![Density driven flow in fractured porous media. Uncertainty quantification via MLMC.
Alexander Litvinenko1
, D. Logashenko2
, R. Tempone1,2
, G. Wittum2
1
RWTH Aachen, Germany, 2
KAUST, Saudi Arabia
litvinenko@uq.rwth-aachen.de
Abstract
Problem: Henry-like saltwater intrusion in fractured media (nonlinear, time-
dependent, two-phase flow)
Input uncertainty: fracture width, porosity, permeability, and recharge (mod-
elled 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 does fracture affect flow?
2.Where is the largest uncertainty?
3.What is the mean scenario and its
variations?
4.What are the extreme scenarios?
5.How do the uncertainties change over
time?
q̂in
c = 0
c = 1
p = −ρ1gy
0
−1 m
2 m
y
x
(2, −0.5)
(1, −0.7)
1 2 3
4 5 6
(left) Henry problem with a fracture.
(right) Mass fraction cm (red for cm = 1)
1. Henry-like problem settings
Denote: the whole domain D, porous medium M ⊂ D, fracture porosity ϕm and permeability Km, salt mass
fraction cm(t, x) and pressure pm(t, x). The flow in M satisfies conservation laws:
∂t(ϕmρm) + ∇ · (ρmqm) = 0
∂t(ϕmρmcm) + ∇ · (ρmcmqm − ρmDm∇cm) = 0
x ∈ M (1)
with the Darcy’s law for the velocity:
qm = −
Km
µm
(∇pm − ρmg), x ∈ M , (2)
where ρm = ρ(cm) and µm = µ(cm) indicate the density and the viscosity of the liquid phase, Dm(t, x) denotes the
molecular diffusion and mechanical dispersion tensor.
The fracture is assumed to be filled with a porous medium, too. The fracture is represented by a surface S ⊂ D,
M ∪ S = D, M ∩ S = ∅.
∂t(ϕfϵρf) + ∇S · (ϵρfqf) + Q
(1)
fn + Q
(2)
fn = 0
∂t(ϕfϵρfcf) + ∇S · (ϵρfcfqf − ϵρfDf∇S cf) + P
(1)
fn + P
(2)
fn = 0
x ∈ S , (3)
where ϵ is the fracture width. The Darcy velocity along the fracture is
qf = −
Kf
µf
(∇S pf − ρfg), x ∈ S . (4)
The terms Q
(k)
fn and P
(k)
fn , k ∈ {1, 2}, are the mass fluxes through the faces S (k) of the fracture:
Q
(k)
fn := ρ(c
(k)
m )q
(k)
fn
P
(k)
fn := ρ(c
(k)
m )c
(k)
upwindq
(k)
fn − ρ(c
(k)
m )D
(k)
fn
c
(k)
m − cf
ϵ/2
x ∈ S (k), (5)
The uncertain width of the fracture, the recharge, and the porosity are modeled as follows, ξ1, ξ2, ξ3 ∈ U[−1, 1],
ϵ(ξ1) = 0.01 · ((1 − 0.01) · ξ1 + (1 + 0.01))/2, (6)
q̂x(t, ξ3) = 3.3 · 10−6 · (1 + 0.1ξ3)(1 + 0.1 sin(πt/40)), (7)
ϕm(x, y, ξ2) = 0.35 · (1 + 0.02 · (ξ2 cos(πx/2) + ξ2 sin(2πy)). (8)
To compute: cm.
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.
Methods: Newton method, BiCGStab, preconditioned with the geometric multigrid method (V-cycle), ILUβ-
smoothers and Gaussian elimination.
Multi Level Monte Carlo (MLMC) method
Spatial and temporal grid hierarchies D0, D1, . . . , DL, and T0, T1, . . . , TL; n0 = 512, nℓ ≈ n0 · 4ℓ, τℓ+1 = 1
2τℓ, number
of time steps rℓ+1 = 2rℓ or rℓ = r02ℓ. Computation complexity is sℓ = O(nℓrℓ) = O(23ℓγn0 · 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.
2. Numerics
Figure 2: The mean value E [cm(t, x)] at t = {7τ, 19τ, 40τ, 94τ}. In all cases, E [cm] (t, x) ∈ [0, 1].
Figure 3: The variance V [cm(t, x)] at t = {7τ, 19τ, 40τ, 94τ}. Maximal values (dark red colour) of V [cm] are 1.9·10−3,
3.4 · 10−3, 2.9 · 10−3, 2.4 · 10−3 respectively. The dark blue colour corresponds to a zero value.
g 0
g 1
- g 0
g 2
- g 1
g 3
- g 2
g 4
- g 3
-6
-4
-2
0
g 0
g 1
- g 0
g 2
- g 1
g 3
- g 2
g 4
- g 3
-16
-14
-12
-10
-8
-6
-4
0 1 2 3 4
10
-3
10
-2
0 1 2 3 4
10
-8
10
-7
10
-6
10
-5
10
-4
Figure 4: (1-2)The weak and the strong convergences, QoI is g := cm(t15, x1), α = 1.07, ζ1 = −1.1, β = 1.97,
ζ2 = −8. (3-4) Decay comparison of (left) E [gℓ − gℓ−1] and E [gℓ] vs. ℓ; (right) V [gℓ − gℓ−1] and V [gℓ]. QoI
g = cm(t18, x1), x1 = (1.1, −0.8).
Level ℓ nℓ, ( nℓ
nℓ−1
) rℓ, ( rℓ
rℓ−1
) τℓ = 6016/rℓ
Computing times (sℓ), ( sℓ
sℓ−1
)
average min. max.
0 608 188 32 3 2.4 3.4
1 2368 (3.9) 376 (2) 16 22 (7.3) 15.5 27.8
2 9344 (3.9) 752 (2) 8 189 (8.6) 115 237
3 37120 (4) 1504 (2) 4 1831 (10) 882 2363
4 147968 (4) 3008 (2) 2 18580 (10) 7865 25418
Table 1: # ndofs nℓ, number of time steps rℓ, step size in time τℓ, average, minimal, and maxi-
mal comp. times.
Estimated the weak and strong convergence rates α and β, and constants C1 and C2 for different QoIs. QoI are so-
lution in a point xi and an integral Ii(t, ω) :=
R
x∈∆i
cm(t, x, ω)ρ(cm(t, x, ω))dx, ∆i := [xi−0.1, xi+0.1]×[yi−0.1, yi+0.1],
i = 1, . . . , 6.
QoI α c1 β C2
cm(x1, 15τ) -1.07 0.47 -1.97 4 · 10−3
I2(cm) -1.5 8.3 -2.6 0.4
ε m0 m1 m2 m3 m4
0.2 28 0 0 0 0
0.1 10 1 0 0 0
0.05 57 4 1 0 0
0.025 342 22 4 1 0
0.01 3278 205 35 6 1
0 10 20 30 40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 10 20 30 40
0
1
2
3
4
5
10 -4
0 10 20 30 40
-8
-6
-4
-2
0
10
-3
0 10 20 30 40
0
0.2
0.4
0.6
0.8
1
10 -6
Figure 5: (1) The coefficient of variance CVℓ := CV (gℓ), (2) the variance V [gℓ], (3) the mean E [gℓ − gℓ−1], (4) the
variance V [gℓ − gℓ−1]. The QoI is g = cm(t, x1). The small oscillations in the two lower pictures are due to the
dependence of the recharge q̂x on the time, cf. (7).
10 -2
10 -1
10
0
10 5
10
10
Complexity comparison of ML and MLMC vs. accuracy ε (horizontal axis) in log-log scale.
Acknowledgements: Alexander von Humboldt foundation and KAUST HPC.
1. D. Logashenko, A. Litvinenko, R. Tempone, E. Vasilyeva, G. Wittum, Uncertainty quantification in the Henry problem using the multilevel Monte Carlo method,
Journal of Computational Physics, Vol. 503, 2024, 112854, https://doi.org/10.1016/j.jcp.2024.112854
2. D. Logashenko, A. Litvinenko, R. Tempone, G. Wittum, Estimation of uncertainties in the density driven flow in fractured porous media using MLMC,
arXiv:2404.18003, 2024](https://image.slidesharecdn.com/posterdensitydrivenwithfracturemlmc-240515105940-588beb87/85/Poster_density_driven_with_fracture_MLMC-pdf-1-320.jpg)
Poster_density_driven_with_fracture_MLMC.pdf
- 1. Density driven flow in fractured porous media. Uncertainty quantification via MLMC. Alexander Litvinenko1 , D. Logashenko2 , R. Tempone1,2 , G. Wittum2 1 RWTH Aachen, Germany, 2 KAUST, Saudi Arabia litvinenko@uq.rwth-aachen.de Abstract Problem: Henry-like saltwater intrusion in fractured media (nonlinear, time- dependent, two-phase flow) Input uncertainty: fracture width, porosity, permeability, and recharge (mod- elled 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 does fracture affect flow? 2.Where is the largest uncertainty? 3.What is the mean scenario and its variations? 4.What are the extreme scenarios? 5.How do the uncertainties change over time? q̂in c = 0 c = 1 p = −ρ1gy 0 −1 m 2 m y x (2, −0.5) (1, −0.7) 1 2 3 4 5 6 (left) Henry problem with a fracture. (right) Mass fraction cm (red for cm = 1) 1. Henry-like problem settings Denote: the whole domain D, porous medium M ⊂ D, fracture porosity ϕm and permeability Km, salt mass fraction cm(t, x) and pressure pm(t, x). The flow in M satisfies conservation laws: ∂t(ϕmρm) + ∇ · (ρmqm) = 0 ∂t(ϕmρmcm) + ∇ · (ρmcmqm − ρmDm∇cm) = 0 x ∈ M (1) with the Darcy’s law for the velocity: qm = − Km µm (∇pm − ρmg), x ∈ M , (2) where ρm = ρ(cm) and µm = µ(cm) indicate the density and the viscosity of the liquid phase, Dm(t, x) denotes the molecular diffusion and mechanical dispersion tensor. The fracture is assumed to be filled with a porous medium, too. The fracture is represented by a surface S ⊂ D, M ∪ S = D, M ∩ S = ∅. ∂t(ϕfϵρf) + ∇S · (ϵρfqf) + Q (1) fn + Q (2) fn = 0 ∂t(ϕfϵρfcf) + ∇S · (ϵρfcfqf − ϵρfDf∇S cf) + P (1) fn + P (2) fn = 0 x ∈ S , (3) where ϵ is the fracture width. The Darcy velocity along the fracture is qf = − Kf µf (∇S pf − ρfg), x ∈ S . (4) The terms Q (k) fn and P (k) fn , k ∈ {1, 2}, are the mass fluxes through the faces S (k) of the fracture: Q (k) fn := ρ(c (k) m )q (k) fn P (k) fn := ρ(c (k) m )c (k) upwindq (k) fn − ρ(c (k) m )D (k) fn c (k) m − cf ϵ/2 x ∈ S (k), (5) The uncertain width of the fracture, the recharge, and the porosity are modeled as follows, ξ1, ξ2, ξ3 ∈ U[−1, 1], ϵ(ξ1) = 0.01 · ((1 − 0.01) · ξ1 + (1 + 0.01))/2, (6) q̂x(t, ξ3) = 3.3 · 10−6 · (1 + 0.1ξ3)(1 + 0.1 sin(πt/40)), (7) ϕm(x, y, ξ2) = 0.35 · (1 + 0.02 · (ξ2 cos(πx/2) + ξ2 sin(2πy)). (8) To compute: cm. 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. Methods: Newton method, BiCGStab, preconditioned with the geometric multigrid method (V-cycle), ILUβ- smoothers and Gaussian elimination. Multi Level Monte Carlo (MLMC) method Spatial and temporal grid hierarchies D0, D1, . . . , DL, and T0, T1, . . . , TL; n0 = 512, nℓ ≈ n0 · 4ℓ, τℓ+1 = 1 2τℓ, number of time steps rℓ+1 = 2rℓ or rℓ = r02ℓ. Computation complexity is sℓ = O(nℓrℓ) = O(23ℓγn0 · 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. 2. Numerics Figure 2: The mean value E [cm(t, x)] at t = {7τ, 19τ, 40τ, 94τ}. In all cases, E [cm] (t, x) ∈ [0, 1]. Figure 3: The variance V [cm(t, x)] at t = {7τ, 19τ, 40τ, 94τ}. Maximal values (dark red colour) of V [cm] are 1.9·10−3, 3.4 · 10−3, 2.9 · 10−3, 2.4 · 10−3 respectively. The dark blue colour corresponds to a zero value. g 0 g 1 - g 0 g 2 - g 1 g 3 - g 2 g 4 - g 3 -6 -4 -2 0 g 0 g 1 - g 0 g 2 - g 1 g 3 - g 2 g 4 - g 3 -16 -14 -12 -10 -8 -6 -4 0 1 2 3 4 10 -3 10 -2 0 1 2 3 4 10 -8 10 -7 10 -6 10 -5 10 -4 Figure 4: (1-2)The weak and the strong convergences, QoI is g := cm(t15, x1), α = 1.07, ζ1 = −1.1, β = 1.97, ζ2 = −8. (3-4) Decay comparison of (left) E [gℓ − gℓ−1] and E [gℓ] vs. ℓ; (right) V [gℓ − gℓ−1] and V [gℓ]. QoI g = cm(t18, x1), x1 = (1.1, −0.8). Level ℓ nℓ, ( nℓ nℓ−1 ) rℓ, ( rℓ rℓ−1 ) τℓ = 6016/rℓ Computing times (sℓ), ( sℓ sℓ−1 ) average min. max. 0 608 188 32 3 2.4 3.4 1 2368 (3.9) 376 (2) 16 22 (7.3) 15.5 27.8 2 9344 (3.9) 752 (2) 8 189 (8.6) 115 237 3 37120 (4) 1504 (2) 4 1831 (10) 882 2363 4 147968 (4) 3008 (2) 2 18580 (10) 7865 25418 Table 1: # ndofs nℓ, number of time steps rℓ, step size in time τℓ, average, minimal, and maxi- mal comp. times. Estimated the weak and strong convergence rates α and β, and constants C1 and C2 for different QoIs. QoI are so- lution in a point xi and an integral Ii(t, ω) := R x∈∆i cm(t, x, ω)ρ(cm(t, x, ω))dx, ∆i := [xi−0.1, xi+0.1]×[yi−0.1, yi+0.1], i = 1, . . . , 6. QoI α c1 β C2 cm(x1, 15τ) -1.07 0.47 -1.97 4 · 10−3 I2(cm) -1.5 8.3 -2.6 0.4 ε m0 m1 m2 m3 m4 0.2 28 0 0 0 0 0.1 10 1 0 0 0 0.05 57 4 1 0 0 0.025 342 22 4 1 0 0.01 3278 205 35 6 1 0 10 20 30 40 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 10 20 30 40 0 1 2 3 4 5 10 -4 0 10 20 30 40 -8 -6 -4 -2 0 10 -3 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 10 -6 Figure 5: (1) The coefficient of variance CVℓ := CV (gℓ), (2) the variance V [gℓ], (3) the mean E [gℓ − gℓ−1], (4) the variance V [gℓ − gℓ−1]. The QoI is g = cm(t, x1). The small oscillations in the two lower pictures are due to the dependence of the recharge q̂x on the time, cf. (7). 10 -2 10 -1 10 0 10 5 10 10 Complexity comparison of ML and MLMC vs. accuracy ε (horizontal axis) in log-log scale. Acknowledgements: Alexander von Humboldt foundation and KAUST HPC. 1. D. Logashenko, A. Litvinenko, R. Tempone, E. Vasilyeva, G. Wittum, Uncertainty quantification in the Henry problem using the multilevel Monte Carlo method, Journal of Computational Physics, Vol. 503, 2024, 112854, https://doi.org/10.1016/j.jcp.2024.112854 2. D. Logashenko, A. Litvinenko, R. Tempone, G. Wittum, Estimation of uncertainties in the density driven flow in fractured porous media using MLMC, arXiv:2404.18003, 2024