Modeling Complex, Multiphase Porous Media SystemsApril 3, 2002
Researchers determined that it was the removal of water from the Venetian water table that caused the framework holding it---the sediments on which Venice rests---to contract, resulting in widespread subsidence, with many buildings sinking by several centimeters.
In the 1970s, scientists figured out why (and how) Venice is sinking. They modeled the sediments on which the watery city rests, and the liquid that fills the pores of the loosely consolidated materials below. The researchers determined that the residents of the city had pumped out too much water, which exacerbated the effects of the rising sea level. What these researchers were describing was a permeable material that was changing because of changes in the liquid it held-a perfect example of deformable porous media.
The term "deformable porous media" basically encompasses a framework---perhaps microns across, perhaps miles---made of a somewhat rigid material containing open spaces. Within the pore spaces is a more pliable material, which can be set in motion. The subsequent changes in pressure---whether brought about by the removal, addition, or motion of a liquid or a gas---can cause deformation in the less flexible lattice holding the fluid.
In the case of Venice, the removal of water from the water table caused the framework holding it---the unconsolidated soils on which Venice is built---to contract; the result has been subsidence across the region and buildings that have dropped by several centimeters. Elsewhere, seismologists have used the concept of deformable porous media to predict where water-pumping activities might induce earthquakes. For example, the removal or addition of water (whether as steam or liquid) from the pore spaces and interstices of a geothermal field might result in subsidence across the entire steam reservoir as pressure on the cracks changes. Such deformation of rocks can create very small earthquakes, as it did in Rangely, Colorado, during experiments with water forced into gas wells in the area. (Feeling earthquakes up to magnitudes 4 and 5 from the pumping, Denver residents got mighty nervous.)
You can also go deeper into the earth and apply the concept to magma fluid flow, but mathematical modeling of deformable porous media does not stop with geophysical applications. Biophysicists apply the idea to the human body, modeling the behavior of cartilage or the deformation of the brain during neurosurgery. Pharmacologists use it to model drug delivery. Materials scientists apply it to injection molding processes. The applications of models of deformable porous media are numerous-but getting the models right can be extremely tricky.
Marc Spiegelman, an associate professor of geology and applied mathematics at Columbia University, says it all starts with the scaling. The scale at which he works is quite large, as Spiegelman models the earth's magma layer. The scale may not be that of a typical deformable porous media case study, but the modeling process he uses is the same.
"The big picture is that you want to model a sponge," Spiegelman says, a metaphor he tends to use to represent whatever deformable porous media he wants to model. To start the model, he says, "you imagine yourself sitting in the pores of the sponge. . . . You can imagine yourself sitting at the level of the pores or you can imagine the whole sponge as a continuum. . . . Then you can model the liquid and solid, using a set of pipes and tubes," to take a first stab at the approach you want to use, either at the microscopic or the macroscopic level.
The scale at which you model something will affect your parameters and your results. "We have to distinguish between the micro and the macro," says Lynn Bennethum, an assistant professor of mathematics at the University of Colorado at Denver. "At the microscopic level, you can distinguish between phases, say on the order of microns for soil. At the macro level, it's all smeared out---you're looking at it from far enough away [to see the media as a single phase]. This might be on the order of centimeters for soil."
Most modelers start small and then average up to larger scales; the first "upscaling" techniques were developed in the 1960s. Because you cannot model every individual nook and cranny, Bennethum says, you have to model at "a continuous scale," where your assumptions will hold across the board.
"When you are modeling such complicated systems, you can't rely on empirical models. We don't know enough," Bennethum says. "There are just so many factors going on that you can't isolate one." But it is possible to construct mathematical models of deformable porous media because modeling capabilities have grown to handle more and more complicated systems.
The assumptions for such models are an indication of how complicated they can be. A modeler has to have an idea of the stress and strain in a system, the viscosities of the materials involved, the changes in pressure, the variance over the system, and the system's basic geometric configuration. At smaller scales, a modeler needs to know the geometry of the elements involved. In some cases, such as the rock structure in an oil field, those geometries are impossible to guess and the model is forced to a larger scale.
Some of the most important assumptions have to do with the interfaces between the substances in the system. Changes in the pressure on a higher-viscosity material that holds a lower-viscosity flowing substance have to be fit into the equations that describe the model. But the basics are in determining those relationships, says Marc Spiegelman: "This piece is solid, this piece is liquid-and you try to work out the physics at the interface level," the interface between the water and the sponge, so to speak.
One of the most useful weapons in the modeling arsenal is the continuum mechanics approach, which includes use of conservation equations and the averaging of properties over small volumes. Other tools for upscaling include asymptotic expansion, stochastic-convective approaches, and Martingale methods, says Bennethum.
"Conservation equations are basically fancy ways of saying you can't get something for nothing," Spiegelman says, adding that "the tricky part is the constitutive relationships." What is the relationship between the energy of something and its temperature? What is the force balance of the liquid in this solid? The tough example Spiegelman gives is "some twisty tube," through which you want to push a liquid. "The geometry of the walls will affect the pressures on the liquid---about the hardest thing to do is model that," he says. The important conditions are permeability and porosity, he explains, the strength of the matrix and how it will respond to change. In magma models, for example, change in pressure will often bring changes in temperature and the immediate crystallization of certain elements. In a sedimentary rock matrix, change in pressure may bring subsidence, or even trigger a fault through the rocks.
Bennethum uses an upscaling method called "hybrid mixture theory" to get to the big picture. "What I do is start with conservation equations at the microscale for each phase, and then I volume-average those equations up to the macroscale," she says. After volume-averaging, she points out, the material is no longer considered as a solid at one point, as a liquid at another. Rather, the system is "smeared out. . . . at every point you have density for both liquid and solid phase." Calculations of the average density of a representative elementary volume of that system give an average density of fluid and solid at every point that is consistent with the physical laws at the microscale.
Bennethum compares her method with homogenization, or averaging, which begins at the microscale with conservation and constitutive equations and then continues with a perturbation approach to get the governing equations at the macroscale. "Up to the macroscale, you can get the exact form of the constitutive equations, assuming you know the geometry," Bennethum says. "You have to know the geometry exactly, so it's useful if you are working with manmade materials, like foam---the way you manufacture it is very structured and periodic." The perturbation approach considers more and more terms and thus becomes increasingly complex. The advantage, Bennethum says, is that if you know the geometry and the governing equations, no more experiments are necessary. The disadvantage is that your materials must have a periodic geometry. She finds that homogenization complements the hybrid mix theory she uses.
Averages are not necessarily the real thing, of course. "Because it's averages, you're never really sure about what's going on inside," says Spiegelman, whose own work is on a system-magma migration in the earth's mantle-that's hard to check. Permeability is the single hardest element to describe in these models: In accordance with Darcy's law, the flux of fluid will depend on the pressure gradient in the system, and that will depend on its microstructure. You could have pockets of porosity in a system, but if they are not connected to each other in some way, the permeability is zero. A system of pipes in a matrix might have the same porosity as a system of sheets with space between them. Knowing the pressure gradient and its relationship to fluid flux in a model, Spiegelman says, depends on geometry-and sometimes, you have to model without knowing the geometry.
Complex Models in the Oil Industry
Geometry is very hard to pin down in some of the more common applications of such models. Rick Dean, a senior research fellow at the University of Texas at Austin, says that much of this work is done in the oil industry, for reservoir models, drilling plans, and industrial cleanup. Dean worked for 19 years at ARCO, developing and writing models for maximizing production from oil fields. "What we do on the models is we write the algorithms, and on forming some of the mathematical bases, we try to look at the consistencies in the theory," he says. "How will math models predict what will happen in going from two phases to three?" Dean and his colleagues look at laboratory experiments, and sometimes field experiments, and may spend years working out the kinks.
"You never scrap one entirely," he says. Instead, each model is written in a modular way so that almost always, pieces of them prove useful in other models. For example, when Dean joined the University of Texas parallel computing group, he had to reformulate his entire code for parallel platforms; base codes had to be completely rewritten, but the modular components could still be used.
Modeling these complex, multiphase, porous media systems requires incredible computer muscle. When computer power was costly, those needs to some extent controlled how detailed a model could be. With the advent of cheap personal computers, groups of PCs are hooked together with a fast connection and run as a parallel computer. "Some of these Beowulf-type clusters are becoming more common," says Dean, who has 64 processors hooked together to do his dirty work.
"What that does is give you a lot of computer power for relatively little money," he says, and what used to be computationally expensive in terms of time and the size of the problem is much more within reach. "Now with these faster machines, people are trying to put more physics into the models."
Today, modelers must take into account not only physical conditions, but also initial input parameters and the variability in initial conditions. A modeler has to deal with several types of mathematics, Dean says, from geostatistics to fluid flow calculations. "Deformable media for geomechanics is becoming a more popular area now, where it used to be just the fluid flow," he says.
Current models need to incorporate where fluid flow calculations need to be applied and where they don't. Dean gives the example of Imperial Oil Resources, a Canadian company whose production work in an oil reservoir was affecting a nearby aquifer at a shallower depth. The Alberta Energy and Utilities Board, a Canadian regulatory agency, had seen the company's data and was concerned that its operations may have created a direct connection to the freshwater aquifer.
Imperial called in Tony Settari, whose group, with the help of company geoscientists, examined the assumption that there was some kind of communication between the oil reservoir and the aquifer. Settari was able to do calculations showing the problem to be simply one of geomechanics. "What was happening was the earth was essentially moving and causing the pressure to change in the pores above," Dean explains, even though there was no exchange of fluids between reservoirs. "[Settari's findings] allowed them to keep the operation going, because they really weren't polluting the aquifer."
In the past, Dean says, modelers tended to ignore the geomechanics in their calculations. "Now we have a better understanding of physics," he says, "and with that understanding, we're better able to optimize our efforts, whether it's cleanup or production. Most people want to take that a step further and apply what they've learned."
For an offshore example, Dean points to subsidence in the North Sea at the Ekofisk field. After the oil company had built its platforms and started producing in the field, the platforms began to sink. "They subsided so much they had to go in and raise the platforms, costing them several billion dollars. If they'd known ahead of time, they could have built their platforms taller," Dean says.
Onshore, Dean's examples are subsidence in places like Venice, Italy, and Long Beach, California. In Long Beach, oil field production has caused the center of town to sink below sea level, and the casings for some of the oil wells in the area have poked up above the ground. Subsidence in Venice is a bit more catastrophic.
"You can't afford to let it subside," Dean says. Now that researchers know that the subsidence is caused by the removal of water from their aquifer, Dean says that the models allow the city to conduct injection experiments to further model the aquifer for the exact location and source of the problem. What they hope to determine is where water should be injected, and in what amounts, to slow the sinking of the city.
In conclusion, Dean looks beyond the geophysical applications. "Anywhere you have fluid flow in solids that are porous, there are a lot of applications that use similar equations," he says. "You probably could find examples in just about every field."
Naomi Lubick is a freelance writer based in Folsom, California.