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.

scholarly journals Axisymmetric viscous gravity currents flowing over a porous medium

2009 ◽  
Vol 622 ◽  
pp. 135-144 ◽  
Author(s):  
MELISSA J. SPANNUTH ◽  
JEROME A. NEUFELD ◽  
J. S. WETTLAUFER ◽  
M. GRAE WORSTER
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Gravity Current ◽  
Gravity Currents ◽  
Constant Input ◽  
Radial Extent ◽  
Vertically Aligned ◽  
Analytical Expressions ◽  
Permeability Structure ◽  
Good Agreement

We study the axisymmetric propagation of a viscous gravity current over a deep porous medium into which it also drains. A model for the propagation and drainage of the current is developed and solved numerically in the case of constant input from a point source. In this case, a steady state is possible in which drainage balances the input, and we present analytical expressions for the resulting steady profile and radial extent. We demonstrate good agreement between our experiments, which use a bed of vertically aligned tubes as the porous medium, and the theoretically predicted evolution and steady state. However, analogous experiments using glass beads as the porous medium exhibit a variety of unexpected behaviours, including overshoot of the steady-state radius and subsequent retreat, thus highlighting the importance of the porous medium geometry and permeability structure in these systems.

Thermal Convective Instabilities in a Porous Medium

1984 ◽  
Vol 106 (1) ◽  
pp. 137-142 ◽  
Author(s):  
M. Kaviany
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Temperature Distribution ◽  
Rayleigh Number ◽  
Saturated Porous Medium ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Onset Of Convection ◽  
Long Time ◽  
Nonlinear Temperature

The onset of convection due to a nonlinear and time-dependent temperature stratification in a saturated porous medium with upper and lower free surfaces is considered. The initial parabolic temperature distribution is due to uniform internal heating. The medium is then cooled by decreasing the upper surface temperature linearly with time. Linear stability theory is applied to the more formally developed governing equations. In order to obtain an asymptotic solution for transient problems involving very long time scales, the critical Rayleigh number for steady-state, nonlinear temperature distribution is also obtained. The effects of porosity, permeability, and Prandtl number on the time of the onset of convection are examined. The steady-state results show that the critical Rayleigh number depends only on the ratio of porosity to permeability and when this ratio exceeds a value of one thousand, the critical Rayleigh number is directly proportional to this ratio.

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.

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.

Entraining gravity currents

2013 ◽  
Vol 731 ◽  
pp. 477-508 ◽  
Author(s):  
Christopher G. Johnson ◽  
Andrew J. Hogg
Keyword(s):  
Richardson Number ◽  
Solution Structure ◽  
Large Scale ◽  
Gravity Current ◽  
Gravity Currents ◽  
Similarity Solutions ◽  
Water Model ◽  
Experimental Measurements ◽  
Entrainment Rate ◽  
Long Time

AbstractEntrainment of ambient fluid into a gravity current, while often negligible in laboratory-scale flows, may become increasingly significant in large-scale natural flows. We present a theoretical study of the effect of this entrainment by augmenting a shallow water model for gravity currents under a deep ambient with a simple empirical model for entrainment, based on experimental measurements of the fluid entrainment rate as a function of the bulk Richardson number. By analysing long-time similarity solutions of the model, we find that the decrease in entrainment coefficient at large Richardson number, due to the suppression of turbulent mixing by stable stratification, qualitatively affects the structure and growth rate of the solutions, compared to currents in which the entrainment is taken to be constant or negligible. In particular, mixing is most significant close to the front of the currents, leading to flows that are more dilute, deeper and slower than their non-entraining counterparts. The long-time solution of an inviscid entraining gravity current generated by a lock-release of dense fluid is a similarity solution of the second kind, in which the current grows as a power of time that is dependent on the form of the entrainment law. With an entrainment law that fits the experimental measurements well, the length of currents in this entraining inviscid regime grows with time approximately as ${t}^{0. 447} $. For currents instigated by a constant buoyancy flux, a different solution structure exists in which the current length grows as ${t}^{4/ 5} $. In both cases, entrainment is most significant close to the current front.

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.

Steady State Solutions to a Forced Nonlinear Oscillator: A Comparison of Results Using a Generalized Harmonic Balance Method and by Numerical Integrations

1993 ◽  
Author(s):  
Ben Noble ◽  
Julian J. Wu
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Initial Point ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Original Equation ◽  
Long Time ◽  
Steady State Solutions ◽  
Generalized Harmonic Balance Method

Abstract Steady state solutions for nonlinear dynamic problems are interesting because (1) the long time behaviors of many problems are of practical concern, and, (2) these behaviors are often difficult to predict. This paper first presents a brief description of a generalized harmonic balance method (GHB) for steady state solutions to nonlinear problems via a nonlinear oscillator problem with a quadratic nonlinearity. Using this approach, steady state solutions are obtained for problems with several parameters: damping, nonlinearity and frequency (subharmonic, superharmonic and primary resonance). These results, plotted in time evolution curves and phase diagrams are compared with those obtained by numerically integrating the original differential equations. The effect of initial conditions on long time solutions is discussed. This investigation indicates that (1) the GHB steady state is an excellent approximate solution to that of the original equation if such a solution is numerically stable, and (2) the GHB steady state simply indicates a region of instability when the numerical solution to the original equation, using a point in that region as the initial point, is unstable.

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 On Steady MHD Thermally Radiating and Reacting Thermosolutal Viscous Flow through a Channel with Porous Medium

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Promise Mebine ◽  
Rhoda H. Gumus
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Boundary Value Problem ◽  
Viscous Flow ◽  
Nonlinear Boundary ◽  
Approximate Solutions ◽  
Nonlinear Boundary Value Problem ◽  
Arrhenius Kinetics ◽  
Flow Through ◽  
Steady State Solutions

This paper investigates steady-state solutions to MHD thermally radiating and reacting thermosolutal viscous flow through a channel with porous medium. The reaction is assumed to be strongly exothermic under generalized Arrhenius kinetics, neglecting the consumption of the material. Approximate solutions are constructed for the governing nonlinear boundary value problem using WKBJ approximations. The results, which are discussed with the aid of the dimensionless parameters entering the problem, are seen to depend sensitively on the parameters.

Two-phase gravity currents in porous media

2011 ◽  
Vol 678 ◽  
pp. 248-270 ◽  
Author(s):  
MADELEINE J. GOLDING ◽  
JEROME A. NEUFELD ◽  
MARC A. HESSE ◽  
HERBERT E. HUPPERT
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Current Model ◽  
Gravity Current ◽  
Capillary Forces ◽  
Gravity Currents ◽  
Flow In Porous Media ◽  
Two Phase ◽  
Residual Trapping

We develop a model describing the buoyancy-driven propagation of two-phase gravity currents, motivated by problems in groundwater hydrology and geological storage of carbon dioxide (CO2). In these settings, fluid invades a porous medium saturated with an immiscible second fluid of different density and viscosity. The action of capillary forces in the porous medium results in spatial variations of the saturation of the two fluids. Here, we consider the propagation of fluid in a semi-infinite porous medium across a horizontal, impermeable boundary. In such systems, once the aspect ratio is large, fluid flow is mainly horizontal and the local saturation is determined by the vertical balance between capillary and gravitational forces. Gradients in the hydrostatic pressure along the current drive fluid flow in proportion to the saturation-dependent relative permeabilities, thus determining the shape and dynamics of two-phase currents. The resulting two-phase gravity current model is attractive because the formalism captures the essential macroscopic physics of multiphase flow in porous media. Residual trapping of CO2 by capillary forces is one of the key mechanisms that can permanently immobilize CO2 in the societally important example of geological CO2 sequestration. The magnitude of residual trapping is set by the areal extent and saturation distribution within the current, both of which are predicted by the two-phase gravity current model. Hence the magnitude of residual trapping during the post-injection buoyant rise of CO2 can be estimated quantitatively. We show that residual trapping increases in the presence of a capillary fringe, despite the decrease in average saturation.

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