Despite some voter confusion and the arrival of a significant snowstorm, vote tallies across Virginia’s 33rd Senate District appeared on track to meet or even exceed the 15 percent turnout rate predicted for Tuesday’s special election, officials said.

In Loudoun County, more than 13 percent of eligible voters had cast ballots by 2 p.m. in the race to fill the Senate seat vacated by Attorney General Mark R. Herring, according to Loudoun Registrar Judy Brown. Some of the county’s eastern precincts reported particularly high turnout rates, between 14 and 19 percent, she said, and other large precincts had yet to report their numbers — representing hundreds of voters who were not yet included in the total.

“It’s certainly surprising,” Brown said. “Once we heard the prediction of bad weather, usually you see that people just aren’t interested enough to come out. But I guess the campaigns have done a good job of convincing people that it’s important to be out to vote.”

The majority of the 33rd Senate District falls in Loudoun County, with 10 precincts also open in Fairfax. By 2 p.m., the Fairfax polls were reporting nearly 15 percent turnout, according to Brian Schoeneman, secretary of the Fairfax County Electoral Board.

Many more voters across both counties had showed up at polling sites but were turned away because they were not residents of the 33rd Senate district, and were not eligible to vote in the special election, officials said.

That sort of confusion is not altogether uncommon, Schoeneman said.

“Whenever there’s a special election, folks get over-eager, and they want to come vote,” he said.

Turnout estimates for special elections are generally modest, Schoeneman said, with typical numbers generally falling between 10 and 20 percent. The last special election in Fairfax — to fill the state Senate seat vacated by former Attorney General Ken Cuccinelli II in 2010 — yielded a 19.8 percent turnout across the county, Schoeneman said.

“We’re almost there already, at 2 p.m. in the middle of a snowstorm,” he said of Tuesday’s election, noting that some precincts had voters waiting in lines as late as noon. By mid-morning, the snowfall across Northern Virginia was steadily mounting, with the bulk of the accumulation forecast for afternoon and evening hours.

“I think for the hard-core folks that are really going to be out to vote, they recognize that potentially control of the state Senate rests with this race,” he said. “So they are going to get to the polls come hell or high water to support their candidates, and that’s borne out by the poll numbers.”