When a fluid is injected into a porous medium saturated with an ambient fluid of a greater density, the injected fluid forms a plume that rises upwards due to buoyancy. In the near field of the ...injection point, the plume adjusts its speed to match the buoyancy velocity of the porous medium, either thinning or thickening to conserve mass. These adjustments are the dominant controls on the near-field plume shape, rather than mixing with the ambient fluid, which occurs over larger vertical distances. In this study, we focus on the plume behaviour in the near field, demonstrating that for moderate injection rates, the plume will reach a steady state, whereby it matches the buoyancy velocity over a few plume width scales from the injection point. However, for very small injection rates, an instability occurs in which the steady plume breaks apart due to the insurmountable density contrast with the surrounding fluid. The steady shape of the plume in the near field depends only on a single dimensionless parameter, which is the ratio between the inlet velocity and the buoyancy velocity. A linear stability analysis is performed, indicating that for small velocity ratios, an infinitesimal perturbation can be constructed that becomes unstable, whilst for moderate velocity ratios, the shape is shown to be stable. Finally, we comment on the application of such flows to the context of CO$_2$ sequestration in porous geological reservoirs.
On wave-driven propulsion Benham, Graham P.; Devauchelle, Olivier; Thomson, Stuart J.
Journal of fluid mechanics,
05/2024, Letnik:
987
Journal Article
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A theory is presented for wave-driven propulsion of floating bodies driven into oscillation at the fluid interface. By coupling the equations of motion of the body to a quasipotential flow model of ...the fluid, we derive expressions for the drift speed and propulsive thrust of the body which in turn are shown to be consistent with global momentum conservation. We explore the efficacy of our model in describing the motion of SurferBot (Rhee et al., Bioinspir. Biomim., vol. 17, issue 5, 2022), demonstrating close agreement with the experimentally determined drift speed and oscillatory dynamics. The efficiency of wave-driven propulsion is then computed as a function of driving oscillation frequency and the forcing location, revealing optimal values for both of these parameters which await confirmation in experiments. A comparison with other modes of locomotion and applications of our model with competitive water sports is discussed in conclusion.
We explore the motion of an axisymmetric gravity current in an anisotropic porous medium in which the horizontal permeability is larger than the vertical permeability. It is well known that the ...classical axisymmetric gravity current supplied by a constant point source of fluid has an unphysical singularity near the origin. We address this by considering a pressure-dominated region near the origin which allows for vertical flow from the source, such that the current remains of finite depth, whilst beyond this region the flow is gravity dominated. At early times the inner pressure-driven region controls the spreading of the current, but at late times the inner region occupies a progressively smaller fraction of the current such that the radius increases as ${\sim }t^{3/7}$, while the depth near the origin increases approximately as ${\sim }t^{1/7}$. The presence of anisotropy highlights this phenomenon, since the vertical permeability maintains an effect on the flow at late times through the pressure-driven flow near the origin. Using these results we provide some quantitative insights into the dominant dynamics which controls CO$_2$ migration through permeable aquifers, as occurs in the context of carbon capture and storage.
We examine the effects of horizontally layered heterogeneities on the spreading of two-phase gravity currents in a porous medium, with application to numerous environmental flows, most notably ...geological carbon sequestration. Heterogeneities, which are ubiquitous within geological reservoirs, affect the large-scale propagation of two-phase flows through the action of small-scale capillary forces, yet the relationship between these small- and large-scale processes is poorly understood. Here, we derive a simple upscaled model for a gravity current under an impermeable cap rock, which we use to investigate the effect of a wide range of centimetre-scale heterogeneities on kilometre-scale plume migration. By parameterising in terms of different types of archetypal layering, we assess the sensitivity of the gravity current to the distribution and magnitude of these heterogeneities. Furthermore, since field measurements of heterogeneities are often sparse or incomplete, we quantify how uncertainty in such measurements manifests as uncertainty in the macroscale flow predictions. Using realistic parameter values, we demonstrate that heterogeneities can enhance plume migration speeds by as much as 200 %, and that uncertainty in field measurements can have dramatic consequences on flow predictions, particularly in post-injection scenarios where the role of capillary forces in heterogeneities is accentuated.
Fault zones have the potential to act as leakage pathways through low permeability structural seals in geological reservoirs. Faults may facilitate migration of groundwater contaminants and stored ...anthropogenic carbon dioxide (CO$_2$), where the waste fluids would otherwise remain securely trapped. We present an analytical model that describes the dynamics of leakage through a fault zone cutting multiple aquifers and seals. Current analytical models for a buoyant plume in a semi-infinite porous media are combined with models for a leaking gravity current and a new model motivated by experimental observation, to account for increased pressure gradients within the fault due to an increase in Darcy velocity directly above the fault. In contrast to previous analytical fault models, we verify our results using a series of analogous porous medium tank experiments, with good matching of observed leakage rates and fluid distribution. We demonstrate the utility of the model for the assessment of CO$_2$ storage security, by application to a naturally occurring CO$_2$ reservoir, showing the dependence of the leakage rates and fluid distribution on the fault/aquifer permeability contrast. The framework developed within this study can be used for quick assessment of fluid leakage through fault zones, given a set of input parameters relating to properties of the fault, aquifer and fluids, and can be incorporated into basin-scale models to improve computational efficiency. The results show the utility of using analytical methods and reduced-order modelling in complex geological systems, as well as the value of laboratory porous medium experiments to verify results.
Gunwale bobbing Benham, Graham P.; Devauchelle, Olivier; Morris, Stephen W. ...
Physical review fluids,
07/2022, Letnik:
7, Številka:
7
Journal Article
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It has been shown experimentally that small droplets, bouncing on a vibrated liquid bath, can "walk" across the surface due to their interaction with their own wave-field. Gunwale bobbing is a ...life-size instance of this phenomena in which a person standing on the gunwales of a canoe propels it by pumping it into oscillation with the legs. The canoe moves forward by surfing the resulting wave-field. After an initial transient, the canoe achieves a cruising velocity which satisfies a balance between the thrust generated from pushing downwards into the surface gradients of the wave-field and the resistance due to a combination of profile drag and wave drag. By superposing the linear wave theories of Havelock (1919) for steady cruising and of Helmholtz for an oscillating source, we demonstrate that such a balance can be sustained. We calculate the optimal parameter values to achieve maximum canoe velocity. We compare our theoretical result to accelerometer data taken from an enthusiastic gunwale bobber. We discuss the similarities and differences between gunwale bobbing and hydrodynamic quantum analogues, and possible applications to competitive sports.
Abstract
Upscaling the effect of heterogeneities in porous media is crucial for macroscopic flow predictions, with numerous applications in energy and environmental settings. In this study, we derive ...simple semi-analytical expressions for the upscaling of multiphase flow in a porous medium with a range of vertical heterogeneities. We use this upscaling to give insight into how the flow transitions between a viscous flow regime, in which macroscopic pressure gradients dominate over heterogeneity-driven capillary forces, and a capillary flow regime, in which these capillary forces dominate and set the saturation distribution of the flow. In particular, by studying the dynamics of flow in an aquifer, we demonstrate that different regions lie within the viscous and capillary flow regimes whilst other regions lie in between these regimes. By modifying the classic Buckley–Leverett problem for fluid displacement we demonstrate where and when the flow transitions between these regimes and how this affects flooding speeds. Then, we discuss the implications of these results in the case of carbon dioxide sequestration, making comparisons with field data.
•Mathematical models for the solidification of silicon.•1D model describes slab casting related to a set of thin-casting experiments.•2D model relates to a wedge casting experiment.•Analytical ...results agree very well with direct COMSOL simulations.
We have developed mathematical models in both one and two spatial dimensions for the solidification of silicon. The one-dimensional model describes slab casting related to a set of thin-casting experiments. The model is fitted to thermocouple data and accounts for various heat transfer mechanisms as well as the latent heat. The model can be used to predict the time taken for the material to completely solidify and the solidification distance (the point where solidification fronts meet which can be observed as a discontinuity in the grain microstructure). Simple approximate analytical results, which agree very well with the full-scale numerical solutions on Matlab and COMSOL, are provided. The two-dimensional model relates to a wedge casting experiment where, again, various heat transfer mechanisms and latent heat need to be accounted for. Experimental data from thermocouples is used to quantify the heat transfer coefficients by fitting to two-dimensional COMSOL simulations. A very simple analytical “Triangle model” is derived by assuming that the solidification fronts move as flat surfaces from each of the two wedge walls and the air surface, independently of each other, as three separate one-dimensional quasi-steady approximations. This model predicts that the area of liquid silicon will diminish as shrinking self-similar triangles. This simplified model provides analytical results for the solidification time and distances which agree very well with the COMSOL simulations.
•Capillary pressure heterogeneity results in significant trapping of CO2.•Capillary heterogeneity trapping depends on field geology and flow conditions.•Upscaled impact of heterogeneity trapping from ...the core to field scale.•Petrophysical and geological uncertainty incorporated into sequestration models.•Capillary pressure hysteresis typically reduces capillary heterogeneity trapping.
A significant uncertainty which remains for CO2 sequestration, is the effect of natural geological heterogeneities and hysteresis on capillary trapping over different length scales. This paper uses laboratory data measured in cores from the Goldeneye formation of the Captain D Sandstone, North Sea in 1D numerical simulations to evaluate the potential capillary trapping from natural rock heterogeneities across a range of scales, from cm to 65m. The impact of different geological realisations, as well as uncertainty in petrophysical properties, on the amount of capillary heterogeneity trapping is estimated. In addition, the validity of upscaling trapping characteristics in terms of the Land trapping parameter is assessed. The numerical models show that the capillary heterogeneity trapped CO2 saturation may vary between 0 and 14% of the total trapped saturation, depending upon the geological realisation and petrophysical uncertainty. When upscaling the Land model from core-scale experimental data, using the maximum experimental Land trapping parameter could increase the expected heterogeneity trapping by a factor of 3. Conversely, depending on the form of the imbibition capillary pressure curve used in the numerical model, including capillary pressure hysteresis may reduce the heterogeneity trapping by up to 70%.
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