In this work, we solved the density driven groundwater flow problem with uncertain porosity and permeability. An accurate solution of this time-dependent and non-linear problem is impossible because of the presence of natural uncertainties in the reservoir such as porosity and permeability. Therefore, we estimated the mean value and the variance of the solution, as well as the propagation of uncertainties from the random input parameters to the solution. We started by defining the Elder-like problem. Then we described the multi-variate polynomial approximation (\gPC) approach and used it to estimate the required statistics of the mass fraction. Utilizing the \gPC method allowed us to reduce the computational cost compared to the classical quasi Monte Carlo method. \gPC assumes that the output function $\sol(t,\bx,\thetab)$ is square-integrable and smooth w.r.t uncertain input variables $\btheta$. Many factors, such as non-linearity, multiple solutions, multiple stationary states, time dependence and complicated solvers, make the investigation of the convergence of the \gPC method a non-trivial task. We used an easy-to-implement, but only sub-optimal \gPC technique to quantify the uncertainty. For example, it is known that by increasing the degree of global polynomials (Hermite, Langange and similar), Runge's phenomenon appears. Here, probably local polynomials, splines or their mixtures would be better. Additionally, we used an easy-to-parallelise quadrature rule, which was also only suboptimal. For instance, adaptive choice of sparse grid (or collocation) points \cite{ConradMarzouk13,nobile-sg-mc-2015,Sudret_sparsePCE,CONSTANTINE12,crestaux2009polynomial} would be better, but we were limited by the usage of parallel methods. Adaptive quadrature rules are not (so well) parallelisable. In conclusion, we can report that: a) we developed a highly parallel method to quantify uncertainty in the Elder-like problem; b) with the \gPC of degree 4 we can achieve similar results as with the \QMC method. In the numerical section we considered two different aquifers - a solid parallelepiped and a solid elliptic cylinder. One of our goals was to see how the domain geometry influences the formation, the number and the shape of fingers. Since the considered problem is nonlinear, a high variance in the porosity may result in totally different solutions; for instance, the number of fingers, their intensity and shape, the propagation time, and the velocity may vary considerably. The number of cells in the presented experiments varied from $241{,}152$ to $15{,}433{,}728$ for the cylindrical domain and from $524{,}288$ to $4{,}194{,}304$ for the parallelepiped. The maximal number of parallel processing units was $600\times 32$, where $600$ is the number of parallel nodes and $32$ is the number of computing cores on each node. The total computing time varied from 2 hours for the coarse mesh to 24 hours for the finest mesh.

1. Density Driven Groundwater Flow with Uncertain
Porosity and Permeability
Alexander Litvinenko1
, Dmitry Logashenko2
, Raul Tempone1,2
, David Keyes2
, Gabriel Wittum2
1
RWTH Aachen, Germany, 2
KAUST, Saudi Arabia
litvinenko@uq.rwth-aachen.de
Abstract
Goal: Accurate modelling of soil and aquifer contamination
Problem: Elder problem (nonlinear and time-dependent, describes a two-
phase subsurface flow)
Input uncertainty: porosity, permeability (model by random fields)
Solution: the salt mass fraction (uncertain and time-dependent)
Method: Polynomial Chaos Expansion, QMC
Deterministic solver: parallel multigrid solver ug4
Questions:
1. Forecast the pollution map in 1-5-10 years
2. Estimate the risk of the pollution concentration exceeding a certain level
3. Where is the greatest uncertainty?
4. What is the mean scenario and its variations?
5. What are the extreme scenarios?
6. How do the uncertainties change with time?
c = 1
c = 0
c = 0
c = 0
600 m
300 m
150 m
2D reservoir D = (0, 600) × (0, 150) and a realisation of the porosity φ(x) ∈ [0.097, 0.115].
Figure 2: Two reservoirs
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.
1. Henry problem settings
The mass conservation laws for the entire liquid phase and salt yield the following equations
∂t(φρ) + ∇ · (ρq) = 0,
∂t(φρc) + ∇ · (ρcq − ρD∇c) = 0,
where φ(x, ξ) is porosity, x ∈ D, is determined by a set of RVs ξ = (ξ1, . . . , ξM, ...).
c(t, x) mass fraction of the salt, ρ = ρ(c) density of the liquid phase, and D(t, x) molecular diffu-
sion tensor.
For q(t, x) velocity, we assume Darcy’s law:
q = −
K
µ
(∇p − ρg),
where p = p(t, x) is the hydrostatic pressure, K permeability, µ = µ(c) viscosity of the liquid
phase, and g gravity. We set ρ(c) = ρ0 + (ρ1 − ρ0)c, and D = φDI, K = KI, K = K(φ).
To compute: c and p.
Methods: Newton method, BiCGStab, preconditioned with the geometric multigrid method (V-
cycle), ILUβ-smoothers and Gaussian elimination.
2. 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) =
X
β∈J
cβ(t, x)Ψβ(θ) ≈ b
c(t, x, θ) =
X
β∈JM,p
cβ(t, x)Ψβ(θ),
where {Ψβ} is a multivariate Legendre basis, β = (β1, ..., βj, ...) a multiindex, J a multiindex set,
Ψβ(θ) :=
Q∞
j=1 ψβj
(θj), and coefficients cβ(t, x) ≈ 1
hΨβ,Ψβi
PNq
i=1 Ψβ(t, θi)c(t, x, θi)wi
3. Numerical experiments
φ(t, x, θ) = 0.1 + 0.05 · c0 ·
θ1x
600
cos
πx
300
+ θ2 sin
πy
150
+ θ3 cos
πx
300
sin
πy
150
(1)
( c0 = 0.01 if z ≤ −100
c0 = 0.10 if −100 z ≤ −50
c0 = 1.0 if −50 z ≤ 0
(2)
Figure 3: 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).
Figure 4: (left) 2.75 years, Var[c](x) ∈ (0, 0.023); (center) 5.5 years, Var[c](x) ∈ (0, 0.055); (right)
8.25 years, Var[c](x) ∈ (0, 0.07).
Figure 5: Five isosurfaces of c after 9.6 years
Figure 6: 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)
Figure 7: Isosurface Var[c] = 0.07, computed via a) QMC and b) gPC response surface of
degree 4, and c) comparison of both isosurfaces
Figure 8: (left) porosity field; (center) isosurface Var[c] = 0.05; (right) isosurface Var[c] = 0.15.
Figure 9: (left) isosurface |cdet − c|0.25; (center) isosurfaces Var[c]0.05 and (right) Var[c]0.12
Acknowledgements: Alexander von Humboldt foundation and KAUST HPC.
References
1. A. Litvinenko, D. Logashenko, R. Tempone, G. Wittum, D. Keyes, Solution of the 3D density-driven groundwater flow problem with
uncertain porosity and permeability, GEM-International Journal on Geomathematics 11, pp 1-29, 2020
2. A. Litvinenko, D. Logashenko, R. Tempone, G. Wittum, D. Keyes, Propagation of Uncertainties in Density-Driven Flow. In: Bungartz,
HJ., Garcke, J., Pflüger, D. (eds) Sparse Grids and Applications - Munich 2018. Lecture Notes in Computational Science and Engineer-
ing, vol 144. Springer, Cham. https://doi.org/10.1007/978-3-030-81362-8_5, 2021
3. A. Litvinenko, D. Logashenko, R. Tempone, E. Vasilyeva, G. Wittum, Uncertainty quantification in coastal aquifers using the multilevel
Monte Carlo method, arXiv:2302.07804, 2023