This paper evinces a great performance of geostatistically estimating the modeling errors and a spatial distribution of permeabilities for two-dimensional groundwater flow. The exact variation in the water levels, which is equivalent to the accurate modeling errors, is formulated from the series solution of the finite element equation. A hypothetical aquifer into twenty piecewise zones, each of which has a constant permeability, is designed to allow assessing the errors and the level-variations in the homogeneous and heterogenous aquifers. Spatial estimation of the error complements and distributed structure identification of inhomogeneous permeabilities are herein carried out. A geostatistical model of the distributed permeabilities is integrated into the physical-based model. The statistical-based parameters in the integrated system are successfully identified from the physical-based observed data. It can be proven that such estimation and identification approaches are powerful and helpful in perfectly modeling.