Terpene-based spotting fluid compositions and processes for freeing differentially stuck pipe are provided. A spotting fluid composition includes a terpene and an acid, such that the acid initiates an exothermic polymerization reaction of the terpene. Another spotting fluid composition includes an invert emulsion having a terpene external phase and a non-miscible organic hygroscopic fluid internal phase. The terpene emulsion spotting fluid composition may be introduced downhole in the vicinity of a portion of a differentially stuck pipe such that the spotting fluid composition contacts a filter cake surrounding the portion of the differentially stuck pipe. An acid may be introduced downhole in the vicinity of a portion of a differentially stuck pipe such that the acid initiates an exothermic polymerization reaction of the terpene of the terpene emulsion spotting fluid composition.