It is a challenge to numerically solve nonlinear partial differential equations whose solution involves discontinuity. In the context of numerical simulators for multi-phase flow in porous media, there exists a long-standing issue known as Grid Orientation Effect (GOE), wherein different numerical solutions can be obtained when considering grids with different orientations under certain unfavorable conditions. Our perspective is that GOE arises due to numerical instability near displacement fronts, where spurious oscillations accompanied by sharp fronts, if not adequately suppressed, lead to GOE. To reduce or even eliminate GOE, we propose augmenting adaptive artificial viscosity when solving the saturation equation. It has been demonstrated that appropriate artificial viscosity can effectively reduce or even eliminate GOE. The proposed numerical method can be easily applied in practical engineering problems.

翻译：在数值求解涉及间断的非线性偏微分方程时存在挑战。针对多孔介质多相流数值模拟器，存在一个长期未解决的网格取向效应（Grid Orientation Effect, GOE）问题，即在某些不利条件下，采用不同取向的网格会得到不同的数值解。我们认为GOE源于驱替前缘附近的数值不稳定性，若未能充分抑制伴随陡峭前缘的虚假振荡，就会导致GOE。为降低甚至消除GOE，我们提出在求解饱和度方程时增加自适应人工粘性。研究表明，适当的人工粘性可有效降低甚至消除GOE。所提出的数值方法可便捷地应用于实际工程问题。