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.

翻译：我们采用多层级蒙特卡洛方法（MLMC）对含裂隙的类亨利盐水入侵问题中的不确定性进行估计。该流动由盐质量分数决定的流体相密度变化驱动。假定裂隙位置固定但开度存在不确定性，其他输入不确定性包括孔隙度场、渗透率场以及补给条件。在本研究中，孔隙度与渗透率呈空间变化特征，补给量具有时间依赖性。针对这些不确定参数的每次实现，通过包含裂隙处解跳跃的非线性时变偏微分方程组（PDEs）对盐质量分数和压力场的演化过程进行建模。这些不确定性将传播至盐浓度分布，而盐浓度是水资源质量的重要特征指标。研究结果表明，与传统蒙特卡洛方法相比，多层级蒙特卡洛方法能够通过平衡离散误差与统计误差降低整体计算成本。我们在不同空间与时间网格层级上评估了多个情景方案，并采用并行运行的确定性求解器ug4计算所有随机情景。