# A Sensitivity Analysis of Simulated Infiltration Rates to Uncertain Discretization in the Moisture Content Domain

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{i}≤ θ ≤ θ

_{e}in a soil. Within a particular bin, this range of moisture content can be found throughout the soil over the vertical domain (Figure 1b). Since the model has only one spatial dimension, this assumption is valid regardless of how sufficiently small the horizontal discretization is. The Green–Ampt equation is transformed and applied to compute the depth infiltration independently in each bin. A process called redistribution, which is invoked at every time step immediately after infiltration, governs the horizontal inter-bin flow along the θ-axis according to the capillary pressure associated with each bin. This process will take into account all the saturated bins but not only restricted to the local neighborhoods of different bins. The infiltration and redistribution are respectively driven by gravitational and relative capillary forces in each bin. During the infiltration, the capillary pressure and hydraulic conductivity become dynamic, and the wetting front as in the Green–Ampt model (Figure 1a) may not exist. The discretized water content domain has also been extended to be affine multi-dimensional [26] for depicting more complicated pore size distributions. It is an intrinsically mass conservative model that can be applied to various soil textures [8,23,27]. However, its suitability is directly related to the uncertain number of bins because the predicted flux is highly nonlinear with respect to the discretization in the moisture content domain. Therefore, this uncertainty plays an important role under different soil conditions in this model. The convergence test for choosing a proper number of bins by Talbot and Ogden [23] is more rigorously analyzed in this work, and its physical meaning, a greater variety of pore sizes leading to a larger infiltration rate, can be naturally explained from this study. It will also be quantitatively estimated how an infinite water content discretization affects the flux variation through an asymptotic analysis by linearly fitting the wetting front. It directly indicates that a particular depth ratio of the deepest to the shallowest bin fronts can maximize the infiltration flux.

## 2. Theory and Methodology

#### 2.1. The Talbot–Ogden Model

_{s}is saturated hydraulic conductivity, θ

_{i}is initial moisture content, θ

_{d}is the maximum moisture content during infiltration, H

_{c}is the effective capillary drive at the wetting front, and F(t) is the cumulative infiltration depth at time t.

_{i}to the effective porosity θ

_{e}there are n bins indexed by j with equal bin width ∆θ. The midpoint value (j − 1)∆θ + ∆θ/2 represented by θ

_{j}is the moisture content of the j-th bin and the depth of its saturated wetting front is z

_{j}. The j-th bin is assumed to be either fully saturated or dry at any depth. The residual moisture content is θ

_{r}. The rightmost saturated bin is θ

_{d}. The bins between θ

_{d}and θ

_{e}are unsaturated but can become saturated later. In the Green–Ampt model, the cumulative infiltration function F(t) is defined as F(t) = z(θ

_{d}− θ

_{i}). If we let f = dF/dt = (θ

_{d}− θ

_{i})dz/dt, the vertical infiltration formula by substituting this expression into Equation (1) is obtained.

_{j}/dt is inversely proportional to z

_{j}if the other parameters are constant for the j-th bin. Hence, bins to the right tend to have greater front depths than the left ones especially in the beginning of infiltration (Figure 3a).

_{j}) of every bin participating in the redistribution [23]. In the redistribution process (Figure 3b), the last deeper bin is defined as the first saturated bin found over the moisture content domain along the negative direction of the θ-axis whose front depth is higher than the shallowest saturated bin. Note that the wetting front depths gradually decrease from left to right after the redistribution since the capillary pressure in the bins on the left acts immediately on water found at depth in the bins to the right in this model [23].

_{d}for both K(θ) and ψ(θ) in Equation (3) are found in [23]. Water tends move downward through the saturated bins with large θ-values, which is the reason that K(θ

_{d}) is chosen for every bin. However, the capillary pressure in a soil with only lower moisture content than the current saturated bin is always satisfied prior to the rightmost saturated bin, which is the reason that ψ(θ

_{d}) is chosen for every bin. The functions K(θ) and ψ(θ) are from Brooks and Corey [28], but other soil hydraulic models [29,30,31] can be used without affecting the analysis and conclusions in this paper.

#### 2.2. Instantaneous Infiltration Rates Analysis

_{d}is assumed to be fixed and independent of the number of bins, which is reasonable since θ

_{d}corresponds to the rightmost saturated bin that is changing during the infiltration; (2). All bins with θ

_{i}≤ θ ≤ θ

_{d}are already saturated. If there are empty bins, it means that the surface water can be absorbed in the next time step so that the instantaneous infiltration rate equals the precipitation rate.

#### 2.2.1. One Bin versus Two Bins

_{1}. Its wetting front is FH, its width is ∆θ

_{1}, and its depth is z

_{1}. Similarly, the two bins in Figure 5 are bin

_{2}(ABJI) and bin

_{3}(BCED) with wetting fronts IJ and DE, respectively. Both their bin widths are equal to ∆θ

_{2}, which means ∆θ

_{1}= 2∆θ

_{2}. Their front depths are z

_{2}and z

_{3}. It is assumed that the entire water content for the two cases is the same. So, the water in bin

_{1}equals that contained in the union of bin

_{2}and bin

_{3}, that is ${z}_{1}\times \Delta {\theta}_{1}={z}_{2}\times \Delta {\theta}_{2}+{z}_{3}\times \Delta {\theta}_{2}$. These two cases can be compared. Let V

_{Onebin}and V

_{Twobins}denote the instantaneous infiltration rates for each case. V

_{Onebin}is calculated by

_{Twobins}is calculated by

_{2}> z

_{1}> z

_{3}, since left bins always have deeper wetting fronts than the bins to their right. Moreover, if z

_{2}= z

_{3}, then splitting one bin into two bins reverts back to the one bin case. Now, the difference between the instantaneous infiltration rates can be computed by subtracting (6) from (7). Thus,

#### 2.2.2. One Bin versus n Bins

_{X}. Its depth and bin width are z

_{X}and ∆θ

_{X}, respectively. Then bin

_{X}is split into n bins, marked by bin

_{1}, bin

_{2}, …, bin

_{n}

_{−1}, and bin

_{n}. All these new bins have the same width ∆θ. Their depths from left to right are z

_{1}, z

_{2}, …, z

_{n−}

_{1}and z

_{n}. It is similar to the previous example to have ${z}_{X}\times \Delta {\theta}_{X}=\left({z}_{1}+{z}_{2}+\dots +{z}_{n}\right)\times \Delta \theta $, n∆θ = ∆θ

_{X}= θ

_{d}− θ

_{i}, and z

_{1}≥ z

_{2}≥ … ≥ z

_{n}. By geometry there exists an index l such that z

_{1}≥ z

_{2}≥ … ≥ z

_{l}≥ z

_{X}≥ z

_{l+}

_{1}≥ … ≥ z

_{n}. The infiltration rate of bin

_{X}is

_{nbins}− V

_{Onebin}≥ 0, more bins in the model result in a greater infiltration rate.

_{1}bins is increased to n

_{2}bins, where n

_{2}and n

_{1}are integers and n

_{2}> n

_{1}. This case resembles the extension from one bin to $\lceil {n}_{2}/{n}_{1}\rceil $ bins. The difference in infiltration rates between these n

_{1}and n

_{2}bins is bounded by ${V}_{{n}_{2}bins}-{V}_{{n}_{1}bins}\approx {V}_{\lceil {n}_{2}/{n}_{1}\rceil bins}-{V}_{Onebin}$. By Equation (11), we can conclude that if a soil texture can be fitted by a finer discretization, its overall conductivity becomes higher in the Talbot–Ogden model. We found the infiltration rate using asymptotic analysis.

#### 2.3. Asymptotic Analyses and Its Physical Meaning

_{d}− θ

_{i}are used first:

_{1}to z

_{n}(Figure 6) after redistribution in every time step. When n → ∞, the effect of the number of bins on the instantaneous infiltration rate in the Talbot–Ogden model is:

_{n}, and the upper bound is 1/z

_{1}.

_{1}corresponds to the deepest saturated bin, whereas z

_{n}corresponds to the shallowest one. For convenience let z

_{i}and z

_{d}denote these two depths, respectively, because of the moisture content they represent. Thus, when n → ∞,

_{i}and the moisture content θ

_{d}of the rightmost saturated bin. These two special bins are actually the deepest and the shallowest ones, respectively. An interpretation is that the whole wetting front is pushed downward by water in the largest saturated porosity θ

_{d}and is prevented from progressing by water in the initial moisture content θ

_{i}. Therefore, the advancement of the wetting front is a compromise between these two bins. Interestingly, the unit of length does not count in the expression, and the quantity $\frac{1}{{z}_{i}-{z}_{d}}ln\frac{{z}_{i}}{{z}_{d}}-\frac{2}{{z}_{i}+{z}_{d}}$ is dimensional and is meaningful in physics. However, its value needs some further research, which is crucial for this model. The following practical case is presented: suppose the two wetting fronts satisfy

_{i}and the effective porosity θ

_{e}are within some distance from each other in depth, the infiltration in the Talbot–Ogden model will eventually become a steady state flow. However, a transformation will make this discussion easier.

_{i}= rz

_{d}with r > 1, then a function D(r) characterizing the difference between V

_{nbins}and V

_{Onebin}, is defined by D(r) = ln(r)/(r − 1) − 2/(r + 1). It follows that

_{d}= θ

_{e}, then the hydraulic models used in inequality (23) are unimportant because K(θ

_{d}) = K

_{s}and ψ(θ

_{d}) = ψ

_{b}always hold. In this situation, hence,

_{d}can grow as a function of time and the rainfall rate, therefore making this upper bound decrease with respect to time.

## 3. Numerical Experiments

_{d}is fixed in Equation (24), what is important is the product of saturated hydraulic conductivity K

_{s}and the bubbling pressure ψ

_{b}. Therefore, 0.0748∙K

_{s}ψ

_{b}gives a bound for discrepancy of infiltration rates. In Table 1 [23], K

_{s}ψ

_{b}decreases from coarser to finer soils. The one exception is between clay loam and silty clay loam. In general, the upper bound of ${\Vert {V}_{nbins}-{V}_{Onebin}\Vert}_{{L}_{1}}$ becomes larger with coarser soils. Significantly, this finding means that if the soil is finer, then the infiltration will not be distinguishable based on a change in the number of bins. However, in coarse textured soil systems, the number of bins is more important than in finer ones. This is the reason why more bins may be required to test the coarser soil: the outcome varies in a wider range.

_{d}is fixed and not too deep nor too shallow, then the value 0.0748∙K

_{s}ψ

_{b}is in fact a certain bound for the variation in infiltration rate. This condition means that if the rainfall rate is around that value or of the same scale for a specific soil system, then the choice of a proper number of bins to make the simulation realistic is needed. Hence, the choice of the number of bins should be determined at least by both the soil and rainfall rates. Note that when z

_{d}is very large in inequality (24), the bound becomes small, which means the Talbot–Ogden model acts as the Green–Ampt model or the Richards model for the steady state flow.

_{e}, which is to the right of the moisture content domain. The depth z

_{d}is very small at this moment so that the change in infiltration rate is more sensitive to the perturbation of the numbers of bins than that in the first pulse. Mathematically, if z

_{d}< 1 cm (the unit is consistent with that in Table 1), the upper bounds given by inequality (24) and Table 1 are relatively easier to approach during the second pulse, thus verifying the analysis in Section 2.

_{1}/RMS

_{2}values indicate that the coarser the soil texture is, the more sensitive the infiltration rate is to the change in the number of bins, especially at the beginning of this change. This phenomenon can be attributed to the relatively large range of infiltration change for coarser soils as shown in Table 1. Note that the root mean square values in Table 3 also satisfy RMS

_{1,2}< 0.0748∙K

_{s}ψ

_{b}.

## 4. Conclusions

_{d}approaches θ

_{e}, the Talbot–Ogden model always generates higher infiltration fluxes than the Green–Ampt model, where K = K

_{s}and H

_{c}= −ψ

_{b}are set.

^{α}, α > 1) to approximate the depths of the wetting fronts.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Physical Notations and Units

Variable | Physical Meaning | Unit |

K_{s} | Saturated hydraulic conductivity | [LT^{−1}] |

K | Hydraulic conductivity | [LT^{−1}] |

ψ_{b} | Bubbling pressure | [L] |

ψ | Capillary pressure | [L] |

H_{c} | Suction head/Effective capillary pressure | [L] |

θ | Moisture content or porosity | [L^{3}L^{−3}] |

θ_{i} | Initial moisture content | [L^{3}L^{−3}] |

θ_{d} | The maximum moisture content during infiltration | [L^{3}L^{−3}] |

θ_{e} | Effective moisture content | [L^{3}L^{−3}] |

θ_{r} | The residual water content | [L^{3}L^{−3}] |

∆θ | Bin width after discretization | [L^{3}L^{−3}] |

f, V | Infiltration rate | [LT^{−1}] |

F | Total water in soil | [L] |

t | Time | [T] |

z | Infiltration depth | [L] |

z_{d} | Front depth of the bin associated with θ_{d} | [L] |

${z}_{{w}_{e}}$ | Front depth of the bin associated with θ_{e} | [L] |

${z}_{{w}_{i}}$ | Front depth of the bin associated with θ_{i} | [L] |

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**Figure 1.**The mechanism in the Talbot–Ogden model. (

**a**) Bins corresponding to different porosity θ-values compared to the wetting front in the Green–Ampt model; (

**b**) Different moisture content in a soil over the vertical domain.

**Figure 3.**The redistribution in the Talbot–Ogden model: (

**a**) pre-redistribution and (

**b**) post-redistribution.

**Figure 9.**Comparison of infiltration rates in (

**a**) sand, (

**b**) silt loam, and (

**c**) sandy clay using different numbers of bins.

**Figure 10.**Infiltration in (

**a**) sandy clay loam, and (

**b**) silt loam with a large increase in the number of bins.

Texture | K_{s} (cm/h) | ψ_{b} (cm) | K_{s}ψ_{b} (cm^{2}/h) | 0.0748∙K_{s}ψ_{b} (cm/h) |
---|---|---|---|---|

Sand | 23.56 | 7.26 | 171.05 | 12.795 |

Loamy sand | 5.98 | 8.69 | 51.18 | 3.828 |

Sandy loam | 2.18 | 14.66 | 31.96 | 2.391 |

Loam | 1.32 | 11.15 | 14.72 | 1.101 |

Silt loam | 0.68 | 20.79 | 14.14 | 1.058 |

Sandy clay loam | 0.30 | 28.08 | 8.42 | 0.630 |

Clay loam | 0.20 | 25.89 | 5.17 | 0.386 |

Silty clay loam | 0.20 | 32.56 | 6.51 | 0.487 |

Sandy clay | 0.12 | 29.17 | 3.50 | 0.262 |

Silt clay | 0.10 | 34.19 | 3.42 | 0.256 |

Clay | 0.06 | 37.30 | 2.24 | 0.168 |

Soil | K_{s} (cm/h) | ψ_{b} (cm) | θ_{r} | θ_{i} | θ_{e} | λ |
---|---|---|---|---|---|---|

Sandy clay | 0.12 | 29.17 | 0.109 | 0.239 | 0.321 | 0.223 |

Silt loam | 0.68 | 20.79 | 0.015 | 0.133 | 0.486 | 0.234 |

Sand | 23.56 | 7.26 | 0.02 | 0.033 | 0.417 | 0.694 |

Type | RMS_{1} (cm/h) | RMS_{2} (cm/h) | RMS_{1}/RMS_{2} (%) | 0.0748∙K_{s}ψ_{b} (cm/h) |
---|---|---|---|---|

Sandy Clay | 0.0171 | 0.0182 | 94.11 | 0.262 |

Silt Loam | 0.1256 | 0.1515 | 82.91 | 1.058 |

Sand | 1.9330 | 2.6853 | 71.99 | 12.795 |

_{1}= The difference of infiltration rates between 25 and 125 bins; Root mean square

_{2}= The difference of infiltration rates between 25 and 250 bins.

**Table 4.**Influence of large number of bins on the infiltration rate compared with 100 bins as the baseline.

Number of Bins (x) | Sand Clay Loam | Silt Loam |
---|---|---|

500 | 8.04% | 17.48% |

8.56% | 24.32% |

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**MDPI and ACS Style**

Liu, L.; Yu, H. A Sensitivity Analysis of Simulated Infiltration Rates to Uncertain Discretization in the Moisture Content Domain. *Water* **2019**, *11*, 1192.
https://doi.org/10.3390/w11061192

**AMA Style**

Liu L, Yu H. A Sensitivity Analysis of Simulated Infiltration Rates to Uncertain Discretization in the Moisture Content Domain. *Water*. 2019; 11(6):1192.
https://doi.org/10.3390/w11061192

**Chicago/Turabian Style**

Liu, Lulu, and Han Yu. 2019. "A Sensitivity Analysis of Simulated Infiltration Rates to Uncertain Discretization in the Moisture Content Domain" *Water* 11, no. 6: 1192.
https://doi.org/10.3390/w11061192