For a multibillion-dollar public-works project called the Alaskan Way Viaduct Replacement, this city brought in a digging machine touted as the largest deployed anywhere: a 7,000-ton behemoth nicknamed "Bertha."

The megamachine's job was to create a 1.7-mile-long tunnel that would allow the state transit authority to reroute a section of Washington State Road 99 underground to replace a 61-year-old traffic-choked viaduct that authorities say is seismically unsafe and must be replaced.

But Bertha has been stuck since early December when the tunnel-boring machine hit an 8-inch diameter, ¾-inch thick steel well-casing after only 1,000 feet of drilling, threatening both a delay in the $3.1 billion project's completion and a multimillion-dollar cost increase.

Now, Bertha's operator and the state transit authority are sparring over the cost of repairs. And the repair work itself is the subject of a lawsuit—with defendants' responses due later this month—from a local watchdog group calling for an additional environmental study.

"Of course I'm concerned that the project is not moving forward," Seattle Mayor Ed Murray said this month. "People are frustrated."

Bertha—longer than a football field and assembled at a cost of $80 million— arrived in April 2013 to a lavish rollout, a Twitter feed (@BerthadigsSR99) and a downtown museum built around a miniature replica.

While the public-relations campaign has helped with transparency, the turn of events could make the machine a symbol of the latest in big regional infrastructure projects—like Boston's overbudget and often-delayed Big Dig, which included replacing the city's central artery with an underground highway—to be beset by cost overruns, completion delays and yearslong headaches for commuters.

Seattle and the Washington State Department of Transportation had spent 10 years hammering out the roadway's redesign, which is being paid for with a mixture of new gasoline taxes, tolls and federal and state funds.

The contractor operating Bertha, Seattle Tunnel Partners, or STP, began digging this past week a rescue tunnel to start a monthslong effort to extract and repair the machine's front "cutterhead." But tunneling won't resume before March 2015, the contractor has told WSDOT.

"I'm apprehensive and skeptical," said Todd Trepanier, a WSDOT official monitoring the project's timetable. "I'll feel better when I see actual dirt moving."

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