There are a couple of methods you could use:
One - take a number of thin sections and then measure the area of the holes under a microscope. You need a number of thin sections to get an average figure.
Two - take a piece of intact core, dry it of all liquids (this includes any liquid hydrocarbons) and weigh it. Then put it in a vacuum jar full of water - run a vacuum to force water into the voids and saturate the sample - weigh the sample again and the weight increase equals the mass of water in the voids 1 gram = 1 cc. This is the method I use on rock core before doing point load tests to get the saturated moisture content. The danger is you can bust the core up in the process of saturating it if it is weak, so I prefer just to leave it in a tub.
Certainly not 5 minute tests and cutting thin sections requires a bit of skill...
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