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Estimation of uncertainties in the density driven flow in fractured porous media using MLMC

Dmitry Logashenko, Alexander Litvinenko, Raul Tempone, Gabriel Wittum

TL;DR

The paper tackles uncertainty quantification for a density-driven flow in a fractured porous medium using Multi-Level Monte Carlo (MLMC) implemented with geometric multigrid on the ug4 platform. By treating porosity, permeability, fracture width, and recharge as uncertain inputs, it propagates these through a nonlinear, time-dependent PDE system to obtain the mean and variance of the salt mass fraction $c_m$. The authors demonstrate that MLMC achieves substantial computational savings compared with classical Monte Carlo, with estimated weak and strong convergence rates around $\alpha\approx1.0$ and $\beta\approx2.0$, and show the fracture width critically influences mixing patterns and hydrogeochemical evolution. The results provide practical guidance on level and sample allocations for MLMC in density-driven, fracture-embedded aquifer problems and establish the method’s applicability to complex, discontinuous transport phenomena.

Abstract

We use the Multi Level Monte Carlo method to estimate uncertainties in a Henry-like salt water intrusion problem with a fracture. The flow is induced by the variation of the density of the fluid phase, which depends on the mass fraction of salt. We assume that the fracture has a known fixed location but an uncertain aperture. Other input uncertainties are the porosity and permeability fields and the recharge. In our setting, porosity and permeability vary spatially and recharge is time-dependent. For each realisation of these uncertain parameters, the evolution of the mass fraction and pressure fields is modelled by a system of non-linear and time-dependent PDEs with a jump of the solution at the fracture. The uncertainties propagate into the distribution of the salt concentration, which is an important characteristic of the quality of water resources. We show that the multilevel Monte Carlo (MLMC) method is able to reduce the overall computational cost compared to classical Monte Carlo methods. This is achieved by balancing discretisation and statistical errors. Multiple scenarios are evaluated at different spatial and temporal mesh levels. The deterministic solver ug4 is run in parallel to calculate all stochastic scenarios.

Estimation of uncertainties in the density driven flow in fractured porous media using MLMC

TL;DR

The paper tackles uncertainty quantification for a density-driven flow in a fractured porous medium using Multi-Level Monte Carlo (MLMC) implemented with geometric multigrid on the ug4 platform. By treating porosity, permeability, fracture width, and recharge as uncertain inputs, it propagates these through a nonlinear, time-dependent PDE system to obtain the mean and variance of the salt mass fraction . The authors demonstrate that MLMC achieves substantial computational savings compared with classical Monte Carlo, with estimated weak and strong convergence rates around and , and show the fracture width critically influences mixing patterns and hydrogeochemical evolution. The results provide practical guidance on level and sample allocations for MLMC in density-driven, fracture-embedded aquifer problems and establish the method’s applicability to complex, discontinuous transport phenomena.

Abstract

We use the Multi Level Monte Carlo method to estimate uncertainties in a Henry-like salt water intrusion problem with a fracture. The flow is induced by the variation of the density of the fluid phase, which depends on the mass fraction of salt. We assume that the fracture has a known fixed location but an uncertain aperture. Other input uncertainties are the porosity and permeability fields and the recharge. In our setting, porosity and permeability vary spatially and recharge is time-dependent. For each realisation of these uncertain parameters, the evolution of the mass fraction and pressure fields is modelled by a system of non-linear and time-dependent PDEs with a jump of the solution at the fracture. The uncertainties propagate into the distribution of the salt concentration, which is an important characteristic of the quality of water resources. We show that the multilevel Monte Carlo (MLMC) method is able to reduce the overall computational cost compared to classical Monte Carlo methods. This is achieved by balancing discretisation and statistical errors. Multiple scenarios are evaluated at different spatial and temporal mesh levels. The deterministic solver ug4 is run in parallel to calculate all stochastic scenarios.
Paper Structure (13 sections, 1 theorem, 45 equations, 7 figures, 4 tables, 2 algorithms)

This paper contains 13 sections, 1 theorem, 45 equations, 7 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Modeling and Problem Settings
  3. Governing equations for the flow
  4. Model problem settings
  5. Stochastic modeling of porosity, permeability and fracture width
  6. Numerical Methods
  7. Numerical methods for the deterministic problem
  8. MLMC Algorithm
  9. Numerical Experiments
  10. The mean value and variance
  11. Estimation of convergence rates $\alpha$ and $\beta$ for various QoIs
  12. Comparison of MC and MLMC methods
  13. Discussion and Conclusion

Key Result

Theorem 2

Consider a fixed $t=t^*$. Suppose positive constants $\alpha,\beta,\gamma > 0$ exist such that $\alpha \geq \frac{1}{2} \hbox{min}(\beta, \gamma \hat{d})$, and Then, for any accuracy $\varepsilon < e^{-1}$, a constant $c_4>0$ and a sequence of realizations $\{m_{\ell}\}_{\ell=0}^L$ exist, such that $\text{MSE} < \varepsilon^2$, where $\text{MSE}$ is defined in eq:MSE, and the computational cost i

Figures (7)

  • Figure 1: Left: Computational geometry $[0,2]\times [-1,0]$, direction of recharge, boundary conditions, location of fracture (marked by black oblique line) and six points, solution in which we use as additional QoIs. Right: The flow streamlines (thin black lines) and the salt mass fraction. The dark red colour indicates a high salt mass fraction $c_m = 1.0$ or close to 1.0, and the dark blue corresponds to $c_m = 0$ (no salt). The thick black curved line indicates the area where the recharge flow meets the density difference induced flow.
  • Figure 2: (top left) The coefficient of variance $CV_{\ell}:=CV(g_{\ell})$, (top right) the variance $\mathbb{V}\left[g_{\ell}\right ]$, (bottom left) the mean $\mathbb{E}\left[g_{\ell} - g_{\ell-1}\right ]$, (bottom right) the variance $\mathbb{V}\left[g_{\ell} - g_{\ell-1}\right ]$. The QoI is $g=c_m(t,\mathbf{x}_1)$. The small oscillations in the two lower pictures are due to the dependence of the recharge $\hat{q}_{\mathrm{in}}$ on the time, cf. \ref{['eq:recharge']}. The time is along the $x$ axis, $t\in[0,47\tau]$
  • Figure 3: The mean value $\mathbb{E}\left[c_m(t,\mathbf{x})\right ]$ at $t=\{7\tau, 19\tau, 40\tau, 94\tau\}$. In all cases, $\mathbb{E}\left[c_m\right ](t,\mathbf{x}) \in [0,1]$. The blue colour corresponds to $\mathbb{E}\left[c_m\right ](t,\mathbf{x})=0$ and the dark red colour to $\mathbb{E}\left[c_m\right ](t,\mathbf{x})=1$.
  • Figure 4: The variance $\mathbb{V}\left[c_m(t,\mathbf{x})\right ]$ at $t=\{7\tau, 19\tau, 40\tau, 94\tau\}$. Maximal values (dark red colour) of $\mathbb{V}\left[c_m\right ]$ are $1.9\cdot 10^{-3}$, $3.4\cdot 10^{-3}$, $2.9\cdot 10^{-3}$, $2.4\cdot 10^{-3}$ respectively. The blue colour corresponds to a zero value.
  • Figure 5: The weak and the strong convergences, QoI is $g:=c_m(t_{15},\mathbf{x}_1)$, $\alpha=1.07$, $\zeta_1=-1.1$, $\beta=1.97$, $\zeta_2 = -8$. The $0y$ axis denotes $\log_2(\mathbb{E}\left[g_{\ell}- g_{\ell-1}\right ])$ on the left and $\log_2(\mathbb{V}\left[g_{\ell}- g_{\ell-1}\right ])$ on the right.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1
  • Theorem 2
  • Remark 4
  • Remark 5
  • Remark 6