Scaling procedures for heterogeneous unconfined aquifers

Abstract

Abstract:Complex flows in heterogeneous confined and unconfined aquifers is a phenomenon that continues to present difficulties in flow mapping and modelling in the field, laboratory, and through numerical simulations. It is often the case with complicated phenomena that transformative scaling and reduction of the problem through symmetry is of great efficacy in the formation of predictive models in both the laboratory and computational settings. A detailed a study of the application of a broad class of Lie scaling transformations on a set of equations representing the groundwater flows in heterogeneous confined and unconfined aquifers has produced a set of scaling relationships between the spatial variables, hydrologic variables, and parameters. The set of scaling transformations preserve the structure of the equations in the sense that the scaling transformations leave the initialboundary value system representing the invariant groundwater flows. This theoretical approach elucidates not only the scaling relationships but also the properties that hydrologic variables and parameters must satisfy in order for calling to be possible. Validation of the theory developed is carried out through a series of four numerical simulations using the USGS MODFLOW-2005 software package. The results of these experiments demonstrate that the derived scaling transformations can effectively form predictive models of large-scale phenomena at small scales with negligible error in many cases. Comments on the limitations of the approach and directions for future research are made in the closing sections.

A comparison of the modern Lie scaling method to classical scaling techniques

“…Type-2 self similarity problems have been seen in engineering and statistical mechanics and are promising for solving difficult problems in hydrology and engineering. Examples can be seen in Polsinelli and Kavvas (2016), Mesa (2014), andGoldenfeld (1992).…”

A comparison of the modern Lie scaling method to classical scaling techniques

“…An early statement of the theorem can be seen in Vaschy (1896Vaschy ( , 1892. Two versions of the theorem were developed (likely) independently by Bertrand (1878) and Riabouchinsky (1911Riabouchinsky ( , 1915. The theorem establishes the existence, number, and composition of dimensionless groups for a given set of variables.…”