Lecture (3) Lecture (3) Uncertainty in Groundwater Flow and Transport Models. 0 50 100 150 200 250 H o rizo n ta lD ista n ce b e tw een W e lls (m ) -5 0 0 D ep th (m ) W e ll 1 W e ll 2 ? K (x ,y ,z)? (x ,y ,z )? C (x ,y ,z)? H=? H=? ? ? ? ? ? ? ? ? ?
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Lecture (3)Lecture (3)
Uncertainty in Groundwater Flow and Transport Models.
0 50 100 150 200 250Horizontal Distance between W ells (m)
-50
0
Dep
th (m
)
W ell 1 W ell 2
? K(x,y,z)?
(x,y,z)?
C(x,y,z)?H=?
H=?
?
???
? ?
?
?
?
Layout of the LectureLayout of the Lecture
• What is uncertainty?What is uncertainty?• Why addressing the uncertainty by the Why addressing the uncertainty by the
stochastic approach?stochastic approach?• How to model uncertainty?How to model uncertainty?
Monte-Carlo sampling.Monte-Carlo sampling.• How to quantify uncertainty? How to quantify uncertainty?
Stochastic Differential Eqs. & Monte Carlo Stochastic Differential Eqs. & Monte Carlo method.method.
• How to reduce uncertainty?How to reduce uncertainty?• Some Applications.Some Applications.
What is Uncertainty?What is Uncertainty?
0 50 100 150 200 250Horizontal Distance between W ells (m )
-50
0
Dep
th (m
)
W ell 1 W ell 2
? K(x,y,z)?
(x,y,z)?
C(x,y,z)?H=?
H=?
?
???
? ?
?
?
?
Classification of Uncertainty:
-Conceptual Model Uncertainty:Darcy’s and Fick’s Laws.
-Geological Uncertainty:Connectivity and dis-connectivity of the layers, geological sequence,boundaries between geological units.
-Parameter Uncertainty: -K, porosity.
-Hydro-geological Uncertainty:Constant head boundaries, impermeable boundaries, Plume boundaries, source area bounaries.
Why Addressing the Uncertainty by the Why Addressing the Uncertainty by the Stochastic Approach?Stochastic Approach?
- The uncertainty due to the The uncertainty due to the lack of information about lack of information about the subsurface structure the subsurface structure which is known only at which is known only at sparse sampled locations.sparse sampled locations.
-The erratic nature of -The erratic nature of the subsurface the subsurface parameters observed parameters observed at field scale.at field scale.
Monte-Carlo SamplingMonte-Carlo Sampling
Uniform random number generator:
Multiplicative Congruence Method developed by Lehmer [1951].
1( )i i-
i i
= MODULO A. B , MN N = /MU N
Ni is a pseudo-random integer, i is subscript of successive pseudo-random integers produced, i-1 is the immediately preceding integer, M is a large integer used as the modulus, A and B are integer constants used to govern the relationship in company with M, Ui is a pseudo-random number in the range {0,1}, and " MODULO" notation indicates that Ni is the remainder of the division of (A.Ni-1) by M.
Uniform Random Number ExampleUniform Random Number Example
1.0,5.0,3.0,9.0.......5,3,9,1,5,3,9:
)3(710737319*8
)9(0109911*8
)1(410414115*8
)5(210252513*8
)3(710737319*8
9)()10()18(
0
1
sequence
remainder
remainder
remainder
remainder
remainder
seedNMODULO N = N i-i
Generation of a Random Variable from any Generation of a Random Variable from any DistributionDistribution
Inverse of Distribution Function.
Transformation Method.
Acceptance-Rejection Method.
Inverse of Distribution FunctionInverse of Distribution Function
)()(
')'()(
UF = F = U
αdαf= αF
1-
α
-
Transformation MethodTransformation Method
Random number generator for normal distribution
(from central limit theory):" Observations which are the sum of many independently operating processes tend to be normally distributed as the number of effects becomes large"
12
21
m/
- m/Uε =
m
i=i
with mean (μ=0) and unit standard deviation (σ=1), Ui is the i-th element of a sequence of random numbers from a uniform distribution in the range {0,1}, and m is the number of Ui to be used.
612
1
- Uε = i
i
If m is 12, a normal distribution with tails truncated at six times standard deviation is produced
σ + εμα = αα
Example of IDF in Discrete 1D Markov ChainExample of IDF in Discrete 1D Markov Chain
Analytical Approaches G reen 's F u n c tion A p p roach
P ertu rb a tion M eth odS p ec tra l M eth od
Num erical ApproachesM on teC arlo M eth od
S o lvin g S D E s
Monte-Carlo MethodMonte-Carlo Method1. Generate a realization of a random field of the parameter under study.2. A classical numerical flow or/and transport model is run on the random field
and a set of results is obtained. 3. Another random field is made and the model is run again, and so on. 4. It's necessary to have a very large number of runs, and the output model
results corresponding to each input is obtained. 5. Statistical analysis of the ensemble of the output can be made to get the
mean, the variance, the covariance or the probability density function for each node with a location in the grid.
0 4 8 12 16 20Distance in the mean Flow direction (m)
0
0.005
0.01
0.015
0.02
0.025
Var (
h)
X-direction
Solute Transport EquationSolute Transport Equation
Dispersion DiffusionAdvection
ij ii j i
C C CVDt x x x
where C is the concentration field at time t, Dij is the hydrodynamic dispersion tensor, i, j are counters, Vi is the component of the Eulerian interstitial velocity in xi direction defined as follows,
j
iji x
K
- = V
where Kij is the hydraulic conductivity tensor, and is the porosity of the medium.
Set-up of the Monte Carlo Transport Set-up of the Monte Carlo Transport Experiment Experiment
.Xc (t)
tx x
y y t(Xo,Yo)
(Xo,Yo) Initial Source Location.
Xc(t) is Plume centroid in X-direction.
2xx(t) is Plume longitudinal variance.
2yy(t) is Plume transverse variance.
Heterogeneous FieldHeterogeneous Field
2 7 12 17 22 27 32 37 42 47
0 20 40 60 80 100 120 140 160 180 200-30
-20
-10
0
0 20 40 60 80 100 120 140 160 180 200-30
-20
-10
0
0 20 40 60 80 100 120 140 160 180 200-30
-20
-10
0
K-field
Flow field
K (m/day)
Single Realization SimulationSingle Realization Simulation