I. Processes that affect transport
Figure 19.2 in S&Z lists both physical and chemical processes that affect transport of dissolved contaminants. The chemical processes include the major classes of reactions that were discussed in previous lectures. The physical processes listed are advection and dispersion.
Advection is the process by which contaminants move at the average linear velocity of groundwater. This was compared to the average velocity of a bus that one would calculate assuming that the bus travels in a straight line from one stop to the next, rather than via the actual route (which is not a straight line).
The advective velocity can be estimated by dividing the flow rate, Q, by the cross-sectional area OPEN FOR FLOW, equal to the total area A times the effective porosity. (Table 3.2 in S&Z compares typical values of total and effective porosity for some rock types. Note that using the total porosity, rather than the effective porosity, will lead to an underestimation of the advective velocity.)
Note also that the advective velocity is not equal to the true microscopic velocity within a pore, since it is computed assuming a straight line path of water rather than the true tortuous path that water takes through a porous medium.
An equation for solute transport by advection can be derived using the mass balance "IN-OUT=CHANGE IN STORAGE" approach that was used to derive the groundwater flow equation. The result (in 3D below, demonstrated only in 1D in the lecture) is
which can be expanded as
and for steady flow reduces to
Pure advection does not occur. Instead there is always some spreading of solute mass relative to that expected for pure advection. This is due to the overall process that we call dispersion. This was compared to the dispersion of runners in a marathon, who all start at the same time but cross the finish line over a period of several hours.
C. Microscopic processes leading to dispersion
Dispersion in porous media is caused, at least in part, by processes that occur at the microscopic scale. Molecular diffusion is governed by Fick's 1st Law, J = -Dd grad C, where Dd is the molecular diffusion coefficient. (Note the mathematical similarity of this to Darcy's Law.) In porous media, diffusion is slower due to the mixture of pores and solids and due to the tortuosity of the diffusion paths. An effective molecular diffusion coefficient for a porous medium is Dd*. (This is the coefficient that is used in parallel with the advective velocity, V.)
Spreading of solutes during groundwater transport is also due to mechanical dispersion, which is the result of tortuosity of the flow paths, velocity variations within individual pores, and velocity variations between pores.
While there is a good basis for using Fick's Law to model molecular diffusion in a porous medium, there is not a fundamental basis for a model for the mechanical dispersion resulting from velocity variations and tortuosity. However, as a first approximation, researchers have assumed that this mechanical dispersion can also be modeled by an equation of the form of Fick's law. Applying a mass conservation approach to Fick's first law yields the (3D) equation
Note the mathematical similarity to the groundwater flow equation. Variations in velocity within and between pores lead to greater spreading in the direction of flow (the longitudinal direction) than in directions perpendicular (or transverse) to the main direction of flow. Thus, the dispersion coefficient D has directional properties and is a tensor like K. Generally DL, the longitudinal dispersion coefficient, has a greater magnitude than the transverse dispersion coefficients DT1 and DT2 for dispersion perpendicular to the mean direction of groundwater flow.
Combining advection and dispersion processes yield the governing equation for advective-dispersive transport.
II. Simple solutions to the solute transport equations
A. Transport by pure advection
Concentration histories (breakthrough curves) and profiles were examined for idealized cases of pure advection with continuous or pulse inputs. The arrival time of the advective front at a given distance (or its position at a given time) can be used to determine V in the case of a continuous input. The time of peak concentration on a breakthrough curve or the position of the peak concentration along a profile can be used to determine V for a pulse input.
B. Transport with dispersion
The sharp breakthrough curves or steep concentration fronts predicted for pure advection are rarely observed. Instead, breakthrough curves and concentration fronts show more gradual changes in concentration with time or distance from the source. Pure diffusion (or dispersion) of a finite source causes spreading to generate a concentration profile that resembles the "normal" (also known as "Gaussian") probability distribution, i.e. a "bell-shaped" curve. This curve can be described by its mean, m, and standard deviation, s. The growth of the spatial standard deviation with time (also with mean travel distance) is determined by the magnitude of the dispersion coefficient D according to the relationship
For the case where t1= 0 and there is no initial spread of the source, the dispersion coefficient can be determined at some later time t by the simplified relationship
Similar relationships between the temporal standard deviation of a breakthrough curve yield
for the analysis of two breakthrough curves at distance x2 and x1 respectively, or for a single breakthrough curve at distance x from the source