Two-dimensional Steady-state Groundwater Flow Research Articles

A reduced-order model for Monte Carlo simulations of stochastic groundwater flow

We explore the ability of the greedy algorithm to serve as an effective tool for the construction of reduced-order models for the solution of fully saturated groundwater flow in the presence of randomly distributed transmissivities. The use of a reduced model is particularly appealing in the context of numerical Monte Carlo (MC) simulations that are typically performed, e.g., within environmental risk assessment protocols. In this context, model order reduction techniques enable one to construct a surrogate model to reduce the computational burden associated with the solution of the partial differential equation governing the evolution of the system. These techniques approximate the model solution with a linear combination of spatially distributed basis functions calculated from a small set of full model simulations. The number and the spatial behavior of these basis functions determine the computational efficiency of the reduced model and the accuracy of the approximated solution. The greedy algorithm provides a deterministic procedure to select the basis functions and build the reduced-order model. Starting from a single basis function, the algorithm enriches the set of basis functions until the largest error between the full and the reduced model solutions is lower than a predefined tolerance. The comparison between the standard MC and the reduced-order approach is performed through a two-dimensional steady-state groundwater flow scenario in the presence of a uniform (in the mean) hydraulic head gradient. The natural logarithm of the aquifer transmissivity is modeled as a second-order stationary Gaussian random field. The accuracy of the reduced basis model is assessed as a function of the correlation scale and variance of the log-transmissivity. We explore the performance of the reduced model in terms of the number of iterations of the greedy algorithm and selected metrics quantifying the discrepancy between the sample distributions of hydraulic heads computed with the full and the reduced model. Our results show that the reduced model is accurate and is highly efficient in the presence of a small variance and/or a large correlation length of the log-transmissivity field. The flow scenarios associated with large variances and small correlation lengths require an increased number of basis functions to accurately describe the collection of the MC solutions, thus reducing significantly the computational advantages associated with the reduced model.

POD-based Monte Carlo approach for the solution of regional scale groundwater flow driven by randomly distributed recharge

We present a methodology conducive to the application of a Galerkin model order reduction technique, Proper Orthogonal Decomposition (POD), to solve a groundwater flow problem driven by spatially distributed stochastic forcing terms. Typical applications of POD to reducing time-dependent deterministic partial differential equations (PDEs) involve solving the governing PDE at some observation times (termed snapshots), which are then used in the order reduction of the problem. Here, the application of POD to solve the stochastic flow problem relies on selecting the snapshots in the probability space of the random quantity of interest. This allows casting a standard Monte Carlo (MC) solution of the groundwater flow field into a Reduced Order Monte Carlo (ROMC) framework. We explore the robustness of the ROMC methodology by way of a set of numerical examples involving two-dimensional steady-state groundwater flow taking place within an aquifer of uniform hydraulic properties and subject to a randomly distributed recharge. We analyze the impact of (i) the number of snapshots selected from the hydraulic heads probability space, (ii) the associated number of principal components, and (iii) the key geostatistical parameters describing the heterogeneity of the distributed recharge on the performance of the method. We find that our ROMC scheme can improve significantly the computational efficiency of a standard MC framework while keeping the same degree of accuracy in providing the leading statistical moments (i.e. mean and covariance) as well as the sample probability density of the state variable of interest.

Multicomponent transport with coupled geochemical and microbiological reactions: model description and example simulations

A reactive transport code (FEREACT) has been developed to examine the coupled effects of two-dimensional steady-state groundwater flow, equilibrium aqueous speciation reactions, and kinetically-controlled interphase reactions. The model uses an iterative two-step (SIA-1) solution algorithm to incorporate the effects of the geochemical and microbial reaction processes in the governing equation for solute transport in the subsurface. This SIA-1 method improves upon the convergence behavior of the traditional sequential iterative approach (SIA) through the inclusion of an additional first-order term from the Taylor Series expansion of the kinetic reaction rate expressions. The ability of FEREACT to simulate coupled reactive processes was demonstrated by modeling the transport of a radionuclide (cobalt, 60Co 2+) and an organic ligand (ethylenediaminetetraacetate, EDTA 4−) through a column packed with an iron oxide-coated sand. The reaction processes considered in this analysis included equilibrium aqueous speciation reactions and three types of kinetic reactions: adsorption, surface dissolution, and biodegradation.

Uncertainty analysis of groundwater flow with DRBEM

The reliability of groundwater flow numerical simulation is affected by the quality of input data of aquifer physical parameters. This paper presents an uncertainty analysis based on a first order approximation of covariances. A two-dimensional steady state groundwater flow problem is carried out by coupling the first-order uncertainty analysis with dual reciprocity boundary element method.

Optimum allocation of monitoring wells around a solid-waste landfill site using precursor indicators and fuzzy utility functions

An optimum monitoring well network (number of wells and their locations) is proposed which enables rapid, redundant and economical detection of contaminants in groundwater around a solid-waste landfill site. The procedure also guarantees detection of the contaminants given data on the probability of detection at different points in the saturated zone. The well selection is accomplished using a two-step procedure: 1. (1) A Monte Carlo simulation of contaminant transport in the unconsolidated shallow saturated zone is conducted. In this zone hydrogeological parameters are variable but their stochastic distributions are known. Three governing equations are solved numerically using the finite-difference method to obtain the travel time distribution of each contaminant: the two-dimensional steady-state groundwater flow equation; the two-dimensional transient convective-dispersion equation for sorptive contaminants; and the sorptive-desorptive isotherm equation. 2. (2) The procedure utilizes “fuzzy” theory, comprising of a set of newly developed mathematical techniques to deal with uncertainty in a wide range of man-machine interface issues, to assist in the design of a monitoring well network. The procedure requires a mathematical description of a four-attribute design problem using fuzzy utility functions and fuzzy weights. An optimum monitoring well network is then defined as the network having maximum total utility, which is evaluated as a fuzzy expectation of weighted arithmetic sums of the four utilities. One result of the simulation is the definition of relationships between the contaminant of interest and precursor materials. The precursor material can then serve as an “indicator” for faster detection of contaminant leaked from solid-waste landfill site. The procedures are applied to a hypothetical solid-waste landfill site under appropriate conditions to obtain the optimum monitoring well network for detection of precursor indicators. Sensitivity analysis of the optimum network was conducted by considering changes in components of the mathematical description of the design problem. Components which were changed include total utility evaluation, quality of uncertainty in weight factor and utility evaluation, weigth level determination, delay time requirement, well number limitation and perfect detection constraint.