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Articles written by Zhang, D.

  1. Zhang, D. and Lu, Z.. "An efficient, high-order perturbation approach for flow in random porous media via Karhunen-Lo\`eve and polynomial expansions" Journal of Computational Physics. 194 (2). 2004. pp. 773--794.

    In this study, we attempt to obtain higher-order solutions of the means and (co)variances of hydraulic head for saturated flow in randomly heterogeneous porous media on the basis of the combination of Karhunen-Loève decomposition, polynomial expansion, and perturbation methods. We first decompose the log hydraulic conductivity Y = ln Ks as an infinite series on the basis of a set of orthogonal Gaussian standard random variables ξi. The coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of log hydraulic conductivity. We then write head as an infinite series whose terms h(n) represent head of nth order in terms of σY, the standard deviation of Y, and derive a set of recursive equations for h(n). We then decompose h(n) with polynomial expansions in terms of the products of n Gaussian random variables. The coefficients in these series are determined by substituting decompositions of Y and h(n) into those recursive equations. We solve the mean head up to fourth-order in σY and the head variances up to third-order in σY2. We conduct Monte Carlo (MC) simulation and compare MC results against approximations of different orders from the moment-equation approach based on Karhunen-Loève decomposition (KLME). We also explore the validity of the KLME approach for different degrees of medium variability and various correlation scales. It is evident that the KLME approach with higher-order corrections is superior to the conventional first-order approximations and is computationally more efficient than the Monte Carlo simulation.


  2. Li, H. and Zhang, D.. "Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods" Water Resources Research. 43 (9). SEP 2007.

    An efficient method for uncertainty analysis of flow in random porous media is explored in this study, on the basis of combination of Karhunen-Loeve expansion and probabilistic collocation method (PCM). The random log transformed hydraulic conductivity field is represented by the Karhunen-Loeve expansion and the hydraulic head is expressed by the polynomial chaos expansion. Probabilistic collocation method is used to determine the coefficients of the polynomial chaos expansion by solving for the hydraulic head fields for different sets of collocation points. The procedure is straightforward and analogous to the Monte Carlo method, but the number of simulations required in PCM is significantly reduced. Steady state flows in saturated random porous media are simulated with the probabilistic collocation method, and comparisons are made with other stochastic methods: Monte Carlo method, the traditional polynomial chaos expansion (PCE) approach based on Galerkin scheme, and the moment-equation approach based on Karhunen-Loeve expansion (KLME). This study reveals that PCM and KLME are more efficient than the Galerkin PCE approach. While the computational efforts are greatly reduced compared to the direct sampling Monte Carlo method, the PCM and KLME approaches are able to accurately estimate the statistical moments and probability density function of the hydraulic head.


  3. Ding, Y., Li, T., Zhang, D., and Zhang, P.. "Adaptive Stroud Stochastic Collocation Method for Flow in Random Porous Media via Karhunen-Loeve Expansion" Communications in Comp. Phys.. 4 (1). 2008. pp. 102--123.

    In this paper we develop a Stochastic Collocation Method (SCM) for flow in randomly heterogeneous porous media. At first, the Karhunen-Lo\`eve expansion is taken to decompose the log transformed hydraulic conductivity field, which leads to a stochastic PDE that only depends on a finite number of i.i.d. Gaussian random variables. Based on the eigenvalue decay property and a rough error estimate of Stroud cubature in SCM, we propose to subdivide the leading dimensions in the integration space for random variables to increase the accuracy. We refer to this approach as \it adaptive Stroud SCM. One- and two-dimensional steady-state single phase flow examples are simulated with the new method, and comparisons are made with other stochastic methods, namely, the Monte Carlo method, the tensor product SCM, and the quasi-Monte Carlo SCM. The results indicate that the adaptive Stroud SCM is more efficient and the statistical moments of the hydraulic head can be more accurately estimated.