If we assume the drop is circular with a radius, r , its approximate volume will be the area of the drop times the parallel plate aperture, e. The width of the drop will be 2 r.
We also introduce an additional factor of two for the two surfaces. For a drop in contact with both parallel plates, Furmidge's sliding criterion defines a critical radius for sliding, r c , which becomes:. Figure shows the critical drop size for vertical parallel plates as a function of aperture and contact angle.
Drops may contact either one wall or both walls of the fracture depending on the height of the drop and aperture of the fracture. We can use Furmidge's relationships to assess whether the drop flow will be drops in contact with one wall or with both walls. If the heights of the critically-sized drops are smaller than the fracture aperture, then we should expect that flow is on one fracture wall only.
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To make a calculation of a critically-sized drop on a single wall, we must make assumptions about the shape of the drop to calculate its height. The heights and shapes of drops are determined by a complicated interaction of surface properties and gravity; however, to get a rough approximation, we may assume that the drop has the form of a spherical cap Figure In this analysis we assume that the liquid-surface contact angle defines the portion of the sphere that forms that cap.
The volume of such a spherical cap is. Inserting this volume expression into Furmidge's sliding criterion allows solution for the critical drop radius, r c , that will induce drop sliding, which is. The height, h c , of a drop that is critically sized for sliding can be derived by trigonometry from the geometry of the spherical cap as:.
Assume that advancing and receding contact angles are a factor of 1. The corresponding drop heights will be 0. Due to the square root relationship in calculating r , the drop size has a reduced sensitivity to the contact angles.
Again, this is a rough approximation subject to experimental testing due to the use of a spherical form for the volume calculation. These analyses suggest that the heights of critical drops on single surfaces will be on the order of a millimeter or less. Hence, we may expect that single-walled drop flow will occur for fractures larger than 1 mm in aperture, and two-walled flow will occur in fractures with less than 0. Larger aperture fractures will be more conducive to flow by drop sliding than smaller fractures for two reasons.
If we consider drops in contact with both walls, the critical volume for drop sliding will be larger for smaller aperture fractures, and the radius of a critical fracture will be larger.
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Larger aperture fractures are also more conducive to flow because they are more likely to contain drops in contact with single walls, and these drops have less resistance to flow because they wet only one surface. Asperities can be expected to play a role in sliding and drop retention. With asperities, the radius of the drop required for sliding will need to be larger for at least two reasons.
Asperities reduce the volume of the drop for a given drop radius; and thus the drop requires a larger radius to slide. Second, the drop will leave behind a volume of water as capillary retention at the asperity's point of contact or the point of least wall separation, if contact is not made. Once drops begin to move they should continue to slide unless they lose volume, the fracture aperture changes, surface properties of the fracture change, or the inclination angle of the fracture changes.
Drops may gain volume by coalescence with small static drops in their path. They may also lose volume by evaporation or imbibition to the matrix along the fracture walls. Drops may also. The movement of water in drops will differ in significant ways to the capillary conceptual model of water movement in the vadose zone and to film flow concepts. The initiation of flow is related to the wetting properties of the surfaces, and once water is moving, wetting resistance may play an equal or more significant role than viscous drag along the fracture walls.
Drops can be immobile; drops will remain in place until they accumulate sufficient volume to overcome wetting resistance, and they become immobile again if they lose mass or fracture conditions change. Once immobile, they will remain immobile until additional mass enters the drop. Darcy's law may not apply if the flow resistance is not primarily viscous and the concept of a potential gradient does not apply to discontinuous drops.
In terms of boundary conditions, drop flows are best described by flow rate or flux boundary conditions, as head boundary conditions do not apply. In this case, there are clear relationships based on conservation of mass for drop velocity, flux, and saturation. Subject to experimental verification, the application of a constant flux at the top of a fracture results in an accumulation, or storage, of drops or mass at the top of the fracture until a drop reaches a mass sufficient to initiate sliding.
At higher fluxes, this flow might occur as a rivulet. At flow fluxes, the mass should emerge from the lower boundary as drops.
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The long-term average fluxes would balance the influx; however, the instantaneous outflow rate could be high or nothing depending on whether or not a drop was emerging at any particular moment. Laboratory experiments of flow in simulated fractures are important for understanding flow processes.
This section presents a brief review of experiments involving flow in simulated fractures. Rasmussen performed experiments on a vertical fracture simulated by two glass panels. The test involved injecting water into a hole in the interior of the fracture under constant head conditions. The sustained injection of the constant-head condition created a region of local saturation primarily downward from the injection point, but also a slight distance upward.
The main use of the experiment was to provide data to compare with numerical models of the air-water interface. The experimental results largely agreed with the analytical predictions of the position of the air-water interface, with some variability caused by air bubbles entrapped in the water-saturated zone.
Because of the constant-head. Fourar et al. The experiments used smooth glass plates and plates with 1 mm beads in the aperture of the fracture to simulate roughness. The fracture materials were glass, the liquid was water, and the gas was compressed air.
Phenomena of liquid drop impact on solid and liquid surfaces - IOPscience
The experimental conditions involved the pressure injection of air into initially water-filled fractures, which would appear to be a better analog for oil reservoir processes than for vadose zone flow. Of particular interest in these experiments is the structure of the phases. The structure of the two phases varied with the gas injection rate. At low gas injection rates, the gas bubbles disperse into the water. With increasing rate, the bubbles start to become unstable and begin to finger. At yet higher gas rates, the gas occupies the major portion of the fracture.
At these higher gas flow rates, the flow geometry of the water varied with the water injection rate.
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At lower water injection rates, the water moved as liquid drops in the gas stream. At higher liquid rates, the water flowed as unstable films on the fracture walls. The expectation for a porous medium was that each phase would occupy its own continuous network of pores, the wetting phase in the smaller pores and the nonwetting phase in the larger. The experiments, on the other hand, showed that only one phase was continuous and the other phase traveled as either bubbles or drops. The occupancy locations of the phases were constantly changing.
Nicholl et al. The test conditions involved slug injections of water at the top of the air-filled fracture. The basic hypothesis of the experiment was that the flow process was one of density inversion, that is, the entry of higher-density water at the top of the flow system would displace the lower-density air, and the geometry of the water distribution would reflect the gravitational instability. The primary observation of the experiment was the breakup of the water invasion front into fingers Figure Being a slug injection, the fingers were not replenished as they moved down the steeply dipping fracture, and they left drained regions behind them.
Smaller fingers were observed to have lower velocities or to stop altogether.
This paper also presents the results of dyed water injections in the top of a natural fracture, which showed clearly the development of fingers. One significant point of this work was to demonstrate that the fingering is primarily the result of instability in the wetting front independent of aperture heterogeneity.