I have $E$ (electric field intensity) in water.
I want to find $J$ (current density).
I don’t know if Ohm's law ($J=\sigma E$) is valid for water.
If it is not valid, how do I may find out $J$?
The Ohm law is approximately(*) valid for water in the local form $J=\sigma \cdot E$ for the free liquid, but not in non local form $I=R \cdot U$, if electrodes and DC current are applied.
The latter is due involved electrochemistry, mainly but not limited to factors as:
Equilibrium potential of electrodes
Potential difference for electrolysis
Kinetics of electrode reactions
Transient electrode effects
Forming electrode layer potentials
Forming potential gradient due concentration gradient
That is why the liquid specific conductivity is measured differentially by AC voltage of frequency typically 1-3 kHz in the analytic technique called conductometry, using an electrode combo with the known geometry and known conversion factor conductance -> conductivity.
It may be integrated into automated titration techniques, following conductivity changes during progressive addition of a solution of known properties.
(*) The topic is rather complex solution chemistry. I say approximately,as there are some minor nonlinearities in driven motion of ions, based on nonlinear interaction of ions ( or rather of their hydrated forms) with water molecules and other ions.
Particularly for water ions, as they disappear and other ones appear in other place, as there is dynamic ion exchange with water molecules.
But in first approximation, it can be taken as linear.