Flow of truncated power-law fluid in fracture channels of variable aperture

A conceptual model is presented for non-Newtonian fluid flow in a rough channel representing a single fracture. The fluid rheology is described via a truncated power-law (TPL) model approximating the Carreau constitutive equation, while the aperture variation along the channel is modeled via a stochastic distribution of assigned mean and variance; the lognormal and gamma distributions are considered, together with a deterministic variation of sinusoidal behavior. The flowrate in a fracture subject to an external pressure gradient is derived under the lubrication approximation for the two limiting cases of a pressure gradient which is i) perpendicular and ii) parallel to aperture variation; these parallel and serial arrangements (PA or SA) provide an upper and lower bound to the fracture conductance. Different combinations of the parameters describing the fluid rheology and the variability of the aperture field are considered for a sensitivity analysis. Results are also compared with those valid for a pure power-law (PL) fluid which provides a relevant benchmark. The channel flowrate shows a direct/inverse dependency upon aperture variability for PA/SA. The difference in flowrate between the PL and TPL models is positively affected by aperture variability and pressure gradient, negatively affected by flow behaviour index, while its sign is positive or negative depending on PA/SA. The influence of the specific pdf adopted for the aperture field is moderate, an increasing function of aperture variability and depends on the third and fourth moment of the distribution. The conductance for a deterministic aperture variation exhibits the same trends as a stochastic variation, with differences from the latter depending on aperture variability and flow arrangement.