Analytical Solutions of Geohydrological Problems

Abstract

PART A: Solutions Introduction to Part A. Abbreviations of solution methods. Classification of groundwater in the saturated zone. Main orientation table. A. Phreatic flow. Orientation table A. Exact solutions in phreatic flow. One-dimensional horizontal phreatic flow. Two-dimensional radial-symmetric horizontal phreatic flow. General two-dimensional horizontal phreatic flow. BI. One-dimensional groundwater flow. Orientation table BI. One-dimensional groundwater flow in an infinite field. One-dimensional groundwater flow in a semi-infinite field. One-dimensional groundwater flow in a finite field. Discontinuities in one-dimensional groundwater flow. BII. Two-dimensional radial-symmetric groundwater flow. Orientation table BII. Two-dimensional radial-symmetric groundwater low in an infinite field. Two-dimensional radial-symmetric groundwater flow outside a circular cylinder. Two-dimensional radial-symmetric groundwater flow inside a circular cylinder. Combinations in radial-symmetric groundwater flow. BIII. General two-dimensional groundwater flow. Orientation table BIII. General two-dimensional groundwater flow in an infinite field. General two-dimensional groundwater flow in an infinite half plane. General two-dimensional groundwater flow in an infinite quarter of a plane. General two-dimensional groundwater flow in in infinite strip. General two-dimensional groundwater flow in a semi-infinite strip. General two-dimensional groundwater flow in a finite strip. Combinations in general two-dimensional groundwater flow. BIV. Three-dimensional spherical symmetry. Orientation table BIV. Three-dimensional spherical-symmetric groundwater flow in an infinite field. Three-dimensional spherical-symmetric groundwater flow outside a sphere. BV. Three-dimensional axial symmetry. Orientation table BV. Three-dimensional axial-symmetric groundwater flow in an infinite field. Three-dimensional axial-symmetric groundwater flow in a semi-infinite field. Three-dimensional axial-symmetric groundwater flow in a finite field. Three-dimensional axial-symmetric groundwater flow inside a circular cylinder. Three-dimensional axial-symmetric groundwater flow outside a circular cylinder. Three-dimensional axial-symmetric groundwater flow with discontinuous ground parameters. BVI. Three-dimensional groundwater flow. Orientation table BVI. General three-dimensional groundwater flow in a semi-infinite or infinite field. General three-dimensional groundwater flow in a finite field. C. Multi-layer systems. Orientation table C. One-dimensional flow in continuous multi-layer systems. Two-dimensional radial flow in continuous multi-layer systems. General two-dimensional flow in continuous multi-layer systems. Multi-layer systems with a vertical flow component in one or more of the aquifers. Discontinuous multi-layer systems. D. Dispersion. Orientation table D. Dispersion in uniform flow. Dispersion in radial-symmetric flow. E. Sharp interface. density flow. Orientation table E. Exact solutions in interface flow. One-dimensional horizontal interface flow. Two-dimensional radial-symmetric horizontal interface flow. General two-dimensional horizontal interface flow. PART B: MATHEMATICAL TOOLS. I. BASIC PRINCIPLES FOR SATURATED GROUNDWATER FLOW. The continuum approach to groundwater flow. Molecular and microscopic level. Macroscopic level. Dispersion. Mathematical description of geohydrological problems. Equations of motion. Stream functions. Pressure and piezometric head. Darcy's law. Relations between stream functions and potential functions. Equations of motion for diffusion and dispersion. Parameters and variables. Groundwater parameters. Density. Compressibility of water. Modulus of elasticity. Viscosity. Ground parameters. Statistical description of the ground. Porosity. Specific surface. Intrinsic permeability. Soil compressibility and modulus of elasticity. Conductivity coefficients. Isotropic conductivity. Anisotropic conductivity. Resistance to flow in layered soils. Coefficients of dispersion. Storage coefficients. Elastic and phreatic storage. Barometric sensitivity. Tidal and phreatic sensitivity. Field tests (general description). Continuity equations and differential equations. General continuity equation (without dispersion). Differential equations for homogeneous water without dispersion. Non-steady flow of compressible groundwater through a compressible porous medium. Non-steady flow of incompressible groundwater in a compressible porous medium. Flow through an incompressible porous medium. Steady flow. Differential equations in other coordinate systems. Differential equations for the stream function psi. Injection term in the differential equation. Differential equations for non-homogeneous groundwater without dispersion. Density flow. Rotational flow. Differential equations for dispersion. Initial and boundary conditions. Linear conditions. General. Initial values. Open boundaries. Impervious boundaries. Common boundaries. Boundary conditions for solute concentrations. Non-linear conditions. Phreatic surface. Interface between fluids of different density. The hodograph. II. ANALYTICAL SOLUTION METHODS. Ordinary differential equations. Direct integration. Variation of parameters. Use of matrix functions for solving problems in multi-layer systems. Partial differential equations. Separation of variables. Laplace transformation. Fundamental properties. Some examples of Laplace transforms. Inverse transforms. Fourier transformations. Fourier series and integrals. Finite Fourier transformations. Infinite Fourier transformations. Hankel transformations. Orthogonal functions. Series and integrals of Bessel functions. Finite Hankel transformations. Infinite Hankel transformation. Conformal transformation. Complex analytic functions. Conformal mapping. The quadratic transformation. The Joukowski transformation. The Schwarz--Christoffel transformation. Successive transformations. Survey of integral transformations. Solutions, derived from known solutions. Superposition. Principles. Method of images. Discharge impulses. Hydrological screens. Product solutions. Periodic flow solutions. Solutions in anisotropic soils. Approximate solutions for phreatic flow and for density flow with interface. Phreatic flow. Interface flow. The reciprocity principle. Description and mathematical proof. The reciprocity principle in multi-layer systems. III. FUNCTIONS. Error functions and related functions. Error function. Polder function. Resistance function. The M-function. Exponential integral and related functions. Exponential integral. Hantush's well function. Bessel functions and related functions. Bessel functions J and Y. Modified Bessel functions I and K. Complex functions. Series of Bessel functions. Integrals of Bessel functions. Gamma function and hypergeometric function. Gamma function and related functions. Hypergeometric function. Elliptic integrals.