During geomagnetic storms, geomagnetically induced currents (GIC) and pipe-soil potential (PSP) generated in a pipeline present a risk to the safe operation of the pipeline. An important factor affecting the size and distribution of GIC and PSP is pipeline insulation leakage points. We obtain the relationship between the resistance of leakage points, their radius, and soil conductivity through modeling and derivation and verify the reliability of this equation by simulation using ANSYS software. Then a pipeline network model including leakage points is established. The influences of different radii and locations of leakage points on GIC and PSP distributions are discussed. The results suggest that medium and even small geomagnetic storms may aggravate corrosion of pipelines and affect their safe operation. This influence is related to the distribution of leakage points, the size of leakage points, and the structure of pipelines at the leakage points. The closer the leakage point is to the end of the pipeline, the greater the impact on the GIC distribution. The larger the radius of the pipeline leakage point, the greater the influence of the leakage point on the GIC distribution of the pipeline. The presence of branches at the leakage points can reduce the influence of leakage points on GIC and PSP distributions.