Avinoam Rabinovich

Vorticity inversion and action-at-a-distance instability in stably stratified shear flow

The somewhat counter-intuitive effect of how stratification destabilizes shear flows and the rationalization of the Miles–Howard stability criterion are re-examined in what we believe to be the simplest example of action-at-a-distance interaction between ‘buoyancy–vorticity gravity wave kernels’. The set-up consists of an infinite uniform shear Couette flow in which the Rayleigh–Fjortoft necessary conditions for shear flow instability are not satisfied. When two stably stratified density jumps are added, the flow may however become unstable. At each density jump the perturbation can be decomposed into two coherent gravity waves propagating horizontally in opposite directions. We show, in detail, how the instability results from a phase-locking action-at-a-distance interaction between the four waves (two at each jump) but can as well be reasonably approximated by the interaction between only the two counter-propagating waves (one at each jump). From this perspective the nature of the instability mechanism is similar to that of the barotropic and baroclinic ones. Next we add a small ambient stratification to examine how the critical-level dynamics alters our conclusions. We find that a strong vorticity anomaly is generated at the critical level because of the persistent vertical velocity induction by the interfacial waves at the jumps. This critical-level anomaly acts in turn at a distance to dampen the interfacial waves. When the ambient stratification is increased so that the Richardson number exceeds the value of a quarter, this destructive interaction overwhelms the constructive interaction between the interfacial waves, and consequently the flow becomes stable. This effect is manifested when considering the different action-at-a-distance contributions to the energy flux divergence at the critical level. The interfacial-wave interaction is found to contribute towards divergence, that is, towards instability, whereas the critical-level–interfacial-wave interaction contributes towards an energy flux convergence, that is, towards stability.

Analytical approximations for effective relative permeability in the capillary limit

We present an analytical method for calculating two-phase effective relative permeability, krjeff, where j designates phase (here CO2 and water), under steady state and capillary limit assumptions. These effective relative permeabilities may be applied in experimental settings and for upscaling in the context of numerical flow simulations, e.g., for CO2 storage. An exact solution for effective absolute permeability, keff, in two-dimensional log-normally distributed isotropic permeability (k) fields is the geometric mean. We show that this does not hold for since log normality is not maintained in the capillary limit phase permeability field (kkrj) when capillary pressure, and thus the saturation field, is varied. Nevertheless, the geometric mean is still shown to be suitable for approximating krjeff when the variance of ln k is low. For high variance cases, we apply a correction to the geometric-average gas effective relative permeability using a Winsorized mean, which neglects large and small Kj values symmetrically. The analytical method is extended to anisotropically correlated log-normal permeability fields using power law averaging. In these cases the Winsorized mean treatment is applied to the gas curves for cases described by negative power law exponents (flow across incomplete layers). The accuracy of our analytical expressions for is demonstrated through extensive numerical tests, using low- and high-variance permeability realizations with a range of correlation structures. We also present integral expressions for geometric-mean and power law average krjeff for the systems considered, which enable derivation of closed-form series solutions for without generating permeability realizations. This article is protected by copyright. All rights reserved.

Oscillatory pumping wells in phreatic, compressible, and homogeneous aquifers

Oscillatory well pumping was proposed recently as a tool for hydraulic tomography. Periodic pumping at a few frequencies is carried out through vertical intervals along the pumping well and the periodic head is measured along a few piezometers. The paper presents an analytical solution for the head field in an unconfined aquifer of finite depth under the common assumptions of a linearized water table condition, different horizontal and vertical constant permeabilities, constant specific storativity and water table drainable porosity, and small well radius to length ratio. The solution provides the expressions of the amplitude and phase of the head as a function of coordinates, frequency, and the problem parameters. The solution simplifies to one pertaining to an upper constant head condition and a rigid aquifer for a wide range of the dimensionless frequency values.