Quasi-implicit schemes for treating velocity-dependent mobilities in underground porous media flow simulation, occurring when modeling non-Newtonian and non-Darcy effects as well as capillary desaturation, are presented. With low-order finite-volume discretizations, the principle is to evaluate mobilities at cell edges using normal velocity components calculated implicitly, and transverse velocity components calculated explicitly (i.e., based on the previously converged time-step); the pressure gradient driving the flow is, as usual, treated implicitly. On 3D hexahedral meshes, the proposed schemes require the same 7-point stencil as that of common semi-implicit schemes where mobilities are evaluated with an entirely explicit velocity argument. When formulated appropriately, their higher level of implicitness however places them, in terms of numerical stability, closer to “real” fully implicit schemes requiring at least a 19-point stencil. A von Neumann stability analysis of these proposed schemes is performed on a simplified pressure equation, representative of both single-phase and multiphase flows, following an approach previously used by the authors to study semi-implicit schemes. Whereas the latter are subject to stability constraints which limit their usage in certain cases where the logarithmic derivative of mobility with respect to velocity is large in magnitude, the former are unconditionally stable for 1D and 2D flows, and only subject to weak restrictionsfor 3D flows.

提出了在模拟非牛顿和非达西效应以及毛细管去饱和作用时发生的地下隐式介质流动模拟中速度依赖运动的拟隐式方案。对于低阶有限体积离散化，原理是使用隐式计算的法向速度分量和显式计算的横向速度分量（即，基于先前收敛的时间步长）来评估单元边缘的迁移率。像往常一样，隐式地处理驱动流量的压力梯度。在3D六面体网格上，提出的方案需要与普通半隐式方案相同的7点模板，在普通半隐式方案中，迁移率是使用完全明确的速度参数进行评估的。如果适当地表述，它们的较高的隐含性将它们置于 就数值稳定性而言，更接近于至少需要19点模版的“真实”完全隐式方案。这些作者提议的方案的冯·诺依曼稳定性分析是在简化的压力方程式上进行的，该方程代表了单相和多相流动，遵循作者先前用来研究半隐式方案的方法。后者受到稳定性约束，在某些情况下会限制其使用，在某些情况下，迁移率相对于速度的对数导数很大，而前者对于1D和2D流则无条件稳定，并且仅对3D流具有弱约束。这些作者提议的方案的冯·诺依曼稳定性分析是在简化的压力方程式上进行的，该方程代表了单相和多相流动，遵循作者先前用来研究半隐式方案的方法。后者受到稳定性约束，在某些情况下会限制其使用，在某些情况下，迁移率相对于速度的对数导数很大，而前者对于1D和2D流则无条件稳定，并且仅对3D流具有弱约束。这些作者提议的方案的冯·诺依曼稳定性分析是在简化的压力方程式上进行的，该方程代表了单相和多相流动，遵循作者先前用来研究半隐式方案的方法。后者受到稳定性约束，在某些情况下会限制其使用，在某些情况下，迁移率相对于速度的对数导数很大，而前者对于1D和2D流则无条件稳定，并且仅对3D流具有弱约束。