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

The approximate solution employed has the same basis as method 3. Here, however, a real two-dimensional approach is
used, as equation (6) is fitted to the concentrations of all observation wells and not only to those measured along the
center line.
In strategy A the first step is an analysis of the concentration distribution based on all measurements made at the
observation wells of the monitoring network. Starting at the well with the highest measured concentration, those
downgradient wells are identified, that best represent the center line of the plume. These then are used to estimate the rate
constant. Only locally measured quantities are used here, i.e. local hydraulic conductivities, hydraulic heads and
contaminant concentrations are “measured” at these wells by reading the model data at the respective nodes of the
numerical grid. The transport velocity va between each pair of center line wells is approximated by:
∆h
va = K ef
(7)
n∆x
with Kef the effective conductivity of local hydraulic conductivities at up and down gradient wells, n the porosity, ∆h the
head difference and ∆x the distance between the wells. According to Rubin (2003) in stationary isotropic twodimensional
domains with gaussian probability density functions of Y = ln(K) the effective conductivity Kef can be
calculated as the geometric mean (cf. Bauer et al. 2006a). The porosity is assumed to be known correctly. With va, λ can
then be calculated for each pair of center line wells using methods A1 – A3. For a set of k center line wells thus k-1 rate
constants are calculated for one method. These are averaged to yield an estimate of the mean degradation rate constant λ.
As an alternative to using only the locally measured conductivities, rate constant estimation is also performed using a
global estimate of Kef, which is obtained from the geometric mean value of all hydraulic conductivities measured at all
observation wells.
In strategy B, method B (equation (6)) is fitted to measured concentrations of all observation wells of the monitoring
network. Both the biodegradation rate constant λ and the source concentration C0 are varied simultaneously to achieve
correspondence of measured and calculated concentrations, as a preliminary analysis (in agreement with results of
Stenback et al. (2004)) showed that this procedure on average yields closer estimates of the true rate constant than fitting
only λ with a single fixed estimate of the source concentration. A least squares criterion for the concentration residuals is
used in the fitting procedure. As in strategy A the flow velocity va is approximated using equation (7). Kef is calculated as
the geometric mean of hydraulic conductivities measured at all wells of the network. The average hydraulic gradient over
the entire site is approximated by fitting a linear trend surface to all head measurements by ordinary least squares
regression.
For methods A2, A3 and B estimates of longitudinal and transverse dispersivities are required. Practical guidance on
estimating these parameters at the field scale is given e.g. by Wiedemeier et al. (1999), where one suggestion is to use 0.1
times the plume length for αL and αT as 0.1 αL. Here, however, an alternative strategy is employed: Macrodispersivities
2
αL and αT are derived from correlation scale, aquifer heterogeneity σ Y and travel distance (Dagan 1984; Hsu 2003;
Rubin et al. 2003). Thus, αL is taken as 7 m, which roughly corresponds to the large time asymptotic limit for the given
conductivity distribution, while αT is taken as the approximate peak value of transverse macrodispersivity, calculated as
0.7 m (which thus also corresponds to the frequently used relationship as αT ≈ 0.1 αL). These values are well within the
ranges of dispersivities commonly used for the field scale modeling of contaminant transport. The true value of the
source width WS is assumed to be known from the site investigation. These approximations and assumptions were made
to ensure that the error introduced in estimated rate constants due to the parameterization of methods A2, A3 and B is as
small as possible.
Results and Discussion
Strategy A - One-dimensional center line approach
The rate constants λA1 - λA3 estimated with equations (3) – (5) are divided by the “true” value used in the numerical
simulations to yield normalized rate constants ΛA1 - ΛA3, which can directly be interpreted as overestimation or
underestimation factors. Results in terms of mean values, medians, standard deviations and coefficients of variation (cv)
as well as the number of realizations (N) used for strategy A are presented in Table 2 and Figure 3. Comparing methods
A1 - A3 for the small source width WS = 4 m yields that all approaches on average result in a distinct overestimation of λ.
For method A1 λ is overestimated on average by a factor of 6.88, while for A2 an mean ΛA2 = 8.24 is observed. Hence,
method A1 performs better than A2. Method A3 yields a slightly lower mean of ΛA3 = 6.82, while the spread of results
for A3 is noticeably larger, as can also be seen by the higher cv. Three main error sources for the observed
overestimation exist:
• neglect of dispersion by method A1, which thus is attributed to the degradation process (λA1 represents a bulk
attenuation rate constant)
6