In this study, we propose a three-dimensional (3D) forward modeling algorithm of surface-to-borehole transient electromagnetic (SBTEM) fields based on an unstructured vector finite-element method to analyze the characteristics of SBTEM responses for complex geoelectrical models. To solve the double-curl diffusion equation for the electric field, we use an unstructured tetrahedral mesh to discretize the model domain and select the unconditionally stable backward Euler scheme to discretize the time derivative. In our numerical experiments, we use a grounded wire as a transmitting source. After validating the algorithm\'s effectiveness, we first analyze the diffusion characteristics and detectability of the electromagnetic field. After that, we focus our attention on the distribution and the cause of zero bands for Ex and dBy/dt components with the hope of guiding future field surveys. Finally, by simulating different models, we analyze the capability of the SBTEM method in detecting typical mineral veins so that we can provide a reference for mineral resource exploration in the deep earth.