An algorithm for the numerical simulation of spontaneous potential (SP) signals based on the finite element method is proposed. The SP signals are passively recorded electrical signals resulting from the physical and chemical processes of various nature occurring inside the well and adjacent formations. The proposed algorithm allows one to simulate the SP signals for the case of axial symmetry of the studied area and an electrochemical SP source. Using sigmoidal functions to describe the distribution of petrophysical parameters in the study area allows one to obtain an analytical expression for the density of the SP source. The correctness of the proposed algorithm is verified on a simplified three-layer model. The dependence of the obtained SP signals on the thickness of the permeable formation and on its salinity is demonstrated.