Comparison of three direct methods of inversion for the identification of transmissivity from source and head data: tests on synthetic aquifers and real data

Model forecast depends on input parameters, which are seldom measured at the proper scale and are often determined by model calibration. The identification of aquifer transmissivity (T) from measurements or estimate of water head (h) and source/sink terms (F) can be obtained with direct methods of solution to inverse problems, i.e., by searching a solution to the balance equation for T. Such methods are claimed to be affected by non-uniqueness and instability and rarely used in practical applications. Three methods (Comparison Model Method, CMM; Differential System Method, DSM; Double Constraint Method, DCM) are tested for their application to different flow situations. CMM and DCM are based on auxiliary models, for a tentative T. CMM and DSM require the knowledge of h over the whole domain; DSM requires at least the T value at one point. Several tests are performed on real data released by prof. Fritz Stauffer (ETH), coming from measurements in an alluvial aquifer. Furthermore, tests are performed on synthetic aquifers, which share geometry and boundary conditions of the real case, but with synthetic T fields showing different degrees of heterogeneity. The results show that these methods are stable (i.e., the errors on estimated T are small, if the errors on h are small enough), but could possibly be ill-conditioned (i.e., the errors on h, required to keep the errors on the estimated T reasonably small, could be excessively small). Moreover, the forward problem (T=>h mapping) behaves as a low-pass filter, smoothing the high-wavelength components of T, whereas the inverse problem (h=>T mapping) as a high-pass filter, which enhances the high-wavelength irregularities of h. Tests on synthetic aquifers show that CMM and DCM smooth the high-wavelength components of the reference T field.