  Measuring the thermal conductivity ($\kappa$) of water at extreme conditions
is a challenging task and few experimental data are available. We predict
$\kappa$ for temperatures and pressures relevant to the conditions of the Earth
mantle, between 1,000 and 2,000 K and up to 22 GPa. We employ close to
equilibrium molecular dynamics simulations and a deep neural network potential
fitted to density functional theory data. We then interpret our results by
computing the equation of state of water on a fine grid of points and using a
simple model for $\kappa$. We find that the thermal conductivity is weakly
dependent on temperature and monotonically increases with pressure with an
approximate square-root behavior. In addition we show how the increase of
$\kappa$ at high pressure, relative to ambient conditions, is related to the
corresponding increase in the sound velocity. Although the relationships
between the thermal conductivity, pressure and sound velocity established here
are not rigorous, they are sufficiently accurate to allow for a robust estimate
of the thermal conductivity of water in a broad range of temperatures and
pressures, where experiments are still difficult to perform.
