Graphs are developed to estimate K for different values of these parameters. This method is validated by comparing the resulting K values with values obtained using existing models and analytical approaches.
Analytical methods
Despite their limiting assumptions, analytical methods for estimating values of K, such as the method of Marston [7, 8] and overburden solutions [5], are useful to mining engineers as they can be used to obtain a preliminary K value. The Marston theory [7] is particularly suitable for sites where arching occurs. This theory accounts for both the weight of the fill and the shear forces between a vertical wall and the backfill at a given depth.

The backfill is bounded by two parallel vertical walls;

A twodimensional plain strain condition is present;

The vertical stress is uniformly distributed along a horizontal plane at all depths;

The adhesion between the rock face and the backfill is equal to the cohesion of the backfill.
The parameters σ_{vz} and σ_{hz} are, respectively, the vertical and horizontal stress at depth z, and δ is the interface friction angle, which is approximately 2/3 of \( \upphi^{\prime } \). Pirapakaran and Sivakugan [11] demonstrated that for a backfill cohesion of zero, t friction angle δ at the interface is approximately 2/3 of \( \upphi^{\prime } \) when K = K _{0}. However, depending on the blastingrelated roughness and irregularities in the stope walls, this parameter could be considered as equivalent to the friction angle of backfill, i.e. \( \upphi^{\prime } \). Under these conditions, the estimated vertical stress obtained from the analytical model approximates the true value. The value of K depends on the material properties and the horizontal displacement of the wall.
 1.
When there is no relative displacement of the walls, the backfill isaid to be at rest (neutral), and K is equal to the atrest earth pressure (K _{0}) given by Jaky [5]:
 2.
When considering an active earth pressure coefficient (K _{a}),
 3.
When considering a passive earth pressure coefficient (K _{p}),
Equation 2, developed by Marston [7], was used by Li et al. [10]. However, these two studies each proposed different definitions of the earth pressure coefficient (K): Marston [7] defined K as the active earth pressure developed by Rankine (K _{a} = 0.33), whereas Li et al. [10] concluded that K _{0} (approximately 0.5) was the most suitable value for vertical walls after they compared the atrest (K _{0}), active (K _{a}) and passive (K _{p}) earth pressures.
Numerical modeling
Figure 5 shows the results of the analytical model for K values of 0.5 and 0.33, based on the findings of Marston [7] and Li and Aubertin [10]. for a narrow vertical stope having a backfill friction angle of 30°. The curve of the numerical model is below the curve of the analytical overburden model and nearly coincides with the curve of the Marston model that has a K value of 0.3. The question that arises is how to determine the value of K that is best suited for adjusting the analytical model to the numerical results for different values of the geomechanical and geometric parameters. The value of K is dependent on both the geomechanical parameters of the backfill and on the geometry of the stope. A suitable K value therefore varies with fluctuations in these parameters. An adjustment is necessary based on the geomechanical and geometric parameters of the backfill. Figure 5 shows that a K value of 0.33 yields a very good fit with the numerical model for a stope measuring 6 × 45 m.
Determination of the earth pressure coefficient based on geomechanical parameters
Several geomechanical parameters affect the distribution of vertical and horizontal stresses in the backfill, including friction angle (\( \upphi^{\prime } \)), cohesion (c), Young’s modulus (E), unit weight (γ) and Poisson’s ratio (ν). The relative importance of the effects of each parameter on K was determined from a sensitivity analysis in which several values were assigned to each of these parameters while keeping the other parameters constant.
Sensitivity analysis
The effect of Young’s modulus on horizontal stress in the backfill appears to be insignificant (Fig. 6). Variations in cohesion, however, produce a range of horizontal stress values of approximately 0.04 MPa. These values appear to be constant below a depth of 40 m (Fig. 7). When cohesion exceeds 0.1 MPa, the results remain similar and the curves are nearly superposed. The friction angle exerts by far the greatest effect on horizontal stress (Fig. 8), as it produces horizontal stress values having a range of 0.15 MPa. The estimated horizontal stress increases gradually with decreasing friction angles (\( \upphi^{\prime } \)), and arching occurs for every value of \( \upphi^{\prime } \). This arching is evident in the stabilization of the curves at the bottom of the stope. Figure 6 indicates that the variation of Young’s modulus in the backfill has negligible effects on the distribution of horizontal stress when compared with the variations in the friction angle. Furthermore, as shown in Fig. 7, a backfill cohesion value of more than 50 kPa above the variation of vertical stress does not have a substantial effect on vertical stress. Friction angle has a much greater effect on the estimated horizontal stress than does the Young’s modulus and cohesion. Therefore, only the friction angle was subsequently used to obtain a corrected value for the earth pressure coefficient.
Analytical and numerical adjustment
The analytical model of Marston [7] can be adjusted based on the results of numerical modeling by varying the value of K in the Marston equation (Eq. 2) so that the analytical and the numerical curves overlap. By gradually changing the value of K, it is possible to obtain nearly perfectly matching curves. This yields the appropriate K value for use with the corresponding values of these geomechanical parameters.
Determination of the earth pressure coefficient based on stope geometry
As discussed above, mass transfer due to shearing and arching is possible only in narrow stopes. It is therefore important to determine the variation of K associated with different stope geometries, including the geometric threshold at which arching—strong or weak—occurs gradually in the backfill. A low H/B ratio likely produces very little arching because very little friction is generated at the backfill–rock interface. In addition, a value of K can be estimated for each H/B value. This permits the determination of several bestfit K values and thus yields more accurate results for determining horizontal stress (σ_{hz}).
It is clear from Fig. 12 that altering the geometry changes sigma xx substantially. Therefore, a single K value cannot be applied. An appropriate K value must be obtained for a given H/B ratio and for given values of geomechanical parameters, such as the friction angle (\( \upphi^{\prime } \)).
A graphic method for application to the Marston solution
Conclusions
This study suggests that the determination of the earth pressure coefficient (K) must account for the stope geometry and geomechanical parameters of the backfill as these parameters can vary markedly even within a given mine. In general, most authors do not adjust the value of K based on their model parameters and instead apply a constant K value.
A sensitivity analysis of the geomechanical parameters demonstrated the relative effects of cohesion, Young’s modulus and friction angle of the backfill on the estimates of horizontal stress; the friction angle (\( \upphi^{\prime } \)) exerts the greatest effect.
Two charts were developed and provide improved estimates of the earth pressure coefficient (K). They are based on the Marston solution and account for both the stope geometry (the H/B ratio) and the geomechanical parameters (the friction angle \( \upphi^{\prime } \)) of the backfill. This method provides more accurate estimates of the earth pressure coefficient and reflects the often heterogeneous characteristics of backfilled stopes.
Notes
Authors’ contributions
AS determined the idea of the paper determined by. YL and AS carried out the numerical modelling. YL, AS and AR wrote the paper. All work supervised by AS and AR. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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