The document summarizes research on quantifying uncertainty in groundwater contamination modeling. It discusses using stochastic methods and surrogate models to estimate how uncertainties in geological parameters, like porosity, propagate and influence quantities of interest in groundwater flow and contaminant transport simulations. Numerical experiments were conducted in 2D and 3D domains using parallel computing to analyze mean concentrations, variances, and other statistics over time under different uncertainty scenarios.

1. Uncertainty quantification of groundwater
contamination
Alexander Litvinenko, joint work with Dmitry Logashenko, Raul
Tempone, Gabriel Wittum and David Keyes
KAUST
https://ecrc.kaust.edu.sa/
Extreme Computing Research Center
Group Seminar at KAUST,
October 8, 2018

2. 4*
The structure of the talk
Major Goal: estimate propagation of uncertainties in the
groundwater flow.
1. Density-driven groundwater flow problem
2. Stochastic modeling and stochastic methods
3. Modeling of porosity
4. Numerical methods
5. 2D numerical experiments
6. 3D numerical experiments
7. Conclusion and best practices
2

3. 4*
Motivation
As groundwater is an essential nutrition
and irrigation resource, its pollution may
lead to catastrophic consequences.
Accurate modeling of the pollution of the
soil and groundwater aquifer is impossi-
ble due to presence of uncertainties in
geological parameters.
Applications: seawater intrusion into coastal aquifers, radioactive
waste disposal, contaminant plumes etc.
3

4. 4*
What do we compute?
The mean and the variance of QoIs
Compute when dangerous concentration in a point achieve a
certain level
How long it takes to achieve a given concentration ?
Estimate the risk that the pollutant concentration exceeds a
certain level
4

5. 4*
Governing equations
∂t(φρ) + · (ρq) = 0, (1)
∂t(φρc) + · (ρcq − ρD c) = 0, (2)
where c is the mass fraction of the salt, the tensor field D
represents the molecular diffusion and the mechanical dispersion of
the salt. We assume the Darcy’s law for q:
q = −
K
µ
( p − ρg), (3)
where p(t, x) is the hydrostatic pressure and g the gravity.
5

6. 4*
Governing equations
Assume permeability
K = KI, where K = K(x) ∈ R, and I ∈ Rd×d
the identity matrix.
Use the linear dependence for the density:
ρ(c) = ρ0 + (ρ1 − ρ0)c,
where ρ0 and ρ1 denote the densities of pure water and the brine,
respectively.
Thus, c ∈ [0, 1] with c = 0 corresponding to the pure water and
c = 1 to the saturated solution.
Assume
D = φDmI,
Dm the coefficient of the molecular diffusion. We neglect the
dispersion.

7. 4*
Computational domains
c = 1c = 0
c = 0
c = 0
600 m
300 m
150 m
Schema of 3 layers 2D reservoir D = (0, 600) × (0, 150) and 3
realisations of the porosity.
φ(x) ∈ [0.05, 0.09], φ(x) ∈ [0.077, 0.11], and φ(x) ∈ [0.097, 0.115].
7

8. 4*
Two 3D reservoirs
(left) D = (0, 600) × (0, 600) × (0, 150) m3.
BC: Zero-flux for the entire fluid phase; concentration: c = 1 in
the red spot, c = 0 otherwise on the top and Neumann-0 at the
other boundaries.
The pressure is set to 0 on the edges between the green and the
blue parts.
8

9. 4*
Mathematical equation
We introduce φ and assume K to be isotropic and dependent on φ:
K = KI, K = K(φ) ∈ R. (4)
The distribution of φ(x, θ), x ∈ D, θ = (θ1, ..., θM, ...). Each
component θi is a random variable depending on random event ω,
for shortness we skip ω and write θ := θ(ω).
Assume (Kozeny-Carman-like equation)
K(φ) = κKC ·
φ3
1 − φ2
, (5)
where a scaling factor κKC .
9

10. 4*
Statistics
The empirical mean
c(t, x) ≈
Nq
i=1
wi c(t, x, θi )
def
=
Nq
i=1
wi ci ,
where Nq - number of quadrature points, wi quadrature weights, ci
are “scenarios”.
The empirical variance
Var[c](t, x) ≈
Nq
i=1
wi (c(t, x) − c(t, x, θi ))2
.
10

11. 4*
gPCE based surrogate
c(t, x, θ) =
β∈J
cβ(t, x)Ψβ(θ),
where {Ψβ} is a multivariate Legendre basis, β = (β1, ..., βj , ...) a
multiindex and J a multiindex set.
Ψβ(θ) :=
∞
j=1
ψβj
(θj ); ∀θ ∈ RN
,
ψβj
(·) are Legendre monomials,
cβ(t, x) ≈
1
Ψβ, Ψβ
Nq
i=1
Ψβ(t, θi )c(t, x, θi )wi ,
11

12. 4*
Numerical stability w.r.t. spatial resolution
Took 200 various scenarios of the porosity.
Compute and compare variances of the mass concentration on
grids with n dofs after t = 5.5 years.
Grid levels n value of QoI
6 132.000 Var[c](x) ∈ [0.0, 0.08]
7 526.000 Var[c](x) ∈ [0.0, 0.08]
8 2.100.000 Var[c](x) ∈ [0.0, 0.08]
Two-level parallelization:
each scenario is computed in parallel on 32 cores,
all scenarios are computed in parallel on 200 nodes.
12

13. 4*
Utilized numerical methods
UG4 is a flexible software system for simulating PDE based models
on high performance parallel clusters (G. Wittum and his group).
Computation of one scenario:
1. Spatial discretization on unstructured grids.
2. Implicit Euler schema in time.
3. Newton method with line search.
4. Solution of linearized systems by BiCGStab with multigrid
preconditioning (V-cycle, ILU-smoothers).
5. Parallelisation is based on the distribution of the domain
between cores.
M scenarios are computed in parallel.
Run on 4-8 spatial grid levels with n = 0.5 . . . 8 Mio grid points.
Used 1 . . . 800 Shaheen nodes, computation time is 2-24 hours,
1000-1800 time steps, modeling time interval 5 − 8 years.
13

14. 4*
2D example with 1 RV and small variance
φ(x, ω) = 0.09 + 0.005ξ(cos(x/300) + sin(y/150)),
where ξ ∼ U[−1, 1], time=5.5 years.
1st row: c(x) ∈ (0, 1) computed via MC (200 simulations) and via
gPCE4 (m = 1, p = 4);
2nd row: Var[c]MC ∈ (0, 0.021), Var[c]gPCE4 ∈ (0, 0.023).
Conclusion: 9 GL points gives almost the same result as 200 MC
points, but are much faster.
14

15. 4*
2D example with 2 RVs and larger variance
φ(x, ω) = 0.1 + 0.01(ξ1 cos(x/1200) + ξ2 sin(y/300)),
where ξ1, ξ2 ∼ U[−1, 1], 1.5 years.
1st row: c(x), computed via MC (1500 simulations) and via gPCE,
c(x) ∈ (0, 1), Var[c]MC ∈ (0, 0.076),
2nd row: Var[c]gPCE5 ∈ (0, 0.068), Var[c]gPCE7 ∈ (0, 0.0714),
Var[c]gPCE9 ∈ (0, 0.0847).
Conclusion: Our surrogate and MC give similar c.
Var[c] computed by surrogate of order 7 is most close to the MC
variance. 15

16. 4*
Difficulties caused by uncertain porosity
Relative small variations in the porosity may result in 3 different
realizations of the mass fraction: with (a) 5 fingers; (b) 4 fingers;
(c) 5 fingers.
(a) (b) (c)
Non-linearity may result in several stationary solutions.
16

17. 4*
Evolution of variance in time
Below we plot Var[c] after 2.75, 5.5 and 8.25 years.
The variance is accumulated and growing.
(a) 2.75year,
Var[c](x) ∈ (0, 0.023)
(b) 5.5 year,
Var[c](x) ∈ (0, 0.055)
(c) 8.25year,
Var[c](x) ∈ (0, 0.07)
Results are obtained with 700 quasi MC samples.
17

18. 4*
3D reservoir
φ(x, θ) = 0.1 + exp(θ1 sin(πx/600) + θ2 sin(πy/600) + θ3 sin(πz/150) + θ1 sin(πx/600)+
+ θ1 sin(πx/600) sin(πy/600) + θ2 sin(πx/600) sin(πz/150) + θ3 sin(πy/600) sin(πz/150)).
(a) (b)
Five isosurfaces of c after 9.6 years
18

19. 4*
Isosurfaces of Var[c] in 3D reservoir
(a) (b)
Var[c] after 4.8 years, N ≈ 8 · 106 grid points.
19

20. 4*
Isosurfaces of Var[c] in 3D reservoir
(a) (b)
Isosurfaces of the variance of the mass fraction after 3 years;
(a) Var[c]0.05; (b) Var[c]0.15.
20

21. 4*
Evolution of probability density function in a point
PDFs at aquifer point (100, 0, −25) after (a) 0.6, (b) 1.2, (c) 1.8,
and (d) 2.4 years.
21

22. 4*
Propagation of the contamination
Evolution of the mean concentration in time after a) 0, b) 0.55, c)
1.1, d) 2.2 years. The cutting plane is (150, y, z). 22

23. 4*
Model: 3D reservoir with three layers
1st row: three layers of the porosity; profiles of c;
2nd row: isosurface |cdet − c|0.25; isosurfaces Var[c]0.05 and
Var[c]0.12.
23

24. 4*
Conclusion
Solved time-dependent, non-linear density driven flow problem
with uncertain porosity and permeability in 2D and 3D
Computed propagation of uncertainties in porosity into the
mass fraction. Computed the mean, variance, exceedance
probabilities, quantiles, risks.
For moderate perturbations, our gPCE cheap surrogate model
successfully replaces expensive MC simulations
Used highly scalable solver on up to 800 nodes
For large variance of porosity, standard sparse grids may fail,
visualization of dependence of QoI on the uncertain
parameters may help
Such QoIs as the number of fingers, their size, propagation
time are unstable for high variability in porosity

25. 4*
Acknowledgement
1. Elmar Zander (TU Braunschweig) for the sglib library.
2. KAUST, Shaheen project k1051, 2.7 Mio hours.
3. KAUST Supercomputing Lab.
4. KAUST Visualization Lab.
THANK YOU FOR YOUR ATTENTION !
25

26. 4*
Literature
1. S. Reiter, A. Vogel, I. Heppner, M. Rupp, and G. Wittum, A massively parallel geometric multigrid solver
on hierarchically distributed grids, Computing and visualization in science 16, 4 (2013), pp 151-164, DOI:
10.1007/s00791-014-0231-x
2. A. Vogel, S. Reiter, M. Rupp, A. N¨agel, and G. Wittum, UG4 – a novel flexible software system for
simulating PDE based models on high performance computers. Computing and visualization in science 16,
4 (2013), pp 165-179, DOI: 10.1007/s00791-014-0232-9
3. A. Schneider, H. Zhao, J. Wolf, D. Logashenko, S. Reiter, M. Howahr, M. Eley, M. Gelleszun, H.
Wiederhold, Modeling saltwater intrusion scenarios for a coastal aquifer at the German North Sea, E3S
Web of Conferences 54, 00031 (2018), DOI:10.1051/e3sconf/20185400031
4. P. Waehnert, W.Hackbusch, M. Espig, A. Litvinenko, H. Matthies: Efficient low-rank approximation of the
stochastic Galerkin matrix in the tensor format, Computers & Mathematics with Applications, 67 (4), pp
818-829, 2014
5. A. Litvinenko, D. Keyes, V. Khoromskaia, B. N. Khoromskij, H. G. Matthies, Tucker Tensor Analysis of
Matern Functions in Spatial Statistics, DOI: 10.1515/cmam-2018-0022, Computational Methods in
Applied Mathematics , 2018.
26

27. 4*
Literature
6. S. Dolgov, B.N. Khoromskij, A. Litvinenko, H.G. Matthies, Computation of the Response Surface in the
Tensor Train data format arXiv preprint arXiv:1406.2816, 2014
7. S. Dolgov, B.N. Khoromskij, A. Litvinenko, H.G. Matthies, Polynomial Chaos Expansion of Random
Coefficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Format,
IAM/ASA J. Uncertainty Quantification 3 (1), pp 1109-1135, 2015
8. A. Litvinenko, H.G. Matthies, T.A. El-Moselhy, Sampling and low-rank tensor approximation of the
response surface, Monte Carlo and Quasi-Monte Carlo Methods 2012, pp 535-551, Springer, 2013.
9. A. Litvinenko, Application of hierarchical matrices for solving multiscale problems, PhD, Leipzig University,
Germany, 2006.
10. A. Litvinenko, H.G. Matthies, Inverse problems and uncertainty quantification, arXiv:1312.5048, 2013.
11. A. Litvinenko, H.G. Matthies, Sparse data representation of random fields, PAMM: Proceedings in Applied
Mathematics and Mechanics 9 (1), pp 587-588, 2009.
27