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A finite element solution of the two-dimensional Richards equation is developed to describe the water movement in an agricultural subsurface drainage system. This solution is partially validated with an exact analytical solution of the Richards equation for a semi-infinite column of soil with constant flux at the soil surface, The analytical solution describes the time evolution of water content profile while the moisture is less than the drain depth. The comparison of both numerical and analytical profiles in two soils, sandy and clayey, has allowed to select one single numerical time step to approach the analytical solution in both soils, with a maximum relative error less than 0.25%. The finite element solution has been applied to describe the hydraulic performance of a drainage system installed in a field of an irrigation district located in the semi-arid northwestern Mexico, taking a seepage-face boundary condition at the pipe drain. The clay textured soil has been hydrodinamically characterized from its granulometric curve, its porosity and from a drainage test on the field. The saturated hydraulic conductivity and a characteristic value of the water pressure have been estimated so that the error between the numerical hydrogram and the observed hydrogram has been minimized. The theoretical hydrogram obtained from the finite element solution describes the mathematical characteristics, like the maximum point, of the experimental hydrogram. The ascending and recession phases of the hydrogram are also well described by the solution. These results validate the treatment of the boundary condition at the pipe drain as a seepage-face. In consequence, the proposed solution can be used for both the soil parameter identification from drainage tests, and the hydraulic performance description of a drainage system.