How much time for a water molecule in pure water to reach the top of a glass of water? I have a curiosity to know how long it will take for a water molecule to move a certain distance in pure water. I can put it simply as how much time for a water molecule to travel the whole length of a typical glass of water if no collective motion is present (therefore very still water with no convective currents).

To simplify things, this is the model:

Suppose I have a completely still and ideally insulated glass of water where the temperature in each part of the fluid is exactly the same (therefore no convective motion). &nbsp;&nbsp;&nbsp;  In addition the top of the glass container is closed and no water air interface is present to avoid evaporation that if present changes the dynamic. &nbsp;&nbsp;&nbsp;In addition the glass of water is big enough so that the dynamic of the water in the core is not influenced by the the limited range of motion of the water near the borders of the glass.
Since it is a liquid, the water molecules move around each others until the original water molecule is at distance r from where it started(A). In absence of gravity the movement is equally probable in all directions, while with gravity I suppose it may be more likely a layer layout of the water molecules making horizontal movements more likely (see elipse shape of the trace).
The questions are (no need to reply to both):
1) How much time to be at distance r in both situations (gravity or not)? (it will depend on temperature, so let's say at room temperature)
2) How much time roughly will it take for a water molecule to reach reach almost the top of a typical glass of water starting from the bottom?    (in absence of convective motion and evaporation)
I am posting here the answer since it has been quickly found thanks to the useful comments. To give an estimate the best way is to look at the self-diffusion coefficient.
This is defined as the diffusion coefficient when the chemical potential gradient equals zero, or more simply, it is the diffusion coefficient in a medium of the same molecules.
According to Ref.1, the self-diffusion coefficient of water at $\pu{25^\circ C}$ is $D = \pu{2.3 \times 10^{−9} m^2 s^{−1}}$.  The diffusion time can be calculated as: $$t \approx \frac{x^2}{2D}   $$
As pointed out by @Karsten Theis, you can use the calculator and insert the water diffusion coefficient D.
For example, if the distance x is $\pu{8cm}$ like a typical glass of water and we are at $\pu{25 ^\circ C}$, then $t \approx \pu{16 days}$!
Instead if the temperature is $\pu{90 ^\circ C}$ then it's around $\pu{5 days}$.
I have drawn a graph that shows how much time to reach the top of the glass depending on temperature:

References:

Manfred Holz, Stefan R. Heila, Antonio Sacco, &quot;Temperature-dependent self-diffusion coefficients of water and six selected molecular liquids for calibration in accurate $\ce{^1H}$ NMRPFG measurements,&quot; Phys. Chem. Chem. Phys. 2000, 2(20), 4740-4742 (https://doi.org/10.1039/B005319H).