scholarly journals Stratified gravity currents in porous media

2016 ◽  
Vol 791 ◽  
pp. 329-357 ◽  
Author(s):  
Samuel S. Pegler ◽  
Herbert E. Huppert ◽  
Jerome A. Neufeld
Keyword(s):  
Porous Media ◽  
Density Distribution ◽  
Vertical Structure ◽  
Gravity Current ◽  
Gravity Currents ◽  
Similarity Solutions ◽  
Density Field ◽  
Vertical Position ◽  
Uniform Density ◽  
Convex Shape

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.

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.

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.

Critical regime of gravity currents flowing in non-rectangular channels with density stratification

2018 ◽  
Vol 840 ◽  
pp. 579-612
Author(s):  
L. Chiapponi ◽  
M. Ungarish ◽  
S. Longo ◽  
V. Di Federico ◽  
F. Addona
Keyword(s):  
Cross Section ◽  
Power Law ◽  
Rectangular Cross Section ◽  
Gravity Current ◽  
Gravity Currents ◽  
Critical Conditions ◽  
Uniform Density ◽  
Ambient Fluid ◽  
Real Numbers ◽  
Positive Real

We present theoretical and experimental analyses of the critical condition where the inertial–buoyancy or viscous–buoyancy regime is preserved in a uniform-density gravity current (which propagates over a horizontal plane) of time-variable volume ${\mathcal{V}}=qt^{\unicode[STIX]{x1D6FF}}$ in a power-law cross-section (with width described by $f(z)=bz^{\unicode[STIX]{x1D6FC}}$, where $z$ is the vertical coordinate, $b$ and $q$ are positive real numbers, and $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FF}$ are non-negative real numbers) occupied by homogeneous or linearly stratified ambient fluid. The magnitude of the ambient stratification is represented by the parameter $S$, with $S=0$ and $S=1$ describing the homogeneous and maximum stratification cases respectively. Earlier theoretical and experimental results valid for a rectangular cross-section ($\unicode[STIX]{x1D6FC}=0$) and uniform ambient fluid are generalized here to a power-law cross-section and stratified ambient. Novel time scalings, obtained for inertial and viscous regimes, allow a derivation of the critical flow parameter $\unicode[STIX]{x1D6FF}_{c}$ and the corresponding propagation rate as $Kt^{\unicode[STIX]{x1D6FD}_{c}}$ as a function of the problem parameters. Estimates of the transition length between the inertial and viscous regimes are also derived. A series of experiments conducted in a semicircular cross-section ($\unicode[STIX]{x1D6FC}=1/2$) validate the critical values $\unicode[STIX]{x1D6FF}_{c}=2$ and $\unicode[STIX]{x1D6FF}_{c}=9/4$ for the two cases $S=0$ and $1$. The ratio between the inertial and viscous forces is determined by an effective Reynolds number proportional to $q$ at some power. The threshold value of this number, which enables a determination of the regime of the current (inertial–buoyancy or viscous–buoyancy) in critical conditions, is determined experimentally for both $S=0$ and $S=1$. We conclude that a very significant generalization of the insights and results from two-dimensional (rectangular cross-section channel) gravity currents to power-law cross-sections is available.

The effect of confining impermeable boundaries on gravity currents in a porous medium

2010 ◽  
Vol 649 ◽  
pp. 1-17 ◽  
Author(s):  
MADELEINE J. GOLDING ◽  
HERBERT E. HUPPERT
Keyword(s):  
Free Surface ◽  
Gravity Current ◽  
Mass Conservation ◽  
Propagation Rate ◽  
Gravity Currents ◽  
Similarity Solutions ◽  
Constant Flux ◽  
Channel Shape ◽  
Dimensional Gravity ◽  
Uniform Cross Section

The effect of confining boundaries on gravity currents in porous media is investigated theoretically and experimentally. Similarity solutions are derived for currents when the volume increases as tα in horizontal channels of uniform cross-section with boundary height b satisfying b ~ a|y/a|n, where y is the cross-channel coordinate and a is a length scale of the channel width. Experiments were carried out in V-shaped and semicircular channels for the case of gravity currents with constant volume (α=0) and constant flux (α=1). These showed generally good agreement with the theory.Typically, we find that the propagation of the current is well described by L ~ tc for some scalar c. We study the dependence of c on the time exponent of the volume of fluid in the current, α, and the geometry of the channel, parameterized by n. For all channel shapes, there exists a critical value of α, αc = 1/2, above which increasing n causes an increase in c and below which increasing n causes a decrease in c, where increasing n corresponds to opening up the channel boundary to the horizontal. The current height increases or decreases with respect to time depending on whether α is greater or less than αc. It is this fact, along with global mass conservation, which explains why varying the channel shape n affects the propagation rate c in different ways depending on α.We also consider channels inclined at an angle θ to the horizontal. When the slope of the channel is much greater than the slope of the free surface of the current, the component of gravity parallel to the slope dominates, causing the current to move with a constant velocity, Vf say, regardless of channel shape n and flux parameter α, in agreement with results for a two-dimensional gravity current obtained by Huppert & Woods (1995) and some initially axisymmetric gravity currents presented by Vella & Huppert (2006). If the effect of the component of gravity perpendicular to the channel may not be neglected, i.e. if the slopes of the channel and free surface of the current are comparable, we find that, in a frame moving with speed Vf, the form of the governing equation for the height of a current in an equivalent horizontal channel is recovered. We calculate that the height of a constant flux gravity current down an inclined channel will tend to a fixed depth, which is determined by the channel shape, n, and the physical properties of the fluid and rock. Experimental and numerical results for inclined V-shaped channels agree very well with this theory.

The effects of drag on turbulent gravity currents

2000 ◽  
Vol 416 ◽  
pp. 297-314 ◽  
Author(s):  
LYNNE HATCHER ◽  
ANDREW J. HOGG ◽  
ANDREW W. WOODS
Keyword(s):  
Analytical Solution ◽  
Laboratory Experiments ◽  
Gravity Current ◽  
Gravity Currents ◽  
Similarity Solutions ◽  
Flow Speed ◽  
Series Solutions ◽  
Accurate Approximation ◽  
New Class ◽  
Power Series Solutions

We model the propagation of turbulent gravity currents through an array of obstacles which exert a drag force on the flow proportional to the square of the flow speed. A new class of similarity solutions is constructed to describe the flows that develop from a source of strength q0tγ. An analytical solution exists for a finite release, γ = 0, while power series solutions are developed for sources with γ > 0. These are shown to provide an accurate approximation to the numerically calculated similarity solutions. The model is successfully tested against a series of new laboratory experiments which investigate the motion of a turbulent gravity current through a large flume containing an array of obstacles. The model is extended to account for the effects of a sloping boundary. Finally, a series of geophysical and environmental applications of the model are discussed.

Particle-driven gravity currents down planar slopes

1999 ◽  
Vol 390 ◽  
pp. 75-91 ◽  
Author(s):  
ROGER T. BONNECAZE ◽  
JOHN R. LISTER
Keyword(s):  
Gravity Current ◽  
Gravity Currents ◽  
Similarity Solutions ◽  
Bottom Friction ◽  
Turbidity Currents ◽  
Scaling Analysis ◽  
Ambient Fluid ◽  
Buoyancy Driven Flows ◽  
Depositional Patterns ◽  
Planar Slope

Particle-driven gravity currents, as exemplified by either turbidity currents in the ocean or ignimbrite flows in the atmosphere, are buoyancy-driven flows due to a suspension of dense particles in an ambient fluid. We present a theoretical study on the dynamics of and deposition from a turbulent current flowing down a uniform planar slope from a constant-flux point source of particle-laden fluid. The flow is modelled using the shallow-water equations, including the effects of bottom friction and entrainment of ambient fluid, coupled to an equation for the transport and settling of the particles. Two flow regimes are identified. Near the source and for mild slopes, the flow is dominated by a balance between buoyancy and bottom friction. Further downstream and for steeper slopes, entrainment also affects the behaviour of the current. Similarity solutions are also developed for the simple cases of homogeneous gravity currents with no settling of particles in the friction-dominated and entrainment-dominated regimes. Estimates of the width and length of the deposit from a monodisperse particle-driven gravity current with settling are derived from scaling analysis for each regime, and the contours of the depositional patterns are determined from numerical solution of the governing equations.

scholarly journals Gravity currents and internal waves in a stratified fluid

2008 ◽  
Vol 616 ◽  
pp. 327-356 ◽  
Author(s):  
BRIAN L. WHITE ◽  
KARL R. HELFRICH
Keyword(s):  
Internal Waves ◽  
Wave Speed ◽  
Numerical Solutions ◽  
Gravity Current ◽  
Gravity Currents ◽  
Stratified Fluid ◽  
Front Speed ◽  
Long Wave ◽  
Conjugate State ◽  
Energy Conserving

A steady theory is presented for gravity currents propagating with constant speed into a stratified fluid with a general density profile. Solution curves for front speed versus height have an energy-conserving upper bound (the conjugate state) and a lower bound marked by the onset of upstream influence. The conjugate state is the largest-amplitude nonlinear internal wave supported by the ambient stratification, and in the limit of weak stratification approaches Benjamin's energy-conserving gravity current solution. When the front speed becomes critical with respect to linear long waves generated above the current, steady solutions cannot be calculated, implying upstream influence. For non-uniform stratification, the critical long-wave speed exceeds the ambient long-wave speed, and the critical-Froude-number condition appropriate for uniform stratification must be generalized. The theoretical results demonstrate a clear connection between internal waves and gravity currents. The steady theory is also compared with non-hydrostatic numerical solutions of the full lock release initial-value problem. Some solutions resemble classic gravity currents with no upstream disturbance, but others show long internal waves propagating ahead of the gravity current. Wave generation generally occurs when the stratification and current speed are such that the steady gravity current theory fails. Thus the steady theory is consistent with the occurrence of either wave-generating or steady gravity solutions to the dam-break problem. When the available potential energy of the dam is large enough, the numerical simulations approach the energy-conserving conjugate state. Existing laboratory experiments for intrusions and gravity currents produced by full-depth lock exchange flows over a range of stratification profiles show excellent agreement with the conjugate state solutions.

scholarly journals Existence and features of similarity solutions for non-Boussinesq gravity currents

2007 ◽  
Vol 226 (1) ◽  
pp. 32-54 ◽  
Author(s):  
C. Ancey ◽  
S. Cochard ◽  
M. Rentschler ◽  
S. Wiederseiner
Keyword(s):  
Gravity Currents ◽  
Similarity Solutions
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