Open-pit slope design using a DtN-FEM: Parameter space exploration

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Abstract

Given the sustained mineral-deposits ore-grade decrease, it becomes necessary to reach greater depths when extracting ore by open-pit mining. Steeper slope angles are thus likely to be required, leading to geomechanical instabilities. In order to determine excavation stability, mathematical modelling and numerical simulation are often used to compute the rock-mass stress-state, to which some stability criterion needs to be added. A problem with this approach is that the volume surrounding the excavation has no clear borders and in practice it might be regarded as an unbounded region. Then, it is necessary to use advanced methods capable of dealing efficiently with this difficulty. In this work, a DtN-FEM procedure is applied to calculate displacements and stresses in open-pit slopes under geostatic stress conditions. This procedure was previously devised by the authors to numerically treat this kind of problems where the surrounding domain is semi-infinite. Its efficiency makes possible to simulate, in a short amount of time, multiple open-pit slope configurations. Therefore, the method potentiality for open-pit slope design is investigated. A regular open-pit slope geometry is assumed, parameterised by the overall-slope and bench-face angles. Multiple geometrically admissible slopes are explored and their stability is assessed by using the computed stress-field and the Mohr–Coulomb failure criterion. Regions of stability and instability are thus explored in the parametric space, opening the way for a new and flexible designing tool for open-pit slopes and related problems.

Introduction

Due to the sustained ore grade decrease in mineral deposits, mine depth is increasing, leading to steeper slopes. A likely consequence of this is geomechanical instability, with adverse effects in terms of lower mineral resources and reserves, loss of equipment, time and risks to workers. A possible control measure for the effects of instability is a precise knowledge of the rock-mass stress-state surrounding the mine. The use of computer simulation by means of numerical methods can significantly help to achieve this purpose.

In general, when applying numerical methods to calculate stresses, a computational domain needs to be established, which corresponds to the spatial region where the computations will be done. However, in the case of a mine (or any excavation) it is not clear a priori how large that computational domain should be. In practice, all the surroundings can be regarded as unbounded, but computers cannot store infinite domains. A popular and heuristic approach to overcome this difficulty is to employ a huge domain, typically a rectangular box, with its external boundaries far away enough from the mine so that they have minimal effect on the results to be computed. The discretisation of such a domain requires a large amount of points, making the numerical method performance inefficient and even inaccurate, mainly due to abuses in boundary conditions. A single simulation of a detailed 3D mine can take up days to yield meaningful results.

A comprehensive survey of numerical approaches to solve problems formulated in unbounded domains is provided in Ref. 1. This kind of methods are classified into four main categories: boundary integral equation methods, absorbing layer methods, infinite element methods, and artificial boundary condition (ABC) methods. All these kinds of methods are well-suited to treat infinite exterior domains, that is, the whole space minus some bounded region, which are relatively simple as they are unbounded in all directions and its boundary is finite. However, in geomechanical applications such as the stress-state in an excavation, the domain is normally assumed to be a half-space minus a bounded region, that is, a semi-infinite domain. This type of domain has some additional difficulties since it is bounded only in some directions and it has an infinite boundary, where some boundary condition is prescribed. The numerical method to be employed should be able to deal with these difficulties, so it may require some adaptations.

Boundary integral equations and the boundary element method (BEM) have been successfully applied to infinite exterior domains and there is a vast literature on the subject. These kind of methods have the advantage of reducing the dimensionality of the problem by one. However, to be applied to semi-infinite domains in a computationally efficient way (avoiding discretisation of an infinite boundary), a half-space fundamental solution is required, which may not be explicitly available in some cases. BEM solutions for elasticity in a half-plane can be found in Refs. 2, 3, where a half-plane fundamental solution is used. In a more recent work Ref. 4, a BEM formulation for the elastic half-space in the axisymmetric case is presented, which uses an axisymmetric fundamental solution for the elastic half-space, given in terms of integrals of the Lipschitz–Hankel type.

Infinite element methods have also been widely used to treat infinite and semi-infinite domains. The advantage of these methods is their simplicity, as the concept of infinite element is the same as that of finite element, except for the infinity of the element domain. However, some issues need to be resolved, such as the right choice of shape functions in order to reflect the asymptotic behaviour of the solution at infinity, as well as the numerical calculation of some integrals over infinite domains. Improved infinite elements based on mapping is proposed in Ref. 5, which are tested on the axisymmetric problem of a point load on an elastic half-space. In a recent work Ref. 6, introduces an approach that combines some features of infinite elements with the numerical manifold method, where the usual shape functions are substituted by infinite patches with the weight functions satisfying the partition of unity. This technique is tested in half-space elasticity problems.

A powerful numerical technique, belonging to the ABC category, is the DtN-FEM (Dirichlet-to-Neumann finite element method), where finite elements are used in combination with the so-called DtN map, defined over an artificial boundary.[7], [8], [9] The main advantage of this approach is that the DtN map provides exact boundary conditions, ensuring continuity of the solution and its derivatives across the artificial boundary, which results in a method with high accuracy. In general, it is possible to apply this type of procedure on the condition that an explicit, analytical closed-form expression for the DtN map exists. Such is the case for most infinite exterior problems.[10], [11], [12], [13] However, applying the DtN-FEM to a semi-infinite domain could be tricky, mainly due to the lack of closed-form expressions for the DtN map in most cases of interest, therefore some approximation of it becomes necessary.[14], [15], [16]

In a relatively recent work,17 the authors presented a DtN-FEM approach for axisymmetric problems in an elastic half-space, based on a suitable approximation of the associated DtN map obtained through a semi-analytical procedure, also developed by the authors in a previous paper.18 The coupling between the DtN map, expressed in series form, and the FEM scheme is done directly on the discretised variational formulation of the boundary-value problem, specifically on the boundary integral terms on a semi-spherical artificial boundary. In order to validate the method, a model problem was used, consisting in a simplified case with exact analytical solution. The relative error between numerical and exact solution was studied in terms of artificial boundary location, number of terms in the DtN map series and mesh size. The method exhibited an excellent performance in terms of flexibility, speed, precision and robustness. The achieved accuracy was satisfactory even for close artificial boundaries, small number of terms in the series and coarse meshes.

This paper aims at exploiting the advantages of this DtN-FEM by solving a problem in open-pit slope design which requires to solve several cases. A homogeneous, isotropic and fully fractured rock-mass is mined with an open-pit having regular axisymmetric slope, and fixed number of benches and overall slope height as well, whereas overall-slope angle and bench-face angle are variable parameters. Regions of geomechanical stability/instability are thus determined in the parametric space by exploring Mohr–Coulomb failure criterion given the geometry-induced stress-field.

Section snippets

Preliminaries

In what follows, an overview of the DtN-FEM approach is given. Full details are found in Ref. 17. Let us consider the lower half-space, described in cylindrical coordinates (ρ,θ,z) as the region in R3 where ρ0 and z<0, or alternatively in spherical coordinates (r,θ,ϕ) as the region in R3 where r>0 and π/2<ϕ<π. Notice that this semi-infinite region is axisymmetric with respect to the z-axis, and in particular it does not depend on the azimuthal angle θ. We assume that an axisymmetric open-pit

Open-pit stability under the Mohr–Coulomb failure criterion

In order to characterise the stability of a particular open-pit geometry, we employ the Mohr–Coulomb failure criterion.21 Given a stress tensor σ defined in every point of Ω, we denote the associated principal stresses as σ1σ2σ3. The Mohr–Coulomb failure criterion is a set of linear equations relating shear stress τ as a function of normal stresses σ, therefore it might describe conditions where isotropic materials are prone to fail given that most of them have a limited range of shear

Parameterised open-pit geometry

In what follows, a regular parameterised geometry of open-pit is assumed, which is schematically described in Fig. 4. The main geometrical parameters are the overall slope height H, the number of benches n, the bench length a, the berm width b, the overall slope angle β and the bench face angle α (cf.  Fig. 4). Notice that the bench height is given by H/n. Additionally, we consider the horizontal distances from the z-axis to the first bench d, and to the last bench L, as indicated in Fig. 4.

The

Discussion and conclusions

A DtN-FEM approach for semi-infinite elastic domains has been applied to study open-pit stability in the axisymmetric case. The method is highly accurate, flexible and efficient, which allowed us to solve multiple open-pit slope configurations in a short amount of time. The open-pit stability was assessed by computing an indicator based upon the Mohr–Coulomb failure criterion, which is positive when the open-pit slope configuration is stable and negative otherwise. Other stability measures

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The first author acknowledges the support of the Departamento de Ingeniería Matemática, Universidad de Concepción. The second author thanks INGMAT R&D Centre for allowing this research to be carried out.

References (27)

  • ZienkiewiczO. et al.

    A novel boundary infinite element

    Internat J Numer Methods Engrg

    (1983)
  • GivoliD.

    Recent advances in the DtN FE method

    Arch. Comput. Method E.

    (1999)
  • GivoliD.

    Exact representations on artificial interfaces and applications in mechanics

    Appl. Mech. Rev.

    (1999)
  • Cited by (3)

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