# Modelling Recharge from Irrigation Developments with a Perched Water Table and Deep Unsaturated Zone

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- develop and test a semi-analytical unsaturated zone model to predict recharge under new irrigation developments with perched water tables. The main attributes being sought are:
- (a)
- a conceptual model representing the main physical processes at the scale of the irrigation district and periods of seasons and years;
- (b)
- continuity of modelling between conditions of perching and non-perching;
- (c)
- a limited number of additional parameters;
- (d)
- availability of field data that could be used to calibrate parameters;
- (e)
- benchmarking against appropriate numerical models;
- (f)
- use of agronomic modelling outputs as input and generates recharge as output to be used in groundwater models; and
- (g)
- a process for estimating recharge under brownfield developments.

- explore the use of even simpler modelling approaches based upon:
- (a)
- superposition of actions; and
- (b)
- conceptual approximants for transfer functions.

## 2. Theory

_{b})

^{−}

^{λ}, ψ > h

_{b}

_{b}

_{b}is the air-entry point, λ is a fitting parameter, and the relative saturation, $\Theta $, is given by

_{r}is the residual volumetric water content and θ

_{s}is the saturation volumetric water content. The relative permeability, K

_{r}, is given by:

_{r}=Θ

^{m},

_{r}by the saturated hydraulic conductivity. These parameters will be different for the different layers and a subscript i will be subsequently used to distinguish layer i. The parameters are assumed to be the same for both vertical and horizontal properties, except for the saturated conductivity. A superscript h and v will be used with the saturated hydraulic conductivity to characterise anisotropy. The values given to the parameters is given in the Methods section.

_{2}, timescale S

_{2}l

_{2}/K

_{s2}

^{v}, horizontal length scale x

_{0}, and vertical flux K

_{s2}

^{v}; where l

_{i}is the thickness of the ith layer; x

_{0}is the half-width of the irrigated area; K

_{si}

^{v}is the saturated vertical conductivity of the ith layer; and S

_{2}is the specific yield for the 2nd layer for the initial dry conditions. The purpose of the non-dimensionalisation is to simplify the range of situations as much as possible using scaling and non-dimensional variables.

#### 2.1. Unsaturated Zone Conditions

_{wf}/dt = (A

_{n}− A

_{o})/(θ

_{n}− θ

_{o})

_{wf}is the depth of the wetting front below the land surface; A

_{n}and A

_{o}are the non-dimensional irrigation accessions for the irrigated and pre-existing agriculture respectively; and θ

_{n}and θ

_{o}are the volumetric water contents above and below the wetting front. Their values are such that the relative vertical hydraulic conductivity equals A

_{n}and A

_{o}, respectively. When the pressure front reaches the water table, the recharge (dimensioned) increases from IA

_{o}to IA

_{n}. Equation (5) can be used to estimate the time delay between the change in land use and the change in groundwater recharge. The above theory, or variants of it, has been used to estimate time delays for changes in non-irrigated agriculture to affect the underlying groundwater. In this paper, we look to adapt this model to the situation, where the soil conductivity of parts of the vadose zone is sufficiently low to not allow the new water flux to move vertically by gravity alone. The simplicity of the model is appropriate for our knowledge of soil properties and input fluxes over representative scales. The parameters in Equations (1)–(4) will change for each layer. A subscript ‘i’ will be used to denote these parameters for layer i.

#### 2.2. Situations Where Perched Water Tables Occur

_{n}to A

_{o}.

- Darcy’s Law is applied to the saturated zone in the upper clay layer. We assume that all of the hydraulic resistance is due to the clay and hence proportional to the thickness of the saturated layer, while the ponded head means a hydraulic gradient greater than one, purely due to gravity.
- The ponded head increases to the stage where the irrigation accession flux can pass through layer 2, or it fills all of layer 1.
- The perching results in a distribution of transit times for a change in irrigation accession rate to reach the water table.
- There may be a difference in magnitude between the irrigation accession and recharge, but only where some of the irrigation accession is returned to the land surface. If not, at equilibrium the recharge rate equals the irrigation accession rate.
- We have found it necessary to no longer consider the wetting front as a sharp transition from pre-development conditions to saturation. The existence of air-entry suction, below which hydraulic conditions the same as saturation occur and the near saturated zone means that the transition can be significant.
- As the wetting front moves through layer 3, the increasing vertical water flux at the base of layer 2 can lead to the wetting front moving more quickly as it moves to the water table.

#### 2.3. Stage 1

_{1}= l

_{1}(θ

_{n}

_{1}− θ

_{o}

_{1})/((A

_{n}− A

_{o})l

_{2})

_{1}is sensitive to θ

_{n}

_{1}, this needs to be calculated for the new flux.

#### 2.4. Stage 2

_{2}= ∫(θ − θ

_{b})dz/(A

_{n}− A

_{o}),

_{b}is the background water content. For layer 1, this background value is the new moisture content and for layer 2, this is the old moisture content. We will return to the calculation of integral (7) later.

#### 2.5. Stage 3

_{s}

_{1}

^{h}l

_{2}

^{2}/(K

_{s}

_{2}

^{v}x

_{0}

^{2})

_{n}/K

_{s2}

^{v}

_{s}

^{1}− θ

_{n}

^{1})/(θ

_{s}

^{2}− θ

_{o}

^{2})

_{1}is the edge of the wetted zone outside of the irrigation field; x

_{0}is the half-width of the irrigated field; and q is the vertical flux into layer 2. Continuity in h and the flux of water (and therefore gradient in h) is assumed to occur at x = 1. At x = x

_{1}, h is zero.

_{1}/l

_{2}

_{sat}

_{sat}is the depth of the saturation front. This assumes that the main hydraulic impedance is in the second layer. Under the wetting front model,

_{wf}/dt = q

- We shall ignore the effect of the ponded head outside the irrigated field on the infiltration into the impeding layer, i.e.,

_{wf}/dt = 1, 1 < x < x

_{1}

- 2.
- We shall assume quasi-steady-state Depuit–Forchheimer equations for this area, which leads to the following equations:

_{1}− x)/sqrt(B), 1 < x < x

_{1}

_{1}= h

_{1}sqrt(B) + 1

_{,}and

_{1}is the head of the perched layer at x = 1.

- 3.
- We shall assume that the head is constant across the irrigated field. Combining Equations (8), (9), (18) and (19) gives:

_{1}/dt = A − q − h

_{1}sqrt(B). 0< x < x

_{1}

- 4.
- We shall assume in early times of ponding that the lateral movement is small, and processes are vertical. We shall also assume that the separation between saturation fronts and wetting fronts is constant. By defining the dimensionless parameter:α = h/z
_{sat,}_{sat}and h increase linearly:z_{sat}= (1 + α)th = α(1 + α)tα = (−(1 + β) + sqrt((1 + β)^{2}+ 4(A − 1)β)/(2β), andq = 1 + α

_{r}(ψ) − 1)l

_{2}),

_{b2}/((q − 1)l

_{2})+ ∫ dψ/((q/(K

_{r}(ψ) − 1)l

_{2})

_{3}, and in Equation (27) from h

_{b}

_{2}to ψ

_{3}, where ψ

_{3}is either (a) the matric potential relating to the pre-development drainage, where the transition zone is entirely within the clay layer or (b) the soil matric suction at the interface with layer 3. We shall assume that φ is constant with respect to q.

_{3}= (1 − φ/α)/(1 + α),

_{o}= α − φ

_{2}in Equation (7).

#### 2.6. Stage 4

_{eq}+ exp(−(t − t

_{0})(1 + sqrt(B))/β)(h

_{0}− h

_{eq})

_{0}and t

_{0}are, respectively, the head and time at which the wetting front breaks through the clay layer Equation (29). The equilibrium head, h

_{eq}, is given by:

_{eq}= (A − 1 − φ)/(1 + sqrt(B))

_{s}= l

_{2}S

_{2}β/((1 + sqrt(B))K

_{s}

_{2}

^{v})

_{s}becomes:

_{s}~ x

_{0}s

_{1}/sqrt(K

_{s}

_{1}

^{h}K

_{s}

_{2}

^{v})

_{eq}from q

_{0}

_{.}Equation (30) and Equation (32). The extent of the wetting outside the irrigation field also increases exponentially to the equilibrium value x

_{1eq}from x

_{10}in parallel with the ponded head (Equation (18) and Equation (32)). Equation (16) implies that the aggregated vertical flux through the clay external to the irrigation field is proportional to the extent of wetting.

_{wf}, is likely to change as the flux at the bottom of layer 2 changes gradually. The change in flux (and associated water content) will move at a speed determined by the group velocity, dK/dθ. This will continue until the change reaches either the wetting front or the capillary fringe (in the case, where the wetting front has already reached the capillary fringe). In the former situation, the time at which the change reaches the wetting front is given by:

_{4}= z

_{wf}/(dK/dθ(θ(t

_{0}))) = z

_{wf}/v

_{g},

_{4}is the time at which the change at the change occurs at the bottom of layer 2 and dK/dθ(θ(t

_{4})) = v

_{g}is the group velocity as determined there.

_{wf}to be calculated by integrating the following equation:

_{wf}= v

_{wf}Δt

_{4}/(1 + v

_{wf}/v

_{g})

_{wf}is the velocity of the wetting front and v

_{g}is the group velocity. If dK/dθ >> ΔK/Δθ, as is the case for our parameterisation of layer 3, this simplifies to:

_{wf}= ∫v

_{wf}dt

_{4}

#### 2.7. Stage 5

_{eq}= (A − 1 − φ)/(1 + sqrt(B)), h > l

_{1}/l

_{2}

_{n}− IA

_{o})

#### 2.8. Theory: Summary

- Irrigation development leads to the formation of a wetting front that moves through layer 1.
- Once the wetting front reaches the interface between layers 1 and 2, the low permeability of layer 2 means that moisture begins to accumulate about the interface.
- The ponded head is zero until the end of Stage 2, increases linearly until end of stage 3 and then exponentially asymptotes to an equilibrium head during stages 4 and 5.
- The flux at the base of layer 2 is zero until the end of stage 3 and then increases exponentially to the irrigation accession rate.
- The recharge rate is zero until the end of stage 4 and then increases exponentially to the irrigation accession flux.
- While the ponded head increases to the stage where the irrigation accession flux passes through layer 2, the value of this may be so high that it intercepts the upper surface.

## 3. Methods

#### 3.1. Model Implementation

_{o}, has been assumed to be 10 mm/year for all experiments. For the two-dimensional experiments, the half-width of the irrigation area is assumed to be 500 m. Pertinent information on the irrigation water balance is from [24].

- 1–6 (1D), 1–4 (2D) are a series of simulations for new developments that cover a range of non-perched and perched situations and illustrate varying degrees of lateral movement. These simulations demonstrate the main processes and the outputs allow benchmarking of the models. For each experiment, the PerTy3 and FEFLOW models are used.
- The experiments 5–10 (2D) are designed, in conjunction with Experiments 1–4 (2D), to explore the effect of a brownfield developments in the vicinity of a development, already at equilibrium. More specifically, the recharge under a greenfield development (1–4(2D)) is compared to brownfield developments either directly adjacent to or 250 m away from a development at equilibrium. The FEFLOW outputs will be compared to the superposition of the two developments.
- Experiments 10–13 (2D) are designed to compare recharge brownfield sites in the vicinity of greenfield sites, that are 5 years old. The FEFLOW outputs will be compared to the superposition of the two developments.

#### 3.2. Transfer Functions and Superposition

_{n}* − IA

_{o}*)

_{o})/(IA

_{n}** − IA

_{o}**)

_{o}is the original drainage rate, IA** is the maximum of IA and the maximum irrigation accession that occurs without rejected recharge.

_{j}(IA*

_{j+1}− IA*

_{j}) TF’

_{j+1})/(IA*

_{p+1}− IA*

_{0})

_{j}is a sequence of modified irrigation accessions that occur from j = 0 to j = p + 1 and TF’

_{j+}

_{1}is the modified transfer function that applies for a change of irrigation accession from IA*

_{j}to IA*

_{j+}

_{1.}

#### 3.3. Approximants

_{6})), t > t

_{5}

_{5}and t

_{6}are fitted parameters. Such models have previously been used [18] for recharge through a deep vadose zone. Such conceptual models have been used widely in surface hydrology to calibrate surface flow models. The linear reservoir model forms a good approximation where the output (recharge) is a linear function of the storage (the mass of water in the unsaturated zone) [18].

## 4. Results

#### 4.1. One-Dimensional Modelling

#### 4.2. Two-Dimensional Modelling

#### 4.3. Modelling of Brownfield Developments

#### 4.4. Approximants

## 5. Discussion

#### 5.1. Accuracy of Modelling

#### 5.2. Brownfield Developments

#### 5.3. Approximants

#### 5.4. Data Requirements

#### 5.5. Further Work

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Conceptualizations of the Loxton-Bookpurnong Irrigation District, showing perched water table under the irrigation district, and groundwater flow to the River Murray. model used to simulate recharge under perched water tables.

**Figure 2.**Used to simulate recharge under perched water tables. The left-hand boundary is a no flow boundary, representing a line of symmetry. The variables are non-dimensionalised, with x = 1 being the outer limit of irrigation and x = x

_{1}being the outer limit of perched water. Below layer 3 is the saturated zone of the aquifer.

**Figure 3.**Figures showing two aspects of the modelling: (

**a**) near-saturated zone between wetting front and saturation front, as described in Equation (26); and (

**b**) changing speed of wetting front, as flux behind increases.

**Figure 4.**Of outputs from one-dimensional outputs from PerTy3 (semi-analyt-ical) (dashed) and FEFLOW (numerical) (solid) models (

**a**) transfer functions; (

**b**) ponded head and (

**c**) normalized drainage volume.

**Figure 5.**Comparison of outputs from two-dimensional semi-analytical and numerical modelling for new accession rate of 200 mm/year: (

**a**) Transfer functions (Total) for B = 0.1, 1, 10; (

**b**) Transfer functions (total, under irrigation field, external to irrigation field) for B = 10. (

**c**) Perched heads for B = 0.1, 1.0, 10 for an increase in IA to 200 mm/year. The head for the FEFLOW model is for x = 0, while that for PerTy3 is an average across the irrigated field. (

**d**) the width of the wetted zone outside the irrigation field for B = 0.1, 1.0, 10, and an increase in IA to 200 mm/year.

**Figure 6.**Functions for brownfield developments: (

**a**) numerical outputs at varying distances from a pre-existing steady-state development and B = 0.1; (

**b**) numerical outputs at varying distances from a pre-existing steady-state development and B = 10; (

**c**) numerical outputs for the total development of a new development followed by another development, 250 m away for B = 1, 10 compared to superposition of numerical outputs for two independent developments, one 5 years after the other.

**Figure 7.**Approximants against numerical solutions for which perching occurs. Solid lines represent numerical solutions and dashed lines approximants. (

**a**) 1D modelling experiments 3, 4 and 6. The approximants are respectively: 1 − exp(− 0.07 × (t + 10)), t < 12; 4: min(0.486,1 − exp(−0.18 × (t − 14))), t > 15; and 6: 1 − exp( − 0.22 × (t + 4.4)), t > 5. (

**b**) 2D Modelling Experiments 2, 3 and 3a. The approximants are respectively: 2: 1 − exp(− 0.12 × (t + 2.5)), t > 6; 3: 1 − exp( − 0.13 × (t − 0.8)), t > 4; and 3a:1 − exp(−0.15 × (t + 1)), t > 5.

**Table 1.**Glossary of parameters, symbols and units used in equations and the equation number, where first used.

Parameter | Unit | Symbol | Equation First Used |
---|---|---|---|

Dimensioned parameters | |||

Thickness of layer i | cm | l_{i} | (6), (10) |

Time scale associated with equilibration process of perched water table | year | t_{s} | (34) |

Soil conductivity for layer i—dimensioned (current, saturated horizontal, saturated vertical) | cm/day | K, K_{si}^{h}, K_{si}^{v} | (10) |

Irrigation accession flux (current, new, old, for step j) | mm/year | IA, IA_{n}, IA_{o}, IA_{j} | (11) |

Half-width of the irrigated area | m | x_{0} | (10) |

Air-entry potential | cm | h_{b} | (1) |

Soil water suction (negative potential) | cm | ψ | (1) |

Recharge to the water table (current, change) | mm/year | R, ΔR | (41) |

Sub-surface drainage | mm/year | D | (40) |

Dimensionless parameters | |||

Depth of the wetting front (wf) or saturation front (sat) below the base of layer 1 | z_{wf}, z_{sat} | (5) | |

Lateral distance from centre of irrigation field | x | (8) | |

Time (current, initial, for wetting front to reach base of layer 1, time for saturation to occur, time for wetting front to reach base of layer 2, time for which flux changes at base of layer 2) | t, t_{0}, t_{1}, t_{2}, t_{3}, t_{4} | (5), (6), (7), (45) | |

Soil volumetric water content for layer i (current, residual, saturated, new, old) | θ_{i}, θ_{ri}, ${\theta}_{si}$, θ_{n}, θ_{o} | (3), (5) | |

Relative permeability for layer i | K_{ri} | (3) | |

Ratio of specific yields for layers 1 and 2 | β | (8) | |

Relative saturation | Θ | (1) | |

Specific yield for the ith layer for the initial dry conditions | Si | (6) | |

Mualem exponent | m_{i} | (4) | |

Coefficient for soil water retention curve | λ | (1) | |

Head of perched water table (current, initial, equilibrium, at edge of irrigation field) | h, h_{0}, h_{eq,} h_{1} | (13), (7), (32) | |

Dimensionless irrigation accession (current, new old) | A, A_{n}, A_{o} | (5) | |

Dimensionless parameters related to thickness of near-saturated zone | φ | (26) | |

Thickness of transitional zone between saturation front and wetting front | Φ | (26) | |

Dimensionless parameter related to the significance of lateral movement | B | (8) | |

Vertical water flux through impeding layer (current, equilibrium, old) | q, q_{eq}, q_{0} | (8) | |

Specific yield of layer 1, relative to soil following passage of wetting front | s_{1} | (35) | |

Width of wetted zone outside irrigation field (current, equilibrium, old) | x_{1}, x_{1eq}, x_{10} | (8), (18) | |

Transfer function for the recharge | TF | (41) | |

Proportionality constant between perched head and depth of saturation front during stage 3 | α | (21) | |

Velocity through layer 3 (group velocity for pressure changes, wetting front velocity) | v_{g}, v_{wf} | (36), (37) | |

Dimensionless parameter depicting lateral movement | B | (8) | |

Fitted parameter for approximants | a | (45) |

Parameter | Symbol | Layer 1 | Layer 2 | Layer 3 |
---|---|---|---|---|

Texture | Sandy Loam | Clay | Sand | |

Saturated volumetric water content (cm^{3}/cm^{3}) | θ_{si} | 0.35 | 0.4 | 0.38 |

Residual water content (cm^{3}/cm^{3}) | θ_{ri} | 0.03 | 0.1 | 0.04 |

Air-entry potential (cm) | h_{bi} | 12.0 | 40.0 | 8.0 |

Mualem exponent | m_{i} | 8.24 | 7 | 6.94 |

Vertical saturated hydraulic conductivity (cm/day) | K_{si}^{v} | 300 | various | 500 |

Anisotropy for saturated conductivity (Horizontal/Vertical) | ~0 (1D) various (2D) | ~0 (1D) 1(2D) | ~0(1D) 1 (2D) | |

Thickness (cm) | l_{i} | 500 | 500 | 1500 (1D) 500 (2D) |

Model Expt Number | Vertical Saturated Hydraulic Conductivity (cm/day) | Anisotropy for Saturated Conductivity (Horizontal/Vertical) | New Irrigation Accession (mm/year) | Parameter | Parameter | Separation from Existing Development (m) |
---|---|---|---|---|---|---|

K^{v}_{si} | IA_{n} | A | B | |||

1 (1D) | 0.0913 | ~0 | 100 | 0.3 | ~0 | |

2 (1D) | 0.0365 | ~0 | 100 | 0.75 | ~0 | |

3 (1D) | 0.0183 | ~0 | 100 | 1.5 | ~0 | |

4 (1D) | 0.00685 | ~0 | 100 | 4.0 | ~0 | |

5 (1D) | 0.146 | ~0 | 400 | 0.75 | ~0 | |

6 (1D) | 0.067 | ~0 | 400 | 1.5 | ~0 | |

1 (2D) | 0.03 | 1000 | 200 | 1.83 | 0.1 | |

2 (2D) | 0.03 | 10,000 | 200 | 1.83 | 1 | |

3 (2D) | 0.03 | 100,000 | 400 | 3.65 | 10 | |

3a (2D) | 0.03 | 100,000 | 200 | 1.83 | 10 | |

4 (2D) | 0.03 | 10,000 | 100 | 0.91 | 1 | |

5 (2D) * | 0.03 | 1000 | 200 | 1.83 | 0.1 | 0 |

6(2D) * | 0.03 | 10,000 | 200 | 1.83 | 1 | 0 |

7(2D) * | 0.03 | 100,000 | 200 | 1.83 | 10 | 0 |

8(2D) * | 0.03 | 1000 | 200 | 1.83 | 0.1 | 250 m from equilibrium development |

9(2D) * | 0.03 | 10,000 | 200 | 1.83 | 1 | “ |

10(2D) * | 0.03 | 100,000 | 200 | 1.83 | 10 | “ |

11(2D) * | 0.03 | 1000 | 200 | 1.83 | 0.1 | 250 m from 5-year-old development |

12(2D) * | 0.03 | 10,000 | 200 | 1.83 | 1.0 | “ |

13(2D) * | 0.03 | 100,000 | 200 | 1.83 | 10.0 | “ |

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## Share and Cite

**MDPI and ACS Style**

Walker, G.R.; Currie, D.; Smith, T.
Modelling Recharge from Irrigation Developments with a Perched Water Table and Deep Unsaturated Zone. *Water* **2020**, *12*, 944.
https://doi.org/10.3390/w12040944

**AMA Style**

Walker GR, Currie D, Smith T.
Modelling Recharge from Irrigation Developments with a Perched Water Table and Deep Unsaturated Zone. *Water*. 2020; 12(4):944.
https://doi.org/10.3390/w12040944

**Chicago/Turabian Style**

Walker, Glen R., Dougal Currie, and Tony Smith.
2020. "Modelling Recharge from Irrigation Developments with a Perched Water Table and Deep Unsaturated Zone" *Water* 12, no. 4: 944.
https://doi.org/10.3390/w12040944