Self-similar gravity currents in porous media: Linear stability of the Barenblatt–Pattle solution revisited

Mathunjwa, Jochonia S.; Hogg, Andrew J.

doi:10.1016/j.euromechflu.2005.09.005

Cited by 18 publications

(23 citation statements)

References 22 publications

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“…Experiments reported by Mitchell & Woods (2006), numerical modelling of Nordbotten & Celia (2006) and stability arguments (cf. Mathunjwa & Hogg 2005) are also consistent with the assumption of a planar, non-fingered interface.…”

Section: The Buoyant Flow Along An Inclined Boundarysupporting

confidence: 73%

The effect of drainage on the capillary retention of CO₂ in a layered permeable rock

Farcas¹,

Woods²

2009

J. Fluid Mech.

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We examine the motion of a buoyant fluid injected into a water-saturated porous rock as it spreads along a thin inclined low-permeability barrier. We account for leakage of the fluid across the barrier once the current is sufficiently deep so that the pressure exceeds the capillary threshold. We show that at some distance from the source, the pressure decreases below this threshold, and all the remaining flux spreads laterally along the barrier. We examine the controls on the partitioning of the flow between the draining flux and the laterally spreading flux and also the controls on the lateral extent of the draining region for the case of an instantaneous release and a maintained release of fluid. We consider the implications of our work for the dispersal of CO 2 plumes which may be sequestered in deep saline aquifers. IntroductionThe injection of CO 2 into deep saline aquifers or depleted oil and gas fields has received considerable interest in the context of global warming and the related challenge of reducing anthropogenic carbon emissions into the atmosphere (Kumar et al. 2005;Nordbotten, Celia & Bachu 2005). In deep saline aquifers, CO 2 is supercritical and has a density of the order 20 %-40 % smaller than that of water, depending on the temperature and pressure. As a result, there is concern that the CO 2 will migrate through the subsurface and back to human environment, perhaps over time scales of tens to hundreds of years. In order to assess this risk, models are being developed to describe the migration of CO 2 plumes through the subsurface (Hesse et al. , 2007Mitchell & Woods 2006;Vella & Huppert 2006), and data are being collected from various field trials using seismic imaging to interpret the dispersal patterns in the field (Bickle et al. 2007). A significant challenge associated with the prediction of such CO 2 transport is the complexity of the subsurface strata through which the CO 2 migrates. This can lead to multiple and tortuous flow paths from the injection point to the surface, with the CO 2 being dispersed laterally over substantial distances but also reaching the surface through different flow paths over a range of different time scales. Data from the Sleipner field in Norway (Bickle et al. 2007) has shown that the gas spreads out in a pine tree-type pattern, spreading laterally beneath low-permeability layers while gradually migrating upwards as it leaks through these layers or fractures which cut across the layers.Some of the models which have been developed to describe the migration of plumes of CO 2 in the subsurface have assessed the buoyancy-driven dispersal through both †

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Section: The Buoyant Flow Along An Inclined Boundarysupporting

confidence: 73%

The effect of drainage on the capillary retention of CO₂ in a layered permeable rock

Farcas¹,

Woods²

2009

J. Fluid Mech.

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“…We also reported the similarity solutions (2.18) corresponding to pure diffusion. As documented in earlier papers (Mathunjwa & Hogg 2006), the solution of the initial value problem tends rather quickly towards the similarity solutions. Even at short times, there is not much difference between the front position computed numerically and that given by (2.19).…”

Section: Summary and Remarksmentioning

confidence: 74%

“…Theoretically, we idealized this setting as a two-dimensional problem. When slope was shallow, we used lubrication theory and followed earlier authors (Huppert 1982a,b;Grundy 1983a;Gratton & Minotti 1990;Lister 1992;Witelski & Bernoff 1998;Mathunjwa & Hogg 2006) by seeking similarity solutions to the Navier-Stokes equations in the low-Reynolds-number and high-capillary-number limit. Slope shallowness implies that the aspect ratio must be order tan θ.…”

Section: Discussionmentioning

confidence: 99%

“…While most analytical results were worked out by seeking similarity solutions to the governing equations, a few results were also obtained using conformal-mapping techniques (Garabedian 1966;van Vroonhoven & Kuijpers 1990). Specific mathematical issues related to the existence of moving boundaries and how the solution to a particular initial value problem approaches the similarity form at infinite times have also been investigated (Ockendon 1978;Grundy & McLaughlin 1982;Grundy 1983b;Witelski & Bernoff 1998;Mathunjwa & Hogg 2006). In all the papers mentioned above, the governing equations were derived from the Navier-Stokes equations using lubrication theory; another approach taken by Hunt (1994) was to use flow-depth averaged equations of motion (i.e.…”

Section: Introductionmentioning

confidence: 99%

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The dam-break problem for viscous fluids in the high-capillary-number limit

2009

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Experiments were undertaken to investigate dam-break flows where a finite volume of highly viscous fluid (glucose with viscosity μ ≈ 350 Pa s) maintained behind a lock gate was released into a horizontal or inclined flume. The resulting sequence of flow-depth profiles was tracked using a three-dimensional visualization system. In the low-Reynolds-number and high-capillary-number limits, analytical solutions can be obtained from the Navier-Stokes equations using lubrication theory and matched asymptotic expansions. At shallow slopes, similarity solutions can also be worked out. While the variation in the front position scaled with time as predicted by theory for both horizontal and sloping flumes, there was a systematic delay in the front position observed. Moreover, taking a closer look at the experimental flowdepth profiles shows that they were similar, but they noticeably deviated from the theoretical similarity form for horizontal planes. For sloping beds, the flow-depth profile is correctly predicted provided that different scalings are used at shallow and large slopes.

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“…Grundy & McLaughlin 1982;Mathunjwa & Hogg 2006), in (B 1)-(B 3), we obtain the Sturm-Liouville eigenvalue problem…”

Section: Appendix a The Transition Towards Horizontal Flowmentioning

confidence: 99%

Fluid injection into a confined porous layer

2014

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We present a theoretical and experimental study of viscous flows injected into a porous medium that is confined vertically by horizontal impermeable boundaries and filled with an ambient fluid of different density and viscosity. General three-dimensional equations describing such flows are developed, showing that the dynamics can be affected by two separate contributions: spreading due to gradients in hydrostatic pressure, and that due to the pressure drop introduced by the injection. In the illustrative case of a two-dimensional injection of fluid at a constant volumetric rate, the injected fluid initially forms a viscous gravity current insensitive both to the depth of the medium and to the viscosity of the ambient fluid. Beyond a characteristic time scale, the dynamics transition to being dominated by the injection pressure, and the injected fluid eventually intersects the second boundary to form a second moving contact line. Three different late-time asymptotic regimes can emerge, depending on whether the viscosity of the injected fluid is less than, equal to or greater than that of the ambient fluid. With a less viscous injection, the flow undergoes a slow decay towards a similarity solution in which the two contact lines extend linearly in time with differing prefactors. Perturbations from this long-term state are shown to decay algebraically with time. Equal viscosities result in both contact lines approaching the same leading-order asymptotic position but with a first-order correction to the distance between them that expands as t 1/2 due to gravitational spreading. For a more viscous injection, the distance between the contact lines approaches a constant value, with perturbations decaying exponentially. Data from a new series of laboratory experiments confirm these theoretical predictions.

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Self-similar gravity currents in porous media: Linear stability of the Barenblatt–Pattle solution revisited

Cited by 18 publications

References 22 publications

The effect of drainage on the capillary retention of CO₂ in a layered permeable rock

The effect of drainage on the capillary retention of CO₂ in a layered permeable rock

The dam-break problem for viscous fluids in the high-capillary-number limit

Fluid injection into a confined porous layer

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Self-similar gravity currents in porous media: Linear stability of the Barenblatt–Pattle solution revisited

Cited by 18 publications

References 22 publications

The effect of drainage on the capillary retention of CO2 in a layered permeable rock

The effect of drainage on the capillary retention of CO2 in a layered permeable rock

The dam-break problem for viscous fluids in the high-capillary-number limit

Fluid injection into a confined porous layer

Contact Info

Product

Resources

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The effect of drainage on the capillary retention of CO₂ in a layered permeable rock

The effect of drainage on the capillary retention of CO₂ in a layered permeable rock