Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.
Philos Trans A Math Phys Eng Sci. 2009 Nov 28;367(1907):4633-54. doi: 10.1098/rsta.2009.0160.
In this paper, we present a second-order accurate adaptive algorithm for solving multi-phase, incompressible flow in porous media. We assume a multi-phase form of Darcy's law with relative permeabilities given as a function of the phase saturation. The remaining equations express conservation of mass for the fluid constituents. In this setting, the total velocity, defined to be the sum of the phase velocities, is divergence free. The basic integration method is based on a total-velocity splitting approach in which we solve a second-order elliptic pressure equation to obtain a total velocity. This total velocity is then used to recast component conservation equations as nonlinear hyperbolic equations. Our approach to adaptive refinement uses a nested hierarchy of logically rectangular grids with simultaneous refinement of the grids in both space and time. The integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time, fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized. The single-grid algorithm is described briefly, but the emphasis here is on the time-stepping procedure for the adaptive hierarchy. Numerical examples are presented to demonstrate the algorithm's accuracy and convergence properties and to illustrate the behaviour of the method.
本文提出了一种求解多相不可压缩多孔介质流的二阶精度自适应算法。我们假设达西定律的多相形式，其中相对渗透率是相饱和度的函数。其余方程表示流体成分的质量守恒。在这种情况下，总速度定义为相速度之和，是无散度的。基本积分方法基于全速度分裂方法，其中我们求解二阶椭圆压力方程以获得总速度。然后，使用该总速度将分量守恒方程重新表示为非线性双曲方程。我们的自适应细化方法使用具有逻辑矩形网格的嵌套层次结构，同时在空间和时间上细化网格。网格层次结构上的积分算法是一个递归过程，其中粗网格随时间推进，细网格推进多个步骤以达到与粗网格相同的时间，然后对不同层次的数据进行同步。简要描述了单网格算法，但重点是自适应层次结构的时间推进过程。给出了数值示例，以证明算法的准确性和收敛特性，并说明该方法的行为。
