ABOVE SALINE GROUNDWATERUDC 532.546.06
V. N.ÉmikhPossible flow variants as a function of the determining physical parameters are analyzed within the framework of the boundary-value problem of horizontal slit drainage in a fresh groundwater fringe above saline groundwater.Kapranov [1] constructed a solution of the Verigin problem of flow in a freshwater fringe to a system horizontal slit drains, which are further called linear drains, for their incomplete flooding. The unique solvability of the system of equations with respect to the mapping parameters related to the velocity hodograph was analytically established; in the calculations, a critical flow regime was found at the interface between fresh and saline groundwater. The present work reveals the physical content of the problem in the general formulation -with complete flooding of the drains.1. Formulation and Solution of the Problem; Mapping Parameters. Stationary plane filtration in a fresh groundwater fringe above immovable saline water with uniform infiltration at rate ε is studied. The supply to the infiltration zone is compensated by outflow into equidistant linear drains of identical width located at the same depth. Because the flow is periodic, it is sufficient to study it within one half-period (Fig. 1).In the case of complete flooding of the drains, the boundary-value problem describing the flow consists of finding its complex potential ω = ϕ + iψ (ϕ is the filtration velocity potential and ψ is the stream function) normalized by the quantity κL (κ is the ground permeability and L is half the distance between the centers of adjacent drains), which, in the filtration region, is an analytic function of the complex coordinate z = x + iy normalized by L under the boundary conditions CM : x = 0, ψ = 0;EN : x = 0, ψ = ε; AG: x = 1, ψ = ε;M DN : y = 0, ϕ = 0; AC: ϕ + y = 0, ψ − εx = 0;(1) EG: ϕ − ρy = const, ψ = ε (ρ = (ρ 2 − ρ 1 )/ρ 1 ).Here ρ 1 and ρ 2 are the densities of the fresh and saline water, respectively. The first condition on EG is a consequence of the assumptions that the saline water is motionless and that the pressure is continuous in passing through this segment. In Fig. 1, the depression curve AC is denoted by digit 1. The problem is solved using the Polubarinova-Kochina method [2], which is based on using the analytical theory of linear differential equations. The method seeks to find the functions Ω = dω/dζ and Z = dz/dζ determined in the half-plane Im ζ 0 of the auxiliary complex variable ζ = ξ +iη (Fig. 2). The velocity hodographw = w x +iw y (Fig. 3) is a circular polygon of the same form as that in the model of an infiltration fringe with tubular drainage [3]. The procedure of constructing the solution is also similar and eventually leads to the following relations:Lavrent'ev Institute
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