2.5. Contaminant Transport and Pollution in Rivers#
Another major challenge in groundwater systems is the transport of contaminants, that is something (potentially toxic) is released into the subsurface and travels with the groundwater, appearing in the water supply at some point later in time (depending on the hydraulic conductivity of the aquifer, and the hydraulic head difference driving the flow). In some respects we have learned the tools you would need to determine how long before a contaminant released into the groundwater would be seen at some point downstream. You could figure out the lines of equipotential, and the hydraulic conductivity, and then calculate the flow velocity, and this would yield time (with some knowledge of the distance travelled).
Contaminant Transport in Groundwater#
Having understood a bit about how the chemistry of groundwater evolves along the flow path, we can consider what happens when we add a contaminant to groundwater and how it is likely to travel and disperse through the groundwater system.
Fig. 2.36 A control volume in groundwater showing solute transport by advection and dispersion, with possible chemical reactions within the box.#
If we consider a control volume in the groundwater where we have flow of a contaminant (or molecule or element – here we will call it a ‘solute’, meaning a component dissolved in groundwater) through that volume, we can apply mass conservation to this box. Solute can move into the box or out of the box, and chemical reactions can create or destroy solute within the box (e.g. mineral dissolution or precipitation). This allows us to consider the net rate of change of the concentration of the solute within the box.
So how does the solute, or contaminant, move through the control volume?
It moves by advection, and by dispersion.
Advection#
Advection causes the wholesale movement of the solute with the flow field at the same velocity as the flow field. So if we have a solute (like a contaminant) in groundwater, and the groundwater is moving at a velocity of 5 meters per year, then if we return one year later, we expect the contaminant to have moved 5 meters away.
Fig. 2.37 The effect of advection on contaminant movement over time from t0 to t2, showing the displacement of the solute plume with the groundwater flow.#
Fig. 2.37 shows the effect of advection on the movement of a contaminant over time from t0 to t2.
In reality what we find is that over this time we often get more than just the movement of the contaminant in the flow field but also the spreading out of the concentration. This is the effect of diffusion and hydrodynamic dispersion.
Dispersion#
Fig. 2.38 Illustration of dispersion processes showing how a contaminant plume spreads due to molecular diffusion and mechanical dispersion.#
Dispersion comprises molecular diffusion and hydrodynamic or mechanical dispersion.
Molecular Diffusion describes the spread of particles through random motion from regions of higher concentration to regions of lower concentration. It is governed by Fick’s First law, where the overall flux due to molecular diffusion is given by a diffusion coefficient (determined in the lab) and the concentration gradient:
(2.25)#\[J = -D \pdv{C}{x}\]where \(J\) is the diffusive flux, \(D\) is the diffusion coefficient, and \(\pdv{C}{x}\) is the concentration gradient.
Mechanical Dispersion reflects the fact that not everything in the porous medium travels at the average water flow speed. Some paths are faster, some slower, some longer, some shorter. This results in a net spreading of the solute plume that looks very much like a diffusive behavior.
Fig. 2.39 Modelling of the mechanical dispersion in porous media, showing variations in flow paths through the pore structure.#
Since mechanical dispersion depends on the flow, it is expected to increase with increasing flow speed. The most common expression for mechanical dispersion is given by:
(2.26)#\[D = \alpha v\]where \(D\) is the mechanical dispersion coefficient, \(\alpha\) is the dynamic dispersivity, and \(v\) is the average linear velocity.
Note that given the very slow rate of molecular diffusion, advection and hydrodynamic dispersion tend to dominate contaminant transport in groundwater.
Fig. 2.40 Summary diagram showing the combined effects of advection, dispersion, and chemical reactions on contaminant transport in groundwater.#
Chemical Reactions#
When solutes flow through a porous medium they can interact with the solid phase. In particular they can sorb and desorb, or precipitate into mineral phases. The net result is a process called retardation that effectively slows the transport of a solute through a porous medium.
The amount of these chemical reactions and how they impact the overall chemical transport depends on the solute, water chemistry, and geochemical makeup of the porous medium.
The Advection-Diffusion-Reaction Equation#
Having discussed advection, dispersion, and chemical reactions individually, we can now consider how they combine in a mathematical framework. In previous lectures, we learned the tools needed to determine how long before a contaminant released into the groundwater would be seen at some point downstream. We could figure out the lines of equipotential and the hydraulic conductivity, and then calculate the flow velocity, which would yield time (with some knowledge of the distance travelled).
However, like most things, it is more complicated than this. The contaminant is not just subject to advection with the flow of the groundwater, but also diffusion (the process by which things move from high concentration to low concentration) and possibly chemical reaction with the porous media. This leads to an entirely different class of equations and models known as advective-diffusive transport models. We will not go through the mathematics of these models in detail, but the standard formula is:
Where \(C\) is the contaminant concentration, and we are interested in how its concentration changes with time \(\left( \frac{\partial C}{\partial t} \right)\) and with distance \(\left( \frac{\partial C}{\partial x} \right)\). Because there are changes with time and with distance, we need to use a partial differential equation. In the partial differential equation above, \(D\) is the diffusion coefficient for the contaminant in water, \(v\) is the flow velocity \(\left( \frac{-K}{\rho} \dv{h}{x} \right)\) (can you figure out why this is the flow velocity from Darcy’s Law?), and \(R\) represents the chemical reactions that change the concentration with time. Diffusion can be molecular diffusion or mechanical dispersion, and the types of chemical reactions that change the concentration vary depending on the contaminant and the geochemical environment.
Extension: Navier-Stokes Equations (Non-Examinable)
Building on what we have discussed about hydraulic head in groundwater, we can consider the Navier-Stokes equations, which are partial differential equations that govern the motion of incompressible fluids. These equations constitute the basic equations for fluid mechanics. They are grounded in the concepts of conservation of mass, conservation of momentum, and conservation of energy.
Conservation of mass means that mass in a control volume (basically a bit of the subsurface fluid – see Fig. 2.43) cannot be created or destroyed. Conservation of momentum means that momentum is not created or destroyed, and changes only because of forces acting on the control volume. Conservation of energy derives from the first law of thermodynamics: work and heat added to the system will result in an increase in the total energy of the system.
Fig. 2.41 A control volume showing the forces and momentum fluxes used in deriving the Navier-Stokes equations from conservation of momentum.#
By understanding the forces on the control volume, we can derive partial differential equations that explain how fluids move through the control volume. The derivation of the Navier-Stokes equations is something we consider in QCES.
Control Volumes and Dispersion#
One way to think about this is using the ‘control volume’ and thinking that flow through advection moves the control volume along the X axis but flow through dispersion (molecular diffusion and mechanical or hydrodynamic dispersion) also moves the flow along in the Y and Z direction (Fig. 2.43).
A brief primer on diffusion
Diffusion is the spontaneous movement of particles from regions of high concentration to regions of low concentration. At the molecular level, this occurs due to random thermal motion of molecules. In groundwater, contaminants diffuse through pore water even in the absence of flow.
Diffusion is governed by Fick’s First Law, which states that the diffusive flux is proportional to the concentration gradient:
where \(J\) is the diffusive flux, \(D\) is the diffusion coefficient, and \(\frac{\partial C}{\partial x}\) is the concentration gradient. The negative sign indicates that diffusion occurs down the concentration gradient (from high to low concentration).
Fig. 2.42 Time series showing the spreading of dye through diffusion at four successive time steps, illustrating how concentration gradients diminish over time.#
Fig. 2.43 A diagram showing how advection moves the control volume along the X axis while dispersion causes movement in the Y and Z directions.#
In beadpacks in the Institute for Energy and Environmental Flows you can see the stretching along the flowpath of an injection of dye (Fig. 2.44).
Fig. 2.44 Dye injection in a beadpack showing stretching along the flowpath.#
The Peclet Number#
Determining the relative importance of advection versus dispersion (so the growing shape of the contaminant plume during subsurface transport) is often done using the dimensionless Peclet Number (Fig. 2.45).
Fig. 2.45 The Peclet Number characterizes the relative importance of advection and dispersion in contaminant transport.#
Fig. 2.46 Normalized dispersion coefficient plotted against Péclet number.#
Contaminant Plume Modeling#
If we assume that the contaminant spreads out in all directions as it moves along the flow of the groundwater, then we can track the shape of the contaminant plume away from the point source. Here the relative movement in the X direction versus the Y and Z direction is a function of the magnitude of the movement due to molecular diffusion versus that due to advective velocity and mechanical dispersion. With a series of simplifying assumptions (isotropic media, flow in the x direction etc.) we can use a gaussian model to track plume movement over time (Fig. 2.47).
Fig. 2.47 Diagram showing the development of a contaminant plume in groundwater, tracking movement in three dimensions over time.#
This is similar to the maths behind following a plume of smoke in the atmosphere (Fig. 2.48, Fig. 2.49).
Fig. 2.48 A smoke plume in the atmosphere showing similar dispersion patterns.#
Fig. 2.49 Plume model diagram illustrating the mathematical framework for tracking dispersion.#
Contaminant Transport Models#
However, like most things, it is more complicated than this. The contaminant is not just subject to advection with the flow of the groundwater, but also diffusion (the process by which things move from high concentration to low concentration) and possibly chemical reaction with the porous media. This leads to an entirely different class of equations and models known as advective diffusive transport models. We wont go through the mathematics of these models but the standard formula is:
Where \(C\) is a contaminant, and I am curious how its concentration changes with time \(\left( \frac{\delta C}{\delta t} \right)\) and with distance \(\left( \frac{\delta C}{\delta x} \right)\). Because there are changes with time and with difference, we need to use a partial differential equation. In the partial differential equation above, \(D\) is the diffusion coefficient for the contaminant in water, \(v\) is the flow velocity \(\left( \frac{-K}{\rho} \dv{h}{x} \right)\) (can you figure out why this is the flow velocity from Darcy’s Law?), and \(R\) represents the chemical reactions that change the concentration with time. In the lecture we’ll talk through each of these terms, and how diffusion can be molecular diffusion or mechanical dispersion, and the types of chemical reactions that change the concentration of common contaminants.
Pollution in rivers has been a growing concern. In general, pollution in rivers is from the release of sewage, chemicals and nutrients into the rivers. The River Trust has a bunch of maps that track pollution levels in rivers across the UK, including the amount of ‘forever chemicals’ (e.g., chemicals that are very hard to degrade so will stay in rivers for a very long time) and the rate of release of sewage. However, the largest source of pollution to rivers is through nutrients from agriculture.
Nutrients and Eutrophication#
Nutrients refer largely to nitrate and phosphate, which are key macronutrients that are needed for photosynthesis. Nitrogen (in the form of ammonium or nitrate) is used in amino acids and proteins while phosphorus is used for DNA and cell walls and lipids. To get plants to grow you need a source of CO2, water, light, nitrogen and phosphorus. This is why nitrogen and phosphorus are often added to soils in the form of fertilizers to maximize plant growth. Currently UK farmers are not penalised for overfertilizing their land, so it is just the cost of the fertilisers that limits the amount that would be used to stimulate growth.
As you add nitrogen and phosphorus run-off from agriculture to rivers you get growth of algae in the rivers, often measured as the chlorophyll abundance (Fig. 2.50).
Fig. 2.50 The addition of nitrate and phosphate leads to exceptional algal growth, shown by chlorophyll abundance.#
And we know that more algal production leads to more respiration which depletes the water in oxygen. This is eutrophication.
Fig. 2.51 The process of eutrophication: excess nutrients lead to algal blooms, which increase respiration and deplete oxygen levels in the water.#
In the lecture we discussed the particular challenge of the River Wye, where intensification of chicken farming has led to mass eutrophication and fish kills along the river. Focusing on cleaning up rivers often focuses on stemming the tide of runoff into the river. But much of the waste is seeping into the groundwater. We can start to ask ourselves how long until it reaches the rivers and the challenges with eutrophication might be over?
Transit Time and Groundwater#
The transit time simply might be considered:
Where the length might be how far a farm is from the river of interest and the velocity could be the rate of flow of a stream or the rate of flow of the groundwater derived from Darcy’s Law. It has long been thought that time scales with length. Next week in Andy’s lectures, he is going to show us why – when it comes to groundwater systems – time scales with length2 (due to the flowpaths in the subsurface and the impact of dispersion).
Groundwater and Surface Water Interactions#
The take home from groundwater is the importance it plays in the overall surface water system!
Fig. 2.52 Diagram showing the network of interactions between groundwater and surface water systems.#