Published in Vadose Zone Journal 3:480-484 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
SPECIAL SECTION: UNCERTAINTY IN VADOSE ZONE FLOW AND TRANSPORT PROPERTIES
Uncertainty in Vadose Zone Flow and Transport Prediction
Robert M. Holta and
Michael J. Nicholl*,b
a Department of Geology and Geological Engineering, University of Mississippi, 118 Carrier Hall, University, MS 38677
b Geoscience Department, University of Nevada, Las Vegas, NV, 89122
* Corresponding author (michael.nicholl{at}ccmail.nevada.edu).
Received 14 February 2004.
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INTRODUCTION
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THIS SPECIAL SECTION of Vadose Zone Journal considers uncertainty in vadose zone flow and transport prediction. The recent emergence of this topic is a consequence of a major change in the motivation for predictive modeling in the vadose zone. Until the 1980s, predictive modeling of vadose zone flow and transport was focused mostly on relatively applied problems in soil science and agricultural engineering. Since that time, the discipline has expanded to include the vastly different problem of groundwater protection, where applications often involve complex geologic environments that include heterogeneous unconsolidated materials (Fig. 1)
and fractured rock (Fig. 2)
. This change in focus has altered not only the questions to be answered by predictive modeling, but also the level of predictive uncertainty that is acceptable to decisionmakers. As a result, there has been substantial growth in research directed at quantifying and reducing the uncertainty associated with predictive modeling of vadose zone flow and transport.
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Fig. 1. An excavated face following a ponded infiltration test ( 50 h) that was conducted in fluvial channel deposits near Albuquerque, NM. Flagged nails in the outcrop are spaced 40 cm apart. Dyes introduced with the infiltrating water illustrate some of the complex processes affecting flow and transport in the vadose zone. Lateral spreading occurs above a capillary barrier (gravel deposits). Dye is transported preferentially along a fracture, while the inclined stratification focuses flow and influences dye transport.
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Fig. 2. A horizontal pavement located approximately 3 m directly beneath a ponded infiltration test in which about 790 L of dyed water were applied to an initially dry fracture network. The grid lines are located at 0.305-m (1-foot) horizontal intervals. Low permeability of the rock matrix in this densely welded tuff unit near Yucca Mountain, Nevada constrained flow to the fracture network. Well-connected near-vertical features dominate the network, with fewer extensive subhorizontal fractures. The infiltrating fluid slug flooded the near-surface network and then fragmented during the transition to unsaturated flow. Here we see a highly nonuniform distribution of the dye tracer (blue). Flow occurred along only portions of some fractures and avoided others altogether, despite obvious connection. There is also evidence of focused flow along some, but not all, of the vertical intersections (from Nicholl and Glass, 2002).
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The papers presented here represent a sampling of current research on the emerging topic of uncertainty in vadose zone flow and transport prediction. The breadth of this topic is illustrated by the eight papers in this special section, which were selected from recent topical sessions at annual meetings of the Geological Society of America (2001) and the American Geophysical Union (2002). To introduce the papers, we begin this preface by placing those papers into the context of "classic" predictive modeling approaches. Afterwards, we discuss two important research needs and conclude the preface with a reminder that uncertainty relates to the entire predictive modeling process.
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PAPERS IN THIS SPECIAL SECTION
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To provide a framework for discussing the papers in this special section, we consider predictive modeling as an iterative process (e.g., Spitz and Moreno, 1996; National Research Council, 2001; Neuman and Wierenga, 2003) in which a set of discrete activities are connected by a set of overlapping feedback loops (Fig. 3)
. The modeling process begins with the definition of a problem (i.e., a question to be answered) and ends with a predicted outcome. The global predictive uncertainty is the cumulative effect of bias and uncertainty that are introduced not only in each discrete modeling activity, but also within the feedback loops. For vadose zone flow and transport, quantitative research into predictive uncertainty has focused mostly on data collection and model input (i.e., verification/calibration in Fig. 3). There has also been substantial effort directed at reducing uncertainty through the development of improved conceptual models. Other areas depicted in Fig. 3 have received much less attention. A thorough discussion of predictive uncertainty in generalized hydrogeologic flow and transport modeling (saturated and unsaturated) can be found in Neuman and Wierenga (2003), where they suggest an integrative strategy that allows the assessment of joint predictive uncertainty resulting from several alternative conceptual models (see also Neuman, 2003). Classifying the papers in this special section with respect to the categories in Fig. 3, we find that two of them address the topic of uncertainty in data collection, five are related to conceptual model uncertainty, and one illustrates the need to address the issue of global uncertainty.
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Fig. 3. Conceptual illustration of the predictive modeling process. Discrete activities are connected by multiple and overlapping feedback loops.
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Data collected in support of predictive modeling include observations of system state variables, system geometry, hydraulic parameters, and sources and sinks. Uncertainty in these data arises from a number of sources, including measurement error and bias, spatial and temporal variations, and incomplete sampling or knowledge. One approach for reducing data uncertainty is to improve our ability to observe state variables and measure hydraulic parameters. Two of the papers in this issue take this approach. Helm-Clark et al. (2004) explore the utility of wireline neutron logging tools to estimate moisture content in unsaturated basalt. Their results support the possibility of making quantitative estimates for both bound and unbound hydrogen. However, they note that inversion of the neutron log response due to local variability in the neutron slowing-down length may lead to spurious results. They also suggest modifications to the wireline tool that may ameliorate this issue. Perfect et al. (2004) present experiments in a large (18 cm high, 10-cm diam.) saprolite core used to consider scaling of the capillary pressure–saturation relation between air–water and water–DNAPL systems. They demonstrate that air-entry measurements can be successfully used to estimate DNAPL entry into a water-saturated system. They also observe that correction for sample height and contact angle had a small influence on their results.
The conceptual model is a qualitative or quantitative description of the essential features of the physical system that are germane to the problem at hand. Although commonly expressed as a deterministic or stochastic mathematical formulation, the conceptual model can also be a verbal description of how the system works. The development of a conceptual model is iterative; new data may be required to formulate the initial conceptual model, or the conceptual model may require adjustment following calibration and verification. Uncertainty can enter the conceptual model through a variety of avenues, including incomplete or erroneous data, uncertain boundary or initial conditions, spatial and temporal heterogeneity, upscaling of parameters, or incomplete understanding. Several papers in this special section consider uncertainty related to conceptual models.
Walvoord et al. (2004) examine the effects of boundary and initial condition uncertainty on the paleohydrologic reconstruction of deep, arid unsaturated zones. They model the development of hydraulic, chemical, and isotopic profiles at the Amargosa Desert Research Site. Walvoord et al. find that the timing of a transition from a wet to a drier climate causes the largest uncertainty in model-predicted contemporary flux rates and that their model results are less sensitive to pretransition initial and boundary conditions. Their results suggest that changes in precipitation, temperature, and vegetation were not necessarily coeval.
Sun and Zhang (2004) develop a solute flux approach for determining solute transport statistics in heterogeneous, nonstationary unsaturated media. They define the transport problem in a Lagrangian framework and then derive general forms of solute flux and travel time statistics. As a special case, Sun and Zhang construct one-particle and two-particle joint probability density functions assuming a lognormal distribution for travel time and a normal distribution for transverse displacement. They demonstrate the impact of nonstationary flow on solute transport with several examples.
Zhu et al. (2004) establish hydraulic-parameter correspondence for various hydraulic parameter models and develop averaging rules for upscaling parameters to relatively large scales (e.g., remote sensing footprints). They investigate correspondence between commonly used soil hydraulic conductivity functions and apply their results to upscaling of hydraulic properties for steady flow in heterogeneous soils. Zhu et al. find that hydraulic parameters correspond well and that the same averaging rules can be applied to different conductivity functions when the surface soil suction is large. They also show that correspondence is poor when surface suction is small.
Conceptual model errors can occur when fundamental physical processes are either excluded from the conceptualization or are inadequately described. Two of the papers in this special section focus on fundamental physical processes that are commonly ignored when formulating conceptual models. Berkowitz et al. (2004) review the influence of the capillary fringe on local flow and transport. In contrast to conceptualizations that assume simple vertical flow across the capillary fringe, they conclude that local flow within the capillary fringe may have a substantial influence on chemical migration and microbial populations. Wang et al. (2004) investigate the occurrence of wetting front instability during the redistribution of flow following rapid infiltration. Wetting front instability fragments a smooth wetting front into gravity-driven fingers, which are a form of preferential flow that is not dependent on material heterogeneity and is excluded from most conceptual models for vadose zone flow and transport. Through theory and experiment, Wang et al. confirm the existence of a critical infiltration depth for the onset of unstable redistribution. This result can be used to reduce uncertainty by constraining the occurrence of gravity-driven fingers.
One paper illustrates the need to assess global uncertainty. Hubbell et al. (2004) estimated liquid flux at a large (39-ha) site by combining an approximately 30-mo record of deep tensiometric data (34–73 m below land surface) with laboratory developed hydraulic properties. Flux estimates across the site ranged over more than four orders of magnitude (0.2–8000 cm yr–1). The authors attribute this variability to amplification of uncertainty associated with mapping the tensiometric data onto the estimated hydraulic properties. In particular, they note the potential for extreme sensitivity in the estimated relationships between water potential and unsaturated hydraulic conductivity.
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IMPORTANT RESEARCH NEEDS
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Most existing research on vadose zone uncertainty has focused on various components of the modeling process (e.g., uncertainty due to heterogeneity, parameter estimation), while other important aspects have received much less attention (e.g., implicit conceptualization and global uncertainty). Implicit conceptualization refers to the unstated or inadvertent alteration of the conceptual model through inclusion of subscale models. Global uncertainty is the cumulative uncertainty that occurs within the predictive modeling process and differs from model sensitivity by acknowledging that uncertainties within individual model components (e.g., parameters, observations, conceptual model) may be amplified or attenuated as they are propagated through the modeling process. In the following, we provide a more complete introduction to these important knowledge gaps.
Implicit Conceptualization
The development of a successful conceptual model is founded on unbiased site-specific data. In reality, much of that data will be collected on the basis of an implicit, if not explicit conceptualization. The choice of data to be collected and the manner in which the data are obtained often reflect preconceived notions about what is important. The worst scenario occurs when data is collected in a manner designed exclusively to meet the input requirements of a specific numerical model. In any case, estimation of most unsaturated hydraulic properties involves implicit conceptualization. Typically, observations are made from a perturbed system, and a nonlinear inversion model is used to infer the target parameter (e.g., unsaturated hydraulic conductivity or the pressure–saturation relationship). The inversion model imposes assumptions regarding process, scale, and geometry. For example, time domain reflectometry and neutron absorption are both accepted means for estimating in situ moisture content. However, neither is a direct measure, and both are dependent on a calibrated mapping function that contains specific assumptions. Another example of implicit conceptualization involves the mathematical representation of constitutive relationships. It is common practice to measure pressure–saturation, estimate conductivity on the basis of a Mualem mapping (Mualem, 1976), and then express it according to the van Genuchten relationship (van Genuchten, 1980). While most practitioners implementing this approach will recognize that uncertainty is introduced through sampling and laboratory procedures, a much smaller number will be aware that both the Mualem and van Genuchten relationships are conceptual models.
Numerical solution techniques may also impose an implicit conceptualization through discretization, poor model resolution, and unanticipated numerical artifacts. For example, grid-based finite-difference approaches constrain the problem geometry, encourage representation of material properties with continuous single-valued functions, and are inherently diffusive. The solution technique may also add implicit terms to the numerical formulation (e.g., Eliassi and Glass, 2001). Discrete simulation approaches are often constrained to a regular grid with limited connectivity and follow logical rules that are linked to physical processes through heuristic means rather than mathematical formalism. Each of these computational expediencies has the potential to impose an implicit conceptualization on the simulation outcome. For both continuum and discrete approaches, imposition of an implicit conceptualization that is at odds with the fundamental processes that control flow and transport can lead to additional uncertainty.
Global Uncertainty
Vadose zone flow and transport predictions are used to support decisions made in the presence of uncertainty. The first step in any modeling effort is to define the problem in terms of a specific question to be answered. The second step is to establish an acceptable level of global uncertainty required of the answer. Combined, these two steps are the most critical, yet often most neglected part of the modeling process. Often, the question asked and the acceptable level of uncertainty are driven by external sources (e.g., regulations, politics, economics) and may not be well-matched to either the physical system or potential engineering solutions. The acceptable level of uncertainty may also be poorly defined or may be set to an unrealistically low value. In reality, many questions asked cannot be answered within acceptable levels of uncertainty. If the question posed cannot be answered with sufficient precision, we must define a simpler problem, revise our expectations, or postpone the decision until adequate research reduces the uncertainty. To emphasize the importance of this component, we note that answering the wrong question leads to biased estimates of global uncertainty and, potentially, to catastrophic decisions.
Once the problem is defined, predictive modeling typically follows an iterative stepwise process similar to that shown in Fig. 3. Feedback loops within the process are designed to reduce error and improve predictions, but may not reduce uncertainty. Uncertainty arises in each step and accumulates as information or results are passed from step to step. Because the modeling process is nonlinear and sensitivities vary between steps, these uncertainties may be amplified or attenuated, increasing or decreasing the global predictive uncertainty. For example, small observation and inversion model errors in hydraulic parameter measurements can lead to order-of-magnitude errors in stochastic model predictions when they are propagated through an inversion model, a geostatistical characterization process, and stochastic flow model (Holt et al., 2003).
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CONCLUDING REMARKS
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Ultimately a holistic approach is required to assess the global uncertainty in our predictive models. Our current approach is too disjointed, as researchers often focus on uncertainties inherent to particular aspects of the modeling process (e.g., parameter measurements, parameter estimation, the development of conceptual models for processes, and numerical modeling approaches) and cannot place their work into a global context. We must remember that we are building components of a bigger tool. Eventually we must examine the tool as a whole and ask the question, Can my component be reliably used in the whole? Identifying and quantifying sources of uncertainty in a particular flow and transport problem is a difficult task. New conceptual and quantitative approaches are needed to characterize the magnitude of global uncertainty inherent in the predictions that we make. The papers in this special section represent a significant contribution to our progress toward understanding uncertainty in vadose zone flow and transport predictions.
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ACKNOWLEDGMENTS
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We would like to thank the contributing authors, reviewers, editor, and technical staff of the Vadose Zone Journal who made this collection possible. We would also like to thank J. Kuszmaul whose comments improved this manuscript. M.J. Nicholl would like to acknowledge support from the U.S. Department of Energy through the Basic Energy Sciences Geoscience Research Program under contract number DE-FG03-01ER15122 and the Environmental Management Science Program under contract number DE-FG07-02ER63499. R.M. Holt would like to acknowledge support from the Environmental Systems Research and Analysis program at the Idaho National Engineering and Environmental Laboratory under DOE Idaho Operations Office Contract DE-AC07-99ID13727 BBWI.
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REFERENCES
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- Eliassi, M., and R.J. Glass. 2001. On the continuum-scale simulation of gravity-driven fingers in unsaturated porous media: The inadequacy of the Richards equation with standard monotonic constitutive relations and hysteretic equations of state. Water Resour. Res. 37:2019–2036.
- Helm-Clark, C.M., R.P. Smith, D.W. Rodgers, and C.F. Knutson. 2004. Neutron log measurement of moisture in unsaturated basalt: Progress and problems. Available at www.vadosezonejournal.org. Vadose Zone J. 3:485–492 (this issue).[Abstract/Free Full Text]
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TOP
INTRODUCTION
PAPERS IN THIS SPECIAL...
