Asymptotic solutions of the seepage exclusion problem for elliptic‐cylindrical, spheroidal, and strip‐ and disc‐shaped cavities

Asymptotic roof boundary layer solutions are obtained for the quasi-linear seepage exclusion problem for elliptic-cylindrical and spheroidal cavities of arbitrary aspect ratio h. The remarkably simple result for ϑmax, the maximum dimensionless potential at the cavity wall, is ϑmax(σ) ∼ 2σ + f0(h) + [f1(h)/σ] + ]f2(h)/σ2 + … where σ=½ αk−1 (α is sorptive number; k is apical total curvature) and the fn are polynomials in h−2 of degree (n + 1). The results agree, as they should, with previous results for the circular cylinder and sphere (h=1) and the parabolic cylinder and paraboloid (h=∞). Connections noted in earlier studies between apical sharpness and the variation of potential along the cavity wall are generalized and made quantitative. The singular (h =0) strip and disc are of especial interest: they indicate behavior near the critical upstream stagnation point on cavities (and impermeable obstacles) with flat strip- and disc-shaped roofs but otherwise (within some limits) arbitrary. For the strip, ϑmax(S1) ∼ 4S12 + 9 - (36/S12) + ⋯, and for the disc, ϑmax(S1) ∼ 2S12 + 6 - (27/S12) + ⋯, where s1=½αl1 (l1 is semiwidth or radius). The central importance of ϑmax is that seepage water enters, or is excluded from, the cavity, depending on whether K0/K1 (K0 is downward steady seepage velocity far from the cavity; K1 is saturated hydraulic conductivity) exceeds or does not exceed ϑmax−1.