nils-otto kitterød femlab conference, oslo october...
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ModellingModelling SubsurfaceSubsurface FlowFlow −−not not whatwhat youyou seesee, , butbut whatwhat youyou imagineimagine
NilsNils--Otto KitterødOtto Kitterød
FEMLAB conference, Oslo October 13. 2005
Outline:
• teaching
• how to make COMSOL MULTIPHYSICS behave according to my concepts
• examples
Why focus on groundwater?
Important for societyGroundwater resources of the world
Major part of the world population are depending on groundwater
resources
The global water balance
~ 30 times more groundwater than water in the lakes
Groundwater flow is a boundary
value problem => measurements
unsaturated zone
saturated zone
Important for nature
Why focus on groundwater?
Groundwater: the liquid of life!interface to human society and science
groundwater flow contaminant transport
numerical simulation
concept ofgeology
Concept of flow:
Henri-Philibert-Gaspard Darcy
Darcy's experiments (1856):
∆h
∆x
A
Q ∝ A ∆h∆x
Darcy’s law:governs the laminar (nonturbulent) flow of fluids in homogeneous, porous media
→ Conservation of momentum
1. Conservation of momentum:
q = − K ∇h, where h = p/(ρg) + z
2. Conservation of mass:
+ N,∂ ( S0 h )
∂t− ∇⋅q= where S0 is specific storage [1/L],
and N is a sink/source term3. Constitutive relations:
e.g. if S0 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, which means 3D → 2Dbut qz ≠ 0
4. Boundary conditions and (if transient) initial conditions
Cross Section of the Gardermoen Delta
After K. Tuttle (1997)
100020003000
200
100
m a.s.l.
West meters East
05001:1
0100020003000
200
180
160
140
120
m a.s.l.
Bedrock
Delta Bottomsets
Delta Foresets
Delta Topsets
??
?
Kettle-hole
Ice-contactslope
1:~15
West meters East
From 1,2 we have: + N.∂ ( S0 h )
∂t∇⋅= K ∇h
where m is thickness of water saturated zonemS0 = S [-] is called storage coefficientmK = T [L2/T] is called transmissivity
For a confined aquifer:
m = top – bottom of aquifer
For an unconfined aquifer:
m = h – bottom of aquifer
H1H2
length of aquiferx
top
bottom m = const.
H1H2
length of aquiferx
bottomm = h(x,t)
From 3, life gets easier: + N,∂ ( mS0 h )
∂t∇⋅= mK ∇h
Remember (2): Q = Aq (L3/T)By using boundary integration in postprocessing, you get the water flow directly!
Transmissivity
Implement the Dupuit-Forchheimer assumption:
Remember (1): h(t0) ≠ 0if bottom = 0
Challenge: estimate Challenge: estimate effectiveeffective parametersparameters
The Boussinesq equation( )hT= ∇ −⋅∇∂∂
t( S h ) N(t)
in a way that the response h is reproduced according to the observations
where: T effective transmissivity (L2/T) S storage coefficient(-) h hydraulic head (L)t time (T)N(t) source/sink (L/T), infiltration or pumping
T = ∫ T(z) dz0
m
The inland glacier ~10500 B.P. (Andersen, B.G.,2000)
Sea level about 200 m higher than present sea level.Average paleo-discharge: ~3000m2/s (Tuttle, K., 1998)
Example: The Gardermoen Delta
The Gardermoenpaleo-delta
15 km
15 km
Gardermoen, grw. obs, eu89
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
11000
4000 5000 6000 7000 8000 9000 10000 11000 12000
UTM-East
UTM
-Nor
th
Paleo-distribution Channels Show Radial Flow:
Trandum Delta Paleo-portal
Helgebostad Delta Paleo-portal
x-profile
””The The GardermoenGardermoen Doughnut”Doughnut”
ii) Balance of mass:
c) QΑ = Nπl2 – Nπr2
d) Qr = – QΑ
2πr
i and ii): d(1/2 kh2) = N2
l2
rdr – r dr
i) Darcy’s law:
= q = – k dhdr
Qr
ma)
Qr = – d(1/2 kh2)dr
b) if m=h, then unconfined aquifer
R2 h2 R1
h1lSteady state modelfor R1 ≤ r ≤ l :
Repeat for l ≤ r ≤ R2
and eliminate l to get oneclosed form expression for h(r)
Plot observations of groundwater head as a function of radial distance from center:
let k = k(r)or H = H(r)
rR2 R1l
h2
h1
N
l
→
WRR(40) 2004http://folk.uio.no/nilsotto/PUBL/analytical_gilbert.pdf
where k = k1 – a(r-R1)
∫ 1r(k0-ar) dr
∫ r(k0-ar) dr
Two (simple) integrals:
(1)
(2)
rR2 R1 l
h2
h1
N
l
→
k1 k2
dh2 N l2
rk dr dr– rk=
Implement the solution as a function in MATLAB:
function[h] = funk_h_lin(radius, R1, R2, h1, h2, N, k_1, k_2)
minimum(hobs – hcalc) gives k1 at R1 and k2 at R2
function[h] = funk_h_lin(radius, R1, R2, h1, h2, N, k_1, k_2)
R2 h2
R1h1
l
Steady State Numerical SolutionandAnalytical Solution:
S=0.01
Precipitation event
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31days from 27.09.2000
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
mm
/d
Transient Simulation
OK !But, what about the real aquifer?
Precipitation event
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31days from 27.09.2000
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00m
m/d
Transient boundary condition:
H1(t) = H1sin(t/day)
Conclusion:1) minor changes in h with N(t)2) not very sensitive to H1
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.
Make solid object by: s1 = geomcoerce('solid',{c});geomplot(s1);and import geometry in Comsol Multiphysics
noflux boundary
h2
h1
N
What about hydraulic conductivity?
Two radial structures with linearly decreasing ksuperimposed on each other. Trandum Delta
Paleo-portal
Helgebostad Delta Paleo-portal
What is best average?
Hydraulic headInput: hydraulic head as geometric mean between the Trandum and Li deltas
arithmetic
geometric
Hydraulic headInput: hydraulic head as harmonic mean between the Trandum and Li deltas
arithmetic
geometric
harmonic
No terminal and railway culvert (drawdown)
Hydraulic head gradient,minimum gradient at ground water dividemax. gradient in ravine areas (landslide) and glacial boundary (kettle lake area)
With terminal and railway culvert (drawdown)
Hydraulic head gradient
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 gradientWith terminal and railway culvert (drawdown)and two injection points of water from the culvert
The majority of the world population depends on groundwater resources
Groundwater resources of the worldImportant for society
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:
[ ] 0)(/])1([ =+∇−⋅∇+∂∂
∂∂
+∂∂
+− gDptc
ctp ρηκρρθζθθξρ
[ ] 0=+∇−⋅∇+∂∂ CcD
tc uθθ
• Salt concentration – Saturated solute transport
compressibility = 0
)( 0cc −+= γρρ• ρ varies with c
0ccs−=γ ρoρf −
where:
from the presentation byLeigh Soutter
Time variable boundary
conditions
What is the effect of tidal changes of 0.1 times thickness of aquifer?
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