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An adaptive time stepping algorithm for IMPES with high order polynomial extrapolation
Applied Mathematical Modelling ( IF 5 ) Pub Date : 2021-03-01 , DOI: 10.1016/j.apm.2020.10.045 Stevens Paz , Alfredo Jaramillo , Rafael T. Guiraldello , Roberto F. Ausas , Fabricio S. Sousa , Felipe Pereira , Gustavo C. Buscaglia
Applied Mathematical Modelling ( IF 5 ) Pub Date : 2021-03-01 , DOI: 10.1016/j.apm.2020.10.045 Stevens Paz , Alfredo Jaramillo , Rafael T. Guiraldello , Roberto F. Ausas , Fabricio S. Sousa , Felipe Pereira , Gustavo C. Buscaglia
Abstract Two-phase flows in oil reservoirs can be modeled by a coupled system of elliptic and hyperbolic partial differential equations. The transport velocity of the multiphase fluid system is related to the pressure through Darcy’s law and it is coupled to a conservation law for the saturation variable of one of the phases. A time step of the classical IMPES (IMplicit Pressure Explicit Saturation) method consists of first solving the elliptic problem for pressure and Darcy velocity, and then updating the saturation with an explicit numerical scheme for conservation laws. This method is very computationally costly, since the time-consuming elliptic solver must be invoked at time intervals defined by the stability limit of the hyperbolic solver. A popular variant is not to update the velocity at all hyperbolic time steps, but to skip a fixed number C of them, with C determined by the user. In this work we propose a more accurate and systematic procedure for time stepping in IMPES codes. The velocity is updated at all transport time steps, though the elliptic solver is only invoked every C steps. In the time steps at which the elliptic problem is not solved, the velocity is extrapolated from previously computed values with polynomials of high degree. Further, we introduce an error estimator that allows for the number C to be adaptively determined without user intervention. The algorithm was tested in several relevant benchmark problems. This allowed for the optimization of its parameters and comparisons with previous variants. The results show that the proposed algorithm is very stable, reliable and time-cost effective. It is also easily implemented in pre-existent IMPES codes.
中文翻译:
一种具有高阶多项式外推的IMPES自适应时间步长算法
摘要 油藏中的两相流可以通过椭圆偏微分方程和双曲偏微分方程的耦合系统建模。多相流体系统的输运速度通过达西定律与压力相关,并与其中一相饱和变量的守恒定律耦合。经典 IMPES(隐式压力显式饱和度)方法的时间步长包括首先求解压力和达西速度的椭圆问题,然后使用显式数值方案更新饱和度以符合守恒定律。这种方法的计算成本非常高,因为必须在由双曲线求解器的稳定性极限定义的时间间隔内调用耗时的椭圆求解器。一个流行的变体是不更新所有双曲线时间步长的速度,但要跳过其中的固定数量 C,C 由用户决定。在这项工作中,我们提出了一种更准确和系统的 IMPES 码时间步长程序。尽管椭圆求解器仅在每 C 步调用一次,但在所有传输时间步中都会更新速度。在椭圆问题未解决的时间步长中,速度是从先前计算的值与高次多项式推断出来的。此外,我们引入了一个误差估计器,允许在没有用户干预的情况下自适应地确定数量 C。该算法在几个相关的基准问题中进行了测试。这允许优化其参数并与以前的变体进行比较。结果表明,所提出的算法非常稳定、可靠且具有时间成本效益。
更新日期:2021-03-01
中文翻译:
一种具有高阶多项式外推的IMPES自适应时间步长算法
摘要 油藏中的两相流可以通过椭圆偏微分方程和双曲偏微分方程的耦合系统建模。多相流体系统的输运速度通过达西定律与压力相关,并与其中一相饱和变量的守恒定律耦合。经典 IMPES(隐式压力显式饱和度)方法的时间步长包括首先求解压力和达西速度的椭圆问题,然后使用显式数值方案更新饱和度以符合守恒定律。这种方法的计算成本非常高,因为必须在由双曲线求解器的稳定性极限定义的时间间隔内调用耗时的椭圆求解器。一个流行的变体是不更新所有双曲线时间步长的速度,但要跳过其中的固定数量 C,C 由用户决定。在这项工作中,我们提出了一种更准确和系统的 IMPES 码时间步长程序。尽管椭圆求解器仅在每 C 步调用一次,但在所有传输时间步中都会更新速度。在椭圆问题未解决的时间步长中,速度是从先前计算的值与高次多项式推断出来的。此外,我们引入了一个误差估计器,允许在没有用户干预的情况下自适应地确定数量 C。该算法在几个相关的基准问题中进行了测试。这允许优化其参数并与以前的变体进行比较。结果表明,所提出的算法非常稳定、可靠且具有时间成本效益。