Solving Free Boundary Problems with Level-Set Methods

Level set methods allow for implicit front tracking which is critical in the solution of many important problems. Shown below is a sample of the results from one of my codes. These figures show the evolution of fronts resulting from plasma enhanced physical vapor deposition onto nano-fibers under various conditions. Each contour represents the front position at 10 second intervals.





















Modeling Natural Gas Production from Hydrate Reservoirs

I serve as the co-coordinator of an international effort to compare the leading methane hydrate reservoir simulators. The National Energy Technology Laboratory (NETL) and the U.S. Geological Survey (USGS) are guiding a collaborative, international effort to compare methane hydrate reservoir simulators. The intentions of the effort are: (1) to exchange information regarding gas hydrate dissociation and physical properties enabling improvements in reservoir modeling, (2) to build confidence in all the leading simulators through exchange of ideas and cross-validation of simulator results on common datasets of escalating complexity, and (3) to establish a depository of gas hydrate related experiment/production scenarios with the associated predictions of these established simulators that can be used for comparison purposes. More information about this effort can be found at:



Flow Through a Fractured Reservoir

I was recently involved in a research project with the National Energy Technology Laboratory (NETL) of the US Department of Energy involving modeling flow through a network of explicit fractures in a porous medium. As part of this effort I worked on the development of a model for simulating flow in a multi-layer fractured reservoir. This involved in extending the model so that it could be used in the future as a tool to aid in the assessment of the impact of carbon dioxide sequestration in naturally existing fractured reservoirs. One example of such reservoirs are exhausted coal beds and/or natural gas reservoirs, into which it has been proposed that the injection of CO2 will not only allow for the sequestration of this green house gas, but will also result in the expulsion of methane from the reservoir. The collection and sale of this clean fuel could help to offset the cost of sequestering the CO2. I am part of a team at NETL that is developing a set of diagnostic tools that could be used to help determine where and how to sequester CO2 so as to minimize the ecological risk to the environment, while maximizing the benefits to man.



Shown below are the results of a simulation on a three layer reservoir where gas is being pumped out of a well in the topmost layer of a reservoir containing on the order or 10,000 explicit fractures covering approximately 20 square miles. As gas is pumped from the well, the pressure decreases in the fractures, with the pressure changes propagating outward from the well through the fracture network in both horizontal and vertical directions.




Thermodynamics of Hydrate Formation/Decomposition in Porous Media

As a result of my sabatical with the Department of Energy, I become involved in efforts to model the thermodynamics of gas hydrates in porous media. Gas hydrates are crystalline structures, belonging to a group of solids known as clathrates, which involve a lattice made up of hydrogen bonded water molecules containing cavities occupied by guest gas molecules. Gas hydrates form under low temperature – high pressure conditions, both above and below the freezing point of water. Under proper conditions the lattice is stabilized by van der Waals forces through the occupation of specific cavities within the lattice by certain types of guest molecules. Among the gases that form hydrates are light alkanes (such as methane and propane), carbon dioxide, nitrogen, chlorine, oxygen, hydrogen sulfide, xenon, and krypton. The type of guest molecule(s) present determines which of three known crystal structures the lattice assumes. For a single component gas, small molecules such as methane stabilize Structure I hydrates, while larger guests such as propane stabilize Structure II, both of which consist of two types of cavities. The stabilization of Structure H hydrates requires the presence of two different sized molecules, with small molecules such as methane occupying two different types of small cavities, and a larger molecule such as neohexane occupying large cavities. Interest in natural gas hydrates is in part due to their presence under two different types of conditions. The first is related to their formation in manmade environments such as pipelines. When hydrates form they tend to do so in aggregates, with a high potential of blocking pipelines if allowed to form unchecked. As a result, the petroleum industry has devoted a large amount of effort towards understanding how to impede their formation in pipelines as well as other related equipment. The second set of conditions are those found in nature, where hydrates form in permafrost regions and in deep oceans, where they are commonly found below the ocean floor in the spaces between sediment pores, acting as a cement holding the sediment together. I have developed a model that is capable of predicting the formation pressure for these compounds under conditions similar to those found on the ocean floor. I am now applying this model to explain/interpret experimental results obtained by our group as part of an effort to better understand the formation and stability of these clathrate compounds. This work is being used by DOE in its evaluation of the potential sequestration of carbon dioxide in the form of hydrates in deep water aquefers and/or in the oceans. For example, the figure below shows the comparison of the the hydrate equilibrium pressures for pure methane and carbon dioxide hydrates, as well as one for a 50-50 mixture as functions of temperature and pore radius.  These calculations along with those involving the enthalpies of formation of these compounds can be used to assess the thermodynamic feasability of the sequestration of carbon dioxide in naturally occuring hydrocarbon hydrates below the ocean floor as well as in other deep aquafers.






Gypsy Moth Population Dynamics

I have also worked on a project relating to the control of gypsy moth populations in hardwood forests. Gypsy moth populations follow "boom and bust" cycles in which there are periods of severe damage to trees separated by times when the pest population is low and does very little if any real damage. During the last severe outbreak of gypsy moths in 1981, over 6 million hectares of mixed oak forests were defoliated, with a cost to Pennsylvania alone of $81 million, $72 million of which was due to loss of timber (the rest was the cost of spraying in an attempt to control the pests). The leading edge of the region where gypsy moths have become established has now entered the West Virginia and the surrounding states.


My research has been directed at constructing mathematical models which can be used to simulate the dynamics of gypsy moth populations, and using these models to evaluate the effectiveness of different pest management strategies. During the last two years, models have been constructed and tested which are useful in applications where the area being studied is on the order of several hundred acres. In studying the effectiveness of different treatment plans, these models showed that under certain conditions, spraying at the wrong time can actually increase the damage done by the gypsy moths over a long time period. Such results need to be studied in more detail to determine how they can be used to help improve current management strategies. Research during the last 2 years has focused on the development of a new model which has shown the ability to accurately predict both the timing and magnitude of future outbreaks (A predictive model for gypsy moth population dynamics with model validation). This is the first model for this important forest pest which has shown this quality of agreement with field data, and which, as a validated model, could be invaluable as both a risk assessment and pest management tool. My most recent work in this area has studied the effects of fractal basin boundaries on predictions over even moderate time scales if data from only a single year is used to make future predictions. For example, the figure shown below depicts the effects of the variation of the carrying capacity of the forest (along the horizontal axis) and that of the initial number of gypsy moths infecting the stand of trees per hectare (vertical axis)  on the behavior of the population.  In this figure, white indicates that the populations take less than 15 years before reaching a stable periodic behavior, while  the eight equally spaced gradations of gray indicate times upto 150 years spent in chaotic transients before reaching the stable periodic behavior. The basin boundary shown here was found to be fractal, indicating that an error of less than 1%  in predicting the initial mass of the gypsy moths infesting a forest is necessary in order to be able to make reasonable predictions over even as small a time as 5 years.




Fluidized Beds

Some time ago I joined an ongoing project in this area involving I. Christie (Dept of Math., WVU), and G.H. Ganser (Dept of Math., WVU). This research involves the investigation of the dynamics of a bed of very fine particles fluidized by having a fluid (such as air) forced through a plate at the bottom of the bed, and has applications in the combustion of coal as well as the manufacture of pharmaceuticals. Due to the nature of the problem, such beds are unstable, and any slight deviation from the equilibrium state results in motion of the particles. The model we are using is made up of a set of hyperbolic partial differential equations, and has lead to the observation of the kidney shaped bubbles observed in experiments. My most recent work in this area is a detailed comparison of the models predictions with the characteristics of experimentally observed bubbles and the overall dynamics of the bed (Validation of a two-dimensional hyperbolic system modelling a fluidized bed). The figure below shows the results of a simulation depicting what is known as "slugging", and is characteristic of narrow fluidized beds. The white regions are gas bubbles, while darker regions contain a higher concentration of particles. The work presented in the above manuscript has substantiated our belief that our simple model has captured the underlying physics of this complicated two-phase flow system. Future work will concentrate on exploring the dynamics of these bubbles, their formation, and the inclusion of other influences which may be of interest to industry.



Convection in Chemical Waves

For a number of years I was involved in a project directed at investigating the effects of convection in hydrodynamic systems on the propagation and dynamics of chemically reacting diffusion fronts with B.F. Edwards (Dept. of Physics, WVU) and D.A. Vasquez (Dept. of Physics, IPFW). We were some of the first investigators to seriously attempt to include the effects of hydrodynamic instability on the dynamics of these chemical fronts. These so called chemical waves are often seen as “cool flames” since they evidence much of the character of combustion fronts with the benefit that they propagate at much slower speeds making them more readily studied experimentally as well as theoretically. I recently completed my work in this area culminating in a paper on nonlinear front evolution in cylinders (Nonlinear Front Evolution of Chemical Waves in Vertical Cylinders) where I compare our theoretical results with experimental ones. The paper shows excellent agreement between the predicted and observed front speed. In addition, the shape of the front and the transition between axisymmetric and nonaxisymmetric fluid flow in the cylinder as its diameter is changed also show excellent agreement. These comparisons are shown in the two figures below.