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

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.

If our initial studies demonstrate that the development of Darcy's law is feasible, then the findings will impact a variety of engineering disciplines, most notably petroleum engineering and groundwater engineering, although other inudstrial porous media flow applications, like wood processing, will also benefit from our research. Petroleum engineers and groundwater engineers are concerned with providing mankind with the basic resources energy and drinking water. Despite major progress in energy efficiency, it is expected that the world's energy demand will be about 40% higher in 2030 compared to 2005 due to the increase in population. Although renewable and nuclear energy are expected to become more proficient, oil, gas, and coal will remain indispensable, providing about 80% of the world's energy. Yet, essentially all of the U.K.'s oil fields (and most of the easily recoverable fields world wide) are declining. Thus so-called enhanced oil recovery techniques become increasingly important. These include, but are not limited to, the injection of surfactants, alternating injection of water and gas, or the injection of CO2. In most cases, three-phases are flowing in the reservoir either because gas is actively injected or already present. The accurate prediction, by means of accurate physical models and fast and robust algorithms, how the three phases, oil, gas, and water, flow in a reservoir is hence of fundamental importance - especially when the volume fraction of one phase is low, the very situation where Darcy's law for three-phase flow fails. Burning fossil fuels releases CO2 into the atmosphere, which causes global warming. One important solution that helps to reduce levels of atmospheric CO2 is to re-inject it into oil and gas fields or deep saline aquifers. Again, situations where three-phase (e.g., brine, CO2 in gaseous and liquid form) flow in the reservoir are possible. Groundwater is a vital resource in all countries and the United Nations have recently published the World-wide Hydrogeological Mapping and Assessment Programme. It shows that most urban groundwater reservoirs are now under stress because of increased populations and subsequent groundwater contamination. Situations where three phases flows occur as well, for example when non-aqueous phase liquids, which are usually highly toxic and carcinogen, have contaminated a groundwater reservoir and are remediated by the injection of air or hot steam. Accurate physical models and numerical algorithms may be even more important in such situation compared to petroleum engineering because the toxic contaminants must be removed to ppm or ppb levels. Thus, in summary, if our initial studies prove to be feasible, then they will have a tremendous impact on a variety of engineering disciplines which will help to foster the economic competitiveness of the U.K., enhance the quality of life and benefit the wider public in the U.K. and world-wide by helping to provide a secure supply of two crucial resources, energy and water.