The theory of poroelasticity concerns with the analysis of a porous medium consisting of an elastic matrix containing interconnected fluid-saturated pores. The theory was proposed by Biot to deal with soil consolidation (quasi-static) and wave propagation (dynamic) problems. The governing differential equations in such problems are derived from constitutive equations, equations of motion, continuity equation and Darcy's law. The theory of poroelasticity is currently applied to a large number of problems in geophysics, petroleum industry, soil mechanics, hydrogeology and biomechanics. Due to the complexity of the solution in such problems, efficient methods are needed to achieve sufficiently accurate solutions. The use of boundary element method (BEM) and the method of fundamental solution (MFS) is recommended in the literature, however, it is well understood that these methods need Green’s functions (or fundamental solutions) which are not available for many problems. In this dissertation exponential basis functions (EBFs) are evaluated as the bases for the solution of problems with porous materials based on Biot's theory (BT). In the presented method, the solution is split into two parts, i.e. homogeneous and particular parts. The homogeneous part of equations is then approximated by linear combination of EBFs. Introduction of the EBFs into the homogeneous governing differential equations leads to a characteristic equation through which the exponents of the EBFs are defined. For many cases, the characteristic equation possesses some multiple roots. This makes the evaluation of the EBFs a rather tedious task. In such situations polynomial functions are added to EBFs. In this dissertation, closed form expressions are found for such bases while explaining the procedure to give insight to the problem. After selection of the EBFs as the bases for the approximation, the unknown coefficients of the series are determined by the satisfaction of the boundary conditions, in a collocation style, through a discrete transformation technique. While using the transformation, the number of EBFs should not necessarily be equal to (or less than) the number of boundary information data. The content of the EBF series plays an important role in reducing the computational error and reaching reasonable accuracy in the results. Therefore, in this study a suitable strategy is proposed for choosing the EBFs based on determining the contribution of the bases to the boundary conditions. The particular part of the solution is constructed by a rather similar approach. In this thesis, EBFs are evaluated for dynamic, quasi-static and static formulations in compressible and incompressible models. The capabilities of the method have been shown by comparing the results with analytical solutions. The results show that method performs excellently in solution of different poroelastic problems with various boundary conditions.