Archie's empirical power laws are strictly valid only for homogeneous, water-wet (WW) rocks deprived of microporosity or substantial clay-exchange cations. When these conditions are not met, non-Archie electrical behavior arises whereby relationships among rock resistivity, porosity, and water saturation no longer exhibit power-law dependence. Currently, such an unreliable behavior of empirical laws can be quantified only through pore-scale modeling of electrical conductivity under specific sets of geometric assumptions and with substantial computation memory requirements. We introduce a new geometric concept to simulate direct-current electrical-conductivity phenomena in arbitrary rock models on the basis of 3D grain and pore objects that include explicit distributions of intragranular porosity, clay-exchange cations, nonwetting fluid blobs, thin films, and pendular rings. These objects are distributed in the pore space following simple heuristic principles of drainage/imbibition that honorcapillary-pressure curves. They provide a simple way to parameterize the 3D pore space and to calculate the electrical conductivity of porous media saturated with two immiscible fluid phases by way of diffusive random walks within the brine-filled pore space. Not only is the random-walk method memory efficient but it also allows the inclusion of clay/brine cation exchange surfaces otherwise not possible with conventional pore-network models. By comparing results stemming from random-walk, pore-network, and percolation simulations, we show the importance of grain surface roughness and thin film thickness, even in water-wet rocks where those factors usually are neglected. For the case of strongly oil-wet rocks, we show that thin films, snap-offs, and pore microgeometry have a primary impact on hysteresis-dominated rock resistivity during imbibition (increasing water saturation). Our simulation method agrees well overall with percolation simulation results and is advantageously unaffected by assumptions concerning site-percolation imbibition.