Effective saturated hydraulic conductivity of two‐dimensional random multifractal fields

Abstract

A means of upscaling the effective saturated hydraulic conductivity, 〈K〉, based on spatial variation in the saturated hydraulic conductivity (K) field is essential for the application of flow and transport models to practical problems. Multifractals are inherently scaling and thus may offer solutions to this dilemma. Random two‐dimensional geometrical multifractal fields (multifractal Sierpinski carpets with a scale factor b = 3) were constructed for iterations i = 1 through 5 using different generator probability values (p). The resulting average mass fractions were normalized and assumed to be directly proportional to K. The objectives were to explore how the frequency distribution of K changes as a function of p and i, how 〈K〉 varies with i for different p values, and how 〈K〉 is related to the generalized dimensions of the multifractal field. Numerical simulations of flow were performed in the multifractal fields, with 〈K〉 computed using Darcy's law. The results showed that 〈K〉 increases with increasing i level and increasing p value. The scaling of 〈K〉 with resolution, 1/bi, followed a power law relationship, similar to that observed for a variety of natural porous media. At the highest resolution (i = 5), ln 〈K〉 was best predicted by the correlation dimension (D2); ln 〈K〉 increased as D2 increased (R2 = 0.991, p < 0.0001). This relationship indicates that 〈K〉 decreases with increasing long‐range spatial correlation among the K values in the field. Furthermore, as 〈K〉 decreases it becomes increasingly dominated by flow channeling. This is because high values of K become more and more clustered as p decreases. This approach may prove useful for the prediction of 〈K〉 from generalized dimensions estimated by multifractal analysis of field measurements of K. The results may also be applicable to the design of sampling strategies for multiple small‐scale slug tests at a given resolution.