A Channel Model for Bi-viscous Fluid Flow in Fractures

In the last decade, the interest towards fluids characterized by a complex rheology has increased in the scientific community. Physicochemical, rheological and fluid mechanical approaches are adopted to characterize the peculiarities of non-Newtonian fluids. These fluids show remarkable properties that can be exploited to improve remediation techniques or optimize industrial operations. While the existence of actual yield stress is still debated, the presence of a plug or pseudo-plug zone is frequent in emulsions, soft glassy materials, jammed non-colloidal suspensions or colloidal gels. Even if simple yield stress models are often faulted because of their acknowledged deficiencies, they allow to study important phenomena without introducing excessive complexity. In this study, the randomness describing the aperture field of a natural rock fracture is coupled with a bi-viscous fluid rheology of parameter $$\epsilon$$ , representing the viscosity ratio; the cases $$\epsilon =0$$ and $$\epsilon \rightarrow {1}$$ represent the Bingham and Newtonian behaviour, respectively. The conceptual model proposed describes the flow of such fluids through a fracture with aperture variable along a single direction, the aperture being constant along the other. The aperture variation is modelled via a generic probability distribution function of assigned mean and variance. Two limit flow cases are considered: (1) parallel arrangement (PA), representing the case of maximum conductance, with the fluid flowing in the direction of channels of constant aperture; and (2) series arrangement (SA), the case of minimum conductance, with flow directed orthogonally to the constant aperture side of the fracture. Results are illustrated for log normal and gamma aperture distributions. The influence of aperture variability and applied pressure gradient on flow rate is investigated for both arrangements. The pressure gradient affects in a nonlinear fashion the flow rate, with a marked increase around a threshold value for both PA and SA. The channel flow rate exhibits a direct dependency upon aperture variability for PA, an inverse one for SA. The shape of the distribution has an impact on model responses: for the PA, the influence is significant but limited to an intermediate threshold range of pressure gradients, while results for the SA are affected in the whole range of pressure gradient. An example application in dimensional form is included.