C. P. Kumar
Scientist ‘F’
C. P. Kumar
Scientist ‘F’National Institute of Hydrology
Roorkee – 247667 (Uttaranchal)
India
Email: cpkumar@yahoo.com
Webpage: http://www.angelfire.com/nh/cpkumar/
Presentation Outline
Groundwater in Hydrologic Cycle
Why Groundwater Modelling is needed?
Mathematical Models
Modelling Protocol
Model Design
Calibration and Validation
Groundwater Flow Models
Types of Terrestrial Water
Types of Terrestrial Water
Ground water
Ground water
Soil
Soil
Moisture
Moisture
Unsaturated Zone / Zone of Aeration / Vadose (Soil Water)
Pores Full of Combination of Air and Water
Zone of Saturation (Ground water)
Groundwater
Important source of clean water More abundant than SW
Linked to SW systems
Sustains flows in streams
pollution
Groundwater Concerns?
Problems with groundwater
Groundwater overdraft / mining / subsidence
Waterlogging
Seawater intrusion
Groundwater
• An important component of water resource systems.
• Extracted from aquifers through pumping wells and supplied for domestic use, industry and agriculture.
• With increased withdrawal of groundwater, the quality of groundwater has been continuously deteriorating.
Management of a groundwater system, means making such decisions as:
• The total volume that may be withdrawn annually from the aquifer.
• The location of pumping and artificial recharge wells, and their rates.
• Decisions related to groundwater quality.
Groundwater contamination by: Hazardous industrial wastes
Leachate from landfills
MANAGEMENT means making decisions to achieve goals without violating specified constraints.
Good management requires information on the response of the managed system to the proposed activities.
This information enables the decision-maker, to compare alternative actions and to ensure that constraints are not violated.
Any planning of mitigation or control measures, once contamination has been detected in the saturated or unsaturated zones, requires the prediction of the path and the fate of the contaminants, in
response to the planned activities.
A tool is needed that will provide this information.
The tool for understanding the system and its behavior and for predicting this response is the model.
Usually, the model takes the form of a set of
mathematical equations, involving one or more partial differential equations. We refer to such model as a mathematical model.
The advantage of the analytical solution is that the same solution can be applied to various numerical values of model coefficients and parameters.
Unfortunately, for most practical problems, because of the heterogeneity of the considered domain, the
irregular shape of its boundaries, and the non-analytic form of the various functions, solving the mathematical models analytically is not possible.
Instead, we transform the mathematical model into a numerical one, solving it by means of computer
We should have a CALIBRATED MODEL of the aquifer, especially, we should know the aquifer’s natural replenishment (from
precipitation and through aquifer boundaries).
Prior to determining the management scheme for any aquifer :
We should have a POLICY that dictates management objectives and constraints
.
Obviously, we also need information about the water demand ( quantity and quality, current and future ),
interaction with other
parts of the water resources system, economic information, sources of pollution, effect of changes on the environment---springs, rivers ,...
The model will provide the response of the aquifer (water levels ,
concentrations, etc.) to the implementation of any management alternative
GROUND WATER MODELING
WHY MODEL?
•To make predictions about a ground-water system’s response to a stress
•To understand the system
•To design field studies
Use of Groundwater models
•
Can be used for three general purposes:
•
To predict or forecast
expected artificial
or natural changes in the system.
Predictive is more applied to deterministic
models since it carries higher degree of
Use of Groundwater models
•
To
describe
the system in order to analyse
various assumptions
•
To
generate
a hypothetical system that
will be used to study principles of
ALL GROUND-WATER HYDROLOGY WORK IS MODELING
A Model is a representation of a system.
Modeling begins when one formulates a concept of a hydrologic system,
continues with application of, for example, Darcy's Law to the problem,
and may
Ground Water Flow Modelling
A Powerful Tool
for furthering our understanding
of hydrogeological systems
Importance of understanding ground water flow models
Construct accurate representations of hydrogeological systems Understand the interrelationships between elements of
systems
Efficiently develop a sound mathematical representation Make reasonable assumptions and simplifications
Introduction to Ground Water Flow Modelling
Predicting heads (and flows) and
Approximating parameters
Solutions to the flow equations
Most ground water flow models are solutions of some form of the ground water flow equation
Potentiometric
“e.g., unidirectional, steady-state flow within a confined aquifer
The partial differential equation needs to be solved to calculate head as a function of position and time, i.e., h=f(x,y,z,t)
h(x,y,z,t)?
The only effective way to test effects of
The only effective way to test effects of
groundwater management strategies
groundwater management strategies
Takes time, money to make model
Takes time, money to make model
Conceptual model
Conceptual model
Steady state model
Steady state model
Transient model
Transient model
The model is only as good as its calibration
The model is only as good as its calibration
Groundwater Modeling
Processes we might want to model
• Groundwater flow
calculate both heads and flow
• Solute transport
– requires information
on flow (velocities)
MODELING PROCESS
TYPES OF MODELS
CONCEPTUAL MODEL QUALITATIVE DESCRIPTION OF SYSTEM "a cartoon of the system in your mind"
MATHEMATICAL MODEL MATHEMATICAL DESCRIPTION OF SYSTEM
SIMPLE - ANALYTICAL (provides a continuous solution over the model domain)
COMPLEX - NUMERICAL (provides a discrete solution - i.e. values are calculated at only a few points)
ANALOG MODEL e.g. ELECTRICAL CURRENT FLOW through a circuit board with resistors to represent hydraulic conductivity and capacitors to represent storage coefficient
Mathematical model
:
simulates ground-water flow and/or
solute fate and transport indirectly by
means of a set of governing equations
thought to represent the physical
processes that occur in the system.
Components of a Mathematical Model
•
Governing Equation
(Darcy’s law + water balance equation)
with head (h) as the dependent variable
•
Boundary Conditions
R x y Q
y
x z
1. Consider flux (q) through REV 2. OUT – IN = - Storage
3. Combine with: q = -K grad hK
q
Law of Mass Balance
+ Darcy’s Law =
Governing Equation for Groundwater Flow
div
q
= - S
s(
h
t)
(Law of Mass Balance)
q
= -
K
grad
h
(Darcy’s Law)
div
(K
grad
h) = S
s(
h
t)
0
General governing equation
for steady-state, heterogeneous, anisotropic conditions, without a source/sink term
*
*
General governing equation for
transient
,
heterogeneous, and anisotropic conditions
Specific Storage
Figures taken from Hornberger et al. (1998)
Unconfined aquifer Specific yield
Confined aquifer Storativity
S = V / A h S = Ss b
b
*
Storage coefficient (S) is either storativity or specific yield. S = Ss b & T = K b
Types of Solutions of Mathematical Models
•
Analytical Solutions
: h= f(x,y,z,t)
(example: Theis equation)
•
Numerical Solutions
Finite difference methods
Finite element methods
The flexibility of analytical modeling is limited due to simplifying assumptions:
Homogeneity, Isotropy, simple geometry, simple initial conditions…
Geology is inherently complex:
Heterogeneous, anisotropic, complex geometry, complex conditions…
This complexity calls for a more
powerful solution to the flow equation Numerical modeling
Numerical Methods
All numerical methods involve
representing the flow domain by a
limited number of discrete points called
nodes.
A set of equations are then derived to
relate the nodal values of the
• Numerical Solutions
Discrete solution of head at selected nodal points. Involves numerical solution of a set of algebraic equations.
Finite difference models (e.g., MODFLOW)
Finite element models (e.g., SUTRA)
Finite Difference Modelling
3-D Finite Difference Models
Requires vertical discretization (or layering) of model
Finite difference models
may be solved using:
• a computer program
(e.g., a FORTRAN program)
Finite Elements:
basis functions, variational principle, Galerkin’s method, weighted residuals• Nodes plus elements; elements defined by nodes
• Nodes located on flux boundaries
• Flexibility in grid design:
elements shaped to boundaries elements fitted to capture detail
• Easier to accommodate anisotropy that occurs at an angle to the coordinate axis
• Able to simulate point sources/sinks at nodes
Involves superposition of analytic solutions. Heads are calculated in continuous space using a computer to do the mathematics involved in superposition.
Hybrid
Analytic Element Method (AEM)
The AE Method was introduced by Otto Strack.
A general purpose code, GFLOW, was developed by Strack’s student Henk Haitjema, who also wrote a
textbook on the AE Method: Analytic Element Modeling of Groundwater Flow, Academic Press, 1995.
What is a “model”?
Any “device” that represents approximation
to field system
Physical Models
Mathematical Models
Analytical
Modelling Protocol
Establish the Purpose of the Model
Develop Conceptual Model of the System
Select Governing Equations and Computer Code Model Design
Calibration
Calibration Sensitivity Analysis Model Verification
Prediction
Predictive Sensitivity Analysis
Presentation of Modeling Design and Results Post Audit
Purpose - What questions do you want the
model to answer?
Prediction; System Interpretation; Generic
Modeling
What do you want to learn from the model?
Is a modeling exercise the best way to
answer the question? Historical data?
Can an analytical model provide the answer?
System Interpretation: Inverse Modeling: Sensitivity Analysis
Generic: Used in a hypothetical sense, not necessarily for a real site
System Interpretation: Inverse Modeling: Sensitivity Analysis
Model “Overkill”?
Is the vast labor of characterizing the system,
combined with the vast labor of analyzing it,
ETHICS
There may be a cheaper, more effective
approach
Conceptual Model
“Everything should be made as simple as possible, but not simpler.” Albert Einstein
Pictorial representation of the groundwater
flow system
Will set the dimensions of the model and
the design of the grid
“Parsimony”….conceptual model has been
Select Computer Code
Select Computer Model
Code Verification
Comparison to Analytical Solutions; Other
Numerical Models
Model Design
Design of Grid, selecting time steps,
boundary and initial conditions, parameter
data set
Steady/Unsteady..1, 2, or 3-D;
…Heterogeneous/Isotropic…..Instantaneous/Continuous
Steady/Unsteady..1, 2, or 3-D;
Calibration
Show that Model can reproduce
field-measured heads and flow (concentrations if
contaminant transport)
Results in parameter data set that best
Calibration Sensitivity Analysis
Uncertainty in Input Conditions
Determine Effect of Uncertainty on
Model Verification
Use Model to Reproduce a Second Set of
Field Data
Prediction
Desired Set of Conditions
Sensitivity Analysis
Effect of uncertainty in parameter values and
Presentation of Modelling
Design and Results
Effective Communication of
Modeling Effort
Postaudit
New field data collected to determine if
prediction was correct
Site-specific data needed to validate
model for specific site application
Model Redesign
Include new insights into system
NUMERICAL MODELING
DISCRETIZE
Write equations of GW Flow between each node Darcy's Law
Conservation of Mass
Define Material Properties Boundary Conditions Initial Conditions Stresses
At each node either H or Q is known, the other is unknown n equations & n unknowns
solve simultaneously with matrix algebra
Result H at each known Q node Q at each known H node
Calibrate Steady State Transient
Validate
Sensitivity
Predictions
MODELs NEED
Geometry
Material Properties (K, S, T,
Φ
e, R, etc.)Boundary Conditions (Head, Flux, Concentration etc.)
Model Design
Model Design
•
Conceptual Model
•
Selection of Computer Code
•
Model Geometry
•
Grid
•
Boundary array
•
Model Parameters
•
Boundary Conditions
•
Initial Conditions
Concept Development
Concept Development
•
Developing a conceptual model is the initial
Conceptual Model
A descriptive representation
of a groundwater system that
incorporates an interpretation of the
geological & hydrological conditions.
Generally includes information about
the water budget. May include
Selection of Computer Code
Selection of Computer Code
•
Which method will be used depends largely
on the type of problem and the knowledge
of the model design.
Model Geometry
Model Geometry
•
Model geometry defines the size and the
shape of the model. It consists of model
Boundaries
Boundaries
•
Physical boundaries
are well defined
geologic and hydrologic features that
permanently influence the pattern of
Boundaries
Boundaries
•
Hydraulic boundaries
are derived from the
groundwater flow net and therefore
“artificial” boundaries set by the model
designer. They can be no flow boundaries
represented by chosen stream lines, or
HYDRAULIC BOUNDARIES
A streamline (flowline) is also a hydraulic boundary because by definition, flow is ALWAYS
parallel to a streamflow. It can also be said that flow NEVER crosses a streamline; therefore it is similar to an IMPERMEABLE (no flow) boundary
BUT
Stress can change the flow pattern and shift the position of streamlines; therefore care must be taken when using a
TYPES OF MODEL BOUNDARY
NO-FLOW BOUNDARY
Neither HEAD nor FLUX is Specified. Can represent a Physical boundary or a flow Line (Groundwater Divide)
SPECIFIED HEAD OR
CONSTANT HEAD BOUNDARY
h = constant
TYPES OF MODEL BOUNDARY (cont’d)
SPECIFIED FLUX BOUNDARY
q = constant
h is determined by the model
(The common method of simulation is to use one injection well for each boundary cell)
HEAD DEPENDANT BOUNDARY
hb = constant q = c (hb – hm)
and c = f (K,L) and is called CONDUCTANCE
Boundary Types
Specified Head/Concentration: a special case of constant head (ABC, EFG)
Constant Head /Concentration: could replace (ABC, EFG)
Specified Flux: could be recharge across (CD)
No Flow (Streamline): a special case of specified flux (HI)
Head Dependent Flux: could replace (ABC, EFG)
Free Surface: water-table, phreatic surface (CD)
Seepage Face: pressure = atmospheric at ground surface (DE)
Boundary conditions in Modflow
Boundary conditions in Modflow
• Constant head boundary
• Head dependent flux
– River Package
– Drain Package
– General-head Boundary Package
• Known Flux
– Recharge
– Evapotranspiration
– Wells
– Stream
Initial Conditions
Initial Conditions
•
Values of the hydraulic head for each active
and constant-head cell in the model. They
must be higher than the elevation of the cell
bottom.
•
For transient simulation, heads to resemble
closely actual heads (realistic).
•
For steady state, only hydraulic heads in
Model Parameters
Model Parameters
•
Time
•
Space (layer top and bottom)
•
Hydrogeologic characteristics
(hydraulic conductivity, transmissivity,
Time
Time
•
Time parameters are specified when
modelling transient (time dependent)
conditions. They include time unit, length
and number of time steps.
•
Length of stress periods is not relevant for
Grid
Grid
•
In Finite Difference model, the grid is
formed by two sets of parallel lines that are
orthogonal. The blocks formed by these
lines are called cells. In the centre of each
cell is the node – the point at which the
Grid
Grid
•
Grid mesh can be uniform or custom, a
uniform grid is better choice when
–
Evenly distributed aquifer characteristics data
–
The entire flow field is equally important
Grid
Grid
•
Grid mesh can be custom when
–
There is less or no data for certain areas
–
There is specific interest in one or more smaller
areas
•
Grid orientation is not an issue in isotropic
aquifers. When the aquifer is anisotropic,
the model coordinate axes must be aligned
with the main axes of the hydraulic
•
Regular vs irregular grid spacing
Irregular spacing may be used to obtain
detailed head distributions in selected areas of the grid.
Finite difference equations that use irregular grid spacing have a higher associated error
Curvature of the water table
Vertical change in head
Variability of aquifer characteristics (K,T,S)
Variability of hydraulic parameters (R, Q)
Considerations in selecting the size of
the grid spacing
Grids
It is generally agreed that from a practical
point-of-view the differences between grid types are minor and unimportant.
Boundary array (cell type)
Boundary array (cell type)
•
Three types of cells
–
Inactive cells through which no flow into or out
of the cells occurs during the entire time of
simulation.
–
Active, or variable-head cells are free to vary
in time.
–
Constant-head cell, model boundaries with
Hydraulic conductivity and
Hydraulic conductivity and
transmissivity
transmissivity
•
Hydraulic conductivity is the most critical
and sensitive modelling parameter.
•
Realistic values of storage coefficient and
Effective porosity
Effective porosity
•
Required to calculate velocity, used mainly
Calibration parameters are uncertain parameters
whose values are adjusted during model calibration.
Typical calibration parameters include hydraulic conductivity and recharge rate.
Calibration Targets
Target with relatively large associated error.
Target with smaller associated error.
• Head measured in an observation well is known as a target.
Targets used in Model Calibration
• The simulated head at the node representing the observation well is compared with the measured head.
• During model calibration, parameter values are
adjusted until the simulated head matches the observed value.
Calibration to Fluxes
When recharge rate (R) is a calibration
H1
H2
q = KI
In this example, flux information
In this example, discharge
Calibration - Remarks
• Calibrations are non-unique.
• A good calibration does not ensure that the model will make good predictions.
• Need for an uncertainty analysis to accompany calibration results and predictions.
• You can never have enough field data.
Uncertainty in the Calibration
Involves uncertainty in:
Parameter values
Conceptual model including boundary conditions, zonation, geometry etc.
Ways to analyze uncertainty
in the calibration
Sensitivity analysis
is used as an uncertainty
analysis after calibration.
Use an inverse model (automated calibration)
to quantify uncertainties and optimize the
Uncertainty in the Prediction
Involves uncertainty in how parameter values
(e.g., recharge) will vary in the future.
Stochastic simulation
Ways to quantify uncertainty
in the prediction
Modeling Chronology
1960’s Flow models are great!
1970’s Contaminant transport models are great!
1975 What about uncertainty of flow models?
1980s Contaminant transport models don’t work.
(because of failure to account for heterogeneity)
“The objective of model validation is to
determine how well the mathematical
representation of the processes describes
the actual system behavior in terms of the
degree of correlation between model
How to build confidence in a model
Calibration (history matching)
“Verification”
requires an independent set of field data
Post-Audit: requires waiting for prediction to occur
KEEPING AN OPEN MIND
Consider all dimensions of the problem before coming to a conclusion.
Considering all the stresses that might be imposed and all the possible processes that might be involved in a
situation before reaching a conclusion.
KEEPING AN OPEN MIND is spending 95% of your TIME DETERMINING WHAT YOU THINK IS HAPPENING and only 5% of your TIME DEFENDING YOUR OPINION.
Groundwater Flow Models
• The most widely used numerical groundwater flow model is
MODFLOW which is a three-dimensional model, originally developed by the U.S. Geological Survey.
• It uses finite difference scheme for saturated zone.
• The advantages of MODFLOW include numerous facilities
for data preparation, easy exchange of data in standard
form, extended worldwide experience, continuous
development, availability of source code, and relatively low price.
• However, surface runoff and unsaturated flow are not
MODFLOW
USGS code
Finite Difference Model
•
MODFLOW 88
•
MODFLOW 96
MODFLOW
(Three-Dimensional Finite-Difference Ground-Water Flow Model)
• When properly applied, MODFLOW is the recognized
standard model.
• Ground-water flow within the aquifer is simulated in
MODFLOW using a block-centered finite-difference approach.
• Layers can be simulated as confined, unconfined, or a
combination of both.
MT3D
(A Modular 3D Solute Transport Model)
• MT3D is a comprehensive three-dimensional numerical
model for simulating solute transport in complex hydrogeologic settings.
• MT3D is linked with the USGS groundwater flow simulator,
MODFLOW, and is designed specifically to handle
FEFLOW
(Finite Element Subsurface Flow System)
FEFLOW is a finite-element package for simulating 3D and 2D fluid density-coupled flow, contaminant mass (salinity) and
heat transport in the subsurface.
HST3D
(3-D Heat and Solute Transport Model)
The Heat and Solute Transport Model HST3D simulates
SEAWAT
(Three-Dimensional Variable-Density Ground-Water Flow)
• The SEAWAT program was developed to simulate
three-dimensional, variable- density, transient ground-water flow in porous media.
• The source code for SEAWAT was developed by
combining MODFLOW and MT3D into a single program
SUTRA
(2-D Saturated/Unsaturated Transport Model)
• SUTRA is a 2D groundwater saturated-unsaturated
transport model, a complete saltwater intrusion and energy transport model.
• SUTRA employs a two-dimensional hybrid finite-element
and integrated finite-difference method to approximate the governing equations that describe the two interdependent processes.
SWIM
(Soil water infiltration and movement model)
• SWIMv1 is a software package for simulating water
infiltration and movement in soils.
• SWIMv2 is a mechanistically-based model designed to
address soil water and solute balance issues.
• The model deals with a one-dimensional vertical soil
profile which may be vertically inhomogeneous but is assumed to be horizontally uniform.
• It can be used to simulate runoff, infiltration,
redistribution, solute transport and redistribution of
VISUAL HELP
(Modeling Environment for Evaluating and Optimizing Landfill Designs)
• Visual HELP is an advanced hydrological modeling
environment available for designing landfills, predicting leachate mounding and evaluating potential leachate
contamination.
Visual MODFLOW
(Integrated Modeling Environment for MODFLOW and MT3D)
• Visual MODFLOW provides professional 3D groundwater
Groundwater Modeling Resources
Kumar Links to Hydrology Resources
http://www.angelfire.com/nh/cpkumar/hydrology.html
USGS Water Resources Software Page
water.usgs.gov/software
Richard B. Winston’s Home Page
www.mindspring.com/~rbwinston/rbwinsto.htm
Geotech & Geoenviron Software Directory
www.ggsd.com
International Ground Water Modeling Center
Ground Water Modelling Discussion Group
An email discussion group related to ground water modelling and analysis. This group is a forum for the communication of all aspects of ground water modelling including technical discussions;
announcement of new public domain and commercial softwares; calls for abstracts and papers; conference and workshop
announcements; and summaries of research results, recent publications, and case studies.
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Visual MODFLOW Users Group
Visual MODFLOW is a proven standard for professional 3D groundwater flow and contaminant transport modeling using MODFLOW-2000, MODPATH, MT3DMS AND RT3D. Visual
MODFLOW seamlessly combines the standard Visual MODFLOW package with Win PEST and the Visual MODFLOW 3D-Explorer to give a complete and powerful graphical modeling environment.
This group aims to provide a forum for exchange of ideas and experiences regarding use and application of Visual MODFLOW software.
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