Stochastic modeling of complex nonstationary groundwater systems Shu-Guang Li a, * , Hua-Sheng Liao b , Chuen-Fa Ni a a Department of Civil and Environmental Engineering, Michigan State University, East Lansing, MI 48824, USA b State Key Laboratory of High Speed Flows, Sichuan University Chengdu, Sichuan, China Received 22 December 2003; received in revised form 22 July 2004; accepted 5 August 2004 Abstract Despitetheintensiveresearchoverthepasttwodecadesinthefieldofstochasticsubsurfacehydrology,asubstantialgapremains between theory and application. The most popular stochastic theories are still based on closed-form solutions that apply, strictly speaking, only for statistically uniform flows. In this paper, we present an efficient, nonstationary spectral approach for modeling complex stochastic flows in moderately heterogeneous media. Specifically, we reformulate the governing stochastic equations and introduce a new transfer function to characterize the propagation of system uncertainty. The new transfer function plays a similar role as the commonly used GreenÕs functions in classical stochastic perturbation methods but is more amenable to numerical solu- tion. The compact transfer function can be used to construct efficiently various spatial statistics of interest, such as covariances, cross-covariances, variances, and mean closure fluxes. We demonstrate the advantages of the proposed approach by applying it toanumberofnonstationaryproblems,includingalarge,complexproblemthatisdifficulttosolvebytraditionalmethods.Inpar- ticular,wefocusinthispaperondemonstratingthenewapproachÕsabilitytocomputeefficientlycovariancesandcross-covariances critical for measurement conditioning, monitoring network analyses, and stochastic transport modeling in the presence of complex mean flow nonstationarities (caused, e.g., by complex trends in aquifer properties, boundary conditions, and sources and sinks). This paper is an extension of our recent work that illustrated the basic approach for modeling nonlocal and nonstationary scale effects and uncertainty propagation in relatively simple situations. Ó 2004 Elsevier Ltd. All rights reserved. 1. Introduction Despite the intensive research over the past two dec- ades in the field of stochastic subsurface hydrology, a substantial gap remains between theory and applica- tions. The most popular stochastic theories are still based on closed-form solutions that apply, strictly speaking, only for statistically uniform flows [8,4]. Sev- eral researchers have recently been investigating general nonstationary flows in heterogeneous media in order to make stochastic modeling practical [30,16,17,39,13,20, 18,5,44,45]. These researchers stressed that stochastic theories must be made much more flexible and efficient before they can be routinely applied to site-specific situ- ations. In particular, Li et al. stressed [23,19] that many recently developed numerical perturbation techniques areverydifficulttoimplementand,contrarytocommon expectation, may actually be more expensive than the classical Monte Carlo approach for problems of mean- ingful sizes. This is so because these techniques still re- quire resolving small-scale heterogeneity and involve computing one or more huge ‘‘nasty’’ transfer function matrices relating the input and output fluctuations (e.g., the correlation functions, GreenÕs functions, or sensitivity derivatives). More quantitatively, Li et al. re- cently estimated that classical grid-based perturbation methods require O(N 2 ) words of memory and between O(N 2 ) and O(N 3 ) floating point operations, where N =O(L/k) d , L being the characteristic problem size, k the scale of heterogeneity, and d the problem dimension 0309-1708/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2004.08.002 * Corresponding author. Tel.: +1 517 432 1929. E-mail address: lishug@egr.msu.edu (S.-G. Li). Advances in Water Resources 27 (2004) 1087–1104 www.elsevier.com/locate/advwatres