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Article written by Sci '14: Travis Dominski, Brett Kolankowski, Andrew Marck, Michal Pasternak, Steve Shamba

Production rate and mine life, play a large role in determining the Project economics. A higher production rate typically allows for lower operating costs, while the subsequent shorter mine life maximizes the Net Present Value of ore extraction. However, a higher production rate requires a greater capital cost, as larger equipment and infrastructure is required. Estimation of production rate is a problem that has been looked at by many scholars. The most well-known scholar to look at the problem was H. K. Taylor who developed the empirical Taylor’s Rule, a rule of thumb that is commonly taught to Mining Engineering students. While the most popular, Taylor’s Rule is not the only method that can be used when estimating production rate. Taylor’s Rule only takes into account tonnage, while other methods use the grade of the ore and financial factors.

Taylor’s Rule (1986)

Taylor's Rule of Thumb for Mine Life http://i152.photobucket.com/albums/s189/brettkol/TaylorsRule-1.jpg

H. K. Taylor has authored several papers looking production rate and mine life. The method he suggested now commonly known as Taylor’s Rule is based on a survey of operating and proposed North American open pit copper porphyry mines. Taylor realized that there was an empirical relationship between mine life and expected tonnage.

Equation for Mine Life and Production Rate

The empirical equation for mine life that Taylor developed is:

$Life=0.2{\sqrt[{4}]{Tonnage}}$

The equation can be used to find the production rate by

$Production(mt/day)={Tonnage \over MineLife\times OperatingDays}$

Assuming a mine operating 350 days a year, Taylor's rule gives the equation

$Production(mt/day)=0.0143\times Tonnage^{(0.75)}$

USBM/USGS Modifications

Modifications of Taylor's Rule http://i152.photobucket.com/albums/s189/brettkol/TaylorsREvisedRule.jpg

Taylor's rule has been modified and tweaked by the United States Bureau of Mines (USBM) and its successor the United States Geological Survey (USGS) to a large and more modern set of data. All the modicications of Taylor's rule use the same general relationship, and just revise the variables.

D.A. Singer, W.D. Menzie and K.R. Long revised Taylor's rule in 1998, based on a data set of 41 open pit gold-silver mines. Their model found that appropriate rates should for open pit gold-silver mines should be significantly higher than Taylor's Rule suggests. Their resulting equation was:

$Production(mt/day)=0.416\times Tonnage^{(0.5874)}$

D.A. Singer, W.D. Menzie and K.R. Long also adapted Taylor's rule for underground massive sulfide deposits in 2000. In this modification it clear that the Taylor's rule overestimates the underground mining rate as it was calibrated to open pit mines. The resulting equation for underground massive sulfide deposits is:

$Production(mt/day)=0.0248\times Tonnage^{(0.704)}$

Long and Singer further studies Taylor's rule in 2001 and calibrated it to 45 open pit copper mines. The open pit copper model proved to have a curve halfway between the 1998 gold-silver curve and Taylor's Rule. Since the mines are the same type as those used for Taylor's rule, it can be seen that the realistic production rate has increased in the decades since Taylor's rule was first developed.

$Production(mt/day)=0.0236\times Tonnage^{(0.74)}$

The latest study on Taylor's rule was completed by Long in 2009. Long's study is the most extensive of all studies looking at the relationship between Capacity and Reserve. The study looked at 342 open pit and 197 underground mines located in the Americas and Australia. Long found that there was a significant difference between the production rate of underground vs. open pit and block caving. The equation found for underground deposits was found to be:

$Production(mt/day)=0.297\times Tonnage^{(0.562)}$

The equation for open pit and block caving deposits was found to be:

$Production(mt/day)=0.123\times Tonnage^{(0.649)}$

Long's 2009 study also found that introducing the variables grade and capital cost played a factor in estimating production rate, however expected tonnage was the primary factor. Long did generate equations involving grade and capital cost for open pit, however the inputs for these equations were not clarified.

Applicability

Many scoping studies use the original Taylor's Rule as a starting point for production rate regardless of the type of mine. It is clear from the USBM/USGS modifications of Taylor's Rule that a better estimate is possible adding the variable of mine type as well as expected tonnage. The table below acts as a guide to selecting an appropriate version of Taylor's rule. The general equation for Taylor's rule is:

$Production(mt/day)=a\times Tonnage^{(b)}$

Mine Type	a	b	Source	# Mines
Unknown	0.0143	0.75	Taylor (1986)	~30
Open Pit - Gold/Silver	0.416	0.5874	Singer, Menzie, Long (1998)	41
Open Pit - Copper	0.0236	0.74	Singer, Long (2001)	45
Open Pit/Block Caving - Other	0.123	0.649	Long (2009)	342
Underground - Massive Sulfide	0.0248	0.704	Singer, Menzie, Long (2000)	28
Underground - Other	0.297	0.562	Long (2009)	197

Other Methods

Wells (1978)

In 1978, H.M. Wells published the paper “Optimization of mining engineering design in mineral valuation”, which proposed maximizing the present value ratio (PVR) in order to find the optimal production rate. PVR was the ratio of PVOUT (the present value of positive cash flows) to PVIN (the present value of negative cash flows). A PVR greater than 1 represented a profitable production rate while a PVR less than 1 was an unprofitable production rate. The optimal production rate is the rate that causes the PVR to be at its maximum.

Lizotte and Elbrond (1982)

Y. Lizotte and J. Elbrond researched optimization of production rates in 1982. They approached the problem using open-ended dynamic programming and created a model for the problem. However they concluded that there were vast difference between their model and realistic mining.

Cavender (1992)

B. Cavender took a theoretical approach to determining appropriate mine life looking at the finance side of mining. He developed three techniques for finding the mine life that optimized the NPV of the project. Cavender looked at cash flow, stochastic risk modeling, and option pricing. Since the model deals with a hypothetical mining and does not include realistic mining constraints, it has little application to real mine design.

Smith (1997)

L.D. Smith in 1997 found that estimating a mines production rate was better determined by a range than a specific point. Smith's paper proposed that an appropriate range of production rates with a upper limit as the rate that resulted in the highest NPV. The lower limit of this range was found to be the rate that best repaid capital costs.

Abdel Sabour (2002)

Effects of various parameters on optimal production rate (Abdel Sabour) http://i152.photobucket.com/albums/s189/brettkol/OptProductionRate.jpg

Using a mathematical model, S.A. Abdel Sabour looked at the effect of various physical, economic and financial factors on the optimal production rate. For the physical factors it was found that the optimal production rate increases with both the tonnage and grade of the deposit. A higher gold price resulted in a higher production rate. The production rate also depended on the expected growth rate of gold prices, with a higher growth rate resulting in a lower production rate. If mining cost growth rate is expected to be high, the optimal production rate is should also be high to avoid higher mining costs in later years. The final factor considered was the cost of capital (discount rate). It was found that for a low (~5%) and high(~35%) cost of capital, the production rate should be low, however between these points of cost and capital, the production rate should be higher.

Abdel Sabour's definition of the optimal production rate is the rate that generates the highest NPV. To obtain the optimal rate this microeconomic theory was applied. Since these optimal production rates look at economic theory rather than engineering design constrains, his work is not useful on it's own to estimate production rate. It can however can be used to tweak the results of other production rate estimates, such as Taylor's rule.

Summary

Taylor's rule is the best way get a preliminary estimate of the production rate and the mine life during mine design. This is due to its simplicity of calculation since it involves only one variable, as well as the real world applicability of the rule since it is built upon real world data. Modifications by the USBM/USGS should be considered when using Taylor's rule, as they tweak it to better suit the type of mine.