Lie Symmetry Classification of the Groundwater Flow Equation with Exponential and Power-Law Depth-Dependent Hydraulic Conductivity in Confined Aquifers

The groundwater flow equation in a confined aquifer whose hydraulic conductivity decay with depth takes the form h_xx+h_zz-f(z) h_z=g(z) h_t, where the coefficient pair (f,g) is fixed by the conductivity model. This paper determines the complete Lie point symmetry algebras for four instances of this equation: the transient and steady-state versions under both exponential conductivity K(z)=K_0 e^(-αz) and power-law conductivity K(z)=K_0 z^(-β). For each equation the second prolongation of the infinitesimal generator is applied, the resulting overdetermined system of determining equations is derived and solved, and the admitted generators are listed. The transient exponential equation admits a four-dimensional solvable algebra spanned by horizontal translation, a coupled depth-shift, time translation, and head scaling. The transient power-law equation also admits a four-dimensional solvable algebra, but the depth-shift is replaced by a self-similar dilation. In the steady-state limit the exponential algebra retains dimension four, with a rotation generator replacing time translation (L≅e(2) ⊕R), whereas the power-law algebra contracts to dimension two. Commutator tables are constructed for every case, and the algebras are classified. The dependence of the symmetry type on the functional form of K(z) is discussed.