Analysis of variance method for the equivalent conductivity of rectangular blocks

We use an analysis of variance (ANOVA) decomposition of the log hydraulic conductivity F (x) = ln K (X) as the sum of an average value F¯ plus main effects and interactions of various orders to evaluate the equivalent conductivity of rectangular blocks in D ‐dimensional space. Some of the ANOVA components are dealt with exactly. The effect of the other components is approximated through calibration to the effective conductivity K eff under ergodic conditions. Our analysis applies to both isotropic and anisotropic lognormal K fields. We evaluate the theoretical findings through analytical and numerical comparisons with exact results and previously proposed approximations. We find that the ANOVA formula is nearly unbiased, although its performance is generally inferior to numerical renormalization. Advantages of the ANOVA formula are that it is less computationally demanding and explicitly shows the dependence of the block conductivity K b on the various components of fluctuation of F inside the block. Such dependence is more complex than previously assumed. Considering the ANOVA formula to be unbiased, we derive the bias of the geometric mean of the Cardwell‐Parsons bounds for three‐dimensional rectangular blocks.