To me the word are not really same because saturated hydraulic conductivity is carried out in the Laboratory because it is very difficult and expensive to carry out unsaturated hydraulic conductivity in the field. therefore unsaturated hydraulic conductivity is the one that will be same as permeability.
Saturated hydraulic conductivity (K) is proportional to permeability (k), but they are not the same quantity. Whereas permeability is an intrinsic property of a porous material (i.e. it only depends on properties such as pore size, tortuosity, and surface area), hydraulic conductivity depends on the properties of the fluid (saturation, viscosity, temperature, and density). Assuming full saturation, k=K*mu/(rho*g), where mu is the dynamic viscosity of the fluid, rho is the fluid density, and g is the gravitational acceleration constant.
Hydraulic conductivity of soil is a measure of the its ability to transmit water when submitted to a hydraulic gradient. The coefficient of permeability (k) represents the soil’s ability to transmit and drain water.
In petroleum engineering, permeability is like the natural ability of the rock matrix to transmit fluids, whereas reservoir conductivity includes the modified (by hydraulic fracturing or any other fracture or faults) ability of the rock matrix to transmit fluids. For example, faults can conduct flow along the fault plane, which is called fluid conductivity through faults.
Hydraulic conductivity is the ability of porous medium(soil) to transmit water under saturated or nearly saturated conditions while permeability is the property of soil to transmit water and air both