Mathematics of geothermal energy

Summary: Geothermal energy can be harnessed for domestic heating or to produce electricity via steam turbine.

“Geothermal” refers to heat from the interior of Earth generated from the forces that led to the planet’s creation and the ongoing slow radioactive decay that continues to generate thermal activity. While Earth’s surface is relatively cool, temperatures increase dramatically with depth, which is known as a region’s “geothermal gradient.” The interiors of continents tend to have lower gradients than “spreading center” regions, where continental tectonic plates are slowly separating. A prime geothermal area is along the Ring of Fire rimming the Pacific Ocean’s eastern, northern, and western coasts.

High geothermal gradients make prime candidates for geothermal energy projects. However, the average gradient is approximately 2.5–3 degrees Celsius per 100 meters. Approximately 6000 kilometers beneath the surface, molten rock reaches temperatures of approximately 5000 degrees Celsius. A small portion of this extreme heat makes its way to the surface as steam through cracks and fissures. Geothermal leakage to the surface leads to dramatic volcanic eruptions as well as to the formation of hot springs and geysers. Geothermal-warmed, mineral-rich waters have long been considered to be sacred or to have healing properties by many people. Geysers such as Old Faithful in Yellowstone continue to attract visitors from around the world.

Mathematicians, geothermal engineers, geologists, and other scientists use mathematical methods to research various aspects of geothermal processes, such as the deformable, porous properties of soil and rock that allow geothermal heat to make its way to the surface. These studies have broad applications in many scientific areas, including the way brains deform during neurosurgery and in industrial injection molding. In other cases, Lagrangian–Eulerian flow models, named for Joseph Lagrange and Leonhard Euler, are used to model characteristics such as precipitation and transport, which have applications for engineering geothermal reservoirs and isolating radioactive waste. Stochastic models for system optimization and control as well as geometric models also help mathematicians understand geothermal heat. Many are working on computer models to update, integrate, and expand the U.S. Geological Survey’s MODFLOW, a three-dimensional finite-difference groundwater flow model first published in 1984 and widely used for research and industrial applications.

Geothermal Heating

As long ago as the nineteenth century, scientists and engineers began to develop geothermal-based applications for chemistry and heating, though there is evidence that even prehistoric people built dwellings around naturally occurring geothermal heat sources. With abundant geothermal resources, Iceland began to emerge by the late 1920s as a world leader in the use of geothermal energy for domestic heating and cooling. Advances since that time have led to the development of geothermal heat pump systems. During cold periods, heat pumps transfer to buildings heat from either the ground (beneath the frost line) or from the bottom of ponds. During warm periods, the process is reversed and heat is taken from buildings and put into the ground or ponds. However, purposeful movement of water on a large scale can have geological consequences. For example, in Venice, the removal of subsurface water resulted in subsidence (settling of loose, porous soil), which lowered some buildings. Adding or subtracting water from one part of a geothermal field can affect all aspects of the field, including system pressure and surface vents. Seismologists use mathematical models describing the behavior of deformable porous rock and soil to predict where events like earthquakes might occur as a result of water-pumping activities.

Geothermal Electricity

Geothermal resources can also be used to produce electricity. The first geothermal electric power plant was built in Larderello, Italy, in 1904. Japan and the United States followed suit in 1910 and 1921, respectively. The spread of geothermal energy has been slow in the decades since. However, because of concerns regarding global warming and a quest to develop nongreenhouse gas (GHG)–emitting energy technologies, geothermal power generation has received more attention.

There are two types of geothermal power plants, both of which rely upon the production of steam to drive the conventional turbines that create electricity. Electricity can be produced directly from steam if the temperatures are at a minimum of 95 degrees Celsius (200 degrees Fahrenheit), and higher outputs are possible after temperatures crest at 175 degrees Celsius (350 degrees Fahrenheit). At the Geysers geothermal power plant in California, steam at a temperature of approximately 235 degrees Celsius (455 degrees Fahrenheit) is used to directly drive turbines. At lower temperatures, geothermal heat can still be used, but it relies upon specialized fluids that have a low boiling point capable of producing high pressures, rather than natural steam.

While the capital costs are high for both types of geothermal electricity, once in production it has several advantages over other forms of electricity generation. Like wind, its fuel costs are negligible. Similar to wind and nuclear power, once constructed, geothermal plants produce far fewer GHG emissions than traditional fossil fuel plants. Geothermal also has advantages over other alternative energy producers. Unlike wind, which is intermittent because of its dependency on weather conditions, geothermal electricity can be relied upon to produce consistent baseload power. Geothermal plants are also less intrusive visually than large wind farms and tend to draw less public attention.

Geothermal also has two key advantages over nuclear generation. Nuclear power plants are dependent upon a finite resource (uranium), and nuclear waste disposal is both controversial and costly. In contrast, geothermal generation depends on a virtually infinite source (heat generated in Earth’s interior), and there are no long-term waste issues.

Popular and government interest in geothermal energy and its advantages over both traditional and alternative electricity generating options led to a 20% increase in global geothermal electricity production between 2005 and 2010. In addition, there has been a 52% increase from 2007 to 2010 in the number of countries developing geothermal resources.

Despite the increasing numbers, geothermal energy production continues to significantly lag behind other electricity sources at the start of the twenty-first century. In part, this lag is the result of a perception that there are a limited number of high-quality geothermal sites that would enable geothermal energy to become a major producer. In addition, there are technical, permitting, and electric transmission issues that drive up capital costs and inhibit substantial expansion.

Bibliography

Bundschuh, Jochen, and Mario Suárez. Introduction to the Numerical Modeling of Groundwater and Geothermal Systems: Fundamentals of Mass, Energy and Solute Transport in Poroelastic Rocks. Oxfordshire, England: Taylor & Francis, 2010.

Dickson, Mary H., and Mario Fanelli. “What Is Geothermal Energy?” International Geothermal Association. http://www.geothermal-energy.org/314,what‗is‗geothermal‗energy.html.

“Geothermal Energy: Tapping the Earth’s Heat.” National Geographic. http://environment.nationalgeographic.com/environment/global-warming/geothermal-profile.

Holm, Alison, Leslie Blodgett, Dan Jennejohn, and Karl Gawell. Geothermal Energy: International Market Update. Washington, DC: Geothermal Energy Association, May 2010.

Idaho National Laboratory. The Future of Geothermal Energy: Impact of Enhanced Geothermal Systems (EGS) on the United States in the 21st Century. Cambridge, MA: MIT Press 2006.

Lerner, K. Lee, and Brenda Wilmoth, eds. “Geothermal Gradient.” In World of Earth Science. Farmington Hills, MI: Gale Cengage, 2003.

National Energy Board. Emerging Technologies in Electricity Generation: An Energy Market Assessment. Ottawa : Her Majesty the Queen in Right of Canada as represented by the National Energy Board, 2006.

U.S. Government Accountability Office. “Renewable Energy: Increased Geothermal Development Will Depend on Overcoming Many Challenges.” May 2006.