Novel mathematical approaches for multiscale modelling of three-phase porous media flow

这是爱丁堡赫瑞瓦特大学数学系（MACS）和石油工程研究所（IPE）的联合竞标，将涉及两个部门的研究人员。两个学系都建立了一个充满活力的跨学科环境，这要归功于目前由PI Dugald Duncan教授主持的弥合差距补助金。所有被命名的研究人员都是爱丁堡工程与数学研究伙伴关系（www.erp.ac.uk）、麦克斯韦数学科学研究所和爱丁堡地下科学与工程合作组织（ECOSSE）的两个主要联合研究机构的成员。这两所大学都是苏格兰资助委员会2200万英镑资助计划的一部分，该计划旨在促进爱丁堡数学和工程领域的跨学科研究。此外，麦克斯韦研究所是470万科学与创新基金的合作伙伴，成立了跨学科的数值算法和智能软件中心（NAIS）。我们的提案旨在巩固现有的和创建新的跨学科合作，将补充和加强MACS和IPE的世界领先的研究概况。我们建议探索新的数学方法对三相（如油、气、水）多孔介质流动进行多尺度建模的可行性。为此，我们将跨越工程学科（即石油、水文学和工业多孔介质应用）、物理和应用数学之间的传统界限，并组建了一个来自不同学科的专家团队。我们的研究将具有高度的推测性和冒险性，因为我们根本不知道是否可以开发出一种适合三相达西定律的替代方案，从而可以以数学上稳健和有效的方式解决，这是在日常工程应用中成功使用所必需的。然而，我们需要另一种选择，因为达西定律在某些流动状态下本质上是无效的，即当一个或多个相作为不连续的斑点和神经节运动时。然而，这种情况正是工程学最感兴趣的，例如，在预测少量有毒地下水污染物或枯竭碳氢化合物储层中残余油的运移时。我们工作的高风险和投机性研究性质不一定会吸引资金，尽管在许多应用中具有重要的基础意义：对于工业支持来说，它过于投机和蓝天，产品的交付时间是几个月，而不是几年，并且由于其新见性，跨学科性和方法范围，将很难从其他来源获得资金。然而，如果我们的初步研究表明，三相流达西定律的替代方案是可行的，其结果将对工程和数学学科产生重大影响：三相流达西定律的可能替代或改进将对提高采收率技术产生根本性影响。这些技术旨在从日渐衰退的油田中提取碳氢化合物，比如英国北海和世界各地的油田。它们也将是改善地下水污染物修复、预测地下二氧化碳储存和各种其他工业多孔介质应用（例如木材加工）的关键。我们的研究还将解决重要和具有挑战性的数学和计算问题，例如开发高效可靠的算法，升级方法，数值稳定性和收敛证明，或误差估计，所有这些都必须能够应对三相流固有的高度非线性物理以及工业应用中出现的数量级参数变化和相关不确定性。

This is a joint bid from the Department of Mathematics (MACS) and Institute of Petroleum Engineering (IPE) at Heriot-Watt University in Edinburgh and will involve researchers across both departments. Both departments have established a virbant cross-disciplinary environment thanks to the current Bridging the Gaps grant held by the PI, Prof. Dugald Duncan. All named investigators are members of two major joint research institutes of the Edinburgh Research Partnership in Engineering and Mathematics (www.erp.ac.uk), the Maxwell Institute of Mathematical Sciences and the Edinburgh Collaboration of Subsurface Science and Engineering (ECOSSE). Both are part of the 22M funding initiative of the Scottish Funding Council to foster interdisciplinary research in mathematics and engineering across Edinburgh. Furthermore the Maxwell Institute is a partner in the 4.7M Science and Innovation Grant to found the interdisciplinary Numerical Algorithms and Intelligent Software Centre (NAIS).Our proposal aims to consolidate existing and create new cross-disciplinary collaborations that will complement and enhance the world-leading research profiles of MACS and IPE. We propose to explore the feasibility of novel mathematical approaches for multiscale modelling of three-phase (e.g. oil, gas, water) porous media flow. For this we will cross traditional boundaries between engineering disciplines (i.e. petroleum, hydrology, and industrial porous media applications), physics, and applied mathematics and have assmebled a team of experts from the different disciplines. Our studies will be highly speculative and adventurous because we simply do not know if a suitable alternative to three-phase Darcy's law can be developed in such a way that it can be solved in the mathematically robust and efficient manner that is required for successful use in daily engineering applications. However, an alternative is required because Darcy's law is intrinsically not valid in certain flow regimes, that is when one or more phases move as discontinuous blobs and ganglia. Yet it is exactly this situation that is of most interest to engineering, for example when predicting the migration of small volumes of toxic groundwater contaminants or residual oil in a depleted hydrocarbon reservoir. The high risk and speculative research nature of our work would not necessarily attract funding despite being of fundamental importance in many applications: It is too speculative and blue sky for industrial support where lead time to product is months, not years and would struggle to be funded from other sources due to its novely, interdisciplinarity, and range of approaches. Yet, if our initial studies show that an alternative to Darcy's law for three phase-flow is feasible, the outcomes will have a major impact on engineering and mathematical disciplines: A possible alternative or improvement of Darcy's law for three-phase flow will have a fundamental influence in improving enhanced oil recovery techniques. Such techniques aim to extract hydrocarbons from declining oil fields such as they are frequent in the U.K. sector of the North Sea and world-wide. They will also be the key to improving the remediation of groundwater contaminants, predicting subsurface CO2 storage, and a variety of other industrial porous media applications (e.g., wood processing). Our research will also tackle important and challenging mathematical and computational issues, such as development of efficient and reliable algorithms, upscaling methods, numerical stability and convergence proofs, or error estimation, all of which must be able to cope with the highly non-linear physics inherent to three-phase flow and the orders-of-magnitude variation in parameter and associated uncertainties arising in industrial applications.