sensor - How to calculate permittivity of a fluid

I am confused about calculating permittivity of a fluid. Permittivity differs from one fluid to another.

$$\epsilon=\epsilon_r\epsilon_0$$

Since it is an electrical property combined with an electrical capacity, it is possible to measure it indirectly in a capacitive sensor. I have used a capacitive sensor to measure the electric relative permittivity factor of a dielectric medium, which can be expressed as a ratio of capacity \$C_x\$ of a capacitor whose space between and around the electrodes is completely filled with the medium, to capacity \$C_0\$ of the same electrodes in vacuum.

$$\epsilon_r= C_x/C_0$$

I know and measured \$C_0\$ value. I am facing problems with calculating \$C_x\$ of the capacitor whose space between and around the electrodes is completely filled with the medium.

I am using method as described below.

I am applying an A.C signal (125 KHz) to the capacitive sensor which is filled with some fluid; in response I am getting an A.C. signal with some phase difference. I am able to measure the amplitudes of the sensor input and sensor output signals and their phase difference also. I am trying to make an equation that will give \$C_x\$ value from above known values (\$V_g\$ (input), \$V_r\$ (output), \$\phi\$ (phase difference)).