# Investigation of the Flux–Concentration Relation for Horizontal Flow in Soils

^{*}

## Abstract

**:**

^{p}

^{+1}is examined. Parameter p is estimated from curve-fitting of the experimentally obtained λ(Θ) data to an analytic expression of the form (1 − Θ)

^{p}where λ is the well-known Boltzmann transformation λ = xt

^{−0.5}(x = distance, t = time). The results show that the equation of (1 − Θ)

^{p}form can satisfactorily describe the λ(Θ) relation for the four porous media tested. The proposed F(Θ) function was compared with the limiting F(Θ) function for linear and Green–Ampt soils and to the actual F(Θ) function. From the results, it was shown that the proposed F(Θ) function gave reasonably accurate results in all cases. Moreover, the analytical expression of the soil water diffusivity (D(Θ)) function, as it was obtained by using the equation for λ(Θ) of the form (1 − Θ)

^{p}, appears to be very close to the experimental D(Θ) data (root mean square error (RMSE) = 0.593 m

^{2}min

^{−1}).

## 1. Introduction

_{in}is the initial volumetric soil water content of the soil column, considered uniform throughout the column; θ

_{0}is the volumetric soil water content at the soil surface (x = 0); q is the soil water flux density at a position x where soil water content is θ; and q

_{0}is the soil water flux density at x = 0, the soil surface, where water is absorbed under constant concentration conditions with θ = θ

_{0}= θ

_{s}(θ

_{s}= volumetric soil moisture at saturation) when the soil water pressure head is H = H

_{0}= 0 cm column of water.

_{in}; the value is 1 at the soil surface, where q = q

_{0}and θ = θ

_{0}[1]. In general, the F is a function of θ, θ

_{in}, θ

_{0}, and time (t) and it is, in most cases, unknown a priori [3].

_{2}being a function of the initial soil water saturation and the soil pore structure index that reflects the shape of the soil water retention curve.

_{i}is the maximum value of λ(Θ) (i.e., when Θ = 0). Parameter λ

_{i}considered to be a characteristic measure of the wetted region in a horizontal absorption experiment, and is influenced only by θ

_{in}for every soil [10,11,12]; p is a fitting parameter.

_{i}and p; by a log-transformation, Equation (6) becomes linear, with the slope equal to p and transect equal to logλi when the λ(Θ) is described by Equation (6). This procedure appears preferable to least squares fitting.

## 2. Theory

- (a)
- t = 0, x > 0, θ = θ
_{in} - (b)
- t > 0, x = 0, θ = θ
_{0} - (c)
- t > 0, x → ∞, θ = θ
_{in}

_{in}, λ → ∞, and θ = θ

_{0}, λ = 0.

_{in}and θ

_{0}.

_{c}function could be estimated by analytical expression [8], since

## 3. Materials and Methods

_{j}(x

_{j}, t*) profile, after a time period t* was elapsed from the beginning of the experiment, was determined by cutting, as quickly as possible, the soil column into small rectangular pieces. Specifically, the soil column was sectioned in 0.01 m increments, and the volumetric water content of each rectangular piece θ

_{j}was determined by using the gravimetric water content and the dry soil bulk density (ρ

_{φ}). The θ

_{j}value corresponds to the center of the soil samples, at distance x

_{j}from the soil infiltration surface.

_{in}were those associated to the air-dry soil, and the condition of the soil infiltration surface was that of a constant pressure head H value (H = 0 at x = 0, with x = 0 denoting the soil infiltration surface). The zero value of H was maintained by a Mariotte device (Scheme 1). A thin wire mesh at x = 0 was installed to keep the soil at rest, and also provide the least possible resistance to the soil water entry.

## 4. Results and Discussion

_{0}, θ

_{in}, ρ

_{φ}, λ

_{i}and S (Equation (18)), together with fitting parameter p and the value of S

_{c}(Equation (14)), are shown for each porous media used in this study. It is easily noticeable that the values of S and λ

_{i}depend strongly on the soil type, and they tend to decrease as soils become finer in texture. The same trend for the values S and λ

_{i}for six different soils were presented from McBride and Horton [10].

_{i}and p) do not represent simple fitting parameters, but are related to the soil’s hydraulic properties. Some researchers insist that parameter λ

_{i}may be considered as an index of the soil’s hydraulic properties [10,11]. Moreover, Shao and Horton [11] correlated parameter p with the parameter n (p = 1/n) of the equation of van Genuchten [17], which describes the soil moisture retention curve.

_{i}(θ

_{0}– θ

_{in}) is equal to (p + 1)

^{−1}. For the Green–Ampt soils (D is a Dirac delta-function of θ), the ratio S/λ

_{i}(θ

_{0}– θ

_{in}) should be unit, thus p would be zero. Similarly, for the linear soils, the expression (p + 1)

^{−1}would be 0.31, and the value of p will be 2.23 [8]. Consequently, the values of p are related to the form of the diffusivity D(θ) function. In order to examine this relationship, Equation (6) is rewritten as in Equation (19):

_{i}) relationship is shown for various values of the parameter p (0 < p < 2.23). From Figure 2, it is shown that all the Θ(λ/λ

_{i}) relations lie in the area with its borders defined by the values of the parameter p (i.e., p $\to 0$; D(θ) Dirac delta function) and p = 2.23 (D(θ) constant). From the investigation of the λ(θ) relations in this study, it is found that p-values fall in the range 0.1 < p < 0.4. Also, Evangelides et al. [6] showed that p-values obtained using data from horizontal infiltration experiments in seven soils fall in the range 0.149 < p < 0.389. In other words, the range of p is narrower than the range $0\le p\le 2.23$.

_{i}, θ

_{0}, and θ

_{in}values for each soil, a re-estimation of sorptivity S

_{c}was performed using Equation (14). From the values of S and S

_{c}presented in Table 1, S

_{c}values are reasonably close to the experimental values of S (S = I/t

^{0.5}) for three out of the four soils examined (absolute values of RE: 1.55% < RE < 3.08%). The relative error value for the SiCL soil appears to be rather high (13.27%). This could be attributed to the long time duration of the experiment (1630 min) and unavoidable soil water evaporation from the upper soil surface. In any case, the overall differences are small, and therefore one may consider that Equation (14) can lead to a quick and reliable way of estimating S from a set of horizontal absorption experimental data.

^{p}

^{+1}to describe the upper and lower theoretical boundaries of F(Θ). As has already mentioned, when p = 0 the value of F = Θ—i.e., it converges to the lower limit (D(θ) function is a delta function). However, the values of F for p = 2.23 (resulted from (p + 1)

^{−1}= 0.31 [8]) differs from the values of F resulting from Equation (13), which is the upper theoretical limit of F(Θ). Specifically, it gives higher values of F(Θ) than the theoretical curve. The fitting of the F(Θ) = 1 − (1 – Θ)

^{p}

^{+1}equation to the theoretical curve (Equation (13)) showed that the equation F(Θ) = 1 − (1 – Θ)

^{2.36}gives very good results of F(Θ) estimation over the entire range of Θ [7]. Therefore, the variation range of the parameter p is between 0 and 1.36, and the resulting shape parameter value for the linear soils ((p + 1)

^{−1}= 0.423) is greater than that presented by Clothier et al. [8].

^{p}

^{+1}seems to be appropriate functional form for describing actual flux–saturation curves of general soils, and its parameter p has a physical meaning, i.e., it represents the shape of the soil moisture profile.

## 5. Conclusions

^{p}can reliably describe the λ(Θ) relationship after the proper selection of the parameter p for a relatively large range of soils. It is also shown that the analytical expressions of the soil hydraulic diffusivity D(θ) and soil sorptivity S approach the experimental ones well. Moreover, for the case of the D(θ) relationship, there is the advantage of obtaining values near saturation, where the classical methodology of Bruce and Klute might be inadequate.

^{p}. Parameter p seems to be strongly related to soil hydraulic properties, and further investigation is needed to find this exact relationship. The analytical F(Θ) relationship, for all soil types investigated, approaches the experimental ones very well, and lies within the limiting F(Θ) function for linear and delta function soils. In addition, the upper and lower limit curves of F(Θ) calculated by the proposed expression were consistent with theoretical curves.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The λ(θ) relationships, as obtained directly from experimental θ(x,t) data for each soil, and the estimated values according to Equation (6).

**Figure 3.**A comparative presentation of the relationship D(θ) as obtained according to the Bruce and Klute [14] method (Equation (16)) and Equation (17) for the loamy (L) soil.

**Figure 4.**Comparative presentation of the relationship F(Θ) for the linear soils and Green–Ampt soils, as well as the F(Θ) calculated according to Equation (15) and the experimental F(Θ) values described in the text for the four soils examined.

**Table 1.**The values of the soil characteristics θ

_{0}, θ

_{in}, ρ

_{φ}, and S, together with the values of the parameters λ

_{i}, p, and S

_{c}for each soil examined. The value of the parameter p was estimated from Equation (6). The value of sorptivity S was obtained directly from the experimental I(t) data (Equation (18)), while S

_{c}comes from Equation (14). RE denotes the absolute relative error between actual S and estimated S

_{c}values for each soil examined.

Porous Media | θ_{0} (m^{3} m^{−3}) | θ_{in} (m^{3} m^{−3}) | ρ_{φ} (g cm^{−3}) ^{1} | λ_{i} (cm min^{−0.5}) | p | S (cm min^{−0.5}) | S_{c} (cm min^{−0.5}) | RE (%) |
---|---|---|---|---|---|---|---|---|

Sand (S) | 0.284 | 0.06 | - | 12.210 | 0.35 | 1.930 | 1.980 | 2.59 |

Sandy loam (SL) | 0.418 | 0.015 | 1.41 | 3.510 | 0.12 | 1.300 | 1.26 | 3.08 |

Loam (L) | 0.465 | 0.022 | 1.12 | 1.545 | 0.1 | 0.644 | 0.634 | 1.55 |

Silty clay loam (SiCL) | 0.511 | 0.028 | 1.22 | 0.656 | 0.18 | 0.309 | 0.268 | 13.27 |

^{1}Dry soil bulk density value for sand soil is not referred to by Poulovassilis [15].

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**MDPI and ACS Style**

Kargas, G.; Londra, P.; Kerkides, P.
Investigation of the Flux–Concentration Relation for Horizontal Flow in Soils. *Water* **2019**, *11*, 2442.
https://doi.org/10.3390/w11122442

**AMA Style**

Kargas G, Londra P, Kerkides P.
Investigation of the Flux–Concentration Relation for Horizontal Flow in Soils. *Water*. 2019; 11(12):2442.
https://doi.org/10.3390/w11122442

**Chicago/Turabian Style**

Kargas, George, Paraskevi Londra, and Petros Kerkides.
2019. "Investigation of the Flux–Concentration Relation for Horizontal Flow in Soils" *Water* 11, no. 12: 2442.
https://doi.org/10.3390/w11122442