Stochastic flow simulation in 3D porous media

Transport in Porous Media, 1989

This paper presents a numerical method for simulating flow fields in a stochastic porous medium that satisfies locally the Darcy equation, and has each of its hydraulic parameters represented as one realization of a three-dimensional random field. These are generated by using the Turning Bands method. Our ultimate objective is to obtain statistically meaningful solutions in order to check and extend a series of approximate analytical results previously obtained by a spectral perturbation method (L. W. Gelhar and co-workers). We investigate the computational aspects of the problem in relation with stochastic concepts. The difficulty of the numerical problem arises from the random nature of the hydraulic conductivities, which implies that a very large discretized algebraic system must be solved. Indeed, a preliminary evaluation with the aid of scale analysis suggests that, in order to solve meaningful flow problems, the total number of nodes must be of the order of 106. This is due to the requirement that Axi "~)q "~ L~, where Ax~ is the mesh size,)q is a typical correlation scale of the inputs, and L~ is the size of the flow domain (i = 1, 2, 3). The optimum strategy for the solution of such a problem is discussed in relation with supercomputer capabilities. Briefly, the proposed discretization method is the seven-point finite differences scheme, and the proposed solution method is iterative, based on prior approximate factorization of the large coefficient matrix. Preliminary results obtained with grids on the order of one hundred thousand nodes are discussed for the case of steady saturated flow with highly variable, random conductivities.

Water Resources Research, 2005

A probabilistic study is attempted to analyze the flow and transport in a threedimensional (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.

2002

Computer Methods in Applied Mechanics and Engineering, 2008

The aim of this paper is to quantify uncertainty of flow in porous media through stochastic modeling and computation of statistical moments. The governing equations are based on Darcy's law with stochastic permeability. Starting from a specified covariance relationship, the log permeability is decomposed using a truncated Karhunen-Loève expansion. Mixed finite element approximations are used in the spatial domain and collocation at the zeros of tensor product Hermite polynomials is used in the stochastic dimensions. Error analysis is performed and experimentally verified with numerical simulations. Computational results include incompressible and slightly compressible single and two-phase flow.

Physical Review E, 2013

Models of porous media are often applied to relatively small systems, which leads not only to system-size-dependent results, but also to phenomena that would be absent in larger systems. Here we investigate one such finite-size effect: anisotropy of the permeability tensor. We show that a nonzero angle between the external body force and macroscopic flux vector exists in three-dimensional periodic models of sizes commonly used in computer simulations and propose a criterion, based on the system size to the grain size ratio, for this phenomenon to be relevant or negligible. The finite-size anisotropy of the porous matrix induces a pressure gradient perpendicular to the axis of a porous duct and we analyze how this effect scales with the system and grain sizes.

Journal of Physics: Condensed Matter, 2007

International Journal for Numerical Methods in Fluids, 2020

Efficient and robust iterative methods are developed for solving the linear systems of equations arising from stochastic finite element methods for single phase fluid flow in porous media. Permeability is assumed to vary randomly in space according to some given correlation function. In the companion paper, herein referred to as Part 1, permeability was approximated using a truncated Karhunen-Loève expansion (KLE). The stochastic variability of permeability is modelled using lognormal random fields and the truncated KLE is projected onto a polynomial chaos basis. This results in a stochastic nonlinear problem since the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables. Symmetric block Gauss-Seidel used as a preconditioner for CG is shown to be efficient and robust for SFEM.

Mathematical and Computer Modelling, 2003

The group teaches subjects leading to a Bachelor of Applied Computing degree and a computing major in the Bachelor of Commerce and Management. In addition, it contributes computing, statistics and mathematics subjects to a wide range of other Lincoln University degrees. In particular students can take a computing and mathematics major in the BSc. The ACMS group is strongly involved in postgraduate teaching leading to honours, masters and PhD degrees. Research interests are in modelling and simulation, applied statistics, end user computing, computer assisted learning, aspects of computer networking, geometric modelling and visualisation. Every paper appearing in this series has undergone editorial review within the ACMS group. The editorial panel is selected by an editor who is appointed by the Chair of the Applied Management and Computing Division Research Committee. The views expressed in this paper are not necessarily the same as those held by members of the editorial panel. The accuracy of the information presented in this paper is the sole responsibility of the authors. This series is a continuation of the series "Centre for Computing and Biometrics Research Report

We developed a stochastic version of MODFLOW, referred to as MODFLOW-STO, for simulating flow in saturated, randomly heterogeneous porous media. The model is on the basis of an innovative combination of Karhunen-Loéve decomposition, polynomial expansion, and perturbation methods. The log conductivity (lnK) field is first decomposed using the Karhunen-Loéve expansion. The head h is then decomposed with a perturbation expansion as the sum of h (m) , m = 0, 1, …, where h (m) represents the m th head in , the standard deviation of lnK. Term h range of flow conditions against the classical Monte Carlo simulations (MCS). Results indicate that MODFLOW-STO is capable of providing accurate solutions and requires much less computation effort as compared to the MCS analysis. With MODFLOW-STO, subsurface flow uncertainty can be quantified under field conditions in an efficient, effective manner.

Journal of colloid and interface science

Journal of Colloid and Interface Science 229, 323–334 (2000) doi:10.1006/jcis.2000.7055, available online at http://www.idealibrary.com on ... On the Geometry and Topology of 3D Stochastic Porous Media