Question 1 of 6: General Underground Mining

Question 1: General Underground Mining (40 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.

1.1 — O'Hara's mine cost-estimating method

Formula and estimation of A, b. O'Hara's method (CIM Bulletin No. 814, 1980) fits capital or operating cost to a single power-law regression, Cost = A·T^b, where T is a scale variable — typically daily mill/mine throughput in tonnes-per-day for a capital-cost term, or a similar capacity measure for an operating-cost term. A is the coefficient (the cost at T = 1, effectively a scale/currency-year constant) and b is the economy-of-scale exponent, almost always well below 1.0 (commonly 0.3–0.7 for mine and mill capital items), which captures the well-known observation that larger operations cost less per unit of capacity than smaller ones. O'Hara estimated A and b by linear regression on log-transformed historical cost data: taking log(Cost) = log(A) + b·log(T) converts the power law into a straight line, so a least-squares fit against a database of real, completed mine/mill capital and operating costs (drawn from CIM/industry cost surveys of the period) yields the slope b and intercept log(A) directly; the exponent, being derived empirically rather than from first-principles engineering, is only valid within the size range of the underlying database and should not be extrapolated far beyond it.

Updating to present-day costs and cost-index publications. A capital or operating cost computed at a fixed historical date is escalated to a later ("present-day") date using the ratio of a published cost index at the two dates: Cost_new = Cost_old×(Index_new/Index_old), applied separately to capital and operating components because they inflate at different rates (equipment/steel vs. labour/electricity). Two publications widely used to facilitate this update in mining cost work are the Marshall & Swift Mining and Milling Equipment Cost Index (tracks mine/mill capital-equipment cost inflation, published regularly in trade press such as Chemical Engineering/Mining Magazine) and the U.S. Bureau of Mines Cost Estimating System / Camm (Bureau of Mines Information Circular) series, which republishes updated capital and operating indices each edition specifically to escalate the O'Hara-family and Camm-family cost equations. In Canadian practice the equivalent Statistics Canada mining capital-expenditure price indices serve the same escalation role.

1.2 — Mine hoist ropes: types, definitions and applications

1.2.1 Round strand. Individual round wires are helically laid into round strands, and the strands are then laid helically around a core (fibre or IWRC). It is the most common, most flexible, lowest-cost construction, used as the standard hoist/friction rope on single- and double-drum hoists and as guide/counterweight rope; its round outer wires give the best sheave-groove grip but the roughest, fastest-wearing outer surface of the families listed here.

1.2.2 Flattened strand. Strands are laid from alternating large and small trapezoidal (flattened) wires, giving a flatter outer profile with more metallic cross-section (hence more strength and abrasion resistance) per rope diameter than round strand. Used on Koepe/friction and multi-rope drum hoists where a longer-wearing, higher-breaking-load rope justifies the higher cost and reduced flexibility — matches the flattened-strand rope specified for the Koepe hoist in Question 2.3.

1.2.3 Locked coil. An inner layer of round-wire strands is overlaid by one or more outer layers of interlocking, shaped (half-round/Z-profile) wires that lock together to form a smooth, essentially solid outer jacket with no exposed strand valleys. It is non-spinning by construction, has the smoothest, most wear- and corrosion-resistant surface of any hoist-rope type, and is the standard choice for deep, single-rope, non-rotating shaft-hoisting (mine-shaft cage/skip hoist ropes and, especially, track/guide-free vertical hoisting) where rope life and rotational stability dominate the selection.

1.2.4 Regular lay. Wires in each strand are laid in the opposite direction to the lay of the strands around the rope (e.g. right-regular-lay = wires lay left, strands lay right), so the visible wire crossings run roughly parallel to the rope axis. This construction resists kinking and unlaying, is easy to handle, and is the standard general-purpose hoisting lay used wherever the rope is not required to resist rotation as strongly as a Lang-lay rope would.

1.2.5 Lang lay. Wires and strands are laid in the SAME direction, so the outer wires run at a shallow diagonal to the rope axis over a longer exposed length per wire. This gives markedly better fatigue and abrasion resistance (each wire contacts the sheave/drum groove over a longer wrap length) than regular lay, at the cost of a strong tendency to unlay/rotate under load — so Lang-lay rope is restricted to applications where BOTH rope ends are restrained from rotating (e.g. non-rotating drum hoist ropes fixed at both drum and conveyance, never a free-hanging single rope with a rotating skip).

1.2.6 Half locked coil. A hybrid between full locked coil and round/flattened strand: only the outermost wire layer is the interlocking Z/half-round profile, with round-wire strand layers beneath it. It recovers most of locked coil's smooth-surface wear resistance at somewhat lower cost and slightly better flexibility, and is used on medium-to-deep single-rope hoists where full locked-coil cost is not justified.

1.2.7 Non-rotating. A multi-layer, opposite-lay-direction construction (e.g. an inner layer laid right, an outer layer laid left) in which the torque generated by each layer under load largely cancels the other's, so the rope does not spin or unlay when suspended free with an unrestrained lower end. This is essential for single, free-hanging hoist ropes (shaft sinking stages, single-rope skip/cage hoisting) where a rotating rope would twist the conveyance and its guides.

1.2.8 6×7, 6×19, 6×37 classifications. The first number is the strand count (6 strands laid around a core); the second is the nominal wire count per strand. 6×7 (few, large wires per strand) is the stiffest, most abrasion-resistant, least flexible — used for guide ropes, haulage/track ropes and other low-bend applications. 6×19 is the general-purpose hoisting classification (moderate flexibility and wear life), matching the rope specified in Question 2.3. 6×37 (many small wires per strand) is the most flexible, best suited to small-diameter sheaves or where the rope must bend repeatedly over drums, at the cost of faster individual-wire wear.

1.2.9 Cores. A fibre core (natural or, more often now, polypropylene) is a flexible, resilient cushion that lets the strands seat and re-lubricate under load; it is light and cheap but crushes/loses diameter under very heavy loading and does not add strength. A wire-strand core (WSC) (an extra strand at the centre in place of fibre) resists crushing better and adds a little strength but transmits less cushioning. An independent wire-rope core (IWRC) (a small wire rope in its own right at the centre) gives the best crush and heat resistance and the highest strength-to-diameter ratio, and is standard on deep, heavily loaded mine hoist ropes where diameter growth/crushing under sheave pressure would otherwise shorten rope life.

1.3 — Kirchhoff's Laws and Atkinson's equation in ventilation circuits

1.3.1 Kirchhoff's Laws. Mine ventilation networks are solved by the direct electrical analogy to Kirchhoff's circuit laws: Kirchhoff's First Law (continuity/nodal law) states that the airflow entering any junction equals the airflow leaving it (∑Q = 0 at every node), the ventilation equivalent of conservation of current; Kirchhoff's Second Law (mesh/loop law) states that around any closed loop in the network the sum of the frictional pressure drops equals the sum of the applied pressures (fan pressures/natural ventilation pressure), ∑H = 0 around the loop, the equivalent of the voltage law. Applied to a mine circuit, the first law forces airway quantities to balance at every split/junction, while the second law forces every parallel path between two common points to settle at the SAME total pressure drop — which is what makes parallel airways share flow unequally when their resistances differ, and is the basis of the Hardy-Cross iterative balancing method used to solve real, multiply-connected mine networks (guess Q's satisfying the first law, then iteratively correct them until the second law is also satisfied to within tolerance).

1.3.2 Atkinson's equation and the Chezy-Darcy connection. Atkinson's equation, H = K·O·L·Q²/A³ (used quantitatively in Question 3.4 below), gives the frictional pressure loss H in an airway of perimeter O, length L and cross-sectional area A carrying airflow Q, with K the Atkinson friction factor (an empirical property of the airway's roughness, shape and Reynolds-number regime, analogous to a Darcy-Weisbach friction factor but expressed per unit of mine-ventilation convenience). It is applied to every individual airway in a mine ventilation network (each drift, raise, shaft segment) to compute its resistance R = K·O·L/A³ so that H = R·Q², the quadratic pressure–quantity relation used throughout fan selection and network balancing (including the fan-characteristic-curve scaling H₂ = H₁·(Q₂/Q₁)² for a fixed airway). Atkinson derived his equation directly from the general Chezy-Darcy (Darcy-Weisbach) pipe-friction equation, H_f = f·(L/D_h)·(V²/2g), by substituting the mine-airway hydraulic radius (A/O) for pipe diameter and folding the Darcy friction factor f, the 2g term and unit-conversion constants into the single empirical factor K; so Atkinson's K is not an independent physical constant but a repackaged Chezy-Darcy friction factor calibrated for mine-airway roughness, shape and duct-effect (turbulence around timber sets, rail, pipes and cars) rather than the smooth-pipe f used in hydraulics.

1.4 — Mine backfill systems

1.4.1 Cemented waste rock fill (CRF). Coarse mine-development or quarry waste rock (broadly graded, typically minus 100–200 mm) is end-dumped or conveyed into an open stope and bound with a low-cement-content slurry (roughly 3–7% cement by dry weight) delivered by pipeline. It gives moderate strength (0.5–2 MPa at 28 days) at low binder cost and is used as a cheap, free-draining fill in large bulk stopes (sub-level open stoping, VCR) where an exposed vertical face rather than a walkable, load-bearing floor is the main requirement.

1.4.2 Mill tailings (hydraulic/classified fill). The fine (typically minus 200 mesh to a few mm) reject stream from the mineral-processing plant is classified (cycloned) to remove the finest, slowest-draining slimes, then pumped underground as a slurry and allowed to settle and drain through a barricade. It is inexpensive because it uses an existing plant waste stream, but its low permeability and fine grading limit placement rate and require careful barricade/drainage design to avoid liquefaction; used with cut-and-fill and other methods where the fill only needs to support subsequent lifts of broken ore, not carry heavy structural/pillar-replacement loads.

1.4.3 Paste fill. The full plant tailings stream (including the fine slimes fraction, typically thickened to 70–85% solids by weight, a non-segregating, non-Newtonian paste) is mixed with a small percentage of binder (2–7% cement/fly-ash/slag) and pumped through a pipeline — not gravity-fed like hydraulic fill — directly to the stope. Because the full tailings stream (not just the coarse fraction) is consumed, paste fill both maximises tailings disposal underground (reducing the surface tailings-storage-facility footprint, a major environmental and closure-liability driver) and, once cured, develops enough strength to serve as a structural pillar substitute in sub-level and post-pillar cut-and-fill mining, at the cost of the highest capital cost (paste plant, positive-displacement pumps, pipeline) of the three systems.

1.5 — Limiting hole deviation when drilling

Hole deviation grows from four interacting causes — collar misalignment, bit "walk" off centreline through variable/jointed/anisotropic rock, rod/steel whip and flex under thrust, and gravity sag over long unsupported drill-string spans — and the practical controls address each in turn. Collaring and starter control: use a rigid collaring guide or starter sleeve/steel to fix the initial angle precisely before the hole develops depth, and check collar angle with an inclinometer before drilling proceeds past the first metre, since an error at the collar compounds over the whole hole length. Steel/rod selection and stiffness: use the largest-diameter, stiffest drill steel/rod the machine and hole size allow (resists whip and sag), keep rod joints tight and in good condition (a worn coupling is a hinge point that concentrates deviation), and for the longer holes described in the question (up to 0.2 m diameter, 10% deviation risk) use stabilisers/reamers spaced along the string to keep the bit centred in the hole rather than relying on the collar alone. Ground-adaptive technique: reduce feed/thrust pressure and rotation speed when drilling across bedding, jointing or contacts between rock of different hardness (the classic cause of bit walk toward the softer material), and where the geology is known to be highly anisotropic, pilot-hole and survey frequently rather than drilling the full pattern blind. Down-hole surveying and correction: for production and long-hole drilling (the larger, deeper holes most exposed to the 10% deviation risk), run periodic down-hole surveys (single-shot or multi-shot gyroscopic/magnetic survey tools) so deviation is caught and, where the equipment allows, corrected with directional steering rather than discovered only when the hole breaks through in the wrong place; this is standard practice on long-hole/VCR rings and raise-bore pilot holes specifically because undetected deviation there can miss the target ring pattern or intersect an adjacent hole.

1.6 — Overhand and underhand stoping

Overhand stoping. Development starts with a haulage drift/sill along the bottom of the block; miners then drill and blast successive slices UPWARD into the overlying ore from that sill (or from stulls/working platforms built up on the broken muck itself), so the pile of broken ore beneath the miners doubles as their working floor and access as the stope rises. Ore is periodically drawn off through the bottom sill to ore passes, leaving enough muck in the stope to keep the floor at a safe working height below the back. It suits steeply dipping, moderately competent ore and walls (the method is the geometric basis of overhand cut-and-fill and shrinkage stoping) and gives good access for ground support installation as each slice is taken, but exposes the crew to an advancing, freshly blasted back on every cycle.

Underhand stoping. Development instead establishes an UPPER sill/drift at the top of the block; miners drill and blast DOWNWARD, working from a solid, already-supported sill or from the previous completed lift, so each new working level is created by removing ore beneath a level that has already been made safe and supported. Broken ore falls or is mucked into the lower drift/ore pass rather than being retained as a floor. Underhand mining is inherently used where the ground above the working face must be secured before it is exposed — classically in weak or caving ground, in post-pillar recovery, and as the "underhand" cut sequence in cut-and-fill mining under backfill (the crew always works beneath a supported back rather than exposing new, unsupported ground overhead) — trading the productivity and simple mucking of overhand mining for materially better face-worker safety in poor ground.