Questions & explanations
1. Compare stope boundary optimization with simple rule-of-thumb methods like mining all ore above a fixed grade.
Simple rule-of-thumb methods like mining all ore above a fixed grade are easy but not always best. They might include low-grade ore that costs more to mine than it is worth, losing money. Or they might leave out high-grade ore that is surrounded by low-grade but could be mined together. Stope boundary optimization uses a computer to try many shapes and find the one that gives the most profit. It considers the exact location of ore blocks and the cost to mine them. This gives a better boundary than a simple grade cut-off. For example, a fixed grade might say mine all ore above 5 g/t, but optimization might include some 4 g/t ore if it is next to high-grade and cheap to mine.
2. Give an example of how changing cut-off grade affects the mine's net present value (NPV).
Suppose a mine has 1 million tons of ore with average grade 5 g/t gold. Gold price is $50/g, mining cost $100/ton, processing cost $50/ton. Break-even grade is 3 g/t. If cut-off is 3 g/t, all ore is processed, total gold = 5 million grams, revenue = $250 million, cost = $150 million, profit = $100 million. If cut-off is raised to 4 g/t, only 800,000 tons of higher-grade ore (say 6 g/t) are processed, gold = 4.8 million grams, revenue = $240 million, cost = $120 million, profit = $120 million. NPV (discounted) might be higher with the higher cut-off because profit comes sooner. Optimization finds the cut-off that gives the highest NPV.
3. How does the factor of safety differ between a Koepe (friction) hoist and a drum hoist?
In a Koepe hoist, the rope runs over a friction wheel and does not wind onto a drum, so the rope sees more bending and wear. Therefore, regulations often require a higher factor of safety for Koepe ropes, like 6.5, compared to drum hoists where the rope winds onto a drum and has less bending stress, so an FoS of 5 or 5.5 may be allowed. The higher FoS for Koepe accounts for the risk of rope slip and fatigue from repeated bending. Drum hoists have a larger rope storage capacity but the rope is subject to winding layers that can cause crushing. Both systems must meet the minimum FoS set by safety codes.
4. Compare NATM with the traditional 'drill and blast' method for a tunnel in soft ground.
In soft ground, NATM is often better than traditional drill and blast. Drill and blast uses explosives to break rock, which can shake the ground and cause collapse in soft ground. NATM digs carefully with machines and immediately supports the tunnel with shotcrete (sprayed concrete) and rock bolts (steel rods). NATM also measures ground movement to adjust support. Traditional methods use heavy steel ribs and thick concrete lining, which cost more. NATM is faster and cheaper in soft ground because it uses the ground's own strength. But in very weak ground, traditional methods might still be needed.
5. Explain the three curves in the convergence-confinement method: ground reaction curve, support characteristic curve, and longitudinal deformation profile.
The ground reaction curve (GRC) shows the relationship between the inward displacement of the tunnel wall and the internal support pressure. As support pressure decreases, displacement increases. The support characteristic curve (SCC) shows how much pressure a support system (e.g., a steel set) provides as it deforms. The longitudinal deformation profile (LDP) shows the displacement of the tunnel wall along the tunnel axis as the face advances. The GRC and SCC are plotted together to find the equilibrium displacement and pressure. The LDP helps decide when to install support relative to the face.
6. Why is the factor of safety higher for a Koepe hoist rope than for a drum hoist rope?
The Koepe hoist rope bends over a friction wheel many times during each trip, causing more fatigue and wear. This bending reduces the rope's strength over time. Also, the rope relies on friction to grip the wheel, so any slip can be dangerous. A higher factor of safety ensures the rope remains strong even after repeated bending. In contrast, a drum hoist rope bends less because it winds onto a drum, so it experiences less fatigue. The higher FoS for Koepe compensates for these extra risks. Safety standards like those from the Mine Safety and Health Administration (MSHA) specify these differences.
7. A mountain range has a crustal thickness of 60 km, while normal crust is 35 km. Using Airy's model, estimate the depth of the mountain root below the normal crust. (Assume crust density 2700 kg/m³, mantle density 3300 kg/m³).
In Airy's model, the root depth r = (ρ_crust / (ρ_mantle - ρ_crust)) * h, where h is the mountain height above normal crust. First, find h: total crust thickness difference = 60 - 35 = 25 km. But part of that is above and part below. Actually, the mountain height above the normal crust is not given directly. However, using the formula, the root depth = (2700/(3300-2700)) * (extra thickness above). But we need the extra height. Alternatively, the total extra thickness is 25 km, and the root is about 4.5 times the height. So if the mountain rises 5 km, root is about 22.5 km. This is approximate.
8. What is the convergence-confinement method used for in tunnel design?
The convergence-confinement method is used to design tunnel supports (like rock bolts and shotcrete). It considers how the rock around the tunnel deforms (converges) inward after excavation and how the support pressure (confinement) resists that deformation. The method plots a ground reaction curve (how much the rock moves for a given support pressure) and a support characteristic curve (how much pressure the support provides as it deforms). The intersection gives the equilibrium point where the support and rock are balanced. This helps choose the right support stiffness and installation time.
9. Give an example of how stope boundary optimization can increase profit.
Imagine a gold mine with two ore zones: one rich zone with 10 grams per ton gold and a poor zone with 3 grams per ton. The cost to mine is $100 per ton, and gold price is $50 per gram. The rich zone value is $500 per ton, so profit is $400. The poor zone value is $150 per ton, profit $50. If the stope includes both, total profit is $450 per ton. But if the stope boundary is drawn to include only the rich zone, the profit per ton is $400, but you mine less waste. However, if you can mine the poor zone cheaply, it adds profit. Optimization finds the boundary that gives the highest total profit.
10. What is cut-off grade in mining?
Cut-off grade is the lowest grade (amount of metal in the rock) that makes it worth mining. If the grade is above the cut-off, the rock is called ore and is sent to the mill. If below, it is waste and is dumped. The cut-off grade is chosen to make the most profit over the life of the mine. It depends on metal price, mining cost, and processing cost. A higher cut-off means only rich ore is mined, but less total ore. A lower cut-off means more ore but lower profit per ton. Cut-off grade optimization finds the best balance to maximize the net present value (NPV), which is the total profit today.
11. Explain the concept of 'opportunity cost' in cut-off grade optimization.
Opportunity cost in cut-off grade optimization means the profit you lose by not mining a block now. If you mine low-grade ore today, you use mill capacity that could have processed high-grade ore later. So the true cost of mining low-grade ore includes the lost chance to process high-grade ore. The optimal cut-off grade considers this: it might be higher than the break-even grade because processing low-grade ore delays high-grade ore. For example, if you have limited mill time, you might set cut-off grade high to process only the best ore first, and save low-grade for later when mill is free.
12. What is linear programming in mine scheduling?
Linear programming (LP) is a math method to find the best way to use limited resources. In mine scheduling, it helps decide how much ore to mine from each area each year to maximize profit. The problem is set up with a goal (like maximize NPV) and constraints (like mining capacity, mill capacity, and ore grade limits). All relationships must be linear, meaning they are straight-line equations. LP uses a computer to solve for the best values. It is widely used because it is fast and gives a clear answer. The result is a schedule that tells how many tons to mine from each block each period.