We consider solutions to two boundary values problems for the Poisson equation on plane domains. We prove several estimates for integrals of solutions using geometric characteristics of domains.
A discussion of a paper with the aforementioned title by H.M. Hathoot, published in this journal (Volume 124, Number 2, March/April 1998), is presented. The discusser maintains that the author's use of the steady problem solution for calculation of water-table drawdown, the corresponding design of drain spacing, and optimization of the mole drain array step are interesting, both practically and mathematically. He questions, however, the assumption of the existence of infinitely high depth of soil above the drain arrays. He maintains that employment of sinks requires a posteriori expressions of the depth of the drain through the depth of the sink and that generally, the two depths do not coincide. In application to the flow pattern, he believes it is unclear for what stages of decrease of the water table the half-sink approximation is valid. Finally, he contends that optimization of drain spacing did not include precise formulation of the mathematical problem. Discussion is followed by closure from the author.
Abstract Riesenkampf's (1938), R‐38 (referred to here as R‐38), analytical solution for steady 2‐D flow from a buried line source in a homogeneous Green‐Ampt soil, with a wetting plume bounded by a free surface (capillary fringe), is compared with Philip's (1969), (P‐69), one for genuinely unsaturated wetting of Gardner's infinite‐extension soil. Conformal mappings are used in R‐38, from which we derived the flow net, pore‐water isobars, isochrones, fields of Darcian velocity and resultant force acting on saturated porous skeleton, fine geometry (shape and size) of the constant‐head contour encompassing a mole‐emitter or leaky‐pipe, as well as the dependence of the total discharge per unit pipe length on uniform pressure in the pipe, capillarity of the soil, radius of the pipe, and saturated hydraulic conductivity. An ovalic “water table” isobar, encompassing P‐69 source, is compared with one of R‐38 for a fixed discharge and saturated conductivity but adjusted sorptive numbers. The Whisler and Bouwer (1970) relation between the static height of capillary rise and sorptive number is shown to give a good match between R‐38 and P‐69 isobars. This allows to use R‐38 in the source vicinity and P‐69 in the far‐field zone. Computer algebra ( Mathematica ) routines are used for visualization of the known and extended R‐38 and P‐69 solutions.
The shape of a condenser of maximal cross‐sectional area at a given capacity is derived in an analytical explicit form. Optimization is performed in the class of practically arbitrary curves by solution of the Dirichlet boundary‐value problem for a complex coordinate and expansions of the Cauchy integral kernels in Chebyshev polynomials. The criterion (area) becomes a quadratic form of the Fourier coefficients and both the necessary and sufficient extremum conditions are rigorously satisfied such that a global and unique extremum is achieved. The resulting curve coincides with the Polubarinova‐Kochina contour of a concrete dam of constant hydraulic gradient, which in its own term coincides with the Taylor‐Saffman bubble. In the limit of high capacitance, the Polubarinova‐Kochina contour tends to the SaffmanTaylor finger, which in its own turn coincides with the Morse‐Feshbach condenser contour of constant field intensity. Thus the contour found is of a minimal breakdown danger in the dielectric between charged surfaces (non‐isoperimetric optimum) and of maximal confined area (isoperimetric extremum).
Seepage through an aquifer, the hydraulic conductivity of which varies vertically, is studied using the Dupuit-Forchheimer approximation (Girinskii's potential) and numerically by FDM-MODFLOW and FEM-HYDRUS-2D. In urban water hydrology, the effect of compaction of the top stratum of an aquifer on the flow rate and the position of the water table (characterized by an integral quantity of the saturated/dry area) is analysed. The area of the saturated zone is evaluated as a function of the conductivity ratio of the two strata, and their thicknesses. Conductivity varying linearly and exponentially is also analytically studied. Boundary-value problems are solved for a linear (nonlinear) ordinary differential equation if the evaporation rate is constant (decreasing exponentially with depth). The MODFLOW and HYDRUS-2D simulations give consistent fields of the piezometric head, pressure head, and Darcian velocity, as well as the water table position, which may have a global minimum (watershed).
A hydropedological study was conducted to investigate the impact of the construction of Al-Khoud recharge dam on soil development in a dry region of Oman. The study involved detailed descriptions of pedons, surface and subsurface soil textural analyses, and double-ring and tension infiltrometer tests in areas inside and adjacent to the dam. The reservoir area of this 25-year-old hydraulic structure has rapidly changed due to spatiotemporally variable deposition of sediments from the dammed water and intensive scraping of the silt cake. This resulted in the formation of multiple micro depressions and a nonflat shape of the reservoir bed. The subsurface soils of the bed showed heterogeneity and complex patterns of sediment deposition as a response to the human-induced changes in the soil development, geomorphology, and hydrological properties of the dam area. In most pedons, a sequence of Stokes' law–generated porous layers indicates ponded conditions of flood events and reservoir filling correlated with the hydrographs of the gaging station that feeds the reservoir. Textural analysis indicates the difference of ponding depth durations and sediment load is highly variable for pedons inside the reservoir and in the off-dam adjacent zones, both upstream and downstream of the embankment. The formation of a unique three-dimensional cascade of silt blocks, sandwiched by sand-filled horizontal and vertical fractures, was discovered. The propagation of silt entrained by infiltrated water into an originally coarse parent alluvium and the ensuing reduction of permeability were detected. Cumulative infiltration and infiltration rate curves and saturated hydraulic conductivities are related to the soil hydraulic-capillary properties.
Abstract Large quantities of water are usually produced in conjunction with oil production in Oman. Because this water is saline and heavily contaminated with oil, it is not suitable for domestic or agricultural uses. A short-duration comparative study on the effects of using treated oily-water and fresh water on soil physical properties has been performed. It was found from this study that the use of treated oily-water caused a sodicity problem, which has adverse effects on the soil physical properties of the soil such as infiltration rate, saturated hydraulic conductivity and pore size distribution. The reduction in the saturated hydraulic conductivity reached up to 43% of the initial value.
A zero-pressure isobar is reconstructed in a potential Laplacian field generated by a sink–source pair that models steady-state Darcian seepage from a ponded soil surface to a cavity draining a homogeneous isotropic half-plane. At sufficiently small flow rates (cavity depths under the soil surface) the obtained contour is almost spherical. For a given "sphere" its depth under a constant-head plane is found providing minimal inflow rate.
The inherent complexity of soil and its interactions with Earth’s diverse spheres, including the atmosphere, biosphere, hydrosphere, lithosphere within the ecosphere, and anthroposphere, requires that soil science specialists and students develop not only a profound understanding of soil science, but also the ability to collaborate across various disciplines to address these complex challenges. Equipping students with the necessary knowledge, skills, and attitudes to tackle the intricate and dynamic issues of the 21st century, spanning soil science, water sciences, hydropedology, geology, agronomy, geotechnical engineering, sedimentation, waste management, recycling, and environmental management, is of paramount importance. In response, innovative pedagogical approaches that integrate classroom learning from diverse soil science courses with practical skills and field-based competencies are needed. This paper suggests merging our own “Soil Skills” (SSK) pedagogical method with the “Soil Judging Contest” (SJC), a teaching approach supported by the American Society of Agronomy and the Soil Science Society of America since 1961. This integration aims to enhance the holistic, harmonized, interdisciplinary, and enthusiastic nature of soil science education. Both the SSK and SJC approaches received positive feedback from students and demonstrated significant improvements in academic performance. Our study begins with an in-depth exploration of the SSK contest, followed by an overview of the pertinent aspects of the SJC. Subsequently, we offer a comparative analysis of the complementarity of these two approaches. Finally, in the concluding remarks, we summarize the strengths of the implemented SSK and outline prospective applications. Our findings underscore the unique advantages of combining SSK and SJC approaches in delivering comprehensive, problem-based, and practical field-learning experiences. This combination approach closely aligns with applied scenarios that demand multidisciplinarity and interdisciplinarity perspectives, preparing students for their future professional careers, and enabling the practical application of their soil science knowledge in real-world contexts.
Abstract A study is made of the longitudinal 2D viscous steady flow and heat flux between two plates. Optimal shape design problems are solved in explicit form and shown to have unique global extrema. Conformal mappings are used to bring the problems into a fixed domain and solve them as Dirichlet boundary value problems in the form of Cauchy integrals and series expansions. For the simplest problem statement the optimum is shown to coincide with the well‐known concrete dam outline of constant hydraulic gradient.
