Andrew I. James

Optimal estimation of residual non–aqueous phase liquid saturations using partitioning tracer concentration data

Stochastic methods are applied to the analysis of partitioning and nonpartitioning tracer breakthrough data to obtain optimal estimates of the spatial distribution of subsurface residual non–aqueous phase liquid (NAPL). Uncertainty in the transport of the partitioning tracer is assumed to result from small-scale spatial variations in a steady state velocity field as well as spatial variations in NAPL saturation. In contrast, uncertainty in the transport of the nonpartitioning tracer is assumed to be due solely to the velocity variations. Partial differential equations for the covariances and cross cpvariances between the partitioning tracer temporal moments, nonpartitioning tracer temporal moments, residual NAPL saturation, pore water velocity, and hydraulic conductivity fields are derived assuming steady flow in an infinite domain [Gelhar, 1993] and the advection-dispersion equation for temporal moment transport [Harvey and Gorelick, 1995]. These equations are solved using a finite difference technique. The resulting covariance matrices are incorporated into a conditioning algorithm which provides optimal estimates of the tracer temporal moments, residual NAPL saturation, pore water velocity, and hydraulic conductivity fields given available measurements of any of these random fields. The algorithm was tested on a synthetically generated data set, patterned after the partitioning tracer test conducted at Hill AFB by Annable et al. [1997]. Results show that the algorithm successfully estimates major features of the random NAPL distribution. The performance of the algorithm, as indicated by analysis of the “true” estimation errors, is consistent with the theoretical estimation errors predicted by the conditioning algorithm.

Estimation of spatially variable residual nonaqueous phase liquid saturations in nonuniform flow fields using partitioning tracer data

Estimates of spatially variable residual NAPL saturations SN are obtained in heterogeneous porous media using first temporal moments of breakthrough curves (BTCs) obtained from multilevel samplers during in situ partitioning tracer tests. An approach is adopted in which the distribution of the log NAPL/water volumetric ratio (Y = ln [SN/(1 − SN∥]) and log hydraulic conductivity (F = ln K) are treated as spatially correlated random fields. A nonlinear Gauss-Newton search technique is used to identify the spatial distribution of Y that minimizes the weighted sum of the deviation of the temporal moment predictions from their measured values and the deviation of the estimate of Y from its prior estimate obtained from the temporal moments of extraction well BTCs. Sensitivities required for the algorithm are obtained using a coupled flow and transport adjoint sensitivity method. In addition to obtaining optimal estimates for the spatial distribution of Y, the method also provides the estimation error covariance. The estimation error covariance can be used to evaluate the information that may be obtained from alternate pumping and monitoring configurations for tracer tests designed to detect NAPL in the subsurface. To this end, we tested the method using two different NAPL distributions (one with a random spatially correlated field and a second that was a block of NAPL) and three different pumping configurations (a double five-spot pattern, an inverted double five-spot pattern, and a line-drive pattern). The results show that measured temporal moments are more sensitive to Y in the double five-spot and inverted double five-spot patterns, and estimates produced in these configurations are slightly superior to those produced in the line-drive pattern.

Numerical approximation of head and flux covariances in three dimensions using mixed finite elements

Abstract A numerical method is developed for accurately approximating head and flux covariances and cross-covariances in finite two- and three-dimensional domains using the mixed finite element method. The method is useful for determining head and flux covariances for non-stationary flow fields, for example those induced by injection or extraction wells, impermeable subsurface barriers, or non-stationary hydraulic conductivity fields. Because the numerical approximations to the flux covariances are obtained directly from the solution to the coupled problem rather than having to differentiate head covariances, the approximations are in general more accurate than those obtained from conventional finite difference or finite element methods. Results for uniform flow example problems are consistent with results from previously published finite domain analyses and demonstrate that head variances and covariances are quite sensitive to boundary conditions and the size of the bounded domain. Flux variances and covariances are less sensitive to boundary conditions and domain size. Results comparing approximations from lower-order Raviart–Thomas–Nedelec and higher order Brezzi–Douglas–Marini [9] finite element spaces indicate that higher order element space improve the estimate of the flux covariances, but do not significantly affect the estimate of the head covariances.

A spatially distributed, deterministic approach to modeling Typha domingensis (cattail) in an Everglades wetland

IntroductionThe emergent wetland species Typha domingensis (cattail) is a native Florida Everglades monocotyledonous macrophyte. It has become invasive due to anthropogenic disturbances and is out-competing other vegetation in the region, especially in areas historically dominated by Cladium jamaicense (sawgrass). There is a need for a quantitative, deterministic model in order to accurately simulate the regional-scale cattail dynamics in the Everglades.MethodsThe Regional Simulation Model (RSM), combined with the Transport and Reaction Simulation Engine (TARSE), was adapted to simulate ecology. This provides a framework for user-defineable equations and relationships and enables multiple theories with different levels of complexity to be tested simultaneously. Five models, or levels, of increasing complexity were used to simulate cattail dynamics across Water Conservation Area 2A (WCA2A), which is located just south of Lake Okeechobee, in Florida, USA. These levels of complexity were formulated to correspond with five hypotheses regarding the growth and spread of cattail. The first level of complexity assumed a logistic growth pattern to test whether cattail growth is density dependent. The second level of complexity built on the first and included a Habitat Suitability Index (HSI) factor influenced by water depth to test whether this might be an important factor for cattail expansion. The third level of complexity built on the second and included an HSI factor influenced by soil phosphorus concentration to test whether this is a contributing factor for cattail expansion. The fourth level of complexity built on the third and included an HSI factor influenced by (a level 1–simulated) sawgrass density to determine whether sawgrass density impacted the rate of cattail expansion. The fifth level of complexity built on the fourth and included a feedback mechanism whereby the cattail densities influenced the sawgrass densities to determine the impact of inter-species interactions on the cattail dynamics.ResultsAll the simulation results from the different levels of complexity were compared to observed data for the years 1995 and 2003. Their performance was analyzed using a number of different statistics that each represent a different perspective on the ecological dynamics of the system. These statistics include box-plots, abundance-area curves, Moran’s I, and classified difference. The statistics were summarized using the Nash-Sutcliffe coefficient. The results from all of these comparisons indicate that the more complex level 4 and level 5 models were able to simulate the observed data with a reasonable degree of accuracy.ConclusionsA user-defineable, quantitative, deterministic modeling framework was introduced and tested against various hypotheses. It was determined that the more complex models (levels 4 and 5) were able to adequately simulate the observed patterns of cattail densities within the WCA2A region. These models require testing for uncertainty and sensitivity of their various parameters in order to better understand them but could eventually be used to provide insight for management decisions concerning the WCA2A region and the Everglades in general.