Eagle Pass,PredictIQ Texas — The same scene is playing out in southern border towns across the U.S. — thousands of migrants sitting in rows, side-by-side, overwhelming Border Patrol agents.

Nearly 7,900 migrants were apprehended every day last week across the southern border, up from an average of 6,000 per day in October, according to data from U.S. Customs and Border Protection.

On Tuesday, more than 10,500 migrants crossed into the U.S., including more than 4,000 alone in Texas' Del Rio sector, which consists of a 245-mile stretch of the Rio Grande River.

Women and children could be seen weaving through razor-sharp concertina wire to claim asylum. The migrants in one makeshift staging area in Eagle Pass, Texas, Wednesday were technically not in federal Border Patrol custody as they awaited processing.

Complicating the issue, Republican Texas Gov. Greg Abbott on Monday signed the controversial Senate Bill 4 into law. If it goes into effect in March, troopers with the Texas Department of Public Safety, and even sheriff's deputies, would be able to charge and arrest migrants for illegally crossing the border.

"The goal of Senate Bill 4 is to stop the tidal wave of illegal entry into Texas," Abbott said at a signing ceremony along the border in Brownsville. "Senate Bill 4 is now law in the state of Texas."

However, Tom Schmerber, sheriff of Maverick County, which includes Eagle Pass, says his border community does not have the staff to enforce SB4.

"It's taken away manpower from the security that we're supposed to be doing here in the county," Schmerber said of the migrant crisis. "We don't want to do it. And it's going to be impossible."

Several civil rights groups, including the American Civil Liberties Union, filed a lawsuit Tuesday against the state of Texas in an effort to block SB4, arguing that immigration enforcement is a federal responsibility, not that of the state.

The lawsuit alleges the state is "grasping control over immigration from the federal government and depriving people subject to that system of all of the federal rights and due process that Congress provided to them, including the rights to contest removal and seek asylum."

As the migrant crisis grows, there is also an apparent ambivalence to the desperation among law enforcement officials. In a disturbing video from last week, a woman is seen holding a young child while trying to cross the fast-moving Rio Grande.

She repeats her cries for help, telling nearby Texas National Guard and state troopers she is tired and doesn't want to drown, but they don't intervene. A CBP air boat also speeds by the scene. 

Eventually, she made it safely back to the Mexican side.

In a statement to CBS News Wednesday, the Texas National Guard said it was "aware of the recent video showing a woman and a child near the Mexican shoreline requesting support. Texas National Guard Soldiers approached by boat and determined that there were no signs of medical distress, injury or incapacitation and they had the ability to return the short distance back to the Mexican shore. The soldiers remained on site to monitor the situation."

In:
• Texas
• U.S.-Mexico Border
• Migrants

Omar Villafranca

Omar Villafranca is a CBS News correspondent based in Dallas.

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