J. Aa. Hansen

Rapid and Numerically stable Simulation of One-dimensional, Transient Water Flow in Unsaturated, Layered Soils

We present a rapid numerical solution for vertical, transient flow of water in unsaturated soil. The model is labeled the moving mean slope model (MMS model), because it uses the slope of the natural log of the hydraulic conductivity (K) versus soil-water potential (ψ) curve as a dynamic parameter. The MMS model is developed from a model for flow in homogeneous, relatively wet soils presented by Wind and von Doorne (1975). The model can simulate transient flow in homogeneous and heterogeneous soils correctly for any rang e of soil-water content. This has been validated against semianalytical solutions and solutions obtained with traditional finite-difference and finite-element models. A Courant number analysis method is presented to make direct comparisons of criteria for avoiding numerical errors for the MMS model compared with traditional finite-difference models. For coarse-textured soils, the MMS model uses about the same computer time as the traditional finite-difference and finite-element models. For soils ranging from fine-textured to medium-textured, the MMS model is one to several orders of magnitude faster than the traditional numerical models.

A SIMPLE, MECHANISTIC MODEL FOR SOIL RESISTANCE TO PLANT WATER UPTAKE

A simple, mechanistic model for soil resistance (Rs) to water flow towards plant roots is presented. This new Rs model is derived from a single root, water flow model and assumes constant soil hydraulic conductivity within the rhizosphere for each combination of depth and time increment in the calculations. The new Rs model takes into account root density, soil hydraulic conductivity, and soil-water capacity. The effect of the soil-water capacity is included by either i) using the slope of the curve representing the unsaturated soil hydraulic conductivity versus the soil-water potential in a log-log coordinate system as a dynamic model parameter or ii) if the unsaturated soil hydraulic conductivity is not measured, by using the slope of the soil-water characteristic curve in a log-log coordinate system as a dynamic model parameter. The latter approach is based on the validity of the so-called Alexander capillary tube model for water flow. The new Rs model is compared with the traditionally used Rs model based on steady-state flow. The two Rs models give different results for both clayey and sandy soils. The differences are mainly due to the effect of the soil-water characteristic curves. The new Rs model is mechanistically more appropriate than the steady-state based Rs model because it includes the effect of the soil-water capacity, i.e., the soils ability to withhold water at given pressure gradient. The new Rs model should be useful in obtaining simple, mechanistic, micro-computer models for plant uptake of water.

AN ACCURATE AND NUMERICALLY STABLE MODEL FOR ONE-DIMENSIONAL SOLUTE TRANSPORT IN SOILS1

Most numerical calculation schemes are either unstable or fairly complicated to solve and program when used for calculating water and solute transport in unsaturated soils. A solute transport model that is both numerically stable and easy to program will represent a useful alternative to existing models. We present an accurate and easily programmed numerical model for one-dimensional transport of solutes in unsaturated or saturated soils at steady or transient water flow. The new approach is based on the classical convection-dispersion concept of solute transport. The model is labeled the moving concentration slope (MCS) model because it uses the slope of the curve representing the natural log of the solute concentration (c) times the ratio of the hydrodynamic dispersion (D) to the water flux (v) plotted versus the convective solute flux (vc) as a governing, time-dependent parameter. The MCS model is a modification of the recently presented moving mean slope (MMS) water flow model and was developed using the mathematical equivalence between the basic flux and continuity equations for water and solute transport. In the MCS model, an integrated version of the solute flux equation is used together with a simple, forward-time discretization of the continuity equation to calculate solute transport. In case of a depth increment larger than 1 cm and a ratio of D to the pore water velocity (u) smaller than 2, it is necessary to correct the MCS model for numerical errors of second order (numerical dispersion), but the corrections are easy to make. Analytical equations for the second and third order numerical errors inherent in the MCS model were derived using Taylor series and validated using method of moments analysis. The magnitude of numerical errors inherent in the MCS model is generally small compared to explicit finite difference (FD) models. Also, the MCS model is very simple to program compared to finite element (FE) and implicit FD models. The MCS model was validated against analytical solutions in the case of steady water flow and against FE models in the case of transient water flow. The MCS solute transport model used together with the MMS water flow model represents a convenient tool for easy and accurate modeling of one-dimensional transport of water and solutes in soils.

Prediction of Cadmium Concentrations in Danish Soils

Detailed studies on the effects of sludge utilization in agriculture have been conducted in Denmark since 1973.

Predicting wetting front advance in soils using simple laboratory derived hydraulic parameters

An approximate simulation of wetting front advance or leaching time is often the necessary and sufficient information required to optimize irrigation strategies or wastewater land treatment systems. For such simulations, we show that only three physically comprehensible and easily measured parameters are necessary, namely the saturated hydraulic conductivity, and the slope and the intercept of the soil-water potential plotted against the natural logarithm to the soil-water content. We derived the new 3-parameter simulation model from a fully linear flow model based on the Richards equation with the additional assumption of zero conductivity at zero soil-water content. A simple analytical solution for water infiltration during saturation at the soil surface, based on the 3-parameter model, gave results that compared well with measured wetting front advances in three Danish loamy and coarse sands and, also, with semi-analytical and numerical solutions from literature. Less agreement between predicted and measured wetting front advance was observed during infiltration into an initially dry forest soil due to extreme soil-water repellency. The presented derivation procedure assuming linear water flow seems promising for obtaining simple, approximate analytical solutions for unsaturated flow problems.

IMPROVED SIMULATION OF UNSATURATED SOIL HYDRAULIC CONDUCTIVITY BY THE MOVING MEAN SLOPE APPROACH

Numerically simple models for soil transport processes based on the use of the slopes of soil parameter curves as dynamic parameters have proved to be a stable and accurate supplement to other models. In this study, the recently presented moving mean slope (MMS) model for vertical, unsaturated water flow is evaluated, using series and numerical analyses. Results show that the MMS model takes into account additional parameters in the estimation of the soil hydraulic conductivity (K) between the calculation depths compared with the arithmetic or geometric mean value considerations used normally. This suggests that the MMS modeling approach implicitly provides a good estimation of the nonlinear change of K with soil depth and explains the observation that it is often possible to use larger depth and time increments in the MMS model than in many of the traditionally used numerical models. The vertical MMS model is modified for horizontal flow, and the corresponding criterion for avoiding numerical errors in the calculations is derived. The agreement between horizontal flow calculations using the simple MMS model and the Crank-Nicolson finite difference model is excellent. A new series expression for the horizontal flow velocity is obtained by using a reformulated, integrated version of the horizontal Darcy flow equation. After a short infiltration time, only the first few terms of the series expression are quantitatively important. This seems convenient for analyzing and mathematically describing infiltration into soils.

A SIMPLE, INVERSE MODEL FOR ESTIMATING NITROGEN REACTION RATES FROM SOIL COLUMN LEACHING EXPERIMENTS AT STEADY WATER FLOW

Many soil nitrogen studies have included estimations of nitrification or denitrification rates in soil column systems based on simple mass balance considerations. This approach does not take into account the effect of the solute dispersion. The present study presents a simple, inverse finite difference model (IFDM) for estimating temporal and spatial variations in nitrification or denitrification rates from measured nitrogen concentration profiles at steady water flow. The model takes into account both convective and dispersive transport of nitrogen by using a Crank-Nicolson finite-difference discretization of a paraphrased convection-dispersion equation for solute transport including a reaction term. A test procedure is suggested and used to find criteria for avoiding numerical errors in the calculations. Significant numerical errors are inherent in the IFDM but can be avoided if the depth increment and the ratio of time increment to depth increment are chosen carefully. The possible applications of the simple IFDM are illustrated by using the model on soil column data for rapid, continuous leaching of nitrate through water-saturated porous media at steady water flow. The IFDM-based analysis showed a large influence of soil composition and temperature on the temporal and spatial variations in nitrate removal rates. The IFDM assumes constant pore water velocity and solute dispersion coefficient. In order to use the IFDM not only for qualitative but also for quantitative predictions, these two parameters need to be carefully measured.

Estimation of the soil-water sorptivity from infiltration in vertical soil columns

This paper presents a simple method for estimating the soil-water sorptivity (S) from vertical infiltration experiments at ponding conditions. The proposed estimation method is based on the Haverkamp et al. (R. Haverkamp, J.-Y. Parlange, J. L. Starr, G. Schmitz, and C. Fuentes. 1990. Infiltration under ponded conditions. 3. A predictive equation based on physical parameters. Soil Sci. 149:292–300.) predictive infiltration equation. This equation is simplified, so that a simple analytical solution is obtained for the sorptivity. The new S estimation method requires measurement of cumulative infiltration versus time and independent measurement of the saturated hydraulic conductivity on the same soil cores as used for the infiltration experiment. Criteria for avoiding numerical instability when using the new estimation method are given. Use of the proposed method is illustrated on infiltration data measured on undisturbed sand cores. The method gave good and robust estimates of the soil-water sorptivity (coefficient of variation > 25% for all S estimates). Horizontal and vertical infiltration tests for homogeneous packed soil columns showed good agreement between the new S estimation method and the classical Philip method for horizontal infiltration at nonponding conditions.

The development of a heat income function for regional solid waste management

A heat income function is developed for the calculation of heat income obtained when heat generated in a solid waste incinerator is sold in the form of hot water to a district heating facility. The heat income function takes into account the variation in the demand for heat over the course of the year and the seasonal variation in the generation of solid waste. The use of the heat income function as a tool in regional solid waste planning is illustrated with an application to the island of Funen in Denmark. A mixed integer programming model is used to find the optimal location of incinerators based on the district heating facility demands at the potential locations.