Transport in saturated soil takes place in that part of soil where pores are completely saturated by water. In the road construction, this usually occurs in the subgrade but rarely in the sub-base. Three principal transport processes are defined:
• Diffusion — pollutants move from compartments with higher concentrations to compartments with lower concentrations, even if the fluid is not moving;
• Advection — pollutants are carried with the flow of the water;
• Dispersion — the pollutants are locally redistributed due to local variations in fluid flow in the pores of the soil or pavement material.
Diffusion will occur as long as a concentration gradient exists. The diffusing mass in the water is proportional to the concentration gradient, which can be expressed as Fick’s first law. In one dimension it is defined as
where F = mass flux of solute (units of M/L2T); Dd = diffusion coefficient (units of L2/T); C = solute concentration (units of M/L3) and dC/dx = concentration gradient (units of M/L4). The negative sign indicates that movement is from areas of higher concentration to areas of lower concentration. In the case where concentrations change with time, Fick’s second law applies. In one dimension it is defined as:
dC _ d2C
~dt = Dd dx2
where dC/dt denotes change of concentration with time.
Diffusion in pores cannot proceed as fast as it can in open water because the ions must follow longer pathways as they travel around grains of road material. To account for this, an effective diffusion coefficient D* is introduced. It is defined as
D* = ш Dd
where w is a dimensionless coefficient that is related to the tortuosity. Tortuosity is defined as the ratio between the linear distance between the starting and ending points of particle flow and the actual flow path of the flowing water particle through the pore space. The value of w is always less than 1 and is usually defined by diffusion experiments.
In the road construction, a dissolved contaminant may be carried along with flowing water in pores. This process is called advective transport, or convection. The amount of solute that is being transported is a function of the solute concentration in the water and the flux of water infiltrating from the pavement surface. For one-dimensional flow normal to a unit area of the porous media, the quantity of flowing water is equal to the average linear velocity times the effective porosity and is defined as
K dh ne dl
where v = average linear velocity (L/T); K = coefficient of permeability (i. e. hydraulic conductivity) (L/T); ne = effective porosity (no units) and dh/dl = hydraulic gradient (no units).
Due to advection, the one-dimensional mass flux, F, is equal to the quantity of water flowing times the concentration of dissolved solids and is given as
F = v neC
One-dimensional advection in the x-direction is, then, defined as
dC _ dC
~dt = ~Vx ~dx
where vx is the velocity of flow in the x — direction. According to this one-dimensional advection equation, the mass transport in homogeneous porous media is represented with a sharp front.
Water in porous media is moving at rates that are both greater and less than the average linear velocity. In a sufficient volume where individual pores are averaged, three phenomena of mass transport in pores are present:
• Asa fluid moves through the pores, it will move faster in the centre of pores than along the edges;
• In porous media, some of the particles in the fluid will travel along longer flow paths than other particles to travel the same linear distance;
• Some pores are larger than others, allowing faster movement.
Due to different velocities of water inside the pores, the invading pollutant dissolved in the water does not travel at the same velocity, and mixing will occur along the flow path. This mixing is called mechanical dispersion, and it results in a dilution of the solute at the advancing edge of flow. The mixing that occurs along the direction of the flow path is called longitudinal dispersion. An advancing solute front will also tend to spread in directions normal to the direction of flow because at the pore
scale the flow paths can diverge. The result is transverse dispersion which is mixing in the direction normal to the flow path. In the road environment, the dispersal of a pollutant having penetrated into the sub-base will usually occur perpendicularly to the road course.
If we assume that mechanical dispersion can be described by Fick’s laws for diffusion and the amount of mechanical dispersion is a function of the average linear velocity, a coefficient of mechanical dispersion can be introduced. This is equal to a property of the medium called dynamic dispersivity, being a times the average linear velocity, vx.
In water flowing through porous media, the process of molecular diffusion cannot be separated from mechanical dispersion. The two are combined to define a parameter called the hydrodynamic dispersion coefficient, D:
Di = ai v + D* (6.7)
Dt = at v + D* (6.8)
where Dl = hydrodynamic dispersion coefficient parallel to the principal direction of flow (longitudinal) (with units of L2/T) and Dt = hydrodynamic dispersion coefficient perpendicular to the principal direction of flow (transversal) (also units of L2/T). ai = longitudinal dynamic dispersivity and at = transversal dynamic dispersivity (both with units of L).
By the combination of the equations above and with proper initial and boundary conditions, the total mass transport of a non-reactive pollutant in two-dimensional saturated porous media can be described by an advection-dispersion equation defined as follows with vx being the velocity of flow in the x-direction, as above:
Often the dispersion and diffusion terms are combined with a “hydrodynamic dispersion coefficient”, Dh=Di + Dt, being used to combine the effects of diffusion and dispersion. Various analytical and numerical solutions of the equation are possible (see, e. g., Fetter, 1993) dependent on the boundary conditions, but will generally involve a distribution of contaminants, with distance from the source and with time, according to a probability function. In practice, the advection-dispersion equation is usually solved by numerical or analytical computer methods such as Hydrus or Stanmod.
