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(CNN) -- Efforts to contain oil gushing from BP's ruptured well in the Gulf of Mexico are becoming more effective, according to Coast Guard Adm. Thad Allen, the federal government's point man for the disaster.
Approximately 25,000 barrels of oil -- slightly more than 1 million gallons -- were collected from the well in the 24-hour period ending at midnight Thursday, Allen told reporters Friday. The total represents an increase of nearly 6,700 barrels -- or roughly 281,000 gallons -- compared with Wednesday's tally.
Federal officials anticipate a total collection capacity of up to 80,000 barrels per day by July.
A team of government and independent scientists announced Tuesday that the flow rate from the well might be as high as 60,000 barrels a day.
On Thursday, Allen said the drilling of two relief wells -- viewed as the ultimate solution to the Deepwater Horizon gusher -- was ahead of schedule, but he shied away from saying the well would be plugged early, citing the complexity of the operation.
One drilling rig is at 10,677 feet below the sea floor and is "starting to close in on the well bore itself," Allen has said, while a backup rig is at 4,662 feet below the sea floor.
But even as oil continues gushing into the Gulf, efforts are now under way to start more effectively compensating the victims of the spill.
Kenneth Feinberg, who was appointed earlier this week by President Obama to manage BP's $20 billion compensation fund, pledged Friday to quickly create a system for processing claims.
"We will have a very transparent methodology in place," Feinberg said. "We'll set up a protocol very quickly so that everybody can examine what's expected."
BP agreed to establish the fund Wednesday after company executives met with Obama in Washington. The company has said repeatedly that it will pay all legitimate claims related to the spill, which has become the worst environmental disaster in U.S. history.
Feinberg, an attorney who was special master of the 9/11 victims fund and advises Obama on Wall Street pay issues, asserted that the fund will be independently managed. He emphasized that he is not aligned with the government or BP.
"I will be running an independent claims facility," he said. "It is my program as an independent force."
BP deserves some credit for the steps it has already taken to process claims, Feinberg said, though it's his job to improve the process.
The company has opened about 25 claims offices, and said this week that it has issued about 25,000 claims checks totaling $63 million.
The political firestorm surrounding BP, however, still shows no signs of slowing.
Lawmakers have said many questions remain after BP CEO Tony Hayward faced a bipartisan barrage of criticism while testifying on Capitol Hill on Thursday.
"It was frustrating, not just to me but to the American people," Rep. Bart Stupak said on CNN's "John King, USA."
In his testimony Thursday, Hayward apologized for the April 20 Deepwater Horizon explosion and continuing oil spill, but insisted that it was "too early" to reach conclusions about whether BP had cut corners on safety.
He repeatedly declined to speculate "on what may or may not have made a difference" in preventing the disaster until ongoing investigations are completed. Pressed for a direct answer on whether BP has made good on a commitment to safety that was pledged when he became CEO three years ago, Hayward would not give a clear yes or no answer.
"We have focused like a laser on safe and reliable operations," he insisted. "We have made major changes."
But Democrats and Republicans alike repeatedly accused the embattled British corporate chieftain of refusing to answer their questions, cutting corners on safety and turning a blind eye to mounting scientific evidence of an unprecedented environmental tragedy.
"It's clear to me you don't want to answer any of our questions," said Rep. Henry Waxman, chairman of the House Energy and Commerce Committee. "I am amazed at this testimony. You're kicking the can down the road. I find that irresponsible."
Rep. Joe Barton, R-Texas, also drew sharp criticism from both sides of the aisle after he opened Republican statements with an apology to Hayward, saying he was "ashamed" that the company has been subjected to what Barton characterized as a $20 billion "shakedown" by the Obama administration.
Barton later backtracked.
"I apologize for using the term 'shakedown' ... and I retract my apology to BP," he said in a written statement. "BP should bear the full financial responsibility for the accident on their lease in the Gulf of Mexico. BP should fully compensate those families and businesses that have been hurt by this accident."
Criticisms of both BP and the federal government have been rising in the weeks since the explosion, with questions arising about regulation and cleanup coordination.
"It is frustrating because it doesn't seem like the left hand knows what the right hand is doing," Louisiana Gov. Bobby Jindal said Thursday. "There is no streamlined system here. This is why we keep stressing that we need to see more of a sense of urgency from the Coast Guard, federal officials and BP. We are in a war here -- we are in a war against this oil that absolutely threatens our way of life."
CNN's Ben Rooney and Alan Silverleib contributed to this report