Abstract A study is made of a steady, two‐dimensional groundwater flow with a horizontal well (drain), which pumps out freshwater from an aquifer sandwiched between a horizontal bedrock and ponded soil surface, and containing a lens‐shaped static volume of a heavier saline water (DNAPL‐dense nonaqueous phase liquid) as a free surface. For flow toward a line sink, an explicit analytical solution is obtained by a conformal mapping of the hexagon in the complex potential plane onto a reference plane and the Keldysh‐Sedov integral representation of a mixed boundary‐value problem for a complex physical coordinate. The interface is found as a function of the pumping rate, the well locus, the ratio of liquid densities, and the hydraulic heads at the soil surface and in the well. The shape with two inflexion points and fronts varies from a small‐thickness bedrock‐spread pancake to a critical curvilinear triangle, which cusps toward the sink. The problem is mathematically solvable in a relatively narrow band of geometric and hydraulic parameters. A similar analytic solution for a static heavy bubble confined by a closed‐curve interface (no contact with the bedrock) is outlined as an illustration of the method to solve a mixed boundary‐value problem.
Formation of soil horizon boundaries are commonly qualitatively described, their formation and time evolution are rarely quantified by adequate mathematical modeling. In this paper, analytical (using conformal mappings) and numerical (involving HYDRUS2D simulations) methods are employed to model 2D infiltration-induced Darcian flows within a layered fine/coarse soil substrate system featuring a wedge-shaped interface, considering both saturated and unsaturated conditions. Initial-boundary value problems to the Richards (in particular, Laplace's) partial differential equations are set in domains with geometry of interfaces more complex than in regular soil profiles. The patterns of 2D distribution of physical characteristics (streamlines, isobars, velocities, etc.) are found in two variants of such a system (loam/gravelly non-capillary material, loam/sand), with comparison of analytical and numerical models. The novelty of the results is in the analysis of concentrating streamlines with maximum water flow rates and fastest travel time of advected marked particles at the tops of the jagged interface of the soil textures and the unsaturation of the underlying coarse-dispersed substrates during pressure infiltration of water in such systems. Also, if the soil surface is subject to small infiltration fluxes, the opposite phenomenon of capillary barrier, i.e. diversion of the infiltration flux from the top of the interface of the coarse substratum, is demonstrated in HYDRUS2D simulations. Prediction of morphological variations of soil profiles with non-horizontal interfaces, which undergo time-factor controlled anthropogenic drivers, is important for theoretical soil hydrophysics and hydrological and soil/geotechnical engineering.
Abstract Transient water table systems under the influence of decaying groundwater mounds are studied in terms of the Dupuit-Forcheimer approximation, its linearization and linear potential theory. Parabolic mounds spreading and shrinking due to gravity and evapotranspiration are derived from the general class defined by Polubarinova- Kochina (1945). The penetration curves are calculated as characteristics of the water table response at prescribed observation wells. The Polubarinova-Kochina solutions for rectangular mounds are used to derive isochrones, a "resting lens" into which a mound transforms, and a distortion picture of reference volumes. These characteristics are obtained from numerical solution of a system of ordinary differential equation. The plume illustrating the advective spread of a contaminant from a reference area is computed as a composition of all path lines passing through this area.
A rigorous, closed form solution for the hydraulic head, sharp interface shape, and position in a confined aquifer bordered on one side by a static interface with intruded sea water and on another side by a constant head boundary is obtained in terms of the Dupuit‐Forcheheimer model. Linear and nonlinear differential equations governing the hydraulic head variation with the horizontal coordinate contain a sink term, which is assumed to have a constant strength over the unintruded part of the catchment and linearly decreasing strength through the section overlying the interface. Solutions of the two equations are conjugated. For the length of the freshwater zone a quartic is obtained and solved. An explicit solvability condition is presented. The interface equation includes the commanding freshwater head, seawater level, density contrast between saline and fresh water, aquifer thickness and conductivity, and intensity of smeared losses, which model abstraction from the aquifer.
An optimal shape design problem for an empty drainage ditch is solved and it is shown that a semicircle intercepting saturated seepage from two constant potential rays is a contour of constant Darcian velocity and of maximal excavated volume at a given flow rate and ditch width. The corresponding family of isobars belongs to the class of conchoids of Nicomedes. The optimal flow is generated by a hydrodynamic dipole placed in a gravitational field.
A channel or axisymmetric dimple filled with a coarse porous material and aimed at a temporary storage of infiltrated water as a perched water table aquifer is studied. The bottom shape is varied based on the criterion of maximal water storage after a certain period of drainage and evaporation. Leakage into the vadose zone through a thin liner occurs with a specific discharge proportional to the pressure drop across the liner. Evaporation through a horizontal shrinking water table is spatially uniform. An ordinary differential equation, which follows from the mass balance condition, is solved either explicitly or numerically. The class of triangular, polynomial, and conical sections is studied. The shape of maximal water retention is calculated for a given initial stored water volume or water table width, evaporation intensity, liner thickness and conductivity, vadose zone pressure beneath the liner, and selected time interval between two sequential infiltration events.
The limiting case of the analytical solution for the downward flow of water in a Green and Ampt soil around a tunnel cavity of critical shape resting on a permeable substratum at infinite depth gives the shape of the cavity and also the shape of the fringe surface of the surrounding tension‐saturated region as parabolas. The cavity shape is compared with the parabolic shape obtained analytically for the water exclusion from a critical tunnel cavity in a Gardner soil of infinite extent where parabolic isobaric surfaces surround the cavity. The results are used to relate the parameters in the two models. They also add confidence for their use in soil hydrology.
The pioneering solution of Zhukovskii for a steady two-dimensional flow of an ideal heavy fluid with a nonlinear free boundary condition is extended to a Darcian flow of groundwater encumbered by an impermeable barrier. The stoss or/and lee sides of the barrier are covered by a macrovolume of a liquid contaminant. Explicit parametric equations of the sharp interface are obtained by inversion of the hodograph domain. Zhukovskii's gas-finger shape is shown to be a particular case of our new class of free surfaces. For a cap of a light liquid, partially covering the roof, from the given cross-sectional area of the cap, the affixes of the conformal mapping are found as a solution of a system of two nonlinear equations. The horizontal width and vertical height of the cap are determined. If the dimensionless incident velocity is higher than the density contrast, then the interface (cap boundary) cusps at its apex. For a relatively small velocity, the interface spreads to the vertices of the barrier, the apex zone remaining blunt shaped. We depict all the relevant domains and plot the flow nets using computer algebra routines.
Seepage through a zoned earth-filled dam with a vertical clay core, two permeable shoulders and toe drain are studied. Seepage through the core and the downstream shoulder are coupled. The flow rate and phreatic surface are found. The hodograph method is used, viz. a conformal mapping of a rectangle in the complex potential plane onto a circular triangle. Numerically, MODFLOW 2005 simulated seepage with the refraction of the streamlines at the interfaces between the core and both shoulders.
A semipermeable membrane (low permeability skin around a well) is modeled by a sink at the center of the well. The sink solution does not exist for thick or resistive skins. Seepage around a low permeability dam contour is modeled by a vortex. The curve, along which the third-type boundary condition holds, is reconstructed from the equation of the contour in polar coordinates. Flow around an impermeable dam in a porous medium containing a semiparabolic interface is studied analytically with the refraction condition, i.e., continuity of the head and normal flux along the interface between zones of contrasting permeability.
