  The total reciprocal space magnetic flux threading through a closed Fermi
surface is a topological invariant for a three-dimensional metal. For a Weyl
metal, the invariant is non-zero for each of its Fermi surfaces. We show that
such an invariant can be related to magneto-valley-transport effect, in which
an external magnetic field can induce a valley current. We further show that a
strain field can drive an electric current, and the effect is dictated by a
second class Chern invariant. These connections open the pathway to observe the
hidden topological invariants in metallic systems.
