Corporate watchdogs are pressuring BP to fire CEO Bob Dudley, who they accuse of trying to keep the oil giant from honoring its obligations to the Gulf Coast in the wake of the 2010 Deepwater Horizon disaster.

On Monday, as the second phase of the BP oil spill civil trial got underway in New Orleans and attorneys argued that the company lied to the feds about the spill, Corporate Action Network started a petition drive calling on BP to "Drop Dudley":

Bob Dudley's company made a promise to the Gulf, and now -- just like a deadbeat dad -- they're trying to avoid their obligations. It's time BP drop Dudley and replace him with someone committed to honoring BP's promises to the men and women whose lives depend on a healthy Gulf coast.

Last week, BP once again asked a federal judge to suspend settlement payments to Gulf residents and businesses who say they were affected by the disaster. The company had twice before asked U.S. District Judge Carl Barbier, who's overseeing the BP trial, to suspend the program, citing corruption in the claims process. Barbier refused those requests.

In making its latest claims suspension bid, BP cited a September report by former FBI director Louis Freeh, who was tapped by Barbier to investigate fraud allegations. Freeh documented problems in the claims program including improper, unethical, and possibly criminal conduct by members of the administrative staff. But while Freeh has called for the settlement program to be fixed, he has not recommended its suspension.

Further upsetting BP's critics, the company launched a massive public relations campaign over the summer to make the case that it's being ripped off by fraudsters -- trial lawyers in particular. One ad that ran in The Washington Post quoted the head of the National Association of Manufacturers saying "the tort system is nothing more than a trial-lawyer bonanza."

BP has paid out about $26 billion so far for restoration, cleanup, and damages related to the Gulf oil disaster, and it set aside $7.8 billion for claims from individuals and businesses that suffered damages from the spill -- an amount that now appears likely to fall billions of dollars short. A report released last week documented severe ecological damages from the spill, with full recovery expect to take a generation or more.

"BP owes the people of the Gulf every last penny for the damage they did to our communities, and it looks like they're trying every strategy to avoid that responsibility," said Steve Murchie, campaign director of the Gulf Restoration Network. 

The second phase of the civil trial will determine how much BP will pay in penalties under the Clean Water Act. A polluter can be required to pay as much as $4,300 per barrel of spilled oil if it is found grossly negligent, which the federal government argues BP was.

The government and the company disagree on the amount of oil spilled. The Justice Department estimates the disaster released 4.2 million barrels (176 million gallons) of oil into the Gulf, which would bring a maximum penalty of $18 billion. But BP wants the judge to use its estimate of 2.45 million barrels (almost 103 million gallons), which would bring a maximum penalty of about $10.5 billion. The company reported $11.6 billion in profits last year.

This phase of the trial is expected to last about four weeks.