Identifying contaminant source information from limited concentration measurements downstream from the source is a crucial step in groundwater pollution investigation for accountability and remediation purposes. However, in reality, the partially known but influential hydraulic conductivity field is always a significant obstacle in this inverse problem. In this work, we apply the restart normal-score Ensemble Kalman filter (NS-EnKF) method to identify the contaminant source and a non-Gaussian conductivity field jointly in a sandbox experiment by only using concentration measurements at a few observation points. As a preliminary step, we verify the restart NS-EnKF in a synthetic case mimicking the sandbox experiment. Some simple tries demonstrate that with a small ensemble size, measurement affected by observation errors and a complex sandbox conductivity lead to filter inbreeding. Consequently, a larger ensemble size, and several inflation methods are analyzed to solve this problem. We found that using a large ensemble size or Bauser's inflation method avoids filter inbreeding. These conclusions are used to analyse the sandbox results to identify the contaminant source and the conductivities using the restart NS-EnKF. The results show that the restart Ns-EnKF with a proper ensemble size or a suitable inflation method is capable to identify the contaminant source and the non-Gaussian conductivities in the sandbox experiment.