• EXPERIMENTAL INVESTIGATION OF SEEPAGE THROUGH HETEROGENEOUS POROUS MEDIA

      Mathieu, James T., Jr.; Yeh, T.-C. Jim; Department of Hydrology & Water Resources, The University of Arizona (Department of Hydrology and Water Resources, University of Arizona (Tucson, AZ), 1988-10)
      Five sand tank experiments were conducted to investigate the behavior of unsaturated flow in heterogeneous porous media and to test the recent stochastic theories of Yeh et al. (1985a, b. c) and Mantoglou et al. (1987a, b, c) on flow through unsaturated porous media. The hydraulic properties @(w) and K(0) of the medium and coarse sand used in the experiments were measured with various laboratory columns. Fourteen medium and coarse sands were alternately layered in the 2.38 m long x 1.12 m high x 0.1 m thick sand tank. Water was infiltrated from a point source for three of five experiments and from a channel source for two experiments. An array of 62 tensiometers were used to record the capillary tension head distribution during each experiment. The wetting front profiles for the first experiment show the stratified sand effects both the development and dissipation of preferential flow paths. The experimental results qualitatively support stochastic theory of saturation dependent anisotropy. Three of the five experiments agree with the stochastic result of Yeh et al. (1985a and b) that an increase in the variance of the capillary tension head (soil becomes drier) is proportional to an increase in the mean tension head.
    • A Geostatistical Inverse Method for Variably Saturated Flow in the Vadose Zone

      Yeh, T.-C. Jim; Zhang, Jinqi; Department of Hydrology & Water Resources, The University of Arizona (Department of Hydrology and Water Resources, University of Arizona (Tucson, AZ), 1995-10-11)
      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 soil -water pressure head, degree of saturation, saturated hydraulic conductivity, and 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 based on 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 Iterative Geostatistical Inverse Method For Steady-Flow In The Vadose Zone

      Zhang, Jinqi; Yeh, T.-C. Jim; Department of Hydrology & Water Resources, The University of Arizona (Department of Hydrology and Water Resources, University of Arizona (Tucson, AZ), 1996-02-01)
      An iterative stochastic 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 parameter 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 is similar to the classical geostatistical approach, which utilizing a linear estimator that depends on the cross- covariance and covariance functions of unsaturated hydraulic conductivity parameters and flow processes. 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. Using an approximate perturbation solution for steady, variably saturated flow under general boundary conditions, the covariances of secondary information and the cross -covariance between the primary and secondary information are derived. The approximate solution is formulated based on a first -order Taylor series expansion of a discretized finite element equation. The sensitivity matrices in the solution are evaluated by an adjoint state sensitivity approach for flow in heterogeneous media under variably saturated conditions. As a result, the nonlinear relationships between unsaturated hydraulic conductivity parameters and flow processes are incorporated in the estimation. Through some numerical examples, the iterative inverse model demonstrates its ability to improve the estimates of the spatial distribution of saturated hydraulic conductivity and pore -size distribution parameters compared to the classical geostatistical inverse approach. In addition, the inconsistency problem existing in classical geostatistical inverse approach is alleviated. The estimated fields of unsaturated hydraulic conductivity parameters and flow fields not only retain their observed values at sample locations, but satisfy the governing flow equation as well.
    • AN ITERATIVE STOCHASTIC INVERSE METHOD: CONDITIONAL EFFECTIVE TRANSMISSIVITY AND HYDRAULIC HEAD FIELDS

      Yeh, T.-C. Jim; Jin, Minghui; Hanna, Samuel; Department of Hydrology & Water Resources, The University of Arizona (Department of Hydrology and Water Resources, University of Arizona (Tucson, AZ), 1995-08-29)
      An iterative stochastic approach is developed to estimate transmissivity and head distributions in heterogeneous aquifers. This approach is similar to the classical cokriging technique and 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.
    • REVIEW OF MODELING OF WATER FLOW AND SOLUTE TRANSPORT IN THE VADOSE ZONE: Stochastic Approaches

      Yeh, T.-C. Jim; McCord, J. T.; Department of Hydrology & Water Resources, The University of Arizona (Department of Hydrology and Water Resources, University of Arizona (Tucson, AZ), 1994-11-10)
      Hydrologic properties of the vadose zone are heterogeneous at many different scales. An accurate prediction of water flow and solute transport in the vadose zone requires detailed information about spatial distributions of the properties. Collecting such detailed spatial distribution of hydrologic properties of geological formations is a formidable task. As a result, hydrologic modelers face a difficult challenge: to make the best prediction with little information. During the past few decades many approaches and theories based on stochastic concepts have been developed in an attempt to overcome this difficulty. These stochastic approaches and theories provide ways not only to predict flow and transport processes in large -scale, heterogeneous vadose zones, but also to assess uncertainties in our predictions. One widely -investigated stochastic approach involves the use of effective flow and transport properties. The effective property approach essentially represents a generalization of the well -known equivalent homogeneous media approach discussed in most hydrology textbooks (e.g., using the arithmetic mean conductivity and harmonic mean conductivity for flow parallel to and normal to stratification, respectively, in layered media). This approach is a valuable tool in many practical situations but it predicts the ensemble behavior of a system which can be quite different from reality. To obtain predictions at higher resolutions than the effective property approach, many heterogeneous approaches have also been developed. This paper presents an overview of the stochastic theories related to both equivalent homogeneous media and heterogeneous approaches, it highlights their applications, and it discusses some of their deficiencies.
    • A REVIEW OF THE SCALE PROBLEM AND APPLICATIONS OF STOCHASTIC METHODS TO DETERMINE GROUNDWATER TRAVEL TIME AND PATH

      Yeh, T.-C. Jim; Stephens, Daniel B.; Department of Hydrology & Water Resources, The University of Arizona (Department of Hydrology and Water Resources, University of Arizona (Tucson, AZ), 1989)
      The groundwater travel time along the fastest path of likely radionuclide transport is a regulatory criterion used to assess the hydrogeologic quality of a high - level radioactive waste repository. Hydrologists and engineers are limited in their ability to define with confidence the fastest path, owing to the heterogeneous nature of geologic materials. Field measurements of hydraulic properties such as in test or observation wells, are inherently averages of properties at scales smaller than the scale of the field measurement. As a result of averaging, subscale information is lost and there is uncertainty in defining the fastest trajectory of groundwater. This scale problem is explained through a review of the continuum and REV concepts in groundwater hydrology. The application of hydrodynamic dispersion concepts is recommended as a means of incorporating the effect of subscale heterogeneity on the fastest groundwater travel time. Sources of uncertainties in predicting groundwater travel time are discussed in the report. The uncertainties are mainly attributed to the heterogeneous nature of geologic formations. The heterogeneity of geologic materials can, however, be characterized quantitatively using geostatistical methods. Important statistical parameters include mean and variance. as well as the spatial correlation structures of the hydrologic properties within the hydrogeologic system. These parameters may he obtained from limited data base. Stochastic methods, reviewed and explained in this report, can take advantage of the geostatistical characterization to predict large -scale groundwater flow and solute transport. Several examples from recent scientific literature are provided to illustrate the application of stochastic methods to the groundwater travel time analysis. Stochastic methods in subsurface hydrology have only recently been evaluated under field conditions for a few locations, and validation of the theories is incomplete, especially in unsaturated fractured rocks. Nevertheless, research efforts should continue to improve the state -of -the art. Geostatistics and stochastic methods will be valuable tools in addressing the groundwater travel time objective
    • Scale issues of heterogeneity in vadose zone hydrology and practical solutions

      Yeh, T.-C. Jim; Department of Hydrology & Water Resources, The University of Arizona (Department of Hydrology and Water Resources, University of Arizona (Tucson, AZ), 1996-08-03)
      Hydrological properties of the vadose zone often exhibit a high degree of spatial variability at various scales due to the heterogeneous nature of geological formations. For laboratory scale problems (i.e., small cores, soil columns, and sand boxes), variation in pore size, pore geometry, and tortuosity of pore channels are the major source of heterogeneity. They are called laboratory-scale heterogeneity. Microstratification, foliation, cracks, and roots are also some possible heterogeneities at this scale. As our observation scale increases to a field, stratification or layering in a geologic formation becomes the dominant heterogeneity, which is often classified as field-scale heterogeneity. At an even larger observation scale, the regional-scale heterogeneity represents the variation of geologic formations or facies. Variations among sedimentary basins are then categorized as the global-scale heterogeneity. Fundamental theories for flow and solute transport through porous media are essentially derived for the laboratory-scale heterogeneity. When we attempt to apply these theories to the vadose zone, comprising heterogeneities of many different scales, we encounter the scale problem. To resolve this problem two approaches have evolved in the past: the system approach and the physical approach. The former approach treats the vadose zone as a low pass filter and its governing principle is determined by the relationship between its input and output histories (e.g., Jury et al., 1986). The latter approach however relies on upscaling the laboratory-scale theories to the vadose zone. While the system approach has been widely used by soil scientists, it is often criticized for its empiricism and the lack of physical principles. Besides, it is known to be limited to nonpoint source problems or those related to the integrated behavior of a system (for example, the average concentration of nitrate in the irrigation return flow at irrigation drains or their breakthrough at the water table beneath an irrigation field). Since this approach requires the knowledge of input and output histories and model calibrations, flow and tracer experiments must be carried out at a given site prior to prediction. Further, a calibrated system model for the vadose zone at a given depth under a given condition is often found unsuitable for different depths and conditions (e.g., Butters et al., 1989; Butters and Jury, 1989; Roth et al., 1991). While such system approaches are practical tools for predicting water flow and pollutant transport through thin vadose zones to the water table or to irrigation drains at agricultural fields, their utility for general hydrogeological problems is limited. Hydrogeological problems involve vadose zones of tens and hundreds of meters in thickness. Input sources to these vadose zones are small compared with the scale of hydrogeological settings. Yet, groundwater hydrologists have to focus on the spatial and temporal evolution of flow and spread of solutes over the vadose zone and regional aquifers (Stephens, 1996). Because of these above- mentioned reasons, the following discussion will concentrate on the physical approach that has been widely used by groundwater hydrologists. Moreover, the discussion will present only the author's point of view about the scale issue and approaches to the heterogeneity in the vadose zone.
    • STOCHASTIC ANALYSIS OF WATER FLOW IN HETEROGENEOUS UNSATURATED SOILS UNDER TRANSIENT CONDITIONS

      Ferrante, Marco; Yeh, T.-C. Jim; Department of Hydrology & Water Resources, The University of Arizona (Department of Hydrology and Water Resources, University of Arizona (Tucson, AZ), 1995-09-08)
      A numerical model for the analysis of uncertainty propagation in flow through unsaturated soils is developed. This model is based on the first -order Taylor series expansion of the discretized Richards' equation, for one -dimensional flow. Soil hydrologic properties, the saturated hydraulic conductivity and the pore size distribution, are assumed to be stochastic processes in space. The surface boundary conditions are considered to be deterministic variable in time or stochastic time series. The purpose of this model is to examine the effect of uncertainty in boundary conditions and heterogeneity on the pressure head and flux variance profiles at various times.
    • Stochastic fusion of information for characterizing and monitoring the vadose zone

      Yeh, T.-C. Jim; Simunek, Jirka; Van Genuchten, Martinus Th.; Department of Hydrology & Water Resources, The University of Arizona (Department of Hydrology and Water Resources, University of Arizona (Tucson, AZ), 2002-03)
      Inverse problems for vadose zone hydrological processes are often being perceived as ill - posed and intractable. Consequently, solutions to inverse problems are often subject to skepticism. In this paper, using examples, we elucidate difficulties associated with inverse problems and the prerequisites for such problems to be well -posed so that a unique solution exists. We subsequently explain the need of a stochastic conceptualization of the inverse problem and, in turn, the conditional- effective -parameter concept. This concept aims to resolve the ill -posed nature of inverse problems for the vadose zone, for which generally only sparse data are available. Next, the development of inverse methods for the vadose zone, based on a conditional -effective -parameter concept, is explored, including cokriging, the use of a successive linear estimator, and a sequential estimator. Their applications to the vadose zone inverse problems are subsequently examined, which include hydraulic /pneumatic and electrical resistivity tomography surveys, and hydraulic conductivity estimation using observed pressure heads, concentrations, and arrival times. Finally, a stochastic information fusion technology is presented that assimilates information from unsaturated hydraulic tomography and electrical resistivity tomography. This technology offers great promise to effectively characterize heterogeneity, to monitor processes in the vadose zone, and to quantify uncertainty associated with vadose zone characterization and monitoring.