We propose a time-variant regularization in affine projection algorithms, where we update the regularization parameter with a gradient method using a momentum term parametrized by a momentum rate. To further improve the convergence properties of the algorithm in transient stages while ensuring a small final misadjustment, we adaptively estimate the momentum parameter. Then, we prove both the weak and strong convergence of the adaptive regularization. We apply the newly proposed algorithm to water quality data for prediction purposes, where we show that the developed algorithm outperforms existing time-varying regularization approaches.