When patients visit primary care clinics, they can be subject to long wait times due to operational inefficiencies and bottlenecks, decreasing patient satisfaction and sometimes leading to worse health outcomes. The existing literature models primary care clinics primarily as agent-based models, which are excellent at tracking individual patients and their movements in a model of a clinic. While agent-based models can detect bottlenecks, a network flow model better detects bottlenecks in the model by correlating changes in patient flow and wait times in the healthcare network. In this paper, a network flow model is constructed, where patients flow along the capacitated edges of a network while receiving treatment at the nodes. This configuration easily identifies bottlenecks by analyzing the flow in and flow out of nodes through metrics such as efficiency and patient wait times. The capacities of the edges for this model are taken from an agent-based model of a case study of a primary care clinic and sampled as random variables. Ensemble runs of the network flow model are created to account for uncertainty in the synthetic data. By changing the topology of the network flow model, bottlenecks are removed, increasing the model efficiency and decreasing patient wait times. Finally, the model is subjected to a sensitivity analysis. The focus in this work is on the method rather than the results per se.