For contaminant transport modeling there will be an initial condition even if you are working with a steady state solution, because there may be a background concentration. The analytical equations are solved, assuming a zero background concentration. The background concentration may then be added to the solution.

Boundary conditions are applied to solve the advection-dispersion equation, which in its most basic, one-dimensional form is expressed as:


where:
C = concentration [M/L³]
t = time [ T ]
D = dispersion coefficient [L²/T ]
= average linear velocity of ground water [L/T ]
x = distance from location of the release of mass, in the flow direction [L]


Dispersion Coefficient is (dispersivity * average linear velocity + molecular diffusion coefficient adjusted for porous medium)

Average linear velocity is the Darcy velocity adjusted to reflect the fact that the ground-water flows only through the connected pores:



First let's explore a simple solution for a slug source in a one-dimensional, uniform flow field with three-dimensional spreading: