Modeling, Analysis, and Computation for Water-Drive Oil Recovery

在石油工业偏微分方程模型的研究中，精确的数学分析和有效的数值方法起着越来越重要的作用。本项目的目标是对修正的Buckley-Leverett（MBL）模型进行数学分析和高阶精度数值方法设计研究，该模型是水驱地下二次采油的偏微分方程模型。当地下石油资源被挖掘时，由于压力差，一定量的石油会自行流出。在流动停止后，通常仍有大量的石油留在地下。一种标准的“二次采油”方法是通过注入井将水泵入油田，迫使石油通过生产井流出。在这个过程中，将产生水和油的混合物。MBL方程模拟了整个油藏，特别是生产井的含油饱和度的演变。本研究项目结合了数学分析、数值方案设计和计算技术。数学分析是基于偏微分方程理论，和正在开发的数值方法是基于国家的最先进的间断伽辽金（DG）计划。将使用实验室实验的数据对结果进行交叉验证。研究者计划开展以下具体研究工作：（1）将成熟的一维MBL模型扩展到二维和三维完全非线性和线性化的MBL模型;（2）确定在小于整个储层的空间域上采用MBL模型所引起的近似误差;（3）设计高阶精度间断Galerkin（DG）方法来数值求解MBL模型;（4）研究MBL模型行波解的非线性渐近稳定性;（5）用实验数据对理论和计算结果进行交叉验证。该项目涉及一名博士后和一名研究生。研究人员还计划开发一个新的研究生课程，内容是水驱采油模型的数值解。

Accurate mathematical analysis and efficient numerical methods play an increasingly important role in studying partial differential equation models in the petroleum industry. The goal of this project is to carry out research in mathematical analysis and design of high-order-accuracy numerical methods for the modified Buckley-Leverett (MBL) model, which is a partial differential equation model for water-drive secondary underground oil recovery. When an underground source of oil is tapped, a certain amount of oil flows out on its own due to pressure difference. After the flow stops, there is typically a significant amount of oil still left in the ground. One standard method of "secondary recovery" is to pump water into the oil field through an injection well, forcing oil out through a production well. In this process there will be a water and oil mixture created. The MBL equation models the evolution of the oil saturation in the entire oil reservoir, in particular, at the production well.This research project combines mathematical analysis, design of numerical schemes, and computational techniques. The mathematical analysis is based on partial differential equation theory, and the numerical methods under development are based on state-of-the-art discontinuous Galerkin (DG) schemes. The results will be cross-validated using data from laboratory experiments. The investigator plans to carry out the following specific research tasks: (1) extend the well-developed 1D MBL model to 2D and 3D fully nonlinear and linearized MBL models; (2) determine the approximation error induced by employing the MBL model on a spatial domain smaller than an entire reservoir; (3) design high-order-accuracy discontinuous Galerkin (DG) methods to numerically solve the MBL model; (4) study the nonlinear asymptotic stability of the traveling-wave solutions of the MBL model; (5) employ experimental data to cross-validate both the analytical and computational conclusions. The project involves a postdoctoral associated and a graduate student in the research. The investigator also plans to develop a new graduate-level course on numerical solutions to water-drive oil recovery models.