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Applicability of the Unit Response Equation to assess salinity impacts of irrigation development in the Mallee region D. Rassam, G. Walker, and J. Knight

CSIRO Land and Water Technical Report No. 35/04 October 2004

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ISSN 1446-6171

Cover Photograph: From CSIRO Land and Water Image Gallery: www.clw.csiro.au/ImageGallery/ File: PDA00066_034.jpg Description: Storm approaching as vehicular ferry crosses the Murray River at Waikerie, SA. 1989. Photographer: Willem van Aken © 2004 CSIRO

Applicability of the Unit Response Equation to assess salinity impacts of irrigation development in the Mallee region David Rassam1, Glen Walker2, and John Knight1

1

CSIRO Land and Water, Long Pocket Laboratory, Indooroopilly, QLD 4048

2

CSIRO Land and Water, Glen Osmond, South Australia

Technical Report No. 35/04 October 2004

Acknowledgements The work contained in this report is collaboration with CSIRO Water for Healthy Country; the work undertaken is a component of the Lower Murray Landscape Futures project. The Water for a Healthy Country National Research Flagship is a research partnership between CSIRO, state and federal governments, private and public industry and other research providers. The Flagship was established in 2003 as part of the CSIRO National Research Flagship Initiative. The authors would like to acknowledge the Water Trade Salinity Impact Evaluation Panel (of MDBC) convened by Bob Newman, and the WATSIEP project consortium led by David Fuller of URS, SA, and consisting of: Nick Watkins and Juliette Woods of AWE, SA, Greg Hoxley of SKM, VIC, and Mat Miles of the Department of Environment and Heritage, SA. The authors greatly appreciate the constructive comments made by Mike Williams of DIPNR, NSW.

EXECUTIVE SUMMARY

SIMRAT and SIMPACT are Geographic Information System (GIS) – based models that relate land use in the Mallee Region to salt load impacts on the River Murray. SIMRAT and SIMPACT are effectively identical in many ways. The main distinction being that the SIMRAT (model and data) has been accredited under the Murray-Darling Basin Salinity Management Strategy (BSMS) to assess the salinity impacts of interstate water trade and to assess South Australian salinity credits under the BSMS. SIMPACT on the other hand is a primary tool that has been used within South Australia to define high, medium and low salinity impact zones for horticultural development in the South Australian Mallee, and has been recently modified to include the salinity impacts of changed land use in dryland areas. The advantage of these tools over traditional groundwater models such as MODFLOW is that they explicitly show the spatial distribution of high-impact zones, they can accumulate the salinity impacts of individual actions, and outputs can be combined easily with other spatial datasets to provide more integrated natural resource analyses. It is for these reasons that SIMPACT will be used in the Water for a Healthy Country and Land technologies Alliance project: ‘Lower Murray Futures’. However, these advantages come at the expense of some modelling assumptions. As part of the BSMS accreditation process, some areas were masked as areas where the SIMRAT model may have unacceptable errors. Reasons for the masking included areas where groundwater gradients were away from or parallel to the River Murray, areas where the thickness of the aquifer was much greater due to irrigation mounds and areas where the River Murray was embedded in large thickness of low permeability Clay. These perceived errors are associated with the applicability of the ‘Unit Response Equation’ (URE) of Knight et al. (2002) and Knight et al. (2004) used in SIMRAT/SIMPACT to relate changed groundwater recharge to changed salt loads to the River Murray. While there were no suggestions of errors in this equation, the masking was based on perceived limitations in the applicability to the real world of the Mallee. The large areas of masking could eventually lead to questioning of credits based on the SIMRAT model, as well as irrigation zoning arising from the use of SIMPACT. In conjunction with the Murray-Darling Basin Commission (MDBC) Water Trading Salinity Impacts Evaluation Panel (WTSIEP), a project was defined to test the criteria used for the masking. The results will be immediately used to revise SIMRAT masked areas and more importantly consolidate policy and zoning decisions based on the SIMPACT model. A key assumption in the model is linearity of the equations, which allow multiple impacts due to different actions to be superimposed. If linearity holds, then this implies that regional gradients, or local gradients induced by large irrigation mounds do not affect the salinity impact of an individual development. Objectives of the current study are therefore to: •

Better describe the assumption of linearity of the URE and the associated ‘superposition principle’ and test limits of applicability.

•

Explain the meaning of hydrological response times and hence present a sensitivity analysis to the relevant model parameters.

•

Provide comparisons of URE outputs with those from the widely used groundwater model MODFLOW. In discussions, it is clear that most hydrogeologists working in the area are familiar with MODFLOW and would trust models whose outputs can be favourably compared to those from MODFLOW.

•

Recommend criteria where the basic URE is likely to be applicable.

CSIRO Land and Water

1

Key findings of the study: 1. MODFLOW analyses have shown that the URE, linearity and superposition principles generally apply for Mallee situations. It needs to be recognised that for most situations in which SIMRAT will be used, there are significant confidence limits for aquifer properties and deep drainage rate under irrigation. An appropriate choice of thickness needs to be used in the URE in areas, where aquifer thickness may vary due to existing large irrigation developments. The MODFLOW simulations show that where groundwater gradients are due to recharge sources, rather than sloping aquifers, the URE can be applied even where groundwater gradients are away from or parallel to the river. A corollary of this is that sites behind an irrigation development are not a low salinity impact zone for an irrigation development, unless by doing so it places the irrigation development sufficiently far from the river. 2. Invoking the concept of ‘Hydrological Response Time’ and non-dimensionalisation of parameters has resulted in simple approaches to compare responses of different aquifer systems of varying properties. For example, by using such a concept, it is possible to develop criteria for when an irrigation development is sufficiently large to reverse gradients near the river or to create such a large groundwater mound, that the URE would not be appropriate to use without modification. 3. Modified forms of the URE are shown to be effective in situations where the simple formula fails; these situations are: a. A no-flow boundary located close to the river. b. A sloping aquifer. c. A meandering river. Recommendations: 1. A modified version of the URE is recommended to be used under the conditions, where: •

The irrigation development is less than four times the distance to a no-flow boundary.

•

The slope of the aquifer is higher than 1.5%.

•

For a meandering river, if the meander is closer than 4 times the shortest distance to the river.

Such areas should be flagged so that a modified URE may be applied by an experienced user. 2. A suitable aquifer thickness should be calculated when applying the URE to aquifers with existing gradients. The non-dimensional plot for the head response may be used to predict the head distribution, and then a weighted mean approach is used to calculate the aquifer thickness. 3. The non-dimensional plot for the head response may be used to predict the size of a new irrigation development such that a certain criteria for head rise will not be violated. 4. Masking of SIMRAT should be reviewed with these analyses.


2

Table of Contents 1 Introduction

5

2 Description of the Mallee region

6

3 Description SIMPACT and SIMRAT

7

4 Linearity and Superposition

7

5 Objectives and modelling experiment design

9

6 Understanding the unit response equation (URE) 6.1

6.2

6.3

6.4

14

Sensitivity Analysis

14

6.1.1 Flux response

14

6.1.2 Head response

16

Non-dimensional Analysis

18

6.2.1 Flux response

18

6.2.2 Head response

20

Predicting aquifer thickness

21

6.3.1 Pulse recharge example

21

6.3.2 Step recharge example

22

Summary

23

7 Comparison with MODFLOW simulations

23

7.1

Flux response

23

7.2

Head response

24

7.2.1 Pulse recharge

24

7.2.2 Step recharge

24

Summary

25

7.3

8 The Principle of Superposition

25

8.1

Two identical sources

25

8.2

Two different sources

26

8.3

Two different sources with time lags

27

8.4

Using the URE for a single source to predict response to a rectangular area

27

8.5

Summary

29

9 Modified forms of the URE

29

9.1

Presence of no-flow boundary

29

9.2

Sloping base aquifer

31

9.2.1 Response for down-gradient case

32

9.2.2 Response for reverse-gradient case

32

Recharge to an aquifer with multiple constant-head boundaries; a meandering river

33

9.3

o

9.3.1 Two orthogonal rivers; a 90 -bend

33

9.3.2 Three orthogonal rivers; a U-shaped meander

35

9.3.3 Effect of discharge edge geometry

36


3

9.4

Summary

10 Stretching the applicability of the URE

37

38

10.1 Flat-base aquifer with groundwater gradient

38

10.2 Presence of groundwater mounds

40

10.2.1 Transient mounds of varying widths and thicknesses

40

10.2.2 New development in front and behind large existing mound

42

10.3 Response of a heterogeneous aquifer

47

10.3.1 Three-layered aquifer; K3 < K2 = K1

48

10.3.2 Three-layered aquifer; K2 < K1 = K3

49

10.4 Summary

49

11 Discussion

50

12 Conclusions and recommendations:

53

Appendix I: Flux distribution along a river

54

I.1

Mathematical formulation

54

I.2

Analysis

55

Appendix II: Flux response with 3 orthogonal constant-head boundaries; a U-shaped meander 57 References


61

4

1

Introduction

The Murray-Darling Basin (MDB) Salinity Audit (1999) showed that the Mallee Region of South Australia was one of the largest salt contributors to the Murray-Darling System. Consequently, there are a number of salinity amelioration approaches being suggested for the region including salt interception schemes, irrigation zoning, improvements in water use efficiency and reduction in dryland recharge through revegetation or introduction of perennials. A number of these require spatial identification of areas with the greatest salinity impacts. Underlying these schemes is accountability designed as part of the MDB Basin Salinity Management Strategy (BSMS, 2001). This requires accounting for both positive and negative salinity impacts of salt amelioration schemes, irrigation and dryland developments and riparian rehabilitation. One of the greater difficulties is that there are many individual decisions made on the delivery of irrigation water and there is a need for the cumulative impact of these to be assessed. Recently, a model, SIMRAT, has been developed within a Geographic Information System (GIS) to assess the impact of irrigation trade. At the heart of the model is the ability to link irrigation development in any given cell to a salinity impact on the river and accumulate these. This model is being used to support the assessment of South Australia’s salinity credits, assessment of the salinity impacts of interstate trade to, from or within the Mallee Region and (using SIMPACT) to support irrigation zoning within the South Australian Mallee. Originally, in the development of iRAT (URS and AWE, 2001), the groundwater function used to link land use to the river corridor was based on a MODFLOW output developed by Watkins and Waclawik (1996). During the development of SIMRAT, this was changed to the Unit Response Equation (URE) developed by Knight et al. (2002). In either case, so-called linearity was assumed that allowed ‘superposition’ i.e. accumulation of impacts. As part of the accreditation under the Basin Salinity management strategy, a number of situations were identified in which there was insufficient documentation to illustrate that SIMRAT could be used. These included: •

Regional groundwater gradients were away from the river

•

Local groundwater gradients induced by groundwater mounds under irrigation areas were away from the river,

•

Areas in which there were two effective aquifers,

•

Areas in which there were large changes in aquifer thickness, and

•

Areas in which the river was embedded in a large thickness of low permeability clay.

Consequently, large areas of the Mallee were masked as areas in which the SIMRAT model as either inappropriate or at least, required some demonstration that it was applicable. Fortunately, most of the interstate water trade occurs within the unmasked regions. Underlying the above questions is not just whether a specific model is appropriate but broader questions regarding what constitutes a high-impact zone. For example, if an irrigation development sits behind a large existing irrigation development for which there is a groundwater mound, some hydrogeologists consider this to have a low-salinity impact as the groundwater gradients are away from the river. This report describes work undertaken under the Lower Murray Futures project of the River Murray node of Water for a Healthy Country. Water for a Healthy Country is a national research program coordinated by CSIRO focusing on water, its use and values. The Lower Murray Futures is a collaborative project involving South Australian and Victorian governments and CSIRO, aimed at understanding how the future Lower Murray landscape may change to address regional natural resource targets, particularly those related to water.


5

An important regional NRM target for the Mallee component of the Lower Murray is that relating to salt loads to the River Murray and the Basin-wide stream salinity target of below 800 EC at Morgan for at least 95% of the time. SIMPACT is being used as part of that project as the model linking land use to the salt load target. Thus, it is important to test whether the model is appropriate for use in that project. The objective of the work described in this report is to: •

Describe the concepts of linearity and superposition and what this implies for the applicability of SIMRAT/SIMPACT and the URE in areas where regional or local gradients are away from the river, and

•

Develop criteria for the applicability of SIMRAT/SIMPACT and the URE for areas of sloping aquifers, changing aquifer thickness, multiple aquifers and meandering rivers, all of which occur in the Mallee Region.

The work is being conducted in conjunction with the Murray-Darling Basin Commission’s WTSIEP Committee, which aims to develop a pilot scheme to assess the salinity impacts of interstate water trade. It was also done in conjunction with the consortium adapting SIMRAT to this application and the reviewer, Hugh Middlemis, contracted under the MDBC accreditation process.

2

Description of the Mallee region

The Mallee region extends across New South Wales (NSW), Victoria (VIC), and South Australia (SA), and covers areas of 19,930 km2, 59,620 km2, and 48,500 km2, respectively (MDBMC, 1991). The region shares its name with the dominant native vegetation, which includes several species of eucalyptus. The main periods of land clearing in SA and VIC were during 1910-1930 and clearing in NSW took place mainly between 1960-1980; most native vegetation was replaced by dryland cropping, pasture, and grazing. Areas of irrigated agriculture occur within a 10 km strip of the River Murray. Much of the Mallee region is underlain by a predominantly marine aquifer, the Murray Group Limestone aquifer. In most parts, the Pliocene-Parilla Sands aquifer is above Murray Group. In SA, discharge from the Murray Group aquifer occurs along the Murray River. However, for much of VIC and NSW groundwater flow lines approximately parallel the river. Cook et al. (2001) provides a more detailed description for the hydrogeology of the area. Mean monthly evaporation exceeds monthly rainfall throughout the year. The deep-rooted Mallee vegetation uses most of the available water thus leaving a very low deep drainage component (water leaking below the root zone). As a result of evapotranspiration, the small amount of salt present in rainfall has concentrated over long periods of time and hence the high soil salinity in the area. The groundwater salinity ranges from 35,000 mg/L. As a result of land clearing, groundwater recharge is increased. The extra recharge that occurs under shallow-rooted vegetation mobilises the stored salts and carries them to the groundwater table. The saline water slowly moves to the River Murray due to low head gradients and long travel distances. However, new irrigation developments can exacerbate the situation as the significant amounts of recharge can mobilise higher amounts of salt in addition to causing head gradients that shorten the travel time to the river.


6

3

Description SIMPACT and SIMRAT

SIMPACT is a GIS framework used to assess the impacts of increased drainage on the Murray River salinity. The model was initially developed to investigate the salinity impacts of new irrigation developments in SA. The impact of a new irrigation development is delayed in the vertical direction as water moves through the unsaturated zone wetting up the soil profile and eventually reaches the water table to become recharge. In addition, it is also delayed horizontally as the water travels from the irrigated area to the river through the saturated zone (the aquifer). The vertical lag time is calculated from knowledge of the drainage rates and the thickness of the unsaturated zone. The horizontal lag time is calculated using the method of Watkins and Waclawik (1996). The SIMRAT model integrates the spatial analytical capacity of SIMPACT II (Miles et al., 2001) with the unsaturated zone method of Cook et al. (2004) and the unit response equation of Knight et al. (2002). The unsaturated zone method calculates recharge from knowledge of deep drainage (passing the root zone), depth to groundwater, clay thickness, and soil moisture contents. It incorporates a lognormal distribution for recharge that accounts for the spatial variability of soil properties, which results in a realistic smooth increase in recharge rate rather than a sudden step increase. The resulting recharge distribution is then fed into the unit response equation to calculate fluxes to the river.

4

Linearity and Superposition

An operator L is considered to be linear if:

L(ah1+h2) = aL(h1) + L(h2)

(1)

The groundwater equation (Equation 2) is linear if the transmissivity is constant. This occurs if: •

The aquifer is effectively confined

•

The aquifer is sufficiently thick, that changes in aquifer thickness can be ignored.

∂h Kh* ∂ 2h N(x, t ) = + ∂t φ ∂x 2 φ

(2)

where h* is the average aquifer thickness, h=H-h* (H is the height of the water table in an unconfined aquifer), t and x are time and spatial variables, respectively, K is the hydraulic conductivity, N is the recharge (source term), and φ is specific yield. The groundwater equation needs to be considered in conjunction with the boundary conditions. The most commonly used boundary conditions are: •

Constant head,

•

Constant flux,


7

•

Flux proportional to the difference between the head and some specified head, often that associated with surface water.

The constant head boundary condition is linear, while the two flux boundary conditions are linear if the transmissivity can be assumed to be constant i.e. under the same assumptions as the groundwater equation. We shall assume until specified otherwise that linearity applies. This implies that if h1 and solutions to the groundwater equation, together with boundary and initial conditions, then ah1+h2 is a solution to the groundwater equation with boundary and initial conditions that correspond to this sum. To better understand the corresponding boundary and initial conditions, suppose h1 equals a constant b1, at a part of the boundary, where a constant head condition holds and h2 equals a constant b2, then ah1+h2 satisfies the groundwater equation together with the constant head boundary condition h = ab1+b2

(3a)

Similarly, if h1 has the initial condition f1 and h2 has the initial condition f2, then ah1+h2 satisfies the groundwater equation, with boundary condition [2] and initial condition af1+f2. This property is very powerful. For the Mallee situation, it allows us to partition the current land use into 3 components: 1. Steady-state pre-development situation, with the corresponding solution hss, 2. Subsequent development, with corresponding solution hd, with zero boundary conditions, and 3. An individual action, with corresponding solution, ha. Under linearity, h = hss + hd + Σ ha

(3b)

i.e. the response to an individual action is independent of previous actions and hence can be accumulated (superposition). This implies that the salt load impact of a development would be independent of either locally induced or regional gradients. Linearity and superposition becomes even more powerful by considering generic situation such as an irrigation development that approximates some set shape e.g. approximates a point. This is the principle of the original iRAT methodology, where MODFLOW was used to generate the generic building blocks. This approach, called a Green Function methodology, has been used for a number of different physical situations and equation types. Knight et al. (2002) have developed the generic solutions for the groundwater equation, focussing on the groundwater discharge to a stream. Situations include different slopes and the effect of groundwater divides. The equation uses a linearised form of the Boussinesq equation. Hence, linearity and superposition are applicable. However, the assumptions underpinning the equation must be respected at all times; the formulation assumes a single horizontal aquifer having a uniform thickness that is at equilibrium; the applied recharge causes a small head rise relative to aquifer thickness such that the associated increase in aquifer transmissivity is negligible.


8

Given the high salinity of the Mallee region and the continuing irrigation developments in the area, a simple model (such as SIMPACT/SIMRAT) that assesses the impacts of increased drainage and associated rise in river salinity is highly desirable. However, linearity never truly exists and hence superposition never truly applies. Therefore, some approximations need to be made when using the model under real-life, non-linear conditions. The question is “under what conditions are these reasonable approximations?” that is, what is the extent of nonlinearity under which the linear model continues to produce acceptable results? There are many sources of non-linearity, they include: •

Factors associated with the hydrogeology of the area (i.e., the aquifer is not 1-layered horizontal, and/or, its thickness is not uniform). In NSW and VIC, gradients exist parallel or away from the river; there are multiple aquifers in areas such as Waikerie, Loxton, and Mildura. The non-linearity introduced by existing gradients becomes more significant if the head difference is large relative to aquifer thickness.

•

Previous land use that causes large groundwater mounds such as the irrigation mounds common in SA.

•

A large recharge is introduced into a thin aquifer (the head perturbations resulting from the applied recharge are significant relative to aquifer thickness and hence the subsequent variations in transmissivity cannot be neglected).

In this study, we investigate the different sources of non-linearity and provide criteria where the basic form of the URE used in SIMPACT/SIMRAT is applicable.

5

Objectives and modelling experiment design

The objectives of the current study are: •

Enhance the understanding of the URE.

•

Verify the URE outputs by comparing them to those obtained from MODFLOW simulations.

•

Better describe the assumption of linearity of the URE and the associated ‘principle of superposition’ and test limits of applicability.

•

Present alternative forms of the basic URE.

•

Recommend criteria where the basic URE is likely to be applicable.

The modelling experiment is designed to address these objectives. Table 1 shows the structure of the modelling experiment; the associated simulations are listed in Table 2. With the exception of simulations S20 to S22 that represent a realistic scenario, other simulations represent hypothetical scenarios that demonstrate the applicability of the URE over a wide range of parameters. We enhance the understanding of the URE in two ways: firstly, by the testing the sensitivity of fluxes and heads to relevant model parameters, and secondly, by explaining the meaning of hydrological response times and hence presenting non-dimensional relationships for fluxes and heads that are independent of aquifer-specific parameters. Simulations S1 to S7 of Section 6 serve this purpose. Most hydrogeologists are familiar with the MODFLOW model, which has become the industry standard modelling tool. Therefore, demonstrating that the URE outputs compare favourably to MODFLOW outputs further enhances the credibility of the URE. In Section 7, we present comparisons of flux and head predictions obtained from the two models (simulations S8 to


9

S10). MODFLOW is also used in Sections 8, 9, and 10 to demonstrate the concept of linearity and to model non-linear scenarios. The various MODFLOW simulations used are briefly described in Table 2. The linearity of the governing differential equation and the applicability of the principle of superposition is demonstrated in Section 8, simulations S11 to S14. Alternative forms of the URE are presented in Section 9; they are used in cases where a noflow boundary is located close to the recharge source, a sloping-base aquifer, and a meandering river. Simulations S15 to S19 show such applications. In Section 10 (Simulations S20 to S25), we test the applicability of the URE under non-linear conditions; the scenarios include groundwater mounds and heterogenous aquifers. Finally, we use the outputs of Sections 9 and 10 to recommend criteria where the basic URE is likely to be applicable.


10

Table 1: Modelling experiment Objectives 6.1

Understand the URE

Verify the URE Illustrate the principle of superposition

Present alternative forms of the URE

6.2

recharge Pulse recharge Step recharge

Simulation numbers* S1 and S2 S3, S4 (pulse); S5 (step) S1; S2 S6 (pulse); S7 (step)

6.3

Predicting aquifer thickness

7.1 7.2 8.1 8.2

Flux response Head response; pulse/step recharge Two identical sources Two different sources

8.3 8.4

Two different sources with lag time Response from a rectangular development

S13 S14

9.1 9.2

Presence of a no flow boundary Sloping base aquifer

9.3

Investigate non-linearity

Section numbers and descriptions 6.1.1 Flux response Sensitivity analysis 6.1.2 Head response; pulse/step recharge 6.2.1 Flux response Non-dimensional analysis 6.2.2 Head response; pulse/step 6.3.1 6.3.2

S2 S8 S9 (pulse); S10 (step) S11 S12

9.2.1

Down gradient

S15 S16

9.2.2

Reverse gradient

S16

Meandering river

S17; S18; S19

10.1 Flat base with gradient

S20

10.2 Groundwater mounds 10.3 Aquifer with 3 layers of varying conductivities K1, K2, and K3

10.2.1 10.2.2 10.3.1

Small transient mound Large mound at equilibrium K30 is: ∞

F2 (a, b, c, t ) = ∫ f2 (y, a, b, c, t )dy 0

∞

∑

=

(2mc + a) exp⎛⎜ − (2mc + a)2 ⎞⎟∞ exp⎛⎜ − (y − b)2 ⎞⎟dy + ⎟∫ ⎠0

4Dt ⎟⎠ ∞ (2mc − a ) exp⎛⎜ − (2mc − a )2 ⎞⎟∞ exp⎛⎜ − (y + b )2 ⎞⎟dy ∑ 4πDt 2 ⎜ ⎜ ⎟∫ 4Dt 4Dt ⎟⎠ m = −∞ ⎝ ⎝ ⎠0 m = −∞

∞

∑

=

m = −∞ ∞

∑

m = −∞


⎜ ⎝

4πDt 2

4Dt

⎜ ⎝

(II.6)

(2mc + a) exp⎛⎜ − (2mc + a)2 ⎞⎟ 4πDt

⎜ ⎝

2

⎡ ⎛ b ⎞⎤ π + Dt 1 erf ⎜ ⎟⎥ + ⎢ ⎟ 2 Dt ⎝ ⎠⎦ ⎣ ⎠

4Dt

(2mc − a ) exp⎛⎜ − (2mc − a )2 ⎞⎟ 4πDt 2

⎜ ⎝

4Dt

⎡ ⎛ b ⎞⎤ ⎟ πDt ⎢1 − erf ⎜⎝ 2 Dt ⎟⎠⎥ ⎣ ⎦ ⎠

(II.7)

58

(2mc + a) exp⎛⎜ − (2mc + a )2 ⎞⎟erf ⎛⎜

∞

∑

=

4πDt

m = −∞

4Dt

b ⎞ ⎟2 πDt ⎝ 2 Dt ⎠

⎟ ⎠

(II.8)

(2mc + a) exp⎛⎜ − (2mc + a)2 ⎞⎟erf ⎛⎜

∞

=

⎜ ⎝

2

∑

⎜ ⎝

2t πDt

m = −∞

4Dt

⎟ ⎠

b ⎞ ⎟ ⎝ 2 Dt ⎠

(II.9)

The instantaneous flux density outward on x=c, y>0 (flux along boundary B2; Figure II-1) is:

⎡ (c − a − 2mc )2 + (y − b )2 ⎤ c − a − 2mc exp ⎢− = ∑ ⎥ 8πDt 2 4Dt m = −∞ ⎦ ⎣ ∞

∂h −T 2 ∂x

x =c

⎡ (c + a − 2mc )2 + (y − b )2 ⎤ c + a − 2mc − ∑ exp ⎢− ⎥ 8πDt 2 4Dt m = −∞ ⎣ ⎦ ∞

−

⎡ (c − a − 2mc )2 + (y + b )2 ⎤ c − a − 2mc exp ⎢− ⎥ ∑ 8πDt 2 4Dt m = −∞ ⎣ ⎦ ∞

⎡ (c + a − 2mc )2 + (y + b )2 ⎤ c + a − 2mc exp ⎢− + ∑ ⎥ 8πDt 2 4Dt m = −∞ ⎦ ⎣ ∞

f2 (y, a, b, c, t ) = +

∞

∑

∞

∑

(2m + 1)c − a exp⎡− [(2m + 1)c − a]2 + (y − b )2 ⎤ 4πDt 2

m = −∞

⎢ ⎣

4Dt

(2m + 1)c + a exp⎡− [(2m + 1)c + a]2 + (y + b)2 ⎤ ⎢ ⎣

4πDt 2

m = −∞

(II.10)

⎥ ⎦

(II.11)

⎥ ⎦

4Dt

The total flux out on both vertical boundaries (B1 and B2) is:

⎡ (mc + a )2 ⎤ ⎛ b ⎞ mc + a F3 = ∑ (− 1) exp ⎢− ⎟ ⎥erf ⎜ 4Dt ⎦ ⎜⎝ 2 Dt ⎟⎠ 2t πDt m= −∞ ⎣ ∞

m

(II.12)

The instantaneous flux density outward on y=0, 0≤x≤c (flux along horizontal boundary B3; Figure II-1) is:

f3 (x, a, b, c, t ) = T


∂h2 ∂y

(II-13) y =0

59

∞ ⎡ (x − 2mc + a )2 + b 2 ⎤ ⎡ (x − 2mc − a )2 + b2 ⎤ b −b exp ⎢− exp ⎢− = ∑ ⎥ ⎥+ ∑ 2 2 4Dt 4Dt m = −∞ 8πDt ⎦ ⎣ ⎦ m = −∞ 8πDt ⎣ (II.14) ∞ ∞ ⎡ (x − 2mc + a )2 + b 2 ⎤ ⎡ (x − 2mc − a )2 + b2 ⎤ b −b exp ⎢− exp ⎢− + ∑ ⎥ ⎥+ ∑ 2 2 4Dt 4Dt m = −∞ 8πDt ⎦ ⎣ ⎦ m = −∞ 8πDt ⎣ ∞

=

∞ ⎡ (x − 2mc + a )2 + b 2 ⎤ ⎡ (x − 2mc − a )2 + b2 ⎤ b b exp exp − − ⎥ (II.15) ⎥ ∑ ⎢− ⎢ ∑ 2 2 4Dt 4Dt m = −∞ 4 πDt ⎦ ⎦ m = −∞ 4πDt ⎣ ⎣ ∞

The total flux out on the horizontal boundary is:

c

F4 (a, b, c, t ) = ∫ f3 (x, a, b, c, t )dx = 0

⎡ (x − 2mc − a )2 ⎤ ⎛ − b2 ⎞ c b ⎟ ⎜ exp exp ⎥dx ⎢− ∑ 2 ⎜ 4Dt ⎟ ∫ 4 Dt m = −∞ 4πDt ⎠0 ⎝ ⎦ ⎣ ∞

⎡ (x − 2mc + a )2 ⎤ ⎛ − b2 ⎞ c b ⎟⎟ ∫ exp ⎢− exp⎜⎜ − ∑ ⎥dx 2 4Dt m = −∞ 4 πDt ⎝ 4Dt ⎠ 0 ⎦ ⎣ ∞

= 2 πDt

⎛ − b2 ⎞ ∞ b ⎛ mc + a ⎞ n ⎟⎟ ∑ (− 1) erf ⎜ ⎜⎜ exp ⎟ 2 4πDt ⎝ 2 Dt ⎠ ⎝ 4Dt ⎠m = −∞

⎛ − b2 ⎞ ∞ b ⎛ mc + a ⎞ n ⎟⎟ ∑ (− 1) erf ⎜ exp⎜⎜ = ⎟ 2t πDt ⎝ 2 Dt ⎠ ⎝ 4Dt ⎠m = −∞

(II.16)

(II.17)

The total flux out is:

f5 (a, b, c, t ) = f3 + f4 =

(mc + a) m mc + a ∑ (− 1) 2t πDt exp⎢− 4Dt ∞

⎡

m = −∞

⎢⎣

⎛ − b2 ⎞ ∞ ⎛ a + nc ⎞ ⎟⎟ ∑ (− 1)n erf ⎜⎜ exp⎜⎜ ⎟⎟ + 4 Dt 2t πDt 2 Dt = −∞ n ⎠ ⎝ ⎠ ⎝

2

⎤ ⎛ b ⎞ ⎟⎟ ⎥erf ⎜⎜ ⎥⎦ ⎝ 2 Dt ⎠

(II.18)

b

(II.19) The equivalent form:

F(a, b, c, t ) = 1 − erf

∞ ⎛ mc − a ⎞ ⎛ mc + a ⎞⎤ ⎫ b ⎧ a n⎡ + ∑ (− 1) ⎢erfc⎜ ⎟ − erfc⎜ ⎟⎥ ⎬ (II.20) ⎨erf 2 Dt ⎩ 2 Dt m =1 ⎝ 2 Dt ⎠ ⎝ 2 Dt ⎠⎦ ⎭ ⎣

converges well except at very large times; for c→∞, it reduces to the two orthogonal rivers case.


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References Cook, PG, Leaney, FW, and Miles, M (2004) Groundwater recharge in the north-east Mallee region South Australia, CSIRO Technical Report 25/04. Hall, FR, and Moench, AF (1972) Application of the convolution equation to stream-aquifer relationships, Water Resources Research, Vol 8, No 2, PP 487-493.

Hantush, MS (1967) Depletion of flow in right-angle stream bends by steady wells, Water Resources Research, Vol 3, No 1, PP 235-240. Knight, JH, Gilfedder, M., and Walker, GR (2002), 'Impact of Irrigation and dryland development on groundwater discharge to rivers: A unit response approach to cumulative impacts analysis', CSIRO Land and Water Technical Report 3/02, Second Edition, August 2002. Knight, JH, Gilfedder, M., and Walker, GR (2004), 'Impact of Irrigation and dryland development on groundwater discharge to rivers: A unit response approach to cumulative impacts analysis', Journal of Hydrology, In Press. McDonald, M.C., and W. Harbaugh, MODFLOW, A modular three-dimensional finite difference groundwater flow model, Open-file report 83-875, Chapter A1, US Geological Survey, Washington DC, 1988. Murray Darling Basin Ministerial Council (1991) Mallee vegetation management in the Murray Hydrological Basin. Prepared by the Mallee Vegetation Management Working group, February, 1991. Murray Darling Basin Ministerial Council (1999) The salinity audit of the Murray Darling Basin, a 100-year perspective. URS Australia and AWE (2001) Interim rapid assessment tool for assessing salinity impacts of interstate water trade. Report to Murray-Darling Basin Commission. URS Australia Pty Limited, Adelaide. Watkins, N. and Waclawik, V. 1996, ‘River Murray water resources management assessment of salt load impacts and drainage hazard for the new irrigation development along the River Murray in South Australia, Department of Mines and Energy.


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