Transient Unsaturated Flow Research Articles

STOCHASTIC FINITE ELEMENT ANALYSIS OF TRANSIENT UNSATURATED FLOW IN POROUS MEDIA

A stochastic perturbation-based finite element formulation for prediction of transient unsaturated flow in porous media was developed and implemented. The stochastic differential equation describing the large-scale transient flow model was implemented using a finite element approach. The system of finite element equations was obtained using Galerkin's method. Six-noded triangular elements were used in the mesh discretization, and Gauss-Legendre quadrature was employed to perform the numerical integrations. The global system of differential equations was evaluated using a finite difference approximation in the time domain. A two-dimensional transient unsaturated flow was simulated using both a stochastic and a deterministic approach. The mean moisture content and capillary pressure head distributions were evaluated as a function of time and depth. Results showed a significant difference between the two approaches. A Fortran 90 computer code was also developed to obtain the stochastic and deterministic finite element solutions. The stochastic perturbation-based finite element formulation is a very attractive alternative to deterministic approaches in terms of cost, efficiency, and accuracy of the results.

Uncertainty analysis of transient unsaturated flow in bounded domain

This paper analyzes the head variance during transient flows in unsaturated porous media. The analysis is based on a first‐order Taylor series expansion of the discretized Richards' equation and stochastic conceptualization of hydrologic properties in space. Effects of boundary conditions and flow conditions (drying or wetting) were examined for different soil models. Results of the study indicate that a constant flux boundary condition affected the pressure head variance distribution close to the surface, originating a decrease of the values at early times during wetting scenarios. On the other hand, a constant head boundary condition strongly affected the variance profiles during both transient and steady flows, also far from the boundary.

An Unsaturated Transient Flow Method for Determining Solute Adsorption by Variable‐Charge Soils

The amount of reactive ions adsorbed during transport in variable‐charge soils is often smaller than those predicted from batch adsorption experiments. This overestimation of adsorption for invading ions by the batch method is likely to be caused by excessive desorption of indigenous, strongly adsorbed ions. We propose an unsaturated transient flow method by which equilibrium adsorption of weakly reactive ions can be determined without causing appreciable desorption of indigenous ions. In our method, water is imbibed into horizontal columns packed with soil that has been mixed with a salt solution at different concentrations to attain adsorption equilibrium. We make use of the observation that during imbibition of water, the antecedent solution is pushed ahead by the invading solution and accumulates in the region beyond the plane of separation, with the solution concentration unchanged. From linear plots of the solute content vs. soil water content in this region, the amount of solutes adsorbed by soil and the equilibrium solution concentration prior to the water imbibition are obtained. The advantages of the method are that adsorption isotherms can be determined under conditions similar to those expected during transport processes in soil, and that it is exempt from uncertainty about attainment of adsorption equilibrium as in the miscible displacement methods. The proposed method is best suited for variable‐charge soils where adsorption of weakly reactive ions is largely due to an increase in the exchange capacity.

Two-dimensional simulation of bromide transport in a heterogeneous field soil with transient unsaturated flow

Summary As a consequence of spatial heterogeneity, solute transport and associated phenomena of soil in the field are typically complex and often poorly described by current one-dimensional models. Therefore, better descriptions should be obtained if the specific variation of the soil is explicitly accounted for. To test this hypothesis, we simulated the two-dimensional transport of a bromide trace in pore-water velocity fields which varied in both space and time. We used data obtained in a field experiment in which we studied bromide transport as input parameters and boundary conditions for the simulation. We conceptualized heterogeneity at two spatial scales. Large regions with constant spatial properties were defined by the measured geometry of the soil’s horizons. On these we superimposed small structures derived from a correlated random field of scaling factors and conditioned on measured hydraulic properties. Simulations with homogeneous and heterogeneous horizons show that both coarse and fine hydraulic variation contribute to characteristic experimental phenomena such as concentration ‘hot spots’, especially near irregularities of horizon boundaries, and early breakthrough of solute leading to large dispersion at the field scale. The comparison between simulation and experimental results demonstrates that these features are reproduced if local hydraulic variation is included in the description of the transport. As expected, the quantitative agreement is limited, since the concentration distribution is strongly sensitive to local hydraulic conditions which were only partly accounted for by the measurements.

Nonstationary stochastic analysis of transient unsaturated flow in randomly heterogeneous media

In this study, a first‐order, nonstationary stochastic model of transient flow is developed which is applicable to the entire domain of a bounded vadose zone in the presence of sink/source. We derive general equations governing the statistical moments of the flow quantities by perturbation expansions. Owing to the mathematical complexity of the equations, in general we need to solve them numerically. The numerical moment equation approach, however, has the flexibility in handling different boundary conditions, flow configurations, input covariance structures, and soil constitutive relationships. The moment equations presented in this study are simpler and easier to solve than those in the literature for transient unsaturated flow. We solve these moment equations by the method of finite differences and demonstrate the developed model through some one‐ and two‐ dimensional examples under various transient conditions.

Boundary Integral Procedures for Unsaturated Flow Problems

Abstract. A novel numerical scheme based on the singular integral theory of the boundary element method. (BEM) is presented for the solution of transient unsaturated flow in porous media. The effort in the present paper is directed in facilitating the application of the boundary integral theory to the solution of the highly non-linear equations that govern unsaturated flow. The resulting algorithm known as the Green element method (GEM) presents a robust attractive method in the state-of -the-art application of the boundary element methodology. Three GEM models based on their different methods of handling the non-linear diffusivity, illustrate the suitability and robustness of this approach for solving highly non-linear 1-D and 2-D flows which would have proved cumbersome or too difficult to implement with the classical BEM approach.

A Schwarz domain decomposition method for solution of transient unsaturated water flow on parallel computers

A parallel Schwarz domain decomposition method for the numerical solution of water flow in soils and aquifers is presented. This method offers the possibility to solve large numerical problems occurring by calculations of water flow in heterogeneous porous media on parallel computer systems. Strongly nonlinear problems requiring fine discretizations in time and space can be handled. The method may easily be incorporated in existing models with only slight changes in the computer code. Communication routines are implemented in such a way that they can be portated on various parallel computer systems, ranging from workstation clusters to massively parallel systems with distributed memory or vector computers with several processors. The method was tested using 3D homogeneous and heterogeneous flow domains with 9216 nodal points. A 3D finite element method was used to solve the Richards equation in an unsaturated flow domain. For runs on an Intel iPSC/860 and Paragon XP/S10 high speedups and considerably reduced execution times compared to the sequential version were achieved.

Hydrodynamic modeling of leachate recirculating landfills

Leachate recirculation is an emerging technology for the enhanced stabilization of active landfills and in-situ treatment of problematic landfills. A variety of researcher have documented the benefits of increasing landfill moisture content and liquid movement through the fill. However, little work has addressed the hydrodynamic characteristics of the recirculating landfill, specifically the effect of leachate flow rate and waste characteristics upon leachate routing. This paper presents the results of mathematical modeling of the horizontal infiltration trench and vertical infiltration well, two commonly employed leachate application methods. The results of transient unsaturated flow modeling are detailed.

Hydrodynamic modeling of leachate recirculating landfills

Leachate recirculation is an emerging technology for the enhanced stabilization of active landfills and in-situ treatment of problematic landfills. A variety of researcher have documented the benefits of increasing landfIll moisture content and liquid movement through the fill. However, little work has addressed the hydrodynamic characteristics of the recirculating landfill, specifically the effect of leachate flow rate and waste characteristics upon leachate routing. This paper presents the results of mathematical modeling of the horizontal infiltration trench and vertical infiltration well, two commonly employed leachate application methods. The results of transient unsaturated flow modeling are detailed.

AN ANALYTICAL SOLUTION FOR PREDICTING SOLUTE TRANSPORT DURING PONDED INFILTRATION

An analytical solution is presented for one-dimensional solute transport in soil during ponded infiltration. The pore-water velocity υ = q/θ in the convection-dispersion transport equation was obtained by assuming that the water flux density q in the soil can be calculated with the Green-Ampt infiltration model, q = a + b/I, in which a and b are constants and I is the cumulative infiltration rate evaluated using Philip's twoterm infiltration equation. The water content θ was approximated by the saturated water content, θ s . Through an appropriate transformation, the solute transport equation for transient unsaturated flow conditions was linearized and solved analytically for a general initial concentration profile given by an arbitrary number of straight line segments. The solution compared well with more complete numerical solutions of the transport problem, as well as with several experimental data sets

Modeling Heavy Metal (Cadmium) Uptake by Soil‐Plant Root System

Among various toxic substances that contaminate plants and enter the food chain, thereby posing health problems to human beings, heavy metals are particularly severe in their action. The behavior of heavy metals present in sewage sludge and industrial wastes that are used in agricultural practices has been the center of much attention during the last decade. Among these heavy metals, cadmium has been identified as the most critical metal due to its human toxicity. The presence of cadmium and its accumulation in the plant root zone also results in decreased crop yield. This study develops a methodology to simulate the transport of cadmium in soil water and its uptake by plant roots. The transient unsaturated water flow equation with a root extraction term is coupled with mass transport equation and mass balance of solute concentration. The model takes into account the root growth with time. The macroscopic root‐soil water flow and solute (cadmium) movement model yields a set of partial differential equatio...

A simplified Newton Iteration Method with linear finite elements for transient unsaturated flow

A simplified Newton iteration method with Galerkin linear finite elements is developed for the solution of the nonlinear transient unsaturated flow equation in “mixed” form. The use of the Galerkin linear finite element method in the spatial discretization produces a recurrence equation in which the interaode hydraulic conductivity is represented by an integrated average expression which generates the exact Darcy flow flux through each element under the steady horizontal flow condition. The numerical computation of the integrated average hydraulic conductivity using Simpson's 1/3 rule leads to an accurate solution even in the presence of a sharp front. In the time discretization using a backward Euler scheme, the standard Newton iteration method is simplified by taking account of the integrated average expression of the hydraulic conductivity and the diffusion‐dominated nature of the flow process. The resulting two‐level scheme is of the same efficiency on a per‐iteration basis as the modified Picard scheme but converges at a faster rate.

Experimental testing of transient unsaturated flow theory at low water content in a centrifugal field : Nimmo, JR Water Resour ResV26, N9, Sept 1990, P1951–1960

Experimental testing of transient unsaturated flow theory at low water content in a centrifugal field : Nimmo, JR Water Resour ResV26, N9, Sept 1990, P1951–1960

On solute transport in a heterogeneous porous formation under saturated and unsaturated water flows

The results of a recent study of solute transport in a hypothetical scale‐heterogeneous soil under transient unsaturated flow conditions (Russo, this issue) suggested that the time‐dependent components of the spatial covariance tensor, based on a single realization of the solute concentration in unsaturated flow, are in relatively good agreement with the components of the spatial covariance tensor, based on the ensemble‐average solute concentration derived by Dagan (1984), for solute transport under steady saturated flow. Furthermore, the asymptotic value of the longitudinal component of the dispersivity tensor λzz, estimated from the horizontally averaged solute concentration in the unsaturated flow domain, was found to be in good agreement with the asymptotic λzz derived by Dagan (1982, 1984) and Gelhar and Axness (1983) for saturated flow. These findings are explored in a general Lagrangian formulation, using small‐perturbation, first‐order approximations. The results of the analysis of the velocity field in the unsaturated flow domain of the hypothetical scale‐heterogeneous soil of Russo (this issue) suggest that there was good agreement between the result of the Lagrangian kinematic analysis and the result of the analysis of the spatial moments of the concentration distribution, in agreement with the stochastic theory of transport in saturated flow. This is explained by the finding that under unsaturated flow conditions there is an increase in the variability of the hydraulic conductivity K (and concurrently an increase in the variability of the velocity Vz), accompanied by a decrease in the correlation scale of K (and concurrently a decrease in the correlation scale of Vz) relative to the saturated conditions. The increase in the variability of the unsaturated conductivity stemmed from the variability in the water content θ and the strong nonlinear dependence of K on θ, while the accompanying decrease in its correlation scale stemmed from the negative correlation between F = log Ks and θ, which persisted for a relatively large separation distance in the longitudinal vertical direction. Consequently, the product between the conductivity variance and its correlation scale is of similar magnitude for both saturated and unsaturated flow regimes. The results suggest that in spite of the complexity of unsaturated flow, as compared with saturated flow, it might be amenable to a similar analysis.

Two-dimensional unsaturated flow in irregularly shaped regions using a finite volume method

To describe the two-dimensional flow of water in unsaturated soil, the governing equation is solved on a mesh constructed from small area elements. A transformation is introduced with which these possibly distorted rectangular elements of the physical plane are mapped into computational squares. Thus, irregularly shaped regions, which present difficulties when attempting to describe their geometry on an orthogonal computational mesh, can be more easily modelled. Using this methodology, here called the finite volume method, numerical results are obtained showing the ability of the method to describe transient unsaturated flow.

Statistical Parameters Characterizing the Spatial Variability of Selected Soil Hydraulic Properties

AbstractThe knowledge of the statistical parameters of the variance, σ2, and the correlation scale, λ, characterizing the spatial structures of the log of the saturated hydraulic conductivity, lnKs, pore size distribution parameter α, and the specific water capacity, C, is required in stochastic modeling in order to understand the overall response of large‐scale heterogeneous unsaturated flow systems. These parameters are estimated assuming second‐order stationarity and an exponential semivariogram model with nugget effect. Methods of ordinary least squares (OLS), maximum likelihood (ML), and restricted maximum likelihood (RML) are used for estimating σ2 and λ, while methods of cross‐validation (kriging) and uncorrelated residuals are used to validate the semivariogram model with estimated σ2 and λ. The objectives of this study were to evaluate the sensitivity of σ2 and λ to the estimation methods and to discuss the implications of the analysis in view of the stochastic modeling. The significance of the results of parameter estimation and model validation in relation to the stochastic modeling of large‐scale transient unsaturated flow is demonstrated with two examples involving the variance of soil‐water pressure head, σ2h, and the vertical component of effective hydraulic conductivity, K*11. Results show that the estimated values of σ2 and λ are highly dependent on the estimation method. Although the majority of the estimated parameters pass the validation tests, the RML estimates of σ2 and λ used in estimating σ2h and K*11 significantly reduce the prediction uncertainties.

Stochastic analysis of unsaturated flow: One‐dimensional Monte Carlo simulations and comparisons with spectral perturbation analysis and field observations

A numerical experiment was designed to study the stochastic behavior of one‐dimensional transient unsaturated flow in a Monte Carlo setting. Soil hydraulic properties, log‐saturated hydraulic conductivity ln Ks, pore size distribution parameter α, and the specific water capacity C are assumed to be statistically homogeneous random fields described by exponential correlation functions with identical correlation lengths. Fifty realizations of each soil hydraulic property, with statistical properties obtained from a field experiment, are generated by using a nearest‐neighbor model. Numerical solutions of the one‐dimensional flow equation are used repeatedly to obtain realizations of soil water pressure head ψ and flux q corresponding to the realizations of In Ks, α, and C. The dependence of ψ and q on the statistical properties of ln Ks, α, and C and the magnitude of the hydraulic head gradient G prevailing at the lower boundary are investigated. In addition, results of Monte Carlo analysis and spectral perturbation analyses are compared with field observations. The greatest variability of ψ and q are observed when soil hydraulic properties are uncorrelated and have large variances and integral scales and when G at the lower boundary is unity. The Monte Carlo and spectral perturbation analyses tend to agree reasonably well for the flow domains in which ergodicity of local soil hydraulic properties is assured. Results of the Monte Carlo and spectral perturbation analyses are also supported by field observations.

Experimental testing of transient unsaturated flow theory at low water content in a centrifugal field

Experimental measurements and theoretical predictions of transient moisture conditions have been compared for a sandy soil approaching hydrostatic equilibrium in a centrifugal field. Starting near saturation, samples were centrifuged at constant speed with a constant suction at the outflow boundary. Water flowed freely out of the sample through a porous plate. Step increases in centrifuge speed produced transient moisture conditions suitable for comparison between experiment and theory. Measurements of electrical conductivity by a direct contact four‐electrode technique indicated the water content according to a calibration based on known moisture conditions at various equilibrium states. A specially modified centrifuge permitted electrical measurements during centrifugation. For comparison, the transient water contents were computed by a finite‐difference solution of Richards' equation (modified by replacing gravitational with centrifugal potential), using soil characteristics measured previously by steady state techniques. The time dependence of water content changes, used as the basis for comparison between experiment and theory, shows agreement which is reasonable given the degree of uncertainty of the measurements. The experiment confirms, within a factor of 4, the validity of Richards' equation for moisture conditions as dry as 25% of saturation, over a hydraulic conductivity range of 5 × 10−11 to 1 × 10−8 m/s, and in a centrifugal field up to about 200 g.

Three-dimensional unsaturated flow in heterogeneous systems and implications on groundwater contamination: A stochastic approach

A stochastic approach for modeling transient unsaturated flow in large-scale spatially variable soils is developed in order to overcome the problem of limited information about the local details of spatial soil variability. It is assumed that local soil properties are realizations of three-dimensional stationary random fields, and a large-scale model representation is derived by averaging the local governing flow equation over the ensemble of realizations of the underlying soil property random fields. The three-dimensionality of the local flow equations and the nonlinear dependence of the local flow output on the local soil properties are considered. The resulting mean representation (structure) is in the form of a partial differential equation in which averaged or effective model parameters occur. These effective model parameters are evalutated using a quasi-linearized fluctuation equation and a spectral representation of stationary processes. The large-scale model structure considers the large-scale effects of soil variability and have relatively few parameters which should be identifiable from a realistic data set. The general stochastic theory is then applied to the case of flow in stratified soil formations, which is of practical importance in applications such as waste disposal control. An important finding of this study is that spatial variability of the hydraulic soil properties produces significant large-scale effects, such as large-scale hysteresis and anisotropy of the effective parameters. These large-scale effects should be considered in field applications such as for predicting the movement of liquid wastes in the unsaturated zone.

Parameter estimation for coupled unsaturated flow and transport

This paper deals with the estimation of soil hydraulic and transport parameters from transient unsaturated flow and tracer experiments using a combined simulations‐optimization approach. Hydraulic properties are defined by a modified form of van Genuchten's (1980) parametric model for two‐phase permeability‐saturation‐pressure relations, and transport properties are defined by an empirical parametric dispersion model. A nonlinear weighted least squares algorithm is used to estimate unknown model parameters by minimizing deviations between concentrations, water contents, and pressure heads obtained from hypothetical infiltration/redistribution/evaporation experiments, and those predicted by solving a numerical model of coupled unsaturated flow and transport. Simultaneous estimation of hydraulic and transport properties is found to yield smaller estimation errors for model parameters than sequential inversion of hydraulic properties from water content and pressure head data followed by inversion for transport properties from concentration data. Effects of random noise in data measurements, soil layering, and choice of improper parametric model on the parameter estimation process are discussed.