^{1}

^{2}

^{1}

^{2}

A method based on a specific power-law relationship between the hydraulic head and the Boltzmann variable, presented using a similarity hypothesis, was recently generalized to a range of powers to satisfy the Bruce and Klute equation exactly. Here, considerations are presented on the proposed similarity assumption, and new analytical support is given to estimate the water density flux into and inside the soil, based on the concept of sorptivity and on Buckingham-Darcy's law. Results show that the new analytical solution satisfies both theories in the calculation of water density fluxes and is in agreement with experimental results of water infiltrating horizontally into sand. However, the utility of this analysis still needs to be verified for a variety of different textured soils having a diverse range of initial soil water contents.

Based on physical laws of similarity applied to the rate of work required for water to wet and move through a soil, Prevedello et al. [

With a similar assumption, but not exactly as expressed by Prevedello et al. [

Although the similarity assumptions used by [

Two identical solutions for the horizontal water infiltration into a soil were presented by [^{3}m^{-3}), and

When

Substituting (

For the cases in which the soil is initially very dry (i.e.,

From (

Equating (

Considering

The solution obtained by [

In this way it is shown that the same soil water profile

Since Prevedello et al. [

If in (

The water flux density can also be evaluated through the Buckingham-Darcy law directly, using the Prevedello et al. [

Notice that this interpretation provides a constant value of

From the soil surface, where the soil is saturated (

The analytical equation to estimate water density flow inside the soil can be determined simply by substituting

To validate experimentally the equations proposed above to quantify the water density into and inside the soil during the horizontal infiltration process for homogeneous sand, we used the same data presented in [

The van Genuchten [

According to van Genuchten [

Data for the saturated hydraulic conductivity and for the parameters of the water retention curve for the marine sand are presented in Table

Saturated hydraulic conductivity

(m·s^{-1}) | (m^{-1}) | (m^{3}·m^{-3}) | (m^{3}·m^{-3}) | |||

0.0001583 | 4.1 | 17 | 0.9412 | 0.3870 | 0.0187 | 0.991 |

From the boundary conditions and the soil parameters (Table ^{3}·m^{-3}, (

Water density flow into the soil (

Measured soil water content profiles (Figure 5 of Prevedello et al. [

is

In light of (^{-1/2}, we have

Cumulative water infiltration analytically estimated from (

To verify the analytical validity of (

Another apparently analogous way to obtain the results of (

Comparing (

Equation (

Considering the boundary conditions of soil water content varying between 0.0187 and 0.387 m^{3}·m^{-3}, to use (^{3}·m^{-3}, to obtain their respective positions from (

Through (

Analogously as before in (

Both (

The water density flow as a function of

It can be noted that for

From the above, it follows that

or

indicating that the gradient decreases with the time, shown in Figure

Matric gradient as a function of

To verify how accurate these estimated results are, the same boundary value problem was solved numerically from Richard’s equation without any similarity assumption, using the procedure of Philip [^{3}·m^{-3}). Results showed that both water density flows at

The new analytical supports to estimate the water density into and inside of the soil during horizontal infiltration agree with the experimental results, with the concept of sorptivity proposed by Philip [

The authors acknowledge Donald R. Nielsen and Klaus Reichardt for their support and guidance in preparing this paper.