This thesis analyzes a numerical method for solving the poroelasticity equations. The model incorporating the poroelasticity equations in this thesis can be applied in intestinal edema, which is a medical condition referring to the accumulation of excess fluid in the spaces between cells of intestinal wall tissue. The model has a dilatation term and can give a comprehensive prediction of pressure and displacement for intestinal edema. I approximate the pressure, displacement and dilatation by the discontinuous Galerkin method, which includes symmetric, nonsymmetric and incomplete interior penalty Galerkin cases. Moreover, in order to solve for the nonsymmetric case, I introduce an additional penalty term in the scheme. Theoretical convergence error estimates derived in a discrete-in-time setting show the a priori error can be bounded by some constant, which is related to the pressure, displacement, dilatation and the mesh size.