Recent laboratory and field studies performed by oil and gas service companies indicate that fracturing polymer may exhibit yield stress after water leak-off. Its rheology may be described by Herschel-Bulkley model. It is reported in the literature that yield stress might be responsible for low fracturing fluid cleanup efficiency. In this paper, we derive an extended analytical Buckley-Leverett type model for the displacement of non-Newtonian Herschel-Bukley fluid (e.g. fracturing fluid) by Newtonian fluid (e.g. hydrocarbon/water). Using this model, we study the effect of yield stress and other rheological parameters on fracturing fluid displacement efficiency. Parametric analysis indicates that high values of yield stress t0, consistency index K' and flow behavior index n' lead to low displacement efficiency. It is found that, for a given set of relative permeabilities, if a pressure gradient is not enough to overcome a critical pressure gradient, non-Newtonian fracturing fluid will not flow and only Newtonian fluid will be produced. It is demonstrated that low displacement efficiency can be overcomed by increasing displacing rate.
Non-Newtonian fluids are extensively used in oil field development. As a result, flow and displacement of non-Newtonian fluid in porous media play an important role in many aspects of petroleum engineering. Generally, oil, gas and water in subsurface reservoir are considered as Newtonian fluid. For heavy oil, an initial pressure gradient is needed to flow. It may be treated as Bingham plastic fluid . Polymer solutions used to enhance oil recovery may be treated as power-law fluids [2–3]. For foam, concentrated fracturing fluid, and some drilling mud, Herschel-Bulkley model may be appropriate to describe the rheological behavior [4–6]. Fig. 1 provides a schematic description of those rheological models.
Literature search indicates that considerable work is done in single-phase non-Newtonian fluid flow [2–3] and multi-phase non-Newtonian flow [6–8] in porous media. Wu et al.[1,7] present analytical models for two-phase flow and displacement of Bingham and power-law non-Newtonian fluid in porous media. Numerical model for multi-phase power-law non-Newtonian fluid flow is presented by Wu et al. and validated by available analytical solutions  and experimental data. Numerical model for multi-phase Herschel-Bulkley non-Newtonian fluid flow is introduced by May et al.  to study the cleanup of polymer after hydraulic fracturing. However, the technical performance of their model is not verified.
To the author's knowledge, there is no analytical model available regarding two-phase flow and displacement of Herschel-Bulkley fluid in porous media. In this paper, an analytical model is provided to study the displacement of Herschel-Bulkley fluid by Newtonian fluid in porous media based on Wu et al.[1,7]'s work. The solution derived in this work can be used to verify numerical models, such as the one presented by May et al. and to study the displacement mechanism of Herschel-Bulkley fluid.