Oil prices surged to start this week after drones attacked Saudi Arabia's oil infrastructure, which knocked half of its output offline. The initial spike in crude prices caused oil stocks to go ballistic.

While crude has since fallen back down to earth, analysts are growing increasingly bullish on oil stocks. That's because the risk of a supply shock has increased, which will likely cause oil prices to trade at a premium price in the near term. Those higher prices will enable oil producers to make more money.

While that rising tide should lift all boats, analysts are most bullish on the producers that they believe have the most exposure to higher crude prices. Here's a look at their top picks.

Two oil pumps with a bright sun in the background

Image source: Getty Images.

A low-cost producer with long-term growth prospects

UBS upgraded shares of U.S. oil giant ConocoPhillips (NYSE:COP) to "buy" this week while giving its stock a $75 price target (about 25% above the current price). The bank believes that ConocoPhillips can outperform its peers even if the recent spike in crude prices doesn't last: It could produce enough free cash flow to fund its capital program and pay its dividend even if oil plunged back to $40 a barrel. Meanwhile, at higher prices, ConocoPhillips can produce excess cash to repurchase its stock. The company already intends on buying back $3.5 billion of its shares this year, a $500 million increase from its initial plan due to higher oil prices.

Another reason UBS likes ConocoPhillips: its long-term growth prospects. The bank noted that the company has enough of an inventory in the Eagle Ford shale to continue increasing output for more than a decade, so it believes ConocoPhillips is a great buy for the long term.

A drilling rig surrounded by oil pumps at sunset

Image source: Getty Images.

These U.S. producers have high exposure to higher oil prices

KeyBanc, meanwhile, upgraded several oil stocks following the attack on Saudi Arabia's oil infrastructure. The bank gave overweight ratings to Whiting Petroleum (NYSE:WLL), Jagged Peak Energy (NYSE:JAG), and Centennial Resource Development (NASDAQ:CDEV). It also boosted their price targets to $14 for Whiting (nearly 50% upside from the current price), $10 for Jagged Peak (almost 30% upside), and $6.50 for Centennial Resources (roughly 40% upside).

The bank noted that all three have high exposure to oil prices and low relative valuations. Whiting Petroleum, for example, is one of the most oil-heavy exploration and production companies in the U.S. at 64% of its expected production mix this year. Add in its natural gas liquids (NGL) output, which trades at oil-linked prices, and 81% of the company's production is liquids. That's the fourth-best in its peer group, giving the company lots of upside to rising crude prices.

Centennial Resource got an additional stamp of approval from an analyst at SunTrust. That bank upgraded it to a buy following the attack, with an even higher price target of $8 per share (implying nearly 70% upside). In that bank's view, Centennial is "one of the better small-cap operators" in the oil patch. That's because it's close to becoming free cash neutral, which will happen even quicker if crude remains above $60 a barrel following the Saudi attack.

Meanwhile, Mizuho got into the action by upgrading shares of Parsley Energy (NYSE:PE) to a buy rating, with a $24 price target (implying nearly 30% upside). In Mizuho's view, Parsley should be able to grow its oil output at a higher rate next year than analysts currently anticipate. That would give it greater upside to potentially higher crude prices.

Lots of upside if oil remains elevated

The attack on Saudi Arabia's oil infrastructure caused the price of the U.S. oil benchmark, West Texas Intermediate, to spike 15%, rising above $63 a barrel at one point. While crude has since slid back below $60, it's still well above where it was before the attack. Thus, energy companies with lots of exposure to the price of oil stand to make more money in the coming months if the current pricing level holds up. And their stock prices could have lots more of room to run, which is certainly what these analysts believe.