J. C. Parker

Parameter estimation for unsaturated flow and transport models. A review

Kool, J.B., Parker, J.C. and Van Genuchten, M.Th., 1987. Parameter estimation for unsaturated flow and transport models -- A review. J. Hydrol., 91: 255-293. This paper reviews the current status of parameter estimation techniques and their utility for determining key parameters affecting water flow and solute transport in the unsaturated (vadose) zone. Historically, hydraulic and transport properties of the unsaturated zone have been determined by imposing rather restrictive initial and boundary conditions so that the governing flow and transport equations can be inverted by analytical or semi-analytical methods. Contrary to these direct methods, parameter estimation techniques do not impose any constraints on the model, on the stipulation of initial and boundary conditions, on the constitutive relationships, or on the treatment of inhomogeneities via deterministic or stochastic representations. While parameter estimation analyses of subsurface saturated flow are increasingly common, their application to unsaturated flow and transport processes is a relatively new endeavor. Nevertheless, a number of laboratory and field applications currently exist that show the great potential of parameter estimation techniques for improved designs and analyses of vadose zone flow and transport experiments. Several practical examples for determining unsaturated soil hydraulic functions and various transport parameters are presented, and advantages and limitations of the estimation process are discussed. Specific research areas in need of future investigation are outlined.

Analyses of infiltration events in relation to determining soil hydraulic properties by inverse problem methodology

At the field scale, because of the relatively large number of observation points required to estimate the spatial distributions of hydraulic properties, application of inversion procedures must be based on a characteristic of the flow that for a given point in the field can be measured with relative ease, and can be used as input for the inversion procedure. Suggesting the use of infiltration data for this purpose, and recognizing the fact that a basic requirement for the design of a transient flow experiment is that the resulting inverse problem be sufficiently well posed to allow solution, this paper addresses the question under which circumstances the resulting inverse problem is well posed, utilizing the concepts of identifiability, uniqueness, and stability. For this purpose, three different models of the soil hydraulic properties were analyzed. The main conclusion of this study is that when infiltration data measured at the Darcy scale are used as input for the inversion procedure, the inclusion of prior information on a single measurable parameter, the saturated conductivity Ks, in the estimation criterion will enhance the likelihood of uniqueness and stability of the inverse solution provided that the structure of the hydraulic model is sufficiently simple. However, since for a particular situation, it is impossible to determine a priori whether the resultant inverse problem is well posed or not, this must be carried out only a posteriori by solving the problem several times with different initial parameter estimates, accompanied by an analysis of the associated estimation errors.

Estimation of soil hydraulic properties and their uncertainty from particle size distribution data

Abstract A unified approach to the estimation of soil hydraulic properties and their uncertainty from particle size distribution data is presented. Soil hydraulic properties are represented by the parametric models of Van Genuchten and/or Brooks and Corey. Particle size distribution data are used to generate theoretical soil-water retention data using a modified form of the model proposed by Arya and Paris, which was calibrated in this study using a data set of 250 soil samples. Parameters in the Van Genuchten model are fitted to the predicted water content — capillary pressure data by nonlinear regression methods and may be optionally converted to equivalent Brooks-Corey retention parameters using an empirical procedure. Saturated conductivity is estimated from particle size data using a modified Kozeny-Carman equation which was developed from the data set of 250 soil samples. Uncertainty in parameter estimates is evaluated using first-order error analysis methods. Application of the proposed methodology to three soils which were not in the calibration set indicated water content-capillary pressure relations can be predicted with reasonable accuracy and precision. Uncertainty in predicted saturated hydraulic conductivity will be rather large making direct measurement of this variable highly desirable.

Comparing Simulated and Experimental Hysteretic Two‐Phase Transient Fluid Flow Phenomena

A hysteretic model for two-phase permeability (k)-saturation (S)-pressure (P) relations is outlined that accounts for effects of nonwetting fluid entrapment. The model can be employed in unsaturated fluid flow computer codes to predict temporal and spatial fluid distributions. Consideration is given to hysteresis in S-P relations caused by contact angle, irregular pore geometry, and nonwetting fluid entrapment effects and to hysteresis in k-S relations caused by nonwetting fluid entrapment effects. An air-water flow experiment is conducted with a 72-cm vertical soil column where the water table is fluctuated to generate scanning S-P paths. Water contents are measured via a gamma radiation system, and water pressures are measured via pressure transducers connected to ceramic tensiometers inserted in the soil column. Computer simulations of the experiment employing the hysteretic k-S-P model and a nonhysteretic k-S-P are compared with measured water contents and pressures. Close agreement is found between experimental water contents and those predicted by a numerical code employing the hysteretic k-S-P relations. When nonhysteretic k-S-P constitutive relations are utilized, there is poor agreement between measured and predicted water saturations of the scanning paths. Only one more parameter is needed to model two-phase hysteretic fluid behavior than to model nonhysteretic behavior. Results of this study suggest that consideration should be given to effects of hysteresis in k-S-P relations to accurately predict fluid distributions.

A numerical model for areal migration of water and light hydrocarbon in unconfined aquifers

Abstract A finite element model has been developed to simulate simultaneous flow of water and light hydrocarbon in an areal flow region of an unconfined aquifer for analyses of hydrocarbon spreading from subsurface leaks or spills and for use in design of free product recovery systems. Vertically integrated governing equations for water and oil flow are employed which assume local vertical equilibrium and negligible gas pressure gradients. Multiple water and free product recovery wells are handled as internal type-I boundary conditions by stipulating air-oil table elevation and free product height with corrections to convert grid averaged nodal heads to actual well bore fluid levels. An automatic updating scheme for well bore correction factors is introduced which ensures consistency of well flux calculations with the global mass balance. Areal model predictions are compared with two dimensional vertical cartesian and radial simulations with multiphase seepage faces for hypothetical trench and well free product recovery systems, respectively. The results indicate that the assumption of vertical equilibrium and lack of explicit treatment of seepage faces in the areal model produce minor loss in accuracy while conferring major reductions in computational effort. Simulations of various spill spreading and free product recovery scenarios with multiple pumping wells are investigated to demonstrate the model capabilities.

Exit condition for miscible displacement experiments

The one-dimensional solute transport is analyzed with the convection-dispersion model, including first and zeroth order irreversible reaction. A simple analytical expression is derived for the residence concentration which depends explicitly on the exit conditions at the end of the soil column or layer. The validity of the flux concentration solution ignoring the finite length of the column is also discussed by relating the exit conditions to the Péclet number.

Field-scale water and solute flux in soils

1 Think Tank Reports.- Flow and Transport Modeling Approaches: Philosophy, Complexity and Relationship to Measurements.- Effective Large Scale Unsaturated Flow and Transport Properties.- Evaluation of Field Properties from Point Measurements.- Evaluating the Role of Preferential Flow on Solute Transport through Unsaturated Field Soils.- 2 Papers.- Spatial Variability of Water and Solute Flux in a Layered Soil.- One and three Dimensional Evaluation of Solute Macrodispersion in an Unsaturated Sandy Soil.- Assessment of Field-Scale Leaching Patterns for Management of Nitrogen Fertilizer Application.- The Effect of Field Soil Variability in Water Flow and Indigenous Solute Concentrations on Transfer Function Modelling of Solute Leaching.- Analysis of Caisson Transport Experiment by Travel Time Approach.- Field Estimates of Hydraulic Conductivity from Unconfined Infiltration Measurements.- Sprinkler Irrigation, Roots and the Uptake of Water.- The Infiltration-Outflow Experiment Used to Detect Flow Deviations.- Spatial Variability of Unsaturated Flow Parameters in Fluvial Gravel Deposits.- Quantification of Deterministic and Stochastic Variability Components of Solute Concentrations at the Groundwater Table in Sandy Soils.- Use of Scaling Techniques to Quantify Variability in Hydraulic Functions of Soils in the Netherlands.- Kriging Versus Alternative Interpolators: Errors and Sensitivity to Model Inputs.- Spatial Averaging of Solute and Water Flows in Soil.- Criteria for Evaluating Pesticide Leaching Models.- Relating the Parameters of a Leaching Model to the Percentages of Clay and other Soil Components.- Prediction of Cation Transport in Soils Using Cation Exchange Reactions.- Transport of a Conservative Tracer under Field Conditions: Qualitative Modelling with Random Walk in a Double Porous Medium.- Mass Flux of Sorptive Solute in Heterogeneous Soils.- Effective Properties for Modeling Unsaturated Flow in Large-Scale Heterogeneous Porous Media.- Transport of Reactive Solutes in Spatially Variable Unsaturated Soils.- A Perturbation Solution for Transport and Diffusion of a Single Reactive Chemical with Nonlinear Rate Loss.- Areal Solute Flux Estimation: Legal Aspects.

Finite element model of nitrogen species transformation and transport in the unsaturated zone

A two-dimensional finite element model based on Galerkins weighted residual approach and incorporating an upstream weighting technique was developed to predict the simultaneous transformation and transport of nitrogen species. The nitrogen cycle used in the analyses assumes first-order rate coefficients for nitrification, denitrification, immobilization and mineralization between the nitrogen species NH+4, NO−3, organic-N and denitrified fractions. The NO−3-N component in the soil solution was assumed to have negligible adsorption on exchange sites while adsorption of NH+4 was assumed to be represented by a Freundlich type nonlinear isotherm (S = kd Cp). The accuracy and validity of the proposed model was examined by comparison with analytical model results and available field data. The results showed improved accuracy and stability with the upstream weighting approach in comparison to the standard weighted residual method. A sensitivity study on the kinetics of the nitrogen cycle showed that both concentration and cumulative mass distribution of different nitrogen species are substantially affected by the nitrification rate but to a lesser extent by the distribution coefficient kd for NH+4 adsorption and less so by nonlinear coefficient p. To evaluate the applicability of the model in actual field situations, data on groundwater nitrogen concentration beneath a functioning drainfield site were compared with the results of numerical simulations. Predicted results agreed with the observed data within the bounds of experimental errors.

Finite element analysis of water flow in variably saturated soil

Abstract A two-dimensional Galerkin finite element model for water flow in variably saturated soil is presented. A fourth-order Runge-Kutta time integration method is employed which allows use of time steps at least 2 times greater than for a traditional finite difference approximation of time derivatives. For short total simulation times computer execution costs for the Runge-Kutta method are greater than for the finite difference approximation due to the start up cost of the Runge-Kutta method, but for longer simulation times the Runge-Kutta method requires considerably less computational effort even when automatic time-step adjustment is used with the finite difference procedure. A comparison of the method of influence coefficients and 2 × 2 Gaussian integration to compute element matrices indicates that the influence coefficient method reduces total execution time to 60% of that required for numerical quadrature. Computed pressure heads using the influence coefficient method and numerical integration are found to be in close agreement with each other even under conditions of highly non-linear soil properties in a heterogeneous domain. Fluxes computed by the two methods are also generally in close agreement except under extremely non-linear conditions when some deviations were observed at short simulation times.

Evaluating the Role of Preferential Flow on Solute Transport Through Unsaturated Field Soils

Our group discussions on “preferential flow in unsaturated soil” dealt with a number of issues believed to be important in the scientific quest for garnering understanding of this ubiquitous phenomenon. The general areas of discussion were flow mechanisms and classification, modes of appearance, methods of observation, understanding of causal mechanisms, experimental evaluation, and the status of modeling efforts.