H13H-01 INVITED 13:40h
The Hydrofacies Approach and Why ln K $\sigma$$^{2}$ $<$5-10 is Unlikely
When heterogeneity of geologic systems is characterized in terms of hydrofacies rather than solely based on K measurements, the resulting flow and transport models typically contain not only aquifer materials but also significant volumes (10-70%) of aquitard materials. This leads to clear, heuristic rationale for the ln K $\sigma$$^{2}$ commonly exceeding 5 to 10, contradicting published data on ln K $\sigma$$^{2}$. I will explain the inconsistencies between commonly held assumptions of low ($<$1-2) ln K $\sigma$$^{2}$ and abundant geologic and hydrologic field data that indicate substantially larger values. The K data commonly cited in support of the low ln K $\sigma$$^{2}$ assumption have been misinterpreted because of unintentional, biased sampling. Geologic fundamentals and field data indicate that ln K $\sigma$$^{2}$ is commonly $>$10 and can easily exceed 20 in typical sedimentary deposits (not surficial soils) at spatial scales on the order of 10$^{1}$ to 10$^{2}$ m. Presence of large ln K $\sigma$$^{2}$ can be paramount in transport models and is often requisite for modeling observed transport phenomena such as preferential flow, extreme tailing, difficult remediation including frequent pump-and-treat failure, and significant, unanticipated mixing of groundwater ages.
H13H-02 INVITED 13:55h
Random Domain Decomposition for Transport in Highly Heterogeneous Aquifers
We discuss a random domain decomposition approach to analyze flow and transport in highly heterogeneous, composite porous media that greatly improves estimates of pressure head statistics. Composite porous media consist of distinct geologic facies. Within a composite medium, hydraulic conductivity and other relevant flow and transport parameters can be represented through a pair of random processes: i) a boundary process that determines facies arrangement and extent and ii) a random process that defines parameters within a given facies. Our primary focus is on efficient use of data to provide a probabilistic description of large-scale boundary processes.
H13H-03 14:10h
Transport Of Reactive Solutes In Bimodal Porous Formations
Solute transport in heterogeneous formations is controlled by hydraulic and geochemical property variations over many spatial scales. In order to describe such variability, multi-Gaussian models have been commonly adopted with the justification that they are consistent with field data at a few experimental sites. However, as soon as more detailed studies on the structure of sedimentary formations were completed, a hierarchical structure emerged in which several modes and correlation lengths combine to create a much more complex and heterogeneous structure. Such a complexity is created by the arrangement of lithofacies units, with size, granulometric and textural properties dependent on the energy of the depositional environment. A way to model such complexity is by multi-indicator models. For the simplest case of a bi-modal formation, the logconductivity field can be described as Y(x)=I(x)Y1(x)+[1-I(x)]Y2(x) where I(x) is an indicator stochastic random function (SRF) while Y1(x) and Y2(x) are SRFs representing small- and large-scale variability of respectively. I(x) is a binary function uniformly distributed which assumes the value 1 with probability P$<$1 and the value 0 with probability (1-P), such that P controls the interplay between small and large scale variability. Rubin (1995) discussed the theoretical foundations of this model providing first order approximations in of the longitudinal and transverse macrodispersion coefficient; most of the work that has been done so far is for non-reactive tracers. In this work we extend the work by Rubin to a reactive solute undergoing non-equilibrium reversible adsorption. We obtained semi-analytical solutions for the effective velocity and macro-dispersion coefficients using the stochastic model for adsorption kinetics suggested by Quinodoz and Valocchi (1993). In doing that we focus on the impact of sorption and spatial conductivity variability on the first two spatial moments assuming spatially constant kinetic rate coefficients. Under the validity of the first order approximation, we conclude: (1) the effective velocity is expressed by a time dependent retardation factor composed by two independent terms; the former is time independent and is related to the mean conductivity contrast between the facies, while the second is controlled by the geochemical parameters; (2) the presence of high conductivity inclusions produces higher asymptotic longitudinal dispersion coefficient than low conductivity inclusions and this effect is enhanced by the kinetics; (3) the early time behavior of the longitudinal dispersion coefficient is mainly influenced by the contrast between the mean hydraulic conductivity of the two facies; (4) the kinetics has a much less dramatic effect upon the plume spreading in the transversal direction. Finally we investigate the trustworthiness of the semi-analytical solution for different value of k and P, comparing it with exact results found through numerical simulations.
H13H-04 14:25h
Upscaling and Uncertainty of Reactive Transport in Heterogeneous Porous Media
Reactive transport models based on the advection dispersion equation (ADE) are inaccurate because they use spatially averaged concentrations to simulate chemical reaction. Using these averaged concentrations for reactants assumes that solutes are completely mixed; i.e. the ADE-predicted concentration is uniform at the unresolved-scale. In reality, dispersion processes create chemical segregation and concentration distributions, and these unresolved scale distributions govern the amount of chemical reaction. Lab scale reactive transport experiments were run by digitally imaging colored dye tracers and colorimetric chemical reactions in clear heterogeneous porous media. Modeling the concentration mean and variance and assuming that reactants can be described by a joint beta distribution, gives a good description of the space-time evolution of our lab-scale product and reactant distributions. Currently, this approach works for instantaneous reactions with equally sorbing reactants.
H13H-05 14:40h
Energy-Based Spatial Weighting Functions and Equivalent Hydraulic Conductivity in Heterogeneous Porous Media.
To improve understanding of property measurements in heterogeneous media, an energy-based weighting function concept was developed [Molz et al., WRR, 39(4), DAN-1, 2003]. In (assumed) homogeneous media, the instrument spatial weighting function (ISWF) depends only on the energy dissipation distribution set up by the measurement procedure, and it reduces to simply inverse sample volume (uniform weighting) for the 1-D parallel flow case (ideal permeameter). For 1-D transient flow in homogeneous media, such as with slug tests, the ISWF varies with position and time, with 95% of the total weighting contained within 115 well radii, even late in the test [Molz et al., Ground Water, in press, 2004]. The present talk deals with the heterogeneous case, which is what one deals with in natural systems. Thus, in actual measurements, the identification of the ISWF is connected to the problem of determining an "equivalent" hydraulic conductivity (Ke), and it would be ideal if the criterion for "equivalence" based on energy-dissipation- rate-weighting would produce the same Ke as that based on the common permeameter test. It can be shown that for 1-D linear and radial flow in heterogeneous porous media, the energy-dissipation based Ke and the usual Ke calculated using assumed homogeneity and Darcy's law are identical. We will explore whether this same equivalence holds in general for 2-D and 3-D heterogeneity. The results to date imply that as one makes K measurements in heterogeneous media at different locations or on different cores of heterogeneous materials, the ISWF will be heterogeneity-dependent, implying that the averaging process resulting in the "equivalent" K value also varies with position. If the testing procedure is transient, then the averaging process varies also with time. This suggests a fundamental ambiguity in the interpretation of hydraulic conductivity measurements in heterogeneous media that may impact how we approach modeling and prediction in a practical sense. Several equivalent K calculations and spatial weighting functions are presented, conclusions drawn and further research suggested.
H13H-06 14:55h
Effective Dispersion in Heterogeneous Media: Comparison between Experiments, Inverse Modeling, and First-Order Theory
Effective dispersion is a measure to characterize dilution and solute mixing in heterogeneous aquifers. Here, we compoute the longitudinal effective dispersion coefficient from the second central temporal moment of a breakthrough curve observed at a point. Standard macrodispersion coefficients, by contrast, describe the spread of concentration observed over the entire cross-section of the domain. We have performed conservative tracer tests in a 14m long sandbox filled heterogeneously with four different types of sands. The filling pattern mimics natural sediments, including larger-scale structures created by the different sand types, and micro-structures within the sand lenses created by the sedimentation applied in the filling procedure. We characterize point-related breakthrough curves and curves averaged over all probes within a measurement plane by their temporal moments. The observed effective dispersion is about 2/3 of the standard macrodispersion. We have fitted the semi-analytical expressions for the second central moments of Dentz et al. (2000) and Fiori and Dagan (2000) to the data, finding good qualitative agreement with physically reasonable parameters. The data were also used to identify the distribution of the conductivity field and the field of an apparent scalar dispersion coefficient by geostatistical inversing. The major patterns of conductivity could be identified, and the dispersion coefficient identified was in the order of local transverse dispersion coefficients. The latter indicates that most of the heterogeneity could be resolved in the inversing procedure.
H13H-07 15:10h
Inverse Stochastic Moment Analysis of Transient Flow in Randomly Heterogeneous Media
Nonlocal stochastic moment equations have been used successfully to analyze transient flow in randomly heterogeneous media conditional on measured values of medium properties. We present a geostatistical inverse algorithm that makes it possible to further condition such analyses on measured values of state variables, notably hydraulic head and flux. Our approach is based on Laplace-transformed recursive finite-element approximations of exact nonlocal first (mean) and second (variance-covariance) conditional moment equations and numerical inversion of their solution. Hydraulic conductivity (or transmissivity) is parameterized geostatistically based on measured values at discrete locations (if available) and unknown values at discrete "pilot points." Prior estimates of these unknown pilot point values are obtained (optionally, subject to the availability of sufficient measured values) by universal kriging. Posterior parameter estimates at pilot points and (optionally) at measurement points (thereby accounting for measurement errors) are obtained by calibrating the conditional mean flow equations against measured values of head and/or flux. The parameters are projected onto a computational grid via universal kriging. Maximum likelihood calibration allows one to estimate not only hydraulic but also (optionally) unknown variogram parameters with or without prior information about the former. The approach yields covariance matrices for parameter estimation as well as head and flux prediction errors, the latter being obtained a posteriori from recursive finite element approximations of the second conditional moment equations. Preliminary results are given for transient flow in a bounded two-dimensional domain.