# A Modified van Genuchten-Mualem Model of Hydraulic Conductivity in Korean Residual Soils

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## Abstract

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## 1. Introduction

## 2. The Relationship between Soil Water Retention and Hydraulic Conductivity

_{b}, n is the parameter on the pore size distribution and m = 1 − 1/n.

_{b}= 2.43 kPa, n = 1.34, q = 0.5). As shown in Figure 1a, in the domain of a sufficiently small p, the pores are almost saturated and the water content varies insignificantly in both the SWRC data and VG fit. In contrast, the slope of the SWRCs near full saturation has a significant effect on integrating the unsaturated HC in Equation (4). As shown in Figure 1b, when Θ approaches 1, the value of 1/p increases to infinity in the smooth SWRCs, such as the VG curves. According to Equation (4), K

_{r}(Figure 1c) is reduced abruptly in the VGM model and is much less than 1.0 (saturated value) near saturation or at very low suction. In particular, the decrease in K

_{r}is drastic in the VGM model, when n < 1.5 for silty soils [7]. Numerical solutions using such HCs can frequently result in oscillations or divergence according to the steep gradient near saturation. Hence, the VGM model tends to estimate the HCs differently, even with extremely small variations in the shape of the SWRCs near saturation and it does not provide a unique HC for a set of SWRC data [14].

**Figure 1.**Comparison of the soil water retention curves and hydraulic conductivities in both van Genuchten-Mualem theory and its modification: (

**a**) soil water retention curves; (

**b**) 1/p-Θ relationships; and (

**c**) hydraulic conductivities.

_{s}in the non-smooth curve (Figure 1a). In the region where p > p

_{s}, the SWRC should be fitted again to the experimental results using a fictitious water content greater than the saturated one. Therefore, the parameters α and n of the SWRCs are different from those of the original fit in the VG SWRC. Furthermore, p

_{s}is determined to be a small value to retain the nonlinear form of the VG SWRC, but is difficult to assess uniquely.

_{b}is the inverse of α in the van Genuchten’s parametrization, which is evaluated from the curve fit of the SWRC data. In Figure 1a, p

_{b}ʹ is an arbitrary estimate from which the curve is extrapolated tangentially on the log p scale to full saturation, which is less than or equal to p

_{b}. Θ

_{b}ʹ is the effective saturation corresponding to p

_{b}ʹ in the VG fit of Equation (2). The VG curve is linearized in the logarithmic p axis by the tangential line from p

_{b}ʹ to p

_{s}. p

_{s}is the value of the matric suction on the tangential line when Θ = 1. The modified VG SWRC has a partially linear shape on the logarithmic scale, where p is between p

_{s}and p

_{b}ʹ and follows the VG curve smoothly in Equation (2), where p > p

_{b}ʹ. The modified VG model on SWRC in Figure 1a is derived as follows:

_{s}. Θ

_{b}and Θ

_{b}ʹ are the effective saturations at p

_{b}and p

_{b}ʹ, respectively, calculated from Equation (2) of the VG model. In Equations (8–11), the modified SWRC is established by two parameters, α and n, of the VG SWRC, but it also needs to define p

_{b}ʹ.

_{s}. In the integration of Equation (4), the function of 1/p is integrated for each interval of Θ. Substituting Equations (8)–(11) into Equation (4), the relative HC is derived as follows:

_{r}= 1 (Equation (14)) according to the definition of Equation (5). Figure 1c shows that the modified HC varies gradually without any abrupt decreases in the small matric suction near saturation. The difference between the SWRCs in Figure 1a is very small but the value of K

_{r}from the modified SWRC is much greater than that of the VGM model.

_{r}is deduced from the parameters, α and n, of the VG SWRC. In addition, p

_{b}ʹ should be determined using a sufficiently small value so that the modified SWRC can fit the actual shape of the SWRC in the experiments equivalently to the original VG model. Figure 2 shows the effect of p

_{b}ʹ on the SWRCs and K

_{r}functions (p

_{b}= 1 kPa, n = 1.2, q = 0.5), in which the maximum value of the p axis is equal to p

_{b}or 1kPa for comparison in a small p value. In Figure 2a, the modified SWRCs were compared with the VG model for three cases: p

_{b}ʹ = p

_{b}(1 kPa), p

_{b}/5 (0.2 kPa) and p

_{b}/50 (0.02 kPa). Note that p

_{s}is 0.29 kPa, 0.081 kPa and 0.0087 kPa, respectively. All modified SWRCs have the same VG curve in the greater matric suction than p

_{b}ʹ. As shown in Figure 2b, the K

_{r}for the VGM model was reduced significantly, by 50%, at a matric suction of 0.001 kPa, and the overall values of K

_{r}are less than the modified HCs. In very small matric suction, the modified VGM model prevents abrupt changes in K

_{r}and retains the saturated K

_{r}of 1.0. For the higher p

_{b}ʹ, the value of K

_{r}is greater than that for p

_{b}ʹ = 0.02 kPa.

_{r}function of Vogel et al. [7], p

_{s}is estimated by 0.2 kPa, and α and n of the SWRCs are assumed to be the same as the original fit in the van Genuchten’s model. The modification by Vogel et al. [7] revealed a non-smooth shape at the break pressure both in the SWRC (Figure 2a) and the HCF (Figure 2b). On the other hand, the proposed K

_{r}functions show a smooth shape, even though the modified SWRCs are not smooth.

**Figure 2.**Effect of linearization by the modified SWRCs on K

_{r}functions (p

_{b}= 1 kPa, n = 1.2, q = 0.5): (

**a**) soil water retention curves; (

**b**) hydraulic conductivities.

_{b}ʹ is defined at a tangential point to extrapolate the SWRC, it is far from the physical parameter or variable, and a parametric study is examined to find p

_{b}ʹ. To fit the water retention behavior equivalently to the VG model, it is better to establish a smaller p

_{b}ʹ. If p

_{b}ʹ is infinitesimal, K

_{r}has the same value as that of the Mualem model based on the VG SWRC, and the numerical stability is no longer preserved. In Figure 2b, K

_{r}at the matric suction of p

_{b}or K

_{r}(p

_{b}) can be a measure to compare the hydraulic conductivities according to p

_{b}ʹ, which decreases from 0.1–0.03 (0.01 for VGM) with decreasing p

_{b}ʹ. Figure 3 shows the change in K

_{r}(p

_{b}) as a function of p

_{b}ʹ when n = 1.1 and 1.5 (p

_{b}= 1 kPa, q = 0.5). When p

_{b}ʹ > p

_{b}/50 (or 0.02 kPa), K

_{r}(p

_{b}) increases remarkably with respect to the logarithmic p

_{b}ʹ. On the other hand, when p

_{b}ʹ < p

_{b}/50, K

_{r}(p

_{b}) varies steadily and converges slowly to the VGMs for very small p

_{b}ʹ(<0.001 p

_{b}). Therefore, p

_{b}ʹ can be assumed to be a threshold value of p

_{b}/50 to consider both the accurate fit of the experimental SWRC and the numerical stability due to the K

_{r}function. In the case of n = 1.5, the difference in K

_{r}(p

_{b}) between the modified and original VGM models is less than in the case of n = 1.1.

## 3. Test Procedure and Sample Conditions

Region | Void Ratio | Dry Unit Weight γ_{d} (kN/m^{3}) | Initial Gravimetric Water Content w (%) | Saturated Hydraulic Conductivity K_{s} (m/s) | Specific Gravity G_{s} | Particles Less Than 75μm #200 | Unified Soil Classification System USCS ** |
---|---|---|---|---|---|---|---|

Asan * | 0.45 | 16.8 | – | 2.47 × 10^{−7} | 2.58 | 0 | SP |

Kimpo * | 0.75 | 14.5 | – | 2.99 × 10^{−7} | 2.65 | 19 | SM |

Okcheon 1 | 0.57 | 16.7 | 19.87 | 1.29 × 10^{−8} | 2.67 | 5 | SW |

Okcheon 2 | 0.77 | 14.9 | 7.93 | 2.38 × 10^{−6} | |||

Seochang | 0.61 | 16.3 | 20.61 | 9.13 × 10^{−7} | 2.64 | 14 | SC-SM |

Yesan 1 | 0.58 | 16 | 19 | 1.39 × 10^{−7} | 2.63 | 2 | SW |

Yesan 2 | 0.78 | 14.2 | 19 | 3.70 × 10^{−7} | |||

Yesan SP | 0.95 | 13.5 | 13.84 | 5.00 × 10^{−7} | 2.68 | 1 | SP |

## 4. Comparison of the HCs of the VGM Model with a Modified Model

_{s}, θ

_{r}, p

_{b}, n) were evaluated to fit the experimental data by RETC code [19] and an additional parameter q was evaluated to fit the HC data. As listed in Table 2, the hydraulic behaviors of the soils are sorted into three groups. Soils with n < 1.5 were classified as group A and the others were classified as group B. Group A was again sorted by two groups of group A-1 and group A-2, in which p

_{b}≥ 5 kPa. The samples belong to each group due to the presence of some fine-grained soil and residual formation environments. The low values of the tortuosity factor can be related to adsorptive water retention and film conductivity [20,21], which is beyond the scope of this study. For each group, the relative hydraulic conductivity, K

_{r}, for the modified VGM model was first compared with that of the VGM model and subsequently with the measured value.

Group | Region | θ_{s} | θ_{r} | p_{b} (kPa) | n | q | R^{2} for Modified HC Fit | R^{2} for VGM HC Fit |
---|---|---|---|---|---|---|---|---|

A-1 (n < 1.5, p_{b} ≥ 5 kPa) | Kimpo | 0.40 | 0.00 | 10.90 | 1.23 | −1.5 | 0.638 | 0.702 |

Seochang | 0.37 | 0.00 | 9.48 | 1.24 | −2.5 | 0.968 | 0.965 | |

Yesan 1 | 0.25 | 0.00 | 16.11 | 1.32 | −3.5 | 0.979 | 0.972 | |

A-2 (n < 1.5, p_{b} < 5 kPa) | Asan | 0.28 | 0.13 | 2.43 | 1.34 | −3.5 | 0.990 | 0.977 |

Okcheon 1 | 0.37 | 0.06 | 2.01 | 1.32 | −2.7 | 0.963 | 0.951 | |

Yesan 2 | 0.25 | 0.05 | 1.89 | 1.33 | −4.3 | 0.990 | 0.941 | |

B (n ≥ 1.5) | Okcheon 2 | 0.33 | 0.14 | 7.86 | 1.50 | −3.7 | 0.916 | 0.880 |

Yesan SP | 0.46 | 0.18 | 1.14 | 1.69 | −3.6 | 0.970 | 0.967 |

_{b}≥ 5 kPa): Kimpo, Seochang and Yesan 1 samples. In the VG fits, n = 1.23–1.32 and p

_{b}= 9.5–16.1 kPa, while θ

_{s}= 0.25–0.40 and θ

_{r}= 0. The three measured SWRCs described by the effective saturation are similar because of the similar n and p

_{b}values. In the modified SWRCs, p

_{b}ʹ = p

_{b}/50 or 0.19–0.32 kPa. The modified VG model fitted differently to the VG model just in the region where the matric suction is less than p

_{b}ʹ. Figure 5a shows the modified VG fit when p < p

_{b}ʹ. In the SWRCs of Figure 5a, the VG model fits the measured data quite accurately on three samples of group A-1 and both models have an equivalent ability to fit the experimental SWRCs.

**Figure 5.**Comparison of the VGM model with the modified model in Group A-1: (

**a**) soil water retention curves (the measured data are shown by points, and the modified VG curve by lines with point when p < p

_{b}ʹ); (

**b**) hydraulic conductivities.

_{r}was calculated using Equations (12)–(15), and the modified models were compared with the VGM HC models on Kimpo, Seochang and Yesan 1 samples (Figure 5b). The K

_{r}function was calculated from the SWRC curve. The parameter q was evaluated to fit HC data as −3.5–−1.5. The SWRCs in Figure 5a incorporates the small difference in a suction less than p

_{b}ʹ but K

_{r}for the modified models is clearly greater than that for the VGM models. Near saturation, K

_{r}= 1 in the modified VGM model, whereas K

_{r}is reduced at a suction less than 0.01 kPa in the VGM model. The difference between the two models is 18%–37% at 0.01 kPa, which increases with decreasing n.

_{b}< 5 kPa) are used to predict the relative HC in Asan, Okcheon 1 and Yesan 2 samples. Both the VG model and the modified VG model can fit the measured SWRCs accurately, in which the parameters of the three samples to fit SWRCs are quite similar; n = 1.32–1.34, p

_{b}= 1.9–2.4 kPa and p

_{b}ʹ = 0.04–0.05 kPa. θ

_{s}, θ

_{r}and q were evaluated to be 0.25–0.37, 0.05–0.13 and −4.3–−2.7, respectively. In Figure 6b, the VGM models for the three samples also showed an abrupt decrease in K

_{r}, which were less than the modified VGM model by ~25% at suction of 0.01 kPa. Because p

_{b}in group A-2 is smaller than that in group A-1, the gradient of the K

_{r}function to the matric suction in Figure 6b is greater than that in Figure 5b.

**Figure 6.**Comparison of the VGM model with the modified model in Group A-2: (

**a**) soil water retention curves (The measured data are shown by points, and the modified VG curve by lines with point when p < p

_{b}ʹ); (

**b**) hydraulic conductivities.

_{b}= 1.1 and 7.9 kPa, θ

_{s}= 0.33 and 0.46, θ

_{r}= 0.14 and 0.18, and q = −3.7–−3.6, respectively. The data on the SWRCs were measured within a matric suction less than 200 kPa, which is far from the actual suction at the residual water content. Therefore, θ

_{r}(Table 2) increased with increasing gradient of the SWRCs or n to optimize the SWRC fit. Near saturation, K

_{r}≈ 1 in both the modified VGM model and the VGM model, where K

_{r}is reduced by 10% at most in group B. Therefore, the VGM model estimates K

_{r}similar to the modified VGM model.

**Figure 7.**Comparison of the VGM model with the modified model in Group B: (

**a**) soil water retention curves (The measured data are shown by points, and the modified VG curve by lines with point when p < p

_{b}ʹ); (

**b**) hydraulic conductivities.

_{r}by the VGM model decreases abruptly near saturation. In such cases, the VGM model induces abnormal oscillation or divergence of the solutions in the numerical analysis near saturation. On the other hand, in group B, i.e., n > 1.5, there was a smaller decrease in K

_{r}in the initial part of the VGM model.

## 5. Comparison of HCs of the Modified VGM Model with the Measured Data

_{r}predicted by both the original and the modified VGM models for group A-1 (n < 1.5, p

_{b}≥ 5kPa), where q was −2.5 on average. The goodness of fit of both models are compared in terms of the correlation coefficient R

^{2}for log K

_{r}(Table 2). Using the modified model, the measured K

_{r}was predicted quite accurately for Yesan 1 sample (n = 1.32, R

^{2}= 0.98 in Table 2). Because the SWRCs of Kimpo (n = 1.23) and Seochang (n = 1.24) samples are similar, as shown in Figure 5a, the measured data of the unsaturated HC were predicted well as similar curves using the modified VGM model (R

^{2}= 0.64 for Kimpo and R

^{2}= 0.97 for Seochang). The VGM model could predict the measured HC equivalently to the modified model (R

^{2}= 0.70–0.97 for group A-1), but underestimated the nearly saturated K

_{r}(~1.0) and subsequently the unsaturated part of the K

_{r}data compared to the modified model.

_{b}< 5 kPa) in Figure 9, the measured K

_{r}was also predicted quite well by the modified model for Asan (R

^{2}= 0.99 in Table 2), Okcheon 1 (R

^{2}= 0.96) and Yesan 2 (R

^{2}= 0.99) samples. The VGM model could predict the measured HC fairly (R

^{2}= 0.94–0.98 for group A-2), but underestimated the saturated and the unsaturated parts of the K

_{r}data compared to the modified model. Regarding the effects of q on K

_{r}, K

_{r}increased with decreasing q. Group A-2, which has a smaller p

_{b}, q (−3.5 on average), has a much stronger effect on HC than Group A-1. Therefore, q contributes significantly to the prediction of K

_{r}. Yesan 2 sample in Group A-2 (Figure 9) is compacted by the looser conditions and shows a higher K

_{r}for both the measurements and predictions than the Yesan 1 sample in Group A-1 (Figure 8). Okcheon 2 sample is also looser than Okcheon 1 sample in Group A-2 (Figure 9) and belongs to Group B (Figure 10).

_{r}reasonably well for Okcheon 2 (R

^{2}= 0.92) and Yesan SP samples. The VGM model could predict similarly to the modified model (R

^{2}= 0.88–0.97 for group B) because n > 1.5, and K

_{r}~ 1.0 in the nearly saturated state. As shown in Equation (4), K

_{r}is multiplied by Θ

^{q}, where Θ is deduced from the SWRC of each group. When q is negative, Θ

^{q}> 1.0 and increases with increasing n. Group B showed the strongest effect of q (−3.6 in average) on HC than Groups A-1 and A-2, because n > 1.5.

## 6. Conclusions

_{b}and n) and Mualem’s HC functions (q), but an arbitrary suction, p

_{b}ʹ, was also estimated to be p

_{b}/50 to determine the tangential point in van Genuchten’s SWRC.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

_{r}is not derived from Equation (A7) but Equation (14) is derived by the definition of the relative HC in Equation (5).

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**MDPI and ACS Style**

Oh, S.; Kim, Y.K.; Kim, J.-W.
A Modified van Genuchten-Mualem Model of Hydraulic Conductivity in Korean Residual Soils. *Water* **2015**, *7*, 5487-5502.
https://doi.org/10.3390/w7105487

**AMA Style**

Oh S, Kim YK, Kim J-W.
A Modified van Genuchten-Mualem Model of Hydraulic Conductivity in Korean Residual Soils. *Water*. 2015; 7(10):5487-5502.
https://doi.org/10.3390/w7105487

**Chicago/Turabian Style**

Oh, Seboong, Yun Ki Kim, and Jun-Woo Kim.
2015. "A Modified van Genuchten-Mualem Model of Hydraulic Conductivity in Korean Residual Soils" *Water* 7, no. 10: 5487-5502.
https://doi.org/10.3390/w7105487