We developed the theory concerning the onset of an abrupt pressure change causing the propagation of compaction waves through a saturated deformable porous medium, yielding the motion of solutes through a variable density Newtonian fluid. A simplified characteristic solution of a one spatial dimension (1D) concerning the formulation of a traveling wave provided the tool to investigate the translation extent of the solute under different application scenarios. A set-up of 1D shock-tube experimental laboratory and limited field experiments confirmed the findings of the characteristic solution. Further elaboration accounted also for the macroscopic mass and momentum balance equations of an elastic porous matrix, and for Forchheimer terms addressing the exchange of inertia through the microscopic fluid-solid interface. The 1D version concerning the fluid, solute and matrix was solved numerically implementing the Total Variation Diminishing (TVD) scheme. The efficiency of extracting solute was assessed on a ratio between pumping using an approximate analytical solution following Darcy's equation, and TVD numerical simulations addressing the emitting of an expansion wave. The latter proved to extract by far more mass for a spectrum of different porous media.