# Calculation of Steady-State Evaporation for an Arbitrary Matric Potential at Bare Ground Surface

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Background and Problem Description

^{−1}], and E is the upward water flux [LT

^{−1}], which is equal to the value of evaporation rate at bare ground surface under the steady-state flow condition. A fixed water table is assumed to be below the ground surface at a distance L (z = −L). Only the vertical flow is of concern here, and the lateral flow is assumed to be secondary and negligible. The soil profile is assumed to be vertically homogeneous, thus soil layering and heterogeneity is not considered at this study. However, soil heterogeneity is an important feature and may be considered in a future investigation on the basis of this study.

_{s}is the saturated hydraulic conductivity [LT

^{−1}] and $\alpha $ is a fitting parameter related to the pore size distribution of soil [L

^{−1}]. The second one was

_{1}= h(z

_{1}) and h

_{2}= h(z

_{2}) are two matric potential heads at two different elevations z

_{1}and z

_{2}, respectively. In the problem studied below, we set z

_{1}= −L (water table) and h

_{1}(−L) = 0; z

_{2}= 0 (ground surface) and h

_{2}(0) = h

_{0}, which is a constant matric potential head at ground surface. Substituting Equation (4) into Equation (6) leads to

_{0}= h

_{0}/a, which are positive, and substituting them into Equation (7) one has:

_{0}in Equation (9). Under this condition, one can employ the following identity [31]:

_{p}in Equation (11) represents the potential evaporation rate hereinafter.

_{p}for a general soil type, and one has to seek help from a numerical root-searching method. Under the special condition that E

_{p}/K

_{s}is much less than 1, one can obtain a closed-form solution for E

_{p}based on Equation (11):

_{p}, as reported by Jury and Horton [10]. However, as the assumption that E

_{p}/K

_{s}is much less than 1 may not hold in actual field conditions, one must be cautious for using Equation (12) for estimating E

_{p}for cases where E

_{p}/K

_{s}is not much less than 1. One question is that of how much error may be introduced for using Equation (12) for a certain E

_{p}/K

_{s}value that is not too much less than 1. To answer this question, one needs to develop an accurate solution for a more general case that E

_{p}/K

_{s}is permitted to vary over a wide range of allowable values, which is developed in the following section.

#### 2.2. New Solutions of Evaporation with Arbitrary Surface Matric Potentials

_{p}/K

_{s}is much less than 1. Our solutions are based on two popular unsaturated hydraulic conductivity models: the modified Gardner [23] model and the Brooks–Corey [26] model in describing the unsaturated zone flow processes.

#### 2.2.1. Calculation of Evaporation Rate E with the Modified Gardner [23] Model

_{0}is positively finite rather than infinite (as in Section 2.1) and it is given as ${\sigma}_{0}={\left(\frac{E/{K}_{s}}{1+E/{K}_{s}}\right)}^{1/N}\frac{{h}_{0}}{a}$. The integration in Equation (9) can be separated into two components: one for $0<{\sigma}_{0}<1$ and one for ${\sigma}_{0}\ge 1$.

_{s}, and a known from the Gardner [23] model. As E is embedded in the definition of σ

_{0}, such a calculation cannot be carried out using a closed-form solution except for the special case that $E/{K}_{s}$ is much less than 1. Rather, a numerical root-searching method such as the Newton–Raphson algorithm [32] may be used.

_{0}is greater or less than 1 before the determination of E, thus one is also unsure whether to use Equation (13) or Equation (14) to perform the computation. To address this issue, we recommend the following steps. Firstly, one should compute E from Equation (13) using the Newton–Raphson method for root-searching [32]. Secondly, after obtaining E, one will check the σ

_{0}value with the obtained E value. If the σ

_{0}value is indeed greater than or equal to 1, then Equation (13) is valid. If the σ

_{0}value is less than 1, then Equation (13) is invalid and one has to use Equation (14) to calculate E.

_{M}. After that, one will repeat the computation of E with (M + 1) terms approximation of Equation (13) or Equation (14), denoted as E

_{M+}

_{1}. Then, one can check the difference of E

_{M}and E

_{M+}

_{1}. If $\left|({E}_{M}-{E}_{M+1})/({E}_{M}+{E}_{M+1})\right|$ is less than a pre-determined small criterion such as 10

^{−6}, one can say that the infinite series of summation can be approximated with the finite M terms series of summation with sufficient accuracy. Our numerical exercises show that the M value is usually around 10–50.

#### 2.2.2. Calculation of Evaporation Rate E with the Brooks–Corey [26] Model

_{v}(negative) [L] is the air-entry value of h (negative). The p-value was assumed to be 1.0 in the original study of Brooks and Corey [26].

## 3. Results

#### 3.1. Check of Applicability of Equations (11) and (12)

_{s}) based on the work of Gardner [23] and Brooks–Corey [26], and Warrick [25] reduced the parameter B in the Brooks–Corey model (Equation (3)) to 0 in order to make the problem analytically amendable, thus it can be regarded as a special case of the Brooks–Corey model that may not be applicable to soils with B value not equaling to or very close to zero.

_{p}/K

_{s}) for water table depths ranging from 10 cm to 1000 cm by Equation (11) are listed in Table 2. A few interesting observations can be made from Table 2. Firstly, when the water table is as shallow as 10 cm, E

_{p}/K

_{s}values for all four soil types are greater than 1.0, thus Equation (12) cannot be used to calculate the evaporation rate as this equation requires that E

_{p}/K

_{s}is much less than 1. Secondly, for a water table depth of 50 cm, Equation (12) is still not applicable as E

_{p}/K

_{s}values are not much less than 1, particularly for the case of Pachappa fine sandy loam, which has an E

_{p}/K

_{s}value of 0.96. Thirdly, for a water table depth of 100 cm, Equation (12) should be applicable for Buckeye fine sand and Yolo Light Clay, but is not recommended for Chino Clay and Pachappa fine sandy loam. Fourthly, for water table depth greater than 300 cm, Equation (12) is applicable for all four soil types as the E

_{p}/K

_{s}values are all less than 0.016.

_{p}/K

_{s}is much less than 1 or greater than 1 depends on the soil properties and the water table depth. For instance, when the water table depth is greater than 300 cm, the E

_{p}/K

_{s}values for the four soil types of Table 2 are all much less than 1. However, for the same soil types, the E

_{p}/K

_{s}values become greater than 1 when the water table depth is as shallow as 10 cm.

_{p}/K

_{s}being much less than 1) and (12) (with the assumption that E

_{p}/K

_{s}being much less than 1), one may use the following formula: $\epsilon =\left|{E}_{11}-{E}_{12}\right|/{E}_{11}$, where E

_{11}and E

_{12}represent E

_{p}calculated from Equations (11) and (12), respectively. The results of discrepancy for five different soil types are listed in Table 3. Previous experimental data suggested the N values to be 2, 3, 4, 4, 5, and the a values to be −20.8 cm, −86.7 cm, −17 cm, −10.9 cm and −44.7 cm, respectively, for clay loam, silty loam, sandy loam, coarse sand and fine sand in Table 3, where the hydraulic properties of soils were measured by Ashraf [35,36], Rijtema [37] and van Hylckama [34].

_{p}/K

_{s}for a range of N and −a/L values based on Equation (11). In Figure 1, six different contours of E

_{p}/K

_{s}ranging from 0.05 to 0.00001 are plotted. This figure may be used to quickly estimate the range of evaporation rate based on the soil type parameters a and N for a given water table depth L. By knowing the range of E

_{p}/K

_{s}, one can subsequently estimate the discrepancy range of the results obtained from Equations (11) and (12) (see Table 3). Such a discrepancy range will allow us to decide if Equation (12) or Equation (11) should be used. In this study, we choose 5% discrepancy as the threshold, meaning that if the discrepancy is greater than 5%, Equation (12) is not recommended and one has to use Equation (11); if the discrepancy is less than 5%, one can use Equation (12) as a good approximation of Equation (11). For instance, when E

_{p}/K

_{s}are 0.05 and 0.01, the discrepancy ratios between Equations (11) and (12) for Buckeye soil (fine sand) are 17.8% and 3.9%, respectively. Then, one can conclude that Equation (12) may be applicable when E

_{p}/K

_{s}is 0.01, but not applicable when E

_{p}/K

_{s}is 0.05. However, for clay loam soil, when E

_{p}/K

_{s}are 0.005 and 0.01, the discrepancy ratios between Equations (11) and (12) are 4.8% and 1.0%, respectively. Therefore, Equation (12) may be applicable for both E

_{p}/K

_{s}of 0.005 and 0.01.

_{p}/K

_{s}values can be regarded as much less than 1, which is an assumption used in the approximation of Equation (12) [10]. The answer depends on the accuracy requirement. For instance, if one can tolerate 5% of approximation error for the final estimation of the evaporation rate, E

_{p}/K

_{s}values less than 0.01 are acceptable.

#### 3.2. Results with Brooks–Corey and Modified Gardner Models

_{v}equaled the Gardner fitting parameters a, and pλ + 2λ + 2 equaled the Gardner fitting parameter N for Chino Clay. The results of evaporation rate calculated by the modified Gardner model and the Brooks–Corey model are shown in Figure 2.

_{p}). If changing the water table depth to 100 cm, the ratios of the E values calculated by the Brooks–Corey and the modified Gardner models are 82%, 88%, 89% and 91% for the matric potential heads of −200 cm, −300 cm, −500 cm and −1000 cm, respectively. If further changing the water table depth to 150 cm, the ratios of the E values calculated by the Brooks–Corey versus the modified Gardner models are 72%, 85%, 89%, 92% for the matric potential heads of −200 cm, −300 cm, −500 cm, −1000 cm, respectively.

_{p}values (corresponding to the −1000 cm surface matric potential head) calculated from these two models are very close to each other. For instance, for the shallower water table depth of 50 cm, the E

_{p}values calculated from both models are essentially the same. The greatest discrepancy of the E

_{p}ratio for the water table depth of 150 cm is only 8%.

#### 3.3. Comparison with Previous Work of Sadeghi et al. [18]

_{SSJ}is the evaporation rate calculated by the solution of Sadeghi et al. [18], and E

_{LZ}is our solutions of Equations (18)–(20). The values of “2 + (2 + p)λ” for four soils in Table 4 are all between 2 and 3. Figure 4 shows that the results of this study are smaller than their counterparts computed from the closed-form solution of Sadeghi et al. [18], with the ratio of E

_{SSJ}/E

_{LZ}varying between 1.43 and 1.28 for the surface matric potential head changing from −150 cm to −1000 cm. This implies that the closed-form solution of Sadeghi et al. [18] may not be accurate enough to calculate the evaporation rates in these soils.

## 4. Discussion

_{v}of the Brooks–Corey model in a fashion of N = pλ + 2λ + 2 and h

_{v}= a [18], the E values obtained from these two models fit reasonably well with some small but consistent discrepancies (Figure 2). Such small discrepancies probably come from different functions employed for describing these two models.

## 5. Conclusions

_{SSJ}/E

_{LZ}varying between 1.43 to 1.28 for the surface matric potential head changing from −150 cm to −1000 cm, where E

_{SSJ}and E

_{LZ}denote the solution of Sadeghi et al. [18] and the solution of this study, respectively. This implies that the closed-form solution of Sadeghi et al. [18] may not be accurate enough to calculate the evaporation rates in some soils.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

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**Figure 1.**Different contours of E

_{p}/K

_{s}as a function of N and −a/L computed using Equation (11) (modified from Liu [38]).

**Figure 2.**The evaporation rate (cm/d) calculated by the modified Gardner [23] model and the Brooks–Corey [26] model versus the surface matric potential (-cm) for the Chino Clay (see Table 1). MG and B–C represent the modified Gardner’s [23] model and the Brooks–Corey [26] model in the figure, respectively.

**Figure 3.**A comparison of the semi-analytical solutions (solid line) calculated with Equations (18)–(20) using the Brooks–Corey model and the results of HYDRUS-1D simulation (dashed-line) for four soils in Table 4.

**Figure 4.**The ratio between Sadeghi’s solution (denoted as E

_{SSJ}where SSJ represents the first letters of last names of three authors of Sadeghi et al. [18]) and our solution of Equations (18)–(20) (denoted as E

_{LZ}, where LZ represents the first letters of last names of this paper) for clay loam and 100 cm depth of water table.

Soil Site/Type | Parameter Value | References |
---|---|---|

Chino Clay | N = 2, a = −23.8 cm | [33] |

Pachappa (fine sandy loam) | N = 3, a = −63.83 cm | [33] |

Buckeye (fine sand) | N = 5, a = −44.7 cm | [34] |

Yolo Light Clay | N = 1.77, a = −15.3 cm | [30] |

L (cm) | 10 | 50 | 100 | 300 | 500 | 1000 |
---|---|---|---|---|---|---|

Chino Clay, E_{p}/K_{s} | 3.27 | 0.40 | 0.124 | 0.015 | 0.0056 | <0.0001 |

Pachappa (fine sandy loam), E_{p}/K_{s} | 7.07 | 0.96 | 0.280 | 0.016 | 0.004 | 0.00045 |

Buckeye (fine sand), E_{p}/K_{s} | 4.00 | 0.29 | 0.023 | 0.0001 | <0.0001 | <0.0001 |

Yolo Light Clay, E_{p}/K_{s} | 2.38 | 0.29 | 0.096 | 0.014 | 0.006 | 0.002 |

**Table 3.**The discrepancy ratio ($\epsilon =\left|{E}_{11}-{E}_{12}\right|/{E}_{11}$) of results calculated from Equations (11) and (12).

E/K_{s} Soil | 0.05 | 0.01 | 0.005 | 0.001 | 0.0001 | 0.00001 |
---|---|---|---|---|---|---|

Buckeye (fine sand) | 17.8% | 3.9% | 2.0% | 0.4% | 0.04% | 0.004% |

clay loam | 4.8% | 1.0% | 0.5% | 0.1% | 0.01% | 0.001% |

silty loam | 9.3% | 2.0% | 1.0% | 0.2% | 0.02% | 0.002% |

sandy loam | 13.6% | 2.9% | 1.5% | 0.3% | 0.03% | 0.003% |

coarse sand | 13.6% | 2.9% | 1.5% | 0.3% | 0.03% | 0.003% |

Soil Type | K_{s} (cm/d) | H_{v} (cm) | λ | θ_{r} (m^{3}m^{−3}) | θ_{s} (m^{3}m^{−3}) |
---|---|---|---|---|---|

Clay loam | 5.52 | −25.9 | 0.194 | 0.075 | 0.390 |

Silty loam | 16.32 | −20.7 | 0.211 | 0.015 | 0.486 |

Sandy loam | 146.6 | −8.69 | 0.474 | 0.035 | 0.401 |

Coarse sand | 504.0 | −4.92 | 0.592 | 0.020 | 0.417 |

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**MDPI and ACS Style**

Liu, X.; Zhan, H.
Calculation of Steady-State Evaporation for an Arbitrary Matric Potential at Bare Ground Surface. *Water* **2017**, *9*, 729.
https://doi.org/10.3390/w9100729

**AMA Style**

Liu X, Zhan H.
Calculation of Steady-State Evaporation for an Arbitrary Matric Potential at Bare Ground Surface. *Water*. 2017; 9(10):729.
https://doi.org/10.3390/w9100729

**Chicago/Turabian Style**

Liu, Xin, and Hongbin Zhan.
2017. "Calculation of Steady-State Evaporation for an Arbitrary Matric Potential at Bare Ground Surface" *Water* 9, no. 10: 729.
https://doi.org/10.3390/w9100729