On near-horizon and limiting values of stripping ratios
Abstract Generated abstract
The paper examines how to determine near-horizon and limiting stripping ratios for the design of open-pit mine boundaries. It derives analytical expressions for near-horizon external stripping coefficients from geometric considerations, identifies simplified formulas for particular deposit geometries, and proposes graphical integration for complex geological and topographic conditions where general analytical calculation is impractical. It also formulates limiting stripping ratios from the condition of equal profitability of open-pit and underground mining, arguing that calculations should account not only for production costs but also for labor-related social costs and differences in ore recovery and exploration expenditure.
Full Text
On Near-Horizon and Limiting Values of Stripping Ratios
B. V. POLYAKOV
(Presented by the scientific seminar of the departments of development of stratified deposits, development of ore deposits, mine construction, safety engineering, and mine ventilation)
In designing the limiting outlines of open pits, it is necessary to establish the ratios of reserves of the useful mineral and of the rock to be removed in different sections and horizons of a deposit. At the same time, it is necessary to determine, with sufficient accuracy, such values of the indicated ratios beyond which open-pit mining ceases to be profitable.
Abstracting from actual practice, one may imagine such a development of operations in an open pit in which the external stripping of the useful mineral is carried out as the ore body is successively mined in horizontal layers of elementarily small height. The ratio of the quantities of waste rock outside the outline of the ore body and of the useful mineral within its limits, whose movement is mutually conditioned in the conduct of open-pit mining at a certain horizon during its successive development, constitutes the near-horizon coefficient of external stripping.
The near-horizon coefficient of external stripping is functionally dependent on the near-horizon heights and the slope angles of the pit walls, the dip angles of the surface of the ore body, the area of the pit bottom, and the coefficient of internal stripping.
The basic mathematical expression for the near-horizon stripping coefficients may be obtained with the aid of Pappus’ second theorem and the theory of limits. In this case it is written in the following form:
\[ n = \frac{ (1+k)\displaystyle\sum_{i=0}^{i=t} H_i \left(\operatorname{ctg}\beta_i+\operatorname{ctg}\delta_i\right)\cdot \Delta l_i }{ F }, \tag{1} \]
where
\(n\) — near-horizon coefficient of external stripping;
\(k\) — near-horizon coefficient of internal stripping;
\(F\) — near-horizon area of the ore body (pit bottom—Fig. 1);
\(i\) — ordinal number of a certain point of the outline of the near-horizon pit bottom;
$t$ — the greatest value of $i$;
$H_i$ — the height of the pit wall corresponding to point $i$ (Fig. 2);
$\beta_i$ — the angle of the pit wall slope at point $i$;
$\delta_i$ — the dip angle of the surface of the ore body at point $i$ (the internal upper angle formed by the lines of intersection of the horizon plane $x - x$ and the surface of the ore body with the plane normal to the contour of the bottom of the pit at point $i$);
$\Delta l_i$ — the segment of the line of centers of gravity of the areas of normal sections of internal stripping corresponding to point $i$.
In particular cases formula (1) is simplified.
- In bulk excavation of an ore body, when the extraction of the mineral is carried out without internal stripping:
\[ n = \frac{ \displaystyle \sum_{i=0}^{i=t} H_i(\operatorname{ctg}\beta_i+\operatorname{ctg}\delta_i)\cdot \Delta l_i }{F}. \tag{2} \]
- On a section with elements of occurrence of the deposit that are persistent along strike and with a calm relief of the ground surface, when the stripping of the ends of the deposit may be neglected:
\[ n = \frac{ H_b(\operatorname{ctg}\beta_b+\operatorname{ctg}\alpha_b) + H_l(\operatorname{ctg}\beta_l-\operatorname{ctg}\alpha_l) }{m}, \tag{3} \]
where
$m$ — the horizontal thickness of the deposit;
$H_b$ and $H_l$ — the horizontal heights of the pit walls on the hanging-wall and footwall sides of the deposit;
$\beta_b$ and $\beta_l$ — the horizontal angles of slope of the pit walls on the hanging-wall and footwall sides of the deposit;
$\alpha_b$ and $\alpha_l$ — the horizontal dip angles of the hanging and footwall sides of the deposit.
The last expression reveals the mathematical essence of the graphical constructions proposed by Candidate of Technical Sciences V. V. Rzhevskii for determining horizontal stripping ratios on transverse profiles of a deposit [3].
- On a section with even relief of the ground surface and concordant dip of the hanging and footwall sides of the deposit, when stripping of its ends may be neglected:
\[ n = \frac{H(\operatorname{ctg}\beta_b+\operatorname{ctg}\beta_l)}{m}, \tag{4} \]
\[ n = \frac{H(\operatorname{ctg}\beta_b+\operatorname{ctg}\alpha)}{m}. \tag{5} \]
The dependences shown in equations 4 and 5 underlie the analytical formulas proposed by Professors P. I. Gorodetskii and B. P. Bogoliubov for determining the limiting depth of pits with steep and inclined $(\alpha \leq \beta_l)$ dip of an “ideal” deposit [2 and 1].
It does not appear possible to calculate the value of the horizontal stripping ratio analytically in the general case.
The difficulty arises from the absence of any regularities in the change of the elevations of the surface contour of the open pit, in the change of the slope angles of its walls, and also of the dip angles of the surface of the ore body. In this connection, in order to determine the near-horizon stripping ratios under complex geological and topographic conditions and with full-periphery stripping, one should resort to approximate graphical integration.
For this purpose, on the corresponding near-horizon plan of the deposit one plots the contour of the bottom of the open pit: \(b_1, b_2, b_3, \ldots\), probable with successive development of the open workings down to the given horizon (Fig. 1). Along the normals to the latter, segments are laid off,
Fig. 1. Plan of the mineral deposit
proportional to the quantities \(H_i(\operatorname{ctg}\beta_i+\operatorname{ctg}\delta_i)\), which are found by means of fairly simple graphical constructions on vertical geological sections of the deposit (Fig. 2). Joining the ends
Fig. 2. Cross section of the deposit
of these segments of the smooth curve gives the contour \(a_1, a_2, a_3,\ldots\). The ratio of the area bounded by the outer contour \(a_1 a_2 a_3 \ldots\) and the inner contours \(b_1 b_2 b_3 \ldots\) to the area of the near-horizon industrial reserves of the mineral is equal to the value of the near-horizon coefficient of external stripping
\[ n=\frac{\text{area } a_1 a_2 a_3\ldots-\text{area } b_1 b_2 b_3\ldots}{f_1+f_2+f_3}. \tag{6} \]
The ratio of the area bounded by the outer contour, less the area of the near-horizon industrial reserves of the mineral, to the area of the near-horizon industrial reserves is equal to the value of the near-horizon coefficient of total stripping \((n_n)\)
\[ n_n=\frac{\text{area } a_1 a_2 a_3\ldots}{f_1+f_2+f_3}-1. \tag{7} \]
Open-pit extraction of a mineral is accompanied by unavoidable costs associated with the removal of barren rocks. The maximum ratio of the quantities of waste rock and mineral extracted at which, under the given mining-geological and economic conditions, open-pit mining of a deposit remains profitable is called the limiting value of the stripping ratio.
In determining the limiting values of stripping ratios, one must proceed from the conditions of equal profitability of open-pit and underground mining of the deposit, determined in accordance with the interests of the most successful development of the national economy as a whole. The basic formula for determining the limiting values of the coefficients of external stripping \((N)\) may be represented in the following form:
\[ N=\frac{U_n\frac{V_0}{V_n}-U_0}{C}\cdot \lambda_0\gamma, \tag{8} \]
where \(V_n\) and \(U_n\) are the national-economic value and the full cost of a certain product obtained in underground mining of the deposit, rub/m;
\(V_0\) and \(U_0\) are the national-economic value of a certain product obtained in open-pit mining of the deposit, and its full cost less the costs associated with the production of external stripping, rub/m;
\(\lambda_0\) is the recovery coefficient of the product of national-economic value \(V_0\) from ore in place in open-pit mining of the deposit;
\(C\) is the full cost of producing external stripping, rub/m\(^3\);
\(\gamma\) is the bulk density of the ore in place, t/m\(^3\).
Establishing the national-economic value of the product obtained in open-pit and underground mining of a deposit, under ordinary design conditions, is not yet possible. But if, in open-pit and underground mining of a deposit, a product of the same type and quality is obtained \((V_0=V_n)\), then in determining the limiting value of the stripping ratio, establish-
…it is not necessary to identify the national-economic value of this product. In this case the limiting stripping ratio is:
\[ N=\frac{U_n-U_0}{C}\cdot \lambda_0\gamma. \tag{9} \]
Determining the full value of the product obtained in open-pit and underground mining of a deposit is complicated by the fact that the question of the efficiency of capital investments has not yet found a proper solution in our economic science. Because of this latter circumstance, in determining the limiting value of the stripping ratio it is still not possible to take sufficient account of differences in the volume of initial capital investments in open-pit and underground mining of a deposit, in the construction periods of the quarry and the mine and their attainment of design capacity, or in the intensity of open-pit and underground mining of the deposit. As for labor productivity and extraction of ore from the subsurface, which usually differ in open-pit and underground mining of a deposit, they can and should be taken into account more fully than is customary in ordinary production cost estimates.
Labor productivity is reflected in the calculated cost of production. But the calculated cost includes only individual wages, and does not include a substantial part of the workers’ income paid by the state. Nor does it take into account the circumstance that housing and communal construction is almost entirely subsidized by the state.
In 1954, expenditures on health care, social security, education and cultural services, industrial training, and various benefits for workers amounted to 25% of the entire state budget. With the introduction of the new law on state pensions, this item of the state budget is increasing still further.
Taking the foregoing into account, the full costs for individual types of work are found from expressions similar to the following:
\[ A=a_k+S_k K, \]
where \(A\) is the value of crude ore in underground mining, taking into account workers’ income in excess of individual wages at the expense of state benefits and allowances, rubles/t;
\(a_k\) is the calculated cost of crude ore in underground mining, rubles/t;
\(S_k\) is the amount of wage-fund costs in the calculated cost of crude ore in underground mining, rubles/t;
\(K\) is the structural coefficient of workers’ incomes in underground mining.
The structural coefficient of workers’ incomes is the averaged ratio of workers’ income in excess of individual wages, received in the form of state benefits and allowances, to individual wages. Taking into account the privileged position of workers in the mining industry, the structural coefficient of the incomes of workers employed in underground and open-pit mining operations should be greater than 0.3, which was the average value of the coefficient \(K\) in the USSR in 1954. For underground work it may be taken as 0.5, and for open-pit work, taking into account its greater safety and hygienic conditions, as 0.4.
The completeness of extraction of a mineral from the depths is reflected in the calculated cost of production only by the amount of depreciation of capital expenditures associated with the construction of the mine. But the economic effect from increasing the extraction of reserves of an ore field is not exhausted by improved utilization of the mine’s fixed assets, since the mineral of the industrial reserve also has value. Expenditures on the search for and exploration of the mineral are financed directly through Prombank and are not reflected in the cost of the mine’s output. Nevertheless, when determining the limiting value of the stripping ratio, they cannot be disregarded. In this case, the expenditures associated with the search for and exploration of the mineral will enter into the calculation as can be seen in formula (10):
\[ N = \frac{ A - B + G\left(\frac{1}{\eta_{\text{п}}} - \frac{1}{\eta_{\text{о}}}\right) }{C} \cdot \eta_{\text{о}} \cdot \gamma, \tag{10} \]
where
\(A\) — cost of crude ore in underground mining, taking into account workers’ income above the individual wage due to state privileges and benefits, rub/t;
\(B\) — the same in open-pit mining, but after deducting the costs of stripping production, rub/t;
\(C\) — cost of stripping, taking into account workers’ income above the individual wage, rub/m\(^3\);
\(G\) — cost of search for and exploration of ore of the industrial reserve, taking into account workers’ income above the individual wage, rub/t;
\(\eta_{\text{п}}\) and \(\eta_{\text{о}}\) — coefficient of apparent extraction of ore of the industrial reserve in underground and open-pit mining;
\(\gamma\) — bulk density of the ore, t/m\(^3\).
Formula (10) is easily derived from formula (9) under the condition that the quality of the ore mass expected in open-pit and underground mining will be the same. This formula, with an acceptable approximation, may be used to determine limiting values of stripping ratios in most cases.
LITERATURE
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Bogolyubov B. P. Expedient boundaries of open-pit operations, Gornyi zhurnal, No. 11, 1956.
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Gorodetskii P. I. Fundamentals of the Design of Mining Enterprises, Metallurgizdat, 1955.
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Rzhevskii V. V. Graphical method for determining stripping ratios, Collection of Scientific Works of MGI, No. 11, Ugletekhizdat, 1953.