What do you take away from this
Very different patterns, yet may share the same variogram (“yet same statistics up to order 2”)
In Multiple-point geostatistics, there is a possible shortfall, what could it be and how can it be solved
- What if a lot of data and NOT relate to the TI?
- Applications with relatively ‘rich’ data sets?
Seems like cumulants can solve this problem.
What are cumulants, what are some characteristics?
They refer to a set of quantities that describe aspects of a random variable’s probability distribution.
Characteristics:
* For a non-Gaussian process, cumulants provide a measure of non-Gaussianity
* For linear process, cumulants may be expressed as higher-order correlations
* Cumulants well defined mathematical objects, that take different form depending on their order
What is a first order cumulant?
The mean
What is a second order cumulant?
A variogram
What are third- to fifth-order spatial cumulants ?
Basically, acquires 3points to 5 points statistics.
What do you conclude from this
Too little nodes create pixelated cumulants
Comment on this
*Red border = ends of high values in the maps
*The borders reflect the shape of the pipe along x and y
*The top part is better detected than the bottom because of drilling density (~300m)
*The same main features and differences between these cumulants maps are presented in the TI cumulant maps, showing how satisfactory is the generation of the cumulant maps using the DDH for this case study
What are Legendre polynomials?
Legendre polynomials are a sequence of orthogonal polynomials that solve Legendre’s differential equation.
What are Legendre cumulants
Legendre cumulants are related to the moments of a probability distribution, transformed via Legendre polynomials.
What mainly affects the quality of the HOSIM realisations?
- Data density (25x25, 50x50)
- Training image
Some HOSIM conclusions?
- Uses no- preprocessing
- Generates complex spatial patterns
- Reproduces any data distribution, high-order spatial cumulants of data
- Data driven (not training image driven)
- Reconstructs the lower-order spatial complexity in data
- Yes, high-order simulations matter to problem solving
Give some background to the Gold deposit case
A critical aspect to consider is that mineral deposits are characterized by spatially complex, non-Gaussian geological properties and multiple-point connectivity of high-grades, features that are not captured by conventional second-order simulation methods.
What did they do at the gold deposit case study?
This paper investigates the benefits of simultaneously optimizing a mining complex where the simulations of the mineral deposit are generated by a high-order, direct-block simulation approach. The optimized life-of-mine (LOM) production schedule is compared to a case in which the same setting is optimized by having the related simulations generated using a second-order simulation method.
Describe the gold deposit (case study). give the # of drill holes, the spacing of drillholes, the training image, the block size.
2,300 drillholes; 35 m x 35 m
Training Image; Based on blasthole data
Block size of 10x10x10 m3 ; 500,000 blocks
Takeaways from this (gold deposit case)
Although the metal quantity is very comparable in both cases, how each method connects these elements in space can be very different, especially at the high-grade values.
Takeaways from this (gold deposit case)
- It is possible to visualize the effect of the maximum entropy property over the second-order simulations, which is enhanced by the fact that the simulation process was performed in the Gaussian space.
- The grades displayed by the simulations generated with SGS are visually more dispersed than those generated using high-order simulations.
Comment on the connectivity graphs from the gold deposit.
- The cut-off applied is 5 g/ton, the focus of the comparison is on the high-grades
- In the NE direction, the second-order realizations are consistently less connected than the high-order realizations for all lags.
- The difference becomes more pronounced in the NE direction and at the 45º dip, with a considerable gap evident between both simulation methods.
- As the high-grade mineralization drives the mine production schedule, this plays a vital role in the optimization of the mining complex.
Comment on these (gold deposit case)
- HOSIM schedule show that the sequence of extraction follows a clear direction [red arrow].
- Note that this trend in the extraction sequence is amplified in the direction where the difference in connectivity is more evident [NE/45°].
- The higher continuity of high grades drives the schedule towards areas with more connected ore materials so that they can be processed together.
Comment on these (gold deposit case)
- These sections show variations in the extension of the ultimate pit limits (UPL).
- The red circles highlight how much larger the UPL for SGS.
o Since, second-order simulations methods represent high-grade material as being more scattered.
o So, pits must be larger to encompass all of the ore to be processed, resulting in a higher waste extraction.
After the optimization is complete, infrastructure, such as ramps, access points, equipment placement and facility locations, reduces ___________________. This optimization opportunity can only be achieved if _____________________________________.
Reduces the flexibility to change the schedule. This optimization opportunity can only be achieved if the degree of connectivity of high grades is correctly modeled.
Takeaways (gold deposit)
- 5% more material mined for HOSIM than in SGS. This difference reaches 8% at the end of the 10th year to ensure a similar throughput at the mill.
- Mining more, in this case, translates to higher waste production, which is quantified by the higher strip ratios.
- This can be explained by the spatial disorder (maximum entropy) that Gaussian-based approaches generate with respect to high-grades.
- Because the SGS realisations results in ore blocks less connected, the optimizer must mine more dispersed mining blocks to provide a consistent feed rate to the mill. This also leads to different total tonnages of materials mined.
- While from HOSIM realisations, the optimizer receives more realistic information regarding spatial grade connectivity. Therefore, the LOM production schedule of HOSIM pursues high grades more efficiently.
Takeaways (gold deposit)
- As the optimizer encounters better-connected zones of high-grade, it can bring their extraction to the same period so they can be processed together. This increases the average feed grade at the mill and recovering more ounces earlier.
- HOSIM shows an ounce’s profile that is consistently higher for the first 17 years; this difference reaches 7% after the 10th year.
- SGS produces, after the 20th year, 2% more gold, but this is not significant due to the effect of discounting and the time value of money.
- Recovering more ounces sooner brings more cash flow earlier to the operation, which positively impacts the net present value (NPV).
- Summing up the joint effects of correctly meeting production targets, mining less waste and producing more gold earlier results in a considerable increase in NPV.
- Overall, by producing more metal and less waste, the LOM production schedule obtained HOSIM generates in a total of 5% higher NPV than SGS, and 16% higher in the initial ten years. The difference is substantial and improves financial returns at the early stages of the development of the mine.
Conclusion and future work for the gold deposit
- High-order simulations are able to provide more realistic connectivity of high-grades
- 5 to 16% higher project value
- More informed mine production schedules can be obtained through high-order simulations
- Future work: Extend the block support framework to integrate correlated elements
