* Chilly weather seen continuing through April * Stock building season gets off to a slow start * Coming up: Baker Hughes rig data, CFTC trade data Friday By Joe Silha NEW YORK, April 18 (Reuters) - U.S. natural gas futures ended higher on Thursday for a third straight day, with the front contract driven to a 21-month high by supportive weekly inventory data and chilly weather forecasts for the next two weeks that should underpin heating demand. A U.S. Energy Information Administration report showed total domestic gas inventories rose 31 billion cubic feet last week to 1.704 trillion cubic feet. Most traders viewed the build as supportive for prices, noting it came in below the Reuters poll estimate of 34 bcf and below the five-year average increase for that week of 39 bcf. "The (EIA) number was slightly below expectations, and the market certainly rallied on it. There's also some cooler weather around that's supporting prices," said Steve Platt, analyst at Archer Financial in Chicago. Front-month gas futures on the New York Mercantile Exchange ended up 18.7 cents, or 4.4 percent, at $4.401 per million British thermal units after climbing late to $4.429, the highest for the front contract since late July 2011. The front contract has gained 6.4 percent in the last three sessions, its biggest three-day run up since mid-January. Gas inventories started the heating season at record highs, but cold late-winter weather, a chilly spring and above-average nuclear plant outages have put a huge dent in inventories. Front-month prices have mostly been in an uptrend since mid-February, rising nearly 35 percent in eight previous weeks and holding another 4 percent gain so far this week. With weather bound to turn milder soon, concerns are growing that the market may be ripe for a pullback, with prices at levels that could further dampen utility demand and increase supply by tempting producers to turn on more wells. There are also concerns that record growth in futures open interest that has accompanied recent price gains means there are a lot of new longs in the market that could rush to take profits when milder spring temperatures finally slow heating needs. Forecaster Commodity Weather Group expects cool weather to continue for most states east of Rockies through next week before the pattern finally starts to moderate as May arrives. INJECTION SEASON OFF TO SLOW START The weekly inventory withdrawal narrowed the deficit relative to last year by 10 bcf to 794 bcf, or 32 percent below last year's record highs at that time. But it increased the shortfall versus the five-year average by 8 bcf, leaving stocks at 74 bcf, or 4 percent, below that benchmark. More chilly weather this month is expected to continue to slow inventory builds and drive stocks further into deficit relative to the five-year average for the next couple of weeks. Early injection estimates for next week's report range from 24 to 48 bcf versus a 43-bcf build during the same week last year and a five-year average rise for that week of 50 bcf. Injections during the April-through-October stock building season on average total about 2 tcf, meaning stocks could head into next winter with about 3.7 tcf in the ground, well above what would be needed to meet even the coldest winter demand but nearly 6 percent below last year's record peak of 3.929 tcf. WHEN WILL OUTPUT SLOW? Traders were waiting for the next Baker Hughes drilling rig report on Friday. The gas-directed drilling rig count rose slightly last week but is still hovering just above the 14-year low of 375 posted two weeks ago. Drilling for natural gas has mostly been in decline for the past 18 months. The count is down about 60 percent since peaking in 2011 at 936, but so far production has not slowed much from the record high hit last year. EIA recently trimmed its estimate for domestic gas production growth in 2013, but still expects output to rise 0.3 percent from 2012's record levels.