1 Introduction

In the last four decades, a clear trend of open pit mines being excavated at increasing depths is apparent (Fig. 1a). As existing mines deepen owing to the increased efficiency of mining equipment and improved exploration techniques and technology, the ore bodies left to be exploited are everywhere deeper (Fig. 1b). Between 1930 and 2000, the depth of the average discovery in Australia, Canada, and the USA increased from surface outcropping to 295 m [1]. As a consequence, ensuring pitwalls as steep as possible has grown in importance since the deeper a mine, the higher the effect of pitwall steepness on the amount of rockwaste to be excavated and therefore mine profitability [2].

figure 1

a Increasing depth of open pit mines over the years, after [3]; b average depth of newly discovered ore deposits, after [1]

Also, there is an increasing trend to excavate pits of significant depth in weak rocks such as saprolites [4]. In weak rocks, the steepness of the pitwalls is lower than the one for pits excavated in competent rock to prevent slope failures. This in turn implies more waste rock is produced. Therefore, any gain in the overall slope angle (OSA) of pitwalls is more important than ever for the economic profitability of open pit mines.

Anecdotal evidence of the fact that a slope profile non-linear in cross-section, i.e., a profile whose inclination varies with depth, is better than a linear one, i.e., a planar profile, was reported as far back as 1890 [5]. In fact, Newman [5] observed that cuttings of concave shape excavated in homogeneous clay layers tended to be more stable than planar ones with the same OSA which in turn are more stable than cuttings of convex shape (Fig. 2a). Many decades later, Hoek and Bray in chapter 12 of the second edition of Rock slope engineering [6] analyze the stability of some concave circular slopes in cross-section. Assuming the slopes to be excavated in homogeneous rock and using the Mohr–Coulomb (M-C) failure criterion to characterize its strength, i.e., cohesion, c, and internal friction angle, φ, they found a higher stability number (which is a dimensionless index capturing the mechanical stability of a slope that will be introduced in Section 3) for circular profiles than for their planar counterparts, i.e., the planar slopes with the same OSA which share the same toe and crest points. They also mention that circular shapes had been considered by [7] for the design of the pitwalls of an iron pit mine in Canada. However, Hoek and Bray strongly caution against the assumption of uniform slope since it completely disregards the fact that the strength of geomaterials tends to vary with depth, being usually lower in the upper part of the excavated slope due to well-known geological processes (e.g., soil deposition, weathering). The first systematic theoretical study on the mechanical properties of concave slope profiles for geomaterials exhibiting some cohesion, applicable to all rocks and clayey soils, appeared in [8]. In that study, the superior stability of logarithmic spiral (logspiral) profiles, with the logspiral being featured by a radius of curvature increasing with the depth of excavation (Fig. 2b), was systematically proven over their planar counterparts having assumed the slopes to be uniform and with strength characterized by c and ϕ. By employing the limit analysis upper bound method, Utili and Nova [8] first systematically determined the optimal logspiral shape, i.e., the shape associated with the highest stability number for several prescribed OSAs, second compared the optimal logspiral slopes to their planar counterparts. They show that logspiral profiles exhibit higher FoS than their planar counterparts for any value of c, and φ considered with the highest gain for inclinations midway \(\varphi\) and the vertical line, i.e., for \(\mathrm{OSA}\sim \frac{\pi }{4}+\frac{\varphi }{2}\). Since then, other researchers [9,10,11] have independently investigated the stability of concave profiles excavated in a uniform cφ geomaterial, employing different methods, namely the slip line method [9], limit equilibrium methods [10], and the finite element method for the assessment of slope stability. They all reached the same conclusion concerning the superior stability of non-linear concave profiles. However, a key limitation of these studies is the assumption of a specific shape, either a circle [10] or a logspiral [8] or a curve stemming from the slip-line field theory and the associated characteristic equations [9], so that the shape claimed to be optimal is found as the shape associated to the highest stability number among curves belonging to a very restricted family. It is obvious that these profiles are instead sub-optimal and the shape of the truly optimal profile cannot be inferred from the aforementioned studies. In fact, in the case of a profile to be excavated in a uniform cφ geomaterial, the optimal shape calculated by OptimalSlope [12] turns out to be partly concave and partly convex (see Fig. 2c) so significantly different from the purely concave shapes considered in [8,9,10,11]. Another perhaps even more important limitation resides in the assumption of uniform slope present in all the aforementioned methods that prevent the application of these findings to real open pit mines which are typically featured by complex lithologies involving multiple rock formations of different mechanical strengths and various geological discontinuities.

Fig. 2
figure 2

Slopes of different shapes: a a concave, planar, and convex slope profile with the same overall slope angle; b profiles excavated into a uniform c, ϕ slope: logspiral optimal profile (gray line) redrawn after [8] and circular optimal profile (black line) redrawn after [10]. Since a circle is a particular case of a logspiral, i.e., a logspiral with a constant radius (rather than variable), an optimal logspiral profile is always more stable than an optimal circular one. c Optimal profile partly concave and partly convex obtained by OptimalSlope for a uniform c, ϕ slope

The search for the optimal shape of a slope profile is a difficult problem of topological optimization since slopes develop very significant irrecoverable (plastic) deformations which cause significant stress redistribution to occur before reaching failure, but the theory of topological optimization developed so far in engineering deals predominantly with elastic media [13]. Neglecting the onset of plastic deformations in a slope, i.e., assuming a purely elastic behavior, is not a viable option since this would severely underestimate the slope resistance to failure hence the slope factor of safety. And the application of plasticity theory to topological optimization is only in its infancy [14]. Therefore current algorithms in the literature for topological optimization are not viable to search for the optimal profile of slopes. The code OptimalSlope exploits the fact that slope failures occur either as a rotational mechanism (a planar failure being a particular type of rotational failure with an infinite radius of curvature) or mechanisms whose kinematics is dictated by the presence of discontinuities (e.g., the interface between two rock layers, a fault, joints, beddings). For a homogeneous slope in a cφ geomaterial, the limit analysis upper bound method allows finding the critical rotational mechanism simply by determining the minimum of an analytical objective function without requiring any discretization of the slope into finite elements [15]. The function is obtained by imposing the energy balance between the external work done by the mass of the candidate failure mechanism and the energy dissipated along its failure surface. The equation has been extended to find the critical mechanism for piecewise linear slope profiles in a uniform layer and then to the case of layered slopes [12]. Also, the formulation has been extended to slopes in rocks obeying the generalized Hoek–Brown (G-H-B) failure criterion [16, 17]. The minimum of the function stemming from the energy balance equation and therefore the critical mechanism is found by OptimalSlope [12]. Since the time the G-H-B criterion was introduced, a strong consensus has gathered in the rock mechanics community that it is a better criterion than M-C to describe the strength of rock masses [18, 19]. But for highly weathered rocks and residual soils M-C is better [4], so there are several cases of mines where the M-C criterion is employed to fit the rock strength data of some layers while the G-H-B criterion for some other layers. OptimalSlope can deal with these situations too, i.e., some rock layers being characterized by the M-C criterion while others by the G-H-B criterion, since the G-H-B parameters of any layer are converted by OptimalSlope into equivalent cφ parameters, employing the method of Renani and Martin [20].

In the excavation of an open pit, several rock layers of different strengths are usually encountered. OptimalSlope can find the optimal profile for any specified lithological sequence (any number of layers can be specified as input with each layer strength characterized by either M-C or G-H-B parameters) without unduly restricting the search to any predefined family of shapes. The optimal slope profile is found as the solution of a mathematical optimization problem where the OSA of the slope, i.e., the inclination from slope crest to toe, is maximized for an assigned stratigraphy, rock strength properties, and a prescribed FoS. Geometric requirements stemming from bench sizes (bench height, face inclination, and minimum berm width) are imposed as constraints binding the maximum local inclination of the sought optimal profile together with any other geometric constraint, e.g., constraints to prevent the occurrence of local failure mechanisms due to geological discontinuities such as faults and joints (see Section 3).

In the next section of the paper, the case study considered to demonstrate the financial and environmental benefits stemming from the adoption of optimal profiles for mine pitwalls is described. Then, in Section 3, the methodology employed is provided followed by results in Sections 4 and 5. In the methodology section, it will be illustrated how the proprietary code OptimalSlope [21] works and interacts with the mining software packages employed to perform strategic pit design.

2 Case Study

The block model of the copper deposit employed as case study has been provided by a mining company collaborating with the Delphos Mine Planning Laboratory at the University of Chile (Fig. 3). Due to a non-disclosure agreement (NDA), its name and location cannot be revealed. All the parameters needed to perform the pit design were taken from [22] except for the mining cost adjustment factor (MCAF), which was estimated for this study. The characteristics of the block model are as follows: cubic blocks of 10 m × 10 m × 10 m, block model dimensions of 2250 m × 2250 m × 360 m, 647,446 blocks in total. From the site lithology and geotechnical properties, two roughly uniform pit sectors were identified in [22] (see Fig. 4). In each pit sector, one representative cross-section was assumed to design the pitwall profile.

Fig. 3
figure 3

View of the east–west section of the block model (after [22])

Fig. 4
figure 4

3D view of the block model and the pit sectors: in red are the blocks in sector 1 and in green the blocks in sector 2. In the middle of the topographic surface, the boundary of the ultimate pit limit (UPL) is visible: the orange curve for the UPL obtained in the case of planar pitwalls and the blue curve for the UPL obtained in the case of optimal pitwalls

The rock geotechnical parameters are reported in Table 1. The rock strength is characterized by the G-H-B failure criterion [16]. The uniaxial compression strength (UCS), the Geological Strength Index (GSI), mi, and the disturbance factor (D) are all provided in the table. The disturbance factor is to accounts for rock weakening due to blasting and stress relaxation. We acknowledge that assuming D = 1 throughout the rock mass is overly conservative since in modern mines blasting is well controlled. Also, the depth of the open pit is such that the amount of stress relaxation is unlikely to cause significant damage. However, we opted to retain the assumption of D = 1 to be consistent with the set of input data from [22], thus accepting a significant degree of conservativism on the value of this parameter.

Table 1 Geotechnical properties after [22]

The values adopted for the economic parameters and metallurgical recovery together with the discount rate assumed were taken from [22] and are listed in Table 2. With regard to the capital costs, in [22] only the cost for the processing plant is accounted for. Here, instead, we wanted to account for all the typical costs needed in an open pit mine project: they are provided in Table 3.

Table 2 Economic parameters and metallurgical recovery taken from [22]
Table 3 Breakdown of the capital costs considered

3 Methodology

The design of pitwalls in open pit mines is an iterative process involving a multidisciplinary team of geologists, geotechnical engineers, and mining engineers [1]. The design process requires iterative steps between the teams [23]. Typically, several boreholes are drilled as part of a site investigation and laboratory tests are performed on the core samples retrieved to characterize the mechanical strength of the geomaterials encountered and the key lithological units. Then, a preliminary simplified design is performed and a pit crest contour is drawn. The pit is then split into sectors in order to design the pitwalls [23]. It is wise to divide the mine into sectors small enough so that a 2D vertical cross-section representative of the lithology of the sector can be determined for each sector. Then for each sector cross-section a pitwall profile needs to be designed as steep as possible and at the same time to satisfy a prescribed FoS against slope failure. The pitwall profiles are then prescribed as input together with the relevant economic and metallurgical data for the mine into a pit optimizer software package to calculate the optimal Ultimate Pit Limit and pushbacks. These are typically based on the Lerch-Grossman algorithm [24] or the more recent pseudo-flow [25]. Both algorithms require to identify precedences between blocks of the block model lying within the pit boundary. How block precedences are built for pitwalls with depth varying inclinations is well described in [26, 27].

The iterative procedure we followed to calculate the UPL is illustrated in Fig. 5. The procedure is the same irrespective of the shape of the pitwall profile adopted, i.e., planar or optimal profile: at the beginning, an initial pit depth (H0) was assumed as equal to the total height of the block model minus the air blocks so HUP L S1,0 = HUP L S2,0 = 370 m with HUP L S1,0, and HUP L S2,0 indicating the pitwall height in sector S1 and S2 respectively. Then, we calculated the representative pitwall profiles for the specified pit depth in each sector: in the case of planar pitwalls employing limit equilibrium method (LEM) analyses by Slide2 [28] whereas for optimal pitwalls employing OptimalSlope (see Section 3.1). Then, we assigned the pitwall profiles into the pit optimizer (Geovia Whittle 4.7.3) and ran it to produce the UPL (the steps entailed are described in Section 3.2). The detail of the procedure followed to integrate OptimalSlope with Geovia is presented in Fig. 6. In the figure, it is also illustrated how to integrate OptimalSlope with three other major commercial mining software packages, namely Datamine, Maptek, and Hexagon Mining. Note that conceptually the process is independent of the software package employed, but the names of the software modules to manipulate the block model, their scripting languages, and some routines within the pit optimizers may differ. For this reason, we verified that the design procedure here employed was applicable in all the software packages mentioned in Fig. 6, namely Geovia Surpac [29] and Whittle [30], Datamine Studio OP [31] and Studio NPVS [32], Maptek Vulcan Open Pit Mine Planning [33] and Hexagon MinePlan3D [34] and Project evaluator [35], to ensure the pit optimization procedure herein described works correctly with each software package. The UPL depths obtained as output of the strategic pit optimization, HUP L S1,1 for sector S1, and HUP L S2,1 for sector S2 were then compared to the values prescribed as input in the slope design process, HUP L S1,0 and HUP L S2,0, respectively. Since they turned out to be different, a second iteration was performed where HUP L S1,1 and HUP L S2,1 were assumed as input for a second slope design process followed by a second run of the pit optimizer. The iterations were then stopped when the UPL depths obtained as output of the pit optimizer ended up being equal to the values prescribed as input in the slope design process for each pit sector. The iterations required to reach convergence are reported in Table 5.

Fig. 5
figure 5

Pit design process: a flow chart illustrating the iterative process followed to determine the ultimate pit limit (UPL) and pushbacks. Note that the process is the same irrespective of the shape of the adopted pitwall profiles. b Slope (pitwall) height versus overall slope angle (OSA) for different FoS values and profile shapes (planar profiles in orange and optimal profiles in blue). Note the curves were derived for the specific geotechnical parameters of the case study mine. A different set of strength parameters would produce different curves; however, the qualitative trend of the curves is valid irrespective of the rock strength values. In the inset a graphical representation of the iterations performed to arrive at the OSA for each pit sector is provided: the gray arrows refer to the slope design process that produces pitwall profiles from an assigned height (Section 3.1), the black arrows refer to the pit optimization process that produces the UPL depth from assigned pitwall profiles (Section 3.2)

Fig. 6
figure 6

Flow chart illustrating how OptimalSlope [21] interacts with major commercial mining software packages, namely Geovia, Datamine, Maptek, and Hexagon Mining to perform a pit design. Going from top to bottom: (1) OptimalSlope produces the optimal slope profile for the UPL of each sector of the pit; (2) the profiles are prescribed for each pit sector in the block model in the standard way by specifying an inclination αi for the row(s) of blocks lying within each segment of the profile of constant inclination (note that there is more than a way slopes can be specified, we chose to do it by an additional block model attribute and grouping blocks in zones since for all the blocks exhibiting the same inclination, so belonging to the same zone, the angle can be assigned with one instruction only); (3) the pit optimizer is run as usual to determine the UPL and pushbacks. In the PIT OPTIMIZER red box, we report the names of the nodes/tasks/panels used to specify the input data and run the pit optimization

3.1 Pitwall Design

Open pit mines are increasingly excavated in complex lithologies where usually different failure mechanisms (e.g., bench failure, interramp failure, overall slope shear failure, failures involving faults and discontinuities) turn out to be dominant in different sectors of the pit so that they all need to be analyzed. In the design of pitwalls, we followed the standard practice of starting with the design of benches and then moved to the overall pitwall profiles [36, 37].

To determine the maximum inclination of each face bench we performed limit equilibrium method Morgenstern-Price analyses using the Rocscience program Slide2 [28] to satisfy the prescribed factor of safety (in this case FoS = 1.1, see Table 4). In general, the amount of backbreak and the effective bench face angle are controlled by the joints and faults intersecting each bench. Here, joints were not considered purely due to the lack of this type of data for the mine analyzed. If data on joints and faults were available, software packages such as SWedge [38] or Frac_Rock [39 ] would allow determining the maximum inclination of each face bench.

Table 4 Acceptability criteria for factor of safety

The height of the benches adopted for the whole mine is 10 m [22]. Consistently with the set of input data from [22], we computed the minimum berm width, bw, using the equation proposed by Call [40]  known as modified Ritchie’s criteria, which has been demonstrated to be effective in field tests in several benched mine slopes [41 ]:

$${b}_{\mathrm{w}}(m) = 4.5(m) + 0.2 *{H}_{\mathrm{bench}}$$

We acknowledge that another equation, bw(m) = 3.5(m) + 0.17 ∗ Hbench, was proposed in [41 ] but it exhibits the undesirable property of being less conservative [42]  so we preferred to use Eq. (1). Berms have multiple objectives that need to be met when establishing a berm width , the most important ones being rockfall retention, capture of debris from bench excavation and operations of the chosen fleet of excavators. With regard to rockfall retention, berm widths are not designed for 100% retention of rockfalls since this would lead to unviably large widths but instead to shelter most of the rockfall. Eq. (1) leads to 70–85% retention of rockfall volumes [18]. Such a percentage may be less than what a mining company would wish for. For this reason, in the last two decades, several rock fall types of research studies were performed utilizing numerical simulations considering potential trajectories on slope designs using either two-dimensional lumped-mass impact models (2DLM), e.g., the Rocscience code RocFall [43], and/or three-dimensional rigid-body impact models (3DRB) such as the code “Trajec3D” [44] . These numerical models use coefficients of restitution to characterize the amount of energy lost due to the inelastic deformations stemming from the collision of a rock fragment bouncing over a berm or bench. Unfortunately, the input parameters are vastly different for 2DLM.  Also 3DRB and they are seldom calibrated with any site-specific rock fall case studies or field test data at the mine feasibility stage, and may remain uncalibrated throughout the operating life of the mine [45] . In conclusion, given the amount of information available at the feasibility stage of mine design , we believe Eq. (1) is adequate to establish minimum berm widths especially given the low bench heights of our case [45] . Finally, note that as pointed out in Read and Stacey [18, see Sect.], Eq. (1) is still largely used by practitioners in North and South America. Another requirement for the chosen berm widths is the ability to capture most debris falling from the bench faces above due to unstable wedges.  The adequacy of the chosen berm widths can be verified via either computer modeling [39] or analytical equations [46] . In [47] , a recent extensive review of the bench design methodologies currently employed by the open pit mine industry is provided.

After the computation of the bench geometries, we computed the geotechnically optimal slope profiles for the cross-sections of the two pit sectors of the mine (Fig. 4) using the proprietary code OptimalSlope [21]. The code requires bench height, bench face inclination, minimum berm width, and road width as input from the user (see Fig. 7) since these geometric data will act as constraints in the search for the optimal profile. Any pitwall profile is defined in OptimalSlope by a discrete set of points in the vertical plane : see the (xi, zi) coordinates in Fig. 8, with zi being values specified according to the bench height (Δz = bench height) input by the user while xi represents the unknown variables to be determined. The search for the optimal profile is constrained to feasible profiles (which lie within the red and blue bounds of Fig. 8). A profile is feasible if \(\frac{{z}_{i}- {z}_{i-1}}{{x}_{i}- {x}_{i-1}}\le \mathrm{tan}{\alpha }_{i\mathrm{max}}\) for every i, i.e., the inclination of each segment of the profile is capped to \({\alpha }_{i\mathrm{max}}\). The \({\alpha }_{i\mathrm{max}}\) values are determined by the code before the optimization algorithm is called on the basis of bench height, bench face inclination, and minimum berm width provided by the user (see Fig. 8b). In case a ramp needs to be included as part of the pitwall profile, a lower \({\alpha }_{i\mathrm{max}}\) value is imposed for the profile segment corresponding to the vertical position of the ramp.

Fig. 7
figure 7

OptimalSlope data input screen

Fig. 8
figure 8

a A generic candidate slope profile. The toe of the profile is at the axes origin (x0, y0), point C is the slope crest. A uniform discretization along the z-direction is adopted. The red and blue lines enclose the region where the profiles are sought. The profile is discretized in n Δz intervals so there are n − 1 unknown to be determined: x1, x2, …xn–1. In the context of open pit mines, a good choice of Δz is to assume Δz equal to the bench height. b Determination of \({\alpha }_{i\mathrm{max}}\) based on bench geometry (input to OptimalSlope).

The optimal pitwall profile is defined as the overall steepest safe profile, i.e., OSA = OSAmax, with OSA being the inclination over the horizontal of the line joining the pitwall toe to the crest (see Fig. 8). OSAmax is determined by OptimalSlope iteratively (see Fig. 9). The main algorithm finds the optimal pitwall shape for an assigned OSA and geometric constraints (\({\alpha }_{i\mathrm{max}}\) values). An initial guess OSA value is first determined for the specified pitwall height, the geotechnical properties of all the layers, and the specified FoS on the basis of a database of stability charts based on [63] built-in OptimalSlope. In the case of rock strength described by G-H-B, the conversion of the H-B parameters to M-C was performed via Eq. (14) in [20]. Note that a few equations for the conversion of the G-H-B parameters into M-C for rock slope stability analyses have been proposed in the literature with [48, 49] being the most prominent. All these relationships depend on the range of confining stress experienced by the slope. Hence, the main challenge in finding equivalent strength criteria is selecting the appropriate range of confinement. However, contrary to what the previous relationships suggest, the appropriate range of confinement was shown in [20] to be insensitive to rock mass strength and instead primarily controlled by slope geometry and only the equation proposed in [20] reflects such a dependency. The FoSi associated with the optimal profile found at the ith iteration is then compared to the target FoStarget: if it is higher a steeper OSA is prescribed at the next iteration; if it is lower a less steep OSA is prescribed. The termination criterion is specified in terms of the percentage difference between FoStarget and FoSi. For the pitwalls of the case study considered, four iterations were enough to obtain a FoS less than 1% different from FoStarget.

Fig. 9
figure 9

Illustration of the iterative procedure used by OptimalSlope to determine the optimal profile for a given pitwall

In the case of a homogeneous slope, i.e., a single uniform rock/soil layer, the optimal profile shape is found by the main algorithm for a prescribed input OSA as the profile associated with the largest stability factor defined as \({N}_{\mathrm{s}}=\frac{\gamma H}{c}\) in the case of the M-C criterion or \({N}_{\mathrm{s}}=\frac{\gamma H}{{\sigma }_{ci}}\) in the case of the G-H-B criterion. \({N}_{\mathrm{s}}\) is a well-known single scalar parameter that was introduced by Taylor [50, 51] in drawing dimensionless stability charts and is the appropriate metric to compare the performance of different slope profiles [21]. In the case of a layered (non-homogeneous) slope, OptimalSlope determines the optimal profile as the profile associated with the maximum of an equivalent \({N}_{\mathrm{s}}\) which is calculated as a physically based weighted average of the γ, H, c or \({\sigma }_{ci}\) parameters of all the slope layers. Note that in a layered slope a failure mechanism may go through several layers of vastly different strength so the length of the failure surface (a curve in 2D) in each layer may affect significantly the overall resistance (i.e., the amount of energy dissipated in the limit analysis energy balance equation). Therefore, OptimalSlope calculates the energy dissipated along the failure surface of each candidate mechanism considered based on the actual length of the failure curve present in each layer ensuring the \({N}_{\mathrm{s}}\) calculated for each mechanism considered is a true reflection of the mechanism stability factor (see Fig. 10). Crucially, to determine the stability factor of any candidate profile shape, all the possible failure mechanisms are considered by OptimalSlope including any failure surface daylighting everywhere above the slope toe (see Fig. 10).

Fig. 10
figure 10

Illustration of the failure mechanisms considered by OptimalSlope for a generic candidate profile in a layered slope with layers of different strength

3.2 Pit Optimization

To assign the pitwall profiles to the pit optimizer Geovia Whittle, we split the block model into “zones” (according to the Whittle terminology) using Geovia Surpac and assigned a slope inclination to each “zone.” The number of zones to be employed depends on the shape of the pitwall profile (see Fig. 6). Then, to compute the ultimate pit limit and pushbacks, first, we ran Whittle to produce the discounted best-case scenario curve in the pit-by-pit graph (see Fig. 15). Next, we employed the so-called Milawa NPV algorithm to generate a specified case scenario curve for an initial set of pushbacks, chosen in correspondence of sharp increases exhibited by the best-case scenario curve. Then, we recomputed the specified case scenario curve a few times exploring the choice of different pit-shells as pushbacks nearby the ones initially selected to make the specified case curve as close to the best-case scenario as possible. Having found the combination of pushbacks that maximizes the NPV of the UPL, we plotted the annual ore production scheduling graph (Fig. 16) and changed the selection of pushbacks if needed to ensure an annual tonnage input to the processing plant as uniform as possible over the mine lifetime. Finally, we chose the UPL in correspondence of the highest point of the plateau exhibited by the specified case curve (Fig. 15). We are aware practitioners may make slightly different choices: for instance, they may decide to pick a pit shell different from the one associated with the peak of the specified case scenario as UPL for various considerations (to maximize the amount of ore extracted or of reserve amount or due to operational reasons) and may not be willing to iterate for the selection of pushbacks and UPL. Moreover, design practices vary between practitioners due to different company objectives and designer experiences. Nevertheless, the key objective of the design exercise performed here is to run a consistent and meaningful comparison between the traditional design based on planar pitwalls and the one here proposed based on geotechnically optimal pitwalls in order to quantify in a rigorous way the financial and environmental gains due to the adoption of geotechnically optimal profiles . To this end, it is logical to adopt the same procedure for the selection of UPL and pushbacks irrespective of the pitwall profile shapes. We also believe that even if design objectives different from the one used here were to be adopted (e.g., compromising some NPV to maximize reserve conservation) and even if the procedure for the determination of UPL and pushbacks was different from the one adopted here (e.g., prioritizing payback time over NPV maximization, using bench discounting), the adoption of optimal profiles for the pitwalls will always bring financial and environmental gains since it  leads to a reduced amount of waste rock excavated whatever the design procedure and corporate priorities .

4 Results

The geometric features of the ultimate pit limits (HUPL and OSA) obtained as the result of each iteration of the design process (Fig. 5a) are reported in Table 5. A total of six iterations was performed for the traditional design based on planar pitwalls, while for the design based on optimal pitwalls, two iterations were enough to reach convergence.

Table 5 Iterations performed to find the optimal ultimate pit limit

Note that to have a consistent and fair comparison between the two designs, we adopted a design procedure envisaging iterations between the geotechnically based design of the pitwall and the pit optimization carried out by Whittle (Fig. 5). However, practitioners may perform fewer iterations or not at all due to time constraints accepting a less optimal pit design. Nevertheless, in the case of no iteration, i.e., looking at the results obtained after iteration 1 in Table 5, we can state that the adoption of optimal pitwalls provides financial gains of the same order of magnitude.

4.1 Pitwall Profiles

The pitwall profiles obtained as a result of the pit design process are plotted in Fig. 11a for pit sector 1 and Fig. 12a for pit sector 2. The factor of safety values of all the pitwall profiles were verified by performing a LEM analysis with the Morgenstern-Price method, which is a rigorous LEM method where all equations of equilibrium are imposed on all slices [52], in Slide2 [28] with non-circular failure surface and other default settings. Preliminary analyses were carried out to make sure the results are independent of the number of slices adopted and of the domain boundaries chosen. The pitwall profiles employed in the Slide2 analyses are reported in Figs. 11b and 12b for the pit design adopting planar profiles and in Figs. 11c and 12c for the pit design adopting optimal profiles together with their FoS. In all the cases, the FoS found is less than 1% from the target value of 1.30 (Table 4). For the optimal pitwall profiles, given the importance of an independent verification of their FoS, we performed additional stability analyses by an explicit finite difference method with shear strength reduction technique (FDMSSR) employing FLAC3D 7.0 [53] having assigned a unit length in the out of plane direction. In the FLAC analyses, the G-H-B criterion was employed together with the “model factor-of-safety” command and default values for the settings affecting the convergence criterion (specified in terms of unbalanced nodal forces) and the detection of slope failure. Details of the algorithm employed by FLAC to calculate the FoS are provided in the FLAC manual [53]: in summary, the standard procedure adopted by FDMSSR as in the case of M-C geomaterials [54] is followed with the G-H-B criterion approximated locally by the M-C criterion, \({\tau }_{\mathrm{max}}={\mathrm{tan}\Phi }_{\mathrm{loc}}+{c}_{\mathrm{loc}}\), where local cohesion, \({c}_{\mathrm{loc}}\), and angle of shearing resistance, \({\mathrm{tan}\Phi }_{\mathrm{loc}}\), are calculated from the local value of the minor principal stress and G-H-B parameters. In each FLAC analysis, \({c}_{\mathrm{loc}}\) and \({\mathrm{tan}\Phi }_{\mathrm{loc}}\) are decreased by dividing them by an increasing reduction factor until active slope failure is detected. The FoS is found as the reduction factor at the verge of slope collapse. Both associated and non-associated with zero dilation analyses were run obtaining practically identical FoS values. The negligible influence of dilatancy on slope stability is good news since its value is not known for the rock types of this mine. The critical failure mechanisms identified by Slide2 and FLAC3D and the associated FoSs are reported in Figs. 11c and 12c. In both mine sectors, the FoSs found are less than 1% from the target value of 1.30. Note that FDMSSR is an entirely different method of slope stability analysis from LEM and from LA (which is employed by OptimalSlope) so the fact that the FoSs found by Slide2 and FLAC3D are so close to the ones determined by OptimalSlope gives confidence to geotechnical practitioners about the trustworthiness of the FoS of the pitwall profiles determined by OptimalSlope. In conclusion, the FoS values of the pitwall profiles found by OptimalSlope were independently verified by two most popular geotechnical software employed for the geotechnical verification of open pit mines, Slide2 and FLAC, confirming that the pitwall profiles determined by OptimalSlope are as safe as their planar counterparts.

Fig. 11
figure 11

UPL pitwalls for sector S1: a comparison between the pitwall profile for traditional design (planar pitwall is in orange) and the optimal design (optimal pitwall is in blue, the black dots represent the xi and zi coordinates obtained as output from OptimalSlope); b failure mechanism (black line) and factor of safety determined by 2D limit equilibrium method (Slide2) for the planar profile; c failure mechanism (black line) and factor of safety determined by 2D limit equilibrium method (Slide2) and shear strain magnitude determined by finite difference method with shear strength reduction (FLAC3D) for the optimal profile

Fig. 12
figure 12

UPL pitwalls for sector S2: a comparison between the pitwall profile for traditional design (planar pitwall is in orange) and the optimal design (optimal pitwall is in blue, the black dots represent the xi and zi coordinates obtained as output from OptimalSlope); b failure mechanism (black line) and factor of safety determined by 2D limit equilibrium method (Slide2) for the planar profile; c failure mechanism (black line) and factor of safety determined by 2D limit equilibrium method (Slide2) and shear strain magnitude determined by finite difference method with shear strength reduction (FLAC3D) for the optimal profile

Figures 11a and 12a allow visual comparison between the planar pitwall profile of a traditional design, demarcated by the orange line, and the optimal pitwall profile, demarcated by the blue line. Considering pit sector S1, the overall height of the profiles is the same. The optimal profile is clearly steeper than the planar one. The area colored in light green (Fig. 11a) highlights the difference between the two profiles: apart from the small difference in the central zone, the optimal profile requires significantly less rock excavation (see the bottom and top third parts). Considering now pit sector S2, the overall heights of the profiles are significantly different with the optimal one being 70 m less deep than the planar one (Fig. 12a). Once again, the optimal profile is steeper than the planar one. Comparing the two profiles, the volumes of rock excavated look similar albeit distributed differently in space. In the upper half of the pitwall, the optimal profile requires a larger excavation to extract mainly ore whereas the planar profile requires extraction of ore to a deeper depth.

4.2 3D Slope Stability Analysis

In Fig. 13a and b the 3D views of the ultimate pit limits obtained for the traditional design based on planar pitwalls and for the design based on optimal pitwalls respectively are plotted. It is apparent that in both cases, a multi-pit shell made by roughly two conical shapes, one distinctly deeper than the other one, is obtained. This is due to the different rock properties and orebody distributions between sectors S1 and S2 with each cone lying within a pit sector.

Fig. 13
figure 13

Ultimate pit limit for: a traditional design and b design with optimal pitwalls. The UPL is multi-pit with the boundary between two conical shells roughly corresponding to the boundary between pit sectors S1 and S2

The FoS of the entire UPL was verified by performing a 3D FDM analysis with FLAC3D 7.0  [53]. Note that the numerical settings adopted in the 3D stability analysis were the same as those employed for the 2D analyses reported in Section 4.1 for the sake of consistency. The critical failure mechanism identified by the software and the FoS contour maps are plotted in Fig. 14. To check for any potential effect of mesh dependency, two analyses were run: one for a coarser (but still fine) mesh and another one for a finer mesh obtained by halving the element sizes of the first mesh. Throughout the entire mine, the minimum FoS found was 1.70 for the first analysis and 1.67 for the second analysis: such a small difference implies the mesh size adopted is sufficiently small so that the resulting FoS is not affected by mesh size for practical purposes. We can then assume convergence of the FoS from the FLAC3D analysis to 1.67, which is significantly higher than the FoS values obtained by the 2D FLAC analyses of Section 4.1, of ~ 1.30 and ~ 1.31 respectively. The main reason for this is due to arching effects that in the case of slope shapes concave in plan view enhance stability [6, 55]. In this case, the very pronounced planar concavity of the UPL acts to constrain movement. This is not taken into account by 2D analyses, that as a consequence are conservative.

Fig. 14
figure 14

a Map of the shear strain increments computed by FLAC3D for the UPL with optimal pitwalls. b Shear strain increments in vertical cross-section A–Aʹ.

4.3 Key Financial Indicators

In Table 6, the key output data for the two design cases are provided. The net present value (NPV) for the design based on optimal pitwalls is around 12 million USD higher than the NPV of the design based on planar pitwalls. Therefore, the adoption of the optimal profiles would lead to an increase of NPV of 34%. Such an increase is to be ascribed to a very substantial decrease of rockwaste volume, around 15%, from 23.7·106 tonnes to 20.7·106 tonnes, while the amount of ore extracted is similar. This implies a reduction of the Stripping Ratio from 0.40 to 0.35. Another metric measuring the financial return of a mine is the internal rate of return (IRR). Adoption of the optimal profiles leads to an IRR of 15.8% instead of the 13.9% obtained from the design with planar pitwalls. Since an IRR of 15% is considered by some companies as a threshold for the viability of a mining project, it can be said that for this mine the adoption of optimal pitwalls could make the difference in terms of the economic viability of the mine project.

Table 6 Pit optimization economic and metallurgical results

4.4 Production Schedule

In Fig. 15, the pit-by-pit graph of the mine is plotted for both the traditional design based on planar pitwalls (Fig. 15a) and the design based on optimal pitwalls (Fig. 15b). The vertical black lines indicate the pitshells selected as pushbacks including the UPL (line most to the right). In both cases, the UPL has been selected as the pushback corresponding to the peak of the discounted specified case scenario curve (green curve). The fact that the specified case scenario curves exhibit a plateau implies the choice provides a robust NPV optimum. In Fig. 16, the amount of tonnage is plotted against the life of the mine in years. It can be seen that an almost uniform amount of ore is to be extracted year after year for both designs. A constant production over time is a highly desirable feature from a logistical point of view.

Fig. 15
figure 15

Pit-by-pit graph showing net present value in USD (left vertical axis) and tonnage of mined ore and wasterock (right vertical axis) plotted against nested pitshell number. Each pitshell corresponds to a different revenue factor (fixed intervals were used): a traditional mine design and b design based on optimal pitwall profiles. The vertical black lines indicate the pitshell selected as pushbacks including the ultimate pit limit (line most to the right)

Fig. 16
figure 16

Ore (yellow) and rock waste (gray) tonnage (left vertical axis) and cashflows in USD (right vertical axis) plotted against time in years for: a traditional mine design and b design based on optimal pitwall profiles

4.5 Environmental Indicators

Recently, several methods have been proposed in the literature to calculate the life cycle assessment for open pit mines. Here, we have computed the energy requested to mine both orebody and waste rock together with the associated carbon footprint for both design types (design with planar pitwalls and with optimal pitwalls) based on [56]. In the Appendix, all the equations employed for these calculations are provided. Note that energy consumption and carbon footprint are calculated for each mined block of the UPLs with block properties such as grade, mass, and distance to a reference point on the surface used to estimate the energy required to produce 1 tonne of mined ore. The energy requirement per tonne is then translated into carbon footprint by using characterization factors and includes scope 1, 2, and 3 emissions associated with drilling, blasting, extraction, loading, and hauling [57]. This approach has been employed in [58] to incorporate sustainability parameters directly into the ore body block model and include them into strategic pit optimization. The characterization factors were taken from [59] having assumed the electricity production mix for Chile where the mine is located. The results are provided in Table 7 below.

Table 7 Energy consumption and carbon footprint for the two design options considered

It emerges that the adoption of optimal pitwalls leads to reductions of carbon footprint and energy consumption of 0.17 million tonnes CO2 eq and 82.5 million MJ respectively over the life of the mine. To provide some context, a reduction of 0.17 million tonnes CO2 eq is equivalent to the carbon sequestered by 2.8 million tree seedlings grown for 10 years and to the greenhouse gas emissions avoided by 35 wind turbines producing electricity for a year [60]. Both carbon footprint and energy usage savings are achieved by a significant reduction of rockwaste excavation, around 15% in volume, with the amount of orebody extracted being very similar in the two design options considered (see Table 6).

5 Conclusions

It is well known that the steepness of the pitwalls of an open pit mine bears a large influence on the volume of rockwaste to be extracted so that any increase of steepness leads to a better stripping ratio and therefore higher profitability and carbon footprint and energy use reductions. In current design practices, mines tend to be designed on the basis of planar pitwalls, i.e., with a constant inter-ramp angle (IRA) or even overall slope angle (OSA), in each sector of the mine. However looking at the final geometry of any pit cross-section, this is anything but planar due to the need to accommodate for benches, step-outs, and roads; therefore, the assumption of constant IRA and/or OSA adopted at the design stage is a simplification which can and should be removed if a better design can be achieved as result. It seems only natural to wonder whether pitwalls of non-linear shape could be used instead of planar ones. In the geotechnical literature, some specific shapes of non-linear profiles have already been proven to be more stable than the planar one ([8,9,10], and [11]) for uniform c-ϕ slopes. The code OptimalSlope [12] determines optimal shapes for both uniform and non-uniform slopes with any number of rock layers, and for geomaterials whose strength is described by either the Mohr–Coulomb criterion or the generalized Hoek–Brown criterion. In this paper, we employed the geotechnically optimal pitwall profiles determined by OptimalSlope to systematically maximize the OSA of the pitwalls of a soon to be opened copper mine. The long-term schedule of the mine was carried out using Geovia Whittle first employing planar pitwalls, secondly adopting the optimal pitwall profiles determined by OptimalSlope.

The adoption of geotechnically optimal profiles led to a 34% higher net present value and very significant reductions of carbon footprint and energy consumption, 0.17 million tonnes CO2 eq and 82.5 million MJ, respectively, due to a 15% reduction of waste rock volume in comparison with the traditional design based on planar pitwalls. The stability of all the pitwall cross-sections determined by OptimalSlope was independently checked by two most popular geotechnical software packages, namely Rocscience Slide2 to perform limit equilibrium method analyses with the Morgenstern-Price method and FLAC3D to perform finite difference method analyses with shear strength reduction. The factor of safety values determined by Slide2 and FLAC turned out to be in very close agreement (less than 1% difference) to the ones determined by OptimalSlope. Also, a 3D stability analysis for the entire ultimate pit was carried out by FLAC3D verifying the resulting FoS was well above the specified acceptability criterion (FoS = 1.3 in this case) across the entire pit.

The case study considered here is of a relatively small pit in a medium-strong rock. It can be expected that the increase of pitwall inclination will be more significant in weaker rocks. Also, we know that the larger the depth of a pit, the more significant the impact of the pitwall steepness is on the economics of the mine. Therefore, we believe the financial and environmental gains obtained for large and deep pits can be even higher. Finally, irrespective of the size of the open pit mine, because a planar slope is a particular case of a curved one, obtained by setting the radius of curvature to infinity along the entire slope, there is a theoretical argument to suggest that adopting planar  pitwall profiles is a suboptimal choice for most rock masses and soil types apart from the case of frictional (zero cohesion) soils.