Computational study of reactive miscible viscous fingering in porous medium

Abstract

Ahydrodynamic instability called viscous fingering (VF) arises when a less viscous fluid displaces a more viscous one in a porous medium. This phenomenon is prevalent in diverse transport scenarios, including applications in the petroleum industry, aquifer contamination, and CO2 sequestration. A chemical reaction may modify the viscosity of the fluids flowing in a porous medium and influence the VF. From macro to micro scales fields utilize VF induced by chemical reactions to enhance mixing. To comprehend chemo-hydrodynamic instability, we examine a reactive displacement involving a second-order chemical reaction, denoted as A + B → C, assuming miscible, Newtonian, and neutrally buoyant fluids. These bio-molecular reactions serve as fundamental components for diverse complex reactions. Moreover, the viscosity profile depends on the viscosities of the reactants and products, determined by Rb = ln(µB/µA) and Rc = ln(µC/µA) where µi is the viscosity of fluid i ∈ {A, B, C}. When reactants and products exhibit viscosity contrasts, a nonlinear interaction arises between chemical reactions and hydrodynamics. This interaction is modeled using a coupled system of partial differential equations that encompasses Darcy’s law and three convection-diffusion-reaction (CDR) equations. Weemploy non-linear simulations (NLS) to investigate reactive VF in radial flow. Our study involves discussing a numerical technique that combines compact finite differences and a pseudo-spectral method. For stable displacements, we report a transient growth in total reaction rate at higher Damk¨ohler numbers (Da) in radial flow, leading to more product formation, a phenomenon absent in rectilinear flow. Addi tionally, we observe an earlier onset of instability and enhanced fluid mixing with increasing viscosity contrast. It also depends on whether the product is high or less viscous than reactants for a constant Rb. Moreover, as the viscosity contrast increases, the mixing process reaches a saturation point, and we identify the existence of frozen fingers in this reactive fluid system at later stages. Further, we extend our analysis for infinitely fast reactions having iso-viscous reactants (Rb = 0) and establish a scaling relation for the onset time of instability depending on the P´eclet number (Pe) and Rc. Further, we conduct the stability analysis using both the approaches, NLS, and linear stability analysis (LSA). Interestingly, the viscosity profile is not modified after the reaction when Rc = Rb. This scenario serves as an equivalent non-reactive case, allowing us to compare VF dynamics when the viscosity profile changes. We establish a phase plane (Rb,Rc) phase plane for a wide range of Da and Pe divided by critical viscosity contrast to induce instability. It states that if the equivalent non-reactive case, is stable, there exists a range of Rc that corresponds to stable for each Rb and vanishes at the critical value of Rb that triggers instability for the equivalent non-reactive cases. Otherwise, the flow remains unstable. The stable zone contracts for larger values of Da and Pe, yet it never disappears, even with Da → ∞. Intriguingly, a region near the line Rc = Rb is identified where flow stability remains unaffected by reaction rate (Da). To explore the influence of flow geometry on reactive VF, we investigate reactive displacements for rectilinear flow in a linear regime. The unsteady base state renders a stability matrix highly non-normal, and hence, the modal analysis like QSSA, may not predict the transient behavior accurately. Therefore, we opt for a non-modal linear stability analysis (NMA) using a propagator matrix approach to assess reactive displacements. As the viscosity contrast increases, an early onset occurs and more amplified perturbations when the reaction generates a less viscous product (Rc < Rb) than the equivalent non-reactive scenario. Conversely, there exist some reactive cases where onset is delayed if Rc > Rb compared to the equivalent non-reactive case (Rc = Rb) for infinitely fast reactions, even with a steeper viscosity contrast. Further, we focus on the reactions having a non-monotonic viscosity profile featuring finite reaction rates with iso-viscous reactants (Rb = 0). For such instances, the unstable zone contracts, and a stable zone develops in the mixing zone. As a consequence of this stable zone, fingering patterns localize and develop either upstream or downstream of the flow, depending on whether the viscosity profile exhibits a maximum or minimum. When Rc > 0, certain reactions exhibit transient growth within the perturbation amplification curve, resulting in secondary instability. Here, we obtain a significant contrast in the early-stage VF dynamics across both flow geometries. For radial flow, we observe the transient growth in perturbation amplification but do not observe secondary instability for the similar non-monotonic viscosity profile having maxima. In rectilinear flow, the velocity consistently facilitates convection towards the interface throughout the process, contrasting with radial flow geometry where it diminishes over time at the interface. Thenumerical technique employed for NMAinthisthesis introduces a new perspective for comprehending time-dependent linear systems inherent in miscible reactive VF. The outcomes obtained align more closely with non-linear simulations than the conventional approach, QSSA. This research contributes a numerical and theoretical framework with potential applications for controlling and enhancing VF in various geophysical processes, including CO2 sequestration, chemical flooding, and reactive pollutant displacement.