create block set of elements bounded by surfaces

I have a tet mesh. I want to calculate the volume enclosed by 5 internal surfaces which are part of the mesh. The surfaces do not completely enclose the volume, they leave some gap. How should I do this?
One idea is to create a new cub file with these surfaces, extend the surfaces so they intersect, webcut, create a closed volume, then (how to) calculate the geometric volume of this volume.
However, I’d prefer to use the given mesh. Because I am also interested in knowing the tet cells which are within this volume. Can I create a material block from these cells?

It might be helpful to have an example. Are you able to send your model?

Here is the cub file, but it is 440 MB:

Is it possible to use exo file which has these surfaces as nodesets? The exo file is only 11.6 MB.

Surface IDs that delineate the volume of interest: 142,155,156,158,159,165,166,170,173,179,182

By the way, there might be a bug in trelis GUI code that makes it crash and close when I click on the Command line window header (by mistake) which then undocks the window and closes trelis.