Formulation of the Governing Partial Differential Equation for Horizontal Velocity as a Function of Vertical Depth with Depth-Dependent Hydraulic Conductivity in Confined Aquifer Systems

In confined aquifer systems, hydraulic conductivity varies with depth due to compaction and lithological changes, but the partial differential equations governing horizontal groundwater velocity under such conditions have not been expressed in a form amenable to systematic analytical treatment. This paper formulates the governing PDEs for two commonly used conductivity-depth relationships: the exponential model K(z)=K_0 e^(-αz) and the power-law model K(z)=K_0 z^(-β). By inserting each model into the mass conservation equation coupled with Darcy’s law and carrying out the required differentiation, both equations are reduced to the common form h_xx+h_zz-f(z) h_z=g(z) h_t. For exponential conductivity, f(z)=α is constant and g(z)=(S_s/K_0 ) e^αz grows without bound; for power-law conductivity, f(z)=β/z is singular at the origin and g(z)=(S_s/K_0 ) z^β grows algebraically. Setting the time derivative to zero produces two elliptic equations: one with constant coefficients (exponential) and one retaining the β/z singularity (power-law). A term-by-term comparison reveals three structural contrasts—in the convective coefficient, the diffusivity growth rate, and the steady-state coefficient type—that govern the applicability of different analytical solution methods. Recovery of the horizontal velocity from the head field via Darcy’s law is stated explicitly.