### 1. Introduction

### 2. Test and Methods

_{3}PO

_{4}) was injected, as the remediation agent. The soil water content was measured in real time by using the soil water content detector in each test hole of equal distance. Data was collected and recorded in the process of experiments.

### 2.1. Test Principle

### 2.2. Test Apparatus

### 2.3. Sample Preparations and Test Procedures

_{3}PO

_{4}) selected in this paper is a kind of inorganic compound, and it is easily soluble in water (its solubility is 28.3 g/100 mL).

### 3. Mathematical Model and Method

##### (1)

$$\frac{\partial \theta}{\partial t}=\frac{\partial}{\partial x}\left[D\left(\theta \right)\frac{\partial \theta}{\partial x}\right]$$*θ*is the water content,

*D*(

*θ*) is the water diffusivity, which is assumed to only related to water content,

*x*is the transport distance of water in soil, and

*t*is the ordinary time. The Boltzmann scaling $x~{t}^{{\scriptstyle \frac{1}{2}}}$ makes transport distance grow as the square root of time.

*α*∈(0, 2),

*α*≠ 1.

##### (3)

$$\frac{\partial \theta}{\partial {t}^{\alpha}}=\frac{\partial}{\partial x}\left[{D}_{\alpha}\left(\theta \right)\frac{\partial \theta}{\partial x}\right]$$*D*

*(*

_{α}*θ*) denotes a fractal water diffusivity. The fractal Richards’ model can be transformed into ordinary differential equation. Two basic forms of diffusivity

*D*

*(*

_{α}*θ*) were proposed: power-law and exponential [45, 46]. The power-law form assumes

*D*

*(*

_{α}*θ*) =

*D*

_{0}

*θ*

*(*

^{n}*n*>0) in a semi-infinite domain. Sun et al. [36] have provided an approximate solution of Richards’ equation with a power-law diffusivity

*D*

*(*

_{α}*θ*) =

*D*

_{0}

*θ*

*, which is given by*

^{n}##### (4)

$${\theta}^{n}=1-\frac{n}{2{C}_{0}}\left[\sqrt{\frac{2{C}_{0}\left(1-g/2\right)}{n+1}}x{t}^{-{\scriptstyle \frac{\alpha}{2}}}+\frac{g}{2}{\left(x{t}^{-{\scriptstyle \frac{\alpha}{2}}}\right)}^{2}\right]$$_{0},

*α*and

*n*. Moreover applications presented by Sun [36] and Fan [37], showed that the fractal Richards’ equation can predict wetting front dynamics for soil examples in a laboratory setting.

_{0},

*α*and

*n*) of the FRE model. The parameter

*n*of the FRE is positively correlated with the change rate of water content in soil [37], and the fractal order

*α*of the FRE can characterize the underlying water transport environment property in heterogeneous soil [47].

### 4. Conclusions

### 4.1. Influence of The Injection Rate on The Speed of Diffusion

#### 4.1.1. Experimental results

*T*

_{1}: the relative time,

*T*

_{2}: the actual time when soil samples at a certain location reaches a certain water content,

*T*

_{3}: the total time spent on this full experiment, and

*T*

_{4}: the total diffusion time of the slowest rate group.

#### 4.1.2. The FRE model fitting results

_{0},

*n*,

*α*) for the FRE, correlation coefficient (R-square) and root mean square error (RMSE) of fitting curve are shown in Table S1. Fig. 4 indicates that FRE can describe the trend of water transport in soil column experiments under three different injection rates, and they fit well at all positions. The R-square and RMSE in Table S1 were computed by pooling data from all curves shown.

*n*of the FRE is positively correlated with the change rate of water content in soil [37]. And, the parameter

*n*in the three groups of experiments is drawn as the change curves in Fig. 5. Fig. 5 shows that the lower injection rate is, the faster the water content of soil changes, that is, the faster the diffusion rate of the simulated agent in the soil column increases, which is consistent with the law reflected in Fig. 4. In addition, the R-square and RMSE in Table S1, reflecting a fitting effect, show that the FRE fits the water transport under a low injection rate more closely than the high group. Especially, in the experimental group (c) with higher injection speed, it can be found that the fitting effect of the forward position (X = 200 mm, 400 mm...) is significantly decreased.

### 4.2. Influence of The Confining Pressure of Soil on The Speed of Diffusion

#### 4.2.1. Experimental results

#### 4.2.2. The FRE model fitting results

_{0},

*n*,

*α*), R-square and RMSE of fitting curve are shown in Table S2.

*α*of the FRE can characterize the underlying water transport environment property in heterogeneous soil, and CHEN Wen [47] pointed out that the smaller the value of

*α*, the more complex the water transport environment would be. Fig. 8 shows curves of parameter

*α*, fitted by FRE in each group, and according to the experimental law shown in Fig. 6, they are also divided into two groups: (a) and (b). Fig. 8 indicates that with the increase of the confining pressure, values of

*α*first decreases (Fig. 7(a)) and then increases (Fig. 7(b)), which is consistent with Fig. 6. This represents that the rate of water transport is related to the complexity of soil environment caused by the confining pressure, which can also be seen from the change of parameter C

_{0}in Table S2.

### 5. Conclusions

_{0},

*α*and

*n*) of FRE model. Therefore, the FRE model can be considered to predict the transport of agent solutions in unsaturated soil, in the actual remediation project.