P. J. Ross

Three-dimensional analysis of infiltration from the disc infiltrometer: 2. Physically based infiltration equation

In situ measurement of soil hydraulic properties may be achieved by analyzing the unconfined efflux from disc tension infiltrometers, once consistent infiltration equations can be derived. In this paper an analytical, three-dimensional infiltration equation is developed, based on the use of parameters with sound physical meaning and adjustable for varying initial and boundary conditions. The equation is valid over the entire time range. For practical purposes, a simplified solution is also derived. The full and simplified equations give excellent agreement with published experimental results and are particularly useful for determining soil hydraulic properties through application of inverse procedures.

Three‐dimensional analysis of infiltration from the disc infiltrometer: 1. A capillary‐based theory

The hydraulic properties of an unsaturated homogenous and isotropic soil can be obtained from the unconfined flux out of a disc infiltrometer into the soil over the depth of wetting. The disc infiltrometer is becoming increasingly popular, but methods of analysis have generally relied on the restrictive assumptions of one-dimensional flow at early times or quasi-steady state flow at large times. We provide an approximate analytical expression for three-dimensional unsteady, unconfined flow out of a disc infiltrometer, and this includes the geometric effect of the circular source but ignores gravity. This physically based solution is tested against data obtained from laboratory experiments on repacked material. The results illustrate that the difference between three-dimensional and one-dimensional flow is linear with time.

Infiltration under ponded conditions: 4. An explicit predictive infiltration formula

An approximation to an implicit infiltration formula presented earlier in this series is developed. At worst, the relative error of the approximation is always less than 1%, and it is much better than that for most cases. The approximation becomes more accurate in both the short- and long-term limits as it becomes exact in each case. Application of the new formula is demonstrated using both laboratory and numerical experiments.

New approximate analytical technique to solve Richards equation for arbitrary surface boundary conditions

A general approximation for the solution to the one-dimensional Richards equation is presented. It applies to arbitrary soil properties and boundary conditions but only uniform initial conditions at current stage. The result is very accurate (within 2%) when the diffusivity is constant, suggesting that the present general formulation is reliable, since the approximation becomes increasingly accurate as the soil-water diffusivity approaches a delta function.

Soil Water Hysteresis Prediction Model Based on Theory and Geometric Scaling

Hysteresis in the water retention characteristic of a soil is important in prediction of soil hydraulic properties and in description of vadose zone flow and transport processes. Although a number of models have been proposed, some are entirely empirical and others are inconvenient to use. We apply a theoretical approach to derive parameters for a more suitable model based on the concept of rational extrapolation. Considering the water retention curve to be described by three parameters, the first defining the shape of the curve and the two others scaling the soil water pressure head and volumetric soil water content, three geometrical scaling conditions are derived. The first condition determines a shape parameter which is identical for all wetting and drying curves and independent of their scanning order; the second defines the relation between the pressure head scale and the water content scale specific to each curve in wetting or drying; and the third condition determines specific water content scale parameters according to the points of departure and arrival of each scanning curve. Equations necessary for the calculation of the different scale parameters are derived. Given the saturated water content, all main, primary and higher order scanning curves can be predicted from knowledge of only one curve, although in practice knowledge of a main or primary curve is desirable. Constraints such as the need for scanning curves to be closed and to lie inside curves of a lower scanning order are automatically satisfied. The method is illustrated using the van Genuchten water retention function but could also be applied to other functional forms. Results agree well with the existing data. It appears that knowledge.

Approximate analytical solution of the nonlinear diffusion equation for arbitrary boundary conditions

A general approximation for the solution of the one-dimensional nonlinear diffusion equation is presented. It applies to arbitrary soil properties and boundary conditions. The approximation becomes more accurate when the soil-water diffusivity approaches a delta function, yet the result is still very accurate for constant diffusivity suggesting that the present formulation is a reliable one. Three examples are given where the method is applied, for a constant water content at the surface, when a saturated zone exists and for a time-dependent surface flux.

Three‐Dimensional Analysis of Infiltration from the Disk Infiltrometer: 3. Parameter Estimation Using a Double‐Disk Tension Infiltrometer

A double-disk tension infiltrometer is developed to measure simultaneously one-dimensional (1-D) and three-dimensional (3-D) infiltration subject to identical initial and boundary conditions. The hypothesis that cumulative three-dimensional infiltration is proportional to the square root of time is incorrect after a few seconds, and in consequence, estimates of sorptivity based on this hypothesis are in error. In contrast, sorptivity calculations using a recently developed 3-D infiltration equation that includes the edge effects of the disk give accurate estimates of sorptivity. The difference between 3-D and 1-D cumulative infiltration is used to calculate the value of an additional infiltration parameter from the double-disk experiment. This parameter, together with the sorptivity, provides the information required to calculate the contribution of gravitational flow during three-dimensional infiltration. It is shown that estimation of hydraulic conductivity using a quasi-steady-state solution may not be justified when 3-D infiltration is dominated by capillary flow effects. Finally, an analysis of the different timescales governing disk infiltrometer experiments under most practical circumstances is provided. A simple expression is given for the time at which the infiltration rate is within a specified amount of the final steady rate.

A note on the error analysis of time compression approximations

The accuracy of the time compression analysis (TCA) is analyzed by comparison with a numerical solution. Both the standard TCA and a new modified TCA are considered for a power law diffusivity and constant surface flux. As expected, the error of the approximations decreases with increasing power, and the error of the modified TCA is about half the error of the standard TCA. In a second part, the errors of the two TCAs are measured using a simple analytical solution instead of a numerical solution. It is shown that the conclusions remain the same for the analytical and numerical solutions. The advantage of using the analytical solution is to obtain simple analytical expressions, showing the influence of parameters. This is done to estimate the maximum error of both TCAs. A practical estimate of the errors can be obtained from equations which only require knowledge of the soil water diffusivity. It appears that for real soils the errors of the TCA are always <1% and thus are a very reliable tool for practical problems. Although not studied systematically, it also appears that gravity effects reduce the errors of the TCA so that the error obtained in the absence of gravity provides a conservative estimate when gravity is present.

Cubic approximation of hydraulic properties for simulations of unsaturated flow

An adequate description of soil hydraulic properties is critical to modeling unsaturated flow. However, flexible descriptions based on the functions normally used may increase simulation times. This disadvantage can be overcome by using piecewise cubic approximation of such descriptions based on a table of function and derivative values. A further advantage of this technique is that the Kirchhofif transform variable can be approximated in the same way, even when it cannot be obtained analytically. Results for simulated infiltration into an aggregated soil indicated that hydraulic properties over the entire matric potential range could be approximated using 8–12 points without causing significant errors in simulations, whether the Kirchhoff transform was used or not. Run times using the original van Genuchten functions were 40% or 280% greater, depending on whether an incomplete beta function was used. The cubic approximation method thus allows more flexible use of the popular van Genuchten hydraulic property functions, since these involve incomplete beta functions unless unsatisfactory restrictions are placed on their parameters.

Calculating parameters for infiltration equations from soil hydraulic functions

Simple equations for predicting infiltration of water into soil are valuable both for hydrological application and for investigating soil hydraulic properties. Their value is greatly enhanced if they involve parameters that can be related to more basic soil hydraulic properties. In this paper we extend infiltration equations developed previously for positive surface heads to negative heads. The equations are then used to calculate infiltration into a sand and a clay for a range of initial and surface conditions. Results show errors of less than three percent compared with accurate numerical solutions. Analytical approximations to parameters in the equations are developed for a Brooks and Corey power law hydraulic conductivity-water content relation combined with either a Brooks and Corey or a van Genuchten water retention function. These are compared with accurate numerical values for a range of hydraulic parameters encompassing the majority of soil types and a range of initial and boundary conditions. The approximations are excellent for a wide range of soil parameters.An important attribute of the infiltration equations is their use of dimensionless parameters that can be calculated from normalised water retention and hydraulic conductivity functions. These normalised functions involve only parameters that it may be possible to estimate from surrogate data such as soil particle size distribution. Application of the equations for predicting infiltration, or their use in inferring hydraulic properties, then involves only simple scaling parameters.