Introduction to Groundwater Modelling
Presentation Outline Groundwater in Hydrologic Cycle Why Groundwater Modeling is needed? Mathematical Models Groundwater Flow Models
Groundwater in Hydrologic Cycle
Types of Terrestrial WaterTypes of Terrestrial Water
Ground waterGround water
SoilSoilMoistureMoisture
SurfaceWater
Unsaturated Zone / Zone of Aeration / Vadose (Soil Water)
Pores Full of Combination of Air and Water
Zone of Saturation (Ground water)
Pores Full Completely with Water
Groundwater
Important source of clean waterMore abundant than SW
Linked to SW systems
Sustains flows in streams
Baseflow
pollution
Groundwater Concerns?
groundwater miningsubsidence
Problems with groundwater
Groundwater overdraft / mining / subsidence
Waterlogging
Seawater intrusion
Groundwater pollution
Why Groundwater Modelling is needed?
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.
• Water can be injected into aquifers for storage and/or quality control purposes.
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 as a thinking tool
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 certainty, while forecasting is used with probabilistic (stochastic) models.
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 groundwater flow associated with various general or specific problems.
Processes we might want to model
• Groundwater flow
calculate both heads and flow
• Solute transport – requires information on flow (velocities)
calculate concentrations
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
PHYSICAL MODEL e.g. SAND TANK which poses scaling problems
Mathematical Models
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.
(Anderson and Woessner, 1992)
Components of a Mathematical Model
• Governing Equation
(Darcy’s law + water balance equation) with head (h) as the dependent variable
• Boundary Conditions
• Initial conditions (for transient problems)
R x y Q
yx
z
1. Consider flux (q) through REV2. OUT – IN = - Storage3. Combine with: q = -KK grad h
q
Derivation of the Governing Equation
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 dependent variable such that they satisfy the governing PDE, either approximately or exactly.
• 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 modelsmay be solved using:
• a computer program (e.g., a FORTRAN program)
• a spreadsheet (e.g., EXCEL)
Groundwater Flow Models
The Two Fundamental Equationsof Ground Water Flow
Basic Form
“First Law of Hydrogeology”
“Second Law of Hydrogeology”
Basic Form
Darcy’s Law:
Average Linear Velocity
Flow Equation: 0dx
hd 2
2
1-D, Steady State
Darcy’s Law
Darcy’s Experiment (1856)
AQxQhQ ,1,
xhAKQ
xhAQ
xQ Q: Volumetric flow rate [L3/T]
h
h
x
Slope = h/x ~ dh/dx
Darcy investigated ground water flow under controlled conditions
hx
h1 h2
h1
h2
x1 x2
K: The proportionality constant is added to form the following equation:
K units [L/T]
: Hydraulic Gradient
A
xhh
A: Cross Sectional Area (Perp. to flow)
Darcy’s Law (cont.)
Other useful forms of Darcy’s LawQA =
QA.n = q
n =
Volumetric Flux
Ave. Linear Velocity
Used for calculating Q given A
Used for calculating average velocity of contaminant transport
Assumptions: Laminar, saturated flow
Introduction to Ground Water Flow Modeling
Predicting heads (and flows) and Approximating parameters
Solutions to the flow equationsMost ground water flow models are solutions of some form of the ground water flow equation
PotentiometricSurface
x
xx
ho
x0
h(x)
x
K q
“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)?
Darcy’s Law Integrated
Flow Modeling (cont.)
Analytical models (a.k.a., closed form models) The previous model is an example of an analytical model
is a solution to the 1-D Laplace equation
i.e., the second derivative of h(x) is zero
With this analytical model, head can be calculated at any position (x) Analytical solutions to the 3-D transient flow equation would give head at any position and at any time, i.e., the continuous function h(x,y,z,t)Examples of analytical models:
1-D solutions to steady state and transient flow equationsThiem Equation: Steady state flow to a well in a confined aquiferThe Theis Equation: Transient flow to a well in a confined aquiferSlug test solutions: Transient response of head within a well to a
pressure pulse
Flow Modeling (cont.)
Common Analytical Models Thiem Equation: steady state flow to a well within a confined aquifer
Analytic solution to the radial (1-D), steady-state, homogeneous K flow equation
Gives head as a function of radial distance
Theis Equation: Transient flow to a well within a confined aquifer Analytic solution of radial, transient, homogeneous K flow equation Gives head as a function of radial distance and time
Flow Modeling (cont.)
Forward Modeling: Prediction
Models can be used to predict h(x,y,z,t) if the parameters are known, K, T, Ss, S, n, b…
Heads are used to predict flow rates,velocity distributions, flow paths, travel times. For example:
Velocities for average contaminant transport Capture zones for ground water contaminant plume capture Travel time zones for wellhead protection
Velocity distributions and flow paths are then used in contaminant transport modeling
1-D, SS Thiem Theis
Flow Modeling (cont.)
Inverse Modeling: Aquifer Characterization Use of forward modeling requires estimates of aquifer
parameters Simple models can be solved for these parameters
e.g., 1-D Steady State:
This inverse model can be used to “characterize” K This estimate of K can then be used in a forward model to
predict what will happen when other variables are changed
ho
h1
Clay
b
x
ho h1
Q Q
Flow Modeling (cont.)
Inverse Modeling: Aquifer Characterization The Thiem Equation can also be solved for K
Pump Test: This inverse model allows measurement of K using a steady state pump test
A pumping well is pumped at a constant rate of Q until heads come to steady state, i.e.,
The steady-state heads, h1 and h2, are measured in two observation wells at different radial distances from the pumping well r1 and r2
The values are “plugged into” the inverse model to calculate K (a bulk measure of K over the area stressed by pumping)
Flow Modeling (cont.)
Inverse Modeling: Aquifer Characterization Indirect solution of flow models
More complex analytical flow models cannot be solved for the parameters
Curve Matching or Iteration
This calls for curve matching or iteration in order to calculate the aquifer parameters
Advantages over steady state solution• gives storage parameters S (or Ss) as well as T (or K)• Pump test does not have to be continued to steady state• Modifications allow the calculation of many other parameters
e.g., Specific yield, aquitard leakage, anisotropy…
Flow Modeling (cont.)
Limitations of Analytical Models Closed form models are well suited to the
characterization of bulk parameters However, the flexibility of forward 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 Modeling in a NutshellA solution of flow equation is approximated on a
discrete grid (or mesh) of points, cells or elements
Within this discretized domain: 1)Aquifer parameters can be set at each cell
within the grid2)Complex aquifer geometry can be modeled3)Complex boundary conditions can be
accounted forRequires detailed knowledge of 1), 2) , and 3)As compared to analytical modeling, numerical modeling is:
Well suited to prediction but More difficult to use for aquifer characterization
Flow Modeling (cont.)
The parameters and variables are specified over the boundary of the domain (region) being modeled
An Introduction to Finite Difference Modeling
Approximate Solutions to the Flow Equation
Partial derivatives of head represent the change in head with respect to a coordinate direction (or time) at a point.e.g., thoryh
h
y
hy
h1
h2
y1 y2
yh
yh
These derivatives can be approximated as the change in head (h) over a finite distance in the coordinate direction (y) that traverses the point
i.e., The component of the hydraulic gradient in the y direction can be approximated by the finite difference h/y
The Finite Difference Approximation of Derivatives
Finite Difference Modeling (cont.)
Approximation of the second derivative The second derivative of head with respect to x represents the change
of the first derivative with respect to x The second derivative can be approximated using two finite differences
centered around x2
This is known as a central difference
h
xx
ha
ho
xo xbxa
xhb
ha-ho
ho-hb
x
xhh
yh
oa
xhh
yh
bo
Finite Difference Modeling (cont.)
Finite Difference Approximation of 1-D, Steady State Flow Equation
Finite Difference Modeling (cont.)
Physical basis for finite difference approximation
y
z
h
x
ha
ho
xo xbxa
xhb
ha-ho
ha-hb
x
xhh
xh
oa
xhh
xh
bo
xhhKzy
qzyQ ii
oaoa
xhhKzy
qzyQ oo
boob
x
Kab: average K of cell and K of cell to the left; Kab: average K of cell and K of cell to the left
2
2boob
aooa
KKKKKK
KaKaKo
Finite Difference Modeling (cont.)
Discretization of the Domain Divide the 1-D domain into equal cells
of heterogeneous K
… …h1 h2 h3 hi-1 hn
x x x x x x x
head specified: and Constant
22
1no
1ii1/2i
1-ii1/2-i
hhx
KKKKKK
…hi hi+1
x x
Solve for the head at each node gives n equations and n unknownsThe head at each node is an average of the head at adjacent cells weighted by the Ks
ho hn+1
Spe
cifie
dH
ead
Spe
cifie
dH
ead
Finite Difference Modeling (cont.)
2-D, Steady State, Uniform Grid Spacing, Finite Difference Scheme Divide the 2-D domain into equally
spaced rows and columns of heterogeneous K
ha ho hb
hd
hc
x
x
x
2
222
dood
cooc
boob
aooa
KKKKKKKKKKKK
Ka
Kc
Kb
Kd
Kd
x x x
Solve for ho
ha ho hb
hcKa
Kc
KbKd
Finite Difference Modeling (cont.)
Incorporate Transmissivity: Confined Aquifers multiply by b (aquifer thickness)
22
222222
ddoodood
ccoocooc
bbooboob
aaooaooa
bKbKTTTbKbKTTTbKbKTTTbKbKTTT
x x x x x
Ko KbKa
ba bo bb
Solve for ho
ha ho hb
hcKa
Kc
KbKd
x x x x x
Ko KbKa
ha ho hb
Finite Difference Modeling (cont.)
Incorporate Transmissivity: Unconfined Aquifers b depends on saturated thickness which is head
measured relative to the aquifer bottom 2
222
dood
cooc
boob
aooa
hhhhhhhhhhhh
Solve for ho
Finite Difference Modeling (cont.)
2-D, Steady State, Isotropic, HomogeneousFinite Difference Scheme
ha ho hb
hd
hc
x
x
x
x x x
Solve for ho
Basic Finite Difference Design
Discretization and Boundary Conditions
Grids should be oriented and spaced to maximize the efficiency of the model
Boundary conditions should represent reality as closely as possible
Basic Finite Difference Design (cont.)
Discretization: Grid orientation Grid rows and columns should line up with as many rivers,
shorelines, valley walls and other major boundaries as much as possible
Basic Finite Difference Design (cont.)
Discretization: Variable Grid Spacing Rules of Thumb
Refine grid around areas of interest
Adjacent rows or columnsshould be no more than twice (or less than half) as wide as each other
Expand spacing smoothly Many implementations of
Numerical models allowOnscreen manipulation of Grids relative to an imported Base map
Basic Finite Difference Design (cont.)
Boundary Conditions Any numerical model must be bounded on
all sides of the domain (including bottom and top)
The types of boundaries and mathematical representation depends on your conceptual model
Types of Boundary Conditions Specified Head Boundaries Specified Flux Boundaries Head Dependant Flux Boundaries