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In this work, a novel method with an adaptive functional basis for reduced order models (ROM) based on proper orthogonal decomposition (POD) is introduced. The method is intended to be applied in particular to hydrocarbon reservoir simulations, where a range of varying boundary conditions must be explored. The proposed method allows us to update the POD functional basis constructed for a specific problem setting in order to match varying boundary conditions, such as modified well locations and geometry, without the necessity to recalculate each time the whole set of basis functions. Such an adaptive technique allows us to significantly reduce the number of snapshots required to calculate the new basis, and hence reduce the computational cost of the simulations. The proposed method was applied to a two-dimensional immiscible displacement model, and simulations were performed using a high resolution model, a classical POD reduced model, and a reduced model whose POD basis was adapted to varying well location and geometry. Numerical simulations show that the proposed approach allows us to reduce the required number of model snapshots by a few orders of magnitude compared to a classical POD scheme, without noticeable loss of accuracy of calculated fluid production rates. Such an adaptive POD scheme can therefore provide a significant gain in computational efficiency for problems where multiple or iterative simulations with varying boundary conditions are required, such as optimization of well design or production optimization.