  GEM are widely used as amplification stages in gaseous detectors exposed to
high rates. The GEM consists of a polyimide foil which is coated by two thin
copper layers. Charges collected into the holes or created during the
amplification process may adhere to the polyimide part inside the holes. This
is often accompanied by a change of the effective gain. The effect is commonly
known as the charging-up effect.
  This work presents a systematic investigation of the effect under
well-controlled and monitored conditions for a single standard GEM foil in
Ar/CO$_2$ (90/10) gas for varying rates of X-ray interactions in the detector
and for different gains of the foil. In order to cover a wide range of
different rates, we apply two different methods. The first one is based on a
current measurement while the second one relies on the analysis of $^{55}$Fe
spectra over time. Both methods give consistent results, showing an initial
increase of the effective gain with time and an asymptotic saturation, which
can be well described by a single-exponential function. We find that the
characteristic time constants extracted from our measurements scale inversely
proportional to the rate of incoming electrons for a given GEM voltage.
Introducing characteristic quantities, which describe either the number of
incoming electrons per hole or the total number of electrons in a hole per
characteristic time, we find consistent numbers for measurements taken at the
same GEM voltages. For measurements taken at different GEM voltages, however,
also the characteristic total number of electrons inside the hole, which is
supposed to take into account the different effective gains, is found to be
higher by a factor of about 3.5 for $U_{\rm GEM}=$ 350V ($n_{\rm tot}=8.8\times
10^8$) compared to 400V ($n_{\rm tot}=2.4\times 10^8$). This hints at a
residual dependence of the charging-up characteristics on the GEM voltage.
