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We investigate the applicability of the well-known multilevel Monte Carlo (MLMC) method to the class of density-driven flow problems, in particular the problem of salinisation of coastal aquifers. As a test case, we solve the uncertain Henry saltwater intrusion problem. Unknown porosity, permeability and recharge parameters are modelled by using random fields. The classical deterministic Henry problem is non-linear and time-dependent, and can easily take several hours of computing time. Uncertain settings require the solution of multiple realisations of the deterministic problem, and the total computational cost increases drastically. Instead of computing of hundreds random realisations, typically the mean value and the variance are computed. The standard methods such as the Monte Carlo or surrogate-based methods is a good choice, but they compute all stochastic realisations on the same, often, very fine mesh. They also do not balance the stochastic and discretisation errors. These facts motivated us to apply the MLMC method. We demonstrate that by solving the Henry problem on multi-level spatial and temporal meshes, the MLMC method reduces the overall computational and storage costs. To reduce the computing cost further, parallelization is performed in both physical and stochastic spaces. To solve each deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion., Comment: 20 pages, 11 figures, 3 tables. arXiv admin note: substantial text overlap with arXiv:2302.07804

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., Comment: 24 pages, 3 tables, 11 figures

As groundwater is an essential nutrition and irrigation resource, its pollution may lead to catastrophic consequences. Therefore, accurate modeling of the pollution of the soil and groundwater aquifer is highly important. As a model, we consider a density-driven groundwater flow problem with uncertain porosity and permeability. This problem may arise in geothermal reservoir simulation, natural saline-disposal basins, modeling of contaminant plumes, and subsurface flow. This strongly nonlinear time-dependent problem describes the convection of the two-phase flow. This liquid streams under the gravity force, building so-called "fingers". The accurate numerical solution requires fine spatial resolution with an unstructured mesh and, therefore, high computational resources. Consequently, we run the parallelized simulation toolbox \myug with the geometric multigrid solver on Shaheen II supercomputer. The parallelization is done in physical and stochastic spaces. Additionally, we demonstrate how the \myug toolbox can be run in a black-box fashion for testing different scenarios in the density-driven flow. As a benchmark, we solve the Elder-like problem in a 3D domain. For approximations in the stochastic space, we use the generalized polynomial chaos expansion. We compute the mean, variance, and exceedance probabilities of the mass fraction. As a reference solution, we use the solution, obtained from the quasi-Monte Carlo method., Comment: 23 pages, 13 figures. arXiv admin note: substantial text overlap with arXiv:1905.01770

Accurate modeling of contamination in subsurface flow and water aquifers is crucial for agriculture and environmental protection. Here, we demonstrate a parallel method to quantify the propagation of the uncertainty in the dispersal of pollution in subsurface flow. Specifically, we consider the density-driven flow and estimate how uncertainty from permeability and porosity propagates to the solution. We take an Elder-like problem as a numerical benchmark and we use random fields to model the limited knowledge on the porosity and permeability. We construct a low-cost generalized polynomial chaos expansion (gPC) surrogate model, where the gPC coefficients are computed by projection on sparse and full tensor grids. We parallelize both the numerical solver for the deterministic problem based on the multigrid method, and the quadrature over the parametric space, Comment: 21 page, 9 Figures, 2 Tables