The Effect of Compressibility and Outer Boundaries on the Onset of Viscous Fingering
Authors: Araque-Martinez Aura N. , Lake Larry W. 
Published: October 2020

Abstract
The knowledge of the effects of instability and heterogeneity on displacements, primarily enhanced oil recovery and carbon dioxide storage, are well known though they remain difficult to predict. The usual recourse to modeling these effects is through numerical simulation. Simulation remains the gold standard for prediction; however, its results lack generality, being case specific. There are also several analytic models for displacements that are usually more informative than simulation results. However, these methods apply to steady-state, incompressible flow.
Carbon dioxide injection for storage uses compressible fluids, and, in the absence of producers, will not approach steady-state flow (Yun et al., 2017). Consequently, it is unlikely that storage will be in reservoirs of open boundaries. Flow of compressible fluid necessitates the use of closed or partially sealed boundaries, a factor that is consistent with compressible flow.
This work deals with the conditions that cause the onset of viscous fingering or Saffman-Taylor (ST) instability. The actual propagation of fingers, a subject of much recent literature, is not discussed here. The original ST formalism of M&gt;1 is highly restrictive: it is for linear flow of nonmixing incompressible fluids in steady-state flow. In this work we relax the incompressible flow restriction and thereby broaden the ST criterion to media that have sealing and/or partially sealing outer boundaries. We use the non-linear partial differential equation for linear flow and developed analytic solutions for a tracer flow analog and also for a two-fluid compressible flow. The analysis is restricted to so-called stabilized flow, and to constant compressibility fluids, but we are not restricted to small compressibility fluids. There is no transition (mixing) zone between displacing and displaced fluids; the two components are locally segregated or the displacement is piston-like. The absence of a transition zone means that the results apply to both miscible and immiscible displacements, absent dispersion or local capillary pressure. We use the product of the fluid compressibility and pressure drop (cfΔP) cut-off to differentiate the compressibility group. Here, small and large compressibility groups correspond to the cases where cfΔP &lt; 0.1 and cf ΔP &gt; 0.1 respectively9-10. In this work, ΔP is defined as the pressure drop within the specific fluid region. Results will be based on proposed analytical solutions compared to numerical flow simulation.
The key contribution here is the addition of compressibility, which additionally makes possible the study of the effect of external boundaries. Absent compressibility, the necessary condition for the growth of a viscous finger is simply the mobility ratio, M&gt;1. It is the objective of this work to study how the ST criterion is affected by the inclusion of compressible flows and in the presence of sealing and partially sealing outer boundaries as in CO2-storage and EOR by gas injection.
Results show that adding compressibility always makes displacements more unstable for steady-state flow, even for favorable mobility ratio (M&lt;1) at extremely large compressibility (e.g. cf &gt; 5e−3 1/psi). For a sealed external boundary, displacements will become more stable as a front approaches an external boundary for all mobility ratios (M) investigated.