This document summarizes a talk on solving density-driven groundwater flow problems with uncertain porosity and permeability coefficients. The major goal is to estimate pollution risks in subsurface flows. The presentation covers: (1) setting up the groundwater flow problem; (2) reviewing stochastic modeling methods; (3) modeling uncertainty in porosity and permeability; (4) numerical methods to solve deterministic problems; and (5) 2D and 3D numerical experiments. The experiments demonstrate computing statistics of contaminant concentration and its propagation under uncertain parameters.

1. Solution of Density Driven Groundwater Flow
with Uncertain Porosity and Permeability
Coefficients
Alexander Litvinenko,
joint work with
Dmitry Logashenko, Raul Tempone, Gabriel Wittum and David Keyes
RWTH Aachen and KAUST
SIAM GS 2019 Conference in Houston, USA

2. The structure of the talk
Major Goal: estimate risks of the pollution in a subsurface flow
How?: we solve density-driven groundwater flow with uncertain
porosity and permeability
Plan:
1. Set up density-driven groundwater flow problem
2. Review stochastic modeling and stochastic methods
3. Modeling of uncertainty in porosity and permeability
4. Numerical methods to solve deterministic problem
5. 2D numerical experiments
6. 3D numerical experiments
7. Conclusion
2

3. Motivation
As groundwater is an essential nutrition and irrigation resource, its
pollution may lead to catastrophic consequences.
seawater intrusion into coastal aquifers,
https://water.usgs.gov/ogw/gwrp/saltwater/salt.html
Saudi Arabia agriculture:
Every dark point is a green
field or farm with diameter
few hundreds meters
Possible applications: seawater intrusion into coastal aquifers,
radioactive waste disposal, contaminant plumes etc.
3

4. What do we compute?
Accurate modeling of the pollution of the soil and groundwater
aquifer is difficult due to presence of uncertainties in geological
parameters.
Use random variables to model input and output uncertainty.
Compute
The mean, variance, and pdf of QoIs
Forecast the pollution map in 1-5-10 years
Estimate the risk that the pollutant concentration exceeds a
certain level
4

5. Governing equations
Use well-known model, see [1,2,3]:
Domain D is filled with two phases: solid porous matrix and
solution of salt in water (brine)
∂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,
ρ = ρ(c) density.
5

6. Governing equations
Assume permeability
K = KI, where K = K(φ) ∈ 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. 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. 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.
8

9. Mathematical equation
We introduce uncertain porosity φ = φ(x, θ), x ∈ D,
θ = (θ1, ..., θM, ...), θi is a random variable.
Assume that permeability (Kozeny-Carman-like equation) is
K(φ) = κKC ·
φ3
1 − φ2
, (4)
where κKC is a scaling factor.
9

10. Statistics
Can apply sampling methods, such as (quasi-)Monte Carlo,
collocations, sparse grids to evaluate:
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
.
Exceedance probability (risks)
P(c > c∗
) ≈
#{c(θi ) : c(θi ) > c∗, i = 1, . . . , Ns}
Ns
.
10

11. gPCE based surrogate
An alternative to sampling is a functional approximation:
We approximate unknown QoI by a surrogate (e.g., gPCE)
c(t, x, θ) =
β∈J
cβ(t, x)Ψβ(θ) ≈ c(t, x, θ) =
β∈JM,p
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, and coefficients
cβ(t, x) ≈
1
Ψβ, Ψβ
Nq
i=1
Ψβ(t, θi )c(t, x, θi )wi ,
11

12. Truncation and approximation errors
By introducing gPCE surrogate, we also introduce truncation and
approximation errors:
Et = c(t, x, θ)−c(t, x, θ)| =
β∈Jc
cβ(t, x)Ψβ(θ), JM,p∪Jc = J .
Additionally, cβ(t, x) ≈ cβ(t, x):
Ea =
β∈JM,p
cβ(t, x)Ψβ(θ) −
β∈JM,p
cβ(t, x)Ψβ(θ)
Et+Ea =
β∈Jc
cβ(t, x)Ψβ(θ)
truncation error
+
β∈JM,p
(cβ(t, x) − cβ(t, x))Ψβ(θ)
approximation error
.
See [Constantine’12, Sinsbeck’15,Conrad’13].
12

13. Utilised numerical methods to solve one realization
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. Ex.1: 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], in 5.5 years.
1st row: c(x) ∈ (0, 1) computed via qMC (200 simulations) and
via gPCE4 (m = 1, p = 4);
2nd row: Var[c]qMC ∈ (0, 0.021), Var[c]gPCE4 ∈ (0, 0.023).
Observed: 9 GL points gives almost the same result as 200 qMC.
14

15. Ex.2: 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], in 1.5 years.
1st row: c(x), computed via qMC (1500 simulations) and via
gPCE, c(x) ∈ (0, 1), Var[c]qMC ∈ (0, 0.076),
2nd row: Var[c]gPCE5 ∈ (0, 0.068), Var[c]gPCE7 ∈ (0, 0.0714),
Var[c]gPCE9 ∈ (0, 0.0847).
Observed: Our surrogate and qMC give similar c.
Var[c] computed by surrogate of order 7 is most close to the qMC
variance. 15

16. Ex.3: 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) 4.5 fingers.
(a) (b) (c)
Non-linearity may result in several stationary solutions.
16

17. Ex.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 qMC samples.
17

18. Ex.5. Elder’s problem with 5 RVs and three layers
φ(x, y, ω) =
0.08 + 0.01 5
i=1 θi sin(ixπ/600) sin(iyπ/150), 120 ≤ y ≤ 150.
0.06 + 0.01 5
i=1 θi sin(ixπ/600) sin(iyπ/150), 50 ≤ y < 120
0.09 + 0.01 5
i=1 θi sin(ixπ/600) sin(iyπ/150), 0 ≤ y < 50
(a) cgPCE; (b) cqMC in t = 5.5 years.
(c) Var[c]gPC (x, t) ∈ [0, 0.0466]; (d) Var[c]qMC (x, t) ∈ [0, 0.0556].
(e) porosity φ ∈ [0.0514, 0.09]; (f) c(x, t, 0) − c(x, t) (difference
between deterministic solution (corresponding to θ = 0) and the
mean value in t = 5.5 years,
18

19. Ex.6: 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
19

20. Ex.7: Isosurfaces of Var[c] in 3D reservoir
(a) (b)
Var[c] after 4.8 years, N ≈ 8 · 106 grid points.
20

21. Ex.8: 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). 21

22. Ex.9: 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.
22

23. Ex.10: Isosurfaces of Var[c] in 3D reservoir
(a) (b)
Isosurfaces of the variance of the mass fraction after 3 years;
(a) surface where Var[c] = 0.05; (b) surface where Var[c] = 0.15.
23

24. Ex.11: 3D reservoir with three layers
1st row: three layers of the porosity; two profiles of c;
2nd row: isosurface |cdet − c|0.25; isosurfaces Var[c]0.05 and
Var[c]0.12.
24

25. Conclusion
Solved time-dependent density driven flow problem with
uncertain porosity and permeability in 2D and 3D
Computed propagation of uncertainties from porosity and
permeability into the mass fraction. Computed the mean,
variance, exceedance probabilities, quantiles and risks
Such QoIs as the number of fingers, their size, shape,
propagation time can be unstable
For moderate perturbations, our gPCE surrogate results are
similar to qMC results
Used highly scalable parallel solver on up to 800 computing
nodes
25

26. Acknowledgement
1. Alexander von Humboldt foundation
2. Elmar Zander (TU Braunschweig) for the sglib library.
3. KAUST, Shaheen project k1051, 2.7 Mio hours.
4. KAUST Supercomputing Lab.
5. KAUST Visualization Lab.
THANK YOU FOR YOUR ATTENTION !
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27. 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.
6. A. Litvinenko, Application of hierarchical matrices for solving multiscale problems, Doctoral Dissertation,
Leipzig University, Germany, http://www.wire.tu-bs.de/mitarbeiter/litvinen/diss.pdf
27

28. Literature
7. 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
8. 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
9. 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.
10. A. Litvinenko, H.G. Matthies, Inverse problems and uncertainty quantification, arXiv:1312.5048, 2013.
11. A. Litvinenko, Y. Sun, M. G. Genton, D. E. Keyes, Likelihood approximation with hierarchical matrices for
large spatial datasets, Computational Statistics & Data Analysis, Volume 137, 2019, pp 115-132, ISSN
0167-9473, https://doi.org/10.1016/j.csda.2019.02.002.
12. A. Litvinenko, A. C. Yucel, H. Bagci, J. Oppelstrup, E. Michielssen and R. Tempone, Computation of
Electromagnetic Fields Scattered From Objects With Uncertain Shapes Using Multilevel Monte Carlo
Method, in IEEE Journal on Multiscale and Multiphysics Computational Techniques, vol. 4, pp. 37-50,
2019. doi: 10.1109/JMMCT.2019.2897490
13. V. Berikov, A. Litvinenko, Semi-Supervised Regression using Cluster Ensemble and Low-Rank
Co-Association Matrix Decomposition under Uncertainties, arXiv preprint
https://arxiv.org/abs/1901.03919, 2019
14. E. Bernholdt, David, R. Ciancosa, Mark, L. Green, David, J.H. Law, Kody, Litvinenko, Alexander, and M.
Park, Jin. Comparing theory based and higher-order reduced models for fusion simulation data. United
States: N. p., 2018. Web. doi:10.3934/BigDIA.2018.2.41. (2018)
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