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：本文采用多级蒙特卡洛方法对含裂隙的类亨利盐分入侵问题中的不确定性进行估计。流动由流体相密度变化所驱动，而密度取决于盐分质量分数。我们假设裂隙位置已知且固定，但其开度具有不确定性。其他输入不确定性包括孔隙度和渗透率场以及补给量。在本研究设定中，孔隙度和渗透率具有空间变异性，补给量则随时间变化。针对这些不确定参数的每个实现，质量分数和压力场的演化通过一组非线性和时间依赖的偏微分方程进行建模，且在裂隙处存在解跳跃。这些不确定性会传递到盐浓度分布中，而盐浓度是水资源质量的重要特征指标。研究表明，与经典蒙特卡洛方法相比，多级蒙特卡洛方法能够通过平衡离散化误差和统计误差来降低总体计算成本。该方法通过在不同时空网格层级上评估多种情景实现这一目标。确定性求解器ug4以并行方式运行，用于计算所有随机情景。