Converging gravity currents over a permeable substrate

2015 ◽  
Vol 778 ◽  
pp. 669-690 ◽  
Author(s):  
Zhong Zheng ◽  
Sangwoo Shin ◽  
Howard A. Stone
Keyword(s):  
Power Law ◽  
Gravity Current ◽  
Gravity Currents ◽  
Numerical Predictions ◽  
Front Position ◽  
Dependent Position ◽  
Self Similar ◽  
Vertical Leakage ◽  
Similar Solutions ◽  
Lock Gate

We study the propagation of viscous gravity currents along a thin permeable substrate where slow vertical drainage is allowed from the boundary. In particular, we report the effect of this vertical fluid drainage on the second-kind self-similar solutions for the shape of the fluid–fluid interface in three contexts: (i) viscous axisymmetric gravity currents converging towards the centre of a cylindrical container; (ii) viscous gravity currents moving towards the origin in a horizontal Hele-Shaw channel with a power-law varying gap thickness in the horizontal direction; and (iii) viscous gravity currents propagating towards the origin of a porous medium with horizontal permeability and porosity gradients in power-law forms. For each of these cases with vertical leakage, we identify a regime diagram that characterizes whether the front reaches the origin or not; in particular, when the front does not reach the origin, we calculate the final location of the front. We have also conducted laboratory experiments with a cylindrical lock gate to generate a converging viscous gravity current where vertical fluid drainage is allowed from various perforated horizontal substrates. The time-dependent position of the propagating front is captured from the experiments, and the front position is found to agree well with the theoretical and numerical predictions when surface tension effects can be neglected.

Self-similar viscous gravity currents: phase-plane formalism

1990 ◽  
Vol 210 ◽  
pp. 155-182 ◽  
Author(s):  
Julio Gratton ◽  
Fernando Minotti
Keyword(s):  
Scaling Laws ◽  
Phase Plane ◽  
Gravity Current ◽  
Classical Problem ◽  
Gravity Currents ◽  
Steady Flows ◽  
Two Phase ◽  
Integral Curves ◽  
Self Similar ◽  
Similar Solutions

A theoretical model for the spreading of viscous gravity currents over a rigid horizontal surface is derived, based on a lubrication theory approximation. The complete family of self-similar solutions of the governing equations is investigated by means of a phase-plane formalism developed in analogy to that of gas dynamics. The currents are represented by integral curves in the plane of two phase variables, Z and V, which are related to the depth and the average horizontal velocity of the fluid. Each integral curve corresponds to a certain self-similar viscous gravity current satisfying a particular set of initial and/or boundary conditions, and is obtained by solving a first-order ordinary differential equation of the form dV/dZ = f(Z, V), where f is a rational function. All conceivable self-similar currents can thus be obtained. A detailed analysis of the properties of the integral curves is presented, and asymptotic formulae describing the behaviour of the physical quantities near the singularities of the phase plane corresponding to sources, sinks, and current fronts are given. The derivation of self-similar solutions from the formalism is illustrated by several examples which include, in addition to the similarity flows studied by other authors, many other novel ones such as the extension to viscous flows of the classical problem of the breaking of a dam, the flows over plates with borders, as well as others. A self-similar solution of the second kind describing the axisymmetric collapse of a current towards the origin is obtained. The scaling laws for these flows are derived. Steady flows and progressive wave solutions are also studied and their connection to self-similar flows is discussed. The mathematical analogy between viscous gravity currents and other physical phenomena such as nonlinear heat conduction, nonlinear diffusion, and ground water motion is commented on.

scholarly journals Influence of heterogeneity on second-kind self-similar solutions for viscous gravity currents

2014 ◽  
Vol 747 ◽  
pp. 218-246 ◽  
Author(s):  
Zhong Zheng ◽  
Ivan C. Christov ◽  
Howard A. Stone
Keyword(s):  
Power Law ◽  
Numerical Solutions ◽  
Gravity Currents ◽  
Similar Solution ◽  
Phase Plane Analysis ◽  
Plane Analysis ◽  
Self Similar Solution ◽  
Theoretical Predictions ◽  
Self Similar ◽  
Similar Solutions

AbstractWe report experimental, theoretical and numerical results on the effects of horizontal heterogeneities on the propagation of viscous gravity currents. We use two geometries to highlight these effects: (a) a horizontal channel (or crack) whose gap thickness varies as a power-law function of the streamwise coordinate; (b) a heterogeneous porous medium whose permeability and porosity have power-law variations. We demonstrate that two types of self-similar behaviours emerge as a result of horizontal heterogeneity: (a) a first-kind self-similar solution is found using dimensional analysis (scaling) for viscous gravity currents that propagate away from the origin (a point of zero permeability); (b) a second-kind self-similar solution is found using a phase-plane analysis for viscous gravity currents that propagate toward the origin. These theoretical predictions, obtained using the ideas of self-similar intermediate asymptotics, are compared with experimental results and numerical solutions of the governing partial differential equation developed under the lubrication approximation. All three results are found to be in good agreement.

Propagation of viscous currents on a porous substrate with finite capillary entry pressure

2016 ◽  
Vol 801 ◽  
pp. 65-90 ◽  
Author(s):  
Roiy Sayag ◽  
Jerome A. Neufeld
Keyword(s):  
Laboratory Experiments ◽  
Gravity Current ◽  
Fluid Pressure ◽  
Analytical Techniques ◽  
Gravity Currents ◽  
Porous Substrate ◽  
Entry Pressure ◽  
Self Similar ◽  
Capillary Entry Pressure ◽  
Theoretical Results

We study the propagation of viscous gravity currents over a thin porous substrate with finite capillary entry pressure. Near the origin, where the current is deep, propagation of the current coincides with leakage through the substrate. Near the nose of the current, where the current is thin and the fluid pressure is below the capillary entry pressure, drainage is absent. Consequently the flow can be characterised by the evolution of drainage and fluid fronts. We analyse this flow using numerical and analytical techniques combined with laboratory-scale experiments. At early times, we find that the position of both fronts evolve as $t^{1/2}$, similar to an axisymmetric gravity current on an impermeable substrate. At later times, the growing effect of drainage inhibits spreading, causing the drainage front to logarithmically approach a steady position. In contrast, the asymptotic propagation of the fluid front is quasi-self-similar, having identical structure to the solution of gravity currents on an impermeable substrate, only with slowly varying fluid flux. We benchmark these theoretical results with laboratory experiments that are consistent with our modelling assumption, but that also highlight the detailed dynamics of drainage inhibited by finite capillary pressure.

Gravity currents over fractured substrates in a porous medium

2007 ◽  
Vol 584 ◽  
pp. 415-431 ◽  
Author(s):  
DAVID PRITCHARD
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Gravity Current ◽  
Gravity Currents ◽  
Limited Extent ◽  
Form Part ◽  
Long Time ◽  
Self Similar ◽  
Steady State Solutions ◽  
A Current

We consider the behaviour of a gravity current in a porous medium when the horizontal surface along which it spreads is punctuated either by narrow fractures or by permeable regions of limited extent. We derive steady-state solutions for the current, and show that these form part of a long-time asymptotic description which may also include a self-similar ‘leakage current’ propagating beyond the fractured region with a length proportional to t1/2. We discuss the conditions under which a current can be completely trapped by a permeable region or a series of fractures.

The Influence of Boundaries on Gravity Currents and Thin Films: Drainage, Confinement, Convergence, and Deformation Effects

2022 ◽  
Vol 54 (1) ◽  
pp. 27-56
Author(s):  
Zhong Zheng ◽  
Howard A. Stone
Keyword(s):  
Thin Film ◽  
Oil And Gas ◽  
Gravity Currents ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Self Similar ◽  
Similar Solutions ◽  
Film Flows ◽  
Gravity Surface

Thin film flows, whether driven by gravity, surface tension, or the relaxation of elastic boundaries, occur in many natural and industrial processes. Applications span problems of oil and gas transport in channels to hydraulic fracture, subsurface propagation of pollutants, storage of supercritical CO2 in porous formations, and flow in elastic Hele–Shaw configurations and their relatives. We review the influence of boundaries on the dynamics of thin film flows, with a focus on gravity currents, including the effects of drainage into the substrate, and the role of the boundaries to confine the flow, force its convergence to a focus, or deform, and thus feedback to alter the flow. In particular, we highlight reduced-order models. In many cases, self-similar solutions can be determined and describe the behaviors in canonical problems at different timescales and length scales, including self-similar solutions of both the first and second kind. Additionally, the time transitions between different solutions are summarized. Where possible, remarks about various applications are provided.

scholarly journals Stability Analysis of Gravity Currents of a Power-Law Fluid in a Porous Medium

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Sandro Longo ◽  
Vittorio Di Federico
Keyword(s):  
Porous Medium ◽  
Stability Analysis ◽  
Linear Stability ◽  
Power Law ◽  
Gravity Currents ◽  
Two Dimensional ◽  
Continuum Spectrum ◽  
Fluid Behavior ◽  
Self Similar ◽  
Flow Domain

We analyse the linear stability of self-similar shallow, two-dimensional and axisymmetric gravity currents of a viscous power-law non-Newtonian fluid in a porous medium. The flow domain is initially saturated by a fluid lighter than the intruding fluid, whose volume varies with time astα. The transition between decelerated and accelerated currents occurs atα= 2 for two-dimensional and atα= 3 for axisymmetric geometry. Stability is investigated analytically for special values ofαand numerically in the remaining cases; axisymmetric currents are analysed only for radially varying perturbations. The two-dimensional currents are linearly stable forα< 2 (decelerated currents) with a continuum spectrum of eigenvalues and unstable forα= 2, with a growth rate proportional to the square of the fluid behavior index. The axisymmetric currents are linearly stable for anyα< 3 (decelerated currents) with a continuum spectrum of eigenvalues, while forα= 3 no firm conclusion can be drawn. Forα> 2 (two-dimensional accelerated currents) andα> 3 (axisymmetric accelerated currents) the linear stability analysis is of limited value since the hypotheses of the model will be violated.

scholarly journals Lubricated viscous gravity currents

2015 ◽  
Vol 766 ◽  
pp. 626-655 ◽  
Author(s):  
Katarzyna N. Kowal ◽  
M. Grae Worster
Keyword(s):  
Gravity Current ◽  
Gravity Currents ◽  
Singular Limit ◽  
Vertical Shear ◽  
Surface Films ◽  
Viscous Coupling ◽  
Self Similarity ◽  
Transient Behaviour ◽  
Fingering Instability ◽  
Self Similar

AbstractWe present a theoretical and experimental study of viscous gravity currents lubricated by another viscous fluid from below. We use lubrication theory to model both layers as Newtonian fluids spreading under their own weight in two-dimensional and axisymmetric settings over a smooth rigid horizontal surface and consider the limit in which vertical shear provides the dominant resistance to the flow in both layers. There are contributions from Poiseuille-like flow driven by buoyancy and Couette-like flow driven by viscous coupling between the layers. The flow is self-similar if both fluids are released simultaneously, and exhibits initial transient behaviour when there is a delay between the initiation of flow in the two layers. We solve for both situations and show that the latter converges towards self-similarity at late times. The flow depends on three key dimensionless parameters relating the relative dynamic viscosities, input fluxes and density differences between the two layers. Provided the density difference between the two layers is bounded away from zero, we find an asymptotic solution in which the front of the lubricant is driven by its own gravitational spreading. There is a singular limit of equal densities in which the lubricant no longer spreads under its own weight in the vicinity of its nose and ends abruptly with a non-zero thickness there. We explore various regimes, from thin lubricating layers underneath a more viscous current to thin surface films coating an underlying more viscous current and find that although a thin film does not greatly influence the more viscous current if it forms a surface coating, it begins to cause interesting dynamics if it lubricates the more viscous current from below. We find experimentally that a lubricated gravity current is prone to a fingering instability.

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