Buoyancy-driven flow in a confined aquifer with a vertical gradient of permeability

We examine the injection of fluid of one viscosity and density into a horizontal permeable aquifer initially saturated with a second fluid of different viscosity and density. The novel feature of the analysis is that we allow the permeability to vary vertically across the aquifer. This leads to recognition that the interface may evolve as either a rarefaction wave that spreads at a rate proportional to $t$, a shock-like front of fixed length or a mixture of shock-like regions and rarefaction-wave-type regions. The classical solutions in which there is no viscosity ratio between the fluids and in which the formation has constant permeability lead to an interface that spreads laterally at a rate proportional to $t^{1/2}$. However, these solutions are unstable to cross-layer variations in the permeability owing to the vertical shear which develops in the flow, causing the structure of the interface to evolve to the rarefaction wave or shock-like structure. In the case that the viscosities of the two fluids are different, it is possible that the solution involves a mixture of shock-like and rarefaction-type structures as a function of the distance above the lower boundary. Using the theory of characteristics, we develop a regime diagram to delineate the different situations. We consider the implications of such heterogeneity for the prediction of front locations during $\text{CO}_{2}$ sequestration. If we neglect the permeability fluctuations, the model always predicts rarefaction-type solutions, while even modest changes in the permeability across a layer can introduce shocks. This difference may be very significant since it leads to the $\text{CO}_{2}$ plume occupying a greater fraction of the pore space between the injector and the leading edge of the $\text{CO}_{2}$ front in a layer of the same mean permeability. This has important implications for estimates of the fraction of the pore space that the $\text{CO}_{2}$ may access.

Onset of natural convection in layered aquifers

The stability of gravitationally unstable, transient boundary layers in heterogeneous saline aquifers is examined with respect to the onset of natural convection. Permeability is assumed to vary periodically across the thickness of the aquifer. We study the interaction between permeability variation and concentration perturbations within the boundary layer. We observe that the instability decreases with an increase in the permeability variance if the boundary layer thickness is large compared with the permeability wavelength. On the other hand, when the boundary layer thickness is smaller than the permeability wavelength, the behaviour of instability as a function of variance depends on the phase of permeability variation. Such behaviours are shown to result from the interaction of two modes of vorticity production related to the coupling of concentration and velocity perturbations with the magnitude and gradient of permeability variation, respectively. We show that these two modes of vorticity production, when coupled with the transient nature of the boundary layer, determine the evolutionary paths followed by the most amplified perturbations that trigger the onset of convection. When the permeability variance is large, we find that small changes in the permeability field can lead to large changes in the onset times for convection.

I. THE PERMEABILITY OF THE WALL OF THE LYMPHATIC CAPILLARY

A technique has been developed for the demonstration of lymphatic capillaries in the ear of the mouse by means of vital dyes and for tests of their permeability under normal and pathological conditions. The lymphatics become visible as closed channels from which the dyes escape secondarily into the tissue. Some of them, cross-connections, with extremely narrow lumen, would seem ordinarily not to be utilized. There is active flow along the lymphatics of the mouse ear under ordinary circumstances. The movement of dye was always toward the main collecting system. The valves of the lymphatics as well as fluid flow prevented distal spread. There was in addition slow migration, apparently interstitial in character, but in the same general direction, of dots of color produced by the local injection of dye. The normal permeability of the lymphatics was studied with dyes of graded diffusibility. Their walls proved readily permeable for those highly diffusible pigments that the blood capillaries let through easily, but retained those that the latter retained. Finely particulate matter (India ink, "Hydrokollag"), they did not let pass. No gradient of permeability was observed to exist along them such as exists along the blood capillaries of certain organs. The observed phenomena of lymphatic permeability, like those of the permeability of the blood capillaries, can be explained on the assumption that the lymphatic wall behaves like a semipermeable membrane.

THE GRADIENT OF PERMEABILITY OF THE SKIN VESSELS AS INFLUENCED BY HEAT, COLD, AND LIGHT

The mounting gradient of permeability along the small vessels of the corium is essentially unaltered by active hyperemia produced by heat, cold, or light. Only when the vascular walls are so damaged that rapid leakage ensues, as shown by the development of edema, does the permeability of the capillary web as a whole approximate that of the venules. It is plain that the normal gradient of vascular permeability depends upon the integrity of the vessel wall. The method of experiment described can be utilized for a study of the functional changes which result in the lesions due to burning and freezing.

THE VESSELS INVOLVED IN HYDROSTATIC TRANSUDATION

The gradient of permeability which exists along the cutaneous capillaries and venules is accentuated and broadened in scope by increasing the venous pressure moderately. Under such circumstances transudation leading to edema takes place most abundantly from the venules. The permeability of the portion of the capillary web that is near the arterioles increases only when the venous pressure rises so high as to approximate that in the arteries. Under such circumstances the gradient of permeability along the small vessels disappears, the capillaries and venules everywhere leaking fluid. The character of the vital staining developing under such circumstances indicates, like the evidence of previous work, that the cause for the gradient is to be sought in a structural differentiation.

THE RELATION OF HYDROSTATIC PRESSURE TO THE GRADIENT OF CAPILLARY PERMEABILITY

The gradient of permeability along the capillaries of voluntary muscle and the capillaries and venules of skin exists independently of the hydrostatic conditions, though influenced by them. Its presence cannot be explained by a graded tonic contraction of the capillaries. The evidence,—like that of previous papers,—points to local differences in the barrier offered by the wall of these vessels as responsible for the gradient.

THE GRADIENT OF VASCULAR PERMEABILITY

The permeability of the venules of the skin of the mouse greatly exceeds that of the capillaries. A mounting gradient of permeability exists along the further portion of the latter. The significance of these facts is discussed with relation to conditions in human skin. The cutaneous venules are differentiated for several functions besides those ordinarily attributed to them, and must be considered as specialized organs.

THE GRADIENT OF VASCULAR PERMEABILITY

A mounting gradient of permeability exists along the capillaries of frog muscle. In chicken muscle on the other hand none has been demonstrated; but the close-knit vascularization is arranged in duplicate in such manner that the blood runs in opposite directions through the capillaries of nearly adjacent fibres. In a flight muscle of the pigeon there exists in addition to this artifice what appears to be a special collecting system of venous capillaries. In the mammalian diaphragm indications of such a system are also to be found, and a gradient of capillary permeability like that in the other skeletal muscles is probably present. These vascular conditions are briefly considered in terms of function.

THE GRADIENT OF VASCULAR PERMEABILITY

A steeply mounting gradient of permeability is demonstrable along the meshwork of capillaries which connects the arterioles and venules of the skin of the frog. The venules incorporated in the meshwork are even more permeable than the capillary meshes giving into them. The presence of the gradient under such differing conditions as exist along frog and mammalian capillaries enables one to rule out certain factors which might be invoked to explain it; and it is not explainable in terms of those influences generally recognized as conditioning exchange between the blood and tissues. Not improbably it results from a structural differentiation along the capillary.