We consider a class of density-driven flow problems. We are particularly interested in the problem of the salinization of coastal aquifers. We consider the Henry saltwater intrusion problem with uncertain porosity, permeability, and recharge parameters as a test case. The reason for the presence of uncertainties is the lack of knowledge, inaccurate measurements, and inability to measure parameters at each spatial or time location. This problem is nonlinear and time-dependent. The solution is the salt mass fraction, which is uncertain and changes in time. Uncertainties in porosity, permeability, recharge, and mass fraction are modeled using random fields. This work investigates the applicability of the well-known multilevel Monte Carlo (MLMC) method for such problems. The MLMC method can reduce the total computational and storage costs. Moreover, the MLMC method runs multiple scenarios on different spatial and time meshes and then estimates the mean value of the mass fraction. The parallelization is performed in both the physical space and stochastic space. To solve every deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion. We use the solution obtained from the quasi-Monte Carlo method as a reference solution.

1. Center for Uncertainty
Quantification
Salinization of coastal aquifers under uncertainties
Alexander Litvinenko1
, Dmitry Logashenko2
, Raul Tempone1,2
, Ekaterina Vasilyeva2
, Gabriel Wittum2
1
RWTH Aachen, Germany, 2
KAUST, Saudi Arabia
litvinenko@uq.rwth-aachen.de
Center for Uncertainty
Quantification
Center for Uncertainty
Quantification
Abstract
Problem: Henry saltwater intrusion (nonlinear and time-dependent, describes a two-phase
subsurface flow)
Input uncertainty: porosity, permeability, and recharge (model by random fields)
Solution: the salt mass fraction (uncertain and time-dependent)
Method: Multi Level Monte Carlo (MLMC) method
Deterministic solver: parallel multigrid solver ug4
Questions:
1. How long can wells be used?
2. Where is the largest uncertainty?
3. Freshwater exceed. probabil.?
4. What is the mean scenario and its
variations?
5. What are the extreme scenarios?
6. How do the uncertainties change
over time?
Figure 1: Henry problem, taken from https://www.mdpi.com/2073-4441/10/2/230
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
diffusion 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. To compute: c and p.
Comput. domain: D × [0, T]. We set ρ(c) = ρ0 + (ρ1 − ρ0)c, and D = ϕDI,
I.C.: c|t=0 = 0, B.C.: c|x=2 = 1, p|x=2 = −ρ1gy. c|x=0 = 0, ρq · ex|x=0 = q̂in.
We model the uncertain ϕ using a random field and assume: K = KI, K = K(ϕ).
Methods: Newton method, BiCGStab, preconditioned with the geometric multigrid method
(V-cycle), ILUβ-smoothers and Gaussian elimination.
2. Solution of the Henry problem
q̂in = 6.6 · 10−2
kg/s
c = 0 c = 1
p = −ρ1gy
0
−1 m
2 m
y
x
D := [0, 2] × [−1, 0]; a realization of c(t, x); ϕ(ξ∗
) ∈ [0.18, 0.59]; E [c] ∈ [0, 0.35]; Var[c] ∈ [0.0, 0.04]
QoIs: c in the whole domain, c at a point, or integral values (the freshwater/saltwater integrals):
QFW (t, ω) :=
R
x∈D I(c(t, x, ω) ≤ 0.012178)dx, Qs(t, ω) :=
R
x∈D c(t, x, ω)ρ(t, x, ω)dx
2.1 Multi Level Monte Carlo (MLMC) method
Spatial and temporal grid hierarchies D0, D1, . . . , DL, and T0, T1, . . . , TL; n0 = 512, nℓ ≈ n0 · 16ℓ
,
τℓ+1 = 1
4τℓ, rℓ+1 = 4rℓ and rℓ = r04ℓ
. Computation complexity is
sℓ = O(nℓrℓ) = O(43ℓγ
n0 · r0)
MLMC approximates E [gL] ≈ E [g] using the following telescopic sum:
E [gL] ≈ m−1
0
m0
X
i=1
g
(0,i)
0 +
L
X
ℓ=1
m−1
ℓ
mℓ
X
i=1
(g
(ℓ,i)
ℓ − g
(ℓ,i)
ℓ−1 )
!
.
Minimize F(m0, . . . , mL) :=
PL
ℓ=0 mℓsℓ + µ2 Vℓ
mℓ
, obtain mℓ = ε−2
q
Vℓ
sℓ
PL
i=0
√
Visi.
100 realizations of QFW (t); Evolution of the pdf of c(t, x), t = {3τ, . . . , 48τ}; pdf of the earliest
time point when c(t, x) < 0.9, x = (1.85, −0.95); mean values E [c(t, x9, y9)]; variances
Var[c](t, x9, y9) on levels 0,1,2,3.
E [gℓ − gℓ−1] (t, x9, y9); V [gℓ − gℓ−1] (t, x9, y9), ℓ = 1, 2, 3, QoI is the integral value over D9 ; 100 realisations of g1 − g0, g2 − g1, g3 − g2, QoI gℓ is the integral value
Qs(t, ω) computed over a subdomain around 9th point, t ∈ [τ, 48τ].
Level ℓ nℓ, ( nℓ
nℓ−1
) rℓ, ( rℓ
rℓ−1
) τℓ = 6016/rℓ
Computing times (sℓ), ( sℓ
sℓ−1
)
average min. max.
0 153 94 64 0.6 0.5 0.7
1 2145 (14) 376 (4) 16 7.1 (14) 6.9 8.7
2 33153 (15.5) 1504 (4) 4 252.9 (36) 246.2 266.2
3 525825 (15.9) 6016 (4) 1 11109.8 (44) 9858.4 15506.9
#ndofs nℓ, number of time steps rℓ, time step τℓ; average, minimal, and maximal computing times on each level ℓ.
ε2
0.1
total cost of MC, SMC 9.5e + 6
total cost of MLMC, S 4.25e + 4
{m0, m1, m2, m3} {7927, 946, 57, 2}
Comparison of MC and MLMC
ε2
m0 m1 m2 m3
1 73 8 1 0
0.5 290 32 3 0
0.1 7258 811 68 1
0.05 29031 3245 274 5
MLMC: number of samples on level ℓ
(1)Weak (α = −2) and (2)strong (β = −3.4) convergences, ℓ = 0, 1, 2, 3. QoI=local integral of c
over (x, y)9 = (1.65, −0.75)-subdomain; (3) decay of absolute and (4) relative errors between
the mean values computed on a fine mesh via MC and via MLMC at (t, x, y) = (12, 1.60, −0.95).
Acknowledgements: KAUST HPC and the Alexander von Humboldt foundation.
1. 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
2. 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, 1-29, 2020
3. A. Litvinenko, A.C. Yucel, H. Bagci, J. Oppelstrup, E. Michielssen, R. Tempone, Computation of electromagnetic fields scattered from objects with uncertain
shapes using multilevel Monte Carlo method, IEEE Journal on Multiscale and Multiphysics Computational Techniques 4, 37-50, 2019
4. 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. LNCSE, vol. 144, pp 121-126, Springer, 2018