Flow in Random Porous Media

Flow in a porous medium with a random hydraulic conductivity tensor K(x) is analyzed when the mean conductivity tensor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overline K$$ \end{document}(x) is a non-constant function of position x. The results are a non-local expression for the mean flux vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overline q$$ \end{document}(x) in terms of the gradient of the mean hydraulic head \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overline \varphi$$ \end{document}(x), an integrodifferential equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overline \varphi$$ \end{document}(x), and expressions for the two point covariance functions of q(x) and ϕ(x). When K(x) is a Gaussian random function, the joint probability distribution of the functions q(x) and ϕ(x) is determined.