Stochastic finite element method for probabilistic analysis of flow and transport in a three‐dimensional heterogeneous porous formation

Abstract

A probabilistic study is attempted to analyze the flow and transport in a three‐dimensional (3‐D) porous formation where the governing parameters are varying randomly in space. It is assumed that the soil parameters, namely, hydraulic conductivity, dispersivity, molecular diffusion, porosity, sorption coefficient, and decay rate, are random fields. A stochastic finite element method (SFEM), which is based on perturbation technique, is developed. The method developed here uses an alternate approach for obtaining improved computational efficiency. The derivatives of the concentration with respect to random parameters are obtained by using the derivatives of local matrices instead of global matrices. This approach increases the computational efficiency of the present method by several orders with respect to standard SFEM. Both accuracy and computational efficiency of this method are compared with that of commonly used Monte Carlo simulation method (MCSM). It is observed that for moderate values of coefficient of variations of the random parameters the mean and standard deviation match reasonably well with MCSM results. Using this method the excessive computational effort required by MCSM can be avoided. In the present study both 1‐D as well as 3‐D problems are solved to show the advantages of SFEM over MCSM. The correlation scale of the random field is found to be an important parameter. For the range of this parameter studied here it is found that as correlation scale increases, the standard deviation increases. The results obtained for two particular problems in this study show that the coefficient of variation of concentration is higher for the 1‐D problem than the 3‐D problem.