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

提出了求解非均质多孔介质中不可压缩混相驱替问题的时间和空间高阶数值方法。溶剂和滞留流体的混合物以符合达西定律的速度作为单一相运动。溶剂浓度满足对流占优的抛物型问题，扩散-弥散矩阵以非线性方式依赖于流体速度。流体压力方程与浓度方程耦合在一起。在一定条件下，混相位移变得物理不稳定，出现粘性指进现象。对粘性手指的数量和位置的准确预测在数值模式的发展中是重要的。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不连续Galerkin方法进行时间积分。对于空间离散，采用了混合有限元、内部罚不连续Galerkin等局部质量守恒方法。在连续求解压力方程和浓度方程的基础上，提出了几种算法。对它们的成本和精度进行了比较。利用Aubin-Lions紧致性定理的一个新推广，在数据和精确解的低正则性假设下，得到了数值解的收敛。通过采样系数并将蒙特卡罗技术与时间和空间离散相结合，考虑了多孔介质渗透率的随机性的影响。该项目中开发的算法也被用于预测粘性手指的开始和生长。研究了影响指进的两个因素：驱替流体粘度与溶剂流体粘度之比的增大，以及纵向和横向弥散的变化。例如，按照目前的技术，美国的大量石油储备被认为是无法开采的。通过改变储集层和碳氢化合物的性质，提高石油采收率(EOR)将有助于生产其中的一些滞留石油。混相驱替是提高采收率的重要技术之一。该项目的主要目标是为不可压缩流体的混相驱替问题提供准确和稳健的数值解。在提高采收率中，这种数值近似可以用来有效地收集剩余的圈闭石油。该项目通过培训至少一名博士生和两名本科生，在促进学习的同时促进发现和理解。此外，首席研究人员组织了一个暑期项目，在该项目中，参与的高中生将学习计算数学及其在多孔介质中复杂流动和传输中的应用。

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.