(Left) A sketch of an axisymmetric gravity current propagating below an impermeable horizontal boundary in a porous medium saturated with an immiscible fluid of higher density and different viscosity. (Right) A representative vertical saturation profile through the current, as indicated by the gray shading on the left.
Contour plots of (a) the saturation function, s 0, and (b) the flux function, , against fB and Λ. The shaded regions indicate the physical limits of (1) weak, (2) strong, and (3) intermediate capillary effects discussed in Secs. IV A–IV C , respectively, as well as the regions for which approximate expressions for s 0 and can be found. The dashed contour lines are plotted using the approximate expressions for s 0 and , given by (33a), (33b), (45a), and (45b) in regions 2 and 3, respectively, and are almost indistinguishable from the actual values there.
Graphs demonstrating how the self-similar height profile of a two-phase gravity current, f(y), varies with the strength of capillary forces. The gray shading indicates how the saturation distribution within the gravity current changes. Strengthening capillary forces, corresponding to decreasing B and Λ, lead to a thicker current, with a more rounded front and lower saturations. The dashed curve when B = 10−3 and Λ = 1 shows the approximate profile in the limit of strong capillary forces, given by the solution of (40) . The single-phase height profile predicted by Lyle et al. 1 is almost identical to the two-phase profile displayed for B = 103 and Λ = 103. Note the difference in vertical scale for each graph.
Contour plot of the dimensionless current height, f, at y = 0.5, normalised by the corresponding height of a single-phase gravity current, f SP (0.5) = 0.348, as a function of Λ and B. The solid curves indicate solutions of the full numerical equations. The circled numbers correspond to regions of (1) weak, (2) strong, and (3) intermediate capillary forces, discussed in Secs. IV A–IV C , respectively. The dashed curves depict the contours calculated using the approximations in regions of strong and intermediate capillary forces. These approximate contour lines have been calculated using constants of proportionality 1 in (38a) in region 2, and 0.5 in (47) in region 3.
Contour plot showing the variation of with Λ and B, where is the value in the single-phase limit. The circled numbers correspond to regions of (1) weak, (2) strong, and (3) intermediate capillary forces, discussed in Secs. IV A–IV C , respectively. In the limit of strong capillary forces, η N = 0.8704(2 + Λ)1/5(B/Λ)1/10, as discussed in Sec. IV B . The contours using this approximation for , 0.6, and 0.8 are shown by the dashed curves.
Graph of the scaled current height profiles in various limits. The dashed curve shows f(y) in the single-phase limit. The solid curve shows g(y), in the limit of very strong capillary forces, where f(y) = [Λ2 B 3(2 + Λ)]−1/5 g(y).
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