Simulating the flow of two fluid phases in porous media is a challenging task, especially when fractures are included in the simulation. Fractures may have highly heterogeneous properties compared to the surrounding rock matrix, significantly affecting fluid flow, and at the same time hydraulic aperture that are much smaller than any other characteristic sizes in the domain. Generally, flow simulators face difficulties with counter-current flow, generated by gravity and pressure gradients, which hinders the convergence of non-linear solvers (Newton). In this work, we model the fracture geometry with a mixed-dimensional discrete fracture network, thus lightening the computational burden associated to an equi-dimensional representation. We address the issue of counter-current flows with appropriate spatial discretization of the advective fluid fluxes, with the aim of improving the convergence speed of the non-linear solver. In particular, the extension of the hybrid upwinding to the mixed-dimensional framework, with the use of a phase potential upstreaming at the interfaces of subdomains. We test the method across several cases with different flow regimes and fracture network geometry. Results show robustness of the chosen discretization and a consistent improvements, in terms of Newton iterations, compared to use the phase potential upstreaming everywhere.

翻译：模拟多孔介质中两种流体相的流动是一项具有挑战性的任务，尤其是在模拟中包含裂缝时。与周围岩石基质相比，裂缝可能具有高度非均质的特性，这会显著影响流体流动，同时其水力开度远小于域内的任何其他特征尺寸。通常，流动模拟器在处理由重力和压力梯度产生的逆流时会遇到困难，这阻碍了非线性求解器（牛顿法）的收敛。在本工作中，我们采用混合维离散裂缝网络对裂缝几何进行建模，从而减轻了与等维表示相关的计算负担。我们通过对对流流体通量进行适当的空间离散化来解决逆流问题，旨在提高非线性求解器的收敛速度。具体而言，将混合迎风格式扩展到混合维框架中，并在子域界面处采用相势上游加权方法。我们在多种不同流态和裂缝网络几何的案例中测试了该方法。结果表明，与在所有位置均使用相势上游加权方法相比，所选离散方案具有鲁棒性，并在牛顿迭代次数方面展现出持续改进。