Chapter 3: Flow and Transport

Ringko Summary, May 8th 2007

CHAPTER 3: FLOW AND TRANSPORT

Prepared by: Pham Ngoc Bao

Hanaki, Aramaki and Kurisu Laboratory

In order to understand the spatial and temporal variability of groundwater quality and quantity, it is necessary to have an insight understanding of the groundwater flow-path and the travel time from site of infiltration to the sampling point. Along the flow-line, chemicals may become sorbed which retards their transport compared to the velocity of water. Moreover, physical processes like diffusion and dispersion cause mixing and smoothen concentration changes. Therefore, in order to achieve a conceptual understanding of the processes, this chapter will focus on hand calculations that provide insight in flow-lines and residence times, in retardation, and in diffusion and dispersion.

Box 1: Groundwater Terminology

Definitions

• Groundwater- the water below the level at which the pore spaces in the soil or rock are fully saturated with water.
• Aquifer- ground water areas capable of yielding water to springs or wells at a flow rate sufficient to serve as a practical source of water.
• UnsaturatedZone- the area above the water table.
• Saturated Zone- the top of the water table.
• Perched Water- water which saturates a zone above a confining layer, such as clay. When such water is shallow (less than 30 feet) and discharges to surface water without serving as a source of drinking water, it would generally not be considered groundwater.

Groundwater Discharge vs. Recharge zones

3.1. Flow in the Unsaturated Zone

Water in the unsaturated zone percolates vertically downward along the maximal gradient of the soil moisture potential when the relief is moderate. The rate of percolation in an unsaturated profile can be derived from a mass balance, dividing the precipitation surplus by the water filled porosity of the soil:

where: is the velocity of water (m/yr)

P is the precipitation surplus (m/yr)

is the water-filled porosity (m3/m3)

Example: see example of tritium transport in a 20m thick unsaturated zone in sandy sediments; page 63, figure 3.1 and 3.2.

3.2. Flow in the Saturated Zone

Consider the landfill site shown in cross section in figure 3.3 (page 65); leachate from the waste is percolating into aquifer and the question is how FAST the various chemicals travel through the subsoil, and WHERE and WHEN they POLLUTE drinking water wells further downstream. In order to answer these questions, we MUST define the flowlines of groundwater in the aquifer and derive the travel time. Normally, the detailed calculation of flow patterns is usually done with numerical models. However, a first guess can be readily obtained by hand calculations.

3.2.1.Darcy’s Law

The groundwater level measured in the well (piezometric level) can be compiled in a map of equipotentials (isohypses), the lines that connect points with the same groundwater elevation of potential (Figure 3.4-page 65). Water flows from a high to low potential and the groundwater flow directions is at the right angles to the isohypses if the aquifer is isotropic, which means that the hydraulic properties are equal in all directions. Hydraulic gradients follow primarily the local relief, but are also influenced by recharge and discharge rates, the groundwater withdrawal.

The rate of flow of water through cross-sectional area A is found to be proportional to hydraulic gradient, but it also depends on the hydraulic conductivity of the subsoil, according to Darcy's law:

where: VD is specific discharge or Darcy flux (m/day)

k is the hydraulic conductivity or coefficient of permeability with dimensions of velocity (m/day). The value of the coefficient of permeability k depends on the average size of the pores and is related to the distribution of particle sizes, particle shape and soil structure. The ratio of permeability of typical sands/gravels to those of typical clays is of the order of 106. A small proportion of fine material in a coarse-grained soil can lead to a significant reduction in permeability.

is the hydraulic gradient

The piezometric level of groundwater gives the potential (h) for groundwater flow. The discharge is obtained by multiplying the specific discharge with the surface area perpendicular to the flow:

Q = VD. A = -kA.

Where: Q is discharge (m3/day)

A is the total subsurface area which includes pore space and grain-skeleton together.

Hydraulic conductivity of sediment can be determined in a permeameter and is calculated with the equation above. Mass balance applies as for the unsaturated zone , and the velocity is:

The water velocity can also be calculated from the discharge and cross section A.

Table 3.1. Hydraulic conductivity and porosity of different sediments

k (m/day) / (fraction)
Gravel / 200-2000 / 0.15-0.25
Sand / 10-300 / 0.20-0.35
Loam / 0.01-10 / 0.30-0.45
Clay / 10-5-1 / 0.30-0.65
Peat / 10-5-1 / 0.60-0.90

3.2.2.Flowlines in the Subsoil

Example of groundwater flowlines in the subsoil

-By specify the shape of the water table with a mathematical formula, it is possible to solve the potential field in the cross section and obtain the flowlines.

-It’s easier to use the water balance to visualize how groundwater flows in aquifers.

-In the rectangular aquifer of figure 3.6, the velocity is nearly equal at all depths along a vertical line. The point along the upper reach where water infiltrates and depth in the aquifer are then related proportionally:

where:

D is the thickness of the aquifer. Water infiltrated at a point xo, upstream of x, is at a given time, found at depth d in the aquifer. Above d flows water that infiltrated between point xo and x, while water at greater depths infiltrated further upstream of xo.

3.2.3.Effects of Non-Homogeneity

Most aquifers are not homogeneous and large differences in hydraulic conductivity are usual. The distribution of flowlines can be calculated by proportioning the volume of flow according to the transmissibility difference (Figure 3.7).

and the velocities in the layers are:

where:

D1, D2 are the thickness of two layers

and : porosities

k1, k2: hydraulic conductivities (m/day)

The discharge through the vertical section can be divided in two portions Q1 and Q2. The ratio of the two is (assuming equal hydraulic gradient in both layers) followed the equation above.

3.2.4.The Aquifer as a Chemical Reactor

-Concentrations of dissolved substances were so far considered as depth related entities which can be measured using depth specific samplers.

-The concentration in a fully penetrating well increases with time to the final value by an exponential function. This function is similar to the one used for an “ideal mixed reactor”.

-With uniform flow (no velocity variations with depth), the averaging in the groundwater well is similar to the mixing in the reactor.

-A difference between a stirred chemical reactor and the phreatic aquifer is that the reactor is well mixed to let all the reactions elapse in the same way everywhere. In the phreatic aquifer, mixing only takes place in the well, while chemical reactions among water and solids are variable in time and distance along the flowlines through aquifer. Nevertheless, the chemical reaction equations are useful when reactions between water and sediment are unimportant.

3.3. Dating of Groundwater

-Calculated groundwater flow patterns and residence times can be verified by groundwater dating. There is a range of dating methods available based on radioactive decay.

-These comprise tritium (3H), 3H/ 3He and 85K for young groundwater of up to about 50 years, radiocarbon (14C) for older groundwater up to 30,000 years and finally isotopes like cosmogenic 81Kr and 36Cl for even older groundwater. Using later methods, groundwater with an age of 400,000 years has been dated.

-For all radioactive decay processes, the rate equals a decay constant, , times the concentration of the reactant. For example, tritium (3H) disintegrates to 3He and at a rate:

Tritium concentration is expressed in tritium units (TU). One TU equals 10-18 mol 3H/mol H, and is equivalent to 3.2 pCi/L from the emitted particles. The decay constant =0.0558/yr. The equation can be integrated from the initial value 3Ho at time t=0 and gives:

ln(3H/ 3Ho)= - .t

The time to let tritium decay to half of its initial concentration is called half-life

t1/2 =ln(0.5)/- = 12.43 years.

3.4. Retardation

The effect of sorption on transport velocity of solutes is illustrated in figure 3.12. In case (A) the solute does not adsorb to sediment grains. Consequently this solute is transported with the velocity of water. In case (B), one out of two solute molecules is adsorbed. Thus, half the mass of chemical is lost from solution and therefore it will travel with half the velocity of water. The solute is retardedcompared to water migration.

3.4.1.The Retardation Equation

-For quantifying retardation, it’s vital to consider the effects of sorption processes on transport.

-Example: see figure 3.13 (page 77) for more information

-The retardation equation is formed by doing the mass balance of solute transport through a cube (figure 3.14-page 77).

-Retardation equation is as followed:

or

The retardation equation shows that the transport velocity is different for different concentrations when the slope of the sorption isotherm is variable.

3.4.2.Indifferent and broadening fronts

The retardation equation yields vc for a specific concentration c and we can evaluate how retarded solutes move compared to water. We can make time-distance graphs for a concentration using:

we can also plot concentration as a function of distance for a fixed time:

or we can chart them as a function of time for a fixed position:

tc = tH2O. Rc

3.4.3.Sharpening fronts

-In example 3.4, the solution with a low concentration flushed the solution with a high concentration. Now we will look at the reverse situation where a high concentration (0.8 mol/l) displaces a low concentration (0.1 mol/L) of the chemical with the convex isotherm.

-In addition to the retardation equation that is based on the slope of the isotherm, we can formulate the retardation equation for a sharp front:

-The sharp front equation is illustrated in figure 3.19. The differences between the initial and the final sorbed and solute concentrations are sufficient for calculating the retardation of the front.

-To discern whether a front is sharpening, we look at the sorption isotherm.

-Sharp fronts are found when chemicals with convex isotherms enter an aquifer, or when several ions or specific complete for the same sorption site, and the displacing species has a stronger affinity for the site than the displaced one.

-Sharpening fronts counteract the tendency of concentration changes to become more diffuse with time and distance.

3.4.4.Solid and Solute Concentration

-Chemical analysts usually express the concentration of substances in solids as concentration per mass of solid, mol/kg, g/kg, ppm by weight, etc. These numbers can be converted to equivalent solute concentrations, by multiplying the bulk density and dividing by the water filled porosity. Example 3.7, the factor / ≈ 6kg/L for sandy aquifers.

-The concentrations expressed per liter pore water always appear surprisingly high compared to those expressed relative to the solid phase. This is a simple consequence of the fact that most of the mass and volume in the aquifer is located in the solid phase.

-The dimensionless ratio of the concentrations, or the slope of the isotherm when the ratio varies, allows for a readily estimate of the mobility of chemicals.

Key questions for discussion:

1. Adsorption vs. Absorption?
2. Dating of groundwater: Empirical vs. Theory?
3. Concept explanations: Indifferent, broadening fronts and sharpening fronts

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