Using mathematical models to assess sediment stability


  • C Kirk Ziegler

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    1. Quantitative Environmental Analysis, 305 West Grand Avenue, Montvale, New Jersey 07645, USA
    • Quantitative Environmental Analysis, 305 West Grand Avenue, Montvale, New Jersey 07645, USA
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The application of mathematical models to a sediment stability study is presented, with an emphasis on using models as a component of an effort to develop, refine, and potentially validate a conceptual site model (CSM) for sediment transport at a study site. The utility of mathematical models is discussed, in which the modeling framework consists of linked hydrodynamic, sediment transport, and contaminant fate and transport models. Benefits and drawbacks of empirically based mechanistic models are presented. An approach for integrating modeling analyses into the development, refinement and validation of a CSM is provided. This approach focuses on a phased study that combines modeling and data-based analyses to test hypotheses related to the CSM and sediment stability. Uncertainty in modeling results is a primary concern in sediment stability studies, and issues related to model uncertainty are discussed. Finally, communication of modeling results to stakeholders is addressed.


This paper is among 9 peer-reviewed papers published as part of a special series, Finding Achievable Risk Reduction Solutions for Contaminated Sediments. Portions of this paper were presented by the author at the Third International Conference on Remediation of Contaminated Sediments held in New Orleans, Louisiana, USA in January 2005.


The focus of a remedial investigation at a contaminated sediment site is typically on the efficacy of various remedial alternatives to reduce ecological and human health risk. At many sites, ecological and human health risk is related to contaminant concentrations in the bioavailable surficial layer of the sediment bed. Sediment transport processes, such as erosion and deposition, have a significant impact on surficial bed concentrations of particle-reactive chemicals such as polychlorinated biphenyls (PCBs) and heavy metals. In particular, significant reductions in contaminant discharges have occurred at many sites since the 1970s. Reductions in active contaminant loading at many sites during the last 30 to 40 y, combined with sediment deposition and erosion processes during that period, have resulted in peak contaminant concentrations typically being buried below the bioavailable layer at a number of locations. Thus, the potential exists for erosional processes to expose the peak concentrations buried in the bed to the bioavailable layer and, hence, increase ecological and human health risk. Therefore, the physical stability of the sediment bed is an issue that must be considered when various methods for risk reduction at a site are evaluated.

The focus of a sediment stability study is to investigate the effects of sediment transport processes on contaminant concentrations in the bioavailable layer of the bed. Because of the complexity of this type of analysis, a weight-of-evidence approach has proven useful. This approach uses multiple lines of evidence, developed from the results of data-based and modeling analyses, to test various hypotheses about sediment stability and develop site-specific conclusions (Figure 1). Typically, the central focus of the investigation is to answer a series of questions related to sediment stability, such as the following:

  • What is the impact of a rare storm on surficial contaminant concentrations?

  • What is the impact of sediment transport processes on the natural attenuation rate?

  • What effects will various remedial alternatives have on the impacts of a rare storm? Conversely, what is the impact of a rare storm on the effectiveness and permanence of proposed remedial alternatives?

This methodology has proven to be useful at a number of contaminated sediment sites (Ziegler 2002). Recent work, however, has indicated that an improved understanding of sediment stability may be achieved by making the conceptual site model (CSM) for sediment transport the focal point of the study. This concept will be explained in more detail below, but the general idea is to develop testable hypotheses from the components of the CSM and use a weight-of-evidence approach to test the validity of those hypotheses; refining and potentially validating a CSM for sediment transport becomes the primary focus of the sediment stability study. The use of a CSM, and associated hypothesis testing, in a contaminated sediment study is a well-established concept that has been discussed by numerous authors. In addition, the US Environmental Protection Agency (USEPA) has incorporated the CSM methodology into various documents related to contaminated sediment site investigations (USEPA 2000, 2002).


Mathematical models have been used to investigate sediment stability and contaminant fate and transport, at a large variety of sites, including Upper Hudson River, New York, USA (Connolly et al. 2000, Ziegler et al. 2000); Grasse River, New York, USA; Watts Bar Reservoir, Tennessee, USA (Ziegler and Nisbet 1995); and Lavaca Bay, Texas, USA (HydroQual 1998). The general structure of a modeling framework consists of the following submodels: hydrodynamics, sediment transport, and contaminant fate and transport (Figure 2). Additional submodels may be needed for sitespecific conditions, such as a wind-wave model for a shallow bay or a bank erosion model for a river.

Figure Figure 1..

Overview of weight-of-evidence approach for sediment stability evaluation.

The sophistication of a mathematical model varies in spatial resolution (1-, 2-, or 3-dimensional) and in the representation of physical and chemical transport processes, as well as the complexity of the modeling framework. In some cases, simply the use of a hydrodynamic model to evaluate the spatial distribution of bottom shear stress during a rare storm may be adequate for a sediment stability analysis. In other cases, it may be necessary to apply the full suite of 3 submodels (hydrodynamics; sediment transport; contaminant fate and transport). When application of a sediment transport model is necessary during a sediment stability study, the structure of that model is of critical importance. Development of a reliable sediment transport model requires that a mechanistic approach be used to formulate erosion and deposition processes. A range of mechanistic formulations, based on laboratory and field data, have been developed to predict the erosion and deposition of cohesive (i.e., flocculating material composed of clay, silt, and organic material) and noncohesive (i.e., sand and gravel) sediments (see Ziegler et al. 2000).

Mathematical models have strengths and weaknesses, which must be considered when considering the utility of a model during a sediment stability study. The benefits of a mathematical model include (1) models can be used to constrain, synthesize, and interpret data; (2) physical and chemical transport processes, some of which cannot be easily measured, can be quantified; (3) quantitative predictions can be realized with a model; and (4) insights about processes affecting sediment stability are developed during the modeling process. The last point is important because even if a modeling study “fails” because of the inability to successfully calibrate the model (which will probably be caused by insufficient site-specific data), it is likely that significant insights about sediment stability will be gained during the model development phase.

The drawbacks, or weaknesses, of a mathematical model include (1) potentially high cost, (2) field studies and data collection needed to support model development and calibration, and (3) the level of uncertainty in model results or predictions may be unacceptable to stakeholders and decision makers. Although the cost of a modeling study is frequently viewed as a potential drawback, application of a model may actually be cost-effective and result in overall project savings, because a model can provide a context for data collection efforts and be used to efficiently use sparse data sets. In most modeling studies, uncertainty of model predictions is the key issue regarding model reliability and utility at a particular site; reducing model uncertainty to a level that is acceptable to all stakeholders can be a challenge. Additional discussion with respect to model uncertainty is provided later in this article.

Figure Figure 2..

General description of mathematical modeling framework.

Because uncertainty exists in modeling results, a question that is often asked by stakeholders and decision makers is the following: Can a useful model be developed and applied at contaminated sediment site? The emphasis is on “useful model” because the utility of a model rests in the ability to reliably use the model as a management tool to guide decisions that are related to sediment stability. Studies conducted at a variety of contaminated sediment sites (e.g., Hudson River, New York, USA, and Lavaca Bay, Texas, USA) demonstrate that useful models can be developed provided that sufficient site-specific data are available for model development, calibration, and validation. In addition, the capabilities of the modeling team are important because the successful development and application of a model requires experience in the areas of numerical analysis, hydrodynamics, sediment transport, and contaminant fate and transport.


Mathematical modeling efforts frequently become the focus of a sediment stability study, primarily because of interest in quantitative predictions of the model. Overreliance on the model must be avoided; do not rely solely on a mathematical model. A preferred approach, which will produce more reliable conclusions about sediment stability, is to use a methodology that consists of 3 interrelated components: (1) CSM for sediment transport, (2) data-based analyses, and (3) mathematical modeling (Figure 3).

As mentioned earlier, the CSM is the focal point of the study. A CSM provides a qualitative understanding of the key processes that control sediment stability within the study area (Figure 4). In addition to erosion and deposition processes, a CSM may incorporate the following components (depending on site conditions): geomorphology of the system, sediment loads from upstream and tributary sources, bank or shore erosion, overbank flow in a river, and hydrodynamic processes that affect bottom shear stress (i.e., currents and waves). An important use of the CSM is that it integrates and synthesizes the results of data-based and modeling analyses so that one can “tell a story” about sediment stability for the system being studied. In addition to being used as a technical tool, the CSM is an effective method for communicating the results of a sediment stability evaluation to stakeholders and decision-makers, which is discussed later in the article.

Conducting a sediment stability study may sometimes appear to be an overwhelming undertaking with a high level of economic risk. Problems that arise on some studies include the following: study objectives and endpoints are not clearly defined, work plans are not efficient, extraneous data are collected, irrelevant analyses are performed, and the weight-of-evidence approach is misunderstood and improperly applied.

Many of these problems can be overcome by using the CSM for sediment transport as the focal point of the study; refinement and potential validation of the CSM is the primary study objective. The basic procedure is to develop hypotheses from the various components of the CSM. Potential validation of the CSM is accomplished by testing each of the hypotheses by use of a weight-of-evidence method in conjunction with results of the data-based and modeling analyses. In addition, the CSM hypotheses can be used to guide the design of field studies and quantitative analyses, resulting in efficient resource utilization. An important question to ask during the design of each study component is the following: What types of data and analyses are needed to test each CSM hypothesis? Application of the data quality objective process developed by USEPA (2000) to the testing of CSM hypotheses testing, which may involve a combination of data-based and modeling analyses, may provide a useful framework that improves the quality of the work product and reduces uncertainty in sediment stability conclusions.

Figure Figure 3..

Three-component framework for assessing sediment stability.

Another beneficial use of a CSM concerns development and application of quantitative models during a sediment stability study. Frequently, this type of study will be conducted in phases, as discussed below. As the phases of the study progress, the CSM can be used to guide the development of models. For example, in the early stages of a study, a provisional CSM may be developed from available data and then used to guide the development of screening-level models. As the study progresses, the CSM may be refined and used to aid in the development and application of more sophisticated models. This iterative process, which tightly couples the CSM and the mathematical model of the study area in a bidirectional mode as shown on Figure 3, provides an effective method for evaluating sediment stability at a site and reducing uncertainty in study conclusions.

In addition to focusing on refinement and potential validation of the CSM, project risk can be reduced by conducting the study in phases. A phased approach to a sediment stability investigation will potentially (1) provide flexibility, (2) reduce project risk, and (3) increase efficiency and cost-effectiveness. During each study phase, analysis results, whether from data-based or modeling analyses, are used to refine the CSM. A decision point is reached at the end of each phase of the study, with this question to be addressed at that time: Can the CSM be validated? If this question is answered affirmatively, then the study is concluded. If the CSM cannot be validated at the end of the current phase, the project proceeds to the next phase of the study, in which additional work is conducted to provide further refinement of the CSM. The ongoing refinement of the CSM during the various study phases makes it possible to answer progressively more complex management questions with increasing certainty, even though complete validation of the CSM may not be achieved by the end of the project.

To illustrate the phased approach, an outline for a 3-phase sediment study is presented. The example study consists of 3 phases.

  • Phase 1—Compile, analyze and synthesize available data; develop provisional CSM.

  • Phase 2—Collect additional data and conduct related data-based analyses; develop hydrodynamic model and use to evaluate potential for bed scour during rare storms; refine CSM.

  • Phase 3—Develop and apply sediment transport model; collect additional data, if needed; refine and potentially finalize CSM.

Figure Figure 4..

Example graphical illustration of a conceptual site model.

Note that the example provided here should not be considered a blueprint for studies at all sites; it is presented to demonstrate the process. Each sediment stability study is unique; the phased approach can be adapted to site-specific conditions and project goals.

The 1st task in phase 1 is to compile, analyze, and synthesize available data. Although available data sets may be relatively sparse at the beginning of the study, the data are probably sufficient to develop a provisional CSM for sediment transport, which is the main goal of this phase. After development of the provisional CSM, testing of sediment stability hypotheses based on components of the CSM is conducted by use of appropriate methods. At the end of phase 1, the decision-point question is asked: Can the CSM be validated? Based on the answer to this question, a decision to proceed or not proceed to the next phase is made.

If the CSM can not be validated during phase 1, then additional data will probably need to be collected during phase 2. Hypotheses from the CSM are used to guide the design of the field studies conducted during this phase. Types of field studies that are typically considered include radioisotope dating of sediment cores (geochronology analysis); multibeam bathymetry, side-scan sonar, and subbottom profiling (bed properties and geomorphology); and site-specific bed erosion properties (e.g., SED flume study). Data from these types of studies can be used to gain insights about geomorphological processes, sedimentation and natural recovery rates, spatial distribution of bed types and properties, and erosional/depositional environments within the study area.

Experience at a number of sites has shown that developing and applying a hydrodynamic model is a useful 1st step in a sediment stability modeling analysis. Although it is proposed that a hydrodynamic model be applied during phase 2 in this example, note that it may be fruitful to develop a hydrodynamic model during phase 1 in many studies. The focus of the hydrodynamic modeling is on the spatial distribution of bottom shear stress throughout the study area, during low-energy periods (e.g., average flow conditions on a river), as well as during high-energy events (e.g., floods or storms) when significant erosion and deposition may occur. Understanding temporal and spatial variability in bottom shear stress is important because this hydrodynamic parameter is a controlling factor for erosion and deposition. An example of the utility of this type of hydrodynamic analysis is to use the bottom shear stress distribution to develop inferences about locations of potential bed scour during a high-energy event; areas where the bottom shear stress exceeds the critical shear stress for erosion are delineated as regions where bed scour may occur.

Results from the data-based and modeling analyses conducted during phase 2 are used to refine the CSM and test the hypotheses. Similar to phase 1, the decision point at the end of phase 2 determines whether or not sufficient evidence exists to validate the CSM. If the CSM is validated, then the study is terminated. The study proceeds to phase 3 if the CSM cannot be validated.

Phase 3 entails the development, calibration, and validation of a sediment transport model, which will provide a quantitative method for evaluating sediment stability over a range of temporal and spatial scales. Typically, development of a sediment transport model will require additional data collection programs, which may include (if not conducted during phase 2) bed erosion properties, side-scan sonar delineation of bed type, sediment loading from upstream and tributary sources, bed-load data, bed elevation changes, and suspended sediment concentrations at various locations within the study area. On completion of the calibration and validation process, the model is used to investigate sediment stability over a range of temporal and spatial scales, from short-term, high-energy events (e.g., a 100-y flood) to long-term, decadal periods. Results from the modeling analyses, in conjunction with analysis of phase 3 data and the results of phases 1 and 2, are used to refine and potentially finalize the CSM. It is possible that the refined CSM may not be completely validated at this point in the study; sufficient understanding of sediment transport processes, however, in the study area may have been achieved so as to adequately address the various management issues. Note that auxiliary site-specific questions related to sediment stability (see Introduction) are also addressed before completion of the study.


As noted earlier in this article, a primary concern with the reliability of mathematical models is the level of uncertainty associated with model predictions. It must be remembered, however, that uncertainty is not limited to modeling analyses; data-based analyses are subject to uncertainty also. Inherent uncertainty in sediment stability analyses, both data-based and modeling, makes it imperative that multiple lines of evidence are used to develop conclusions because this approach tends to reduce the uncertainty that is associated with any single analysis.

Optimized use of a mathematical model requires strict adherence to the scientific method as well as explicit recognition of and accounting for uncertainty. Sources of uncertainty in a modeling analysis include (but are not limited to) (1) temporal and spatial variability in site data, (2) spatial and temporal resolution of the model (e.g., numerical grid resolution), (3) algorithms or formulations used to represent physical and chemical transport processes, (4) site-specific model parameters, (5) boundary conditions, and (6) extrapolation to conditions beyond limits of model calibration and validation (e.g., 100-y flood simulation).

Accounting for model uncertainty can be accomplished through a bounding-estimate analysis that is constrained by the calibration and validation data. The basic approach for a bounding-estimate analysis consists of 3 steps. First, quantify or estimate uncertainty in site data and model inputs. The next step is to establish upper- and lower-bound estimates of critical or controlling model parameters and inputs. When the upper- and lower-bound parameters are established, care must be taken to ensure that the bounding estimates represent a realistic range. Arbitrary adjustment of parameters beyond values that are consistent with site-specific conditions must be avoided. An indication that realistic bounds are exceeded is that the bounding parameters produce results that significantly deviate from the original calibration simulation. Constraining the bounding parameters through the calibration data sets is important because failure to do so may produce bounding-estimate simulations that significantly overestimate model uncertainty. The 3rd step is to perform upper- and lower-bound simulations for the various scenarios being considered in the modeling study.

The results of the bounding-estimate simulations are presented as a range instead of single result (either a specific number or spatial/temporal distribution that is produced by the conventional approach). With the bounding-estimate approach, the true answer presumably resides within the upper- and lower-bounds of the range; the bounding simulations represent the realistic range of model results. The bounding-estimate results are used as lines of evidence in the sediment stability analysis. Thus, this approach increases the reliability of sediment stability conclusions because the effects of uncertainty are explicitly incorporated into the modeling analysis.


Mathematical modeling of hydrodynamics, sediment transport, and contaminant fate and transport can be quite complex because it incorporates a diverse range of technical disciplines. Typically, a long and steep learning curve is associated with modeling; it can take a number of years of intense study and training for a person to develop into a competent modeler. The complexity of modeling, as well as the long learning curve, poses a significant hurdle for the most important step in a sediment stability study: communication of results to stakeholders and decision-makers. Without effective communication of modeling and data-based analysis results, the utility of a sediment stability study is minimized or negated.

A primary impediment to effective communication is the lack of familiarity that many stakeholders and decision-makers have with modeling. As might be expected, being unfamiliar with a complex technical area can create uncertainty and distrust in the results of the various analyses. Overcoming misunderstanding, uncertainty, and distrust of modeling needs to be the main goal for modelers attempting to communicate modeling results to stakeholders.

To develop confidence in a modeling study, stakeholders must be incorporated into the model development process from the earliest stages of the project. Frequent communication, both informal and formal, between the modeling group and stakeholders must occur throughout all phases of the study. Interactions between the modelers and stakeholders are necessary so that the stakeholders understand, or are at least apprised of, the technical basis for various decisions made during the model development and calibration process. The stakeholders need to be engaged in the modeling process throughout the entire project. Frequent and steady engagement with the modeling group will increase stakeholder familiarity and understanding, while decreasing their level of distrust and uncertainty. The end result will be clearer communication channels between the modelers and stakeholders.

Although incorporating stakeholders into the modeling process is helpful, the ultimate responsibility for effective communication still resides with the modeling group. Communicating the results of a sophisticated modeling analysis to a group of stakeholders, some of whom may have limited technical backgrounds, can be a daunting task. An approach that may prove useful is to utilize the CSM to communicate model results. By focusing on the CSM, the modeling group should be able to tell a story about sediment stability in the study area that all of the stakeholders can understand. As discussed earlier in this article, the CSM and mathematical model should be developed and refined by use of an iterative approach that stresses the dependency between these 2 components of the assessment process (Figure 3). This iterative approach, wherein the CSM and model are progressively refined during the various phases of the study, should be incorporated into the communication process so that stakeholders can better understand the development and application of the model.

Model results, through simple and clear graphics, are used to support various components of the CSM. Three-dimensional graphical plots and animation of model results are 2 methods that can be very effective for presenting model results. In the end, the ideal situation is the development of a “conversation” between the modelers and stakeholders in which the focus is not on model details or specific results but rather on sediment stability issues within the study area.