Applied numerical modeling of saturated / unsaturated flow and ...

Table 3
Calculation of contaminant plume lengths for first order and Michaelis–Menten degradation kinetics
Method Formula Equation
1 L1 ¼ Dx ¼ − va
k lnðCðxÞ=C0Þ (7)
2
lnðCðxÞ=C0Þ
L2 ¼ Dx ¼ 2aL
1−ð1 þ 4kaL=vaÞ 0:5
( " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#
)
(8)
3, 4 L3;4 ¼ Dxi; for CðxÞ
5 LMM ¼ Dx ¼ va
degradation of a contaminant is not limited by electron acceptor availability and the microbial
density is assumed to be constant with time, Eq. (6) applies (Simkins and Alexander, 1984):
dC
dt
C
¼ −kmax
C þ MC
C. Beyer et al. / Journal of Contaminant Hydrology 87 (2006) 73–95
where kmax is the maximum degradation rate [M L − 3 T − 1 ] and MC is the MM half-saturation
concentration [ML − 3 ]. This approximation may be applicable when aquifer sediments have been
exposed to contaminants for several years (Bekins et al., 1998). The integral form of Eq. (6) can be
rearranged to yield equation (5) of Table 2. According to Robinson (1985), equation (5) is the most
reliable of several different formulations of the integrated MM model for estimation of kmax and
MC. Both parameters are estimated by a linear least squares fit of Δx/[va(C0−C(x))] vs. ln(C0/C
(x))/(C0−C(x)). Thus kmax is obtained as the reciprocal of the intercept of the linear function and
MC as its slope multiplied by kmax. Application of method 5 assumes advective transport only (see
also Parlange et al., 1984). As Robinson (1985) points out, application of a linearized integrated
MM model for least squares estimation of its parameters may be problematic, because measured
concentrations C(x) appear in the dependent as well as in the independent variable. A preliminary
examination of several nonlinear least squares approaches for fitting the MM parameters to the
concentration vs. distance data showed that the parameters determined by method 5 were on
average more accurate than those obtained by other methods.
4.2. Estimation of contaminant plume lengths
C0
kmax
−exp Dxi
2aL
1−
1 þ 4kaL
va
Given a first order degradation rate constant λ (by estimation with one of the four center line
methods presented above), equations (1)–(3) (Table 2) can be rearranged to calculate the length of
the steady state contaminant plume (see Table 3). The plume length here is defined as the largest
distance between the source and the concentration isoline for concentration CPL [ML − 3 ]. Equation
(7) gives this distance for the purely advective case of method 1, equations (8) and (9) correspond
with methods 2 and 3, respectively. As the rate constant estimated with method 4 is corrected for
dispersion, this process has to be accounted for when the plume length is calculated. Therefore
equation (9) is also used for calculation of plume lengths based on λ4. A steady state plume length
can also be calculated using the MM parameters estimated using method 5. Rearrangement of
equation (5) of Table 2 yields equation (10) and gives the distance L MM at which concentrations fall
erf
WS
4 ffiffiffiffiffiffiffiffiffiffiffi p ¼ 0 (9)
aTDxi
MCln C0
CðxÞ þ C0−CðxÞ (10)
79
ð6Þ