Horizontal miscible displacements through porous media: the interplay between viscous fingering and gravity segregation

Abstract

We consider miscible displacements in two-dimensional homogeneous porous media where the displacing fluid is less viscous and has a different density than the displaced fluid. We find that the dynamics evolve through nine possible regimes depending on the viscosity ratio, strength of density variations and the strength of the background flow, as characterized by the Péclet number. At early times the interface is dominated by longitudinal diffusion before undergoing a transition to a slumping regime where vertical flow is important. At intermediate times, vertical flow and diffusion can be neglected and there are three different limiting solutions: a fingering limit; an injection-driven gravity-current limit; and a density-driven gravity-current limit. Finally at late times, transverse diffusion becomes important and there is a transition from an apparent shutdown regime to a viscously enhanced Taylor-slumping regime. In each of the regimes, the dominant scalings are identified and reduced-order models for the evolution of the concentration field are developed. Lastly, three case studies are considered to illustrate the dominant physical balances in the geophysically relevant setting of geological $\textrm {CO}_2$ storage.

Buoyancy segregation suppresses viscous fingering in horizontal displacements in a porous layer

“…Axisymmetric perturbations decay faster than θ-dependent perturbations, with the error decaying proportional to T −1 as has been found previously for unconfined gravity currents (Grundy & McLaughlin 1982;Mathunjwa & Hogg 2006). For a different stability argument in the case of axisymmetric perturbations, see the Appendix of Nordbotten & Celia (2006).…”

“…At later times, it is generally assumed that the fluids segregate vertically due to buoyancy, and a stable interface between the input and ambient fluids develops, which seems to be corroborated by laboratory experiments (Nordbotten & Celia 2006;Pegler, Huppert & Neufeld 2014;Guo et al 2016). It has also been shown that the combination of buoyancy and mixing between the fluids can suppress the Saffman-Taylor instability at long times (Tchelepi 1994;Riaz & Meiburg 2003;Nijjer, Hewitt & Neufeld 2022). However, for sharp-interface gravity currents in confined porous media, stability of the gravity current solution has not yet been confirmed and the flow physics that suppresses viscous fingering is not fully understood.…”

“…The set-up is shown in figure 1. A similar geometry was studied by Nordbotten & Celia (2006) and Guo et al (2016). Fluid of density ρ and viscosity μ i is injected into a porous medium of thickness H 0 , horizontal permeability k x , and porosity φ.…”

“…We are interested in analysing the stability of and convergence to these solutions in § § 3, 4 and 5. In the present section, we derive a single ordinary differential equation that governs the axisymmetric self-similar interface shape, and we recall an important analytical solution of this equation from Nordbotten & Celia (2006) and Guo et al (2016). For axisymmetric flow, the pressure at the top boundary, P 0 (η), can be eliminated from the problem as follows.…”

“…The stability of confined gravity currents is different because the displacement of the more viscous ambient fluid influences the physics. For these confined flows, stability has been established for perturbations that do not depend on the transverse or azimuthal coordinate (Nordbotten & Celia 2006). The stability analysis of both unconfined gravity currents and axisymmetric variations to confined gravity currents is simplified significantly because the driving pressure gradient in the input fluid can be eliminated from the problem.…”