The quasi-static linear poroelasticity model is widely applied in science, geophysics, biomechanics, and engineering. The simulations of this model exhibits locking phenomena and may depend on some model parameters. In this work, we follow the idea proposed in [Feng, Ge, and Li, IMA J. Numer. Anal., 2018,330–359] to transform the linear poroelasticity model into a four-field problem. The new four-field problem has a built-in mechanism to circumvent the locking phenomena existing in the original problem. We first prove that the inf-sup condition holds uniformly independent of the model parameters for the four-field problem and design several coupled, decoupled, and BDF2 algorithms in time. After that, we establish the analysis of the unconditionally optimal convergence and parameter robustness for the proposed algorithms. Finally, numerical examples are provided to investigate the convergence and parameter robustness of the proposed algorithms, which have no locking phenomena and are consistent with our theory. In addition, we also apply the algorithms to simulate brain edema, which is aligned with the experiment results.