The term modelling in this context refers to the use of computer-based numerical methods to obtain approximate solutions to the coupled equations of groundwater flow and solute transport. Groundwater flow simulations require an understanding of geology and the hydraulics of groundwater flow as well as a command of numerical simulation methods. When solute movement is to be simulated, the complexity of the problem is increased. The description of the flow regime may not have the resolution needed to support transport analysis; chemical, physicochemical and biochemical mechanisms must be represented in the governing equations.
2. Model Application Process
A numerical model is a representation of the real world by discrete volumes of materials. These volumes are called cells in the finite difference method and elements in the finite element method (see below).
The first step in the model application process is the definition of the problem during a thorough investigation. Literature has to be reviewed, data has to be collected and preliminary analyses have to be made.
The subsequent steps are:
- Development of a conceptual model
- Selection of a numerical code
- Assignment of properties and boundary conditions to a grid
- Calibration and sensitivity analysis
- Model execution and interpretation of results
In order to determine a unique solution to the second order partial differential equation governing the flow of fluid through porous media, boundary conditions have to be defined. They have great influence on the computation of flow velocities and heads within the model area. Three types of boundary conditions are commonly specified (Table 1).
Table 1: Common boundary conditions
h (x,y,z,t) = constant
constant (where c is also a constant)
The USGS has published a chapter on “System and Boundary Conceptualization in Ground-Water Flow Simulation” (Reilly 2001).
In groundwater modelling the three most common solution methods are: analytical, finite difference and finite element. Each method solves the governing equation of groundwater flow and storage but differ in their approaches, assumptions and applicability to real-world problems (US Army Corps of Engineers 1999).
(1) Analytical Methods:
Analytical methods use classical mathematical approaches to resolve differential equations into exact solutions. They provide quick results to simple problems and require assumptions of homogeneity.
(2) Finite difference Methods:
Finite difference methods solve the partial-differential equations by using algebraic equations to approximate the solution at discrete points in a rectangular grid. The points in the grid, called nodes represent the average for the surrounding cell. The grid can be one-, two- or three-dimensional. Many codes, such as MODFLOW use the finite difference solution method. The overall size of the grid (i.e. total number of nodes) should be adequate to define the problem but not so large as to cause excessive run preparation and computation requirements.
(3) Finite element Methods:
The solution for each element between adjacent nodes is defined by means of a "basis function". The function actually serves as a spatial interpolation funtion between the calculated heads at the nodal points. The finite element codes allow for flexible placement of nodes which can be important during the definition of irregular boundaries. FEMWATER is a common code using the finite element solution method.
In three dimensional models, different model layers allow for the simulation of flow in separate hydrographic units, leakage between aquifers, and vertical flow gradients. If there are significant vertical head gradients, two or more layers should be used to represent a single hydrostratigraphic unit (Anderson and Woessner 1992).
Numerical errors can be introduced depending on the aspect ratio of the cells. The aspect ratio is the maximum dimension of a block or element divided by the minimum dimension. An aspect ratio of one is usually ideal for minimizing numerical errors.
Before the beginning of the simulation, values need to be defined for the dependent variables (initial conditions). For steady state models (no time variation), initial conditions need only approximately match the natural system because the solution for each dependent node can be found eventually through repeated iteration. In contrast, transient models (time variation included) require initial conditions closely matching natural conditions at the beginning of the simulation.
The term "time-stepping" refers to the discretization of the flow equation through time and is used in transient simulations.
During model calibration potentiometric surfaces (represented by groundwater heads) or concentration values are compared with field measurements. A common method for model calibration is manual trial and error. This method of calibration is labor intensive. Depending on the type of the modelling project, automated calibration can be used. This method uses an objective function, such as minimization of the sum of the squared differences between observed and computed heads, to govern an automatic iterative adjustment of values.
During the simulation process the extent of the model, the conceptualization of the flow system and mathematical representation of the boundaries has to be checked and evaluated. The USGS has published guidelines for evaluating ground water-flow Models (Reilly, Harbaugh 2004).
After calibration a sensitivity analysis can be performed. A sensitivity analysis is a quantitative evaluation of the influence on model outputs from variation of model input parameters. It can be used to aid in model construction by identifying inputs requiring more definition.
3. Data Needs and Requirements
Site specific Information has been gained on:
· Unconfined and confined aquifers - Ground-water flow and storage changes
· Faults and other barriers -- Resistance to horizontal ground-water flow
· Fine-grained confining units and interbeds
· Confining units - Ground-water flow and storage changes
· Rivers - Exchange of water with aquifers
· Drains and springs - Discharge of water from aquifers
· Ephemeral streams - Exchange of water with aquifers
· Reservoirs - Exchange of water with aquifers
· Recharge from precipitation and irrigation
· Wells - Withdrawal or recharge oat specified rates
Information on key model input parameters are also provided in ASTM (1999).
4. Weblinks and Guidelines
ASTM (1999): RBCA Fate and Transport Models: Compendium and Selection Guidance (http://www.epa.gov/oust/rbdm)
Harbaugh, A.W., Banta, E.R., Hill, M.C., and McDonald, M.G. (2000), MODFLOW-2000, the U.S. Geological Survey modular ground-water model -- User guide to modularization concepts and the Ground-Water Flow Process: U.S. Geological Survey Open-File Report 00-92, 121 p. ftp://water.usgs.gov/pub/software/ground_water/modflow/doc/ofr00-92.pdf
USGS Modflow Factsheet
Guidelines and Information on Modflow 2000
Cooley, R.L. (2004): A theory for modeling ground-water flow in heterogeneous media: U.S. Geological Survey Professional Paper 1679, 220 p
Environmental Contaminants Encyclopedia
Nordstrom, D.K. (2002): Aqueous redox chemistry and the behavior of iron in acid mine waters, in Wilkin, R.T., Ludwig, R.D., and Ford, R.G., eds, Proceedings of the Workshop on Monitoring Oxidation-Reduction Processes for Ground-water Restoration, Dallas, Texas, April 25-27, 2000: Cincinnati, OH, U.S. Enviromental Protection Agency, EPA/600/R-02/002, p. 43-47.
Reilly, T.E. (2001): System and Boundary Conceptualization in
Ground-Water Flow Simulation. Techniques of Water-Resources Investigations of
US Army Corps of Engineers (1999): Groundwater Hydrology. Engineer Manual, 1110-2-1421
Reilly, T.E., Harbaugh, A.W. (2004): Guidelines
for Evaluating Ground-Water Flow Models. Scientific
Investigation Report. 2004-5038.
5. Available Groundwater-Software, Web links
Center for Subsurface Modeling Support (CSMOS)- Free Public Domain Ground-Water and Vadose Zone Models
United States Geological Survey (USGS) Ground-Water Software
EPA’s Online Course: Modeling of Subsurface Transport of Petroleum Hydrocarbons http://www.epa.gov/ATHENS/learn2model/index.html
Summary of selected computer programs produced by the USGS for simulation of Ground-Water Flow and quality (1994)