High Order in Time and Space Numerical Methods for Solving the Miscible Displacement Problem

求解混相位移问题的高阶时空数值方法

• 批准号: 1318348
• 负责人: Beatrice Riviere
• 金额: $ 22.98万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1318348&HistoricalAwards=false
• 关键词: Order; Time; Space; Numerical; Methods

High order numerical methods in time and space are proposed to solve the incompressible miscible displacement problem in heterogeneous porous media. A mixture of solvent and resident fluids moves as a single phase with a velocity that follows Darcy's law. The solvent concentration satisfies a convection-dominated parabolic problem, with a diffusion-dispersion matrix that depends on the fluid velocity in a non-linear fashion. The fluid pressure equation is coupled with the concentration equation. Under certain conditions, the miscible displacement becomes physically unstable and the phenomenon of viscous fingering occurs. Accurate prediction of the number and location of the viscous fingers is important in the development of a numerical model. Additional numerical challenges include the nonlinear coupling between the pressure and concentration equations, and the unboundedness of the diffusion-dispersion matrix. The investigator and her team propose to use a discontinuous Galerkin method for the time integration. For the spatial discretizations, locally mass conservative methods such as mixed finite element methods and interior penalty discontinuous Galerkin methods are utilized. Several algorithms, based on solving the pressure and concentration equations consecutively, are formulated. Their cost and accuracy are compared. Convergence of the numerical solution is obtained under low regularity assumptions on the data and exact solution using a new generalization of the Aubin-Lions compactness theorem. The effects of randomness in the permeability of the porous media are taken into account by sampling the coefficients and combining the Monte Carlo technique with temporal and spatial discretizations. The algorithms developed in this project are also used to predict the onset and growth of viscous fingers. Two factors contributing to fingering are investigated: the increase of the ratio of the displaced fluid viscosity to the solvent fluid viscosity, and the variation of longitudinal and transverse dispersions.The miscible displacement problem occurs in several applications, including environment and energy. For instance, a large amount of the oil reserve in the U.S. is deemed unrecoverable by current technology. Enhanced Oil Recovery (EOR), by changing the properties of the reservoir and the hydrocarbons, will help produce some of this trapped oil. Miscible displacement is one important technique used in EOR. The main goal of this project is to provide accurate and robust numerical solutions to the miscible displacement problem for incompressible fluids. In EOR, this numerical approximation can be used to efficiently harvest the remaining trapped oil. This project advances discovery and understanding while promoting learning through the training of at least one Ph.D. student and two undergraduate students. In addition, the principal investigator organizes a Summer program in which participating high school students learn about computational mathematics and its applications to complex flow and transport in porous media.

提出了求解非均质多孔介质中不可压缩可混溶驱动问题的时间和空间高阶数值方法。溶剂和驻留流体的混合物作为单相以遵循达西定律的速度移动。溶剂浓度满足对流为主的抛物型问题，与扩散-分散矩阵，取决于流体速度在一个非线性的方式。 流体压力方程与浓度方程耦合。 在一定的条件下，混相驱油变得物理不稳定，出现粘性指进现象。 准确预测粘性指状物的数量和位置在数值模型的开发中是重要的。其他的数值挑战包括压力和浓度方程之间的非线性耦合，以及扩散-弥散矩阵的无界性。 研究人员和她的团队建议使用间断Galerkin方法进行时间积分。 对于空间离散，局部质量守恒的方法，如混合有限元方法和内部惩罚间断伽辽金方法。几个算法，连续求解压力和浓度方程的基础上，制定。比较了它们的成本和精度。数值解的收敛性得到低正则性假设下的数据和精确解使用一个新的推广的奥宾-狮子紧性定理。考虑到随机性的影响，在多孔介质的渗透率的采样系数和时间和空间离散相结合的Monte Carlo技术。 在这个项目中开发的算法也被用来预测粘性手指的发病和增长。 研究了驱替液粘度与溶剂粘度之比的增大以及纵向和横向分散度的变化对指进的影响。 例如，美国的大量石油储量被认为是目前技术无法开采的。通过改变储层和碳氢化合物的性质，提高石油采收率（EOR）将有助于开采一些被困石油。混相驱是提高采收率的一项重要技术。 该项目的主要目标是为不可压缩流体的混溶驱替问题提供精确和鲁棒的数值解。 在EOR中，这种数值近似可以用于有效地收获剩余的被困油。该项目通过培训至少一名博士来促进发现和理解，同时促进学习。学生和两名本科生。此外，首席研究员组织了一个夏季计划，参与其中 高中学生学习计算数学及其在多孔介质中复杂流动和输运中的应用。