In this paper a finite volume (FV) numerical method is implemented to solve a Biot consolidation model with discontinuous coefficients. Our studies show that the FV scheme leads to a locally mass conservative approach which removes pressure oscillations especially along the interface between materials with different properties and yields higher accuracy for the flow and mechanics parameters. Then this numerical discretization is utilized to investigate different sequential strategies with various degrees of coupling including: iteratively, explicitly and loosely coupled methods. A comprehensive study is performed on the stability, accuracy and rate of convergence of all of these sequential methods. In the iterative and explicit solutions four splits of drained, undrained, fixed-stress and fixed-strain are studied. In loosely coupled methods three techniques of the local error method, the pore pressure method, and constant step size are considered and results are compared with other types of coupling methods. It is shown that the fixed-stress method is the best operator split in comparison with other sequential methods because of its unconditional stability, accuracy and the rate of convergence. Among loosely coupled schemes, the pore pressure and local error methods which are, respectively, based on variation of pressure and displacement, show consistency with the physics of the problem. In these methods with low number of total mechanical iterations, errors within acceptance range can be achieved. As in the pore pressure method mechanics time step increases more uniformly, this method would be less costly in comparison with the local error method. These results are likely to be useful in decision making regarding choice of solution schemes. Moreover, the stability of the FV method in multilayered media is verified using a numerical example.