Fig. 6. Relative Euclidean ( L 2 ) and maximum ( L ∞ ) norms of the upscaled solutions corresponding to the different degrees of coarsening for the Gardner–Basha model (left) and the Mualem–van Genuchten model (right); r denotes the ratio of upscaling.

F ig . 2. Conductivity of an aggregate pair, k eq ( h ), as derived from observed water flow and as fitted with the Mualem–van Genuchten model with tortuosity τ = 5. The conductivity of the aggregate interior k agg ( h ) is fitted with τ = 0.5.

Fig. 7. Relative Euclidean ( L 2 ) and maximum ( L ∞ ) norms of upscaled solutions corresponding to the different degrees of spatial heterogeneity for the Gardner–Basha model (left) and the Mualem–van Genuchten model (right); σ denotes the standard deviation of the logarithm of the fine-scale

Fig. 3. Comparison of the water pressure head ( h ) between the coarse-scale models using the upscaling method (UM) and the upscaling method with pre-calculation (UM-p) and the fine-scale model for the one-dimensional ponded infiltration problem at time t = 1, 4, and 8 h using the Mualem–van

Fig. 5. Comparison of the water pressure head ( h ) between the coarse-scale models using the upscaling method (UM) and the upscaling method with pre-calculation (UM-p) and the fine-scale model for the one-dimensional upward evaporation problem at time t = 1, 4, and 8 h using the Mualem–van

Fig. 14. Comparison of the water content derived from the fine-scale model (left) and the coarse-scale model using the upscaling method with pre-calculation (UM-p) (right) at time t = 1, 2, and 5 h for the two-dimensional downward infiltration problem with the Mualem–van Genuchten model

Fig. 15. Comparison of the water content derived from the fine-scale model (left) and the coarse-scale model using the upscaling method with pre-calculation (UM-p) (right) at time t = 1, 2, and 5 h for the two-dimensional downward infiltration problem with the Mualem–van Genuchten model

Fig. 19. Comparison of the water content derived from the fine-scale model (left) and the coarse-scale model using the upscaling method with pre-calculation (UM-p) (right) at time t = 1, 2, and 5 h for the two-dimensional upward evaporation problem with the Mualem–van Genuchten model

Fig. 22. Comparison of the water content derived from the fine-scale model (left) and the coarse-scale model using the upscaling method with pre-calculation (UM-p) (right) at time t = 1, 10, and 50 h for the two-dimensional infiltration problem with mixed boundary conditions with the Mualem–van

Fig. 7. Scatter plot of 1500 combinations of (A) θ r – n , (B) K s – l vg , (C) θ r –σ, and (D) K s – l k parameters sampled for the one-step outflow experiment using the SCEM-UA algorithm in the case of Scenario II for the Mualem–van Genuchten model (A–B) and Kosugi model (C–D

infiltration problem with the Mualem–van Genuchten model and correlation lengths λ x = λ z = 4 m.

Fig. 1. Univariate posterior probability distributions for the soil hydraulic parameters (A) θ r , (B) α, (C) K s , and (D) l vg in the Mualem–van Genuchten model (1980) using observed outflow dynamics during the a transient one-step outflow experiment. The symbol p (β) along the y -axis

infiltration problem with the Mualem–van Genuchten model and correlation lengths λ x = 16 m and λ z = 2 m.

evaporation problem with the Mualem–van Genuchten model and correlation lengths λ x = 16 m and λ z = 2 m.

infiltration problem with mixed boundary conditions with the Mualem–van Genuchten model and correlation lengths λ x = 16 m and λ z = 2 m.

Fig. 13. The random lognormal saturated conductivity K s ε ( x , z ) fields for the Mualem–van Genuchten model: the isotropic case with λ x = λ z = 4 m (left) and the anisotropic case with λ x = 16 m and λ z = 2 m (right). The ratios of maximum to minimum are 1.2811 × 10 6

Fig. 9. Cumulative distribution functions (CDF) of Pareto-optimal solutions for layers 1–3 from the L 3 scenario at John Wayne Hill (JWH) for the Mualem–van Genuchten model (MVG) hydraulic parameters (a) residual water content, θ r ; (b) saturated water content, θ s ; (c) air entry, α; (d) n