Cut-off grade estimation

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The distinction between ore and waste cannot be made without a cut-off grade

Cut-Off grade is the minimum grade required in order for a mineral or metal to be economically mined (or processed). Material found to be above this grade is considered to be ore, while material below this grade is considered to be waste. Although in a variety of units can be used, the cut-off grade is usually expressed as:

  • g/t (grams per tonne)
  • $/t (dollars per tonne)
  •  % (percent metal)

The cut-off grade can be determined through a variety of methods, each of varying complexity. Cut-off grades are selected to achieve a certain objective, such as resource utilization or economic benefit. Dividing these objectives even further gives way to specific goals such as the maximization of total profits, immediate profits, and present value. It is important to recognize that the cut-off grade is not simply calculated to a definitive answer. It is in fact a strategic variable that has major implications on mine design. The cut-off grade is adapted as the economic environment changes with regard to metal prices and mining costs, and is therefore a constantly changing. Metal value is not the only factor affecting the profitability of a block. The presence of unwanted (often hazardous) material in a block may increase the processing cost. This should also be considered when classifying waste rock and ore. [1]

Contents

Calculating the Cut-Off Grade

Break-Even and Internal

The most simplistic way to determine cut-off grade is through the break-even and internal method. This method is easy to use and involves minimal background information. It is preferred as a first estimate, and is most effective in the early planning phases of a project. Although the limited number of parameters makes this method a good first step, it is not ideal to use this method in the long term, since it does not take production capacity or discount rates into consideration. The internal cut-off grade assumes that the block in question must be mined and determines whether it should be sent to the mill. The break-even cut-off grade assumes that the block does not have to be mined and classifies the material as ore or waste to mine or ignore respectively. Both methods utilize the following formula:


\mathbf{G=(c+m_o-m_w)/(y(s-r))}


Parameters
mo Cost to mine ore
mw Cost to mine waste
r Refinery unit cost
c Cost to process ore
y Metal recovery
s Unit metal sale price


For the Internal (Milling) cut-off grade, mw = mo. Material below this grade should not be processed whether or not it has already been mined.


\mathbf{ Gmill=c/(y(s-r))}


For the Break-even (Mining) cut-off grade, mw = 0. Material below this grade should not be mined.


\mathbf{ Gmine=(c+m_o)/(y(s-r))}


These cut-off grades are the fastest way to classify ore and waste, however, in a more detailed analysis they should not be relied on.


Lane’s Method

Lane's Method was presented by Ken Lane in 1988 when he identified three distinct stages of cut-off grade application; mining, concentrating, and processing. He related these steps to the concept of Net Present Value to create a method of cut-off grade determination that maximizes present value. [2]Lane’s Method offers a more accurate and complex way to calculate cut-off grade. This method also requires more information, so it may not be the first choice for a preliminary calculation. Although there are many intermediate grades determined through Lane’s Method, there is ultimately only one output cut-off grade. This grade is known as the optimum cut-off grade and it is used to maximize present value. [3] The optimum cut-off grade gives the advantage of considering discount rates and process capacities, however, these inputs may change. In the early stages of the design process the production rate and concentrator capacity have not yet been determined, therefore Lane's method is most accurate and effective during the production (operational) phase of the project.


Parameters
y Recovery
C Concentrator capacity
R Refining capacity
f Fixed costs
s Sale price
c Concentrator costs
r Refining Costs
V Present value
d Discount rate

The grades; gm, gc, and gr represent the milling, concentrating and refining cut-off grades respectively.

\mathbf{g_m=c/(y(s-r))}

\mathbf{g_c=c+(f+dV)/C)/(y(s-r)}

\mathbf{g_r=c/(y(s-r-(f+dV)/R))}


Using the parameters:

Qm Mine at capacity
Qc Concentrator Feed
Qr Refinery Product

The following intermediate grades relate each of the previous grades:


\mathbf{g_mc = f(Qm,Qc,gradedistribution)}

\mathbf{g_rc = f(Qr,Qc,gradedistribution)}

\mathbf{g_mr = f(Qm,Qr,gradedistribution)}


The values for each can be found through the following steps:

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1. Mine at capacity and concentrator cut-off grade are used to find concentrator feed. 2. Concentrator feed and concentrator cut-off grade are used to find refinery product. 3. Mine at capacity and mine cut-off are used to find refinery product


[4]

Using all of the intermediate cut-off grades, the optimum cut-off grade can be determined.

Gmc.png
Grc.png
Gmr.png


The median value of Gmc,Grc, and Gmr is the optimum cut-off grade.

Polymetallic Deposits

Polymetallic deposits are orebodies which produce or are capable of producing multiple metals. To calculate the cut-off grade of a polymetallic deposit, an equivalent cut-off grade method is used. This method uses the gross recoverable value (GRV) of each individual metal as well as an equivalent grade factor, which is a ratio of the GRV of each metal to the GRV of the metal being used for the equivalent grade. GRV is calculated by taking the product of reasonable metal price, in $/weight, and the estimated recovery of the metal.

The following formula can be used to classify the equivalent grade for a mining unit:


\mathbf{G_e = g_1 + f_2g_2 + f_3g_3 +…}


Where Ge is equivalent grade of a specified metal (in either g/t or %, depending on the metal), fn is the ratio of the grade of metal n to the grade of the specified metal, and gn is the grade of grad of metal n.

Equivalent grade is necessary for polymetallic deposits in which the value of a block may be influenced by more than one valuable mineral. It simplifies the value of the block so that cut-off grade analysis can be performed, and a decision of whether or not to mine the block can be made. [5]

Mineral Classification

Cut-off grades are used to determine the point at which material becomes economically valuable. Therefore, they are necessary to define the amount of ore in a deposit. The classification of ore depends entirely on cut-off grade. Increasing the cut-off grade places a stricter requirement on what can be considered ore. This decreases the amount of material that can be considered ore, and with it the reserves and life of mine.

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This table shows the impact of cut-off grade on mineral reserves and average grade. Note the changes in tonnage and grade as the cut-off grade changes.


Grade-Tonnage Curves

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An example of a grade tonnage curve

Grade-Tonnage curves are a visual representation of the impact of cut-off grades on mineral reserves. The Grade Tonnage Curves display the tonnage above the cut-off grade and average grade of a deposit relative to cut-off grade. As the criteria for ore classification becomes more selective, the tonnage above the cut-off grade of the deposit decreases. Conversely, as the cut-off grade is lowered, the tonnage of the deposit increases. This is simply because the standard used to distinguish between ore and waste has become less selective. As the cut-off grade increases, so too does the average grade of the ore mined. The curves ultimately show how the average grade and tonnage of a material delivered to a certain process are dependent on the cut-off grade selected. The revenue generated by sales can be calculated by using the following equation:

\mathbf{Revenue=T_+c*x_+c*r*V}

Where T+c is the tonnage and x+c is the average grade of the material above the cut-off grade, r is the ratio of valuable product recovered during processing, and V is the market value of the material being sold. The tonnage of the mined material that will not be processed is also determined by the cut-off grade selected. Grade-tonnage curves are applicable throughout various stages of deposit evaluation. During exploration, for example, they can be a significant tool used to estimate the general size of a resource in tons of material of metal, using exploration data to generate the estimates. [6] The grade and tonnage data used to create the curves are compiled with several assumptions, which should be taken into consideration when using the curves. Such assumptions are that the deposit is correctly classified, the data represents the complete in situ resource, and that the data represents a single deposit or several small deposits that are considered to be a single deposit. Certain sources of error that can be prevalent when creating the curves include mixed geological environments, which may also be poorly known, the use of multiple mining methods, and incomplete estimates/errors in data recording. [7] A significant issue with relying on the grade-tonnage curves is that the continuity of the geology is not taken into consideration, ie. the constructed curves do not discriminate the data based on where in the mineralized section it was recorded. This could lead to issues related to the grade distribution of the material that is mined. [8]


Adjusting the Cut-Off Grade

The cut-off grade determined in the design phase of a project may become outdated as production continues. It is therefore important to reevaluate the cut-off grade over time. However, the mine plan should not be completely altered for short term gain. Tactical short term objectives must not supersede strategic long term objectives for operations to be successful in the long term.[9] If the metal price increases, it may seem intuitive to lower the cut-off grade since the low grade material can be mined at a profit. However, the better decision is often to increase the cut-off grade instead. Processing the higher grade material will yield an increase in the amount of valuable mineral produced. This is assuming that the processing capacity does not change; only the ore being processed does. The extra product can then be sold at this increased price. There may also be several objectives involved the selection of cut-off grade. A Hill of Values can be used to find the optimal grade in this case.

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A Hill of Value is used to evaluate cut-off strategies with regard to multiple objectives


Interprocess Options

In the case of multiple processing options, cut-off grade is used to determine which process or destination the ore will be sent to, by using the following equation:

\mathbf{U_1(x_c) = U_2(x_c)}

Where U1 is the utility of sending ore to process 1, and U2 is the utility of sending ore to process 2. Utility is defined as the value (economic or otherwise) of selecting a process. This means that if U1(x) is greater than U2(x) when x is greater than the cut-off grade (xc), then that material with x greater than xc should be sent to process 1 as it will be financially beneficial. When the cut-off grade is estimated, all variables that might affect the outcome must be controlled, and the following equation is used to facilitate this:

\mathbf{U(x) = U_dir(x_c)+U_opp(x)+U_oth(x)}

Where U(x) is utility of sending a material of grade x to a specific process, Udir(x) is the profit or loss incurred from processing a specific amount of ore at a certain process, Uopp(x) represents the opportunity cost of adding a unit of a material of a specific grade to the process, and Uoth(x) represents unquantifiable factors that must be considered in determining cut-off grade.

Stockpiling

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Stockpiling can be used to delay the processing of ore

Stockpiling is the storage of unprocessed ore, which can be sent to the mill for processing at a later time. Typical scenarios when stockpiling is utilised are when an abundance of high grade material is mined early in the mine life, or when lower grade material is mined and it would be more financially beneficial to sell the low grade material at a later time when the sale price of the material would be potentially higher. Generally speaking, stockpiling is used to level out the grade distribution of the material that is processed over the life of the mine. Stockpiling can have a number of benefits, including an increase in the useful life of a processing facility and balancing the financial return throughout the mine life. Alternatively, stockpiling might also have negative impacts, which potentially include heightened environmental risks, reduced metallurgical recovery, and extra costs associated with re-handling the material. [10]


References

  1. Rendu, JM. An Introduction to Cut-Off Grade Estimation
  2. Minnitt, R. Cut-off grade determination for the maximum value of a small Wits-type gold mining operation (2004)
  3. Thorley, Ursula. MINE 341 Lecture 10: Cutoff Grade (2013)
  4. Lane, Ken. Commercial Aspects of Choosing Cutoff Grades (1979)
  5. Thorley, Ursula. MINE 341 Lecture 10: Cutoff Grade (2013)
  6. Silva, F. Soares, A. Grade Tonnage Curve: How Far Can It Be Relied Upon?
  7. Grunsky, E. GRADE AND TONNAGE DATA FOR BRITISH COLUMBIA MINERAL DEPOSIT MODELS, BC Ministry of Energy and Mines (1995)
  8. Sinclair, Alastair. Applied Mineral Inventory Estimation (2002)
  9. Hall, B. "Short-Term Gain for Long-Term Pain” – How Focussing on Tactical Issues can Destroy Long-Term Value (2006)
  10. Rendu, JM. An Introduction to Cut-Off Grade Estimation
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