Groundwater Modeling - 1 Groundwater Hydraulics Daene C. McKinney.

Groundwater Modeling - 1

Groundwater Hydraulics

Daene C. McKinney

Models …?

Input(Explanatory

Variable)

Model(Represents the

Phenomena)

Output(Results – Response

variable) Run off

InfiltrationEvaporation

ET

Precipitation Soil Characteristics

Models and more models …

Input(Explanatory

Variable)

Model(Phenomena)

Output(Results)

Inflow Data

Basin Water Allocation

Policy

Response to the Policy

Inflow Data

Basin Objectives and

Constraints

Optimum Policy

Precip. & Soil Charact.

Mimic Physics of the Basin

Runoff

SimulationModel

OptimizationModel

HydrologicSimulation

Predict Response to given

design/policy

Identify optimal design/policy

Source for Input data of other models

Modeling Process

• Problem identification (1)– Important elements to be modeled – Relations and interactions between them– Degree of accuracy

• Conceptualization and development (2 – 3)– Mathematical description– Type of model – Numerical method - computer code– Grid, boundary & initial conditions

• Calibration (4)– Estimate model parameters– Model outputs compared with actual outputs– Parameters adjusted until the values agree

• Verification (4)– Independent set of input data used – Results compared with measured outputs

Problem identificationand description

Model verification & sensitivity analysis

Model Documentation

Model application

Model calibration & parameter estimation

Model conceptualization

Model development

Data

Present results

1

2

3

4

5

6

7

Tools to Solve Groundwater Problems• Physical and analog methods

– Some of the first methods used.

• Analytical methods – What we have been discussing so far– Difficult for irregular boundaries, different

boundary conditions, heterogeneous and anisotropic properties, multiple phases, nonlinearities

• Numerical methods– Transform PDEs governing flow of

groundwater into a system of ODEs or algebraic equations for solution

Conceptual Model• Descriptive representation of

groundwater system incorporating interpretation of geological & hydrological conditions

• What processes are important to model?

• What are the boundaries?• What parameter values are

available?• What parameter values must

be collected?

What Do We Really Want To Solve?

• Horizontal flow in a leaky confined aquifer

• Governing Equations• Boundary Conditions• Initial conditions

Ground surface

Bedrock

Confined aquiferQx

K

xyz

h

Head in confined aquifer

Confining Layer

b

Flux Leakage Source/Sink Storage

Finite Difference Method

• Finite-difference method– Replace derivatives in governing equations with

Taylor series approximations– Generates set of algebraic equations to solve

1st derivatives

Taylor Series

• Taylor series expansion of h(x) at a point x+Dx close to x

• If we truncate the series after the nth term, the error will be

xxx x

First Derivative - Forward • Consider the forward Taylor series expansion of a function

h(x) near a point x

• Solve for 1st derivative

xxx

x

xxx

x

First Derivative - Backward • Consider the backward Taylor series expansion of a function

f(x) near a point x

• Solve for 1st derivative

xxx

x

xxx

x

Finite Difference Approximations

x

x x

1st Derivative(Backward)

1st Derivative(Forward)

i

Grids and Discretrization • Discretization process • Grid defined to cover domain• Goal - predict values of head at

node points of mesh– Determine effects of pumping– Flow from a river, etc

• Finite Difference method– Popular due to simplicity – Attractive for simple geometry

i,j

i,j+1

i+1,j

i-1,j

i,j-1

x, i

y, j

Domain

Mesh

Node point

D x

D y

Grid cell

Three-Dimensional Grids• An aquifer system is divided into rectangular blocks by a grid. • The grid is organized by rows (i), columns (j), and layers (k),

and each block is called a "cell"• Types of Layers

– Confined– Unconfined– Convertible

Layers can be different materials

i, rows

j, columns

k, layers

1-D Confined Aquifer Flow

• Homogeneous, isotropic, 1-D, confined flow

• Governing equation

• Initial Condition

• Boundary Conditions

Ground surface

Aquifer

x

yz

hB

Confining Layer

b

hA

Dx

i = 0 1 2 3 4 5 6 7 8 9 10

Node

Grid Cell

Derivative Approximations• Need 2nd derivative WRT x

li ,1

ix,

lt,

li ,1

x

li,

x

Derivative Approximations• Governing Equation

• 2nd derivative WRT x

• Need 1st derivative WRT t

Forward Backward

li ,1

ix,

lt,

1, li

li ,1

1, li

x

t

li,

Which one to use?

Time Derivative• Explicit

– Use all the information at the previous time step to compute the value at this time step.

– Proceed point by point through the domain.

• Implicit– Use information from one

point at the previous time step to compute the value at all points of this time step.

– Solve for all points in domain simultaneously.

Explicit Method

• Use all the information at the previous time step to compute the value at this time step.

• Proceed point by point through the domain.

• Can be unstable for large time steps.

li ,1

1, li

li ,1

1, li

x

tli,

FD Approx.Forward

Explicit Method

l+1 time levelunknown

l time levelknown

1-D Confined Aquifer Flow

• Initial Condition

• Boundary Conditions

Ground surface

Aquifer

x

yz

hB

Confining Layer

b

hA

Dx

i = 0 1 2 3 4 5 6 7 8 9 10

Node

Grid CellL

Dx = 1 m

L = 10 m

T=bK = 0.75 m2/d

S = 0.02

Explicit MethodGround surface

Aquifer

hB

Confining Layer

b

hA

Dx

i = 0 1 2 3 4 5 6 7 8 9 10

Node

Grid CellConsider: r = 0.48

r = 0.52 Dx = 1 mL = 10 mT = 0.75 m2/dS = 0.02

Explicit Results (Dt = 18.5 min; r = 0.48 < 0.5)

Explicit Results (Dt = 20 min; r = 0.52 > 0.5)

What’s Going On Here?• At time t = 0 no flow• At time t > 0 flow• Water released from storage

in a cell over time Dt

• Water flowing out of cell over interval Dt

Ground surface

Aquifer

hB

Confining Layer

b

hA

Dx

i = 0 1 2 … i-1 i i+1 … 8 9 10

Dx

Grid Cell i

r > 0.5Tme interval is too large Cell doesn’t contain enough water Causes instability

Implicit Method• Use information from one

point at the previous time step to compute the value at all points of this time step.

• Solve for all points in domain simultaneously.

• Inherently stable

li ,1

ix,

lt,

1, li

li ,1

1, li

x

tli,

1,1 li1,1 li

1,1 li 1,1 li

FD Approx. Backward

Implicit Method

l+1 time levelunknown

l time levelknown

2-D Steady-State Flow

• General Equation

• Homogeneous, isotropic aquifer, no well

• Equal spacing (average of surrounding cells)

jy,

ix,x

y

)4,1( )4,2( )4,3( )4,4(

)3,1( )3,2( )3,3( )3,4(

)2,1( )2,2( )2,3( )2,4(

)1,1( )1,2( )1,3( )1,4(

)0,1( )0,2( )0,3( )0,4(

)5,1( )5,2( )5,3( )5,4(

)4,0(

)3,0(

)2,0(

)1,0(

)4,5(

)3,5(

)2,5(

)1,5(

)4,5(

)5,1(Node No. Unknown heads

Known heads

2-D Heterogeneous Anisotropic Flow

j+ 1

j-1

j

i-1

i i+ 1

i+ 1 /2

j+ 1 /2

j-1 /2

x

y

Q x ,i+ 1 /2 Q x ,i-1 /2

Q y ,j+ 1 /2

Q y ,j-1 /2

x

y

n o d e ( i ,j) i-1 /2

ce ll ( i ,j)

Tx and Ty are transmissivities in the x and y directions

2-D Heterogeneous Anisotropic Flow• Harmonic average transmissivity

Transient Problems

MODFLOW

• USGS supported mathematical model• Uses finite-difference method• Several versions available

– MODFLOW 88, 96, 2000, 2005 (water.usgs.gov/nrp/gwsoftware/modflow.html)

• Graphical user interfaces for MODFLOW:– GWV (www.groundwater-vistas.com)

– GMS (www.ems-i.com)

– PMWIN (www.ifu.ethz.ch/publications/software/pmwin/index_EN)

– Each includes MODFLOW code

What Can MODFLOW Simulate?

1. Unconfined and confined aquifers2. Faults and other barriers3. Fine-grained confining units and

interbeds 4. Confining unit - Ground-water flow

and storage changes 5. River – aquifer water exchange6. Discharge of water from drains

and springs7. Ephemeral stream - aquifer water

exchange8. Reservoir - aquifer water exchange9. Recharge from precipitation and

irrigation 10. Evapotranspiration 11. Withdrawal or recharge wells12. Seawater intrusion