Fluid invasion of an unsaturated leaky porous layer

We study the flow and leakage of gravity currents injected into an unsaturated (dry), vertically confined porous layer containing a localized outlet or leakage point in its lower boundary. The leakage is driven by the combination of the gravitational hydrostatic pressure head of the current above the outlet and the pressure build-up from driving fluid downstream of the leakage point. Model solutions illustrate transitions towards one of three long-term regimes of flow, depending on the value of a dimensionless parameter D, which, when positive, represents the ratio of the hydrostatic head above the outlet for which gravity-driven leakage balances the input flux, to the depth of the medium. If D⩽0, the input flux is insufficient to accumulate any fluid above the outlet and fluid migrates directly through the leakage pathway. If 0<D⩽1, some fluid propagates downstream of the outlet but retains a free surface above it. The leakage rate subsequently approaches the input flux asymptotically but much more gradually than if D⩽0. If D>1, the current fills the entire depth of the medium above the outlet. Confinement then fixes gravity-driven leakage at a constant rate but introduces a new force driving leakage in the form of the pressure build-up associated with mobilizing fluid downstream of the outlet. This causes the leakage rate to approach the injection rate faster than would occur in the absence of the confining boundary. This conclusion is in complete contrast to fluid-saturated media, where confinement can potentially reduce long-term leakage by orders of magnitude. Data from a new series of laboratory experiments confirm these predictions

Topographic controls on gravity currents in porous media

We present a theoretical and experimental study of the propagation of gravity currents in porous media with variations in the topography over which they flow, motivated in part by the sequestration of carbon dioxide in saline aquifers. We consider cases where the height of the topography slopes upwards in the direction of the flow and is proportional to the nth power of the horizontal distance from a line or point source of a constant volumetric flux. In two-dimensional cases with n>1/2, the current evolves from a self-similar form at early times, when the effects of variations in topography are negligible, towards a late-time regime that has an approximately horizontal upper surface and whose evolution is dictated entirely by the geometry of the topography. For n<1/2, the transition between these flow regimes is reversed. We compare our theoretical results in the case n=1 with data from a series of laboratory experiments in which viscous glycerine is injected into an inclined Hele-Shaw cell, obtaining good agreement between the theoretical results and the experimental data. In the case of axisymmetric topography, all topographic exponents n>0 result in a transition from an early-time similarity solution towards a topographically controlled regime that has an approximately horizontal free surface. We also analyse the evolution over topography that can vary with different curvatures and topographic exponents between the two horizontal dimensions, finding that the flow transitions towards a horizontally topped regime at a rate which depends strongly on the ratio of the curvatures along the principle axes. Finally, we apply our mathematical solutions to the geophysical setting at the Sleipner field, concluding that topographic influence is unlikely to explain the observed non-axisymmetric flow

Fluid invasion of an unsaturated leaky porous layer

We study the flow and leakage of gravity currents injected into an unsaturated (dry), vertically confined porous layer containing a localized outlet or leakage point in its lower boundary. The leakage is driven by the combination of the gravitational hydrostatic pressure head of the current above the outlet and the pressure build-up from driving fluid downstream of the leakage point. Model solutions illustrate transitions towards one of three long-term regimes of flow, depending on the value of a dimensionless parameter D, which, when positive, represents the ratio of the hydrostatic head above the outlet for which gravity-driven leakage balances the input flux, to the depth of the medium. If D⩽0, the input flux is insufficient to accumulate any fluid above the outlet and fluid migrates directly through the leakage pathway. If 01, the current fills the entire depth of the medium above the outlet. Confinement then fixes gravity-driven leakage at a constant rate but introduces a new force driving leakage in the form of the pressure build-up associated with mobilizing fluid downstream of the outlet. This causes the leakage rate to approach the injection rate faster than would occur in the absence of the confining boundary. This conclusion is in complete contrast to fluid-saturated media, where confinement can potentially reduce long-term leakage by orders of magnitude. Data from a new series of laboratory experiments confirm these predictions

Stratified gravity currents in porous media

We consider theoretically and experimentally the propagation in porous media of variable-density gravity currents containing a stably stratified density field, with most previous studies of gravity currents having focused on cases of uniform density. New thin-layer equations are developed to describe stably stratified fluid flows in which the density field is materially advected with the flow. Similarity solutions describing both the fixed-volume release of a distributed density stratification and the continuous input of fluid containing a distribution of densities are obtained. The results indicate that the density distribution of the stratification significantly influences the vertical structure of the gravity current. When more mass is distributed into lighter densities, it is found that the shape of the current changes from the convex shape familiar from studies of the uniform-density case to a concave shape in which lighter fluid accumulates primarily vertically above the origin of the current. For a constant-volume release, the density contours stratify horizontally, a simplification which is used to develop analytical solutions. For currents introduced continuously, the horizontal velocity varies with vertical position, a feature which does not apply to uniform-density gravity currents in porous media. Despite significant effects on vertical structure, the density distribution has almost no effect on overall horizontal propagation, for a given total mass. Good agreement with data from a laboratory study confirms the predictions of the model

Salt fingering staircases and the three-component Phillips effect

Understanding the dynamics of staircases in salt fingering convection presents a long-standing theoretical challenge to fluid dynamicists. Although there has been significant progress, particularly through numerical simulations, there are a number of conflicting theoretical explanations as to the driving mechanism underlying staircase formation. The Phillips effect proposes that layering in stirred stratified flow is due to an antidiffusive process, and it has been suggested that this mechanism may also be responsible for salt fingering staircases. However, the details of this process, as well as mathematical models to predict the evolution and merger dynamics of staircases, have yet to be developed. We generalise the theory of the Phillips effect to a three-component system (e.g. temperature, salinity, energy) and demonstrate the first regularised nonlinear model of layering based on mixing-length parameterisations. The model predicts both the inception of layering and its long-term evolution through mergers , whilst generalising, and remaining consistent with, previous results for double-diffusive layering based on flux ratios. Our model of salt fingering is formulated using spatial averaging processes and closed by a mixing length parameterised in terms of the kinetic energy and the salt and temperature gradients. The model predicts a layering instability for a bounded range of parameter values in the salt fingering regime. Nonlinear solutions show that an initially unstable linear buoyancy gradient develops into layers, which merge through a process of stronger interfaces growing at the expense of weaker interfaces. Mergers increase the buoyancy gradient across interfaces, and increase the buoyancy flux through the staircase.Comment: 24 pages, 8 figure

Singularity of Lévy walks in the lifted Pomeau-Manneville map.

Since groundbreaking works in the 1980s it is well-known that simple deterministic dynamical systems can display intermittent dynamics and weak chaos leading to anomalous diffusion. A paradigmatic example is the Pomeau-Manneville (PM) map which, suitably lifted onto the whole real line, was shown to generate superdiffusion that can be reproduced by stochastic Lévy walks (LWs). Here, we report that this matching only holds for parameter values of the PM map that are of Lebesgue measure zero in its two-dimensional parameter space. This is due to a bifurcation scenario that the map exhibits under variation of one parameter. Constraining this parameter to specific singular values at which the map generates superdiffusion by varying the second one, as has been done in the previous literature, we find quantitative deviations between deterministic diffusion and diffusion generated by stochastic LWs in a particular range of parameter values, which cannot be cured by simple LW modifications. We also explore the effect of aging on superdiffusion in the PM map and show that this yields a profound change of the diffusive properties under variation of the aging time, which should be important for experiments. Our findings demonstrate that even in this simplest well-studied setting, a matching of deterministic and stochastic diffusive properties is non-trivial