1. INTRODUCTION
1.3. The word learning challenge
In this section, we present some solutions to saturated flow and transport problems that are encountered in the practice of subsurface hydrology. We begin with well hydraulics, an understanding of which is important to the design of pump-and-treat systems, and dis-cuss several models of transport of soluble plumes.
4.3.1 Flow to a Single Well
Darcy's law describes the average water flux through a porous medium when the local hydraulic gradient is known. To determine discharges we use the law of conservation of mass for the water, also known as the continuity equation. For the analysis of wells, it is assumed that the flow toward the well caused by the well pumping is radially symmetric.
In this situation, it is convenient to use the equation of continuity in radial coordinates (r, θ). For steady-state state flow toward a single well in a confined aquifer without recharge, the equation reads
Q 2πKB冢ddhr冣 (4.36)
where K and B aquifer conductivity and thickness, respectively. The equation of conti-nuity for well flow in an unconfined aquifer is similar to Eq. (4.19): namely,
Q 2πrKh冢ddhr冣 (4.37)
with B replaced by the variable flow depth h. Implicit in Eq. (4.36) and (4.37) is the Dupuit-Forchheimer assumption, the implication of which is that the flow in the aquifer can be assumed to be practically horizontal.
Equations (4.36) and (4.37) are used to obtain solutions for the steady-state discharge to a well in confined and unconfined aquifers, respectively, if the piezometric heads (con-fined aquifers) or water-table elevations (uncon(con-fined aquifers) h1and h2are known at two radial distances r1and r2, respectively:
Q 2πKB(h2 h1)/ln(r2/r1) 2πT(h2 h1)/ln(r2/r1) (4.38) where T KB aquifer transmissivity, and
Q πKD(h2 2 h1
2)/ln(r2/r1) (4.39)
The solutions given in Eqs. (4.38) and (4.39) assume that the well penetrates to the impermeable bottom of the aquifer and that there is no recharge into the aquifer. They also assume an infinitely large aquifer with no interaction with surface streams or imperme-able boundaries. Well solutions are often given in terms of the drawdown s as a function of the radial distance r. The drawdown is defined as
s H h (4.40)
where H is the elevation of the original piezometric surface before pumping at the well starts. The distance R for which h H and s 0 is called the radius of influence of the well. Using the concepts of drawdown and radius of influence, the steady-state flow equa-tion in a confined aquifer can be rewritten as
s 2 Q πT Ln
R r
(4.41)
By combining the solutions for the steady state flow in confined and unconfined aquifers, one can derive a relationship between the drawdown calculated from confined conditions (assuming constant in space and time aquifer thickness equal to H), and that estimated for unconfined conditions (when the change in aquifer thickness caused by pumping is taken into account):
sUNC H 兹H苶2苶苶 2苶s苶CON苶F苶H (4.42) Using this formula, known as Jacob’s correction, one can initially assume constant aquifer thickness B H in calculations and use “confined” aquifer solutions to calculate drawdown, then correct the drawdown using Jacob’s correction. This approach is particu-larly useful when dealing with transient flow. For transient flow conditions in an aquifer with constant flow thickness, the transient drawdown is given by
s (r,t) 4 Q
πT兰u∞exp[x dx x] 4QπT W [u(r,t)] (4.43) where
u(r,t) 4 r2 T
S
t (4.44)
and S storativity (for confined aquifers) or porosity (for unconfined aquifers) and W(u) is known in subsurface hydrology as well function and in mathematics as exponential inte-gral. This function is tabulated in almost every groundwater hydrology textbook. It also is available in many engineering mathematics software packages, such as Mathematica©, as a library function.
4.3.2 Superposition and Convolution
For a time-variable pumping rate, the principle of convolution can be used to estimate the transient drawdown. This approach is strictly valid for linear systems: i.e., systems in which the response (drawdown) is a linear function of the excitement (pumping rate). The linearity assumption is strictly valid for confined aquifers only; however, as long as the drawdowns do not exceed 20% of the initial aquifer thickness, it also may be used for unconfined aquifers. Using the convolution approach, the transient drawdown for a pump-ing rate changpump-ing in a step-wise fashion is given by
s(r,t) 4π1
Tk
冘
1n(QK QK 1)W(u(r,∆tK 1)) (4.45) where the drawdown is estimated at time t, tn t tn 1, QKis the pumping rate for tK 1 t tK,tO 0, Q0 0.0, and ∆tK 1 t – tK 1. When several wells are present, the superposition approach is used to estimate the cumulative drawdown by adding the drawdown contributions from all the wells:
s(t) 4π1
TL
冘
m 1QLW(u(rL,t)) (4.46)
where rLis the distance between the point of interest (where the drawdown is estimated) and well L. When several wells are pumping at variable rates, the superposition and convolution approaches are used simultaneously. The superposition principle also can be used to super-impose the drawdown on the natural (ambient) flow conditions. Using this principle leads to
h(x, y, t) H(x, y, t) s(x, y, t) (4.47) where h transient potentiometric surface that combines ambient conditions and well impact, H potentiometric surface under natural (ambient) conditions, and s transient drawdown.
4.3.3 Interception Wells
With respect to contaminant transport in the subsurface, interception wells are used to trap the contaminant plume within the well flow field. It is assumed in this analysis that there is an ambient steady-state uniform flow through the aquifer. The combination of well-related flow and ambient uniform flow satisfies the conditions of two-dimensional poten-tial flow in a horizontal plane, where the discharge described by stream function y is relat-ed to potential φ, or the piezometric head h. The extent of the aquifer through which water travels to the well and is captured by it is called the capture zone. The derivation of the analytical solution for steady-state flow capture-zone uses the following assumptions:
(1) ahomogeneous, isotropic, infinitely large aquifer, (2) uniform flow, (3) no leakage, (4) aquifer storativity or specific yield neglected, (not relevant for steady-state analysis), (5) hydrodynamic dispersion neglected, (6) the Dupuit assumption applies, and (7) the well is fully penetrating and pumping at a constant rate. Three important parameters are used in delineating the capture zone: namely, the stagnation point, the upgradient maxi-mum width of the capture zone, and the equation for the capture zone boundary.
For a confined aquifer, the distance from the well to the stagnation point (measured in the direction of the uniform flow) is
xSTAG 2 Q
πTwI (4.48)
where Qw well discharge, T aquifer transmissivity KB (K aquifer permeability, B aquifer depth), and I natural hydraulic gradient: i.e., the gradient responsible for the ambient steady-state uniform flow in the aquifer. The upgradient divide, defined by the maximum width of the capture zone far upgradient of the well, for the confined aquifer is given by
wDIV Q TI
w (4.49)
and the equation of the dividing streamline is
x (4.50)
The procedure for delineating the capture zone consists of the following steps: (1) esti-mate the location of the stagnation point (xSTAG, 0), (2) estimate the maximum width of the capture zone wDIV, and (3) vary y between zero and wDIV/2 and use the capture zone bound-ary to estimate the boundbound-ary location (x,y).
4.3.4 Partially Penetrating Wells
Performance of wells that penetrate only partially through the bearing strata is discussed in this section. The simplest case consists of a well that is barely penetrating into an
semi-y
tan冢π 2πQTwIy冣
infinite porous medium so that the aquifer flow is three-dimensional and spherically sym-metric. In this case, the following relationship applies between the flow into the partially penetrating well Qpand the flow to a fully penetrating one Q:
Q
where B is the aquifer thickness, rwis the well radius, and R is the radius of influence of the partially penetrating well. Because, in general, rw B, then the equation above indi-cates that the spherical flow to a partially penetrating well is highly inefficient compared with simple radial flow: i.e., for the same drawdown in the well, it results in a significantly smaller pumping rate.
In the general case of partial penetration, one may consider the total drawdown sT, which consists of the drawdown equivalent to that of a fully penetrating well s and addi-tional head loss because of the partial penetration of the well ∆s:
sT s ∆s (4.51)
Additional head loss for a well penetrating from the top (or the bottom) of the aquifer is estimated as follows: thickness (Fig. 4.2). For the well centrally positioned in the aquifer, the following formu-la is used:
Thus, when the pumping rate is defined for a well, we calculate the drawdown correc-tion∆s and add it to the full penetration drawdown s. However, when the drawdown is given for a well, we have to recalculate the pumping rate. In this case, the true pumping
FIGURE 4.2 Partially-penetrating well.
rate is given by
where s drawdown defined at the well, Q pumping rate estimated using s, and Qp actual pumping rate
4.3.5 Well Duplets
Well duplets, each of which consists of one pumping and one recharge well, are frequent-ly used as a means of injecting and removing aquifer mitigation solutes, such as co-sol-vents, surfactants, or both. Typically, the recharge well is positioned directly upgradient from the discharge well, and the magnitudes of pumping and injection rates are the same.
In this case, the two wells form a flow circulation cell: i.e., all the injected water is pumped out by the discharge well. In the case of co-solvent flushing, it is important to understand what region of the aquifer is subject to the mitigation: i.e., what the boundary is of the circulation cell. This boundary is defined by the upgradient and downgradient stagnation points (xSTAG, 0) and (xSTAG, 0) and the cell boundary equation. For the x-axis parallel to the direction of ambient flow and the origin of the coordinate system located at the mid-point between the two wells, the two stagnation points, (xSTAG, 0) and (xSTAG, 0) are given as the roots of the quadratic equation
q
where qo ambient flux, 2d distance between the wells, and Qw pumping/injection rate. The boundary of the circulation cell is defined by
q The circulation cell is symmetric with respect to the y axis. The cell delineation pro-cedure consists of estimating the locations of the stagnation points and varying x between zero and xSTAGand using the cell-boundary equation to solve for y. This implicit equation can be solved by any calculation software, such as Mathematica© or MS Excel.
4.3.6 Transport Equations
The following general form of mass transport equation in the saturated zone is derived assum-ing one-dimensional advective and three-dimensional diffusive-dispersive transport in the aqueous phase, linear partitioning of a compound between the three phases (water-soil-(D)NAPL), and first-order degradation in the aqueous phase. For this conditions, we have
R ∂∂
x i冢冢αiv 冣∂∂CxWi冣 v∂∂Cx λC1W W (4.58)
where CW aqueous phase concentration, v pore-water velocity, and θW volumetric moisture content, and αi longitudinal, transverse horizontal, and transverse vertical
θDW
∂CW
macrodispersivities, and D Millington-Quirk dispersion coefficient, and λ first-order degradation rate in the aqueous phase, and R retardation factor,
R 1 kS
where kSOIL water-soil partitioning coefficient, kSOIL S/CW, S weight/weight con-centration of absorbed compound in soil, kNAPL NAPL – water partitioning coefficient, kNAPL CNAPL/CW, CNAPL concentration of compound in the NAPL phase, γB bulk den-sity of dry soil, and θNAPL volumetric NAPL content.
Assuming that the diffusive fluxes are negligible compared the macrodispersive flux-es, the transport equation can be simplified to yield
∂C
Closed-form solutions are available for a variety of flow, boundary, and initial conditions.
Van Genuchten and Alves (1982) presented a good summary of these solutions. Some of the most useful solutions are presented below.
4.3.7.1 One-dimensional transport with step change in concentration–no degrada-tion. This simple case has the initial condition C(x,0) 0 for x 0, and it is subject to the following boundary conditions: C(0,t) Co, t 0 and C(∞,t) 0, t 0. The solution of the transport equation for these conditions is given by
C(x,t) C 2
O冢Erfc2(xαxvRvtR)t1/2 expαxxErfc2(xαxvRvtR)t1/2冣 (4.61)
4.3.7.2 One-dimensional transport with step change in concentration and first-order degradation. The initial and boundary conditions are the same as in Sec. 4.7.1.
The solution is given by
C(x,t) C
where Erfc complementary error function.
4.3.7.3 Continuous point injection, 2-D dispersive transport, no retardation, and no degradation. A tracer is continuously injected at a rate Q (per unit depth of the aquifer)
x vR冢1 4λvRRαx冣1/2t
with a concentration Cointo a uniform flow field from a point (x 0, y 0). Let the uniform velocity be vx. The asymtotic solution, i.e., for t→∞, is given by
C(x,y)
where Ko modified Bessel function of the second kind and of 0th order (Bear, 1972).
The time-dependent solution is
W (t, b) leaky well function (see, for example, Hunt, 1983, p. 100).
4.3.7.4 Point slug injection into a uniform flow field—3-D transport and retardation.
In this case, a slug of contaminant of the mass M CoV is injected at point (0,0,0). The transient distribution of concentration is described by
C(x, y, z, t)
4.3.7.5 Continuous injection from a finite-sized source with retardation and degrada-tion. In this case, consider transport from a rectangular source that is perpendicular to the direction of flow. The source width is Y, and its depth below the water table is Z.
The transient concentration distribution in the presence of retardation and degradation is given by
4.4 FLOW AND TRANSPORT IN UNSATURATED