The present invention is directed to a system, apparatus and methods for determining influx of formation fluids into a well borehole during drilling. The methods of the present invention comprise determining the dielectric constant and/or conductivity of the fluid in the annulus about a drill pipe in a well borehole, comparing the determined dielectric constant and/or conductivity to a reference value and inferring that influx has occurred when the difference between the dielectric constant and/or conductivity and the reference value is greater than a predetermined limit. The dielectric constant and/or conductivity is determined by transmitting electromagnetic energy between about 10 KHz and about 10 GHz through the fluid in the annulus about the drill pipe from a transmitter located in the drill pipe section, receiving a portion of that energy which has travelled through the fluid but not through the surrounding formation in a receiver located in the drill pipe, and determining the dielectric constant and/or conductivity as a function of at least one of the phase and the amplitude of the received energy.