Abstract. The present work explores a class of analytical solutions of moisture movement in unsaturated porous media characterized by an exponential dependence of the hydraulic conductivity and the moisture content on water pressure. The Green's function method is used to derive a general analytical model pertaining to multidimensional nonsteady infiltration in a semi-infinite flow domain with arbitrary initial conditions, boundary conditions, and root-uptake forcing functions and for various simple source geometry. The general solution is expressed in integral form from which particular analytical solutions pertaining to cases of surface and subsurface irrigation, evaporation, root uptake, and moisture redistribution can be easily deduced from the general analytical model. The model offers the analyst significant flexibility in deriving results and analyzing infiltration phenomena of practical interest. New explicit solutions have been obtained for onedimensional infiltration under various prescribed time-dependent flux boundary conditions and for two-and three-dimensional moisture redistribution. For constant initial or boundary conditions, the multidimensional solution is essentially the product of two or three time-dependent terms with each term being a function of only one space variable. IntroductionThe analysis of nonsteady infiltration has largely been carried out using numerical methods mainly because of the difficulty in solving analytically the governing nonlinear differential equation. Although numerical methods are powerful in solving complex nonlinear problems, analytical results provide general insights and concisely identify the relationships among the variables from which rational approximations and simplifications can be derived. They can also be useful for checking numerical schemes. During the past few decades, many analytical solutions for steady and nonsteady infiltration have been developed. Most of these solutions use the exponential hydraulic conductivity model first proposed by Gardner [1958]: TheoryThe which models an exponentially decreasing uptake as well as a constant uptake with b -0 over a defined depth z -< d. Analytical solutions to (2) can be obtained only for some special functions of the hydraulic conductivity. Such a special function is the exponential form of the hydraulic conductivity (1), where a, the sorptive number, is a measure of the importance of gravity relative to capillarity. It is small in fine-textured soils where capillarity is dominant and large in coarse-textured soils where gravity is dominant [Philip, 1969]. Philip [1984] stated that a -• ranges between 0.2 and 5 m; and fitted values of ko and a for some 17 soils is given by Bresler [1978].In order to linearize the right-hand side of the differential equation, a similar assumption on the variation of the moisture content must be taken:

An analytical model which provides an approximate description of scale‐dependent transport is presented. The model is based on the advection‐dispersion equation but with the dispersion coefficient dependent on the travel time of the solute from a single input source. The time dependence of the dispersion coefficient can assume arbitrary functional forms. The governing equation, which includes a time‐varying dispersion coefficient, linear equilibrium adsorption, and first‐order reaction, is reduced to the heat diffusion equation after a series of transformations. Analytical solutions pertaining to a time‐varying mass injection in an infinite medium with arbitrary initial distribution of concentration and functional dependence of the dispersion coefficient can be easily derived from the model. Since the analytical solution is in an integral form, different temporal variation of the dispersion coefficient over different time intervals can also be incorporated. It is shown that the solution reduces to the well‐known result for the case of constant dispersion and mass injection. In this study, particular solutions for various dispersion functions and mass injection scenarios are presented. These include linear, exponential, and asymptotic variation of the dispersion functions and instantaneous as well as continuous mass injection. The analytical results could model the transport of solute in a hydrogeologic system characterized by a dispersion coefficient that varies as a function of travel time from the input source. It could provide a modeling solution to solute transport problems in heterogeneous media and be used as a suitable model for the inversion problem, especially since more than one fitting parameter is available to fit the field tracer data which exhibit a scale effect.

[1] An approximate nonlinear solution of the one-dimensional Boussinesq equation is presented using the traveling wave approach. The solution pertains to a semi-infinite phreatic aquifer with a uniform water table that is subject to a time-varying water level at the stream-aquifer boundary. The advantage of the traveling wave method is in its versatility in handling transient boundary conditions while preserving the inherent nonlinearity of the physical phenomenon. The nonlinear solution is of a simple logarithmic form and describes the position of the water table as a function of time. It yields the exact solution for the special case of uniform water level rise at the boundary. Algebraic expressions that quantify the main flow processes are derived from the basic solution. These include the stream-aquifer exchange flow rates, bank storage and depletion, front position and propagation speed, and an improved working relationship for aquifer parameter estimation. A comparison with two exact solutions and numerical solutions of the Boussinesq equation validates the accuracy of the approximation and highlights the limitation of the method in specific flow conditions. The traveling wave model performs best for sharp front movements and monotonic water table profiles and provides excellent estimates of the flow rate and volume at the inlet boundary. The accuracy of the solution deteriorates for fluctuating inlet conditions and worsens in cases when there is a sharp reversal of flow conditions.

[1] This work derives the fracture flow equation from the two-dimensional steady form of the Navier-Stokes equation. Asymptotic solutions are obtained whereby the perturbation parameter is the ratio of the mean width over the length of the fracture segment. The perturbation expansion can handle arbitrary variation of the fracture walls as long as the dominant velocity is in the longitudinal direction. The effect of the matrix-fracture interaction is also taken into account by allowing leakage through the fracture walls. The perturbation solution is used to obtain an estimate of the flow rate and the fracture transmissivity as well as the velocity and the pressure distribution in fractures of various geometries. The analysis covers eight different configurations of fracture geometry including linear and curvilinear variation as well as sinusoidal variation in the top and bottom walls with varying horizontal alignment and roughness wavelengths. The zeroorder solution yields the Reynolds lubrication approximation, and the higher-order equations provide a correction term to the flow rate in terms of the roughness frequency and the Reynolds number. For sinusoidal and linear walls, the mathematical analysis shows that the zero-order flow rate could be expressed in terms of the maximum to minimum width ratio. For equal widths at both ends of the fracture, the first-order correction is zero. For sinusoidal fractures, the flow rate decreases with increasing Reynolds number and with increasing roughness amplitude and frequency. The effect of leakage is to create a nonuniform flow distribution in the fracture that deviates significantly from the flow rate estimate for impermeable walls. The derived flow expressions can provide a more reliable tool for flow and transport predictions in fractured domain.

Abstract. The Green's function method is used to derive a general analytical model pertaining to multidimensional nonsteady infiltration toward a shallow water table with arbitrary initial conditions, boundary conditions, and root uptake forcing functions and for various simple source geometries. The general Green's function solution allows the derivation of particular analytical solutions pertaining to cases of surface and subsurface irrigation, evaporation, root uptake, and moisture redistribution. The model assumes that the hydraulic conductivity function is an exponential function of the water pressure and a linear function of the moisture content. One-dimensional solutions are obtained for timedependent distributions of the flux at the land surface and of root uptake in the subsurface. Two-and three-dimensional solutions are also derived for point and line sources in the domain, for a strip, a rectangle, and a disc source at the land surface, and for two-and three-dimensional moisture redistribution and root uptake. Most of the solutions are a product of two terms; one term is a function of depth while the other term is a function of time and the lateral coordinates. Numerical results show the effect of the water table and soil type on the two-dimensional moisture movement from a strip source.