There is such a diagram--- it is the flux of energy in the linearized gravitational field (if you use full GR, you get complications with defining the energy). Unlike the electromagnetic case, where the electric field carry the bulk of the energy and the momentum of the charge carriers is negligible, in the gravitational case, it is the opposite.
You can also imagine electromagnetic circuits in which you accelerate very massive spheres which are very lightly charged, and use these as current carriers, and in this case, the momentum of the current carriers will not be negligible.
EDIT: To clarify, there are gravitational fields created by moving water which surround the pipe, like the electric and magnetic fields surround the current-carrying wire. There is an energy flow in these gravitational fields, which carries energy, just like the Poynting flux does. These effects are negligible for ordinary materials at ordinary density. Nearly all the energy flux (all but the tiny negligible fraction in the gravitational field) is carried by the water in the pipe, but the momentum in the water is not analogous to the Poynting vector, it is analogous to the electron momentum, which also carries a small amount of energy in a current carrying wire.