Nils Otto Kitterød
FEMLAB conference, Oslo October 13. 2005

• teaching
• how to make COMSOL MULTIPHYSICS behave according to my concepts
• examples

Important for society
Groundwater resources of the world
Major part of the world population are depending on groundwater resources

Important for nature unsaturated zone saturated zone
The global water balance
~ 30 times more groundwater than water in the lakes
Groundwater flow is a boundary value problem => measurements

Groundwater: the liquid of life!
interface to human society and science groundwater flow concept of geology numerical simulation contaminant transport

Concept of flow: Darcy's experiments (1856):
∆ x
∆ h
Henri-Philibert-Gaspard Darcy
Darcy’s law: Q
∝
A
∆ h
∆ x
→
Conservation of momentum
A governs the laminar (nonturbulent) flow of fluids in homogeneous, porous media

1. Conservation of momentum: q
= −
K
∇ h,
2. Conservation of mass: where h = p/(
ρ g) + z
∂ (
S
0 h )
∂ t
= − ∇⋅ q
+
N, where S
0 is specific storage [1/L], and N is a sink/source term
3. Constitutive relations: e.g. if S
0 is a function of h
→
Richards’ equation (not today!) or if
ρ is a function of concentration of solute (later)
Here: Dupuit-Forchheimer assumption:
∂ h
∂ z
= 0, but q z
≠
0 which means 3D
→
2D
4. Boundary conditions and (if transient) initial conditions

m a.s.l.
200
180
160
Delta Foresets
Delta Bottomsets
140
120
Bedrock
3000
West
200
100
West
3000
2000 meters
2000 meters
Delta Topsets
Kettlehole
Ice-contact slope
1:~15
?
?
?
1000
1000 500
East
0
East
0 m a.s.l.
1:1
After K. Tuttle (1997)

From 1,2 we have:
From 3, life gets easier:
For a confined aquifer: m = top – bottom of aquifer
For an unconfined aquifer: m = h – bottom of aquifer
∂ ( S
0 h )
∂ t
= ∇⋅
K
∇ h
+
N.
∂ ( m S
0 h )
∂ t
= ∇⋅ m K
∇ h
+
N, where m is thickness of water saturated zone mS
0
= S [-] is called storage coefficient m K = T [L 2 /T] is called transmissivity
H
1
H
2 top bottom x m = const.
length of aquifer
H
1
H
2 m = h(x,t) bottom x length of aquifer

Implement the Dupuit-Forchheimer assumption:
Remember (1): h(t
0
)
≠
0 if bottom = 0
Transmissivity
Remember (2): Q = Aq (L 3 /T)
By using boundary integration in postprocessing, you get the water flow directly!

0 m
T
∫
T(z) dz in a way that the response h is reproduced according to the observations
( S h )
t
T
h
N(t)
The Boussinesq equation where: t
T
S h effective transmissivity (L 2 /T) storage coefficient(-) hydraulic head (L) time (T)
N(t) source/sink (L/T), infiltration or pumping

15 km
15 km
The inland glacier
~10500 B.P.
(Andersen, B.G.,2000)
Sea level about 200 m higher than present sea level.
Average paleodischarge: ~3000m 2 /s
(Tuttle, K., 1998)
The Gardermoen paleo-delta

Trandum Delta
Paleo-portal
Helgebostad Delta
Paleo-portal

Paleo-distribution Channels Show Radial Flow:
11000
10000
9000
8000
7000
6000
5000
4000 x-profile Trandum Delta
Paleo-portal
Helgebostad Delta
Paleo-portal
3000
2000
1000
4000 5000 6000 7000 8000 9000 10000 11000 12000
UTM-East Gardermoen, grw. obs, eu89

Steady state model for R
1
r
l : i) Darcy’s law: a)
Q m r = q = – k b) Q r
= – d( 1 /
2 kh 2 ) dr ii) Balance of mass: dh dr c) Q
Α
= N
π l 2 – N
π r 2 d) Q r
= –
Q
Α
2
π r if m=h, then unconfined aquifer i and ii): d( 1 /
2 kh 2 ) =
N
2 l 2 r dr – r dr h
R
2
2 l h
1
R
1
Repeat for l ≤ r ≤ R
2 and eliminate l to get one closed form expression for h(r)

Plot observations of groundwater head as a function of radial distance from center: h
2
→ r
R
2 l l
N
R
1 h
1 let k = k(r) or H = H(r)

dh 2
N l 2 rk dr
– k r dr where k = k
1
– a(r-R
1
)
Two (simple) integrals:
∫ 1 r(k
0
-ar) dr
∫ r
(k
0
-ar) dr
(1)
(2) h
1
R
1 k
1
N
→ l l R
2 k
2 h
2 r
WRR(40) 2004 http://folk.uio.no/nilsotto/PUBL/analytical_gilbert.pdf
Implement the solution as a function in MATLAB: function[h] = funk_h_lin(radius, R1, R2, h1, h2, N, k_1, k_2)

function[h] = funk_h_lin(radius, R1, R2, h1, h2, N, k_1, k_2) minimum(h obs
– h calc
) gives k
1 at R
1 and k
2 at R
2
R
2 h
2 l h
1
R
1

Radial Duit-Forchheimer Solution where k=k(r)
Interpolation Functions:
R
1
R
2 k
1 k
2

Steady State Numerical Solution and
Analytical Solution:
Transient Simulation
Precipitation event
18.00
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 days from 27.09.2000
S=0.01

Precipitation event
18.00
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 days from 27.09.2000
Transient boundary condition:
H
1
(t) = H
1 sin(t/day)
Conclusion:
1) minor changes in h with N(t)
2) not very sensitive to H
1
OK !
But, what about the real aquifer?

190
190
Trandum Delta
Paleo-portal
Helgebostad Delta
Paleo-portal
180
188
188
188

Digitize boundary by using: GINPUT Graphical input from mouse.
[X,Y] = GINPUT(N) gets N points from the current axes and returns the X- and Y-coordinates in length N vectors X and Y.

Spline interpolation

h
2
N h
1 noflux boun dary
Make solid object by: s1 = geomcoerce('solid',{c}); geomplot(s1); and import geometry in Comsol Multiphysics

What about hydraulic conductivity?
Two radial structures with linearly decreasing k superimposed on each other.
What is best average?
Trandum Delta
Paleo-portal
Helgebostad Delta
Paleo-portal

arithmetic mean ?
1/T
1
1/T
2
Glacial portals

geometric mean ?
Glacial portals

harmonic mean ?
1/T
1
1/T
2
Glacial portals

Difference in hydraulic conductivity between harmonic and geometric mean

Hydraulic head
Input: hydraulic head as arithmetic mean between the Trandum and Li deltas arithmetic

Hydraulic head
Input: hydraulic head as geometric mean between the Trandum and Li deltas geometric arithmetic

Hydraulic head
Input: hydraulic head as harmonic mean between the Trandum and Li deltas harmonic geometric arithmetic

Hydraulic head gradient, minimum gradient at ground water divide max. gradient in ravine areas (landslide) and glacial boundary (kettle lake area)
No terminal and railway culvert (drawdown)

Hydraulic head gradient
With terminal and railway culvert (drawdown)
To protect the preservation area, the water balance of the area has to be maintained because the hydraulic gradient is the driving force of the gully processes.

Hydraulic head gradient
With terminal and railway culvert (drawdown) and two injection points of water from the culvert

The C omso l mes hing r outin es m akes l ife m ore p ractic al!

Plot cross-section before and after Airport construction

Note change in head gradient towards ravine area

Important for society Groundwater resources of the world
The majority of the world population depends on groundwater resources
Coastal aquifers : possible contamination of groundwater by seawater intrusion

Density driven flow:
2-way coupling between flow & transport
• Density dependent fluid flow - Darcy’s Law expresses in terms of pressure p :
ρ
[
ξ
( 1
− θ
)
+ ζθ
]
∂ p
∂ t
+ θ
∂ ρ
∂ c compressibility = 0
∂ c
∂ t
+ ∇ ⋅
[
− ρ κ
/
η ∇
( p
+ ρ gD )
]
=
0
• Salt concentration – Saturated solute transport
∂ c
∂ t
+ ∇ ⋅
[
−
D
∇ c
+ u C
]
=
0
•
ρ varies with c
( c
c
0
)
where:
c s f
c
0 o from the presentation by
Leigh Soutter

The Henry Problem

Dm = 1.6295 m2/d

Dm = 0.57024 m2/d

Decrease influx of fresh water from 5.702 m 3 /s to 0.05702 m 3 /s

What is the effect of tidal changes of 0.1 times thickness of aquifer?
Time variable boundary conditions

COMSOL MULTIPHYSICS is useful for teaching and business and it’s fun!