  We conducted a study on measuring $W^+W^-$ production and on the sensitivity
limits at $95\%$ Confidence Level on thirteen anomalous couplings obtained by
dimension-8 operators which are related to the anomalous quartic
$WW\gamma\gamma$ couplings. We consider the main $e^-p \to e^-\gamma^*\gamma^*p
\to e^-W^+W^-p$ reaction with the sub-process $\gamma^*\gamma^* \to W^+W^-$ at
the Large Hadron electron Collider (LHeC) and the Future Circular
Collider-hadron electron (FCC-he). For the LHeC, energies of the $e^-$ beams
are taken to be $E_e =60$ and 140 GeV and the energy of the $p$ beams is taken
to be $E_p = 7$ TeV. For the FCC-he, energies of the $e^-$ beams are taken to
be $E_e =60$ and 140 GeV and the energy of the $p$ beams is taken to be $E_p =
50$ TeV, respectively. It is interesting to notice that the LHeC and the FCC-he
will lead to model-independent limits on the anomalous quartic $WW\gamma\gamma$
couplings which are one order of magnitude stringent than the CMS Collaboration
limits, in addition to being competitive with other limits reported in the
literature.
