HOUSTON — Tropical Storm Harvey sent devastating floods pouring into the nation’s fourth-largest city Sunday as rising water chased thousands of people to rooftops or higher ground and overwhelmed rescuers who could not keep up with the constant calls for help.

The incessant rain covered much of Houston in turbid, gray-green water and turned streets into rivers navigable only by boat. In a rescue effort that recalled the aftermath of Hurricane Katrina, helicopters landed near flooded freeways, airboats buzzed across submerged neighborhoods and high-water vehicles plowed through water-logged intersections. Some people managed with kayaks or canoes or swam.

Volunteers joined emergency teams to pull people from their homes or from the water, which was high enough in places to gush into second floors. The flooding was so widespread that authorities had trouble pinpointing the worst areas. They urged people to get on top of their houses to avoid becoming trapped in attics and to wave sheets or towels to draw attention to their location.

Judging from federal disaster declarations, the storm has so far affected about a quarter of the Texas population, or 6.8 million people in 18 counties. It was blamed for at least two deaths.

As the water rose, the National Weather Service issued another ominous forecast: Before the storm that arrived Friday as a Category 4 hurricane is gone, some parts of Houston and its suburbs could get as much as 50 inches of rain. That would be the highest amount ever recorded in Texas.

Some areas have already received about half that amount. Since Thursday, South Houston recorded nearly 25 inches, and the suburbs of Santa Fe and Dayton got 27 inches.

“The breadth and intensity of this rainfall is beyond anything experienced before,” the National Weather Service said in a statement.

Average rainfall totals will end up around 40 inches for Houston, weather service meteorologist Patrick Burke said.

The director of the Federal Emergency Management Agency, Brock Long, predicted that the aftermath of the storm would require FEMA’s involvement for years.

“This disaster’s going to be a landmark event,” Long said.

Rescuers had to give top priority to life-and-death situations, leaving many affected families to fend for themselves.

Tom Bartlett and Steven Craig pulled a rowboat on a rope through chest-deep water for a mile to rescue Bartlett’s mother from her home in west Houston. It took them 45 minutes to reach the house. Inside, the water was halfway up the walls.

Marie Bartlett, 88, waited in her bedroom upstairs.

“When I was younger, I used to wish I had a daughter, but I have the best son in the world,” she said. “In my 40 years here, I’ve never seen the water this high.”

It was not clear how many people were plucked from the floodwaters. Up to 1,200 people had to be rescued in Galveston County alone, said Mark Henry, the county judge, the county’s top administrative post.

Houston’s George R. Brown Convention Center was quickly opened as a shelter. It was also used as a shelter for Katrina refugees in 2005.

Gillis Leho arrived there soaking wet. She said she awoke Sunday to find her downstairs flooded. She tried to move some belongings upstairs, then grabbed her grandchildren.

“When they told us the current was getting high, we had to bust a window to get out,” Leho said.

William Cain sought shelter after water started coming inside his family’s apartment and they lost power. “I live in a lake where there was once dry land,” he said.

Some people used inflatable beach toys, rubber rafts and even air mattresses to get through the water to safety. Others waded while carrying trash bags stuffed with their belongings and small animals in picnic coolers.

Houston Mayor Sylvester Turner said authorities had received more than 2,000 calls for help, with more coming in. He urged drivers to stay off roads to avoid adding to the number of those stranded.

“I don’t need to tell anyone this is a very, very serious and unprecedented storm,” Turner told a news conference. “We have several hundred structural flooding reports. We expect that number to rise pretty dramatically.”

The deteriorating situation was bound to provoke questions about the conflicting advice given by the governor and Houston leaders before the hurricane. Gov. Greg Abbott urged people to flee from Harvey’s path, but the Houston mayor issued no evacuation orders and told everyone to stay home.

The governor refused to point fingers on Sunday.

“Now is not the time to second-guess the decisions that were made,” Abbott, a Republican, said at a news conference in Austin. “What’s important is that everybody work together to ensure that we are going to, first, save lives and, second, help people across the state rebuild.”

The mayor, a Democrat, defended his decision, saying there was no way to know which parts of the city were most vulnerable.

“If you think the situation right now is bad, and you give an order to evacuate, you are creating a nightmare,” Turner said, citing the risks of sending the city’s 2.3 million inhabitants onto the highways at the same time.

Jesse Gonzalez, and his son, also named Jesse, used their boat to rescue people from a southeast Houston neighborhood. Asked what he had seen, the younger Gonzalez replied: “A lot of people walking and a lot of dogs swimming.”

“It’s chest- to shoulder-deep out there in certain areas,” he told television station KTRK as the pair grabbed a gasoline can to refill their boat.

The Coast Guard deployed five helicopters and asked for additional aircraft from New Orleans.

The White House announced that President Trump would visit Texas on Tuesday. He met Sunday by teleconference with top administration officials to discuss federal support for response and recovery efforts.

The rescues unfolded a day after Harvey settled over the Texas coastline. The system weakened Saturday to a tropical storm.

On Sunday, it was virtually stationary about 25 miles northwest of Victoria, Texas, with maximum sustained winds of about 40 mph, the hurricane center said.

Harvey was the fiercest hurricane to hit the U.S. in 13 years and the strongest to strike Texas since 1961’s Hurricane Carla, the most powerful Texas hurricane on record.