Solution of the One-Dimensional Consolidation Equation for Saturated Clays Using a Spectral Method

The nonlinear, one-dimensional consolidation equation of fully saturated clays interbedded in an aquifer, derived by Gibson and others in 1967, is solved using a spectral method. This equation considers the variation of soil compressibility and permeability during consolidation and recasts Darcy's law so that the relative velocity of the soil skeleton and the pore fluid are related to the excess pore fluid pressure gradient. The spectral solution presented herein uses the matrix representation with Chebyshev collocation to compute the spatial derivative of functions that depend on void ratio, vertical hydraulic conductivity, and the vertical gradient of effective stress. A fourth-order Runge-Kutta algorithm is used to solve the derivative of the void ratio with respect to time. The spectral method requires neither the linearization of the originally nonlinear equation nor the convergence of iterative processes of traditional numerical methods such as finite differences and finite elements. The solution identifies temporal changes in void ratio within the clay lenses occurring in the aquifer system. The compaction is calculated from void ratio changes accumulated throughout the simulated time periods. Laboratory data were used to obtain the mean value for the soil grain density and depth-dependent profiles for aquifer compressibility, hydraulic conductivity, and initial vertical distribution of void ratio for each clay lens. The vertical gradient of the effective stress, needed in the consolidation equation, was derived and the resulting expression was evaluated by using the depth-dependent void ratio profile and drawdown data from a well hydrograph. Compactions and expansions of the clay lens resulting from temporal variations in drawdown due to ground-water withdrawals and recharge periods were simulated for two observation wells in the Santa Clara Valley, California. The solution of the one-dimensional consolidation equation generated temporal changes in void ratio that closely matched measured compaction in both observation wells, demonstrating that this equation and the numerical solution presented could be used to simulate subsidence.