Solute and Contaminant Transport

We now look at the transport of materials in soil systems. Not only do water and liquids move, but so also do a variety of chemical and biological constituents. In this chapter, the emphasis will be on flow processes involving water that is carrying different types of solute. The solutes of interest can be harmful or they can be beneficial. The same chemical species could be desirable when contained within one region and undesirable if it escapes to another—such as from the root zone to the ground water. Both conservative and reactive tracers will be discussed. A conservative tracer is assumed to move freely with the soil water and is non-reactive, non-volatile, and non-absorbing. Of course, this is only an ideal case, and all materials carried with water will react in some way with the solid phase. The degree of interaction depends on the solute, the soil, and the flow regime. However, if there is little interaction, the solute can often be treated as a conservative tracer. Examples are tritium and, to a lesser degree, bromide. Other materials are only slightly soluble, readily react with the solid phase, or perhaps can change into alternative phases that are clearly non-conservative. In some cases, whether a solute can logically be considered as a conservative or non-conservative tracer depends on the time scale and where the process is occurring—for example, perhaps the reaction rates are low, and for a short time scale the process is conservative; alternatively, perhaps the reaction rate is driven by whether oxygen is available or whether a specific microbe or catalyst is present at a particular time and place. Most of the discussion is directed toward “miscible displacement” processes. For a miscible displacement process, the invading fluid mixes freely with the fluid that is being driven out. An example is the displacement of water of a concentration differing from that of the antecedent water. Miscible displacement of a low molecular weight alcohol with water is another example. Many problems of environmental concern are included, such as leaching of nitrates from the soil surface to the groundwater.

Soil Water Flow

The definitions of hydraulic head and soil water potential in chapter 1 assumed equilibrium. However, the primary motivation was to develop a background useful to describe dynamic systems. If a system is in equilibrium, no flow will occur; otherwise, flow will occur from regions of high to low hydraulic head. The primary flow equation will be Darcy’s law. When Darcy’s law is combined with conservation of mass, the result is a continuity equation that can have several different forms. We will refer to all of those forms generically as soil water flow equations. Generally, for unsaturated conditions, the soil water flow equation is called the Richards equation. A starting point is to examine two classical relationships from fluid dynamics, the Bernoulli and the Poiseuille laws. Bernoulli’s law relates the total potential for ideal fluids and is commonly derived in introductory physics and fluid mechanics texts (see Serway, 1990). Assumptions include an ideal fluid (non-viscous), which is one that is incompressible and which exhibits steady and irrotational flow. For these conditions, the sum of gravitational, pressure, and inertial energy at positions S1and S2 are the same along any streamline For a real fluid, viscosity causes a loss of energy as friction that must be overcome. Additionally, for most problems of interest in soils, the velocity head will be negligible compared with the pressure and gravitational terms.

Multidimensional Water Flow in Variably Saturated Soils

Chapters 4 and 5 dealt with one-dimensional rectilinear flow, with and without the effect of gravity. Now the focus is on multidimensional flow. We will refer to two- and three-dimensional flow based on the number of Cartesian coordinates necessary to describe the problem. For this convention, a point source emitting a volume of water per unit time results in a three-dimensional problem even if it can be described with a single spherical coordinate. Similarly, a line source would be two-dimensional even if it could be described with a single radial coordinate. A problem with axial symmetry will be termed a three-dimensional problem even when only a depth and radius are needed to describe the geometry. The pressure at a point source is undefined. But more generally, three-dimensional point sources refer to flow from finite-sized sources into a larger soil domain, such as infiltration from a small surface pond into the soil. Often, the soil domain can be taken as infinite in one or more directions. Also, a point sink can occur with flow to a sump or to a suction sampler. In two dimensions, the same types of example can be given, but we will refer to them as line sources or sinks. Practical interest in point sources includes analyses of surface or subsurface leaks and of trickle (drip) irrigation. The desirability of determining soil properties in situ has provided the impetus for a rigorous analysis of disctension and borehole infiltrometers. Also, environmental monitoring with suction cups or candles, pan lysimeters, and wicking devices all include convergent or divergent flow in multidimensions. There are some conceptual differences between line and point sources and one-dimensional sources. For discussion, consider water supplied at a constant matric potential into drier surroundings. For a one-dimensional source, the corresponding physical problem includes a planar source over an area large enough for “edge” effects to be negligible. For two dimensions, the source might be a long horizontal cylinder or a furrow of finite depth from which water flows. For three dimensions, the source could be a small orifice providing water at a finite rate or a small, shallow pond on the soil surface.

Saturated Flow

Saturated conditions generally exist below a water table, either as part of the permanent groundwater system (aquifer) or in the vadose zone as perched water. For isotropic and steady-state conditions, such systems can be modeled by Laplace’s equation. Because it is linear, Laplace’s equation is much easier to solve than the variably saturated forms of Richards’ equation and, hence, provides a convenient place to begin. Analyses of water flow for drainage and groundwater systems borrow heavily from the classical (and old!) work in heat conduction, hydrodynamics, and electrostatics. This section presents analytical solutions for subsurface drainage and well discharge in fully penetrating confined aquifers (the solutions are the same). Included are the definition of stream functions and demonstrations of the Cauchy–Riemann relations. A comparable numerical solution is presented, and also for the ponded drainage and well discharge, and the results compared with the analytical solutions. A more complex example is then presented concerning drainage below a curved water table. These results are followed by travel-time calculations relevant to solute movement from the soil surface to a drainage system. A short section covering analytical techniques with three-dimensional images is then given, followed by a section covering additional topics, which includes a complex image example (two dimensional) and some relationships for Fourier series. Consider a point source in a two-dimensional x—y plane, as in figure 3-1. The origin corresponds to a source that is assumed to be an infinite line perpendicular to the x—y plane. If the steady flow rate is Q, then the conservation of mass results in . . . Q = Jr(2πr) (3-1) . . . where Jr is the Darcian flow in the r direction and evaluated at a polar radius r. The dimensions of Q are [L2T-1] corresponding to a volume of flow per unit time from a unit length of the line perpendicular to the x—y plane. Values of Q are taken to be positive for water entering the system.

One-Dimensional Infiltration and Vertical Flow

This chapter addresses one-dimensional infiltration and vertical flow problems. Traditionally, infiltration has received more attention than other unsaturated flow procedures, both for empirical formulations and for applications of Richards’ equation. Rarely is infiltration the only process of interest, and from an overall point of view it is only one example of soil water dynamics. Here, we will first emphasize systems for which analytical (or quasi-analytical) solutions can be found. These include the Green and Ampt solution (1911), which adds gravity to the simplified analysis discussed in chapter 4. Then a linearized form of Richards’ equation will be examined, followed by the perturbation of the horizontal problem of Philip leading to his famous series solution. Although the closed-form and quasi-analytical solutions are convenient for calculations and discussing the physical principles, generally, the nonlinearity of Richards’ equation precludes such convenient forms. However, numerical approximations can be used. The conventional numerical methods applied in water and solute transport are based on finite differences and finite elements. Because of its greater simplicity, we will emphasize finite differences and build on the methodology from the saturated-flow example in chapter 3. Richards’ equation is a parabolic partial differential equation reducing to an elliptical form for steady-state cases. The analyses and methods parallel developments for techniques developed primarily for the linear diffusion equation. Many texts exist for numerical methods; one to which we refer is by Smith (1985). Ideally, numerical methods give solutions that are as accurate as the input warrants or as necessary for application. In some cases, results may be easier or more accurate than the evaluation of a complex analytical expression. Clearly, infiltration is of limited duration, with drainage and redistribution occurring over much longer time frames. We will visit briefly some steady-state examples, including layered profile and upward flow from a shallow water table. Other examples include modeling plant water uptake from the profile and drainage of initially wet profiles. The rapid increase in computational power and availability of computers make solutions feasible and routine for problems that were very tedious or time consuming only a few years ago. This is particularly true of the one-dimensional numerical solutions.

The Soil System

Soil exists at the boundary between the atmosphere and the Earth’s subsurface. It plays a critical role in the hydrologic cycle, in addition to serving as the location of most human activity. An examination below the Earth’s surface generally reveals a profile similar to that shown in figure 1-1 A. The first zone encountered is the soil zone. This soil has developed from parent material through biological and other factors of weathering. If time is sufficient, then horizons will have formed with differing physical and chemical properties. At greater depths the soil merges with additional unconsolidated material. Eventually, at still greater depths, bedrock is encountered. The dimensions of these various zones are highly variable. For example, the soil profile may exist on bedrock that is partially exposed at the soil surface. Conversely, the unconsolidated layer can be hundreds of meters thick, as is the case in many alluvial basins. The subsurface can also be described in terms of water regimes that exist.The hydrologic profile consists of the vadose zone and the phreatic zone. The vadose zone is from the ground surface to the permanent water table, and includes the root zone, the soil profile, and the capillary fringe, which is a tension-saturated zone bordering the water table. The water at the water table is at atmospheric pressure; above the water table the pressure is less than atmospheric pressure and below the water table it is greater. The system is unsaturated above the capillary fringe, meaning that not only is the water under tension, but that some of the pore space is filled with air. The extent of the capillary fringe is dependent on the porous material. Generally, itextends a few centimeters for coarse material, or perhaps a meter for fine materials. A more complete depiction would include further saturated regions in the vadose zone, such as those due to surface infiltration or due to impeding layers that result in a perched water. Historically, the term groundwater was used to denote water beneath the permanent water table, but it is now commonly used to describe all subsurface water.

One-Dimensional Absorption

In this chapter we address one-dimensional absorption. Absorption denotes movement of water (or other liquid) into a soil under the influence of capillarity without the effects of gravity. Although important for horizontal flow conditions, there is more interest in the results and principles relevant to the early stage of infiltration and in general relationships descriptive of the physical principles for all unsaturated systems. At the outset, two simplified systems will be considered. Included is the classical problem of linear diffusion into a semi-infinite domain. Then the Boltzmann similarity transform will be applied, confirming results from the simplified solutions and leading to methods for finding soil-water diffusivity and Philip’s quasi-analytical solution. Finally, simultaneous water flow will be considered as a two-phase process. Figure 4-1 shows water introduced into a horizontal column of soil at a matric potential hwet. The value of hwet is maintained as zero or negative by the “mariotte” device to the left. The initial condition is that the matric potential is hdry with hdry < hwet ≤ 0. A porous plate at x = 0 allows water to come into the system but prevents air from flowing from the soil back into the water supply. The right-hand end allows air to freely escape the system as the water displaces the air. Vertical movement in the soil column is ignored. We make key simplifying assumptions that the conductivity is a constant K = Kwet in the wet part of the column and K = 0 for the dry part. Furthermore, we assume a sharp division between the wet and dry part at xf. On the supply side of the column (0 < x < xf), the water content is a constant (θ = θwet) and, to the dry side, the initial value is maintained (θ = θdry). These are equivalent to the “Green–Ampt” assumptions used in chapter 5, when gravity will be included as a driving force.