Stochastic uncertainty analysis for solute transport in randomly heterogeneous media using a Karhunen-Loève-based moment equation approach

This is the published version. Copyright American Geophysical Union[1] A new approach has been developed for solving solute transport problems in randomly heterogeneous media using the Karhunen-Loève-based moment equation (KLME) technique proposed by Zhang and Lu (2004). The KLME approach combines the Karhunen-Loève decomposition of the underlying random conductivity field and the perturbative and polynomial expansions of dependent variables including the hydraulic head, flow velocity, dispersion coefficient, and solute concentration. The equations obtained in this approach are sequential, and their structure is formulated in the same form as the original governing equations such that any existing simulator, such as Modular Three-Dimensional Multispecies Transport Model for Simulation of Advection, Dispersion, and Chemical Reactions of Contaminants in Groundwater Systems (MT3DMS), can be directly applied as the solver. Through a series of two-dimensional examples, the validity of the KLME approach is evaluated against the classical Monte Carlo simulations. Results indicate that under the flow and transport conditions examined in this work, the KLME approach provides an accurate representation of the mean concentration. For the concentration variance, the accuracy of the KLME approach is good when the conductivity variance is 0.5. As the conductivity variance increases up to 1.0, the mismatch on the concentration variance becomes large, although the mean concentration can still be accurately reproduced by the KLME approach. Our results also indicate that when the conductivity variance is relatively large, neglecting the effects of the cross terms between velocity fluctuations and local dispersivities, as done in some previous studies, can produce noticeable errors, and a rigorous treatment of the dispersion terms becomes more appropriate

Evaluation of the applicability of the dual-domain mass transfer model in porous media containing connected high-conductivity channels

This is the published version. Copyright American Geophysical Union[1] This paper evaluates the dual-domain mass transfer (DDMT) model to represent transport processes when small-scale high-conductivity (K) preferential flow paths (PFPs) are present in a homogenous porous media matrix. The effects of PFPs upon solute transport were examined through detailed numerical experiments involving different realizations of PFP networks, PFP/matrix conductivity contrasts varying from 10:1 to 200:1, different magnitudes of effective conductivities, and a range of molecular diffusion coefficients. Results suggest that the DDMT model can reproduce both the near-source peak and the downstream low-concentration spreading observed in the embedded dendritic network when there are large conductivity contrasts between high-K PFPs and the low-K matrix. The accuracy of the DDMT model is also affected by the geometry of PFP networks and by the relative significance of the diffusion process in the networkmatrix system

Simulation assessment of the direct-push permeameter for characterizing vertical variations in hydraulic conductivity

This is the published version. Copyright American Geophysical Union[1] The direct-push permeameter (DPP) is a tool for the in situ characterization of hydraulic conductivity (K) in shallow, unconsolidated formations. This device, which consists of a short screened section with a pair of pressure transducers near the screen, is advanced into the subsurface with direct-push technology. K is determined through a series of injection tests conducted between advancements. Recent field work by Butler et al. (2007) has shown that the DPP holds great potential for describing vertical variations in K at an unprecedented level of detail, accuracy and speed. In this paper, the fundamental efficacy of the DPP is evaluated through a series of numerical simulations. These simulations demonstrate that the DPP can provide accurate K information under conditions commonly faced in the field. A single DPP test provides an effective K for the domain immediately surrounding the interval between the injection screen and the most distant pressure transducer. Features that are thinner than that interval can be quantified by reducing the vertical distance between successive tests and analyzing the data from all tests simultaneously. A particular advantage of the DPP is that, unlike most other single borehole techniques, a low-K skin or a clogged screen has a minimal impact on the K estimate. In addition, the requirement that only steady-shape conditions be attained allows for a dramatic reduction in the time required for each injection test

Spatial connectivity in a highly heterogeneous aquifer: From cores to preferential flow paths

This is the published version. Copyright American Geophysical Union[1] This study investigates connectivity in a small portion of the extremely heterogeneous aquifer at the Macrodispersion Experiment (MADE) site in Columbus, Mississippi. A total of 19 fully penetrating soil cores were collected from a rectangular grid of 4 m by 4 m. Detailed grain size analysis was performed on 5 cm segments of each core, yielding 1740 hydraulic conductivity (K) estimates. Three different geostatistical simulation methods were used to generate 3-D conditional realizations of the K field for the sampled block. Particle tracking calculations showed that the fastest particles, as represented by the first 5% to arrive, converge along preferential flow paths and exit the model domain within preferred areas. These 5% fastest flow paths accounted for about 40% of the flow. The distribution of preferential flow paths and particle exit locations is clearly influenced by the occurrence of clusters formed by interconnected cells with K equal to or greater than the 0.9 decile of the data distribution (10% of the volume). The fraction of particle paths within the high-K clusters ranges from 43% to 69%. In variogram-based K fields, some of the fastest paths are through media with lower K values, suggesting that transport connectivity may not require fully connected zones of relatively homogenous K. The high degree of flow and transport connectivity was confirmed by the values of two groups of connectivity indicators. In particular, the ratio between effective and geometric mean K (on average, about 2) and the ratio between the average arrival time and the arrival time of the fastest particles (on average, about 9) are consistent with flow and advective transport behavior characterized by channeling along preferential flow paths

Relative importance of dispersion and rate-limited mass transfer in highly heterogeneous porous media: Analysis of a new tracer test at the Macrodispersion Experiment (MADE) site

This is the published version. Copyright American Geophysical Union[1] A single-well injection-withdrawal (SWIW) bromide tracer test was conducted to further investigate transport processes at the Macrodispersion Experiment (MADE) site on Columbus Air Force Base in Mississippi. The bromide breakthrough curve is highly asymmetric and exhibits an early time high-concentration peak followed by an extended period of low-concentration tailing. Comparisons of results simulated by advection-dispersion (AD) and dual-domain mass transfer (DDMT) models with the field data show that the DDMT model more accurately represents the magnitudes of both the early high-concentration peak and the later low-concentration tail. For both the AD and DDMT models, the match with field data is enhanced by incorporating hydraulic conductivity information from new direct-push profiling methods. The Akaike information criterion for the DDMT models is much smaller than that for the AD models in both the homogeneous and heterogeneous cases investigated in this work. The improved match of the DDMT model with the SWIW test data supports the hypothesis of mass transfer processes occurring at this highly heterogeneous site

Importance of a sound hydrologic foundation for assessing the future of the High Plains Aquifer in Kansas

This is the published version. Copyright National Academy of SciencesSteward et al. (1) assess the hydrologic and agricultural future of the High Plains Aquifer. We have many concerns about hydrologic aspects of their study and describe the most significant here. The authors state “…the lines of recharge plus storage in Fig. 1C very closely approximate the recent data points of metered groundwater pumping….” That is not correct, as is clear from a comparison of reported pumping data (diamonds) and the authors’ calculated groundwater use (solid line) for the SW region. There is a systematic deviation (authors’ calculated use is increasing, whereas reported metered pumping data are decreasing), which persists even when uncertain pre-1990 pumping data are neglected. The authors’ groundwater use is also markedly inconsistent with common experiences in western Kansas (2). The 2020–2025 (SW) and 2025–2030 (NW) peaks in the authors’ groundwater use are simply a product of their logistic function representation (maximum use at normalized thickness of 0.5) and are in dramatic contrast to recorded pumping trends. Given that calculated groundwater use is input into the agricultural models, we question all of the agricultural projections. The authors provide no objective basis for accepting the logistic function as an accurate tool for projecting water level declines. The comparisons in their table S1 do little to substantiate the use of the function given that the authors (i) adjust two parameters per well; (ii) adjust parameters at each well independently of the other 1,600 wells; and (iii) in aggregate, only assess the first 30% of depletion. A number of alternative functions could be found that would produce similar agreement with existing data but markedly different future projections. We note the circularity of including extrapolated 2060 values in the dataset used to develop logistic curves that are then used to make future projections. The authors state “…and measurement points were added at 1930 and 2060 from a linear extrapolation of observations while keeping these points within the saturated aquifer.” We are concerned about the sensitivity of future projections to inclusion of 1930 and 2060 “measurements” and to the process (unexplained) for “keeping these points within the saturated aquifer.” The authors state that “We computed recent recharge rates to preserve conservation of mass….” That cannot be correct, as is clear from a comparison of reported pumping data (diamonds) and the authors’ calculated change in storage plus recharge (solid line) for the SW region in their figure 1C; a conservation of mass calculation would produce a line through the center of mass of the reported 1981–2009 data. The calculated recharge values appear to have been adjusted in an unexplained manner. Given that, we also question the significance of the match obtained for the groundwater-supported corn plot in their figure 3A. The comparisons in their table S3 do little to substantiate the authors’ recharge estimates because of the above concerns and the lack of consistency with more recent process-based modeling investigations (3, 4). We conclude that this is an interesting, but highly flawed, mathematical exercise that has little bearing on future conditions in the High Plains Aquifer in western Kansas

Limits of applicability of the advection-dispersion model in aquifers containing connected high-conductivity channels

This is the published version. Copyright American Geophysical Union[1] The macrodispersion model from stochastic transport theory is demonstrated to be of limited utility when applied to heterogeneous aquifer systems containing narrow connected pathways. This is so even when contrasts in hydraulic conductivity (K) are small and variance in ln K is less than 0.10. We evaluated how well an advection-dispersion model (ADM) could be used to represent solute plumes transported through mildly heterogeneous three-dimensional (3-D) systems characterized by a well-connected dendritic network of 10 cm wide high-K channels. Each high-K channel network was generated using an invasion percolation algorithm and consisted of ∼10% by volume high-K regions. Contrasts in K between the channels and matrix were varied systematically from 2:1 to 30:1, corresponding to ln K values ranging from 0.04 to 1.05. Simulations involved numerical models with 3-D decimeter discretization, and each model contained 2–4 million active cells. Transport through each channel network considered only the processes of advection and molecular diffusion. In every case, the temporal change in the second spatial moment of concentrations was linear, with R2 values ranging from 0.97 to 0.99. The third spatial moment, or alternatively, the skewness coefficient values, indicated significant tailing downstream of the plume center. For each case, a corresponding ADM was used to simulate transport through the system. The corresponding ADM employed the effective mean hydraulic conductivity that reproduced the total discharge through the channel network system under an identical ambient gradient. Dispersivity values used in the ADM were obtained from the temporal change in the second spatial moments of concentrations for the plumes in the channel network systems and ranged from 0.014 m to 0.85 m. The results indicate that as the conductivity contrast between the channels and matrix increased, the simulated plumes in the channel network system became more and more asymmetric, with little solute dispersed upstream of the plume center and extensive downstream spreading of low concentrations. Distinctly different spreading was found upstream versus downstream of the plume center. The ADM failed to capture this asymmetry. Comparison of each plume in the channel network system with the corresponding plume produced using the corresponding ADM showed a maximum correlation of only 0.64 and a minimum fractional error of 0.29 for cases in which the log K variance was ∼0.20 (ln K variance was ∼1.0). At early times the correlations were as low as 0.40. The greatest correlation occurred at late times and for cases in which a wide source was considered

Stochastic uncertainty analysis for unconfined flow systems

This is the published version. Copyright American Geophysical Union[1] A new stochastic approach proposed by Zhang and Lu (2004), called the Karhunen-Loeve decomposition-based moment equation (KLME), has been extended to solving nonlinear, unconfined flow problems in randomly heterogeneous aquifers. This approach is on the basis of an innovative combination of Karhunen-Loeve decomposition, polynomial expansion, and perturbation methods. The random log-transformed hydraulic conductivity field (lnKS) is first expanded into a series in terms of orthogonal Gaussian standard random variables with their coefficients obtained as the eigenvalues and eigenfunctions of the covariance function of lnKS. Next, head h is decomposed as a perturbation expansion series Σh(m), where h(m) represents the mth-order head term with respect to the standard deviation of lnKS. Then h(m) is further expanded into a polynomial series of m products of orthogonal Gaussian standard random variables whose coefficients image are deterministic and solved sequentially from low to high expansion orders using MODFLOW-2000. Finally, the statistics of head and flux are computed using simple algebraic operations on image A series of numerical test results in 2-D and 3-D unconfined flow systems indicated that the KLME approach is effective in estimating the mean and (co)variance of both heads and fluxes and requires much less computational effort as compared to the traditional Monte Carlo simulation technique

Geostatistical analysis of centimeter-scale hydraulic conductivity variations at the MADE site

This is the published version. Copyright American Geophysical Union[1] Spatial variations in hydraulic conductivity (K) provide critical controls on solute transport in the subsurface. Recently, new direct-push tools were developed for high-resolution characterization of K variations in unconsolidated settings. These tools were applied to obtain 58 profiles (vertical resolution of 1.5 cm) from the heavily studied macrodispersion experiment (MADE) site. We compare the data from these 58 profiles with those from the 67 flowmeter profiles that have served as the primary basis for characterizing the heterogeneous aquifer at the site. Overall, the patterns of variation displayed by the two data sets are quite similar, in terms of both large-scale structure and autocorrelation characteristics. The direct-push K values are, on average, roughly a factor of 5 lower than the flowmeter values. This discrepancy appears to be attributable, at least in part, to opposite biases between the two methods, with the current versions of the direct-push tools underestimating K in the highly permeable upper portions of the aquifer and the flowmeter overestimating K in the less permeable lower portions. The vertically averaged K values from a series of direct-push profiles in the vicinity of two pumping tests at the site are consistent with the K estimates from those tests, providing evidence that the direct-push estimates are of a reasonable magnitude. The results of this field demonstration show that direct-push profiling has the potential to characterize highly heterogeneous aquifers with a speed and resolution that has not previously been possible