Concentration statistics for mixing-controlled reactive transport in random heterogeneous media.

Uncertainty in the distribution of hydraulic parameters leads to uncertainty in flow and reactive transport. Traditional stochastic analysis of solute transport in heterogeneous media has focused on the ensemble mean of conservative-tracer concentration. Studies in the past years have shown that the mean concentration often is associated with a high variance. Because the range of possible concentration values is bounded, a high variance implies high probability weights on the extreme values. In certain cases of mixing-controlled reactive transport, concentrations of conservative tracers, denoted mixing ratios, can be mapped to those of constituents that react with each other upon mixing. This facilitates mapping entire statistical distributions from mixing ratios to reactive-constituent concentrations. In perturbative approximations, only the mean and variance of the mixing-ratio distribution are used. We demonstrate that the second-order perturbative approximation leads to erroneous or even physically impossible estimates of mean reactive-constituent concentrations when the variance of the mixing ratio is high and the relationship between the mixing ratio and the reactive-constituent concentrations strongly deviates from a quadratic function. The latter might be the case in biokinetic reactions or in equilibrium reactions with small equilibrium constant in comparison to the range of reactive-constituent concentrations. When only the mean and variance of the mixing ratio is known, we recommend assuming a distribution that meets the known bounds of the mixing ratio, such as the beta distribution, and mapping the assumed distribution of the mixing ratio to the distributions of the reactive constituents.

Stochastic evaluation of mass discharge from pointlike concentration measurements.

The contaminant mass discharge crossing a control plane is an important metric in the assessment of natural attenuation at contaminated sites. For risk-assessment purposes, the mass discharge must be estimated together with a level of uncertainty. We present a conditional Monte Carlo approach that allows estimating the statistical distribution of mass discharge. The approach is based on conditioning multiple realizations of the hydraulic conductivity field on all data available. We jointly determine a first-order decay coefficient in each realization, leading to conditional statistical distributions of all estimated parameters and the total mass discharge. The resulting statistical distribution of contaminant mass discharges can be used in the assessment of risks at the contaminated site. The method is applied to data of hypothetical test cases, which gives the opportunity to compare estimation results to the true field. As concentration data, we account for pointlike measurements obtained in multi-level sampling wells. The obtained empirical distribution of mass discharge crossing the multi-level sampling fence could be well fitted by a log-normal distribution.

Interpolation of steady-state concentration data by inverse modeling.

In most groundwater applications, measurements of concentration are limited in number and sparsely distributed within the domain of interest. Therefore, interpolation techniques are needed to obtain most likely values of concentration at locations where no measurements are available. For further processing, for example, in environmental risk analysis, interpolated values should be given with uncertainty bounds, so that a geostatistical framework is preferable. Linear interpolation of steady-state concentration measurements is problematic because the dependence of concentration on the primary uncertain material property, the hydraulic conductivity field, is highly nonlinear, suggesting that the statistical interrelationship between concentration values at different points is also nonlinear. We suggest interpolating steady-state concentration measurements by conditioning an ensemble of the underlying log-conductivity field on the available hydrological data in a conditional Monte Carlo approach. Flow and transport simulations for each conditional conductivity field must meet the measurements within their given uncertainty. The ensemble of transport simulations based on the conditional log-conductivity fields yields conditional statistical distributions of concentration at points between observation points. This method implicitly meets physical bounds of concentration values and non-Gaussianity of their statistical distributions and obeys the nonlinearity of the underlying processes. We validate our method by artificial test cases and compare the results to kriging estimates assuming different conditional statistical distributions of concentration. Assuming a beta distribution in kriging leads to estimates of concentration with zero probability of concentrations below zero or above the maximal possible value; however, the concentrations are not forced to meet the advection-dispersion equation.