Philip Broadbridge

Infiltration Theory for Hydrologic Applications

Here is your state-of-the-art guidebook through soil infiltration theory in response to hydrologic problems. By focusing on the theoretical basis of physically based infiltration functions and their application, Infiltration Theory for Hydrologic Applications presents an in-depth review of current issues and concerns. For scientists wishing concise and robust equations that can be applied in models for a variety of objectives.

Nonclassical symmetry solutions and the methods of Bluman–Cole and Clarkson–Kruskal

Nonclassical symmetry solutions of physically relevant partial differential equations are considered via the reduction methods of Bluman and Cole and Clarkson and Kruskal. Consistency conditions will be provided to show that, if satisfied, these two methods are equivalent in the sense that they lead to the same symmetry solutions. The Boussinesq equation and Burgers’ equation are used as illustrative examples. Exact solutions, one of which is new, will be presented for Burgers’ equation obtained from the Bluman and Cole method, yet not obtainable by Clarkson and Kruskal’s method.

Sorptivity and macroscopic capillary length relationships

The relationship of sorptivity S and macroscopic capillary length λc is explored. Sorptivity is the proportionality constant between cumulative intake, expressed as a length, and the square root of time for sorption into an unsaturated soil. The capillary length is defined as the integral of the unsaturated hydraulic conductivity with respect to matric potential normalized by the difference between the conductivity at the two limits of integration. The approximation b = λc(θwct - θdry)(Kwct - Kdry)S−2 was found previously to have bounds of 0.5 and π/4, with 0.55 as a good overall approximation. Those results were for two diffusivity functions, an exponential D = D0 exp (Bθ) and for D = D0(θb - θ)−2. We now extend the investigation to four additional cases, two for which S and λc are analytical and two described by the empirical hydraulic functions of Brooks and Corey (1964) and by van Genuchten (1980). The representative value b = 0.55 was found to be generally correct. For those functions with a finite D value at the wettest value, a refinement from the 0.55 value is offered. The results are believed of value for future parameter estimation problems, including two previous applications related to time of ponding and disk infiltrometers.

Exact solutions for vertical drainage and redistribution in soils

We solve a versatile nonlinear convection-diffusion model for nonhysteretic redistribution of liquid in a finite vertical unsaturated porous column. With zero-flux boundary conditions, the nonlinear boundary-value problem may be transformed to a linear problem which is exactly solvable by the method of Laplace transforms. In principle, this technique applies to arbitrary initial conditions.The analytic solution for drainage in an initially saturated semi-finite column is compared to previously available approximate analytic solutions, obtained by assuming constant diffusivity, as in the Burgers equation, or by neglecting diffusivity, as in the hyperbolic model. Contrary to popular opinion, the hyperbolic model has more than one shock-free solution in a semi-infinite medium z ⩾ 0, as opposed to an infinite medium z ∈ ℝ. However, both the Burgers equation and an improved hyperbolic model underestimate diffusivity at high liquid contents and consequently overestimate the curvature of the soil liquid content profile.

Systematic construction of hidden nonlocal symmetries for the inhomogeneous nonlinear diffusion equation

We consider a class of inhomogeneous nonlinear diffusion equations (INDE) that arise in solute transport theory. Hidden nonlocal symmetries that seem not to be recorded in the literature are systematically determined by considering an integrated equation, obtained using the general integral variable, rather than a system of first-order partial differential equations (PDEs) associated with the concentration and flux of a conservation law. Reductions for the INDE to ordinary differential equations (ODEs) are performed and some invariant solutions are constructed.

Exact solvability of the Mullins nonlinear diffusion model of groove development

The Mullins equation for the development of a surface groove by evaporation–condensation is yt=yxx/1+y2x. It is pointed out that this is the equation of the potential for the field variable Θ satisfying the nonlinear diffusion equation Θt=∂x[Θx/1+Θ2]. The latter has already been solved exactly, with boundary conditions corresponding exactly to those specified by Mullins. The depth of a groove at a grain boundary is predicted exactly without first making the linear (small‐slope) approximation. For some types of initial data, the Cauchy problem may be solved for some related equations.

Exact transient solutions to nonlinear diffusion-convection equations in higher dimensions

The complete Lie algebra of classical infinitesimal symmetries of the nonlinear diffusion-convection equation in two and three dimensions is presented. Except for some cases involving constant diffusivity, a complete reduction to an ordinary differential equation is not possible. However, closed-form solutions are obtained for special forms of both the 2D and 3D nonlinear diffusion-convection equations, using a symmetry reduction and an additional physical constraint. This extends the small list of closed-form transient solutions already known.

Integrable forms of the one‐dimensional flow equation for unsaturated heterogeneous porous media

The equation for the horizontal transport of a liquid in an unsaturated scale‐heterogeneous porous medium is ∂θ/∂t=λ(x)∂/∂x[C(θ)∂θ/∂x] −λ’(x)E(θ)∂θ/∂x−λ‘(x)∫(C+E)dθ. A systematic search for Lie–Backlund symmetries leads to the requirement that C=a(b−θ)−2, as in the homogeneous (λ=1) case. More generally, (λ,E) may be ((1+mx)α, (1/α− (3)/(2) )C) or (exp(mx),−3C/2). In these cases the transport equation may be linearized and solved exactly. Examples of more complicated heterogeneous extensions are presented for the integrable nonlinear diffusion equations and for Burgers’ equation.

Grain boundary grooving by surface diffusion: an analytic nonlinear model for a symmetric groove

The fourth-order nonlinear boundary-value problem for the evolution of a single symmetric grain-boundary groove by surface diffusion is modelled analytically. A solution is achieved by partitioning the surface into subintervals delimited by lines of constant slope. Within each subinterval, the advance of the surface is described by an integrable nonlinear evolution equation. The model is capable of incorporating the actual nonlinearity arbitrarily closely. The surface profile is determined for various values of the central groove slope including the limiting case of a groove which has a root that is vertical. Such a solution exists only because of the nonlinearity.

Calculation of humidity during evaporation from soil

Abstract The relative humidity at the surface of a soil is calculated as a function of time from a realistic, exactly solvable model for unsaturated fluid flow. When the evaporation rate is constant, as in the atmosphere-controlled phase, the surface relative humidity suddenly drops at a particular time, even though the humidity–time function is differentiable. This sudden transition still occurs, but is smoother, when more realistic radiation-type boundary conditions are introduced and when the soil-water diffusivity approaches zero continuously at zero water content. In these cases, solutions are obtained by a numerical method of lines which has been validated against the analytically solvable model. A gradual decrease in surface water content is not inconsistent with a sharp, step-like decrease in relative humidity. This is due to the universal exponential Gibbs–Boltzmann relationship between relative humidity and soil-water potential.