We study the problem of drag reduction (DR) in a lubricated conduit, in which a thin layer of water is injected in the near-wall region and facilitates the transport of a core of highly viscous oil. In the present investigation, the flow instance is a channel flow, and consequently we have one thin layer of low-viscosity fluid lubricating each wall. We run direct numerical simulations (DNS) of this flow instance, respecting the protocol of the constant power input (CPI) approach. This approach, which prescribes that the flow rate is adjusted according to the current pressure gradient, so as to keep constant the power injected into the flow, closely mimics real transport pipelines. A phase-field method (PFM) is used to describe the dynamics of the liquid-liquid interface. As this technique is tailored toward the transport of very viscous fluids like oils, we study the drag reduction performance of the system by keeping fixed the lubricating fluid properties (water) and by considering two different types of oil characterized by different viscosities, 10 and 100 times more viscous than water, respectively. As in real instances the presence of impurities and surfactants – which act by locally reducing the local value of the surface tension – is inevitable, we consider, for each type of transported oil, a clean and a surfactant-laden interface. For all four tested configurations, we unambiguously show that significant DR can be achieved. Reportedly, compared to the single-phase case, we observe a reduction of the mean pressure gradient down to \(p_x/p_{x,sp}=0.25\) for the largest viscosity oil. By analyzing the features of turbulence in the lubricating layer, and the close interaction with the perturbations induced by the oil-water interface deformation, we elucidate the physical mechanisms leading to DR and we underline the effects of viscosity ratios and of surfactants.