T.-C. Jim Yeh

Analytical solutions for one‐dimensional, transient infiltration toward the water table in homogeneous and layered soils

Analytical solutions describing the transient soil water pressure distributions during one-dimensional, vertical infiltration toward the water table through homogeneous and two-layer soils are derived. Exponential functional forms K = Kseαψ and θ = θr + (θs − θr)eαψ are used to represent the hydraulic conductivity and pressure relation and the soil water release curve. Steady state profiles are used as initial conditions. Hydraulic behavior of the soils during wetting and drainage processes is discussed in terms of the pressure head and moisture content profiles and temporal variation of the specific discharge. The solutions provide a reliable means of comparing the accuracy of various numerical methods, especially in very dry layered soils.

An Iterative Stochastic Inverse Method: Conditional Effective Transmissivity and Hydraulic Head Fields

An iterative stochastic approach is developed to estimate transmissivity and head distributions in heterogeneous aquifers. This approach is similar to the classical cokriging technique; it uses a linear estimator that depends on the covariances of transmissivity and hydraulic head and their cross covariance. The linear estimator is, however, improved successively by solving the governing flow equation and by updating the covariances and cross-covariance function of transmissivity and hydraulic head fields in an iterative manner. As a result the nonlinear relationship between transmissivity and head is incorporated in the estimation, and the estimated fields are approximate conditional means. The ability of the iterative approach is tested with some deterministic and stochastic inverse problems. The results show that the estimated transmissivity and hydraulic head fields have smaller mean square errors than those obtained by classical cokriging even in the aquifer with variance of transmissivity up to 3.

The effect of water content on solute transport in unsaturated porous media

The effect of water content on NaCl transport in unsaturated porous media was investigated under steady state flow conditions for water contents ranging between full saturation and 15% by volume. The experiments were conducted in a 25 cm column packed with homogeneous sand. Results of the experiments indicate that solute transport in unsaturated porous media is subject to greater velocity variations and slower solute mixing than one in saturated media. As a result, NaCl breakthrough curves (BTCs) show earlier initial arrival and greater tailing and variance as the average water content decreases. These results suggest that transport processes in our experiments have not fully developed to the Fickian regime at lower water contents. Because the classical convection- dispersion equation does not adequately describe the movement of solutes under the pre- Fickian regime, a mobile-immobile model was employed to reproduce the BTCs obtained under unsaturated conditions. In general, the results indicate that at lower water contents the medium has a greater fraction of immobile water, higher dispersion, and slower mass transfer between the mobile and immobile regions. A power law relationship between dispersion and water content-normalized velocities was found to exist for our experiments and other experiments reported in the literature using different porous media. Thus we suggest dispersivity is not only a function of properties of the media but also of water content.

A geostatistically based inverse model for electrical resistivity surveys and its applications to vadose zone hydrology

[1] A sequential, geostatistical inverse approach was developed for electrical resistivity tomography (ERT). Unlike most ERT inverse approaches, this new approach allows inclusion of our prior knowledge of general geological structures of an area and point electrical resistivity measurements to constrain the estimate of the electrical resistivity field. This approach also permits sequential inclusion of different data sets, mimicking the ERT data collection scheme commonly employed in the field survey. Furthermore, using the conditional variance concept, the inverse model quantifies uncertainty of the estimate caused by spatial variability and measurement errors. Using this approach, numerical experiments were conducted to demonstrate the effects of bedding orientation on ERT surveys and to show both the usefulness and uncertainty associated with the inverse approach for delineating the electrical resistivity distribution using down-hole ERT arrays. A statistical analysis was subsequently undertaken to explore the effects of spatial variability of the electrical resistivity-moisture relation on the interpretation of the change in water content in the vadose zone, using the change in electrical resistivity. Core samples were collected from a field site to investigate the spatial variability of the electrical resistivity-moisture relation. Numerical experiments were subsequently conducted to illustrate how the spatially varying relations affect the level of uncertainty in the interpretation of change of moisture content based on the estimated change in electrical resistivity. Other possible complications are also discussed. INDEX TERMS: 0903 Exploration Geophysics: Computational methods, potential fields; 1869 Hydrology: Stochastic processes; 1875 Hydrology: Unsaturated zone; 1866 Hydrology: Soil moisture; 3260 Mathematical Geophysics: Inverse theory; KEYWORDS: geostatistical inverse model, electrical resistivity tomography, vadose zone, resistivitymoisture relation, spatial variability, sequential/successive linear estimator

Bacteriophage and microsphere transport in saturated porous media: Forced-gradient experiment at Borden, Ontario

A two-well forced-gradient experiment involving virus and microsphere transport was carried out in a sandy aquifer in Borden, Ontario, Canada. Virus traveled at least a few meters in the experiment, but virus concentrations at observation points 1 and 2.54 m away from the injection well were a small fraction of those injected. A simplified planar radial advection-dispersion equation with constant dispersivity, coupled with equilibrium and reversiblefirst-order mass transfer, was found to be adequate to simulate the attachment and transport process. During the experiment a short-duration injection of high-pH water was also made, which caused detachment of previously attached viruses. For simulating this detachment and associated transport, the same transport and mass- transfer equations were used; but all rate parameters were varied as groundwaterpH changed from 7.4 to 8.4 and then back to 7.4. The physicochemical parameters obtained fromfitting breakthrough curves at one sampling well were used to predict those at another well downstream. However, laboratory-determined parameters overpredicted colloid removal. The predicted pattern and timing of biocolloid breakthrough was in agreement with observations, though the data showed a more-disperse breakthrough than expected from modeling. Though clearly not an equilibrium process, retardation involving a dynamic steady state between attachment and detachment was nevertheless a major determinant of transport versus retention of virus in thisfield experiment.

An Iterative Geostatistical Inverse Method For Steady-Flow In The Vadose Zone

An iterative geostatistical inverse approach is developed to estimate conditional effective unsaturated hydraulic conductivity parameters, soil-water pressure head, and degree of saturation in heterogeneous vadose zones. This approach is similar to the classical cokriging technique, and it uses a linear estimator that depends on covariances and cross covariances of unsaturated hydraulic parameters, soil-water pressure head, and degree of saturation. The linear estimator is, however, improved successively by solving the governing flow equation and by updating the residual covariance and cross-covariance functions in an iterative manner. As a result, the nonlinear relationship between unsaturated hydraulic conductivity parameters and head is incorporated in the estimation and the estimated fields are approximate conditional means. The ability of the iterative approach is demonstrated through some numerical examples.

A Geostatistical Inverse Method for Variably Saturated Flow in the Vadose Zone

A geostatistical inverse technique utilizing both primary and secondary information is developed to estimate conditional means of unsaturated hydraulic conductivity parameters (saturated hydraulic conductivity and pore size distribution parameters) in the vadose zone. Measurements of saturated hydraulic conductivity and pore size distribution parameters are considered as the primary information, while measurements of steady state flow processes (soil-water pressure head and degree of saturation) are regarded as the secondary information. This inverse approach relies on the classical linear predictor (cokriging) theory and takes the advantage of the spatial cross correlation between the soil-water pressure head and each of the following: degree of saturation, saturated hydraulic conductivity, and a pore size distribution parameter. Using an approximate perturbation solution for steady, variably saturated flow under general boundary conditions, the cross covariances between the primary and secondary information are derived. The approximate solution is formulated on the basis of a first-order Taylor series expansion of a discretized finite element equation. The sensitivity matrix in the solution is evaluated by an adjoint state sensitivity approach for flow in heterogeneous media under variably saturated conditions. Through several numerical examples the inverse model demonstrates its ability to improve the estimates of the spatial distribution of saturated hydraulic conductivity and pore size distribution parameters using the secondary information.

An inverse model for three-dimensional flow in variably saturated porous media

Geostatistical theory has shown promise in dealing with issues of stability, uniqueness, and identity of estimates inherent in inverse problems of subsurface flow. Here the geostatistical method is extended to three-dimensional, unsteady flow in variably saturated porous geological media (the vadose zone) that are modeled using the Richards equation and the van Genuchten-Mualem constitutive equations. The saturated hydraulic conductivity, α, and n parameters of this relationship are treated as spatially correlated, statistically independent, stochastic processes for representing heterogeneity of porous media. For given covariance functions of the parameters the adjoint-state sensitivity method is used to calculate first-order approximations for covariances of capillary pressure and moisture content and cross covariances between capillary pressure, moisture content, and the hydraulic properties. These covariances and cross covariances are then used in a successive linear estimator (SLE) to estimate the conditional means of the heterogeneous hydraulic property fields based on measurements of pressure and moisture content data. A sequential conditioning approach for our SLE was also applied to data sets collected at different sampling times during a transient infiltration event. This approach has the benefit of reducing the size of the matrices and so helps avoid numerical stability problems. On the basis of our study, pressure and moisture content data sets collected at later times of an infiltration event or during steady state flow were found to provide better estimates (smaller mean-square error compared to the true field) of the hydrological parameters of the vadose zone than data from very early times.

Stochastic analysis of solute transport in heterogeneous, variably saturated soils

Statistical moments of solute plumes from small sources in variably saturated, heterogeneous porous media are analyzed by using a newly developed, efficient high-resolution Monte Carlo technique. In agreement with previous theoretical work, it is illustrated that the prediction of such solute plumes is associated with large uncertainties for dimensionless travel times, t′, exceeding 40, particularly predictions of plumes in highly heterogeneous soils (σy2>2). Uncertainty about the travel path of the plume center contributes significantly to overall concentration uncertainty as flux fields become more variable. It is shown that the concentration coefficient of variation at the center of the plume initially increases but stagnates or decreases at later times. For highly heterogeneous soil flux conditions or for the common case of soils with strongly anisotropic conditions, analytical models underestimate transverse spreading of the mean concentration plume at any given time, while overestimating longitudinal spreading. At identical mean plume displacement distances, analytical models underestimate both transverse and longitudinal spreading and overestimate the variance of solute flux (breakthrough curve) by up to a factor 4. As an alternative to the statistical analysis of solute flux, we propose to analyze statistical properties of time associated with peak solute flux and with first exceedance of a given solute flux level.

Cokriging estimation of the conductivity field under variably saturated flow conditions

A linear estimator, cokriging, was applied to estimate hydraulic conductivity, using pressure head, solute concentration, and solute arrival time measurements in a hypothetical, heterogeneous vadose zone under steady state infiltrations at different degrees of saturation. Covariances and cross-covariances required by the estimator were determined by a first-order approximation in which sensitivity matrices were calculated using an adjoint state method. The effectiveness of the pressure, concentration, and arrival time measurements for the estimator were then evaluated using two statistical criteria, L1 and L2 norms, i.e., the average absolute error and the mean square error of the estimated conductivity field. Results of our analysis showed that pressure head measurements at steady state flow provided the best estimation of hydraulic conductivity among the three types of measurements. In addition, head measurements of flow near saturation were found more useful for estimating conductivity than those at low saturations. The arrival time measurements do not have any significant advantage over concentration. Factors such as variability, linearity, and ergodicity were discussed to explain advantage and limitation of each type of data set. Finally, to take advantage of different types of data set (e.g., head and concentration), a computationally efficient estimation approach was developed to combine them sequentially to estimate the hydraulic conductivity field. The conductivity field estimated by using this sequential approach proves to be better than all the previous estimates, using one type of data set alone.