Abstract: We address a generalised three-dimensional$\mathrm{ \alpha }$ -Muskat model that comes from the fluid interface problem given by two incompressible fluids with different densities in the stable regime. We establish local-in-time wellposedness when $\alpha \in [0,1)$ and also prove global-in-time existence for strong solutions when $\alpha \in [0,12)$ with initial data controlled by explicit constants. We obtain maximum principles for the $L^{\mathrm{\infty }}$-norms of both the solutions and their gradients, and we further acquire the corresponding decay rates of these $L^{\mathrm{\infty }}$-norms. Finally, we give some convergence results for strong solutions as $\alpha \to 0^{+}$.