  It is known that gap opening depends on the disc's viscosity; however,
eccentricity damping formulas have only been derived at high viscosities,
ignoring partial gap opening. We aim at obtaining a simple formula to model
$e$-damping of the type-I regime in low viscosity discs, where even small
planets may start opening partial. We perform high resolution 2D locally
isothermal hydrodynamical simulations of planets with varying masses on fixed
orbits in discs with varying aspect ratios and viscosities. We determine the
torque and power felt by the planet to derive migration and eccentricity
damping timescales. We first find a lower limit to the gap depths below which
vortices appear; this happens roughly at the transition between type-I and
type-II regimes. For the simulations that remain stable, we obtain a fit to the
observed gap depth in the limit of vanishing eccentricities that is similar to
the one currently used in the literature but is accurate down to
$\alpha=3.16\times 10^{-5}$. We record the $e$-damping efficiency as a function
of the observed gap depth and $e$: when the planet has opened a deep enough
gap, a linear trend is observed independently of $e$; at shallower gaps this
linear trend is preserved at low $e$, while it deviates to more efficient
damping when $e$ is comparable to the disc's scale height. Both trends can be
understood on theoretical grounds and are reproduced by a simple fitting
formula. Our combined fits yield a simple recipe to implement type-I
$e$-damping in $N$-body for partial gap opening planets that is consistent with
high-resolution 2D hydro-simulations. The typical error of the fit is of the
order of a few percent, and lower than the error of type-I torque formulas
widely used in the literature. This will allow a more self-consistent treatment
of planet-disc interactions of the type-I regime for population synthesis
models at low viscosities.
