Clay materials are ubiquitous in the Earth's continental and oceanic crusts. They are characterized by a large specific surface area, and, in contact with water, they have remarkable adsorption, catalytic and containment properties. Clay materials also exhibit a high electrical conductivity response associated with their large surface. However, sedimentary clay minerals have a complex microstructure and electrochemistry that are not fully understood and make the quantitative petrophysical interpretation of geoelectrical measurements difficult. In this study, we have developed a new mechanistic model to better understand and predict the complex conductivity of clay materials from their microstructure and electrical double layer properties. For the conductive component, our model considers ion electromigration in bulk water, clay electrical diffuse layer, and interlayer space, if any. Concerning polarisation, it takes into account ion back-diffusion in the Stern layer and in the interlayer space, if any, and also Maxwell-Wagner effect. Water and clay surface conductivities, formation factor, cementation exponent and sample electrically connected porosity can be extracted from the comparison of the model to experimental data. This study is a step forward to better understand and quantify the complex conductivity of clays observed during electrical and electromagnetic measurements, from laboratory to geological formation scales.