A pair of theoretical physicists at the Federal University of Pernambuco (UFPE) identified an incredibly simple solution to a problem which the oil industry has faced for decades: the formation of what is called viscous fingers, fluid extensions created when engineers inject water into porous rocks to push the oil stored in them towards the extraction shaft.
Ideally, the water would work like a piston, uniformly displacing the more viscous liquid. What happens, however, is that minute differences in pressure on the boundary between the two fluids tend to amplify rapidly and lead to the creation of little tentacles of water, known as viscous fingers, which advance into the oil and push only part of it. “The water creates preferred channels that sweep a very limited area of the reservoir,” says mechanical engineer Márcio Carvalho, of the Pontifical Catholic University of Rio de Janeiro (PUC-Rio), who researches methods for advanced oil recovery for Chevron, Petrobras and Repsol. According to Carvalho, the viscous fingers are one of the main factors that, in some cases, restrict extraction to just 30% of a reservoir’s potential.
Oil companies have partially solved the problem by adding polymers to the water to increase its viscosity. This stabilizes the interface with the oil, reducing the formation of viscous fingers. “But this process is costly and involves complicated logistics,” says Carvalho.
Physicists José Américo Miranda and Eduardo Dias, both from UFPE, aided by Carvalho and Enrique Alvarez-Lacalle, of the Polytechnic University of Catalonia, Spain, have proposed a solution that, in principle, seems to be more practical. In an article in the journal Physical Review Letters, they reported that the viscous fingers can be completely eliminated simply by controlling the rate at which water is injected into the rocks.
To simulate the way fluids flow through the pores of rocks in the laboratory, the researchers who study viscous fingers use an apparatus known as a Hele-Shaw cell, which is relatively easy to build. First, they fill the millimetric space between two parallel glass plates with a viscous fluid, such as mineral oil. Next, they create a small hole in the top plate and, through it, continuously inject a less viscous fluid, such as water. Soon, the circular shape of the water bubble gives rise to an increasingly complicated radial pattern of fingers. “The fingers split into two, into four, and so forth,” explains Miranda.
Ana Paula CamposThis technique has been used since 1958 when the British physicists Philip Saffman and Geoffrey Taylor showed that the mathematical equations that describe the formation of viscous fingers in Hele-Shaw cells are the same that explain their emergence in porous rocks. But it was not until 2009 that several groups of researchers began to develop effective methods for manipulating the formation of fingers in the cells.
Miranda and Dias were investigating the mathematics behind one of these methods when they discovered a simple approach to tackle the problem and reduce the size of the fingers. The calculations suggested that the viscous fingers would disappear if, instead of injecting a constant flow of water, as is usually done, the initial injected volume was small, increasing linearly over time (see information graphic).
“The solution could be a very complicated mathematical function, but it’s not,” recalls Miranda, “it’s simply a straight line.” The result seemed too good to be true. At first, the pair suspected that the approximations made in the calculations would no longer apply as the force of the injected water increased over time.
The theory was put to the test in Carvalho’s laboratory. He and Dias carried out tests in a Hele-Shaw cell, controlling injected water flow using a computer. “Injecting a linearly increasing volume really stabilized the interface,” says Miranda. Computer simulations carried out by Alvarez-Lacalle confirmed the solution to the problem.
The experiments and simulations suggest that viscous fingers will not form even if water injection becomes very strong. “As long as the water pump is efficient, the process improves over time,” explains Miranda.
This is not the first study to show how viscous fingers can be eliminated. This year, two other groups of physicists announced different methods capable of preventing the formation of viscous fingers — in one of them, the upper glass plate of the Hele-Shaw cell is tilted; in the other, the upper plate is replaced by an elastic membrane. There is, however, no obvious way to apply these approaches to oil extraction, unlike the solution proposed by the Brazilians.
“There are technical challenges to implementing our solution too, such as how to control water pumping,” said Carvalho. “But these challenges can be overcome.” Even if this is controlled, unexpected effects of the passage of liquids through porous rock can arise and hinder the extraction of oil, according to physicist Albert Tufaile of the University of São Paulo.
Avoiding the formation of viscous fingers is important not only in the oil sector, but for fluids in general. Several growth processes in nature are governed by mathematical equations similar or identical to those that describe viscous fingers. Miranda believes that the solution he and his colleague have found could be adapted to prevent the uncontrolled growth of semiconductor crystals in the microelectronics industry. Or even assist in planning the administration of drugs to prevent the spread of tumors in the body, since part of the tumor grows like a tree that branches.
DIAS, E. O. et al. Minimization of viscous fluid fingering: a variational scheme for optimal flow rates. Physical Review Letters. v. 109 (14). Oct. 5, 2012