A. W. Warrick

Numerical approximations of darcian flow through unsaturated soil

The choice of appropriate intrablock approximations for the unsaturated hydraulic conductivity is a critical step for the numerical solution of Richards equation. Forms commonly used include arithmetic, geometric and harmonic averaging. The choice is most often based on convenience and custom. This study presents a framework for comparing any scheme for weighting the unsaturated hydraulic conductivity against the correct Darcian flow across a block for steady conditions. Such comparisons are carried out for a number of commonly used averages, and errors are shown to be alarmingly high in some cases. A scheme for approximately matching the correct Darcian flux is illustrated by two infiltration examples. In the first, the arithmetic and weighted schemes are superior to the geometric average when compared to a quasi-analytical solution. For the second example, the weighted solution is slightly better than either the arithmetic or geometric averaging schemes. These examples also illustrate that an acceptable mass balance should not be used as a sole criterion for success as it can probably be attained with most schemes. The conclusions are believed relevant for algorithms which are pressure head based, water content based or other choices and for multidimensional as well as one-dimensional flow problems.

An analytical solution to Richards' equation for time-varying infiltration

An analytical solution is developed for Richards equation for time-varying infiltration. The infiltration rates must be specified as piecewise constants with time. The answer is expressed as a sum of two terms, the first is a function of the instantaneous infiltration and the second an integral which accounts for the water distribution within the profile for the previous infiltration history. The solutions assume specific forms for the soil water diffusivity and hydraulic functions used earlier by Broadbridge and White (1988).

Finite element methods for modeling water flow in variably saturated porous media: Numerical oscillation and mass-distributed schemes

The finite element method is an attractive numerical method for modeling water flow in variably saturated porous media due to its flexibility in dealing with complicated geometries. It is well known that the conventional mass-distributed finite element method suffers from numerical oscillations at the wetting front, especially for very dry initial conditions. Routinely, mass-lumped procedures are used to eliminate them. This paper proposes a physical interpretation of the finite element method applied to the water flow problem. With the finite element method, mass conservation is applied at the element level. The water storage and the flux within each element are split into several components in the function space, each of which corresponds to one component of the boundary flux of the element. However, even though physical laws are correctly applied at the element level, it is shown that the traditional mass-distributed scheme can still generate an incorrect neighboring node response due to the highly nonlinear properties of water flow in unsaturated soil and cause numerical oscillation. We propose two new mass-distributed schemes which are free of the numerical oscillation, and reduce the smearing near the wetting front, at slightly increased CPU time.

Alternative analyses of hydraulic data from disc tension infiltrometers

Hydraulic conductivity values were compared based on alternative analyses of data from disc tension infiltrometers. The first method was based on a single disc and tension and depends on the estimate of sorptivity and steady state flow. A second method used steady state flow measurements for two different disc radii, 52 and 118 mm. A third method used a single disc with multiple tensions from which steady state flow was obtained at three or more tensions which could be at the same location. A fourth method used a single disc with two tensions from which steady state flow was obtained at two tensions. Finally, values based on soil cores were compared. The results show reasonable agreement between methods for the hydraulic conductivity with the largest differences for data collected for zero tension. For the most part, there were no significant differences in hydraulic conductivity due to the disc infiltrometer radius. A single-disc method with multiple tensions (more than 3 points) and large disc radius gave results which were the most stable, accurate, and repeatable.

Improvement of the prediction of soil particle size fractions using spectral properties

Abstract Estimation of soil particle size fractions (percentage of sand, clay and silt) with sparse data can be improved by taking into account the spatial correlation with other variables. Reflectance of the near infrared band (0.76–0.90 μm) is used as an auxiliary variable in the prediction of soil particle size fractions on a 100 m × 100 m grid in a 2200 m × 1300 m field. The results of kriging using the textural information alone are compared with those of cokriging with the auxiliary variable. Cokriging gives a higher correlation coefficient and a lower mean squared error between estimates and measurements than kriging. The most significant improvement of the estimation is in areas of the field with sparse data for the estimated variables. The relative improvement is up to 90% in terms of reduced kriging variance and up to 33% for the mean squared error between actual measurements and estimated values. Only 17–25% of the original observations of texture are needed to obtain relatively accurate estimates when cokriging with reflectance data.

Downward water flow through sloping layers in the vadose zone: analytical solutions for diversions

Abstract Steady downward water flow through a vadose zone of sloping strata of different materials is studied. The resulting streamlines are mostly vertical but are deflected at the strata interfaces depending on the related unsaturated hydraulic conductivity functions and the flow rate. Updip or downdip deflection can occur for either fine over coarse or coarse over fine material. This phenomenon is of particular interest as related to flow regimes and the diversion capacity around buried wastes, and in the engineering design of capillary barriers. Assuming that the pressure head distribution is only a function of the normal distance from the sloping interface, which allows a one-dimensional analysis, we calculated pressure profiles, streamlines and deflection distances. The calculations confirm that a considerable downdip deflection can occur for low flow rates and a finer-textured soil overlying a coarser material. Updip deflections are most likely to be significant for high flow velocities, which occur for artificial recharge or humid climates.

Heterogeneity, plot shape effect and optimum plot size

By introducing heterogeneity indices, an empirical equation is proposed for characterizing the heterogeneity of non-isotropic fields. The formula is an extension of Fairfield Smiths (1938) empirical law describing heterogeneity in isotropic fields. Based on these indices, criteria are provided for choosing optimum plot shapes in terms of minimizing the sample variance and cost. Sample plots having their largest dimension in the direction with the largest index will give more accurate results (less variable) than plots with other shapes. Relations between the optimum plot size and relative cost versus variogram parameters are given for several variogram models. These relations indicate that variograms with small effective ranges have a very profound effect on the optimum plot sizes.

Solubility effects on surfactant-induced unsaturated flow through porous media

Abstract It has been demonstrated that local surface tension depression due to the presence of an insoluble surfactant, myristyl alcohol, can result in capillary pressure gradients that cause significant flow in unsaturated porous media. We hypothesized that the effects of a soluble surfactant, butanol, would differ from those of myristyl alcohol despite the fact that both give a similar reduction in the surface tension of water. We investigated this hypothesis through a series of horizontal column experiments and unsaturated flow modeling. We found that both surfactants induced water flow from regions of high surfactant concentration to regions of low concentration but that the shapes of the resultant moisture content profiles were fundamentally different. Because surface tension depression from myristyl alcohol was confined to the original source zone, we were able to simulate the system using a standard unsaturated flow model by assigning separate sets of hydraulic functions to the initially clean and source zones. Modeling showed that the redistribution of water due to the initial surfactant-induced capillary pressure gradient created moisture content induced capillary pressure gradients which acted to balance the system over time. The moisture content profile within the butanol system indicated a propagation of the zone of reduced surface tension. Because the surface tension of water is a function of solute concentration, surfactant-induced capillary pressure gradients are expected to remain and influence the flow system as long as concentration gradients exist.

Surfactant effects on unsaturated flow in porous media with hysteresis: horizontal column experiments and numerical modeling

Many organic compounds depress the surface tension of water relative to their aqueous concentration. These surface-active organic solutes have been shown to cause water flow in unsaturated porous media. Flow in these systems typically occurs from contaminated (high concentration) regions toward cleaner (lower concentration) regions. That flow is characterized by significant drainage and rewetting associated with the advance of the solute front. In the literature, modeling of unsaturated flow and transport in systems with solute concentration-dependent surface tensions has been limited to simulations without hysteresis in the hydraulic functions. However, experimental evidence is also presented which shows that hysteresis is a factor. We modified the hysteretic unsaturated flow and transport numerical model HYDRUS 5.0 to include concentration-dependent effects of a mobile organic solute. The moisture content-pressure head and unsaturated hydraulic conductivity functions were scaled for concentration-dependent surface tension and viscosity, respectively. The modified model successfully simulated data from surfactant-induced flow in 1-D horizontal column experiments. The effects of hysteresis were shown to be important to accurately simulate flow in these systems. For example, at final steady state, hysteretic simulations predicted uniform concentration and pressure profiles, but moisture contents that varied with distance. The final simulated moisture contents ranged from 0.12 to 0.25 along the column. In contrast, a non-hysteretic simulation would predict uniform distributions of concentration, pressure and moisture content. Dispersivity also had an effect on flow simulations. Lower dispersivity values caused sharper surfactant concentration gradients, which resulted in larger capillary pressure gradients and higher fluxes near the solute front. The modeling approach used here is expected to be applicable to many organic compounds of environmental interest that depress surface tension and thereby are capable of inducing unsaturated flow in porous media.

Algebraic models for disc tension permeameters

Four infiltration models for disc tension permeameters were compared to field and simulated data. Comparisons were made in order to assess the alternative values of sorptivity and steady state flow rates. Model 1 was the conventional analysis with assumptions of one-dimensional flow at small times and Woodings steady state flow rate at large times. Model 2 was a modified Horton equation defined to fit one-dimensional infiltration at small times with an exponential decay to a steady state value for large times. Model 3 assumed linear diffusion for a soil with constant diffusivity ignoring gravity. Model 4 was based on “linearized” flow with gravity. The number of unknown parameters to be estimated ranged from one to four. Sorptivity data using models 2 and 3 generally agreed with the conventional method (model 1) both for the field and simulated data. For the steady state infiltration rate, all of the models gave generally consistent results. Model 2 with four parameters was the best fitting model to field and simulated data. The simulated data allowed comparisons of results based on the total time over which the parameters were evaluated. Results were sensitive to the time of evaluation up to 0.3–1.3 hours for the sandy loam soil.