Modelling
1.
Summary
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
-
Reporting
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
|
Boundary Type |
Formal Name |
Mathematical designation |
|
Type 1 Specified Head |
Dirichlet |
h (x,y,z,t) = constant |
|
Type 2 Specified Flow |
Neumann |
|
|
Type 3 Head-dependent flow |
Cauchy |
|
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
·
Evapotranspiration
·
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)
Liu, G.,
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
http://water.usgs.gov/pubs/fs/FS-121-97/fs-121-97.pdf
Guidelines and Information on Modflow 2000
http://water.usgs.gov/nrp/gwsoftware/modflow2000/modflow2000.html
Cooley, R.L. (2004): A theory for modeling
ground-water flow in heterogeneous media: U.S. Geological Survey Professional
Paper 1679, 220 p
http://pubs.er.usgs.gov/pubs/pp/pp1679
Environmental Contaminants Encyclopedia
http://www.nature.nps.gov/hazardssafety/toxic/index.html
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.
http://water.usgs.gov/nrp/proj.bib/Publications/nordstrom_epa.pdf
Reilly, T.E. (2001): System and Boundary Conceptualization in
Ground-Water Flow Simulation. Techniques of Water-Resources Investigations of
the
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
http://www.epa.gov/ada/csmos/models.html
United States Geological Survey (USGS) Ground-Water Software
http://water.usgs.gov/software/ground_water.html
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)
http://water.usgs.gov/ogw/pubs/Circ1104/c1104.pdf
6.
Literature
|
Stefan Gödeke