Move over, BP (NYSE: BP), Transocean (NYSE: RIG), and Halliburton (NYSE: HAL). Since April, when the companies were involved in a massive Gulf of Mexico tragedy, the trio has sucked up most of our attention to Big Oil. But it now looks like Chevron (NYSE: CVX) might assume center stage, at least for a while.

You're likely aware that the second-largest U.S. oil company is involved in a 17-year-old lawsuit originally filed on behalf of indigenous tribes in Ecuador. The suit maintains that the Ecuadorians suffered severe health problems and other difficulties when Texaco -- which Chevron bought in 2001 -- failed to remediate an environmental mess that occurred when it operated a consortium in the country between 1965 and 1990.

Plaintiffs' weapons
Key to the arsenal of the plaintiffs' team during the past two years has been a detailed report ostensibly written by engineer Richard Cabrera, an "independent expert" appointed by the Ecuadorian court, along with a 2009 documentary film called Crude. Cabrera's final report estimates the damages for which Chevron should be liable at $27 billion.

But, as both Fortune and Forbes have described in their most recent issues, Gibson, Dunn & Crutcher, Chevron's outside law firm, has filed 11 Section 1782 actions. Through these filings, those involved in overseas litigation can subpoena testimony and documents from U.S. witnesses.

The firm pulls out its files
Of Gibson Dunn's total filings, at least part of all 11 have been granted by the judges involved, a good average in any sport. The filings were targeted at cities where the plaintiffs' team members or consultants worked, including San Diego, Denver, Houston, Atlanta, Newark, N.J., New York City, Washington, D.C., and Asheville, N.C. The search ultimately was intended to elicit proof that Cabrera's report had actually been written by the plaintiffs' attorneys. According to Fortune, at least one of the judges who have looked at the 1782's findings thus far has noted the appearance of "fraud."

A couple of specifics: An expert used earlier by the plaintiffs testified that two reports that he had supposedly written claiming contamination at the sites of four former Texaco wells were not his work and didn't contain his conclusions. And Fortune maintains that a Chevron-hired forensic linguist has written a report concluding that Cabrera's report was largely written in English and subsequently translated into Spanish. Cabrera does not speak English.

The battle rages
After more than a decade and a half, the plaintiffs clearly intend to continue fighting. In a release responding to the Fortune article, their primary contention is that fraud in the U.S. legal system doesn't necessarily equate to fraud in "Chevron's preferred forum of Ecuador."

So, the litigation rocks along, and likely will continue apace, perhaps for years. Chevron, of course, isn't the only member of Big Oil to face difficulties in South America. You'll surely recall Venezuelan President Hugo Chavez's nationalization of energy a few years ago. The Castro wannabe's action affected France's Total (NYSE: TOT), along with, yes, Chevron, plus ExxonMobil (NYSE: XOM), and ConocoPhillips (NYSE: COP).

But whatever ultimately occurs with Chevron and Ecuador, the big company is active and successful around the world, including in countries such as China, Australia -- where it is operating the huge Gorgon gas project -- Nigeria, and Brazil.

Just this week, my colleague Seth Jayson wrote an article on Chevron's solid cash flow. I endorse his assessment and suggest strongly that Fools closely watch this big company with the key role in the world's oil and gas production.  

Fool contributor David Lee Smith doesn't own shares in any of the companies mentioned. Chevron and Total are Motley Fool Income Investor selections. The Fool owns shares of ExxonMobil. Try any of our Foolish newsletter services free for 30 days.

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