148.
We study two asymptotic regimes of unstable miscible displacements in porous media, in the two limits, where a permeability-modified aspect ratio,
RL=
L/
H(
kv/
kh)
1/2, becomes large or small, respectively. The first limit is known as transverse (or vertical) equilibrium, the second leads to the problem of non-communicating layers (the Dykstra–Parsons problem). In either case, the problem reduces to the solution of a single integro-differential equation. Although at opposite limits of the parameter
RL, the two regimes coincide in the case of equal viscosities,
M=1. By comparison with high-resolution simulation we investigate the validity of these two approximations. The evolution of transverse averages, particularly under viscous fingering conditions, depends on
RL. We investigate the development of a model to describe viscous fingering in weakly heterogeneous porous media under transverse equilibrium conditions, and compare with the various existing empirical models (such as the Koval, Todd–Longstaff and Fayers models).
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