LOS ANGELES — Two months ago, Disney released a sequel to Where’s My Water?, a hit smartphone game about a showering alligator. Hopes were high: Disney had pointed to the original game as evidence of overdue traction in mobile gaming.

It flopped.

For the Walt Disney Company — where theme parks, TV, merchandise and films deliver more than $6 billion in annual profit — the failure of one smartphone game, even an important one, has no financial consequence. But mobile games are a major growth opportunity, and analysts say Where’s My Water? 2 underscores the degree to which Disney is encountering new challenges in a shifting marketplace.

In particular, mobile game publishers are rapidly moving from apps that cost 99 cents per download to free apps that make money by selling virtual goods and upgrades. This “freemium,” Zynga-style model can be much more profitable. Of the 100 highest-grossing iPhone games in September, analysts note, 94 percent were free with in-game purchases — titles like Candy Crush Saga from King, based in London.

Disney has found some new hits of this kind, including Star Wars: Tiny Death Star, a game that recently hit No. 1 on the Apple app store charts. (Players help the Empire complete its Alderaan-destroyer.)