We present a new set of analytical solutions for analysing pumping tests in a well near a vertical fault zone, dyke or vein. We consider a fault zone with finite width, storativity and hydraulic conductivity, which can be anisotropic, and without limitation of the diffusivity contrast between the three aquifer domains (pumped aquifer, fault zone and opposite-side aquifer). This configuration, not described before, also considers flow transience within the fault zone. Drawdown solutions were developed for the three-aquifer domains based on an unconventional application of the well-image theory. Their comparison with numerical modelling results was very satisfactory. Drawdown behaviour at the pumping well is discussed for contrasted properties between the three-aquifer domains. Original analytical solutions for transient flow along both sides of the fault zone, and net transient flow from the fault itself, are proposed and discussed, with particular attention to flow reversal, from the pumped compartment to the fault zone, which occurs where the fault zone is the most transmissive. Finally, we extend the solutions to the case of a well intersecting and pumping a vertical fracture of finite length located near the same vertical geological discontinuity.