The Saffman-Taylor Instability using Complex Fluids in Tapered Hele-Shaw Cells

Abstract

This thesis research concerns controlling viscous fingering instability when a less-viscous fluid pushes another more-viscous one in a porous medium. This instability is called the Saffman-Taylor instability and has been extensively studied, primarily for simple Newtonian fluids. The resultant of such growing and wavy interfacial perturbation is the deterioration of the efficiency of industrial processes (e.g., Enhanced Oil Recovery). For Newtonian fluids, it has already been proved that adding a gap gradient to a Hele-Shaw cell is an efficient method to suppress this interfacial instability. In our work, we focus on achieving a total sweep displacement with complex (yield-stress) fluids. We demonstrate the viscous fingering instability suppression using converging cells, whereby we add a negative depth gradient by tapering the upper plate of both radial and rectangular Hele-Shaw cells. Performing experiments in our homemade rectangularly tapered Hele-Shaw cell, we observe that a converging cell, implying a permeability gradient, can be used to inhibit the viscous fingering instability of complex yield-stress fluids. We investigate, in particular, the impact of the gap gradient (α) and the injection flow rate (Q) on the stabilization of the interface of three complex yield-stress viscous fluids. For a fixed cell gradient, our experimental results show that a full sweep is achieved at a low flow rate, whereas a partial displacement with fingering is obtained when the flow rate is over a critical value. Furthermore, we develop a theoretical linear stability analysis generalized for common complex fluids possessing a power-law varying viscosity and yield stress. From this analysis, we establish a theoretical stability criterion that we tested which depends on the cell geometry (α, the gap gradient, and W, the cell’s width), the interface’s gap thickness, and velocity (h0 and U0, respectively), the fluid’s viscosity (μ), surface tension (γ) and contact angle (θc). Using the experimental values of h0 and U0 at the interface, we calculate the value of our theoretical criterion and obtain a good agreement to separate both stable and unstable experimental displacements. We also carry out similar experimental and theoretical investigations using two complex yield-stress viscous fluids for radially tapered Hele-Shaw cells. We obtain a stability diagram for one of them depending on the gap gradient and the injection flow rate (α v.s Q). Theoretically, we derive a linear stability analysis starting from an effective Darcy’s law and the continuity equation replacing the constant viscosity, μ, by an effective viscosity μeff respecting the Herschel-Bulkley law. From the two governing equations, we obtain a criterion depending on three important parameters. First, the fluid’s rheology and characteristics μeff and the different constants, γ and θc. Then, the gap gradient (α) and lastly, the interface radial position, gap thickness, and velocity (r0, h0 and U0, respectively). Once again, using the experimental values of radius of the interface, gap thickness at the interface, and velocity of the interface, we compare our theoretical stability criterion to the experiments. We found good agreement between the two, but we observed a slight discrepancy which is expected due to our assumptions. As a consequence, we investigate the impact of one major assumption we made in our first linear stability analysis in the radial geometry concerning the Bingham number (ratio of the yield stress to the viscous stress). To do so, we develop a second more complex linear analysis in a radially tapered Hele-Shaw cell. In this new derivation, we obtain another stability criterion by avoiding to neglect the Bingham number. As in our first theoretical derivation, the stability is dependent on the gap gradient (α), the interface position, gap thickness and velocity (r0, h0 and U0), as well as the fluids’ viscosity, surface tension and contact angle. We immediately compare this second criterion to our experimental results. We are able to improve the agreement between them for low values of gap gradient. However, for higher values of α, it seems that neglecting or not the Bingham number does not affect the overall agreement between theoretical and experimental results, meaning other assumptions made during our stability analyses still undermine our work.