Crash Course to Drilling:
Regular drill rigs are typically used to install deep foundation and drill drilled shafts/piers. It is also used commonly to drill pre-bored holes for driven piles for structures like retaining wall along the roadside. A typical drill rig uses Kelly drilling method to drill the hole. Kelly drilling is a rotary drilling method where the shaft is attached to a short rotary drill tools like an auger, core barrels or buckets. The conventional auger head size goes by an increment of 6 inches, starting from mere 6 inches to as big as 12 feet in diameter. Of course, with a larger diameter, a more powerful machine will be needed to drill the holes as the drill rigs work by rotating and pushing the auger down into the ground at a rate of a few feet at a time and then pulling it out of the ground to spin off the dirt/soil that the drilling head had accumulated. This process is repeated until the desired depth of the hole is achieved. Here is a video by Liebherr showing how a Kelly drilling method looks like.

In the above picture, the process of an uncased dry drilling with the installation of the rebar cages is clearly shown.
1 – The temporary case is first placed into the ground to keep the hole opening from collapsing.
2 – The auger proceeds to drill into the expected depths.
3 – The rebar is inserted using the hoisting machinery of the rigs.
4 – Concrete is being placed into the pile and at the same time, the temporary casing is slowly removed when the concrete fluid reaches a certain depth where it can hold the soil pressure to prevent the hole from collapsing in.
5 – Shows how a completed pile looks like.
This process is suitable for most types of soil, whether it be wet soil, dry soil, rock bed soil or sandy soil. A typical dry drilling will be just using the auger to start drilling at the location of the holes. However, a conservative way of drilling, which stills plays a big part in today’s drilling industry, is that a temporary casing will also be used along with dry drilling. The purpose of having a casing, whether it be permanent or temporary casing depending on the geotechnical report and the structural engineer’s design, is to stabilize the hole and prevent collapse of the wall of the holes before concrete is poured in. There are many ways to insert a casing into the hole, you can either use a casing oscillator which vibrates the casing into a sandy soil, displacing the soil around the hole before the drilling start. Another way which is more common is just simply hammering the casing into the top of the hole and start drilling. Once a deeper hole is drilled, another casing in put into the deeper depth and the process repeats itself. This is pretty much the same for rock drilling as well, but instead of using a regular earth auger, a rock auger will be utilized. A cased drilling is typically seen more often on soil report that showcases high water table but a decent blow count on the Standard Penetrometer Test (SPT).
Wet drilling or slurry drilling is used when the soil condition has low blow counts on SPT and a high water table. In slurry drilling, a mixture call Bentonite is mixed in with water as the hole is being drilled. The bentonite mix with water to create a slurry like mixture, and this mixture will keep the water of the holes from seeping in until concrete is being poured into the hole. Now this may seem like a lot of water and slurry product will be used for each hole drilled, but there is a way of working around it. In this industry, typically half of the slurry water used would be reused for the next holes to minimize waste and cost.
Here is a picture by pilebuck.com showing how the slurry helps in preventing the wall from caving:

A & B – slurry is added as the hole is drilled
C – depending on the soil condition, settling time may be required for the filter cake to form to keep the wall intact
D – then, settled sand are cleaned up with a cleanout bucket
E – rebar can the be placed and concrete is placed using a tremie or concrete pump, and the slurry mixture will be pushed out by the concrete to be reused or disposed of.
Here is a video by Bob Moore Construction explaining how slurry drilling works:
and with this information, the crash course into drilling is concluded.