Question 4 of 4: Open Pit Limits and Development Planning

Question 4: Open Pit Limits and Development Planning (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Figure 4.1 is a hand-drawn section on a 20 m grid.

Find. (a) the floor elevation and position, wall daylight points and ore/waste quantities of the pit that satisfies SR = 2.0±0.1 while mining the most ore; (b) a development plan, sketched on the section, that keeps the stripping ratio approximately constant.

Approach. Because ρ_ore=ρ_waste, the mass stripping ratio equals the ratio of cross-sectional areas per metre of strike. A trial pit has a horizontal floor at El. $z_f$ from $x_L$ to $x_L+20$ m, with 45° walls rising from each floor edge until they meet the ground. Its total area is integrated column by column (ground minus pit bottom), and its ore area is the part of each column lying between the two vein contacts. Floor elevation and floor position are searched together in 0.5 m steps; the answer is the deepest pit whose SR is still inside the 1.9–2.1 band, since a deeper pit holds more ore.

1. Where to put the floor. At every level the best floor position hugs the vein. Its east edge sits at or near the hanging-wall contact, and the extra floor width beyond the vein's own ≈12.5 m goes on the west side. That is because the ground rises steeply to the east (to El. 470 at x ≈ 32,176), so every metre of floor pushed east adds more wall waste than a metre pushed west onto the flat at El. 450.
2. Scan floor elevation. The lowest stripping ratio achievable at each floor level (best floor position, 20 m floor) climbs steadily with depth:

   Lowest-SR pit at each floor level (per m strike)

   Floor El. (m)	Floor xL	Ore (m²)	Waste (m²)	SR
   446	32,115.0	78.0	111.7	1.43
   444	32,116.5	107.0	186.0	1.74
   442	32,117.5	133.8	275.0	2.06
   440	32,118.5	159.0	379.9	2.39

   The ratio passes the band's upper edge (2.1) between El. 442 and 440, and no floor at El. 441.5 or deeper can be placed with SR ≤ 2.1. At El. 442 the floor position that captures the most ore while staying inside the band is x = 32,119–32,139.
3. Final pit limits. $$\boxed{z_{floor}\approx 442\ \text{m},\ \ x=32{,}119\text{--}32{,}139\ (20\ \text{m floor}),\ \ \text{west wall daylights at }(32{,}104.8,\ 456.2),\ \ \text{east wall at }(32{,}161.4,\ 464.4)}$$
4. Ore and waste quantities (per m strike length). $$A_{pit}=416.2\ \text{m}^2,\qquad A_{ore}=134.6\ \text{m}^2,\qquad A_{waste}=416.2-134.6=281.7\ \text{m}^2 \implies SR=\frac{281.7}{134.6}=2.09$$ At $\rho=2{,}700\ \text{kg/m}^3=2.7\ \text{t/m}^3$: $$\boxed{ore \approx 363\ \text{t/m strike}, \quad waste\approx 761\ \text{t/m strike},\quad SR\approx 2.09}$$

b) Development plan maintaining an approximately constant stripping ratio

Approach. Deepening beyond the final pit is uneconomic at this SR target. Taking the floor from El. 442 to 440 adds only 24.9 m² of ore for 99.7 m² of waste, a marginal ratio of about 4.0, because both 45° walls widen as the pit deepens while the vein adds only its own ≈12.5 m of width per metre of depth. So the constant-SR requirement applies to the sequence in which the final pit is mined. If it is mined as nested stages, each simply deepened in turn, the stripping ratio rises stage by stage. To hold it constant, waste has to be stripped ahead of the ore.

1. Natural (unsmoothed) stage ratios. Using nested pits on the same floor position (x = 32,119–32,139): stage 1 to El. 446 mines 79.7 m² of ore and 146.9 m² of waste (SR 1.84); stage 2, from El. 446 to 444, adds 28.3 m² of ore and 58.3 m² of waste (SR 2.06); stage 3, from El. 444 to 442, adds 26.6 m² of ore and 76.5 m² of waste (SR 2.87). Mined this way, the ratio climbs from 1.8 to 2.9.
2. Constant-SR schedule. Strip waste in every stage at the final-pit ratio of 2.09: $$W_1=2.09\times79.7=166.7,\quad W_2=2.09\times28.3=59.2,\quad W_3=2.09\times26.6=55.7\ \text{m}^2$$ Stage 1 therefore removes 166.7−146.9 = 19.8 m² more waste than its own pit contains. That extra waste is taken from the upper benches of the stage-3 pushback on both walls, mostly on the higher, hill-side east wall. By the end of stage 2 the pre-strip totals 166.7 + 59.2 − 205.2 = 20.7 m², and stage 3 finishes the remaining pushback waste. $$\boxed{SR\approx2.1\ \text{in every stage; pre-strip}\approx 20\ \text{m}^2/\text{m of stage-3 wall waste during stages 1\textendash 2}}$$
3. Beyond the final pit. The vein continues down-dip to El. 371 and, presumably, along strike. Along strike the same staged design is repeated panel by panel, each panel at SR ≈ 2.1, keeping the mine-wide ratio constant. The deeper ore (marginal SR ≈ 4 or more) is left for a later, separately justified pushback or for underground extraction.