World oil demand and advanced oil recovery techniques have made the rehabilitation of previously abandoned oil wells economically attractive. This requires relatively fast mapping of the shape and location of the down-hole well structures. Practical factors prohibit the use of visual and other range sensors in this situation. Here, the feasibility of using a robotic manipulator for tactile mapping is studied. A method is developed that requires only robot joint encoders and avoids the use of any force or tactile sensors, which are complex and unreliable in such a hostile environment. This paper addresses the general problem of intelligent tactile exploration of constrained internal geometries when time is critical. It is assumed that the time required to move a manipulator to acquire a new touch point outweighs computational time. This approach models the down-hole structures with geometric primitives and focuses on exploration efficiency by intelligently searching for new touch points to build the geometric models. The algorithms developed here are shown in simulations and laboratory experiments to substantially reduce the data acquisition effort for exploration with a tactile manipulator.