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

• Author(s): Oyombe Aluala; Vincent Marani
• Paper ID: 1718920
• Page: 1572-1576
• Published Date: 16-06-2026
• Published In: Iconic Research And Engineering Journals
• Publisher: IRE Journals
• e-ISSN: 2456-8880
• Volume/Issue: Volume 9 Issue 12 June-2026

Abstract

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.

Keywords

Lie Point Symmetry, Infinitesimal Generator, Determining Equations, Confined Aquifer, Depth-Dependent Conductivity, Commutator Table, Lie Algebra Classification

Citations

IRE Journals:
Oyombe Aluala, Vincent Marani "Lie Symmetry Classification of the Groundwater Flow Equation with Exponential and Power-Law Depth-Dependent Hydraulic Conductivity in Confined Aquifers" Iconic Research And Engineering Journals Volume 9 Issue 12 2026 Page 1572-1576 https://doi.org/10.64388/IREV9I12-1718920

IEEE:
Oyombe Aluala, Vincent Marani "Lie Symmetry Classification of the Groundwater Flow Equation with Exponential and Power-Law Depth-Dependent Hydraulic Conductivity in Confined Aquifers" Iconic Research And Engineering Journals, 9(12) https://doi.org/10.64388/IREV9I12-1718920