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 either as a rarefaction wave which 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 which 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 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 CO 2 plume occupying a greater fraction of the pore space between the injector and the leading edge of the CO 2 front in a layer of the same mean permeability. This has important implications for estimates of the fraction of the pore space which the CO 2 may access.
The interaction of gravitationally driven, free-surface flows of viscous fluid with topographic features is investigated theoretically. The motion is studied in the regime where the depth of the flow is much smaller than the streamwise extent of the topography. A lubrication model of the motion is developed, integrated numerically and analysed asymptotically. For small mounds, it is shown that the flow surmounts the obstacles, but for larger mounds the flow is deflected around it and can form dry zones in its wake into which fluid does not flow, as well as forming deeper ponded regions upstream. Which of these phenomena prevails is shown to depend upon the amplitude of the mound height and the thickness of the oncoming flow relative to the streamwise length scale over which the topography varies. By using numerical and asymptotic results, we demonstrate that relatively wide mounds lead to the development of deep ponds of material upstream, which may lead to flow overtopping if the mound is not sufficiently high. These insights can be used to inform the design of barriers that defend built infrastructures from lava flows, and it is shown how this model can also provide an upper bound on the force exerted by the flow on them.
We study the migration of a tracer within an injection-driven flow in a horizontal aquifer in which the permeability varies with depth. The permeability gradient produces a shear and this leads to lateral dispersion of the tracer. In the high permeability regions, the tracer moves substantially faster than the mean flow and eventually enters the nose region of the flow where the depth of the current is less than the depth of the aquifer. Depending on the influence of (i) the viscosity contrast between the injected fluid and the original fluid, and (ii) the vertical permeability gradient, the nose of the current may be of fixed shape or may gradually lengthen with time. This leads to a variety of patterns of dispersal of the tracer, which may either remain in the nose or cycle through the nose and be left behind. Our results illustrate the complexity of the migration of a tracer in a heterogeneous aquifer which has important implications for interpreting the results of tracer tests as may be proposed for monitoring $\text{CO}_{2}$ or gas injected into subsurface reservoirs.
The flow of viscous fluid injected from a point source into the space between two horizontal plates initially filled with a second fluid of lesser density and different viscosity is studied theoretically and numerically. The volume of the dense input fluid increases with time in proportion to tα. When the fluid has spread far from the source, lubrication theory is used to derive the governing equations for the axisymmetric evolution of the interface between the fluids. The flow is driven by the combination of pressure gradients associated with buoyancy and pressure gradients associated with the input flux. The governing equation is integrated numerically, and we identify that with a constant input flux, the flow is self-similar at all times with the radius growing in proportion to t1/2. In the regimes of injection-dominated and gravity-dominated currents, we obtain asymptotic approximations for the interface shape, which are found to agree well with the numerical computations. For a decreasing input flux (0 < α < 1), at short times, the flow is controlled by injection; the current fills the depth of the channel spreading with radius r ∼ tα/2. At long times, buoyancy dominates and the current becomes unconfined with the radius growing in proportion to t(3α+1)/8. The sequence of regimes is reversed in the case of an increasing input flux (α > 1) with buoyancy dominating initially while the pressure associated with the injection dominates at late times. Finally, we consider the release of a fixed volume of fluid (α = 0). The current slumps under gravity and transitions from confined to unconfined, and we obtain asymptotic predictions for the interface shape in both regimes.
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