**Branch :**Civil Engineering

**Subject :**Soil Mechanics

## Finite Difference Solution of the Governing Consolidation Equation

Numerical methods (fi nite difference, fi nite element, and boundary element) provide approximate solutions to differential and integral equations for boundary conditions in which closed-form solutions.

(analytical solutions) are not possible. We will use the fi nite difference method here to fi nd a solution to the consolidation equation because it involves only the expansion of the differential equation using.

Taylor’s theorem and can easily be adopted for spreadsheet applications.

Using Taylor’s theorem

where (i, j) denotes a nodal position at the intersection of row i and column j. Columns represent time divisions and rows represent soil depth divisions. The assumption implicit in Equation (9.38) is that the excess porewater pressure between two adjacent nodes changes linearly with time. This assumption is

reasonable if the distance between the two nodes is small. Substituting Equationsin the governing consolidation equation and rearranging, we get

Equation is valid for nodes that are not boundary nodes. There are special conditions that apply to boundary nodes. For example, at an impermeable boundary, no fl ow across it can occur and, consequently, ≠u/≠z 5 0, for which the fi nite difference equation is

and the governing consolidation equation becomes

To determine how the porewater pressure is distributed within a soil at a given time, we have to establish the initial excess porewater pressure at the boundaries. Once we do this, we have to estimate the variation of the initial excess porewater pressure within the soil. We may, for example, assume a

linear distribution of initial excess porewater pressure with depth if the soil layer is thin or has a triangular distribution for a thick soil layer. If you cannot estimate the initial excess porewater pressure, you can guess reasonable values for the interior of the soil or use linear interpolation. Then you successively

apply Equation (9.40) to each interior nodal point and replace the old value by the newly calculated value until the old value and the new value differ by a small tolerance. At impermeable boundaries, you have to apply Equation .

The procedure to apply the fi nite difference form of the governing consolidation equation to determine the variation of excess porewater pressure with time and depth is as follows:

1. Divide the soil layer into a depth–time grid (Figure ). Rows represent subdivisions of the depth, columns represent subdivisions of time. Let’s say we divide the depth into m rows and the time into n columns; then Dz 5 Ho/m and Dt 5 t/n, where Ho is the thickness of the soil layer and t is the total time. A nodal point represents the ith depth position and the jth elapsed time. To avoid convergence problems, researchers have found that a 5 CvDt/(Dz)2 must be less than 1 2 . This places a limit on the number of subdivisions in the grid. Often, the depth is subdivided arbitrarily and the time step Dt is selected so that a , 1 2 . In many practical situations, a 5 0.25 usually ensures convergence.

2. Identify the boundary conditions. For example, if the top boundary is a drainage boundary, then the excess porewater pressure there is zero. If, however, the top boundary is an impermeable boundary, then no fl ow can occur across it and Equation applies.

3. Estimate the distribution of initial excess porewater pressure and determine the nodal initial excess porewater pressures.

4. Calculate the excess porewater pressure at interior nodes using Equation and at impermeable boundary nodes using Equation (9.42). If the boundary is permeable, then the excess porewater pressure is zero at all nodes on this boundary.