World oil demand and advanced oil recovery techniques have made it economically attractive to rehabilitate previously abandoned oil wells. 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 robotic tactile mapping is studied. A method is developed that only uses the robot joint encoders and avoids any force or tactile sensor, which are complex and unreliable in such a hostile environment. This paper addresses the general problem of intelligent tactile exploration of constrained internal geometries where 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 hardware experiments to substantially reduce the data acquisition effort for exploration with a tactile manipulator.