Efficient Approach to Solving Transient Unsaturated Flow Problems. I: Analytical Solutions

Abstract

Richard's equation governs the migration of moisture in the soil under unsaturated conditions. Although this differential equation provides a rigorous approach to simulating important infiltration problems, obtaining analytical and numerical solutions to this equation has been a particularly challenging task. This is largely due to the highly nonlinear nature of the soil hydraulic properties, including the moisture retention curve and the hydraulic conductivity function. Whereas analytical solutions of Richard's equation have been reported for problems involving steady-state conditions and simple hydraulic models, solutions for transient conditions have rarely been obtained. However, such analytical solutions would be particularly valuable, for example, to validate the accuracy of numerical schemes, as well as to facilitate parametric evaluations. A series of analytical solutions of Richard's equation for unsaturated flow under transient conditions have been developed as part of this study. The solutions involve a variety of initial and boundary conditions. The analytical solutions in this study could be obtained after expressing the governing equation as the addition of advective and diffusive flow components. The solutions consider logarithmic and linear models to represent the soil moisture retention and the hydraulic conductivity functions, respectively. Solutions are also provided for special cases in which either the advective or the diffusive components dominate the flow process, as well as for the steady-state cases. A parametric evaluation was found to provide insight into important characteristics of infiltration problems. In particular, relevant features of an unsaturated flow problem can be explained by evaluating the trends in its advective and diffusive flow components.

Transient unsaturated shaft resistance of a single pile during water flow

“…Cavalcante and Zornberg (2017) [6] proposed a model considering one fitting parameter for the soil-water retention curve (SWRC) and a linear function concerning volumetric water content () for the hydraulic conductivity function. The study solved Richard's equation -which governs the unsaturated flow through porous media by a rigorous approach -analytically for a one-dimensional flow.…”

“…The soil-water retention curve (SWRC) that correlates the soil suction and the soil moisture, defined by Cavalcante and Zornberg (2017) [6], allows analytical transient solutions for Richard's equation, which models the infiltration into unsaturated soil. This model is advantageous because it suits parametric evaluations of the moisture during water infiltration in the soil.…”

Evaluation of antecedent rainfall effects in the analysis of the probability of transient failure in unsaturated slopes

“…The points of the SWRC and the physical indices [i.e., θs and θr] (Tab. 1), both found experimentally, were implemented in constitutive models [ψ(θ) and kz(θ)] proposed by Cavalcante & Zornberg (2017) [8]. The constitutive models allowed the authors to propose an analytical solution to solve the Richards equation, as follows Where δ = fitting hydraulic parameter (M -1 LT ² ); θ = volumetric water content (L 3 /L³); θr = residual volumetric water content (L 3 /L³); θs = volumetric water content at saturation (L 3 /L³).…”

“…Subsequently, was applied the analytical solution proposed by Cavalcante & Zornberg (2017) [8] to solve the Richards equation. The proposal involves a modification of the flow equations by expressing the governing equation as the addition of advective and diffusive flow components and corresponds to the equation as follows…”

“…ξ = positive constant dependent on porous soil constitution, history and magnitude of applied stresses and hysteresis during soil wetting and drying; ψ(z, t) = total transient suction derived from the constitutive soil model proposed byCavalcante & Zornberg (2017) [8]; Sr (z, t) = degree of transient saturation expressed in terms of moisture content; γnat = natural specific soil weight (LM -; and z = thickness of the sliding mass (L).…”