Conceptual Model

A descriptive representationof a groundwater system that incorporates an interpretation of the geological & hydrological conditions. Generally includes information about the water budget. May include information on water chemistry.

a set of equations that describesthe physical and/or chemicalprocesses occurring in a system.

Mathematical Model

R x y Q

y

x

z

1. Consider flux (q) through REV2. OUT – IN = - Storage3. Combine with: q = -KK grad h

q

Derivation of the Governing Equation

*)()()( Rt

hS

z

hK

zy

hK

yx

hK

xszyx

Rt

hS

y

hT

yx

hT

xyx

)()(

Rt

hS

y

hhK

yx

hhK

xyx

)()(

2D confined:

2D unconfined w/Dupuit assumptions:

Storage coefficient (S) is either storativity or specific yield.S = Ss b & T = K b

General 3D equation

Types of Boundary Conditions

1. Specified head

2. Specified flow (including no flow)

3. Head-dependent flow

From conceptual model

to mathematical model…

Toth Problem

Laplace Equation

2D, steady state

02

2

2

2

z

h

x

h0

x

h0

x

h

0z

h

h = c x + zo

Cross section through an unconfined aquifer.

Water table formsWater table formsthe upper boundary conditionthe upper boundary condition

b

h

ocean ocean

groundwater divide

“Confined” Island Recharge Problem

R

x = 0 x = Lx = - L

We can treat this system as a “confined” aquifer if we assume that T= Kb.

datum

T

R

y

h

x

h

2

2

2

2

Arealview

Water table is the solution.Water table is the solution.

Poisson’s Eqn.

2D horizontal flow through an unconfined aquifer where T=Kb.

b hocean ocean

groundwater divide

R

x = 0 x = Lx = - L

datum

Unconfined version of the Island Recharge Problem

K

R

y

h

x

h 22

22

2

22

Water table is the solution.Water table is the solution.

(Pumping can be accommodated by appropriatedefinition of the source/sink term.)

2D horizontal flow through an unconfined aquifer underthe Dupuit assumptions.

02

2

2

2

y

h

x

h

02

2

2

2

z

h

x

h

Vertical cross section through an unconfinedaquifer with the water table as the upper boundary.

2D horizontal flow in a confined aquifer; solutionis h(x,y), i.e., the potentiometric surface.

02

2

2

2

y

v

x

v

2D horizontal flow in an unconfined aquiferwhere v= h2. Solution is h(x,y), i.e., the water table.

All three governing equations are the LaPlace Eqn.

t = 0

t > 0

BC:h (0, t) = 16 m; t > 0h (L, t) = 11 m; t > 0

datum

0 L = 100 mx

IC: h (x, 0) = 16 m; 0 < x < L(represents static steady state)

t

h

T

S

x

h

2

2

Reservoir Problem

1D transient flow through a confined aquifer.

Solution techniques…

Three options:

• Iteration

• Direct solution by matrix inversion

• A combination of iteration and matrix solution

Examples of Iteration methods include:

Gauss-Seidel Iteration

Successive Over-Relaxation (SOR)

02

2

2

2

y

h

x

h

022

2

1,1,

2

,1,,1

y

hhh

x

hhh jiijjijijiji

41,1,,1,1

,

jijijijiji

hhhhh

Let x=y=a

4

11,1,

1,1,11

,

m

jim

jimji

mjim

ji

hhhhh

Gauss-Seidel Formula for 2D Laplace Equation

General SOR Formula

)( ,1

,,1

,mji

mji

mji

mji hhhh

Relaxation factor= 1 Gauss-Seidel< 1 under-relaxation>1 over-relaxation, typically between 1 and 2 (e.g., 1.8)

T

Rahhhhh

mji

mji

mji

mjim

ji 44

211,1,

1,1,11

,

Gauss-Seidel Formula for 2D Poisson Equation

SOR Formula

)( ,1

,,1

,mji

mji

mji

mji hhhh

Relaxation factor= 1 Gauss-Seidel< 1 under-relaxation>1 over-relaxation

(Eqn. 3.7W&A)

m

m+1

m+2

m+3

solution

(Initial guesses)

Iteration fora steady state problem.

Iteration levels

n

n+1

n+2

n+3

Steady state

t

t

t

Initial conditions(at steady state)

Transient Problemsrequire time steps.

Time levels

t

hh

T

S

x

hhh ni

ni

ni

ni

ni

1

211

)(

2

Explicit Approximation

t

h

T

S

x

h

2

2

t

hh

T

S

x

hhh ni

ni

ni

ni

ni

1

2

11

111

)(

2

Implicit Approximation

Or weighted average

• Explicit solutions do not require iteration but are unstable with large time steps.

• We can derive the stability criterion by writingthe explicit approx. in a form that looks like the SORiteration formula and setting the terms in theposition occupied by omega equal to 1.

• For the 1D governing equation used in the reservoirproblem, the stability criterion is:

1)(

22

xS

tT <

T

xSt

2)(5.0

<or

Implicit solutions require iteration

or direct solution by matrix inversion.

tIterationplanes

n

n+1

m+2

m+1

m+3

Solution by iteration

• Boundary conditions always affecta steady state solution.

• Initial conditions should be selected to represent a steady state configuration of heads.

Modeling “Rules”