The total stress increases when additional vertical load is first applied. Instantaneously, the pore water pressure increases by exactly the same amount. Subsequently there will be flow from regions of higher excess pore pressure to regions of lower excess pore pressure causing dissipation. The effective stress will change and the soil will consolidate with time. This is shown schematically.
On the assumption that the excess pore water drains only along vertical lines, an analytical procedure can be developed for computing the rate of consolidation.
Consider a saturated soil element of sides dx, dy and dz.
The initial volume of soil element = dx.dy.dz
If n is the porosity, the volume of water in the element = n.dx.dy.dz
The continuity equation for one-dimensional flow in the vertical direction is
Only the excess head (h) causes consolidation, and it is related to the excess pore water pressure (u) by
h = u/gw.The Darcy equation can be written as
The Darcy eqn. can be substituted in the continuity eqn., and the porosity n can be expressed in terms of void ratio e, to obtain the flow equation as
The soil element can be represented schematically as
If e0 is the initial void ratio of the consolidating layer, the initial volume of solids in the element is (dx dy dz ) / (1 + e0), which remains constant. The change in water volume can be represented by small changes De in the current void ratio e.
The flow eqn. can then be written as
or
This is the hydrodynamic equation of one-dimensional consolidation.
If av = coefficient of compressibility, the change in void ratio can be expressed as
De = av.(-Ds') = av.(Du) since any increase in effective stress equals the decrease in excess pore water pressure. Thus,
The flow eqn. can then be expressed as
or
By introducing a parameter called the coefficient of consolidation, 
, the flow eqn. then becomes
This is Terzaghi's one-dimensional consolidation equation. A solution of this for a set of boundary conditions will describe how the excess pore water pressure u dissipates with time t and location z. When all the u has dissipated completely throughout the depth of the compressible soil layer, consolidation is complete and the transient flow situation ceases to exist.