Dr. James M. Martin-Hayden

1
Analytical and Numerical Ground Water Flow
Modeling
An Introduction
Dr. James M. Martin-Hayden Associate Professor
[phone] Jhayden_at_Geology.UToledo.edu
2
Ground Water Flow Modeling

• 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 (
  a necessity)
• Understand the limitations of the mathematical
  representation
• Understand the limitations of the interpretation
  of the results

3
Ground Water Flow Modeling

• Avoid the Black Box Approach
• The temptation is to plug in numbers and have a
  model spit out answers, especially with fancy
  preprocessors and postprocessors
• This will not provide a sound appreciation for
  the accuracy of the results or how reliable the
  predictions will be
• Modeling is intimidating equations, vectors,
  partial differential equations, huge computer
  code
• However, it is surprising how straightforward the
  finite difference modeling process is

4
Ground Water Flow Modeling

• Goals
• Gain an appreciation of the Modeling Process
• Develop finite difference equations from first
  principles with a minimum of higher level
  mathematics
• Implement the finite difference models with a
  spreadsheet
• Learn what is involved in setting up a finite
  difference model
• In general make the modeling process transparent

5
The Two Fundamental Equationsof Ground Water Flow
First Law of Hydrogeology

• Basic Form Full Form

Second Law of Hydrogeology
Basic Form Full Form
6
Darcys Law

• Darcys Experiment (1856)

Darcy investigated ground water flow under
controlled conditions
h1
h2
A
Dh
A Cross Sectional Area (Perp. to flow)
Q Volumetric flow rate L3/T
Dx
Q
K The proportionality constant is added to form
the following equation
Hydraulic Gradient
h
Slope Dh/Dx dh/dx
h1
Dh
h2
K units L/T
Dx
x
x1
x2
7
Darcys Law (cont.)

• Other useful forms of Darcys Law

• Used for calculating Q given A

Q

• Volumetric Flux

A

• Ave. Linear
• Velocity

• Used for calculating average velocity of
  contaminant transport

Q
q


A.n
n

• Assumptions Laminar, saturated flow

8
3-D Darcys Law

• Hydrogeology is Three Dimensional

• Vertical flow is important in field
  investigations
• Vertical flow of contaminants
• Delineation of recharge and discharge
• Ground-water/surface-water interactions
• Interactions between aquifers

vh
vy
vx
vz

• Finite difference models are constructed in 3
  (or 2) dimensions
• Vectors quantities divided into 3 orthogonal
  components
• Many calculations done in each dimension
  separately
• Components are then summed to calculate
  resultants and
• Used for particle tracking and display of
  results (post processing)

9
3-D Darcys Law

• Velocity Vectors and Components

• The Velocity Vector

• Horizontal
• Component (Vh)

• Vertical
• Component (Vz)

Average linear ground water flow velocity
10
3-D Darcys Law (cont.)
(i.e., the velocity vector scaled by porosity,
a scalar)

• Volumetric Flux Vector ?
• The Velocity Vector

a.k.a. Specific discharge or Darcy flux
11
3-D Darcys Law (cont.)

• Hydraulic Gradient and Components
• Hydraulic gradient vector

• Horizontal
• Component

• Vertical Component

12
3-D Darcys Law (cont.)

• Horizontal and Vertical Hydraulic Gradients

• Horizontal Component, in field applications
• Represented as a single vector perpendicular to
  flow lines
• Approximated using a 3 point problem or a
  contour map of the piezometric surface
  (horizontal component only)

Ds385m
0.0104

• Vertical component
• Often taken as positive downward
• Can be approximated using a well nest

Dh4.00m
hs 291.26m
hd 290.74
Dh0.52m
Dz 5.84m
Vz
Finite difference approximation
0.0890
13
3-D Darcys Law (cont.)

• Vector representation of Darcys Law

• Calculation of horizontal components, vh and qh

• Assumptions
• Aligned anisotropy minimum hydraulic
  conductivity (Kz) is vertical
• A reasonable assumption for horizontally layered
  (or fractured) aquifers
• Hydraulic conductivity is typically more than an
  order of magnitude greater in the horizontal
  direction for these aquifers (Khgt10x Kz)
• Horizontal isotropy of K KxKyKh
• Vertical anisotropy is typically much greater
  than the horizontal anisotropy

• Calculation of vertical components, vz and qz

14
3-D Darcys Law (cont.)

• Generalization of Darcys Law to 3-D
• Anisotropic K
• Anisotropy aligned with coordinate axes

• Three vector components

Volumetric Flux
Hydraulic Conductivity
Hydraulic Gradient
Vector
Tensor (3x3 matrix)
Vector

• Shorthand representation

15
3-D Darcys Law (cont.)

• 3-D Velocity Calculation
• Divide flux by porosity (n), a scalar.

• Three vector components

Velocity Vector
Hydraulic Conductivity
Hydraulic Gradient
Vector
Tensor (3x3 matrix)
Vector
Scalar

• Shorthand representation

16
Alternative form of 3-D Darcys Law

• U velocity (in some texts)
• T Transmissivity (TKb)

U T?h
ne b

• Have you seen this equation somewhere?

17
Horizontal (2-D) Darcys Law
10m

• Horizontal ground water flow
• Common characteristics of aquifers and resulting
  assumption
• Aquifers tend to be much more extensive in the
  horizontal direction than in the vertical
• Hydraulic conductivity tends to be higher in the
  horizontal direction

• Thus vertical flow is often neglected (Dupuit
  assumption, 1863)
• This allows the use of Transmissivity as a
  measure of the aquifers ability to transmit
  water, TKhb

• 2-D Darcys Law, flow per unit width of aquifer

18
The Ground Water Flow Equation

• Mass Balance ?Objective of modeling represent
  hf(x,y,z,t)
• A common method of analysis in sciences
• For a system, during a period of time (e.g., a
  unit of time),
• Assumption Water is incompressible
• Mass per unit volume (density, r) does not change
  significantly
• Volume is directly related to mass by density
  Vm/r
• In this case water balance models are essentially
  mass balance models divided by density

Mass In Mass Out Change in Mass Stored
Volume In Volume Out Change in Volume Stored

• Dividing by
• unit time gives
• If Q is a continuous function of time Q(t) then
  dv/dt is at any instant in time

Qi Qo DVw /Dt
Qi(t) Qo(t) dVw/dt
19
The Flow Equation (cont.)

• Example 1 Storage in a reservoir
• If Qi Qo, dVw/dt 0 ? no change in level,
  i.e., steady state
• If Qi gt Qo, dVw/dt gt 0 ?filling If Qi lt Qo,
  dVw/dt lt 0 ? -emptying
• E.g., Change in storage due to
• linearly varying flows

Qi
Qo
dVw/dt
Qi(t) Qo(t) dVw/dt
Q
Q1
Qi m1tQ1
Qo m2tQ2
Q2
dV/dt 0
dV/dt gt 0
dV/dt lt 0
t
0
20
The Flow Equation (cont.)

• Example 1 Storage in a reservoir
• If Qi Qo, dVw/dt 0 ? no change in level,
  i.e., steady state
• If Qi gt Qo, dVw/dt gt 0 ?filling If Qi lt Qo,
  dVw/dt lt 0 ? -emptying

Qi
Qo
dVw/dt
Qi(t) Qo(t) dVw/dt
V
Q
dV/dt gt 0
dV/dt lt 0
Q1
dV/dt 0
Qi m1tQ1
Qo m2tQ2
Q2
dV/dt 0
dV/dt gt 0
dV/dt lt 0
t
t
0
21
The Flow Equation (cont.)

• Example 1 Storage in a reservoir
• If Qi Qo, dVw/dt 0 ? no change in level,
  i.e., steady state
• If Qi gt Qo, dVw/dt gt 0 ?filling If Qi lt Qo,
  dVw/dt lt 0 ? -emptying
• Example 2 Storage in a REV (Representative
  Elementary Volume)
• REV The smallest parcel of a unit that has
• properties (n, K, r) that are representative
• of the formation
• The same water balance can be used to examine the
  saturated (or unsaturated) REV

Qi
Qo
dVw/dt
Qi(t) Qo(t) dVw/dt
22
The Flow Equation (cont.)

• Mass Balance for the REV (or any volume of a
  flow system)
• aka, The Ground Water Flow Equation

Qi Qo dVw/dt
Dz
DVw/Dt
Dy
For saturated, incompressible, 1-D flow
Dx
23
The Flow Equation (cont.)

• External Sources and Sinks (Qs)

Differential Form
24
The Flow Equation (cont.)

• 3-D, flow equation
• Summing the mass balance equation for each
  coordinate direction gives the total net
  inflow per unit volume into the REV
• Add a source term Qs/V volumetric flow rate per
  unit volume injected into REV

qz
qx
Change in volume stored
qy
Net inflow

• Substitute components of q from 3-D Darcys law

per unit volume per unit time
25
The Flow Equation (cont.)

• Specific storage and homogeneity
• Due to aquifer compressibility change in
  porosity is proportional to a change in head
  (over a infinitesimally small range, dh)

• Ss Specific Storage, a proportionality constant

• Assumption K is homogeneous over small
  distances, i.e., K ? f(x,y,z)
• Compressibility of water is much less than
  aquifer compressibility

• This gives the equation on which ground water
  flow models are based hf(x,y,z,t)

26
The Flow Equation (cont.)

• Flow Equation Simplifications

• Steady State Flow Equation
• If inflow out flow, Net inflow 0
• Change in storage 0

• Two dimensional flow equation
• Horizontal flow (Dupuit assumption)
• Thus dh/dz 0

• 2-D, horizontally isotropic flow equation
• KxKyKh
• TKh b units L2/T, SSs b
• Ah horizontal area of recharge
• Ss/KhS/T ? hydraulic diffusivity

• Isotropic, 2-D, steady state flow equation
• (without source term), a.k.a. The Laplace
  Equation

• Our job is not yet finished, hf(x,y,z,t)?

27
Introduction to Ground Water Flow Modeling

• 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

• The partial differential equation needs to be
  solved to calculate head as a function of
  position and time, i.e., hf(x,y,z,t)

• e.g., unidirectional, steady-state flow within a
  confined aquifer

28
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
  equations
• Thiem Equation Steady state flow to a well in a
  confined aquifer
• The Theis Equation Transient flow to a well in a
  confined aquifer
• Slug test solutions Transient response of head
  within a well to a
• pressure pulse

29
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

30
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

31
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
ho
h1
Q
b
Q
h1
Clay
x
32
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)

33
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

34
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

35
Flow Modeling (cont.)

• Numerical Modeling in a Nutshell
• A solution of flow equation is approximated on a
  discrete grid (or mesh) of points, cells or
  elements

• The parameters and variables are specified over
  the boundary of the domain (region) being modeled

• Within this discretized domain
• Aquifer parameters can be set at each cell within
  the grid
• Complex aquifer geometry can be modeled
• Complex boundary conditions can be accounted for
• Requires 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

36
An Introduction to Finite Difference Modeling

• Approximate Solutions to the
• Flow Equation

• The Finite Difference Approximation of
  Derivatives

• Partial derivatives of head represent the change
  in head with respect to a coordinate direction
  (or time) at a point.e.g.,

• These derivatives can be approximated as the
  change in head (Dh) over a finite distance in the
  coordinate direction (Dy) that traverses the
  point
• i.e., The component of the hydraulic gradient in
  the y direction can be approximated by the finite
  difference Dh/Dy

h
h1
Dh
h2
Dy
y
y1
y2
37
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
ha
ha-ho
ho
ho-hb
hb
Dx
Dx
xo
xb
xa
x
Dx
38
Finite Difference Modeling (cont.)

• Finite Difference Approximation of
• 1-D, Steady State Flow Equation

?
39
Finite Difference Modeling (cont.)

• Physical basis for finite difference approximation

h

ha
ha-ho
ho
ha-hb
hb
Dx
Dx
xo
xa
xb
Dz
Dx
Ka
Ko
Ka
Dy
Dx

• Kab average K of cell and K of cell to the left
  Kab average K of cell and K of cell to the left

40
Finite Difference Modeling (cont.)

• Discretization of the Domain
• Divide the 1-D domain into equal cells of
  heterogeneous K

Ko
K1
K2
K3
Ki-1
Ki
Ki1
Specified Head
h1
h2
h3
hi-1
hn
hi
hi1
ho
hn1
Specified Head



Dx
Dx
Dx
Dx
Dx
Dx
Dx
Dx
Dx

• Solve for the head at each node gives n equations
  and n unknowns
• The head at each node is an average of the head
  at adjacent cells weighted by the Ks

41
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

Kc
hc
Dx
Ka
Kb
Kd
ha
ho
hb
Dx
Solve for ho
Kd
hd
Dx
Dx
Dx
Dx
42
Finite Difference Modeling (cont.)

• Incorporate Transmissivity Confined Aquifers
• multiply by b (aquifer thickness)

Kc
hc
Ka
Kb
Kd
ha
ho
hb
Solve for ho
ba
bo
bb
Ko
Kb
Ka
Dx
Dx
Dx
Dx
Dx
43
Finite Difference Modeling (cont.)

• Incorporate Transmissivity Unconfined Aquifers
• b depends on saturated thickness which is head
  measured relative to the aquifer bottom

Kc
hc
Ka
Kb
Kd
ha
ho
hb
Solve for ho
ha
ho
hb
Ko
Kb
Ka
Dx
Dx
Dx
Dx
Dx
44
Finite Difference Modeling (cont.)

• 2-D, Steady State, Isotropic, Homogeneous
• Finite Difference Scheme

hc
Dx
ha
ho
hb
Dx
Solve for ho
hd
Dx
Dx
Dx
Dx
45
Finite Difference Modeling (cont.)

• Spreadsheet Implementation
• Spreadsheets provide all you need to do basic
  finite difference modeling
• Interdependent calculations among grids of cells
• Iteration control
• Multiple sheets for multiple layers, 3-D, or
  heterogeneous parameter input
• Built in graphics x-y scatter plots and basic
  surface plots

A
B
C
D
E
1
2
3
4

46
Finite Difference Modeling (cont.)

• Spreadsheet Implementation of 2-D, Steady State,
  Isotropic, Homogeneous Finite Difference
• Type the formula into a computational cell
• Copy that cell into all other interior
  computational cell and the references will
  automatically adjust to calculate value for that
  cell
• Note Boundary cells will be treated differently

(C2D3C4B3)/4
47
Finite Difference Modeling (cont.)

• A simple example
• This will give a circular reference error
• Set? ToolsOptions Calculation to Manual
• Select? ToolsOptions Itteration
• Set?Maximum Iteration and Maximum Change
• Press F9 to iteratively calculate

A
B
C
D
E
10
10
10
Lake 1
10
1
2
10
1
Lake 2
3
10
1
4
7
5
2
5
River
48
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

49
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

50
Basic Finite Difference Design (cont.)

• Discretization Variable Grid Spacing
• Rules of Thumb
• Refine grid around areas
• of interest
• Adjacent rows or columns
• should be no more than
• twice (or less than half)
• as wide as each other
• Expand spacing smoothly
• Many implementations of
• Numerical models allow
• Onscreen manipulation of
• Grids relative to an imported
• Base map

51
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

52
Basic Finite Difference Design (cont.)

• Specified Head Boundaries
• Boundaries along which the heads have been
  measured and can be specified in the model
• e.g., surface water bodies
• They must be in good hydraulic connection
• with the aquifer
• Must influence heads throughout layer being
  modeled
• Large streams and lakes in unconfined aquifers
• with highly permeable beds
• Uniform Head Boundaries Head is uniform in
  space, e.g., Lakes
• Spatially Varying Head Boundaries e.g., River
• heads can be picked of of a topo map if
• Hydraulic connection with and unconfined aquifer
• the streambed materials are more permeable than
  the aquifer materials

53
Basic Finite Difference Design (cont.)

• Specified Flux Boundaries
• Boundaries along which, or cells within which,
  inflows or outflows are set
• Recharge due to infiltration (R)
• Pumping wells (Qp)
• Induced infiltration
• Underflow
• No flow boundaries
• Valley wall of low permeable sediment or rock
• Fault

54
Finite Difference Modeling (cont.)

• No Flow Boundary Implementation
• A special type of specified flux boundary
• Because there are no nodes outside the domain,
  the perpendicular node is reflected across the
  boundary as and image node

C
A
B
D
E
1
hA2
2
(A2 2B3 C4)/4
A3
hB3
hA3
hB3
C1
(B1D12C2)/4
hC4
4
E3
(2D3E2 E4)/4
5
C5
(B52C4D5)/4
55
Finite Difference Modeling (cont.)

• No Flow Boundary Implementation
• Corner nodes have two image nodes

B
C
D
E
(2B1 2A2)/4
A1
2
E1
(2D12E2)/4
3
E5
(2D52E4)/4
4
A5
(2A42B5)/4
5
56
Finite Difference Modeling (cont.)

• No Flow Boundary Implementation
• Combinations of edge and corner points are used
  to approximate irregular boundaries

57
Finite Difference Modeling (cont.)

• Head Dependent Flux Boundaries
• Flow into or out of cell depends on the
  difference between the head in the cell and the
  head on the other side of a conductive boundary
• e.g. 1, Streambed conductance
• hs stage of the stream
• ho Head within the cell
• Ksb K of streambed materials
• bsb Thickness of streambed
• w width of stream
• L length of reach within cell
• Csb Streambed conductance
• Based on Darcys law

• 1-D Flow through streambed

Qsb
ho
hs
L
bsb
w
Qsb
58
Finite Difference Modeling (cont.)

• Head Dependent Flux Boundaries
• e.g. 2, Flow through aquitard
• hc Head within confined aquifer
• Ho Head within the cell
• Kc K of aquitard
• bc Thickness of aquitard
• Dx2 Area of cell
• Cc aquitard conductance
• Based on Darcys law

• 1-D Flow through aquitard

ho
Qc
hc
Dx
bc
Dx
Qc
59
Case Study

• The Layered Modeling Approach

60
Finite Difference Modeling (cont.)

• 3-D Finite Difference Models
• Approximate solution to the 3-D flow equation
• e.g., 3-D, Steady State, Homogeneous Finite
  Difference approximation
• 3-D Computational Cell

61
Finite Difference Modeling (cont.)

• 3-D Finite Difference Models
• Requires vertical discretization (or layering) of
  model

K1
K2
K3
K4
62
Implementing Finite Difference Modeling

• Model Set-Up, Sensitivity Analysis, Calibration
  and Prediction

• Model Set-Up
• Develop a Conceptual Model
• Collect Data
• Develop Mathematical Representation of your
  System
• Model set-up is an Iterative process
• Start simple and make sure the model runs after
  every added complexity
• Make Back-ups

63
Implementation

• Anatomy of a Hydrogeological Investigation
• and accompanying report
• Significance
• Define the problem in lay-terms
• Highlight the importance of the problem being
  addressed
• Objectives
• Define the specific objectives in technical terms
• Description of site and general hydrogeology
• This is a presentation of your conceptual model

64
Implementation

• Anatomy of a Hydrogeological Investigation
  (cont.)
• Methodology
• Convert your conceptual model into mathematical
  models that will specifically address the
  Objectives
• Determine specifically where you will get the
  information to set-up the models
• Results
• Set up the models, calibrate, and use them to
  address the objectives
• Conclusions
• Discuss specifically, and concisely, how your
  results achieved the Objectives (or not)
• If not, discuss improvements on the conceptual
  model and mathematical representations

65
Developing a Conceptual Model

• Settling Pond Example

• A company has installed two settling ponds to
  (Significance)
• Settle suspended solids from effluent
• Filter water before it discharges to stream
• Damp flow surges

• Questions to be addressed (Objectives)
• How much flow can Pond 1 receive without
  overflowing? ?Q?
• How long will water (contamination) take to reach
  Pond 2 on average??v?
• How much contaminant mass will enter Pond 2 (per
  unit time)? ?M?

This is a hypothetical example based on a
composite of a few real cases
66
Conceptual Model (cont.)

• Develop your conceptual model

Water flows between ponds through the saturated
fine sand barrier driven by the head difference
Pond 1
Pond 2
W 1510 ft
Outfall
Overflow
Elev. 658.74 ft
Elev. 652.23 ft
Dx 186
Sand
67
Conceptual Model (cont.)

• Develop your mathematical representation (model)
  
• (i.e., convert your conceptual model into a
  mathematical model)
• Formulate reasonable assumptions
• Saturated flow (constant hydraulic conductivity)
• Laminar flow (a fundamental Darcys Law
  assumption)
• Parallel flow (so you can use 1-D Darcys law)
• Formulate a mathematical representation of your
  conceptual model that
• Meets the assumptions and
• Addresses the objectives

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Conceptual Model (cont.)

• Collect data to complete your Conceptual Model
  and to Set up your Mathematical Model
• The model determines the data to be collected
• Cross sectional area (A w b)
• w length perpendicular to flow
• b thickness of the permeable unit
• Hydraulic gradient (Dh/Dx)
• Dh difference in water level in ponds
• Dx flow path length, width of barrier
• Hydraulic Parameters
• K hydraulic tests and/or laboratory tests
• n estimated from grainsize and/or laboratory
  tests
• Sensitivity analysis
• Which parameters influence the results most
  strongly?
• Which parameter uncertainty lead to the most
  uncertainty in the results?

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Implementing Finite Difference Modeling

• Testing and Sensitivity Analysis
• Adjust parameters and boundary conditions to get
  realistic results
• Test each parameter to learn how the model reacts
• Gain an appreciation for interdependence of
  parameters
• Document how each change effected the head
  distribution (and heads at key points in the
  model)

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Implementation (cont.)

• Calibration
• Fine tune the model by minimizing the error
• Quantify the difference between the calculated
  and the measured heads (and flows)
• Mean Absolute Error Minimize?
• Calibration Plot
• Allows identification of trouble spots
• Calebration of a transient model
• requires that the model be calibrated
• over time steps to a transient event
• e.g., pump test or rainfall episode
• Automatic Calibration allows
• parameter estimation
• e.g., ModflowP

MW28d?x
Calculated Measured
x
Calculated Head
x
x
x
x
x
o
o
Measured Head
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Implementation (cont.)

• Prediction
• A well calibrated model can be used to perform
  reliable what if investigations
• Effects of pumping on
• Regional heads
• Induced infiltration
• Inter aquifer flow
• Flow paths
• Effects of urbanization
• Reduced infiltration
• Regional use of ground water
• Addition and diversion of drainage

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Case Study

• An unconfined sand aquifer in northwest Ohio
• Conceptual Model

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Case Study

• An unconfined sand aquifer in northwest Ohio
• Surface water hydrology and topography

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Boundary Conditions
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