Lagrange's equations for seepage flow in porous media with a mixed Lagrangian-Eulerian description

Most conventional numerical models employ partial differential equations (PDEs) to describe seepage flow problems and use weighted residual and finite difference solution techniques to solve the PDEs. These PDEs are established in view of a spatial point, which mathematically stems from the infinitesimal concept. An alternative approach to such problems is developed. It applies an energy approach, i.e., the Lagrange's equations, to the representation of the seepage flow system, instead of directly resorting to the PDEs. The Lagrange's functional is established on a representative volume element (RVE) by integrating the energy of the RVE. Following a Lagrange formulation, the variation of the functional is conducted with regard to appropriate generalized coordinates. Then the resulting integral equations are considered with the description from the Lagrangian frame into the Eulerian frame for an improved accuracy. Afterwards, the equations are numerically discretized with a cell-centered finite volume method. Finally, two seepage front estimation schemes are presented-one scheme is implemented by local mesh refinement and the other scheme by seepage front movement. The resulting model is a true energy formulation, developed without reference to the partial differential momentum equations. Numerical examples are demonstrated and show that the model generates physically sound results.

Automated summary

This study presents an alternative approach to modeling seepage flow problems using an energy method and Lagrange's equations instead of traditional partial differential equations (PDEs). By integrating the energy of a representative volume element (RVE), a Lagrange's functional is established and then numerically discretized using a cell-centered finite volume method. This true energy formulation avoids the limitations of PDEs and produces physically sound results.