Validation and Verification of Moving Boundary Models of Land Building ProcessesVaughan R. Voller
National Center for Earth-surface DynamicsCivil Engineering, University of Minnesota
Wax Lake Solid Crystal Growing in undercooled melt
Tetsuji Muto, Wonsuck Kim, Chris Paola, Gary Parker, John Swenson, Jorge Lorenzo Trueba, Man Liang
Fans Toes Shoreline
MovinG Boundaries in the Landscape
1km
Examples
Badwater Deathvalley
Sediment Fans
Sediment Delta
sediment
h(x,t)
x = u(t)
0q
bed-rock
ocean
x
shoreline
x = s(t)
land surface
An Ocean Basin
The Swenson Analogy: Melting vs. Shoreline movementSwenson et al, Eur J App Math, 2000
Physical Process
Isolate Key
Phenomena
Experiment
Pheno
men
ologic
al
Assum
ption
s 2
2
x
h
t
h
Model
Approximation
Assumptions
NumericalSolution
Validation: If assumptions forAnalytical solution are consistent withPhysical assumptions In experimentCan VALIDATE phenomenological assumptions
Verification: Comparisonof numerical and analytical predictions VERIFY Numerical Approach
The Modeling Paradigm
Limit CaseAssumptions
01
12
212
21
2
shsh
shshsh
erfe
erfsh
Analytical Solution
CASE OF CONSTANT BASE LEVEL and Bed Rock
• The delta progrades into standing water.• The rate of progradation slows in time as deeper water is invaded.• The bedrock-alluvial transition migrates upstream.
bedrock basement
sediment feed
below-capacity transport
capacity transport
constant base level
deltaic deposit
shoreline
bedrock-alluvial transition
foreset
topset
Slide from MUTO and PARKER---Muto Experiments
Experiments and image analysis by Tetsuji Muto and Wonsuck Kim, In slot flume
q0
h
)t(sx)t(s,x
h
t
h212
2
ts1 ts2
dt
dss
x
hiii 2
2
0h)iv(
0qx
h)i(
1sh)ii(
A mathematical model based on the Swenson Stefan Analogy with Fixed base slope and sea level
Note 4 conditions2 for the 2nd order equations2 for the 2 moving boundaries
dt
dss
x
hiii 2
2
1sh)ii(
Similarity Solution
)t(sx)t(s,x
h
t
h212
2
21
2t
x
21
2t
h
0h)iv(0qx
h)i(
212
2
02
,d
d
d
d
21
21
21
21
22
21
21
21
21
2
212
1
20
erferfe
erferfeq
21
21
21 ts ba 21
21
22 ts sh
20
0
2212
21
sh
shbash
shbash q
erferfe
erferfq
sh
0
0
1212
2
q
erferfe
eq
sh
ba
shbashsh
ba
q0
h
ts1
0qx
h)i(
1sh)ii(
)t(sx,x
h
t
H12
2
To develop numerical solution write problem in terms of Total Sediment Balance (enthalpy). Then there is NO need to treat shoreline conditions making for an easier numerical solution
xL
LHh,LhH if0
“Latent Heat”
Amount of sediment that needs to be providedTo move shoreline a unit distance (L = 0 in sub-aerial)
Numerical Solution
q
k=k-1
k-1 k
q
x
)k(hkh 1
i-1 i i+1
ONLAP CONDITION
x
ihih
x
ihih
x
tiHiH new )1()()()1()()(
h
q q
On-lap update—if Update on-lap node flag
0
)i(L)i(H)i(h
1<L<0
)t(sx,x
h
t
H12
2
Physical Process
Isolate Key
Phenomena
Experiment
Pheno
men
ologic
al
Assum
ption
s 2
2
x
h
t
h
Model
Approximation
Assumptions
NumericalSolution
Validation: If assumptions forAnalytical solution are consistent withPhysical assumptions In experimentCan VALIDATE phenomenological assumptions
The Modeling Paradigm
Limit CaseAssumptions
01
12
212
21
2
shsh
shshsh
erfe
erfsh
Analytical Solution
0
500
1000
1500
2000
2500
3000
3500
4000
-2000-1500-1000-50005001000
seaward landward
Experiments
Analytical Solution
Get Fit by choosing diffusivity
Bed porosity fixed at 30%
Experiment vs. Analytical: VALIDATION
Two Consistency Checks
1. Compare physical and Predicted surfaces
A little more concaved than we would like (experiment may be better modeled byNon-linear diffusion)
2. Across a range of experiments best fit diffusivity should scale withwater discharge
y = 0.0729x - 9.819
R2 = 0.9797
4
6
8
10
12
14
150 200 250 300 350
Reasonable
Physical Process
Isolate Key
Phenomena
Experiment
Pheno
men
ologic
al
Assum
ption
s 2
2
x
h
t
h
Model
Approximation
Assumptions
NumericalSolution
The Modeling Paradigm
Limit CaseAssumptions
01
12
212
21
2
shsh
shshsh
erfe
erfsh
Analytical Solution
Verification: Comparisonof numerical and analytical predictions VERIFY Numerical Approach
0
500
1000
1500
2000
2500
3000
3500
4000
-2000-1500-1000-50005001000
NUMERICAL VS. ANALYTICAL: Verification
An Interesting Limit Case
q0
t)(q
ts
2
No- on-lap
A horizontal fluvial surface coinciding with sea level
In a Two-Dimensional plan view this limit case gets a little more interesting
Current: Towards a CAFÉ Delta Model (Voller, Paola, Man-Ling)
The simulation shows a “particle” solution of the filling model. This is based onthe introduction, probabilistic movement, and deposition of particles in the domain. IT can be shown that this is a solution of the discrete equations associated with a Finite Element Model of the governing equations. Cellular RULES can be introducedby linking the probability of particle movement to the path taken. Thereby modelingchannels and vegetation.
Can make physical arguments that a suitable Background model is the filling of a thin-cavity (Hele-Shaw cell)
CAFÉ—Background deterministic (PDE) model solved with Finite Elements Superimposed with a Cellular (rule based Model)
Some ExamplesUniform Probs
High MiddleProb
High Edge
Efi Research Question: How is CADFE model based on a “normal” PDERelated to a “fractional derivative PDE”
Saltwater intrusion occurs when saltwater from the Gulf moves into areas that have formerly been influenced by freshwater. As saltwater intrudes into a fresh marsh, the habitat will be altered as the plants and organisms that once thrived in the freshwater marsh cannot survive in saltwater. If the intrusion of saltwater is gradual enough, plants and organisms that can survive in a saltwater habitat begin to invade and grow, eventually establishing a brackish marsh. If saltwater vegetation does not replace the freshwater plants, the area will become exposed mud flats, and they are likely to revert to open water. This process is common in an abandoned delta lobe where the discharge of the river decreases or even in areas of the modern delta where freshwater is diverted or maintained within existing channels.
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12 14
Depth
Mineralization
M = 0.5 = 1DC-A = 1DB-A =0.2
CB=1=CA
CB=2=CA
CB=1.5>CA
