This paper introduces the general theory of water exclusion from, or entry into, subterranean holes from steady uniform downward unsaturated seepage. Buried holes serve as obstacles to the flow and so increase water pressure at parts of the hole surface. When downward seepage is fast enough and/or the hole is large enough, water pressure increases to the point where a seepage surface forms and water enters the hole. Contrary to the conventional picture drawn from capillary statics, hydrodynamics shows that the larger the hole the more vulnerable it is to water entry. Cavity shape is important also. Applications include optimal design of configurations of tunnels and underground repositories (e.g., for nuclear wastes) against entry of seepage water. The theory also embraces the disturbance of seepage flows by buried impermeable obstacles such as stones and structures. The quasi‐linear exclusion problem for circular cylindrical cavities is solved. Both exact solutions and simple asymptotic results are found, and graphs and tables presented. The moisture field about the cavity exhibits upstream and downstream stagnation points and retarded regions: “roof‐drip lobes” in which the moisture content and downward flow velocity are augmented by water essentially deflected from the cavity roof, and the “dry shadow” region of reduced moisture content and flow velocity, essentially sheltered by the cavity. A practical consequence is that we can establish, for any given combination of downward seepage velocity, cavity radius, saturated hydraulic conductivity K1, and sorptive number α, whether or not seepage water will enter a cylindrical cavity.
1. Introduction. The underlying problem is to deduce the shape of a drum or plane uniform membrane from the knowledge of its spectrum of eigenvalues ωn = i√λn. It has been shown by Kac(3) that some progress is possible on establishing the leading terms of the asymptotic expansion of the trace function for small positive t. In particular, for a simply connected membrane Ω bounded by a smooth convex curve Γ for which the displacement satisfies the wave equation ∇2ø = ∂2ø/∂t2 and Dirichlet conditions on Γwhere |Ω| = area of Ω, L = length of Γ, and the constant ⅙ is determined (apart from a factor) on integrating the curvature of the boundary; moreover, if Ω is permitted to have a finite number of smooth convex holes then the constant becomes ⅙(1–r), where r = number of holes.
An exact analog exists between steady quasi‐linear flow in unsaturated soils and porous media and the scattering of plane pulses, and the analog carries over to the scattering of plane harmonic waves. Numerous established results, and powerful techniques such as the Watson transform, are thus available for the solution and understanding of problems of unsaturated flow. These are needed, in particular, to provide the asymptotics of the physically interesting and practically important limit of flows strongly dominated by gravity, with capillary effects weak but nonzero. This is the limit of large s, with s a characteristic length of the water supply surface normalized with respect to the sorptive length of the soil. These problems are singular in the sense that ignoring capillarity gives a totally incorrect picture of the wetted region. In terms of the optical analog, neglecting capillarity is equivalent to using geometrical optics, with coherent shadows projected to infinity. The paper deals specifically with steady infiltration from circular cylindrical and spherical cavities. The asymptotic methods prove remarkably accurate, even far from the limit. The results replace, and explain, previous semiempirical estimates of the limiting behavior. One notable result is that the depth of the penumbra (effectively wetted region) for the cylinder is 128 times the depth for the sphere, confirming and supplementing previous studies. An odd byproduct is that we correct a long‐standing classical result in scattering theory. The scope for extending these methods to flows in other geometries, to heterogeneous soils, and generally to linear convection‐diffusion processes, is indicated briefly.
The quasi‐linear problem of water exclusion from, or entry into, spherical cavities from steady uniform downward unsaturated seepage is solved. Both exact solutions and simple asymptotic results are found. These are qualitatively similar to those given previously for circular cylindrical cavities, exhibiting such features as the dry shadow and the roof‐drip lobes. A major practical result of the analysis is the function ∂max(s), the dependence of the maximum potential (at the roof apex) on the dimensionless quantitys = ½ αl, with α the sorptive number and l cavity radius. ∂max(s) is almost indistinguishable from ∂max(½s) for circular‐cylindrical cavities. This implies that physically the most relevant configuration parameter is total curvature at the apex of the cavity surface. Our results enable us to establish, for given values of downward seepage velocity, cavity radius, saturated hydraulic conductivity K1 and of α, whether or not seepage water will enter a spherical cavity.
The inverse eigenvalue problem for vibrating membranes (4), may also be examined in three or more dimensions. Let us suppose that λn are the eigen values of the problemwhere Ω is a closed convex region or body in En and S is the bounding surface of Ω. The basic problem is to determine the precise shape of Ω on being given the spectrum of eigenvalues λn. In analogy with the membrane problem, it is clear that the trace function may be constructed in identical fashion; thuswhere G(r, r', t) is the Green's function of the diffusion equationand satisfies the Dirichiet condition G(r, r', t) = 0, r∈S, and the initial condition G(r, r', t) → δ(r–r') as t → 0.
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