ES&T FEATURES
Groundwater contaminant transport modeling Its theory and the role it plays in evaluating, containing, and remedying contamination are discussed
George F. Pinder Department of Civil Engineering Princeton University Princeton, N.J. 08544 The last decade has seen an extraordinary growth in the development and use of groundwater trans-
port modeling. Whereas models in the 1960s were devoted almost exclusively to problems associated with groundwater supply, current modeling efforts are very often motivated by a desire to simulate subsurface contaminant movement. Moreover, advanced computer technology has helped the groundwater model evolve
from a scientific curiosity to an important and widely used engineering tool. Groundwater flow Because groundwater contaminant transport is neither readily observed nor easily measured, the lay public views it as something approaching the
FIGURE 1
The saturated and unsaturated zones3 How groundwater moves in these zones . . .
a
Degree of water saturation is indicated by density of shading.
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Environ. S c i . Technol., V o l . 18, No. 4, 1984
. . . and how it moves in the presence of a vertical flow component (Ah = h2—hi)
FIGURE 2
Species transport
(a) Convection of afluid with a constant concentration source of Co: front moves a distance x, - v,t where tisefapsed time
metaphysical. Yet, because of the enormous impact this phenomenon has on the long-term viability of po table water supplies, contaminant transport is of tremendous scientific and practical importance. Groundwater can be regarded as water occupying subterranean intergranular void spaces. Near the land surface, this water normally shares the available void space with air; this region is called the unsaturated zone (Figure 1). With depth, the propor tion of water to air increases until the pore space is entirely saturated with water. While the water pressure in the unsaturated zone is normally less than atmospheric, in the saturated zone, it is generally greater than at mospheric. The level that represents the depth at which atmospheric pres sure is encountered is designated as the water table, as shown in Figure 1. The elevation (h) of the water table is, in a sense, a measure of the fluid potential, and is represented by the height to which water will rise in a well that penetrates the saturated zone. Because groundwater velocity in that zone is proportional to the fluid potential gradient, the direction of the maximum slope of the water table generally coincides with the direction of t h e h o r i z o n t a l c o m p o n e n t of groundwater flow. The proportionality constant in this relationship is the ratio of the hydrau lic conductivity of the groundwater reservoir to its porosity. The hydrau lic conductivity (Ky) is a measure of the friction losses incurred by ground water in transit, while the porosity (φ) is a measure of the intergranular void
(c) Adsorption is added to convection anô dispersion; the front is retarded and dispersion is iess effective
(b) Same as {a), but with dispersion added: note changes in the front
space. Combining these terms, we ob tain a relationship describing flow in the areal plane, or Darcy's law
where (χι, X2, X3) are principal axes and dh/dxj can be thought of as the slope of the water table in the horizon tal Xi directions. Summation is as sumed over repeated subscripts. Although the concept of horizontal groundwater flow is easily understood when the water table is viewed as a potential surface, the vertical flow component requires additional expla nation. The vertical groundwater flow velocity is also proportional to the flu id potential gradient. In fact, Equa tion l holds in this case as well; one simply allows the indexes i and j to range over the interval 1-3. Now, however, one must discard the water table concept and think of h as the hydraulic head. The hydraulic head is readily measured as the eleva tion of the fluid level in a well which admits groundwater only over a small vertical increment. This is illustrated in Figure l. Thus, the vertical gradi ent is established in the field by locat ing two such wells very near to each other in such a manner that they have access to the groundwater reservoir at two different vertical increments sep arated by a distance ΔΧ3. One then observes the change in hydraulic head Ah over this interval, i.e., K33 dh _ V3 _
K 33 Ah
— ^
φ
θΧ3
^ )
φ
Δχ3
In addition to the momentum bal
ance relationship expressed by Equa tions l and 2, a complete mathemat ical description of groundwater flow requires a mass conservation princi ple. For the case of saturated ground water of density p, mass conservation is given by:
£ F~ (*νϋ°) - °· at (ΦΡ) +dxi i = 1,2, 3
(3)
where t is time. Although the first term in Equation 3 could be impor tant for problems of water supply in which reservoir storage effects can be significant, this term can be safely neglected for most problems involving contaminant transport. The reason for this is the dramatic difference be tween the rate of propagation of a pressure front, which is described by the first term in Equation 3 and ex pressed in terms of ft/s, and the fluid velocity which is on the order of ft/d. Neglecting this term and combining Equations 1 - 3 , we obtain the classical equation of groundwater flow:
έΝΗ i=l,2,3
(4)
y >
Solving Equation 4 requires the ad ditional specification of boundary conditions expressed in terms of the dependent variable h. This informa tion, along with the parameters Kij and , is obtained from field investi gations. Once h is obtained through the solution of Equation 4 for the en tire region of interest, one can readily Environ. Sci. Technol., Vol. 18, No. 4, 1984
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FIGURE 3
Defining discrete elements
Triangular finite-element net
Finite-difference net
Collocation finite-element net
Boundary element net
Note: Each node represents one equation per independent variable, except in the case of collocation, in which each collocation point represents one equation The boundary element, collocation, and finite-element methods offer flexibility in geometric representation.
return to Equation 1 or 2 to retrieve the fluid velocity. Solute transport Although knowledge of the average velocity of fluids through pores Vj is necessary for the description of contaminant movement, it is not sufficient. The average pore velocity Vj does not account for the small-scale, rather tortuous pore-level behavior of the fluids. This highly complicated porelevel fluid flow pattern is important in contaminant transport because it tends to spread the solute in the fluid while the solute is being convected through the porous medium. The mass flux J„i is, therefore, given in terms of the mass concentration of a species a per unit volume of the fluid phase, c„, as J«i = ViCa - Dij —- c„
(5)
where Dy is the dispersion coefficient and the gradient of the fluid density is assumed to be small. The first term on the right side of Equation 5 describes the convection of species a, and the second defines the dispersion. Molec110A
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ular diffusion is incorporated into the functional form of D;J. The conservation of mass for species a is given as: jr ( « c j + / - (4>Saù = - fa at dx,
(6)
coefficient, which describes the adsorption of species on the solid phase. The commonly used transport equation is the combination of Equations 5-8: — [φοα+ at
(1 - 0 ) p s K D „ c „ ] + Τ - (V|C„)
where f„ is the transfer of species from the liquid to the solid phase. An equation analogous to Equation 6 can also be written for the solid phase, that is, the soil grains and rock matrix; by expressing the concentration of species on the solid phase as c?„ we obtain | [ 0 -)
