Designing an effective drilling plan using an optimisation algorithm

Often the first budget item to be cut in times of austerity is exploration drilling. We will not go into the wisdom or consequences of this decision, but rather look at a means to optimise the money available for ongoing drilling. Most geologists who have had to do drill hole layouts will have come up with any number of tools or process automations through scripts or macros that help them design a reasonable drill plan in an acceptable time frame. One such approach is described in this article.

## Drilling

Drilling costs are generally broken down into mobilisation, set-up, unit drilling costs (per metre), logging, sampling, sample transport and assaying and storage. Logging, sampling and the other downstream activities tend to be set at fixed prices unless contractors and ad hoc assay contracts are being used. The biggest costs are therefore the drill collar set-up (pad or drilling station construction) and the drilling itself.

As such, if you have a constrained drilling budget then a balance between the number of drill pads and set-ups, as well as the drill metres required to achieve the campaign objectives, needs to be found. Drilling the deposit to achieve an objective specification, such as infill drilling, is typically constructed around:

- targeted drilling to increase geological understanding
- confirming continuity and minimising geological risk through additional data and variations of this theme, or
- increasing and upgrading resources.

Designing a drill plan without a clear goal in mind, or some sort of practical constraint on what can be completed and what is an acceptable outcome, can result in a plan that will just burn money and not achieve the intended purpose.

A manual drill campaign planning approach (even with personal customisations and tools) can be a painful and laborious process. Even with a clear understanding of the drilling objectives, it can be a significant challenge to generate a reasonable drill plan in the timeframe given. Once the object of the drilling has been decided on, such as upgrading an Indicated Resource to Measured, and the parameters that will limit the drilling (maximum hole length, minimum angles of intersection and drill hole spacing), the process of drill planning begins.

## Drill planning

The general approach to drill planning is to ignore the limiting factor of the drill pad or drill bay construction (often you have limited options from the get-go) and to iteratively lay out holes in section and 3D while considering the existing drilling and spacing requirements. This basic approach seems to revolve around working in plan, cross-section or longitudinal (long) section and positioning the pierce points, then iteratively switch back and forth until the drilling layout is formed. An example of this is given in Figure 1, where two phases of extensional drilling have been planned in long section.

The challenges inherent with this process are that it can be time consuming and subject to individual bias. Combined with irregular or complicated drill spacing (Figure 2), this can be a very difficult exercise. Further challenges come from ensuring that the drill plan meets the requirements for maximum length of hole, rig capacities, intersection angles and local and average pierce point spacing.

A manual, iterative process is not able to optimise the entire drill plan. It is limited to a hole-by-hole decision-making process that, due to the time required to complete, cannot be iteratively refined. The manual process, especially for short-term drilling that requires rapid turnaround on the planning, is subject to limited refinement beyond a default layout. This is not to say that the resulting plan is not achievable or effective; it might just not be the optimal use of the available budget or compromise the objective of the program.

## A how-to guide

Consider the situation where the budget is severely constrained, drill set-up positions are limited and there is little time to complete the plan (all of us have been there at some point in our careers).

My initial approach to this situation was to develop a tool that laid out a theoretical intersection grid within the plane of the deposit and then randomly selected traces between the provided drilling positions and the intersections. While this approach was quick, it did not always provide the best solution for a drill plan. The ‘best plan’ being defined as a solution that gave the most spatial coverage for the least number of drill metres and therefore the least expensive with the best return.

An interim output of this initial random drilling selection is shown in Figure 3. In this example, there are nine surface drilling positions and a steeply dipping narrow tabular deposit. Based on the non-ideal minimum intersection angles (45°) between the planned hole and the deposit local orientation, as well as selecting only the shortest holes, the roughly eye-shaped drilling footprints were calculated. It was from these restricted regularly spaced intersection points that the initial drill plan was selected.

This random selection approach results in hundreds of possible drilling configurations that may or may not be the best possible solution. The question then was: how do I, with a limited drilling budget and no time, select the best possible configuration from all these options?

In many respects, this problem resembles the travelling salesman problem (TSP) which is defined as:

*The TSP asks the following question: ‘Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?’ *(Wikipedia, 2016)

We can restate the TSP for drill planning as follows:

*What drilling configuration will give me the shortest set of holes that can be drilled by the equipment available, have the highest possible intersection angle and, on average, give the best infill coverage for the budget available, all the while staying a minimum distance from existing or newly planned drill holes?*

Due to the complexity of the problem it requires either tedious work hours or some sort of algorithm. There are any number of optimisation algorithms out there (hill climbing, genetic algorithms, etc). For this solution, simulated annealing was chosen as the optimisation approach as it was the simplest to implement in the time frame available.

## How optimisation algorithms work

In general, an optimisation algorithm randomly selects an initial solution. It then randomly selects a second solution and if it is better than the first, it switches to the second, and repeats until the algorithm finds a maximum solution. But these can be trapped at local maxima. These solutions are locally good but not the best. This is schematically illustrated in Figure 4.

Simulated annealing avoids this ‘trap’ by adding a random component to the decision to switch.

Relying heavily on the work of Geltman (2014), a simulated annealing algorithm was developed. The basic process was summarised by Geltman (2014) as:

- First, generate a random solution.
- Calculate its cost using some cost function you’ve defined.
- Generate a random neighbouring solution.
- Calculate the new solution’s cost.
- Compare them:

a. If cnew < cold: move to the new solution.

b. If cnew > cold: maybe move to the new solution. - Repeat steps 3-5 above until an acceptable solution is found or you reach some maximum number of iterations.
- When calculating if a solution is better it calculates an acceptability probability (Ap)
*A**p**=e*^*((old−new)/T)*.

Key to the calculation is the use of (T), or the annealing temperature, which is a decrementing value starting at 1 that allows the process to move from a random solution to an optimal or minimised solution.

If the acceptability probability is equal to zero, then the new solution is infinitely worse and is rejected. If the Ap value is equal to one then the new solution is infinitely better and is automatically selected. For values between zero and one, the Ap value is compared to a random value (Rv). If it is greater than the Rv, the new solution is kept, otherwise it is rejected.

In this manner, at decreasing T values, a solution must be dramatically better than the current solution to be kept, except that the random function means that a potentially worse local solution can be selected. This is the key to not getting stuck at local maxima.

## The working solution

My solution uses an implementation of this simulated annealing algorithm. Key to its function is the ability to calculate the intersection angle between the drill hole and the orebody to ensure the steepest and therefore the most optimum intersection angle possible. It also allows control over the drilling dip, maximum hole length, and total drilling budget as defined by a total number of metres to be drilled.

Further inputs are the initial geological model, any existing holes and any existing mining voids. By using the existing drilling and mining data, the tool ensures that the planned drilling does not come within a specified distance of existing holes (unnecessary over-drilling) or of existing voids (safety).

Key to the implementation is the definition of the cost function (eg average hole length) against which each of the randomly constructed drilling plans are measured. It must be noted that the cost function is not a direct financial cost, but rather a mathematical expression of value as a function of one or more parameters. Table 1 shows the six possible optimisation targets (‘cost functions’) that can be selected. Minimising the standard deviation of the planned hole spacing will result in holes clustering around each other at the minimum specified spacing. Maximising the standard deviation will force the holes to disperse as widely as possible within the constraints of the minimum allowable intersection angle, drill rig operating parameters and maximum hole length.

The tool also allows for the simultaneous design of both an underground and surface drilling program (provided they share a budget for drilling metres).

We have used this tool to efficiently lay out underground drill programs with limited drilling positions in both the hanging wall and footwall of deposits. The tool effectively constrains the ‘drillable’ area, ensures that the drilling spacing is correct and that no unrealistic upholes are planned – as had happened previously during manual layout in the same areas. An example of an underground drilling plan is show in Figure 5, where 23 holes have been laid out on a 5 m × 5 m intersection grid with no hole steeper than 20° up and no closer than 30 m from any existing or planned hole.

The tool has also shown itself to be very effective at laying out grade control drilling rapidly and reasonably, as shown in the synthetic example in Figure 6. The total time to layout these 200 plus holes was less than ten minutes.

## Conclusion

The implementation of this algorithm allows the user to test many possible – but potentially suboptimal – drill hole layouts within a few minutes to produce a robust and achievable drilling plan within the specified budget, for the desired objective and constrained by the configuration or layout specification.

There are some refinements still required. The algorithm needs to be further tested and the ranking of the various cost functions (spacing, average length, etc) needs to be adjusted as it is currently hardcoded in the order shown in Table 1. Further tweaking is required to minimise the number of drilling stations used and to allow the individual drilling rig controlling parameters to be set on the collar rather than having simple global defaults.

While the tool can test an existing drilling plan in general terms by working with the specified budget and planned collar positions, the next stage is to be able to load the drill plan as the first ‘random’ scenario to the optimisation process and use the simulated annealing algorithm to test the robustness of the manual design.

This tool is aimed at aiding a geologist to generate the optimal design that is going to maximise the available drilling budget according to the resource classification or grade control requirements of that specific site.

**References**

Geltman K E, 2014. The Simulated Annealing Algorithm [online]. Available from: http://katrinaeg.com/simulated-annealing.html [Accessed: June 2017].

Wikipedia, 2016. Travelling salesman problem [online]. Available from: https://en.wikipedia.org/wiki/Travelling_salesman_problem [Accessed: June 2017].