A downhole apparatus, such as a wellbore packer, is provided with a swellable member and a fluid supply assembly. The fluid supply assembly is to receive fluid and expose the swellable member to the fluid to cause expansion of the swellable member, and comprises a support structure for supporting the swellable member on the body. In a preferred embodiment, the support structure defines a chamber and is configured to allow fluid to flow and access the swellable member. A method of use and method of sealing a wellbore is described.