Although the question has already been answered by Mr Agricola, being what I am I have to throw my ten pennyworth in.
Please take attention of Pumping Formulae, O’Donahue 1914 just uploaded which should be of great interest and worth the download.
I would like opportunity to make a few points.
First, it is not the depth of the shaft but the volume of water to be raised that dictates the power of the engine and size of pumps, rising mains etc.
Small mines with a small ‘make’ of water could have been kept in fork with a single bucket pump and say 6 inch rising main. This was very common in Wales and would have been driven by a waterwheel.
The more extensive and deeper mines typically of the West Country required a different type of system comprising a single bucket pump at the shaft sump, followed by a series of plunger lifts and an engine powerful enough to do the job.
As one progresses up the shaft the volume of water required to be lifted would increase because water would be draining from the side levels, so the size of the pumps and rising mains would increase in diameter. (Ref, inventory of Engine Shaft, Frongoch Mine, central Wales)
Now the thing to remember is that it is the weight of the pump rods that raises the water by displacement on a system of plunger pumps, and that access weight of rods would be balanced out by the balance bobs at each lift.
To go further. To displace a certain weight of water would require a rod that is heavier, but only by a margin sufficient to do the job otherwise the motion of the rods would be violent and also wasteful of fuel consumed by the engine.
Distribution of the balance bobs throughout the depth of the shaft is important otherwise if it where all at surface the surface balance bob and associated fulcrum would be carrying all of the weight and would certainly fail.
The whole subject is quite complex and I hope that the uploaded document is useful.
My avatar is a poor likeness.