## 1 Introduction  
### Background  
Incompatible biphasic flow often occurs in the petrochemical and oil industries. When two liquids with different densities touch one another in a horizontal tube, they incline to be affected by gravitational force. The heaviest phase generally stays below and the lightest phase flows as a separate layer over the top, creating a stratified flow regime. Controlled process conditions can also yield a core annular flow (CAF) regime when the difference in densities of the fluids is not very high. The heavier liquid (usually water) forms a thin lubricating annulus that sheathes the viscous core so that the core cannot touch the pipe wall. This is an alternative pipeline transportation technology that is beneficial for highly viscous fluids like unconventional heavy oils and viscous petrochemicals. The lubricating water can significantly reduce the requirement of pumping energy when compared to similar requirements for pumping viscous fluid alone through the pipe. In fact, it is comparable to the power consumption for pumping only water. A considerable amount of research has been undertaken to find a reliable method for designing such multiphase pipe flows.  
In practice, sufficient knowledge of pressure gradients or frictional pressure losses in pipes is needed to develop an energy-efficient transportation system (e.g., to determine the optimal size of pipes and pumps that can control various flow conditions throughout the lifetime in the field). [PERSON] et al. [1] introduced a friction loss model for pumping heavy oils in a lab-scale horizontal pipeline with the application of an idealized CAF technology. Although this model could predict a large CAF dataset with acceptable precision, it failed to do so for the self-lubricated flow (SLF) of bitumen froth (which represented a commercial-scale application of this water-lubricated flow technology). [PERSON] et al. [2] investigated the SLF phenomenon to develop their own empirical model based on data generated from the lab- and pilot-scale experiments. A 35-km long SLF pipeline was designed, commissioned, and operated based on this model in Athabasca by Syncraude Canada Ltd. The SLF involved intermittent water lubrication with the oil-rich core frequently touching the pipe wall. Meanwhile, for CAF, this kind of contact was negligible, and the lubrication was continuous. [PERSON] et al. [3] applied CAF in a pilot-scale pipeline. Although proper attention was paid to eliminating wall-fouling, it was a natural consequence of water lubrication and could not be excluded from the large-scale water-lubricated pipeline transportation of viscous oils. Based on the data produced from CAF experiments, both with and without wall-fouling, a new semi-mechanistic two-parameter model was proposed to assist with friction losses. The model was claimed to perform better than similar models. However, it failed to provide satisfactory results for the water-assisted flow (WAF) of unconventional heavy oils [4; 5; 6]. WAF refers to large-scale applications of CAF that involve wall-fouling (Figure 1). It is a commercially applicable mode of the flow technology. One of the most significant technical challenges facing the industrial application of WAF is the necessity of a model that can reliably predict frictional pressure losses. Previously proposed models for various modes of water lubrication are not necessarily applicable to WAF pipelines. Applications of existing analytical models to different WAF datasets produce unreliable results, with errors as high as 500% [5]. This is because most of these models are empirical and were developed using system-specific data. An exception is the phenomenological model proposed by [PERSON] et al. [7]. It is probably the best analytical model for WAF systems. A concise description of the model is included in Section 3.1.