What is the area of a hole in an irregular surface?

Question

When calculating area of a hole in an irregular surface for water flow calculations, what defines that area?

I need to calculate the amount of water passing through holes in surfaces in a fixed time, knowing the pressure on either side. This is mostly a straight-forward process (discounting the discharge coefficient, but that is a different matter), but the formulas depend on the area of the hole. What I know about the hole is its boundary. If the surface is planar, this is obvious enough and can be calculated via Green's theorem.

But when the hole is in a non-planar surface, it becomes much more problematic. What does the area of the hole even mean? The area of the surface? The hole is exactly where the surface does not exist. Outside the hole there is some surface, which I could presumably find information about (though at a serious increase in complexity), but where the hole is, there is nothing. I cannot even be sure how the surface where the hole was cut was shaped. That information was lost with the hole. "Yeah, there was a kilometer long capped tube there! Good thing that was where the cut-out needed to be." Besides, the water doesn't know how this mythical surface was shaped either, so its behavior will not be influenced by that. Whatever definition of area is appropriate here, it cannot be dependent on the exact shape of some surface.

The areas of all possible surfaces filling the hole is clearly bounded below, so there is some minimal area. Presumably this would be the best choice. That is a mathematical rather than physical question, but I want to be sure I am properly handling the physics instead just jumping on an idea.

Answer 1

An engineering problem such as this calls for some "standardization". A similar problem arises in calculating the drag on bodies of arbitrary shape. What we do is calculate the "projected area" (area of the projection of the body on a plane perpendicular to the bulk flow) for use in the standard formula for drag. Then all the complexities regarding the actual shape of the body (because two bodies of different shapes can have the same projected area, for e.g. a cylinder and a sphere of the same diameter) gets dumped into the "drag coefficient" which is to be empirically determined.

You could adopt this procedure and similarly define a "projected area of the holes" and use it inside standard formulas which contain coefficients that must be empirically determined. If the holes are randomly created by a particular physical/manufacturing process so that their statistical properties remain the same from one specimen to another, then there is hope that the drag coefficient may need to be determined only once and it will be nearly constant for all the specimens.