ONE-DIMENSIONAL CONSOLIDATION THEORY
Derivation of Governing Equation:
We now return to our experiment described in Section 9.3 to derive the theory for time rate of settlement using an element of the soil sample of thickness dz and cross-sectional area dA 5 dx dy (Figure).
We will assume the following:
1. The soil is saturated, isotropic, and homogeneous.
2. Darcy’s law is valid.
3. Flow only occurs vertically.
4. The strains are small.
We will use the following observations made:
1. The change in volume of the soil (DV) is equal to the change in volume of porewater expelled (DVw), which is equal to the change in the volume of the voids (DVv). Since the area of the soil.
One-dimensional fl ow
through a two-dimensional
is constant (the soil is laterally constrained), the change in volume is directly proportional to the change in height. 2. At any depth, the change in vertical effective stress is equal to the change in excess porewater pressure at that depth. That is, 'srz 5 'u. For our soil element in Figure, the infl ow of water is v dA and the outfl ow over the elemental thickness dz is 3v 1 1'v/'z2dz4 dA. Recall from Chapter 6 that the fl ow rate is the product of the velocity and the cross-sectional area normal to its (velocity) direction. The change in fl ow is then 1'v/'z2dz dA. The rate of change in volume of water expelled, which is equal to the rate of change of volume of the soil, must equal the change in fl ow. That is,
where h is the height of water in the burette.
Partial differentiation of Equation with respect to z gives
Recall [Equation ] that the volumetric strain εp 5 'V/V 5 'e/ 11 1 eo 2, and therefore
Substituting Equation and simplifying, we obtain
The one-dimensional fl ow of water from Darcy’s law is
where kz is the hydraulic conductivity in the vertical direction. Partial differentiation of Equation with respect to z gives
The porewater pressure at any time from our experiment in Section is
where h is the height of water in the burette. Partial differentiation of Equation with respect to z gives
By substitution of Equation, we get
We can replace kz mvgw by a coeffi cient Cv called the coeffi cient of consolidation.
The units for Cv are length2/time, for example, cm2/min. Rewriting Equation (9.30) by substituting Cv, we get the general equation for one-dimensional consolidation as
This equation describes the spatial variation of excess porewater pressure (Du) with time (t) and depth (z). It is a common equation in many branches of engineering. For example, the heat diffusion equation commonly used in mechanical engineering is similar to Equation except that temperature, T, replaces u and heat factor, K, replaces Cv. Equation is called the Terzaghi one-dimensional consolidation equation because Terzaghi (1925) developed it.