Applicability of the Unit Response Equation to assess salinity impacts of irrigation development in the Mallee region

Transcription

1 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. 35/4 October 4

2 Copyright and Disclaimer 4 CSIRO To the extent permitted by law, all rights are reserved and no part of this publication covered by copyright may be reproduced or copied in any form or by any means except with the written permission of CSIRO Land and Water. Important Disclaimer: CSIRO Land and Water advises that the information contained in this publication comprises general statements based on scientific research. The reader is advised and needs to be aware that such information may be incomplete or unable to be used in any specific situation. No reliance or actions must therefore be made on that information without seeking prior expert professional, scientific and technical advice. To the extent permitted by law, CSIRO Land and Water (including its employees and consultants) excludes all liability to any person for any consequences, including but not limited to all losses, damages, costs, expenses and any other compensation, arising directly or indirectly from using this publication (in part or in whole) and any information or material contained in it. ISSN Cover Photograph: From CSIRO Land and Water Image Gallery: File: PDA66_34.jpg Description: Storm approaching as vehicular ferry crosses the Murray River at Waikerie, SA Photographer: Willem van Aken 4 CSIRO

3 Applicability of the Unit Response Equation to assess salinity impacts of irrigation development in the Mallee region David Rassam 1, Glen Walker, and John Knight 1 1 CSIRO Land and Water, Long Pocket Laboratory, Indooroopilly, QLD 448 CSIRO Land and Water, Glen Osmond, South Australia Technical Report No. 35/4 October 4

4 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 3 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.

5 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. () and Knight et al. (4) 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

6 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.. 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.. 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. CSIRO Land and Water

7 Table of Contents 1 Introduction 5 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) Sensitivity Analysis Flux response Head response Non-dimensional Analysis Flux response Head response 6.3 Predicting aquifer thickness Pulse recharge example Step recharge example 6.4 Summary 3 7 Comparison with MODFLOW simulations Flux response 3 7. Head response Pulse recharge Step recharge Summary 5 8 The Principle of Superposition Two identical sources 5 8. Two different sources Two different sources with time lags Using the URE for a single source to predict response to a rectangular area Summary 9 9 Modified forms of the URE Presence of no-flow boundary 9 9. Sloping base aquifer Response for down-gradient case Response for reverse-gradient case Recharge to an aquifer with multiple constant-head boundaries; a meandering river Two orthogonal rivers; a 9 o -bend Three orthogonal rivers; a U-shaped meander Effect of discharge edge geometry 36 CSIRO Land and Water 3

8 9.4 Summary 37 1 Stretching the applicability of the URE Flat-base aquifer with groundwater gradient Presence of groundwater mounds Transient mounds of varying widths and thicknesses New development in front and behind large existing mound Response of a heterogeneous aquifer Three-layered aquifer; K3 < K = K Three-layered aquifer; K < K1 = K Summary Discussion 5 1 Conclusions and recommendations: 53 Appendix I: Flux distribution along a river 54 I.1 Mathematical formulation 54 I. Analysis 55 Appendix II: Flux response with 3 orthogonal constant-head boundaries; a U-shaped meander 57 References 61 CSIRO Land and Water 4

9 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, 1). 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, 1), 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. (). 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. CSIRO Land and Water 5

10 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 8 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. Description of the Mallee region The Mallee region extends across New South Wales (NSW), Victoria (VIC), and South Australia (SA), and covers areas of 19,93 km, 59,6 km, and 48,5 km, 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 and clearing in NSW took place mainly between ; most native vegetation was replaced by dryland cropping, pasture, and grazing. Areas of irrigated agriculture occur within a 1 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. (1) 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 <5 mg/l to >35, 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. CSIRO Land and Water 6

11 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., 1) with the unsaturated zone method of Cook et al. (4) and the unit response equation of Knight et al. (). 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(ah 1 +h ) = al(h 1 ) + L(h ) (1) The groundwater equation (Equation ) 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 = t φ * ( t) h N x, + x φ 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, () CSIRO Land and Water 7

12 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 h 1 and solutions to the groundwater equation, together with boundary and initial conditions, then ah 1 +h 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 h 1 equals a constant b 1, at a part of the boundary, where a constant head condition holds and h equals a constant b, then ah 1 +h satisfies the groundwater equation together with the constant head boundary condition h = ab 1 +b (3a) Similarly, if h 1 has the initial condition f 1 and h has the initial condition f, then ah 1 +h satisfies the groundwater equation, with boundary condition [] and initial condition af 1 +f. 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 h ss,. Subsequent development, with corresponding solution h d, with zero boundary conditions, and 3. An individual action, with corresponding solution, h a. Under linearity, h = h ss + h d + Σ h a (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. () 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. CSIRO Land and Water 8

13 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. With the exception of simulations S to S 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 CSIRO Land and Water 9

14 S1). MODFLOW is also used in Sections 8, 9, and 1 to demonstrate the concept of linearity and to model non-linear scenarios. The various MODFLOW simulations used are briefly described in Table. 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 1 (Simulations S to S5), 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 1 to recommend criteria where the basic URE is likely to be applicable. CSIRO Land and Water 1

15 Table 1: Modelling experiment Objectives Section numbers and descriptions Simulation numbers* Flux response S1 and S 6.1 Sensitivity analysis 6.1. Head response; pulse/step S3, S4 (pulse); S5 (step) recharge 6..1 Flux response S1; S Understand the URE 6. Non-dimensional analysis 6.. Head response; pulse/step S6 (pulse); S7 (step) recharge 6.3 Predicting aquifer thickness Pulse recharge S 6.3. Step recharge 7.1 Flux response S8 Verify the URE 7. Head response; pulse/step recharge S9 (pulse); S1 (step) 8.1 Two identical sources S11 Illustrate the principle of 8. Two different sources S1 superposition 8.3 Two different sources with lag time S Response from a rectangular development S14 Present alternative forms of the URE 9.1 Presence of a no flow boundary S15 9. Sloping base aquifer 9..1 Down gradient S Reverse gradient S Meandering river S17; S18; S Flat base with gradient S Investigate non-linearity 1. Groundwater mounds 1..1 Small transient mound S1 1.. Large mound at equilibrium S; S3; S4 1.3 Aquifer with 3 layers of varying K3<K=K1 S5 conductivities K1, K, and K K<K1=K3 *Refer to Table for simulation details CSIRO Land and Water 11

16 Table : Parameters and model simulation details Model parameters Simulation numbers a (m) K (m/day) phi h (m) URE Model details MODFLOW models S1 1, 5, 1, and.5 4 S, 5, 1, and.5 4 S3 1, 1 and S4 1, 1.5,.5, and.1 5 S and 1.5 and S6 1, S7 1, S8 45 and S9, S S11 S1 S13 45 and and and dimensional domain; No-flow boundary at 5, m and 5, m; pulse recharge to a one cell (5x5 m) applied for.1 year 1-dimensional domain; No-flow boundary at 5, m; pulse recharge to a -m strip applied for.1 year 1-dimensional domain; No-flow boundary at 3, m; step recharge to a one cell (1x1 m) As S8, but two identical recharge sources are applied simultaneously on two locations As S8, but two different recharge sources are applied simultaneously on two locations - CSIRO Land and Water 1

17 Simulation number Model parameters Model details a (m) K (m/day) phi h (m) URE MODFLOW simulation S14 8, to 13, & * - # 1-dimensional domain; No-flow boundary at, m; pulse S15 1, & 4 recharge to a one cell $ 1-dimensional domain; No-flow boundary at, m; pulse S recharge at m from downstream boundary (down gradient); at m from upstream boundary (reverse gradient) S S18 1, 5, S S dimensional,-m square domain; two constant-head boundaries opposite to two no-flow boundaries -dimensional,-m square domain; circular domain radius =, m; constant-head boundaries all around - 1-dimensional domain; No-flow boundary at 1, m; pulse recharge, at 5 m from downstream boundary (down gradient); at 5 m from upstream boundary (reverse gradient) # 1-dimensional domain; No-flow boundary at, m; pulse S1 1, ; 18 4 recharge; 3 cases of varying initial head distributions * -dimensional 5,-m square domain with constant-head S 5.1 boundary on one side; recharge to an area 3, 3, m; simulation time 3.8 years; 1 mm/year continuous recharge * Use the output from S as initial conditions; recharge to an S3 3.1 area m square (in front of mound) * Use the output from S as initial conditions; recharge to an S4 3,5.1 area m square (behind mound) S5 5 Varies across layers # 1-dimensional 3-layered domain; No-flow boundary at, m; pulse recharge * Eq., Section.3 (Knight et al., ); # Eq. 4, Section.4 (Knight et al., ); $ Eq. 6, Section.5 (Knight et al., ) CSIRO Land and Water 13

18 6 Understanding the unit response equation (URE) In this section, we present the basic form of the unit response equation of Knight et al. (), which is used in SIMPACT. A sensitivity analysis is conducted to investigate how various aquifer parameters affect pressure head changes between an irrigation development (a recharge source) and the river, and how these parameters affect discharge fluxes to the river. A non-dimensional analysis shows unique relationships between head response and dimensionless time, and, flux response and dimensionless time. All the scenarios investigated in the following sections treat the river as an integral discharge edge, that is, the response is always reported along the entire length of the river. Appendix I discusses the flux distribution along the river for the case of a pulse recharge. 6.1 Sensitivity Analysis Flux response The cumulative flux resulting from a recharge applied to a semi-infinite system similar to that shown in Figure 1 was given by (Knight et al., ): a F ( a,t) = erfc (4) ( Dt) where (a) represents the distance separating the recharge source from the river (the constant head boundary), (t) is a time variable, and (D) is the diffusivity (D=Kh*/φ, where K is the saturated hydraulic conductivity, h* is the average height of the water table, and φ is the specific yield of the layer where the water table exists. The formulation of Equation 4 is based on the Boussinesq equation (see Knight et al. (), Appendix A). The formulation of the Boussinesq equation is based on the continuity equation, which conserves mass balance (assuming density differences are negligible). The conceptualisation shown in Figure 1 assumes that the river fully penetrates the aquifer and that all the applied recharge eventually enters the river. A similar solution was presented by Hall and Moench (197). a Recharge x h * Constant head boundary (river) Figure 1: Recharge to a semi-infinite aquifer CSIRO Land and Water 14

19 It is apparent in Equation 4 that the flux to the river is dependent on the distance (a) and the diffusivity (D); the diffusivity is a function of specific yield, conductivity, and aquifer thickness (the product of the latter two is the transmissivity). This means that if we double K and φ, or, double K and halve h, the response will be identical, i.e., the three parameters are perfectly correlated. Hence for the sensitivity analysis we only need to change D and a. We investigate the response for a=1, and, m, and, D=365, and 73, and 1,46, m /year; Figures and 3 show the responses for the six combinations of (a) and (D); for both values of (a), the response increases as the diffusivity increases. As the time scales in Figures and 3 differ, it is difficult to see whether or not the sensitivity to (D) changes for different values of (a). -1 Time (years) Normalised flux a = 1, m D = 365, D = 73, D = 1,46, Figure : Effect of diffusivity (D, m /yr) on response; a=1, m -1 Time (years) Normalised flux a =, m D = 365, D = 73, D = 1,46, Figure 3: Effect of diffusivity (D, m /yr) on response; a=, m CSIRO Land and Water 15

20 It is easier to examine the sensitivity to (D) for the two (a) values when plotted on the same graph with two different x-axes (one for each time scale). Figure 4 shows that varying the diffusivity had identical impacts for both (a) values. Note that the maximum value of the upper x-axis is four folds higher that of the lower x-axis; the ratio between the squares of the two (a) values is also equal to 4. We will see how we can scale the time scale in order to obtain a unique response curve that is independent of varying aquifer properties. -1 Time (years) Normalised flux Series with lines and open markers: upper x-axis Series with solid markers: lower x-axis Time (years) D = 365,; a= D = 73,; a= D = 1,46,; a= D = 365,; a=1 D = 73,; a=1 D = 1,46,; a=1 Figure 4: Effect of diffusivity (D, m /yr) on response; a=1, and, m 6.1. Head response Pulse recharge The pressure head rise (h) resulting from a pulse recharge applied to a semi-infinite system similar to that shown in Figure 1 was given by (Knight et al., ): R h(x,a, t) = φ π 1 Dt exp ( x a) ( x + a) 4Dt exp 4Dt (5) where R is the recharge volume over a finite width per unit length parallel to the river (m ). Figure 5 shows the pressure heads at 4, m from the river due to a pulse recharge applied at 1, m from the river; it is notable that increasing the hydraulic conductivity (or more generically the diffusivity D), cause the head pulse to occur at an earlier time. Note that the head (y-axis in Figure 5) is equal to (h*+h). CSIRO Land and Water 16

21 x=.4a h= m a=1 km k=1 k=1.5 Head (m) Time (years) Figure 5: Pressure head rise due to a pulse recharge for different conductivities (k, m/day) Figure 6 shows how the pressure heads at the same location respond to changes in specific yield..8 Head (m) x=.4a h= m a=1 km phi=.1 phi=.5 phi= Time (years) Figure 6: Pressure head rise due to a pulse recharge for different specific yields CSIRO Land and Water 17

22 6.1.. Step recharge The pressure head rise (h) resulting from a continuous (step) recharge is given by: R h(x,a,t) = φ t D π exp ( x a) x a x a t ( x + a) 4Dt D erfc Dt exp πd 4Dt x + a x + a + erfc D Dt (6) where R is the recharge rate over a finite width per unit length parallel to the river (m /t). Figure 7 shows how the pressure head increases with time below the recharge source (a=45 m; h*=1). The long-term head response is more sensitive to variations in hydraulic conductivity Head (m) k=5; phi= k=1; phi=.5 k=5; phi= Time (years) Figure 7: Head rise due to step recharge for different conductivities (k, m/day) and specific yields 6. Non-dimensional Analysis 6..1 Flux response Dimensionless analysis is a very powerful tool for establishing unique relationships that are independent of aquifer parameters and recharge. It is very useful to express the response in terms of a dimensionless time scale, i.e., normalise time with respect to a quantity which is a function of relevant system variables, in this case (a) and (D) as was shown in the previous section (Figure 4). Knight et al. () showed that the response is a function of a /D, the hydrological response time. The hydrological response times for the six cases investigated in the previous section are listed in Table 3. When we normalise the time scale for each case with respect to its CSIRO Land and Water 18

23 hydrological response time (to obtain dimensionless time τ = t/(a /D); time normalised with respect to the hydrological response time), we see that all responses collapse to one characteristic response curve shown in Figure 8. Table 3: Hydrological response time (a /D) a K = 5 K = 1 K = D = 365, * D = 73, * D = 1,46, * 1, , * φ=.5; h=1 m; K (m/day); D (m /year); a (m) Normalised (dimesionless) time τ Normalised flux % response τ = % response τ = 3.1 Characteristic respone curve -. Figure 8: Non-dimensional characteristic response curve Figure 8 shows that the response is highly non-linear at early stages; 8% of the response is achieved when τ = 7.81 and 9% of the response is achieved when τ = 3.1. Beyond that stage the response increases marginally (99% of the response is reached when τ = 3,188; 99.5% of the response is reached when τ = 1,737). This is because the region is unbounded (semi-infinite) in the positive x-direction; initially half of the flux goes in the positive direction, and returns very slowly towards the stream (Knight et al., ). Figure 9 shows the actual time required to achieve an 8% response for three diffusivities and values of (a) of up to 5, m. It is apparent in Figure 9 that the response time varies nonlinearly with the distance to the river, and varies linearly with the diffusivity (as D is increased by an order of magnitude, the response time increases by one cycle on the log scale). CSIRO Land and Water 19

24 1 1 Time to 8% response (years) D = 36,5 D = 365, D = 3,65, Distance to river 'a' (m) Figure 9: Time for 8% response for various (a) and (D, m /yr) values 6.. Head response Pulse recharge Similarly, one can apply the same technique to pressure head response. Using Equation 5 and normalising as shown in Figure 1, we get dimensionless head response curves for various distances from the source to the river; each of those curves is unique; v is the recharge volume for a finite width per unit length parallel to the river (m ), T is the transmissivity, and t is time...16 x =.5 a x =.75 a x =.5 a x = a htt/va τ (dimensionless time) Figure 1: Characteristic head response curves for pulse recharge CSIRO Land and Water

25 6... Step recharge Using Equation 6 and normalising as shown in Figure 11, we get dimensionless head response curves for various distances from the source to the river; each of those curves is unique; v is the recharge rate for a finite width per unit length parallel to the river (m /year) ht/va.4. x = a x =.75 a x =.5 a x =.5 a τ (dimensionless time) Figure 11: Characteristic head response curves for step recharge 6.3 Predicting aquifer thickness In many cases, new irrigation developments are introduced into areas that have previous developments. It is vital to know the effect that existing developments had on aquifer transmissivity especially if they are significantly large and recent groundwater data is not available. The non-dimensional relationships presented in the previous section can be used to calculate actual heads for any situation; we present two examples demonstrating how pressure heads can be calculated using the non-dimensional relationships shown in Figures 1 and Pulse recharge example In this example, we estimate the pressure head in an aquifer at 1, m from a river 1.3 years after recharge is applied at, m from the river (x =.5 a). The pulse recharge of 1 m/year is applied for a period of.1 year over a strip 4m wide; aquifer properties are as follows: CSIRO Land and Water 1

26 a=, m; K=5 m/day; aquifer thickness h*=1 m; phi=.5; T = = 18,5 m /year; D = T /.5 = 365, m /year τ (1.3 years) = t D/a = / = htt From Figure 1, for τ= , =. 565 va The recharge applied in this case is: v = =.4 m h = =.195m Pressure head at 1, m (after 1.3 years) = = m 6.3. Step recharge example In this example, we show how one can predict the size of an irrigation development that would result in a certain increase in aquifer thickness. Here, we wish to predict the size of an irrigation development that would cause the aquifer thickness under its centre to increase by 5% of the initial thickness (long-term steady state). The new development centre is located at a distance a=, m from the river; aquifer properties are as follows: K=5 m/day; aquifer thickness h*= m; Phi=.5; T=36,5 m /year From Figure 11: ht/va =.95 (under the centre of the development, x=a; long-term, τ=1) h =.95 va/t Our criterion: (h) =.5h* (h is the increase in head) h =.5h* =.95 va/t Hence: v =.5 T h* /.95a = 19.1 m /year For a recharge of 1 mm/year, this represents an irrigation strip about km wide (width here is the dimension orthogonal to the river); the calculations shown here are per unit length parallel to the river CSIRO Land and Water

27 6.4 Summary The formulation of the URE is based on the Boussinesq equation and hence conserves mass balance. The response time varies nonlinearly with the distance (with the square of the distance) to the river, and linearly with the diffusivity. Non-dimensional analysis resulted in unique relationships for flux and head that are independent of aquifer parameters; they can be used to predict flux and head responses due to application of recharge to aquifers with any set of parameters. The non-dimensional relationships for head may be used to predict the appropriate aquifer thickness used in the basic URE when applied to areas that have existing irrigation developments. 7 Comparison with MODFLOW simulations Predictions of head and flux responses obtained from the URE are validated against those obtained from MODFLOW simulations. Recharge is applied to one cell (usually measuring 5x5 m or 1x1 m). When larger -dimensional domains are used, coarser meshes are used; they are refined in the vicinity of the recharge area. Pulse recharge is applied for a period of.1 year. The no-flow boundary is situated at a large distance from the constant-head boundary when the basic form is the URE is used for comparison. Stringent pressure head tolerances (1-6 m) and a high number of time steps per stress period are used to ensure accuracy. 7.1 Flux response The results obtained from Equation 4 were compared to MODFLOW simulations. The problem shown in Figure 1 was modelled; a 3, m wide domain was used to minimise the effect of the no-flow boundary, which is inevitable in MODFLOW (the effect of the no-flow boundary will be discussed in a later section). A pulse recharge was applied to a single grid cell. The cumulative efflux from the constant head boundary (the river) was normalised with respect to the total applied recharge (rate area duration) and compared with results obtained from Equation 4. Figure 1 shows that the results of the URE compare favourably to estimates obtained from MODFLOW. It is notable that as (a) increases, the long-term analytical response departs from the MODFLOW estimates; this is attributable to the influence of the no-flow boundary, whose presence is inevitable in MODFLOW simulations. We demonstrate this phenomenon by repeating the simulation and using a much wider flow domain (5, m compared to 5, m); Figure 1 shows that as the no-flow boundary is located at a much larger distance, perfect long-term agreement is achieved between the numerical and the analytical solutions. The effect of no-flow boundaries will further be investigated in Section 9.1. CSIRO Land and Water 3

28 -1 Time (years) Normalised flux a = 15 m a = 45 m -. Lines refer to Modflow results Markers refer to URE results Domain width=5, m Domain width=5, m Figure 1: Comparison of URE and MODFLOW; semi-infinite aquifer 7. Head response Pressure head rises predicted from Equations 5 and 6 were also compared to those obtained from MODFLOW simulations. Two different cases were considered Pulse recharge a =, m; phi=.5; K=5 m/day; h*=1 m; a pulse recharge was applied over a 4-m strip for a duration of 1/1 year (3.65 days). The pressure head rise is investigated at 1, m from the river (x=.5a). 7.. Step recharge a = 45 m; phi=.5; K=5 m/day; h*=1 m; recharge rate is 1 m/year and was applied over a 1-m strip for a duration of 1 years. The pressure head rise is investigated at 45 m from the river (x=a). Figure 13 shows good agreement was obtained for both cases. CSIRO Land and Water 4

29 Head due to step recharge (m) Analytical; step recharge MODFLOW; step recharge Analytical; pulse recharge MODFLOW; pulse recharge Head due to pulse recharge (m) Time (years) 1 Figure 13: Comparisons of pressure head response obtained from URE and MODFLOW 7.3 Summary The URE produced results compared favourably to those obtained from MODFLOW. The no-flow boundary in MODFLOW should be located at a large distance from the recharge source when comparisons are made with the basic form of the URE. 8 The Principle of Superposition The responses resulting from the application of multiple recharge sources are additive. That is, the response to two identical sources acting instantaneously at two different locations is equivalent to the mathematical average of the two responses resulting from the same sources acting individually in the same locations. 8.1 Two identical sources Figure 14 shows excellent agreement with a MODFLOW simulation; the URE result shown in Figure 14 is an average of the two responses shown in Figure 1. CSIRO Land and Water 5

30 -1 Time (years) Normalised flux a1 = 45m; a = 15m Analytical; average of responses Modflow Figure 14: Combined response from two identical recharge sources 8. Two different sources Figure 15 shows that the principle holds when different recharge values are applied. In this case, it is necessary to calculate the cumulative flux from each source then add them up. 1 1 Cumulative flux (m 3 ) different sources at 45m & 15m Analytical Modflow Time (years) Figure 15: Combined response from two different recharge sources CSIRO Land and Water 6

31 8.3 Two different sources with time lags The same principle may be extended to estimate fluxes resulting from different sources with time lags. Figure 16 shows the individual responses resulting from each source along with the combined cumulative efflux through the constant-head boundary using the URE. -1 Time (years) Normalised flux a1 = 15m; Ti = Cumulative flux (m 3 ) -. a = 45m; Ti = 1 year Resultant cumulative flux Figure 16: Combined response from two different recharge sources with a time lag 8.4 Using the URE for a single source to predict response to a rectangular area Knight et al. () provided an expression for the groundwater flux to a river resulting from a recharge over a rectangular area (Section.3, Knight et al., ) as shown in Figure 17. b a Recharge h * Constant head boundary (river) Figure 17: Recharge over a rectangular area; a semi-infinite aquifer CSIRO Land and Water 7

32 We can use the principle of superposition to obtain a solution for the rectangular case recharge using Equation 4. Figure 18 shows how we can split the continuous recharge into 6 pulses; we then use Equation 4 to obtain a response for each pulse, then using the principle of superposition and obtain one average response. Figure 19 shows that the results compare well to the solution for a recharge over a rectangular area. b a Recharge h * Constant head boundary (river) Figure 18: 6 pulses over a rectangular area; a semi-infinite aquifer -.35 Time (years) Analytical: Unit step to an area Normalised flux Analytical: Average of 6 pulses Analytical solution without a no-flow boundary a = 8, m; b = 13, m Figure 19: Comparison of unit step response to an area and average of multiple responses CSIRO Land and Water 8

33 8.5 Summary The governing differential equation of the URE is linear, and hence the principle of superposition generally applies. That is, individual responses may be added to obtain a cumulative response. True linearity never exists since the applied recharge itself is putting the system into a state of non-equilibrium (i.e. increasing the head in parts of the aquifer thus resulting in a variable transmissivity). However, if these head perturbations are small relative to the aquifer thickness, we ignore their effect and assume that linearity holds. It is very rare to have an ideal aquifer that is perfectly homogenous (i.e. having a uniform thickness) and is at steady state. An aquifer with a variable transmissivity may be the result of a transient condition, and/or, an existing head gradient, and/or, a groundwater mound from previous developments. We must use our judgement to ensure that linearity holds (i.e. the head difference due to any combination of those conditions is small relative to the aquifer thickness). In such cases we must estimate an average thickness that is likely to produce an optimal result. 9 Modified forms of the URE The assumptions that underpin the basic form of the URE may not hold in many practical situations. Such conditions include: the presence of a no-flow boundary, a sloping aquifer, and a meandering river. In this section, we will show alternative forms of the URE that cater for such situations. 9.1 Presence of no-flow boundary Knight et al. () investigated the effect of a no-flow boundary located at a distance (c) from the constant-head boundary (see Figure ). c a Recharge h * Constant head boundary (river) No flow boundary Figure : Recharge to a finite aquifer Figure 1 shows that when c=a, Equation 4 no longer provides an adequate solution. The analytical solution that incorporates the no-flow boundary (Section.4, Equation 4, Knight et al., ) compares well with the MODFLOW results. CSIRO Land and Water 9