In this dissertation, we investigate the flow of fluid in porous media. The mathematical drainage models are developed in a domain that is partitioned into a saturated portion and a unsaturated portion. The boundary value problem of elliptic equation and parabolic equation on the time dependent domain which represents the model of pressure in the saturated domain were posted with given inflow and outflow measurements. By change of variables, we formulate the problem with time dependent coefficients on a fixed domain at every time t. We prove the existence and uniqueness of the weak solution of the problem by the variational method. The continuity of the solution with respect to permeability was also discussed. In order to estimate the permeability of existing pavement system, we pose a minimization problem to find the permeability k minimizing the error between calculated outflow and observed outflow. Furthermore, the numerical analysis and computer simulation results are presented for both elliptic and parabolic problems.