Stochastic groundwater models in practice

Abstract

Question 1: Why have there not been many real-world applications of stochastic theories and approaches, despite the significant progress in developing such rigorous theories and approaches for studying fluid flow and solute transport in heterogeneous media?Among the reasons for this state of affairs I count the following:There is a perception among groundwater hydrologists that real-world applications of stochastic concepts and models require numerous reliable measurements of spatially-variable hydrogeologic quantities, most notably permeability, on relatively small scales in three dimensions. The collection of such data is considered to be expensive and time consuming. Uniform equivalent values of aquifer parameters (e.g. transmissivity, storativity) are routinely estimated on the basis of pumping test data by means of simple graphical or computational techniques. There are presently no analogous methods to estimate geostatistical aquifer parameters (such as mean, integral scale and variance of log transmissivity); the feasibility of developing such methods has not been explored until quite recently (e.g., Neuman et al. 2004) and remains to be demonstrated on real data. The future will bring about a major improvement in our ability to image the subsurface using a combination of geophysical, hydrogeologic and other related techniques. A recent development has been the use of three-dimensional geostatistical inversion to assess the spatial variability of parameters, such as permeability and porosity, on the basis of multiple cross-hole pressure interference tests (Vesellinov et al.

Quantifying flow and remediation zone uncertainties for partially opened wells in heterogeneous aquifers

“…Discussions regarding to the limitations of MCS have been made in many previous studies (e.g. Neuman, 1999, Kunstmann andKinzelbach, 2000;Zhang, 2002, Feyen et al, 2003aBallio and Guadagnini, 2004;Dagan, 2004;Neuman, 2004;Li et al, 2003, 2004a, b, Ni and Li, 2005.…”

An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems

“…Direct calculation shows the estimator (60) for the mean η̄ converges subject to the estimate (61) where σ is the standard deviation of η. As the mean η̄ is unknown, convergence may be conveniently judged by examining the fluctuation sizes between successive estimates; the fluctuation of ηN with lag k is given by (62) where σ 2 is the variance of η, and the log-fluctuation of the standard deviation is therefore (63) when N ≫ k.…”

“…Certain pitfalls must be avoided when using MC methods: for example, the pseudorandom variates used for simulation can fail in important ways [53]. These methods are not easily conditionalized [61], and their slow convergence has led to interest in acceleration techniques [13,30]. Transformation of the underlying probability space using importance sampling was suggested in [49], using a spatially constant random conductivity field with one-dimensional flow, but to the best of our knowledge it has not been considered for nontrivial stochastic flow problems.…”

Effects of uncertainty of lithofacies, conductivity and porosity distributions on stochastic interpretations of a field scale tracer test

“…Proper depiction of the spatial variability of hydraulic attributes (e.g., hydraulic conductivity and porosity) on the basis of commonly available information at scales which are relevant for groundwater flow and transport modeling is a key issue in stochastic hydrogeology (e.g., Dagan 2004;Neuman 2004). Gégo et al (2001) showed that stochastic simulations with uniform conductivity assigned to different (randomly distributed) litho-facies do not necessarily provide a good representation of the uncertainty associated with the system.…”