Zhiming Lu | Search USGS

USGS2007

Stochastic uncertainty analysis for solute transport in randomly heterogeneous media using a Karhunen‐Loève‐based moment equation approach

A new approach has been developed for solving solute transport problems in randomly heterogeneous media using the Karhunen‐Loève‐based moment equation (KLME) technique proposed by Zhang and Lu (2004). The KLME approach combines the Karhunen‐Loève decomposition of the underlying random conductivity field and the perturbative and polynomial expansions of dependent variables including the hydraulic head, flow velocity, dispersion coefficient, and solute concentration. The equations obtained in this approach are sequential, and their structure is formulated in the same form as the original governing equations such that any existing simulator, such as Modular Three‐Dimensional Multispecies Transport Model for Simulation of Advection, Dispersion, and Chemical Reactions of Contaminants in Groundwater Systems (MT3DMS), can be directly applied as the solver. Through a series of two‐dimensional examples, the validity of the KLME approach is evaluated against the classical Monte Carlo simulations. Results indicate that under the flow and transport conditions examined in this work, the KLME approach provides an accurate representation of the mean concentration. For the concentration variance, the accuracy of the KLME approach is good when the conductivity variance is 0.5. As the conductivity variance increases up to 1.0, the mismatch on the concentration variance becomes large, although the mean concentration can still be accurately reproduced by the KLME approach. Our results also indicate that when the conductivity variance is relatively large, neglecting the effects of the cross terms between velocity fluctuations and local dispersivities, as done in some previous studies, can produce noticeable errors, and a rigorous treatment of the dispersion terms becomes more appropriate.

Water Resources Research↗

USGS2006

Stochastic uncertainty analysis for unconfined flow systems

A new stochastic approach proposed by Zhang and Lu (2004), called the Karhunen‐Loeve decomposition‐based moment equation (KLME), has been extended to solving nonlinear, unconfined flow problems in randomly heterogeneous aquifers. This approach is on the basis of an innovative combination of Karhunen‐Loeve decomposition, polynomial expansion, and perturbation methods. The random log‐transformed hydraulic conductivity field ( lnK S ) is first expanded into a series in terms of orthogonal Gaussian standard random variables with their coefficients obtained as the eigenvalues and eigenfunctions of the covariance function of lnK S . Next, head h is decomposed as a perturbation expansion series Σ h ( m ) , where h ( m ) represents the m th‐order head term with respect to the standard deviation of lnK S . Then h ( m ) is further expanded into a polynomial series of m products of orthogonal Gaussian standard random variables whose coefficients h i 1 , i 2 ,..., i m ( m ) are deterministic and solved sequentially from low to high expansion orders using MODFLOW‐2000. Finally, the statistics of head and flux are computed using simple algebraic operations on h i 1 , i 2 ,..., i m ( m ) . A series of numerical test results in 2‐D and 3‐D unconfined flow systems indicated that the KLME approach is effective in estimating the mean and (co)variance of both heads and fluxes and requires much less computational effort as compared to the traditional Monte Carlo simulation technique.

Water Resources Research↗