Steady two-dimensional gravity-driven seepage in homogeneous porous lumps is studied with the help of conformal mappings and boundary value problem technique. The Terzaghi flow pattern for a trapezoidal dam exposed to a heavy rainstorm is analysed. For a semi-circular massif, the influence of impervious bed inclination is studied. Recharge–discharge distributions, hinge points, gradients along the lump contour as well as the total flow rate exhibiting water–bearing capacity of the unit are found in explicit form. Generalizations for non-isobaric boundary conditions are discussed.
Computer algebraic routines are applied for determination of the phreatic surface from standard boundary-value problems for ordinary differential equations. The method does not require iterative steps as other methods do and therefore may be readily used by engineers. The problem of seepage from (into) an unconfined semiinfinite aquifer into (from) an adjacent reservoir that has a sudden change of water level is revised. Comparisons with the Polubarinova-Kochina series expansion are done. Superelevation of the water table in a wetting regime compared to a drainage regime is quantified by the value of sorptivity (desorptivity). The absolute values of sorptivity and desorptivity diverge as the amplitude of the reservoir level change increases. A problem of a steady flow into (from) a constant head well from (into) an unconfined leaky aquifer is also examined. The water table elevation, well rate, and volume of the cone of depression (injection) are calculated.
Steady two-dimensional gravity-driven seepage in homogeneous porous lumps is studied with the help of conformal mappings and boundary value problem technique. The Terzaghi flow pattern for a trapezoidal dam exposed to a heavy rainstorm is analysed. For a semi-circular massif, the influence of impervious bed inclination is studied. Recharge–discharge distributions, hinge points, gradients along the lump contour as well as the total flow rate exhibiting water–bearing capacity of the unit are found in explicit form. Generalizations for non-isobaric boundary conditions are discussed.
An explicit closed-form analytical solution giving the water-table depression is obtained for the steady-state problem of seepage to a water-bearing substratum from two reservoirs separated by a strip of land over which water loss by evaporation occurs at a constant rate. Two hydrodynamic situations can occur. In the first, there is only drainage into the substratum and all evaporating water originates from the reservoirs. In the second, the substratum, while accepting water draining from the reservoirs, also supplies some water to the evaporating surface. The method of boundary-value problems is used through the conformal mapping of a strip in the Zhukovskii plane to an auxiliary half-plane and reconstruction of the complex physical coordinate by the Signorini formula. The two hydrodynamic regimes are demarcated in the solution with the shape of the water table obtained in a closed form. The results of calculations show the effect of the evaporation rate, the hydraulic conductivity of the soil, the pressure in the water-bearing substratum, the distance between reservoirs, and the depth to the substratum. The method of boundary-value problems is also used to obtain an analytical solution for the water-table depression caused by evaporation from a strip in an infinite extent of land above a water-bearing substratum under pressure.
Steady, Darcian, one‐phase, phreatic surface flow of groundwater into a horizontal well with a pancake lens of light nonaqueous phase liquid (LNAPL) accumulated in the water table trough is studied by the method of complex analysis. A sharp interface model assumes groundwater capped by two isobaric limbs (groundwater–vadose zone interfaces) of a free surface with an in‐between cambered segment of an immiscible LNAPL‐water interface, along which pressure is hydrostatically increasing with the depth of the LNAPL “channel.” The complex potential polygon is mapped onto an auxiliary half plane where the complex physical coordinate of the flow domain is represented in terms of singular integrals as a solution of the Keldysh‐Sedov problem. The shapes of semi‐infinite “wings” of the water table contacting the vadose zone gas and of a finite length LNAPL‐groundwater interface are found from parametric equations that involve the sink strength and location with respect to the pancake surface, the ordinate of the lowest trough point, and the volume of LNAPL accreted in the lens. Critical conditions, corresponding to the lens contour cusping toward the sink, are found. The Riesenkampf solution contains a free parameter, which is fixed by specifying either a point on the free surface or the volume of the trough‐intercepted LNAPL.
