# Effects of Decaying Hydraulic Conductivity on the Groundwater Flow Processes in a Managed Aquifer Recharge Area in an Alluvial Fan

^{1}

^{2}

^{3}

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^{5}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Laboratory Experiments

#### 2.2. Numerical Simulation

^{−1}], h is the hydraulic head [L], t is time [T], $\mathit{\epsilon}$ is the porosity (void space) [1], $\mathrm{\nabla}$ is the Nabla (vector) operator [L

^{−1}],

**q**is the Darcy velocity of fluid [L T

^{−1}], Q is the bulk source/sink term of flow [L

^{3}T

^{−1}], Q

_{EOB}is the correction sink/source term of the extended Oberbeck–Boussinesq approximation [T

^{−1}], ${\mathit{k}}_{\mathit{r}}$ is the relative permeability [1],

**K**is the tensor of hydraulic conductivity [LT

^{−1}], ${\mathit{f}}_{\mathit{\mu}}$ is the viscosity relation function [1], $\mathit{\chi}$ is the buoyancy coefficient [1], and e is the gravitational unit vector [1].

**K**(

**z**) was described by using the exponential decay model [35,36]. Assuming a locally isotropic condition, the depth-dependent

**K**(

**z**) is given by:

**K**

_{0}is the

**K**at the ground surface. The

**K**

_{0}adopted in the numerical model was the same as that measured in the sand tank. A

_{1}is the decay exponent that dictates the rate of decrease in

**K**with depth, z

_{s}(x) is a function of the ground surface elevation, and z is the elevation of the aquifer bottom.

**K**(x) was assumed to be:

_{2}is the decay exponent that dictates the rate of decrease in K with length, and x

_{u}(z) is a function of the vertical boundary on the upper reaches side.

#### 2.3. Scenario Definition

_{1}) have been reported in the literature, mostly ranging from 0.003 to 0.5 [20,33,35,36]. Here, we set the decay exponents as 0.01, 0.05, and 0.1 m

^{−1}. The corresponding K fields and their relationships to depth are shown in Figure 3 and Figure 4, respectively. The artificial recharge rate through the infiltration basin was set to 5 and 0.2 m/d in the injection wells. The water level in the right and left water chambers was the same as that in scenario A.

_{2}was assumed as 0.005, 0.01, and 0.02 m

^{−1}here. The corresponding K fields and their relationships with length are shown in Figure 5 and Figure 6, respectively. The artificial recharge rate and water level in the water chambers were the same as those in scenario B.

## 3. Results

#### 3.1. Groundwater Flow Patterns

^{−1}and length-decay exponent was 0.005 m

^{−1}). The flow directions were almost horizontal from upstream to downstream direction. The velocities ranged from 3 to 4.2 m/d and increased along the flow paths. The left-side and right-side figures in Figure 7 and Figure 8 also show that the velocities decreased with the increase of decay exponents (both with depth and length). Taking Figure 7a–c as an example, the flow velocities decreased from 3–4 to 0–0.6 m/d when the depth-decay exponents increased from 0.01 to 0.1 m

^{−1}. The flow fields became increasingly complex when driven by the same artificial recharge rate with the increase of decay exponents. That is, the flow directions changed from a horizontal to vertical direction, especially near the artificially recharged areas. The antidromic flow area, where the groundwater flows in the opposite direction of the ambient groundwater flow, became larger with the increasing decay exponents, which can be more obviously seen in Figure 8f.

#### 3.2. Groundwater Age

^{−1}. As shown in Figure 10a,b, age mainly ranged from 0 to 2711 s when the length-decay exponents were set to 0.005 m

^{−1}. Furthermore, the increasing decay exponents can also increase the groundwater age and expand the degree of influence of the artificial recharge on groundwater age distribution. Taking left side figures in Figure 9 as an example, with the increase of depth-decay exponents, the spatial variability of groundwater age becomes increasingly obvious.

#### 3.3. Residence Time Distributions

^{−1}), showed the fitted trend line for FDs to RTDs toward a quadratic polynomial law and the frequency of different residence time showed less distinction. With the increase of depth-decay exponents, when the depth-decay exponent was set to 0.05 m

^{−1}, the fitted trend line for FDs to RTDs was divided into two steps: the early step toward a logarithmic behavior and the later step toward a power behavior. When the depth-decay exponent was set to a sufficiently large value (A

_{1}= 0.1 m

^{−1}), the two-step fitted trend line merged into one and toward a logarithmic behavior. Furthermore, similar results could be obtained with the K decay in the length direction. With the increase of the length-decay exponents, the fitted lines for FDs to RTDs exhibited a quadratic polynomial behavior and then was divided into two steps. Decay of K in different directions could affect the RTD behavior obviously.

#### 3.4. Ambient Groundwater Flow Paths

#### 3.5. Artificially Recharged Water Lens

_{2}= 0.002 m

^{−1}, Figure 17e), the aquifer was almost filled with artificially recharged water, and the left recharge boundary became a discharge boundary. The right panels of Figure 16 show that the thickness of the artificially recharged water lens increased with the increasing depth-decay exponent, with the up-interface moving to the shallow layer and the down-interface to deep layer. Similar results can also be seen in right panel of Figure 17 when the decay of K was set in the length direction and artificially recharged through injection well IW1. With the increase of length-decay exponents, the interface of the artificially recharged–ambient groundwater interface gradually moved to the shallow layer, and the thickness of the artificially recharged water lens increased. Furthermore, with the increase of decay exponents, increased travel times could be seen in every scenario.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic diagram of the sand tank and the experimental apparatus; (

**b**) side view of the laboratory-scale numerical model setup with prescribed boundary conditions. The boundaries on the right and left sides were set to a constant head boundary condition. The 19 red points in (

**a**) are the pressure transducers for the hydraulic head data acquisition. The three black points in (

**a**) are the injection wells. The red line in (

**b**) on the top boundary was set to a flux boundary condition. The three red points in (

**b**) are injection wells.

**Figure 2.**Variation of hydraulic conductivity along the distance from apex to apron zone in the Tailan river basin.

**Figure 3.**Hydraulic conductivity fields with different depth-decay exponents. (

**a**) the decay exponent was set to 0.01 m

^{−1}, (

**b**) the decay exponent was set to 0.05 m

^{−1}, (

**c**) the decay exponent was set to 0.1 m

^{−1}.

**Figure 5.**Hydraulic conductivity fields with different length-decay exponents. (

**a**) the decay exponent was set to 0.005 m

^{−1}, (

**b**) the decay exponent was set to 0.01 m

^{−1}, (

**c**) the decay exponent was set to 0.02 m

^{−1}.

**Figure 7.**Flow fields when artificially recharged through the infiltration basin (left side figures) and injection well IW3 (right side figures) with different depth-decay exponents. The depth-decay exponents in (

**a**) and (

**b**) were set to 0.01 m

^{−1}. The depth-decay exponents in (

**c**) and (

**d**) were set to 0.05 m

^{−1}. The depth-decay exponents in (

**e**) and (

**f**) were set to 0.1 m

^{−1}. The blue lines in the figures represent the water table.

**Figure 8.**Flow fields when artificially recharged through the infiltration basin (left side figures) and injection well IW3 (right side figures) with different length-decay exponents. The length-decay exponents in (

**a**) and (

**b**) were set to 0.005 m

^{−1}. The length-decay exponents in (

**c**) and (

**d**) were set to 0.01 m

^{−1}. The length-decay exponents in (

**e**) and (

**f**) were set to 0.02 m

^{−1}. The blue lines in the figures represent the water table.

**Figure 9.**Groundwater age driven by depth-decaying hydraulic conductivity and artificial recharge through the infiltration basin (left side) and injection well IW3 (right side). The depth-decay exponents in (

**a**) and (

**b**) were set to 0.01 m

^{−1}. The depth-decay exponents in (

**c**) and (

**d**) were set to 0.05 m

^{−1}. The depth-decay exponents in (

**e**) and (

**f**) were set to 0.1 m

^{−1}. The blue lines in the figures represent the water table.

**Figure 10.**Groundwater age driven by artificial recharge through the infiltration basin (left side figures) and injection well IW3 (right side figures) with different length-decay exponents. The length-decay exponents in (

**a**) and (

**b**) were set to 0.005 m

^{−1}. The length-decay exponents in (

**c**) and (

**d**) were set to 0.01 m

^{−1}. The length-decay exponents in (

**e**) and (

**f**) were set to 0.02 m

^{−1}. The blue lines in the figures represent the water table.

**Figure 11.**Groundwater age at the bottom of the aquifer driven by artificial recharge through injection wells at different locations and different decay exponents.

**Figure 12.**Frequency distribution of RTDs of artificially recharged water (injection well IW1) with different decay direction and decay exponents of hydraulic conductivity. “D-0.01” means the depth-decay exponent was set to 0.01, and “L-0.005” means the length-decay exponent was set to 0.005. The red lines are fitted trend lines.

**Figure 13.**Frequency distribution of RTDs of ambient groundwater with different decay direction and decay exponents of hydraulic conductivity. “D-0.01” means the depth-decay exponent was set to 0.01, and “L-0.005” means the length-decay exponent was set to 0.005. The red lines are fitted trend lines.

**Figure 14.**Ambient groundwater flow paths associated with travel time produced by artificial recharge and different depth-decay exponents. The left panel of figures were produced by the infiltration basin and the right panel of figures were produced by injection well IW3. The depth-decay exponents in (

**a**) and (

**b**) were set to 0.01 m

^{−1}. The depth-decay exponents in (

**c**) and (

**d**) were set to 0.05 m

^{−1}. The depth-decay exponents in (

**e**) and (

**f**) were set to 0.1 m

^{−1}. The blue lines in the figures represent the water table.

**Figure 15.**Ambient groundwater flow paths associated with travel time produced by artificial recharge and different length-decay exponents. The left panel of figures were produced by injection well IW2 and the right panel of figures were produced by injection well IW1. The length-decay exponents in (

**a**) and (

**b**) were set to 0.005 m

^{−1}. The length-decay exponents in (

**c**) and (

**d**) were set to 0.01 m

^{−1}. The length-decay exponents in (

**e**) and (

**f**) were set to 0.02 m

^{−1}. The blue lines in the figures represent the water table.

**Figure 16.**Artificially recharged water lens associated with travel times (TTs) produced by the infiltration basin (left panel of the figures) and injection well IW3 (right panel of the figures) with different depth-decay exponents. The depth-decay exponents in (

**a**) and (

**b**) were set to 0.01 m

^{−1}. The depth-decay exponents in (

**c**) and (

**d**) were set to 0.05 m

^{−1}. The depth-decay exponents in (

**e**) and (

**f**) were set to 0.1 m

^{−1}. The blue lines in the figures represent the water table. The yellow points in the figures are the locations of injection wells. The red lines are flow paths of artificially recharged water. The rainbow lines are flow paths of artificially recharged water.

**Figure 17.**Artificially recharged water lens associated with travel times (TTs) produced by the infiltration basin (left panel of the figures) and injection well IW1 (right panel of the figures) with different length-decay exponents. The length-decay exponents in (

**a**) and (

**b**) were set to 0.005 m

^{−1}. The length-decay exponents in (

**c**) and (

**d**) were set to 0.01 m

^{−1}. The length-decay exponents in (

**e**) and (

**f**) were set to 0.02 m

^{−1}. The blue lines in the figures represent the water table. The yellow points in the figures are the locations of injection wells. The rainbow lines are flow paths of artificially recharged water.

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**MDPI and ACS Style**

Wu, P.; Zhang, L.; Chang, B.; Wang, S. Effects of Decaying Hydraulic Conductivity on the Groundwater Flow Processes in a Managed Aquifer Recharge Area in an Alluvial Fan. *Water* **2021**, *13*, 1649.
https://doi.org/10.3390/w13121649

**AMA Style**

Wu P, Zhang L, Chang B, Wang S. Effects of Decaying Hydraulic Conductivity on the Groundwater Flow Processes in a Managed Aquifer Recharge Area in an Alluvial Fan. *Water*. 2021; 13(12):1649.
https://doi.org/10.3390/w13121649

**Chicago/Turabian Style**

Wu, Peipeng, Lijuan Zhang, Bin Chang, and Shuhong Wang. 2021. "Effects of Decaying Hydraulic Conductivity on the Groundwater Flow Processes in a Managed Aquifer Recharge Area in an Alluvial Fan" *Water* 13, no. 12: 1649.
https://doi.org/10.3390/w13121649