. Classical Ore Reserves Methodsss...
GEOS GEOSTA TATI TIST STIC ICS S CO COUR URSE SE
CLASSICAL ORE RESERVES ESTIMATION METHODS
Dr. Arifudin Idrus Department of Geological Engineering Gadjah Mada University E-mail:
[email protected]
Definition of terms
Global reserves refers to the mean grade of the reserves to be mined over the life time of the mine. Local reserves refers to the mean grade of reserves to be mined over short time increaments eg. year by year. Local ore reserves are used to produce the mining schedule.
Definition of terms
Global reserves refers to the mean grade of the reserves to be mined over the life time of the mine. Local reserves refers to the mean grade of reserves to be mined over short time increaments eg. year by year. Local ore reserves are used to produce the mining schedule.
Definition of terms
Resources usually based on geological interpretation only. Mining parameters have not (or only partly) applied. Reserves incorporate all aspect of the impact of mining on the geological interpretation, interpretation, especially ore loss and dilution.
Requirements of an ore reserve method On th the e ph phiilos oso oph phiica call sid ide e
they must be consistent with the nature of the orebody being evaluated. i.e., they must be geol ge olog ogyc ycal ally ly co cont ntro roll lled ed and must follow on form the geo og ca n er erpre a on. ey mus no c a e e geology. Computational methods are not geological predictors. they should be simple in that most people associated with the project should be sufficiently comfortable with the methodology.
On the philosophical side
They should be robust. The method should yield the correct answer over a wide renge of data and should not be such that small changes in the data can yield . They should be understandable. If the method cannot understand what is happening then this can create uneasiness with the method and result in uncertainly. They should de defendable when confronted by peers. Or reserve calculation must be defendable with conviction.
On the philosophical side
They should be consistent with the data density. If only limited data is available the method should reflect this. They must reflect themining method realistically. “It is only in comics and films that superman mines ore o es Most orebodies are mined by mortals. The assumptions used shouldbe clearly stated.
On the practical side the computations should be: 1. Rapid 2. Reliable 3. Easily checked The reliability of reserve computations depends on:
The accuracy and completeness of our knowledge of the orebody under study. The data density and reliability of the base data. The assumptions for interpreting the variables under study. The relevance of the mathematical methodology used.
Selecting an Ore Reserve Method depends on:
The The The The
geology of the mineral deposit. densit of data. purpose of computations. degree of accuracy required.
At the mine design stage More complete calculations are required. (Global Estimates)
More complete calculations are required. (Local Estimates)
Types of Ore Reserve Method
Classical method. Geostatistical method.
CLASSICAL ORE RESERVE METHODS In carrying out classical ore reserve calculations we work from the highest data density to the least data density. Consider an orebody drilled on section. The sectional data is densest. The interpretation should be carried out on section first. All methods are based on computing solids with their bases in the plane.
For sectional data (and vertical longitudinal sections)
Interpret data in the plane of the section.
Measure the area in the plane.
Use a horizontal distance for extending to a volume.
For plan data 1. Interpret data in he plane of the plan. 2. Measure are in plan. 3. Use a vertical distance for extendig to a volume. Often the vertical distance is the bench height. Often ore reserves are calculated by generating longitudinal sections in the plane of the dip of the mineral body.
For inclined longitudinal sections
interpret data in the plane of the sections. use true thickness for he inclined plane.
Generally in reserve calculations the true strike and true If data in not presented with true dip and strike it needs to be corrected to allow for this. This is achieved by carrying out geometrical corrections. All ore reserve methods involve. 1. Weighting data (ie. Weighting assay data). 2. Extending data to obtain volume.
VOLUME CALCULATONS
1. Trapezoidal rules The trapezoidal rule – assumes the area consists of a sequence of trapezoids Area =
1
2
2
+
2
3
2
+•••
Where the area of a simple trapezoid is
S =
(a + b )h 2
a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
h
2. Simpsons rule Assumes the boundaries of each strip are best represented by parabolas passing through consecutive points. Area =
1 3
(
∑
h a1 + 2
odd
+ 4∑even + an )
Volume calculations Volume = Area x thickness.
Tonnage Calculations Tonnes = area x thickness x tonnage factors.
This is an improtant factor in the calculation of tonnage.
WEIGHTING OF DATA This is done by various methods including:Simple arithmetic methods. Weighting by width or thickness, length, area, specific gravity.
Some weighting examples to calculate the mean grade. 1.
Simple Arithmetic Methods mean =
g 1 + g 2 + g 3 + ••• g n n
Assumption all blocks are equal. 2. Thickness weighted All blocks are equal in area and have the same SG. mean = t g + t g + t g + ...t g 1
1
2
2
3
3
t 1 + t 2 + t 3 + ...t n
n
n
3. Area weighted All blocks have constant thickness and weight factor but different area. mean = A1 g 1 + A2 g 2 + A3 g 3 + ... An g n
A1 + A2 + A3 + ... An 4. Volume weighted The assumption here is that all blocks have SG. mean =
V 1 g 1 + V 2 g 2 + V 3 g 3 + ...V n g n V 1 + V 2 + V 3 + V n
5. Tonnage weighted the assumption is that the tonnage and grade of blocks are different. mean = g + 2 g 2 + 3 g 3 + ... n g n 1 1
T 1 + T 2 + T 3 + T n
CASE 1A An averaging method (simplest case) Consider a plan view of 14 drillholes
4
5 9
13
6
10
7 11
8
12
14
Each drillholes has a thickness ti and a grade gi
Drillhole
Thickness
Grade
1
t1
g1
2
t2
g2
3
t3
g3
Measure the area A
4
t4
g4
Then tonnes = t A SG
.
.
.
grade = G
.
.
.
.
.
.
14
t14
g14
t =
∑ t i
14
G=
This method is accurate in uniform deposits, where there is a very small difference in thickness.
∑ g
i
14
In this calculation thickness is not considered important
CASE 1B
Consider the previous example of the 14 drillholes. In the folowing calculation it is thickness varies from point to point.
An average method (thickness weighting).
Drillhole
Thickness
Grade
Product
1
t1
g1
t1g1
2
t2
g2
t2g2
3
t3
g3
t3g3
4
g4
4g
.
.
.
.
.
.
.
.
.
.
.
.
14
t14
g14
t14g14
Σti
Σtigi
Average grade
t g ∑ G = ∑ t i
i
i
Tonnes are calculated by using local thickness. If the blocks are all the same area, and only the thickness changes, then
Tonnes = A [t1 + t2 + … t14] SG Example p 9-10
A PLAN METHOD USING POLYGONS
Area of influence polygon boundary Drillhole Ore zone limit Line segment between drillholes (construction aid)
If the polygons have different tonnages. Then use a tonnage weighted method.
CROSS SECTIONAL METHODS The orebody is devided into geological section along the lines of drilling. Two methods are used. . A step change. In plan a. Gradual change
b. Step change The calculation of volumes may use the following formulae. (1) End Area Formula
A1 + A2 L 2
V =
A1 + A2 V = L For several sections
L 2
V = A1 + 2 A2 + 2 A3 + ... An (2) Wedge Formula – where one end tapers to a line.
V =
A 2
(3) Cone formula – where where one and tapers to a cone
V = L
A 3
L
(4) The frustum formula
Note The frustum formula is inaccurate in wedge like orebodies
V =
L 3
( A + A 1
2
+ A1 A2 )
L
1
A2
(5) The prismoidal formula
L
V = ( A1 + 4 Am + A2 )
6
This is better for ore bodies which pinch and swell. Am = mean area between section i.e., auxiliary. Sections need to be constructed.
COUNTOURING METHODS (Isoline methods) Contours are curved lines which join all points of equal value. Data is used to construct contours by interpolation between point of known values. Various techniques of interpolating data may be used. (Specific techniques of interpolation are discussed later).
Contouring method As an example of the interpolation technique consider the method of finding the volume of the following:-
Section
The volume is calculated by measurin each area within the contour interval and using volume calculation procedures previously discussed. The average ore grade can be computed by constructing contour maps and by weighting each area by its contour grade.
plan
Contouring method To achieve this use
g 0 + g =
g A + A + A + A 2 2 ... 0 n 1 2 2
where go is the minimum grade of the ore g is the constant grade interval between contours * Ao is the area of the body with grade go plus g and * higher A1 area of orebody with grade go plus g and higher * * A2 area of the orebody with grade go plus 2g and higher etc
Contouring method The method reqiures data which has 1. A sufficient number of dat. 2. Appropriate data density 3. Appropriate distribution of data. When data is unvenly distributed there can be problems. (These problems will be discussed later). The map produced shows the areas of rich and poor ore.
Contouring method As an example consider the data
G =
g 0 a0 +
g 2
( A + 2 A + 2( A + A ) + ( A + A )) 0
1
A0
g3 A3
A2 1
21
A3
1
A2
g3 2
g2
2
g1 A0 g0
22
31
32
Contouring method The method of contouring should be used only in deposits of orderly changing thickness and grade. It is not useful in very complex, discontinous orebodies. It is partucularly useful in orebodies where thickness and grade decrease from the centre to the periphery
THE METHOD OF POLYGONS In this method all factors determined for a certain point of a minerals body extend half way to the adjoining and surrounding points forming an area of influence. 1. For regular drillholes
THE METHOD OF POLYGONS 2. Stagged drillholes (face centered)
A note on the use of the FRUSTUM FORMULA A1 = area M1 = metal = grade x area A2 = area M2 = metal = grade x area The volume is V =
L
( A + A 1
+ A1 A2 )
2
V =
The metal M =
1 3
( M + M
Then
1
For the prismoidal method
2
6
+ M 1 M 2 )
M = G =
m v
G =
1 6
( A + 4 A + A ) 1
2
3
( M + M + M )
M V
Note: the formula are only approximate
1
2
3
EXAMPLE OF POLYGONAL ORE RESERVES
The area of influence method of calculating ore reserves is as following: 1.
Difine for each drillhole, a boundary enclosing the area closest to that drillhole. This is done by constructing lines which are to the line segment between the two drillhole location points.
2.
Each area so defined is treated as a polygon of constant grade and thickness, ie. The grade and thickness of the single drillhole inside the polygon.
3.
The reserves are determined simply by adding the tonnes and metal derived for each polygon.
This method is ussualy applied on a plan basis as shown below:
Area of influence Polygon boundary Drillhole Ore zone limit Line segment between Drillholes (construction aid)
EXAMPLE OF POLYGONAL ORE RESERVE
In some instances, the same method may be apllied on a cross Sectional basis. Each polygon is assigned the average grade Of samples inside the polygon. The polygon thickness is the Cross-section width (ie. Mid-way to adjoining section).
Ore zona limit
Area of influence Polygon boundary Drillhole
INVERSE DISTANCED METHODS Consider the problem of estimating the grade of a block from the surrounding data. Eg. Data
INVERSE DISTANCED METHODS Ore method of solving this problem is to use a method based on the distance of the samples from the block. The most common distance weighted methods are:1.Inverse distance. 2.Inverse distance squared. 3.Inverse distance cubed. The following examples show the application of these techniques.
The general formulae Inverse Distance 1
G =
d 1
g 1 + 1
1
d 2
g 2 + ...
1
+
1
+ ...
1
d n
Inverse Distance cubed
g n
1
1
G=
n
2
G=
1
1
g 1 + 2 g 2 + ... 2 g n d 12 d 2 d n 1
d 12
+
1
d 22
+ ...
1
d 13
Inverse Distance Squared 1
1
1
d n2
1
g 1 + 3 g 2 + ... 3 g n d 13 d 2 d n
+
1
d 23
+ ...
1
d n3
INVERSE DISTANCED METHODS
Estimation of block grade:
n
Z V = *
− r
Z ( xi )λ i
Where:
λi =
d i n
−
d i
n =1 i= 1
n
∑ λi= 1 i= 1
Example-Inverse distanced square (IDS)
0,64% 0,48%
d3=66m d4=78m
0,69% d2=52m d1=32m
d5=92m
0,43%
d6=64m 0,75%
0,53%
