By Malgorzata Peszynska
Methane hydrate is an ice-like crystalline substance (gas clathrate) made of water molecules encasing a molecule of methane. Low temperatures and high pressure favor the hydrate’s existence (e.g., entrapment of methane molecules in the ice cages), making hydrate deposits abundant in deep ocean sediments and in Alaska (see Figure 1). Upon disturbance of these favorable conditions, methane might escape to the atmosphere from hydrate-bearing sediments, contributing to the overall balance of other greenhouse gases. Methane hydrate is considered a “smoking gun” in environmental and (paleo-)climate studies [8]. Hydrates are also known for the nuisance they cause plugging up wells and pipes that carry other hydrocarbons to their destinations. They are a significant drilling hazard and contributed to the Deepwater Horizon explosion and oil spill. Researchers study the presence of hydrates due to their impact on slope stability of submarine formations [4]. If you haven’t yet heard about methane hydrate, it has likely not caused any recent high-profile disasters.
However, exciting (non-disastrous) scientific news related to the study of gas hydrates is also possible, and mathematicians can play a role. Geophysicists have been studying hydrates for a long time, but many scientific mysteries surrounding their origin and evolution remain. Computational models exist, but require interdisciplinary collaboration and access to first-rate information and data to be meaningful (see Figure 2). Computational and applied mathematicians can build virtual laboratories for hydrates, but their real impact requires investment in the language, methodology, priorities, and funding models of the allied fields.
For example, the hydrate “lives” in its host rock. How did it get there? Basin modeling, which predicts hydrate deposit formation from upward migration of gas over timescales of thousands of years and spatial scales of hundreds of meters, provides the answer. Comprehensive basin models [5] offer convincing scenarios of gas hydrate formation (see Figure 1). Notwithstanding the enormous uncertainty in basin modeling, these models are very complex and sensitive. While many mathematicians are versed in Stefan nonlinear free-boundary problems, it takes an investment to understand, analyze, and build multicomponent models in which the temperature is at the backstage rather than at the center (where mass fractions rule). Further, computational models for hydrates are quite sensitive to their data due to both their singularity and the relatively narrow envelope of physically-meaningful solutions. To work around the sensitivity, one must analyze the models and their computational counterparts.
Figure 1. This two-dimensional simulation is inspired by a one-dimensional case study [2], while geometry is modeled after that of Hydrate Ridge. The red-to-yellow colors show the increase of methane hydrate saturation (Sh) caused by the migration of methane upwards from the bottom of the reservoir into the hydrate zone. The blue lines delineate rock types; initially, they differentiate between fine-grained (clay) and coarse-grained (silt) sediment. Fractures form at the bottom over time. The uneven migration and hydrate formation are due to the heterogeneity of the rock and the complicated dependence of thermodynamics on the host rock. Image credit: Malgorzata Peszynska.
The study of gas hydrates is relatively uncharted territory for computational mathematics. While investigating solutions to the comprehensive, coupled, multiphase multicomponent models is out of reach for well-posedness analysis, one can study subproblems that focus on the primary difficulties while freezing the model’s other elements. This strategy works particularly well for basin modeling at large timescales in which the hydrate evolution (almost) fits in the traditional framework of free boundary problems. However, there is a snag. Unlike freezing of water (melting of ice), which always occurs around 0°C, the phase behavior for hydrate formation (dissociation) is associated with variable pressure and temperature conditions. This challenge requires a special convex analysis construction [3, 7]. In addition, while transport by diffusion is unlikely to have singular solutions (similarly to the temperature in the Stefan problem), strong advection fluxes lead to large discontinuous deposits observed in nature and require a very weak notion of solutions to the partial differential equations.
Another challenge accompanies the burden (and beauty) of multiple scales. In many places around the world, the presence of hydrates has been inferred rather than confirmed. Unlike the free gas deposits visible in seismic images of the subsurface, the hydrate cannot be easily seen because its density is close to that of water, necessitating modeling to assist the interpretation of well core data [6]. Furthermore, the response of sediment to seismic waves depends on the microscopic (pore-scale) distribution of the hydrate and whether it cements the distance between host rock grains, thus altering the elastic response. Without detailed models at the pore-scale, the unusual pockmark or chimney-like morphology of some deposits is hard to explain. However, at and below the pore-scale, the mathematical modeling of methane hydrate is still an art rather than a science. While some phase-field and molecular dynamics models have emerged [9], they are still far from the comfortable framework in which we prove theorems.
Figure 2. Interdisciplinary objectives and multiple scales in gas hydrate studies and modeling. The molecules pictured in the middle include water (H20), methane (CH4), and carbon dioxide (CO2); the latter is present because of its relevance to methane production and carbon sequestration by the process of molecule exchange [10]. Image credit: Malgorzata Peszynska.
Going down in pore-scales from centimeter to meter or kilometer to micron or nanometer has become fairly established in petroleum engineering and hydrology. Geoscientists and imaging experts design flow experiments and carry out X-ray computed tomography imaging that displays the evolution of fluids in the void space between rock grains in either the fully-opaque three-dimensional porous media or two-dimensional micromodels, which look like pinball structures. The virtual laboratory for hydrates at the pore-scale is nearly there. Unfortunately, hydrates are unstable in standard conditions, rendering a bank of images and experimental data both costly and difficult to acquire. First-principles modeling can therefore play a significant role.
Methane hydrate is also of interest as a potential energy source. In the last decade, the U.S. (at the Ignik-Sikumi well in Alaska), Japan, Korea, and India have implemented and explored hydrate projects [1]. However, technological difficulties and many open questions remain — the models used for production require a very different timescale compared to basin modeling, and demand another degree of complexity.
In summary, the overarching challenge in studying hydrates is arguably not just the complexity of the problem itself, but rather the ability to simplify and extract subproblems so as to move forward and make progress without compromising the results. For applied and computational mathematicians, the process of translating the hydrate model so it can fit into a mathematical framework amenable to analysis and simulation can be very rewarding. However, it requires vigilance from all participants of the interdisciplinary team. As we make simplifying assumptions that enable intricate mathematics, we must be careful not to render the model inapplicable. When implementing numerical schemes for the comprehensive model, it is hard to know if the visible discontinuities are real, attributable to deficiencies of the scheme, or due to poor resolution of the phase behavior data or solver. Hence, it is essential for all team members to understand each other’s objectives, respectfully acknowledge the existence of knowledge gaps, and carefully fill them in. Progressing from basin to production timescales—or from sediment depths to the scale of microcracks and gas chimneys—requires more resources across several fields. Observations and data are needed, so modeling can at least be guided even if validation is impossible.
Acknowledgments: This research was partially supported by NSF grants DMS-1522734 "Phase transitions in porous media across multiple scales" and DMS-1115827 "Hybrid modeling in porous media."
References
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[7] Peszynska, M., Showalter, R.E., & Webster, J.T. (2015). Advection of methane in the hydrate zone: model, analysis and examples. Math. Meth. Appl. Sci., 38(18), 4613-4629.
[8] Ruppel, C.D., & Kessler, J.D. (2017). The interaction of climate change and methane hydrates. Rev. Geophys., 55(1), 126-168.
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[10] White, M., & Suk Lee, W. (2014). Guest Molecule Exchange Kinetics for the 2012 Ignik Sikumi Gas Hydrate Field Trial. In Offshore Technology Conference. Houston, TX.
Malgorzata Peszynska is a professor of mathematics at Oregon State University. She received her Ph.D. from the University of Augsburg in 1992, and has held various positions at the Polish Academy of Sciences, Warsaw University of Technology, Purdue University, and The Institute for Computational Engineering and Sciences at the University of Texas at Austin.