Groundwater Modeling


  1. Introduction

    A significant step in ground water flow and contaminant transport modeling began with advancements in personal computer technology in the early 1980's. In the last ten years, the use of computers in ground water investigations has become decentralized, inexpensive, and fast. Today, ground water modeling and graphics packages are so user-friendly that even non-modelers can easily apply them. One problem, however, is that there is no commonly agreed-upon methodology to evaluate ground water model applications. Faced with decision-making based on model applications in numerous water quality issues, regulatory personnel need guidance for objective model evaluation. This is an approach to evaluation of mathematical ground water model applications for regulators and for model users.

  2. Background

    Ground water models are mathematical representations of ground water systems and include assumptions and simplifications made for various specific purposes. Models were basically developed for the hydrogeological processes of flow, transport, and transformation, and have many specific applications. The purposes of these applications have increased enormously, parallel to the advancements in computer software technology. For example, there are models developed specifically for estimating leachate generation at a waste facility, evaluation of various remedial activities, risk assessment, biodegradation, waste classification, etc.

    Modeling plays a significant role in risk assessment when used for estimating contaminant concentrations for exposure assessment. Ideally, the contaminant concentration values are obtained through field sampling or monitoring. In many cases, however, sampling and monitoring are not feasible options. Also, many risk assessment applications may involve case studies in which the contaminant is released only at the source location, and the concentration values at the receptor points need to be estimated. Modeling is the most effective method to make such estimations involving complexities and uncertainties. Therefore, the validity of the results of any risk assessment effort based on modeling is directly related to the effectiveness of a model application in representing a ground water system.

    A ground water model application can be considered to be two distinct processes (Figure 1): The first process is model development resulting in a software product, and the second process is application of that product for a specific purpose.

    Figure 1: Main steps involving model development and model application.

    Model Development

    Model development involves the following main steps:

    1. Consideration of hydrogeological processes

      This step involves observations of processes which may affect ground water flow and mass balance, contaminant mass balance and transport, and contaminant mass transformation Figure 2.

      Figure 2: Major processes involving contaminant fate and transport in modeling.

    2. Laws and mathematical formulations

      In this step a hydrogeological phenomenon is represented mathematically based on certain simplifying assumptions (Figure 3.A). Examples are Darcy’s Law for ground water laminar flow, Fick’s Law for dispersive transport, and Monod Functions for contaminant transformation by biodegradation. These general expressions represent an infinitesimally small portion of a given system under ideal conditions.

    3. Transfer to differential equations

      Since the general expressions of laws, for example Darcy's Law, can only represent an infinitesimally small portion of a given system, their application to real scale systems necessitates the conversion of these expressions into differential equations in order to consider spatial and temporal variations. Incorporation of these complex variations in space and time yields differential equations (Figure 3.B).

    4. Replacement of differential equations with approximations

      Differential equations must be solved or approximated mathematically in order to convert them into formulas which can be used for hydrogeological calculations (Figure 3.C). In most cases, an exact solution of a differential equation representing a hydrogeological phenomenon is not feasible, and it is better to approximate the solution. Approximations to solutions of differential equations necessitate further assumptions which may include simplifications, truncations, and rounding-off.

      The degree of simplification is dependent on the model developer’s purpose in representing the system's spatial and temporal variations. For example, some models are intended to address the spatial variations of systems such as heterogeneities, anisotropies, as well as temporal variations such as transient regimes. In these cases, the model developer must use very complex numerical approximations (Figure 3.C). Numerical models divide ground water systems into smaller and hydrologically representable units by a method called spatial discretization. They also divide the time period for model simulation into smaller time segments to address temporal variation in the ground water systems.

      If, however, the model is intended for preliminary level simulation of simple systems without spatial and temporal complexities, the model developer can use analytical approaches (Figure 3.C) or fugacity approach. Analytical approaches are essentially exact solutions of differential equations through reliance on certain simplifying assumptions. For example, if the model developer assumes that boundaries of the ground water system are located in infinitely remote areas and that the ground water system is homogeneous and isotropic, the model user may analytically resolve the pertinent differential equations to form simple equations for flow and contaminant transport. Similarly, if the model developer assumes that environmental media, including ground water, are made up of homogeneous and isotropic compartments and that each compartment can be simulated by a single equation, the model user may adopt fugacity approach to simulate the fate and transport of a contaminant through the interfaces between environmental compartments.

      Following the adoption of an approximation, a computer programmer converts the resulting formulas into computer codes, resulting in software. After the software is developed, the model should be supported by empirical validation, and made available to the market with appropriate documentation of its theoretical basis and with instructions for use.

      Figure 3: Main steps in model development.

    Model Application

    The second process is the application of the models developed to ground water systems for specific purposes of a model user. This process is quite variable and dependent on the purpose and operational modes of the user. In general, a model application includes the formation of a conceptual model, model selection, data collection, model run, interpretation, and post-audit processes.

    1. Conceptual Model

      The conceptual model (Figure 1) consists of statements regarding the model application’s specific objective, and the hypothesis which explains how this objective is to be achieved. Formulation of a realistic and acceptable conceptual model is the most significant step in a model application. Without an acceptable conceptual model, a model user cannot simulate a ground water system and cannot achieve the objectives of modeling regardless of sophistication of the model selected and data collected.

      The following example is to explain the significance of adopting an acceptable conceptual model. Assume that the owner/operator of the hypothetical property shown in Figure 4 would like to demonstrate that the disposal of a waste at an unlined landfill (location shown as contaminant source in Figure 4) would not pose a threat to ground water quality . If the property owner/operator attempts to achieve this objective by demonstrating through modeling that a contaminant derived from the waste will be subject to 100-fold attenuation in the vadose and saturated zone, this approach would not be an appropriate one because the hypothesis that there will be 100-fold attenuation does not always ensure that the waste would not pose a threat to ground water quality. In California, such disposal even when a 100-fold attenuation is assumed to exist may not be in compliance with the applicable water quality standards. Appropriate water quality standards, given in the Regional Water Quality Control Boards’ Water Quality Control Plans must be the basis for any such demonstration.

      Figure 4:
      A conceptual site with a contaminant source
      (no specific scale)

    2. Model Selection

      The second major step, model selection (Figure 1) involves the review and selection of one or several of the models available on the market that best fits the conceptual model according to the user’s specific needs and according to the complexities of the system. Table 1 shows the general types of models with their ability to represent ground water systems and data needs. A more thorough list of different models with their capabilities and limitations is given by van der Heijde and Elnawawy (1993).

      • Space is divided into blocks or other geometric units by spatial discretization.
      • Each unit is given a node.
      • Each node is treated as a separate subsystem in order to incorporate spatial variability in model-related parameters.
      • Both steady-state and transient cases and their combinations can be simulated through separately calculated time steps.
      • Each discrete unit is assigned particular parameter values.
      • Data need is very high.
      • Temporal variations in hydraulic heads, and boundary conditions for flow and contaminant must be specified.
      • Simple analytical models treat entire system as a single unit.
      • Semi-analytical and analytical-element models divide systems into hydrologic units.
      • Simple analytical models treat the entire time period for simulation as a single time step.
      • Semi-analytical and analytical element models can divide simulation into multiple time steps.
      • Varies depending on the level of discretization of space or time.
      • Much more simplistic than numerical models.
      • The environment is divided into environmental media as compartments.
      • Each compartment, including ground water compartment, is treated as a single unit.
      • Temporal variations may be simulated only among the compartments, not within them.
      • For example, transient conditions occurring at the boundary between vadose zone and saturated zone can be simulated, but transient conditions within each zone cannot be simulated.
      • Little, if any, actual data are needed.
      • Uses generic or default data supplied by the model.

      There are many different types of models available to simulate different ground water systems for various purposes. Selection of an appropriate model is the second important factor in success of model application. Model selection must be based specifically on the purpose of modeling and complexity of the system, not on the availability of data (Konikow, 1986). As an example of the significance of model selection, assume that the property owner/operator mentioned above has an acceptable objective and hypothesis, namely the estimation of the fate of the contaminant through natural transport and transformation processes. The next step is to select an appropriate model for ground water simulation.

      The selection criteria for a model which can successfully simulate the contaminant fate and transport will depend on the complexity of the ground water system. For example, the ground water systems shown in Figures 5.A through 5.D are significantly different from each other with respect to the complexity of their hydrologic and hydrogeologic features. In the case shown in Figure 5.A, the system consists of a single water table aquifer which is fairly homogeneous and isotropic. The flow regime has a non-horizontal component,as well as the predominant horizontal component. If the system is expected to be in steady-state condition for the simulation time period, a simple analytical model can be used to simulate a contaminant plume which may form at shallow portions of the aquifer. If, however, transient conditions are expected (e.g. change into conditions similar to those shown in Figure 5.B), the use of more advanced semi-analytical or analytical element models or numerical models may be necessary. Ideally, a model application which involves three-dimensional simulations, especially for transient conditions, should be carried out by using numerical models. Cases involving highly complex systems with heterogeneities, anisotropies and multi-aquifer systems coupled with transient conditions in hydrologic regimes and boundary conditions (Figures 5.C and D) should only be simulated through numerical models.

      Figure 5: Complexity of ground water systems
      A: Hydrogeologically simple system (homogeneous and isotropic conditions)

      B: Same system in Figure 5.A going through transient hydrologic conditions

      C: Hydrogeologically complex system (heterogeneous and anisotropic conditions)

      D: Same system in Figure 5.C going through transient hydrologic conditions

    3. Data collection and Model Run

      After model selection, the model user incorporates data into the model, runs the model, interprets the model results, and conducts post-audit processes which may include calibration, sensitivity analysis, model re-runs, and uncertainty analysis (Figure 1).

      Data needs for a model application will depend on the purpose of model application and complexity of the ground water system. If the model application's purpose is only to serve as a preliminary screening level investigation for contaminant fate and transport and the modeled system consists of a single aquifer which is homogeneous, isotropic, and under steady-state conditions, as shown in Figure 5.A, data needs are minimal. However, if the ground water system indicates transient conditions within the simulation time period (Figure 5.B, C and D), the data needs would significantly increase due to changes in hydraulic heads (for 2-dimensional simulations) or potentiometric surfaces (for 3-dimensional simulations), and boundary conditions. Simulation of ground water systems having spatial and temporal variability demands increased amounts of data. The data quantity and frequency depend on the degree and types of heterogeneities and anisotropies, and frequency of transient conditions, as well as the specific purposes of modeling. For example, the data needs for simulating the single aquifer in Figure 5.A would be much less than that for simulating the multiple aquifer system shown in Figure 5.C and D due to stratifications and different hydrologic units. While the model user may only need a single set of samples to determine and demonstrate the system parameters and their gradients for the case shown in Figure 5.A, the model user would have to increase the quantity and frequency of sampling three-dimensionally for the case shown in Figure 5.C and D

  3. State Water Resources Control Board staff's approach

    The State Water Resources Control Board (SWRCB) has not issued any specific regulatory requirements for model development, model application, or model evaluation, nor do we specify models. We use the general methodology for the evaluation of ground water model applications presented above, and we have provided training to the Regional Water Quality Control Boards (RWQCBs). The SWRCB and RWQCBs have been using this approach to evaluate ground water models applications for demonstrations by dischargers.

    We recognize that because most ground water model applications are relevant to approximating uncertainties in contaminant fate and transport, validation and verification of these models may not always be possible, as was addressed by Oreskes et al. (1994), Bredehoeft and Konikow (1993), and Schwartz et al. (1990). Also, although useful for parameter estimation in model applications, calibration is not an absolute tool for model evaluation, as addressed by Konikow (1986), Schwartz et al. (1990) and Oreskes et al. (1994). Instead of relying on calibration, this approach to model evaluation is based on the application’s ability to adequately represent the ground water system for the purpose of model application. Such evaluation involves an assessment of the adequacy of the steps in a model application. Adequacy is considered to be technical ability of the model application to achieve the stated objective which ensures compliance with applicable policy, statute, and regulations.

    There may be other methods for evaluation of ground water model applications. If you know any such methods or have questions or concerns about the contents of this section and the method described in the article listed above, please contact us to convey your opinion.

  4. Conclusions

    • SWRCB regulations and policies do not specifically address ground water modeling or evaluation of groundwater model applications. The SWRCB and RWQCB have evaluated the adequacy of a ground water model applications using a stepwise approach regarding:
      • Conceptual model;
      • Model selection; and
      • Data collection and model run.
    • We are open to suggestions regarding other methods for the evaluation of ground water models.
  5. Literature Cited

    • Bredehoeft, J.D., & Konikow, L.F., 1993, Groundwater models: Validate or invalidate. Ground Water, v.31, n.2.
    • Cleary, R.W., 1994, Groundwater pollution and hydrogeology. The Princeton Groundwater Course.
    • Fetter, C.W., 1993, Contaminant hydrogeology. Macmillan Publishing Company.
    • Freyberg, D.L., 1988, An exercise in groundwater model calibration and prediction. Ground Water, v.26, n.3.
    • Konikow, L.F., 1986, Current issues predictive modeling of ground water. U.S. Geological Survey Newsletter, p.4.
    • Gelhar, L.W., Welty, C., & Rehfeldt K.R., 1992, A Critical Review of Data on Field-Scale Dispersion in Aquifers. Water Resources Research, v.28, no.7, p.1955-1974.
    • Oreskes, N., Shrader-Frechette, K., & Belitz K., 1994, Verification, validation, and confirmation of numerical models in the Earth Sciences, Science, v.263, p.641-645.
    • Schwartz, F.W. & others (Committee on Ground Water Modeling Assessment), 1990, Ground water models scientific and regulatory applications. National Academy Press, Washington, D.C.
    • Smith, L. & Schwartz, F.W., 1980, Mass Transport, 1. A stochastic analysis of macroscopic dispersion. Water Resources Research, v.16(2), p. 303-313.
    • van der Heijde, P.K.M., images/spacer.gif O.A., 1993, Compilation of groundwater models. EPA/600/R-93/118.