simulations

In addition to simulating grade, categorical variables such as lithology, alteration, or mineralization can also be simulated.  In reality, most simulations of ore deposits are a combination of geologic and grade simulations.  For example, the deposit discussed in the simulation of NPV is actually the combination of a simulation of estimation unit (a combination of lithology, alteration, and mineralogy) and grade.  In total there were 5 estimation units.  To create the final simulation, the grade of each unit was simulated and an indicator simulation of estimation unit was used to determine which of the 5 possible grades should be assigned at each simulation node.   

Creating an indicator simulation that provides a reasonable representation of the deposit geology can be a difficult task.  Of course it is important to correctly model the indicator variogram associated with the unit and to capture any spatial anisotropies.  However even if this step is performed correctly, it can be difficult develop a simulation that is both believable and representative of the deposit geology.  

We have found that a few simple steps can be taken to improve the quality of indicator simulations.  First, in the fringes of the deposit where the geologic unit is known, add extra data to help control the simulation.  These fake data are only added in areas that are more than one search radius away from any known geologic contact.  Without these additional data the simulation will occasionally introduce nonsense values.  Second when using a program such as SISIM from GSLIB, always move the data points to the nearest simulation node.  This ensures that the data will be respected.  Finally when creating the simulation use ordinary rather than simple Kriging.  In cases where the nugget effect is important and simple Kriging is used, the simulation will introduce points that are not sensible.  Use of ordinary Kriging restricts the simulation to the data in the search neighborhood and therefore better reproduces the properties of the deposit near the simulation node.

The following two images show simulations created when the data are and are not moved to the simulation nodes.  It is clear that moving the data greatly improves the quality of the simulation.

The conditional simulation technique provides a method of generating multiple equiprobable representations of the spatial distribution of geology and/or grade.  Each representation or realization is conditioned to the available data and reproduces the model of spatial correlation (the variogram model) and the statistical distribution of the grades.  By examining the differences among the various realizations of the deposit, detailed evaluations of uncertainty can be developed.  Furthermore the reduction in uncertainty associated with additional drilling can be predicted.

The sequence of images to the right show five realizations of block grades.  When the simulation was performed, the project was at a fairly early stage and significant additional drilling was required to reduce the uncertainty in the resource estimate.  To asses whether the planned drilling added value to the project,, the drill program was simulated (the planned drillholes passed through a detailed simulation) and a new simulation of the deposit was performed.

Each of the simulations was evaluated against the proposed mining plan to define the project NPV.  Since each simulation was composed of 30 realizations, an estimate of the project NPV distribution before and after the drilling was defined.  These two distributions of NPV are shown on a normal probability plot in the adjacent figure.   The dashed line shows the project NPV distribution at a 10% discount rate given the existing data.  When the new data are incorporated, the NPV distribution moves to the blue line.  As shown the blue line is shifted to the left and variability is much reduced.  The line shifts to the left by an amount equal to the cost of the additional drilling.  The benefit of this additional drilling is the reduction in variance or increase in certainty of the ultimate project NPV.  

These curves show that it is possible to spend too much on drilling.  As the amount spent on drilling increases, the blue line moves further to the left and becomes steeper.  At the limit, the project NPV becomes perfectly known, but the project has no value.  Clearly it is preferable to accept some risk rather than destroying the value of the project by over drilling.  The question then becomes how should the risk be evaluated.  

One way to look at risk is through the risk premium concept.  If an investor can invest in two projects, he will only invest in the riskier project if there is the possibility of a higher return.  The difference in the expected rate of return is the risk premium.  The risk premium can be assigned in this example by setting the p10s equal for the 2 projects.  The p10 is the NPV value for which 10% of the projects are less and 90% are greater.  In the above graph it is seen that the p10 for the project with the additional drilling (blue line) is greater than the NPV for the current project (dashed line).  The NPV distribution is strongly a function of the discount (interest) rate.  Through trial and error it is seen that if the additional drilling is not performed a 13.6% discount rate is required to set the two projects equal.  The risk premium associated with not performing the drilling is therefore 3.6%.  Evaluating the project with a 13.6% discount rate strongly reduces the project NPV and shows that the additional drilling is money well spent.