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User Manual for the Layered R

1. we is Z o E g A1 M A1 E N A L e A3 e s a b c Figure 4 Movement of the water table and its effects on the connections between the model nodes 34 REPRESENTATION OF THE SEEPAGE FACE In an unconfined aquifer it is possible that the water level drops inside the pumped well to levels that are significantly below the location of the water table node adjacent to the pumped well A seepage face will occur as discussed in Section 2 3 The numerical model must include enough numerical layers at least two to represent the seepage face In addition the nodes of these layers adjacent to the pumped well must not be in contact with the casing of the well Figure 5 shows an example of a seepage face occurring in a partially penetrating cased well This figure shows that at least four numerical layers must exist in the model to include the effects of a seepage face In this particular example the upper most node is used to include the casing of the well the node of the second layer adjacent to the pumped well is used to represent the seepage face The existence of the node of the third layer is necessary to maintain the abstraction and the node of the last numerical layer is included to represent the part of the aquifer that is not penetrated by the well Note that the increase of the number of numerical layers improves the representation of the effects of the seepage face on t
2. T T are the transmissivity values of the aquifer in the radial and circumferential directions respectively L T H is the numerical value of the groundwater head at the central node L H is the numerical value of the groundwater head at the central node at the end of a previous time step L dis the recharge at a selected node L T 7 and Tare the vertical flow per unit area resulting from the existence of the upper layer and the vertical flow per unit area resulting from the existence of the lower layer respectively 1 LT OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 The flow moving across the face separating two numerical layers can be determined by considering mass conservation at this face Figure 2 shows three layers of a numerical model Considering the upper and the intermediate layers this flow is given by 2 K Ky K m K m q x H px H Equation 9 Where K and K are the vertical hydraulic conductivities of the upper and the intermediate layer respectively L T H andH are the head values at nodes in the upper layer and in the intermediate layer respectively L m and m are the thicknesses of the upper layer and the intermediate layer respectively L Similarly the quantity of flow moving across the face separating the intermediate and the lower layers is given by 2 Ky Ky K m K m d H H Equation 10 Where Ky m and H are the vertical hydraulic
3. The water hydraulic conductivity specified in the Well01 dat input file affects both the water balance results and the speed of the model convergence The larger this value is the slower is the convergence Setting the value of this parameter to 2000 m day in this exercise is acceptable since the calculated total released water is equal to the specified abstraction rate The user can experiment with this parameter value to speed up the time required for the model to converge and to examine the effects of this parameter on the calculated water balance 46 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 P WaterBalance01 out Notepad File Edit Format View Help from well 233358 water from ss 1255 77 Total water m3 d 1256 from well 0134972 water from ss 1255 99 Total water m3 d 1256 from well 00609065 water from ss 1255 99 Total water m3 d 1256 from wel 00366035 water from ss 1256 Total water m3 d 1256 from wel 00246755 water from ss 1256 Total water m3 d 1256 from wel 00176911 water from ss 1256 Total water m3 d 1256 from wel 00131736 water from ss 1256 Total water m3 d 1256 from wel 0010061 water from ss 1256 Total water m3 d 1256 from wel 000782199 water from ss 1256 Total water m3 d 1256 from wel 000616089 water from ss 1256 Total water m3 d 1256 from wel 000490015 water from ss 1256 Total water m3 d 1256 from wel 000392665 water from ss 1256 Total water m3 d 1256 from wel 00031648
4. 7 1 CONTOURO01 OUT OUTPUT FILE Section 7 10 Section 7 11 Contour01 out output file can be used to plot drawdown values contour lines within one aquifer layer or one horizontal slice of an aquifer layer The first two columns of this file hold the Cartesian coordinates of the grid nodes projected on a horizontal plan These two columns are followed by a number of columns equal to the number of horizontal slices constituting the aquifer layers Table 15 The first column after the X and Y columns gives drawdown values at nodes located in the uppermost layer The last column gives the drawdown values at nodes located in the lowermost layer 34 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Table 15 Contour01 out output file from a model simulating groundwater flows in an aquifer with three layers Column 1 Column 2 Column 3 Column 4 Column 5 X Y Drawdown value at Drawdown value at Drawdown value at coordinate coordinate anode located in the anodelocatedinan anode in the uppermost layer intermediate layer lowermost layer 0 05 0 086603 0 828006 0 827996 0 103213 0 07339 0 127115 0 782464 0 779814 0 103211 0 10772 0 18658 0 736923 0 731634 0 103208 0 15811 0 273861 0 691382 0 683458 0 103202 0 23208 0 401973 0 645841 0 635288 0 103189 0 34065 0 590016 0 600301 0 587133 0 103163 0 5 0 866025 0 554762 0 539005 0 103111 0 7339 1 27115 0 509225 0 490929 0 103009 7 2 NODECA01 0UT NODEC
5. Path Press on this command to select it and then press the button Edit The Edit System Variable will popup as shown in Figure 11 Add the full path of the selected directory C MLWell for example after typing a semi colon do not delete existing info Once the path of the selected directory is set as described above create a batch file a text file containing a series of commands in your working directory Figure 12 shows an example of a batch file Name this file Run bat for example Type the name of the executable followed by a string holding the full path of the working directory Figure 12 Close the batch file and double click its name to run the model Note that you can edit the batch file by right clicking its name and selecting the command Edit 52 RUNNING THE MODEL FROM A DOS BOX If the model is run using the batch file and an error occurs the DOS box launched by the batch file terminates instantly In such cases the user will not be able to investigate the cause of the model termination To overcome this problem the model can be run from the command line in a console window This window do not terminate if an error occurs To start a command window select Run from the Windows start menu and type cmd in the drop down list box as shown in Figure 13 Move to the root directory and then to the working directory i e where the model executable is placed Type the name of the executab
6. Representation of the seepage face Inclusion of rivers 3 The numerical solution 3 1 3 2 3 3 3 4 3 5 3 6 3 7 3 8 Finite difference equations Definition of the hydraulic parameter values at a grid node Representation of the water table Representation of the seepage face Representation of well losses Solving the numerical equations The successive over relaxation method SOR Convergence criteria Time stepping 4 Capabilities of the radial flow model 4 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 Multiple layers and multiple numerical grid lines in each layer Confined and unconfined conditions Impermeable and fixed head boundaries Heterogeneity Variable pumping rate and recovery phases Gradually increasing pumping rate Seepage face Well Losses Well casing 4 10 Partially penetrating well 4 11 Recharge 4 12 Additional abstraction points 4 13 Rivers 5 Running the model 5 1 Installing the executable ii Last modified 2008 02 08 10 18 14 14 14 14 14 14 15 15 15 15 15 15 15 16 17 17 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 5 2 Running the model from a dos box 17 5 3 Model run time 19 6 Model input files 20 6 1 The philosophy of model input files 20 6 2 Clock0I dat input file 22 6 3 InputO1 dat input file 23 6 4 OutputO1 dat input file 26 6 5 PumpingOl dat input file 27 6 6 Recharge01 dat input file 28 6 7 river01 dat input file 28 6 8
7. 9321 8636 114 113 113 112 112 111 110 109 107 106 104 101 99 95 92 88 85 81 77 73 69 66 63 60 57 55 52 50 48 45 43 361 945 459 886 201 379 388 191 761 061 055 718 0302 9934 6295 9914 1587 2372 3389 5674 9938 6514 5345 6173 8654 2525 7592 3776 1022 9321 8636 Figure 36 SeepageFlowFile01 dat showing the seepage flows calculated at the different time steps 1 00E 04 6 00E 00 1 00E 03 Time days 1 00E 02 1 00E 01 1 00E 00 6 50E 00 Numerical results with no seepage face 7 00E 00 Numerical results with seepage face 7 50E 00 8 00E 00 Drawdown m 8 50E 00 9 00E 00 9 50E 00 Figure 37 Comparison between the simulated results with and without a seepage face 57 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 References MANSOUR M M 2003 Flow to wells in unconfined aquifers using mesh refinement and object oriented technology Ph D Thesis The University of Birmingham England RUSHTON K R and REDSHAW S C 1979 Seepage and groundwater flow John Wiley and Sons Chichester 58
8. D L U u f Equation 11 Let V denote an approximate solution to Substituting in Equation 11 and using the components of the approximation as soon as they are updated The Gauss Seidel method The iteration method in this case is written as 11 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Dv Ly 2 Uy f D L v 2 uy f Equation 12 y 2 D L Uy D L f Figure 7 An example of a sparse matrix where the stars illustrate a given number in the matrix and blank areas illustrate numbers equal to zero The correction for the Gauss Seidel iteration is given by the difference between yin and y let denote this difference When this correction is added to v9 the new value of v is expected to become closer to the exact solution In addition if this correction is enlarged by multiplying it by a factor that has a value greater than one it will result in a value of yl n 1 1 E that is even closer to the exact solutionu v can therefore be determined from vr 2 y 4 Q where is called the relaxation factor From Equation 12 let v R D L uy t D L f then the new approximation of v can be written as v y ol R v The last equation can be rearranged to take the final form of the SOR scheme as follows y oR I ov The value of ranges between 1 0 and 2 0 and a value of 1 6 is typically adopted in the numerical models 3 7 CONVERGENCE CRITERIA The basic equat
9. Help Layer Number 1 Slice Number 1 dangle 0 785398 0 01 0 01 0 01 0 01 0 01 Slice Number 2 dangle 1 5708 Q 10 10 10 10 Slice Number 2 35619 10 10 10 10 10 10 10 Slice Number 10 10 10 Slice Number 10 10 10 Slice Number 10 10 10 Slice Number 10 10 10 Slice Number 10 10 10 Figure 29 Example of the krFile01 txt input file Open KtFileO1 txt input file using a text editor Do not allow the text editor to wrap lines This file has the same format as KrFileO1 txt As in the previous step the circumferential hydraulic conductivity values at the nodes located along Slice 1 angle 0 785 rd will be reduced to 0 01 Change the values of the first data line from 10 to 0 01 However as explained in Section 3 2 of the manual this means that the circumferential hydraulic conductivity values between the nodes at this line and the nodes located on line at angle 1 5708 rd are equal to 0 01 m day The circumferential hydraulic conductivity values between these nodes and those located at angle 6 28 rd must also be reduced This is done by changing the values of the last data line from 10 to 0 01 An example of the modified file is shown in Figure 30 50 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 P KtFile01 txt Notepad File Edit Format View Help Layer Number 1 Slice Number 1 dangle 0 785398 0 01 0 01 0 01 0 01 Slice Number dangle 10 10 10 10 Slice Number 10 10 10 Sl
10. Modified Well01 dat file nennen nennen 43 Figure 20 Modified Clock01 dat file ee eeeeeeeesceceeesnceeeesseeeeeeseaeeecessaeeeeseeceeeesneeerees 44 Figure 21 Modified SolverO1 dat file 20 eee eeessceceessneeeceeseeeceeesaceeceseaeeeeseseeeeeeneeeeees 44 Figure 22 Modified Output01 dat file essen enne ener 45 Figure 23 DOS box showing the status of the model run eee 46 Figure 24 Water balance calculated at the end of the first 20 time steps of the first abstraction phases see teet Nm ee Ru oet e Dececeadea ep ceder e epee a 47 Figure 25 Water balance calculated at the end of the first 20 time steps of the recovery phase EP E E 47 Figure 26 Drawdown time series produced at an observation borehole located at 10 m from the central well eere Reese criam e eliede penne 48 Figure 27 Comparison between the simulated results and those calculated using the Theis Solution henan re hapa heo e et CO ERE NS EE a 48 Figure 28 Plan of the two permeability zone aquifer esee 49 Figure 29 Example of the krFileO1 txt input file seen 50 Figure 30 Example of the KtFile01 txt input file eeeeeeeeeeennn 51 iv OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Figure 31 Example of the SsFileO1 txt input file eene 51 Figure 32 Comparison between the numerical results produced at the observation borehole located i
11. U OH OH UOH OH OH H U OOOO IOI ROKR U tr kr te AX Layer Number L Layer Thickness 10 0 Horizontal hydraulic conductivity 10 0 vertical hydraulic conductivity 1 Specific storage 0004 specific yield 0 2 Number of grid lines 1 Figure 18 Modified second section of InputO1 dat file SETTING THE VALUES OF THE CENTRAL ABSTRACTION BOREHOLE PARAMETERS InputO1 dat I Well01 dat Clock01 dat Solver01 dat Output01 dat The characteristics of the central abstraction borehole are specified in Well01 dat file Open this file using a text editor The first parameter value to specify in this file is the value of the water hydraulic conductivity which theoretically is equal to infinity This can be left as defined in the dummy file Borehole depth must be set to a value equal to the depth of the lowest gridline which is in contact with the abstraction borehole Since there is only one gridline located in the middle of the layer which is 10 m thick the borehole depth must be set to 5 0 m Change borehole depth value to a value of 5 m Pump depth is the depth of lowest gridline that is in contact with the pump As in the previous point this must be equal to 5 0 m in this model since there is only one gridline Note that pump depth cannot be greater than that of the borehole depth if the numerical grid is made of four gridlines for example and the borehole depth is set to 6 25 m the pump can be located at 1 25
12. conductivity the thickness and the head value at a node in the lower layer respectively Note that groundwater heads in the terms of the left hand side of Equation 8 are all expressed at the new time This is the implicit numerical form of the basic flow equation and requires that the set of equations produced using Equation 8 be solved iteratively as will be discussed in Section 3 6 32 DEFINITION OF THE HYDRAULIC PARAMETER VALUES AT A GRID NODE It is important to clarify the following basic assumptions used in the model A horizontal hydraulic conductivity is specified at each grid node This hydraulic conductivity applies between this node and its adjacent node in the positive radial or circumferential direction For example the horizontal hydraulic conductivity specified at Node A Figure 3 is used to calculate the radial conductance between nodes A and B The horizontal hydraulic conductivity specified at node B is used to calculate the radial conductance between nodes B and C Similarly the horizontal hydraulic conductivity specified at Node A is used to calculate the circumferential conductance between A and D Figure 3 e The specific storage specific yield and vertical hydraulic conductivity specified at a node represent the characteristics of the materials within that node OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 KTheta at A is used to calculate the conductance between A and D i r e e e A B C K
13. each time step Figure 36 shows that the seepage flow is decreasing with time This is because in this particular example only one seepage node has occurred and the saturated thickness of this node is decreasing with time The seepage face process can be improved by increasing the number of nodes along the face of the well Figure 37 shows a comparison between the simulated results obtained at the abstraction borehole with and without the occurrence of the seepage face sx C WINDOWS system32 cmd exe jol Layer Borehole Free Surf Link Done Read DD Read Coef Solver xro xx No xx Solving Iterations Phase Time step Time 19370 15435 14475 14032 13752 13545 13368 13195 13007 14655 14618 5091 13426 8232 13126 12912 1 1 i 1 a 1 1 1 0 00012589 3 0 000158489 6 666199526 6 666251189 6 666316228 6 666398167 6 666561187 0 000630 57 6 666794328 6 661 6 66125893 0 00158489 Figure 35 Runtime information displayed in a DOS box 56 0 0001258 3 0 000158489 6 666199526 6 66625118 9 6 666316228 6 666398167 6 666561187 6 666636957 6 666794328 98 00125893 4 6615848 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 P SeepageF lowFile01 out Note A File Edit Format View Help 361 945 459 886 201 379 388 191 761 061 055 7218 0302 9934 6295 9914 1587 2372 3389 5674 9938 6514 5345 6173 8654 2525 7392 3776 1022
14. either parallel to the north south direction or the east west direction However because the radial flow model deals poorly with linear features these boundaries must be treated with care once they are included These boundaries can be thought of as a rectangle with the position of its lower left and upper right corners having to be defined Once included in the model the conductance values of all nodes falling outside this rectangle are set to zero This is illustrated in Figure 14 An example when only one boundary has an effect on the model is also shown in Figure 14 The locations of the rectangle lower left and upper right corners are specified in this example so that only the right hand side of this rectangle intersects the model area UR Corner North Sansa North DP oundary Condition 222722 Boundary Condition JP x specified on line 12 PA is Ke y specified on line 12 C X f v K Y UR Corne C f j e A i East a West East 4 7 West 2 Jj j 7 3 4 f a LL Corner A a WY 2 7 Internal boundaries caused by 2 7 Internal boundaries caused by my D the rectangle defined on line 20 o the rectangle defined on line 20 72 p gt gt P Pz zzz South South LL Corner Figure 14 Defining internal boundaries The right figure shows an example where only one side of the rectangle causes internal boundaries 24 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Table 4 Main section
15. in the basic flow equation results in the appearance of the transmissivity terms which replaces the hydraulic conductivity terms in Equation 2 The term representing the differentiation in the vertical direction also disappears and is replaced by two source terms on the right hand side of the equation as follows T h Ty h _ oh D 38 S x 4474 Equation 3 where h r Gt is the groundwater head value within a layer and at time t L T r O and T r are the transmissivity values in the radial and circumferential directions respectively E TH S r 0 is the storage coefficient of the layer Dimensionless q r 8 is an external source term per unit surface area L Y and q r 0 and qp r 0 are leakage rates from layers above and below L Ti OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 LLLLLLLLLLLLLLLLLLLLLL DER MNA LLL ELL Ay LL r E Lld uut di Well casing Partially penetrating bc n Jp Nee ie 0 T XI Bond E ans t Piezometer J l P 12 g aes Ss 2E a Nt NE j Pornot a ms EA i p Upper E Layer EU Ai l T I J 8 T eae J af e 522 4 Intermediate Pd Layer o Numerical grid in a ry Lower P ee 7 c ww vertical plane KN Layer ie ar gt gt pe rr 77777777777 qe Figure 2 Schematic representing a multi layered aquifer 2 2 AQUIFERS UNDER UNCONFINRD CONDITIONS THE INCLUSION OF THE FREE SURFACE The free surf
16. its operational abstraction rate instantly at the start of pumping This behaviour can be represented in the model by increasing the abstraction rate progressively in an exponential form until the specified abstraction rate is reached 4 7 SEEPAGE FACE In unconfined aquifers a seepage face may occur at the abstraction well The seepage face is represented in the model by fixing the head value at a seepage node to the value of its elevation This satisfies the seepage face condition as discussed in Section 2 3 48 WELL LOSSES Well losses can be simulated at the central abstraction well A factor that multiplies the hydraulic conductivity values of the nodes located at the well face is used to include the well losses Section 3 5 Once this factor is specified in the model it is used in all abstraction phases however this factor has no influence on the hydraulic conductivity values when simulating the groundwater heads during the recovery phases 49 WELL CASING Well casing can be included in the model if there is more than one layer in the aquifer or if the single layer aquifer is represented using more than one numerical grid Casing is represented in the model by disconnecting the layer or layers that are in contact with the casing and replacing this connection with an impermeable boundary Section 3 4 The casing has clear effects on the cone of depression which forms a right angle with the casing 4 10 PARTIALLY PENETRATING WELL G
17. m 3 75 m or 6 25 m but not at a greater depth than 6 25 m Casing depth is the depth of the lowest gridline that is in contact with the solid casing This depth must be always less than the borehole depth Since there is only one 42 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 gridline in this example there could not be solid casing and the casing depth must be set to 0 0 All other parameters except the number of abstraction phases and the corresponding durations and abstraction rates of these phases are not used Set the number of abstraction phases to 2 Set the abstraction rate of the first phase to 1256 m day and the duration of this phase to 10 days 1 Set the abstraction rate of the second phase to 0 0 m day and the duration of this phase to 10 days this is the recovery phase The file should now look like Figure 19 P Well01 dat Notepad File Edit Format View Help Hydraulic conductivity representing water m day 10000 Borehole depth m 5 0 Pump depth m Se Casing depth m 0 0 Set N gradual delivery at the start of each phase Time after which pump delivers full abstraction rate days 0 00208 Include well Losses N well losses factor 0 Nbr 2 of Abstraction Phases duration days Figure 19 Modified Well01 dat file SETTING THE PARAMETER VALUES TO DISCRETISE THE SIMULATION TIME InputO1 dat m Well01 dat Clock01 dat Solver01 dat Output01
18. pumping test results This allows The estimation of the hydraulic parameters of an aquifer The determination of the yield and efficiency of a pumped borehole The design of pumping tests in terms of the determination of abstraction duration time required to monitor the groundwater head recovery and the positioning of observation boreholes The radial flow model also simulates the effects of rivers on pumping test results This allows investigating the river aquifer interaction Ultimately the radial flow model helps improving the conceptual model of the aquifer system The manual starts by describing the basic flow equations used to build the radial flow model and their conversion into numerical form Sections 2 and 3 are dedicated for this purpose The users may want to apply the model directly and to read these sections later In this case they may proceed to Sections 4 to 8 However it is highly advisable that users familiarise themselves with the background of the model because this helps them improve their understanding of the model behaviour and the meaning of the produced results and consequently the analysis of the pumping test results Section 4 describes the capabilities of the radial flow model Section 5 describes how to install and run the model Section 6 describes the model input files Section 7 describes the model output files Section 8 gives three tutorials that are designed to help the user to contro
19. solver01 dat input file 29 6 9 WatertableO1 dat input file 30 6 10 Well01 dat input file 30 6 11 Possible additional input files 32 7 Model Output Files 34 7 Contour01 out output file 34 7 2 NodeCAO01 out NodeCRO01 out NodeCStorage01 out and NodeCZ01 out output files 35 7 3 NodeRadius01 out output file 35 7 4 ReleasedWaterOl out 35 7 5 ResultO1 out output file 35 7 6 TimeSeries01 out output file 36 7 WaterBalance01 out output file 36 7 8 wellNodeCAO1 out WellNodeCRO1 out WellNodeCStorage01 out and WEIINodeCZO01l out output files 37 7 9 WellResult01 out output file 37 7 10 WTReIWater01 out output file 38 7 11 WTResult01 out output file 38 8 Tutorials 39 8 1 Introduction 39 8 2 Tutoriall Using a Basic Model to Simulate Groundwater DRAWDOWN values in a Homogeneous Aquifer 39 8 3 Tutorial2 Using a Two dimensional R 0 Model to Simulate Groundwater Flows in a Heterogeneous Aquifer 49 8 4 Tutorial 3 Using the Model to Simulate Groundwater Flow in an Unconfined Aquifer 53 References 58 iii OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 FIGURES Figure 1 Three dimensional cylindrical control element cceeesseeceeeeneeeeeeeeeeceeeeneeeeeens 3 Figure 2 Schematic representing a multi layered aquifer eene 4 Figure 3 The definition of hydraulic parameters at grid nodes essess 9 Figure 4 Movem
20. specifying multiple sections of solid well casing at different depths In this case the first number on line 8 is the number of gridlines to disconnect from the abstraction well The subsequent numbers that must follow the first number are the numerical gridline numbers to disconnect For example if two aquifer layers are discretised using two numerical gridlines in the upper layer and three gridlines in the second layer and if the first gridlines of the two 30 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 layers as well as the last gridline of the lower layer are cased out line 8 must read 3 1 3 5 indicating that there are three cased sections and these affect gridlines 1 3 and 5 The value specified on line 2 is the hydraulic conductivity value of water Theoretically this value must be infinity and therefore must be set to a large number Note that this value may cause difficulties to the numerical solution This value must be therefore large enough to produce an even distribution of drawdown values along the face of the central well and to maintain the water balance but without compromising the run time of the model The flag on line 10 controls the gradual increase of pumping rate from zero to the value specified in Well01 dat input file Section 4 6 Set this flag to Y for gradual increase of abstraction rate The time required to reach the full abstraction rate is specified on line 12 Well loss effects can b
21. the abstraction phase stop time in a logarithmic pattern The timing of abstraction for all the boreholes specified in the model is given with respect to the time at which the first abstraction borehole starts pumping These rates are defined within a model input file However the simulation time is always given with respect to the abstraction borehole at the centre of the radial flow model For example this may start pumping later than a non central borehole The time difference between the onset of abstraction at the borehole which starts pumping first and the simulation time is defined on line 2 of the input file Table 3 The logarithmic pattern start time is defined on line 4 of the input file The number of time steps in each ten fold increase in the time step Figure 8 is defined on line 8 of the input file Because the abstraction rates are usually available in the form of time series the abstraction phases stop times are defined along with the abstraction rates in the Well01 dat input file discussed in Section 6 10 It is possible that the time steps become undesirably large after a certain number of time steps The user can control the size of the time step using the parameter defined on line 6 This parameter is a target value up to which the time stepping increases logarithmically Once the simulation time becomes equal to the value of this parameter the time step becomes constant and equal to the value of this parameter for the r
22. to the number of numerical time 35 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 steps Each section has the same format as the input file discussed in Section 6 11 and shown in Table 13 7 6 TIMESERIESO1 OUT OUTPUT FILE This file is used to plot drawdown time series at selected observation boreholes This file starts with three header lines that give the radii angles and depths of the observation boreholes where drawdown values are monitored These observation boreholes are specified in the Output01 dat input file discussed in Section 6 4 The three header lines are followed by a section of output data that gives the phase number the time since the phase started days the time since the simulation started days then a set of columns giving the drawdown time series m at the selected observation boreholes The number of these columns is equal to the number of selected observation wells and is specified in Output01 dat Table 16 TimeSeries01 out output file Column 1 Column2 Column3 Time Colum Column5 Phase Time since since simulation Drawdown time Drawdown time number phase started started series at observation series at observation well 1 well 2 1 1 26E 05 1 26E 05 7 40E 02 7 40E 02 1 1 58E 05 1 58E 05 1 57E 01 1 57E 01 1 2 00E 05 2 00E 05 2 47E 01 2 47E 01 1 2 51E 05 2 51E 05 3 43E 01 3 43E 01 1 3 16E 05 3 16E 05 4 42E 01 4 42E 01 1 3 98E 05 3 98E 05 5 39E 01 5 39E 01 1 5 01E 05 5
23. 0 0 eee eeesseeeeesseeeecesnaeeeesseneecessaeeeeseeeecessaeeeeenaes 28 Table 9 RiverO1 dat input file ession en eap e e ea a eaaa 29 Table 10 Solver01 dat input file eseonsoessssssessseseesseessssseessereesssressssssesssceeesseeessseseersreesss 30 Table 11 WaterTable01 dat input file 0 eee eeeseeeceeenneeeeeseeeecessaaeeeceseeecesenaeeeeenaes 30 Table 12 Well01 dat input Tile ie th er EE e HE boe digne 31 Table 13 Template of possible additional input files txt eese 33 T ble 14 Listof Output files tet REN re ER Ga es 34 Table 15 Contour01 out output file from a model simulating groundwater flows in an aquifer with three layers zuo rere eene nete eir ee Reader e 35 Table 16 TimeSeriesO1 out output file sse enne 36 Table 17 WaterBalance01 dat output file eeeesesseeeeseeeeeeeeeeenerene nene 37 Table 18 Template of output files of the hydraulic parameters calculated at the central well C 37 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 1 Introduction This user manual introduces the finite difference radial flow model developed by the British Geological Survey and the University of Birmingham The cylindrical shape of the three dimensional model grid is best suited to simulate groundwater flows converging to a pumped borehole The main application of the radial flow model is therefore to simulate groundwater flows to analyse
24. 002 4 641589e 002 3 216254e4000 623413e 002 5 623413e 002 3 408089e 000 812921e 002 6 812921e 002 3 599925e4000 254042e 002 8 254042e 002 3 791759e4 000 000000e 001 1 000000e 001 3 983588e 000 211528e 001 1 211528e 001 4 175412e 000 467799e 001 1 467799e 001 4 367231e 000 III 2I I d 2 I2 qm 3 c3 14 l 5 16 18 JL 2 152 lo i2 d 2 as L3 14 L5 1 6 18 qu ICI LE Figure 26 Drawdown time series produced at an observation borehole located at 10 m from the central well Time days 0 001 0 01 0 1 1 10 o Theis solution Numerical results Drawdown m Figure 27 Comparison between the simulated results and those calculated using the Theis solution 48 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 83 TUTORIAL2 USING A TWO DIMENSIONAL R 0 MODEL TO SIMULATE GROUNDWATER FLOWS IN A HETEROGENEOUS AQUIFER 8 3 1 Description of Problem It is possible to represent the spatial variations of the aquifer characteristics using a two dimensional numerical model Simulating groundwater flows in a two zone aquifer is used as an example to build a radial flow model that represents the aquifer heterogeneity One part of the aquifer is assumed to have the same characteristics of the aquifer used in Tutorial 1 The same steps used in Tutorial 1 to build the model can be used to start this tutorial Figure 28 shows a planar view of the two zone aquife
25. 01E 05 6 29E 01 6 29E 01 1 6 31E 05 6 31E 05 7 10E 01 7 10E 01 1 7 94E 05 7 94E 05 7 76E 01 7 76E 01 1 1 00E 04 1 00E 04 8 28E 01 8 28E 01 7 1 WATERBALANCEO01 0UT OUTPUT FILE This file holds the total amount of water m released from the aquifer the central well and the water table nodes at each time step Water released from the central well is termed Water from well water from the aquifer node is termed Water from SS water from the water table nodes is termed Water from SY Table 17 These components of total water are calculated in m If confined aquifers are studied this file will not include the component of water released from water table nodes At the end of each line in the file the total water is calculated in m day For each abstraction phase this rate of water must be equal to the abstraction rate specified in file Well01 dat Section 6 10 The rate of water calculated at the end may be different from the abstraction rate at the early time steps only If this is the case the difference represents the water released from the well well storage If there are differences at all time steps these differences represent the water released from the rivers if included If there are no rivers in the model these differences indicate lack of water balance and the numerical model must be checked 36 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Table 17 WaterBalance01 dat output file for a c
26. 1 Total water 27796e 005 from 0 000316145 water from ss 0 000388974 Total water 28289e 005 from 0 000255916 water from ss 0 000328776 Total water 28602e 005 from 0 000207902 water from ss 000280794 Total water 28921e 005 from 0 000169377 water from ss 0 000242305 Total water 29281e 005 from 0 000138302 water from ss 0 000211259 Total water 29573e 005 from 0 000113131 water from ss 0 000186131 Total water m3 30006e 005 from 9 26719e 005 water from ss 0 000165706 TOtal water m3 d 7 3034e 005 from 7 59969e 005 water from ss 0 000149059 Total water m3 d 7 30622e 005 OOoODOOOdOUOCD O0t0 Ww m m w ww IN CN Sal Sd os pod ojos Figure 25 Water balance calculated at the end of the first 20 time steps of the recovery phase COMPARING THE SIMULATED RESULTS TO THOSE CALCULATED USING THE THEIS SOLUTION Open TimeseriesO1 out output file using a text editor A part of this file is shown in Figure 26 The first column gives the phase number The second column gives the phase time The third column gives the simulation time Phase time plus the reference start time The fourth column gives the drawdown values If there are more than one observation wells the number of columns written to this file after the third column is equal to the number of observation wells The fourth column gives the drawdown value at the first observation well specified in the file OutputOl dat The subsequent columns give the d
27. 13 Template of possible additional input files txt Line Core of the input file 1 Layer Number 1 2 Slice Number 1 Angle 2 0944 3 PPPPPPPPPPPPPPPPPPPPPPPPP Z 4 Slice Number 2 Angle 4 18879 5b e 5 pPpppppppppppPpPpppPpPpppPppppp 6 Slice Number 3 Angle 6 28319 5 7 PPPPPPPPPPPPPPPPPPPPPPPpp 8 9 Slice Number 1 Angle 2 0944 10 PPPPPPPPPPPPPPPPPPPDPPPPPPI 11 Slice Number 2 dAngle 4 18879 T 7 5b 5 D PPPPPPPPPPPPPPPPPPPPPPPPP S 3 13 Slice Number 3 Angle 6 28319 E 3 14 PPPPPPPPPPPPPPPPPPPPPPDPPpP Z 4 15 16 Slice Number 1 Angle 2 0944 17 PPpPPPPPPPPPPPPPPDPPPPPPPPPI 18 Slice Number 2 Angle 4 18879 19 PPPPPPPPPPPPPPPPPPPPPPPPPI GS 20 Slice Number 3 Angle 6 28319 3 21 PPPPPPPPPPPPPPPPPPPPPPPPDP 22 23 Layer Number 2 24 Slice Number 1 Angle 2 0944 25 PDPPPPPPPPPPPPPPPPPPPPPDPPP Z 26 Slice Number 2 Angle 4 18879 5b e 27 PPPPPPPPPPPPPPPPPPPPPPPPP s 28 Slice Number 3 Angle 6 28319 E 29 PPPPPPPPPPPPPPPPPPPPPPPpp d 30 5 F 31 Slice Number 1 Angle 2 0944 4 32 PPPPPPPPPPPPPPPPPPPPPPPPPYI_ 33 Slice Number 2 Angle 4 18879 E 34 IPPPPPPPPPPPPPPPPPPPPPPPPP S 35 Slice Number 3 Angle 6 28319 2 Ss 36 PDPPPPPPPPPPPPPPPPPPPPPPPDP Z 33 OR 07 029 Draft 0 1 7 Model Output Files Last modified 2008 02 08 10 18 The possible use of each of the output files listed in Table 14 together with its format are described in detail in the following subsections Table
28. 14 List of output files WTRelIWaterOl out WTResult01 out Output file name Description Relevant section Contour01 out File used to plot drawdown contour lines Section 7 1 NodeCAO1 out Files that hold the conductance and storage Section 7 2 NodeCRO1 out coefficient values calculated between or at NodeCStorage01 out and aquifer grid nodes These coefficient values NodeCZO1 out are produced once at the start of the numerical simulation NodeRadius01 out File that holds the radii of the aquifer nodes Section 7 3 ReleasedWater01 out File that holds the amount of water released Section 7 4 from each grid node Result01 out File that holds the drawdown values at the Section 7 5 aquifer nodes TimeSeriesO1 out File used to plot drawdown time series at Section 7 6 selected observation boreholes WaterBalance01 out File that holds the total amount of water Section 7 7 released from the aquifer the central well and the water table nodes WellNodeCA01 out Files that hold the conductance and storage Section 7 8 WellNodeCRO1 out coefficient values calculated between or at WellNodeCStorage01 out and the central well nodes These coefficient WellNodeCZO1 out values are produced once at the start of the numerical simulation WellResult01 out File that holds the drawdown values at the Section 7 9 well nodes File that holds the amount of water released from each water table nodes File that holds the drawdown values at the water table nodes
29. 21153e 805 1 21153e 605 E E 2m 5413 5486 554 5614 5681 5748 5816 5883 5951 6618 6086 6154 7596 7389 7090 6918 6776 665 6555 6465 6385 121153 121153 14678 6 14678 177828 6 177828 215443 215443 261616 261616 316228 316228 383119 383119 464159 464159 562341 562341 681292 681292 825404 825404 Seaoqoogogoog BOSNIA AONE OSOSGeaeaggggggg aO GO JO C1 COD IR e fk fk fk fk ju j fk ponk d RR P fk fk R panh Figure 23 DOS box showing the status of the model run CHECKING THE WATER BALANCE To check the water balance open Waterbalance01 out output file using a text editor Figure 24 shows the water balance calculated at the end of the first twenty time steps of the first abstraction phase Figure 25 shows the water balance calculated at the first twenty time steps of the Note second phase the recovery phase The first column gives the amount of water m day released from the pumped well The second column gives the amount of water m day released from the specific storage of the aquifer The third column gives the sum of released groundwater from both the well and the aquifer m day and for each phase this must be equal to the abstraction rate specified for this phase In an unconfined aquifer an extra column that gives the amount of water released from the water table is added before the last column of the file shown in Figure 24 and Figure 25
30. 3 4 shows the numerical representation of the seepage face and Section 3 5 deals with the numerical representation of well losses In addition Sections 3 6 3 7 and 3 8 detail the method utilised to solve the mathematical equations the convergence criterion and the temporal discretisation of the simulation period respectively 3 1 FINITE DIFFERENCE EQUATIONS The principle of the finite difference method is to replace the continuous problem domain by a discretised domain by laying a mesh consisting of an array of nodes over the area under consideration In the current model the distance between nodes increases logarithmically in terms of r but they are equally spaced in the transformed coordinate system The hydraulic properties are represented by an average value between the nodes See Section 3 2 Further the finite difference method relies on one numerical value of the head in the centre of each block as illustrated in Figure 1 and assumes that this head is an average value of the true head in this block The basic flow equation Equation 3 can be transformed into numerical form using different methods such as Taylor s expansion or by considering a mass balance for the block or using integrated finite differences for example The numerical form of Equation 3 is 1 Nu TENE r Hiis r gz T Bo m T Hi r Aa r i4l i r i4l p E ae ap dum cp deccm AB C 67m en 6 j ejl ijk 6 j44 i jak Ap ijk d d d Equation 8 Where
31. 4 Investigating the formation of a seepage face MODIFYING INPUTO1 DAT INPUT FILE Change the horizontal hydraulic conductivity to 5 0 MODIFYING OUTPUTO1 DAT INPUT FILE Set the radius where the time series results are produced to 0 004 The above two steps are needed to produce results that are not affected by a seepage face at the abstraction borehole Run the model and plot the results written to Timeseries01 out MODIFYING INPUTO1 DAT INPUT FILE Change the number of grid lines in Layer 1 to 7 MODIFYING WELLO1 DAT INPUT FILE Change the borehole depth to 46 4 Change the pump depth to 46 4 Change the hydraulic conductivity value representing water to 1000 m day 55 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 8 4 5 Running the model and checking the resuls Follow Section 5 instructions to install and run the model executable The repetitive calculations of groundwater heads and seepage flow discussed in Section 3 4 is clear in Figure 35 which shows the runtime information displayed in a DOS box This figure shows that the seepage face occurs after 0 00063 days Time step 8 This is shown by the 3 additional groundwater head and seepage flow simulations that were required before the seepage flow stabilised to a certain value in Time step 9 refer to Section 3 4 The seepage flow is produced in an output file called SeepageFlowFile01 out shown in Figure 36 one line for
32. 8 The river stage can be time variant however one river stage value is allocated at all the nodes belonging to the same river at a given time River stage time series are included in River01 dat input file from line 11 onward The number of data points in this time series is specified on line 9 Note that the entry locations of these data are not fixed to lines 9 and 11 This depends on the number of nodes constituting the river Specified on line 5 and whether the specification of their locations requires the use of more than one line lines 7 8 etc in this input file If more than one river is included in the file the input section starting from line 3 must be repeated a number of times equal to the number of rivers Table 9 River01 dat input file Core of River01 dat input file Number of rivers 1 River 1 Number of river nodes 1 Radius m Angle rd Riverbed thickness m Riverbed kv m day 50 0 1 3 6 Number of data in water elevation time series 1 1 2 3 4 5 6 7 8 9 Water elevation time series Time days Elevation m 0 6 8 SOLVERO0I DAT INPUT FILE The solver of the numerical equations can be activated or deactivated using the flag on line 2 of Table 10 To include aquifer heterogeneity parameter values a set of input files See Section 6 11 need to be changed The user may need to run the model with the solver deactivated to produce these files The point successive over relaxation metho
33. 87e 005 39687e 005 94289e 005 50832e 005 10607e 005 74573e 005 4322e 005 16584e 005 1256 1256 1256 from from from from from from from from from from from from from from from from from from from from from from from Ww w uw w m u y m u y m u y m y y m u y m u y m n y y n y gm n n y m n m m nu m wm w w n m m n m m n m w Ip pPrrIMHNUuuSISuunueOOOO WlIOuUOoDDOOOOOOD0Oooo0o00n0o00 Figure 33 Tutorial 3 calculated water balance calculated CHECKING THE RESULTS Open the Timeseries01 out output file Using a plotting tool plot the drawdown values using either the phase time column or the simulation time column and compare with the drawdown values calculated using the Neuman solution This comparison is shown in Figure 34 The discrepancy between the results at the early times is due to well storage These results seem to compare well at the later times 54 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Time days 1 00E 04 1 00E 03 1 00E 02 1 00E 01 1 00E 00 0 00 E 00 9 Numerical results 1 00E 01 a o Neuman solution 2 00E 01 3 00E 01 4 00E 01 Drawdown m 5 00E 01 6 00E 01 7 00E 01 8 00E 01 Ld Figure 34 Comparison between the simulated results and those calculated using the Neuman solution Part 2 8 4
34. 9 water from ss 1256 Total water m3 d 1256 from wel 00025626 water from ss 1256 Total water m3 d 1256 from wel 000208246 water from ss 1256 Total water m3 d 1256 from wel 000169721 water from ss 1256 Total water m3 d 1256 from wel 000138646 water from ss 1256 Total water m3 d 1256 from wel 000113474 water from ss 1256 Total water m3 d 1256 from wel 30152e 005 water from ss 1256 TOtal water m3 d 1256 from wel 63398e 005 water from ss 1256 Total water m3 d 1256 i i i OOo DOO0D OOOOOdOOO0O0o0odoruo0to E WaterBalance01 out Notepad File Edit Format View Help from 0 233358 water from ss 0 23343 Total water m3 d 7 24294e 005 from 0 0134969 water from ss 0 0135693 Total water m3 d 7 24515e 005 from 0 0060903 water from ss 0 00616277 Total water m3 d 7 24721e 005 from 0 00366 water from ss 0 00373251 Total water m3 d 7 25079e 005 from 0 0024672 water from ss 0 00253977 Total water m3 d 7 25644e 005 from 0 00176877 water from ss 0 00184136 Total water m3 d 7 25896e 005 from 0 00131702 water from ss 0 00138964 Total water m3 d 7 26233e 005 from 0 00100576 water from ss 0 00107842 Total water m3 d 7 26632e 005 from 0 000781854 water from ss 0 00085453 Total water DEDI 7 26757e 005 from 0 000615744 water from ss 0 000688453 Total water m3 d 7 27086e 005 from 0 000489671 water from ss 0 000562437 Total water 27661e 005 from 0 000392321 water from ss 00046510
35. British Geological Survey NATURAL ENVIRONMENT RESEARCH COUNCIL ge BGS a User Manual for the Layered R Theta Numerical Model Groundwater Management Programme Research Report OR 07 029 LLL LL adi Well casing Partially penetrating T d Water table nodes borehole Lee TR M NV f VU V pi N LN Y E t m Piezometer A P ONERE Ye p3 2 2 id p pg XM j PORS B Upper p NEN Layer eee pop rape yy m e EN Intermediate Layer ALE E i on Sit eee Numerical grid in a ome Lower fo vertical plane on Layer A rm uu o 77777 7 md rrr The National Grid and other Ordnance Survey data are used with the permission of the Controller of Her Majesty s Stationery Office Licence No 100017897 2005 Keywords Numerical model Radial flow Pumping test Front cover Three dimensional view of the Layered R Theta numerical model Bibliographical reference MM MANSOUR AG HUGHES AEF SPINK 2007 User Manual for the Layered R Theta Numerical Model British Geological Survey Research Report OR 07 029 68pp Copyright in materials derived from the British Geological Survey s work is owned by the Natural Environment Research Council NERC and or the authority that commissioned the work You may not copy or adapt this publication without first obtaining permission Contact the BGS Intellectual Property Rights Section British Geologi
36. Department for International Development and other agencies The British Geological Survey is a component body of the Natural Environment Research Council British Geological Survey offices Keyworth Nottingham NG12 5GG 0115 936 3241 Fax 0115 936 3488 e mail sales bgs ac uk www bgs ac uk Shop online at www geologyshop com Murchison House West Mains Road Edinburgh EH9 3LA 0131 667 1000 Fax 0131 668 2683 e mail scotsales bgs ac uk London Information Office at the Natural History Museum Earth Galleries Exhibition Road South Kensington London SW7 2DE 020 7589 4090 020 7942 5344 45 Fax 020 7584 8270 email bgslondon bgs ac uk Forde House Park Five Business Centre Harrier Way Sowton Exeter Devon EX2 7HU 01392 445271 Fax 01392 445371 Geological Survey of Northern Ireland Colby House Stranmillis Court Belfast BT9 SBF 028 9038 8462 Fax 028 9066 2835 e mail gsni detini gov uk Maclean Building Crowmarsh Gifford Wallingford Oxfordshire OX10 8BB 01491 838800 e mail hydro bgs ac uk Fax 01491 692345 Columbus House Greenmeadow Springs Tongwynlais Cardiff CF15 7NE T 029 2052 1962 Fax 029 2052 1963 Parent Body Natural Environment Research Council Polaris House North Star Avenue Swindon Wiltshire SN2 1EU 01793 411500 Fax 01793 411501 www nerc ac uk OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Foreword The finite difference radial flow model is develo
37. RO01 OUT NODECSTORAGE01 0UT AND NODECZO01 0UT OUTPUT FILES These files hold the values of the hydraulic parameters calculated between or at the aquifer nodes NodeCAO0l out NodeCRO1 out and NodeCZO0l dat files hold the hydraulic conductance values calculated between the grid nodes in the circumferential radial and vertical directions respectively NodeCStorageO0l out holds the storage coefficient calculated at the grid nodes The values of these parameters are calculated at the start of the numerical simulation The values of these files may not be therefore representative of the hydraulic parameter values at later times These files have the same format as the input files discussed in Section 6 11 and shown in Table 13 7 3 NODERADIUS01 0UT OUTPUT FILE This file holds the radii at which the aquifer grid nodes are located This file has the same format as the input files discussed in Section 6 11 and shown in Table 13 7 4 RELEASEDWATEROI OUT File that holds the amount of water in m released from each grid node during each time step This file is composed of a number of sections equal to the number of numerical time steps Each section has the same format as the input file discussed in Section 6 11 and shown in Table 13 7 5 RESULTO01 OUT OUTPUT FILE This file holds the drawdown values in metres calculated at the aquifer nodes at each time step This file is composed of a number of sections equal
38. a Figure 9 Control Panel Window System Properties System Restore Automatic Updates Remote General Computer Name Hardware Advanced Environment Variables You must be logged on as an Administrator to make most of these changes Performance User variables For majm Visual effects processor scheduling memory usage and virtual memory Variable Value INCLUDE C Program Files Microsoft Visual Studio LIB C Program Files Microsoft Visual Studio Path C Program Files IDM Computer Solutio TEMP C Documents and Settings majm Local TMP C Documents and Settings majm Local User Profiles Desktop settings related to your logon System variables Startup and Recovery Variable System startup system failure and debugging information ogram Files Insightfullsp PATHEXT COM EXE BAT CMD VBS BE J5 Settings PROCESSOR A x86 PROCESSOR ID x86 Family 15 Model 2 Stepping 9 Genu PROCESSOR LE 15 M Figure 10 The System Properties dialog box Figure 10a The Environment Variables dialog box 18 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Edit System Variable Variable name Path Variable value ter SolutionsiLlltraEdit 32 gentu E TS Figure 11 The Edit System Variable dialog box P run bat Notepad File Edit Format View Help MLWell exe C MMyDirectoryMMyradialProjectNMyRun Figure 12 Example of a batch file Type the na
39. ace boundary condition is described by a non linear equation given in three dimensional r 0 z coordinates by ope x dsd9 Ko Os 09 _ ad or n Oror r 0006 az E Equation 4 Where s r 0 z is the drawdown at a given position and at time t L 9 is the free surface position at time t L Ne 1s the porosity of the aquifer Dimensionless and N is the infiltration LTH The derivation of this equation assumes that the vertical z axis is increasing vertically downward The direct implementation of the above equation in the numerical system is not possible because of its non linearity The complexity of the above equation is reduced by assuming that the hydraulic gradients are small so that their multiplications yield terms with even smaller values that can be ignored The moving water table equation becomes dg dt 1 S k ds dz Equation 5 where g is a function that represents the location of the water table L S is the specific yield Dimensionless The term k ds dz is a vertical flow term equal to S 0g 8t The water table is taken into account by adding this term to the basic flow equation and applying it at the water table nodes Rushton and Redshaw 1979 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 2 5 REPRESENTATION OF THE SEEPAGE FACE In unconfined aquifers it is possible that the water level inside the well drops below the elevation where the free surface meets the walls of the boreholes If
40. al water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 water from SY 1242 98 water from sY 1248 water from sy 1251 water from SY 1252 water from sy 1252 water from sy 1252 water from SY 1252 water from sY 1252 water from sY 1252 water from sy 1252 water from sY 1252 water from sy 1252 water from SY 1252 79 water from sY 1252 8 water from sy 1252 81 water from sy 1252 water from sY 1252 water from sy 1252 water from sy 1252 water from SY 1252 water from sy 1252 water from sY 1252 86 water from sY 1252 86 water from ss 13 0162 water from 03756 water from 4 52092 water from 61553 water from 33865 water from 26547 water from 24657 water from 23925 water from 23362 water from 22779 water from 22147 water from 21461 water from ss 3 20722 water from ss 3 19936 water from ss 3 19115 water from 3 18276 water from 3 17444 water from 3 16647 water from 3 15916 water from 3 15276 water from 3 14744 water from ss 3 14322 water from ss 3 14004 well well well well well well well well well 87777e 005 32946e 005 70605e 005 42069e 005 23355e 005 05851e 005 86953e 005 65736e 005 41832e 005 14898e 005 84712e 005 51089e 005 1403e 005 73718e 005 3064e 005 855
41. an infinitesimal control volume which in cylindrical coordinates takes the shape shown in Figure 1 Using the transformation functiona In r the form of this equation becomes similar to the form of the basic flow equation written in Cartesian coordinates The new form is given by K 0h K h d h oh Pa paar TOS Y r oa r 98 z t Equation 2 The derivation of the above equation includes an intrinsic assumption that is the water density is constant over space and time OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 i j kl e Ar P d ps i 1 j k gi Pa AP aa C o P deer 6 LN ELE MM Ljk n uu LjLk e mes e 2 X cA P y iljk e NUM s N Mae i j l k i j k l Figure 1 Three dimensional cylindrical control element Equation 2 is three dimensional equation that provides an accurate solution of the problem The conversion of this equation into finite difference form yields a numerical system that is computationally demanding An approximation is usually introduced to minimise the run time required for the model to reach the solution The approximation is derived by first representing the porous medium vertically by a set of layers and integrating the head over the thickness of each layer and second considering the effect of overlying layers as providing vertical flows leakages through their horizontal interfaces Figure 2 The integration of the head values over the thickness of the layer
42. boundary Cy for yes and N for no if yes include XLL YLL XUR YUR Li 1000000 0 1000000 0 1000000 0 1000000 0 Figure 17 Modified main section of InputO1 dat file In the second section of the Input01 dat file Adjust the layer thickness to a value of 10 m Adjust the horizontal hydraulic conductivity value to 10 m day Because there is only one layer in the model and the aquifer is confined the vertical hydraulic conductivity has no effects on the simulated groundwater heads and its value can be left to the one specified in the dummy file Adjust the specific storage value to 0 0004 m 41 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Because the aquifer is under confined conditions the specific yield parameter is not used and its value can be left as specified in the dummy file It is possible to model a specific layer by more than one numerical grid line However this will make the constructed grid a two dimensional R Z grid if more than one gridline is included This is not necessary in this specific exercise so the Number of grid lines can be left to a value of 1 Hint Specifying more than one gridline to represent a layer helps plotting the drawdown contour lines in a vertical section within that layer Most often the value of this parameter is specified as 1 The second section of the input files must now look like Figure 18 P Input01 dat Notepad File Edit Format View Help HREH HHHOH H
43. cal Survey Keyworth e mail ipr bgs ac uk You may quote extracts of a reasonable length without prior permission provided a full acknowledgement is given of the source of the extract Maps and diagrams in this book use topography based on Ordnance Survey mapping NERC 2007 All rights reserved BRITISH GEOLOGICAL SURVEY GROUNDWATER MANAGEMENT PROGRAMME RESEARCH REPORT OR 07 029 User Manual for the Layered R Theta Numerical Model M M Mansour A G Hughes and A E F Spink British Geological Survey 2 The University of Birmingham Keyworth Nottingham British Geological Survey 2007 BRITISH GEOLOGICAL SURVEY The full range of Survey publications is available from the BGS Sales Desks at Nottingham Edinburgh and London see contact details below or shop online at www geologyshop com The London Information Office also maintains a reference collection of BGS publications including maps for consultation The Survey publishes an annual catalogue of its maps and other publications this catalogue is available from any of the BGS Sales Desks The British Geological Survey carries out the geological survey of Great Britain and Northern Ireland the latter as an agency service for the government of Northern Ireland and of the surrounding continental shelf as well as its basic research projects It also undertakes programmes of British technical aid in geology in developing countries as arranged by the
44. certain values of u In this tutorial the model simulated drawdown values are compared to those produced by the Theis solution at a selected observation borehole The conceptual model must therefore satisfy the Theis conditions The selected aquifer is homogeneous 10 metres deep with a horizontal hydraulic conductivity value of 10m day The aquifer elastic storage value is 0 0004 m which yields a storage coefficient of 0 004 The applied recharge is zero m day The aquifer is pumped at its centre from a borehole with radius of 0 001 m and at a rate of 1256 m day It has impermeable outer boundaries located at a radius of 10000 m from its 39 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 centre The drawdown values will be monitored over a short period of time that during which there will be no boundary effects The conceptual model of this example is illustrated in Figure 15 Central abstraction borehole Radius 0 001 m A Observation borehole Transmissivity 100 m2 d Storage coefficient 0 004 Aquifer radius 10000 m Figure 15 Illustration of Tutorial 1 conceptual model 8 2 2 Building the model There are nine basic input files required to run the model See Table 1 Section 6 1 The user can use any text editor to create these files However to save the user the efforts of creating these files the numerical model produces a set of dummy files if the inpu
45. d SOR used to solve the numerical equations Section 3 6 stops when the maximum residual water balance becomes less than a user defined value This value is specified on line 4 of this input file Under some circumstances the numerical solution may become too complicated so that the residual water balance error may not reduce to a value less than the value specified on line 4 The iterative process must be terminated in this case for the current time step but without terminating the numerical solution The user can specify a maximum number of iterations the numerical solution is allowed to undertake This is the parameter value specified on line 6 of this input file When this number is exceeded the iterative process is stopped for the current time step The SOR depends on a factor to speed up the process of reaching the required numerical solution Section 3 6 The value of this parameter is specified on line 8 of this input file 29 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Table 10 Solver01 dat input file Core of the input file Activate solver Y for yes Y Allowable error m3 day 0 001 Maximum number of iterations 500000 Omega 1 6 on QN tA A U N e e 69 WATERTABLE01 DAT INPUT FILE When a water table node becomes too close to its underlying aquifer node it disconnects from it and connects to the subsequent aquifer node This is discussed in Section 3 3 of this manual The dista
46. d aquifer nodes move downward due to pumping as shown in Figure 4b The conductance between the two aquifer nodes is then based on the smallest saturated thickness between these nodes elevation of W1 minus the elevation of the base of the layer as shown in Figure 4b This depth is always smaller than the initial layer thickness and reduces towards zero with time At the same time horizontal conductances between the water table nodes W1 and W2 are modified The conductance between the free surface nodes is calculated as the mid point saturated thickness half the elevation of W2 minus the elevation of W1 multiplied by the hydraulic conductivity value of the layer where the water table nodes are located If adjacent water table nodes occur in different layers W1 in Layer 2 and W2 and Layer 1 for example the conductance term uses the average hydraulic conductivity of the layers When the distance between the water table node and the base of a layer becomes smaller than a user defined value the water table disconnects from the underlying aquifer node which becomes inactive and connects to the next lower aquifer node W1 disconnects from A1 and connects to A3 in Figure 4c and A1 becomes inactive Since the saturated thickness at the node which OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 becomes inactive is small at this time Figure 4 the de activation of the node does not lead to groundwater head oscillations Wi we W2
47. dat Clock01 dat file is used to set the time steps of the simulations Open this file using a text editor The first parameter value to specify in this file is the value of a reference start time days This is not used in this tutorial and the value of this parameter can be left as Zero Set the value of the start time to 0 001 days Set the value of the time after which the time step stops increasing in logarithmic fashion to 1 The time step after 1 day will be equal to 1 43 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Change the number of time steps in each ten fold increase to 12 There will be 12 time steps between time 0 001 and 0 01 days another 12 time steps between 0 01 and 0 1 days and 12 time steps between 0 1 and 1 days Clock01 dat should now look like Figure 20 P Clock01 dat Notepad File Edit Format View Help Real Start Time Cdays Start Time Cdays 0 001 Time after which the time step becomes constant this is also the constant time step days Number of time steps per cycle 12 Figure 20 Modified Clock01 dat file SETTING THE CONVERGENCE CRITERIA Input0l dat wellO1 dat Clock01 dat I Solver01 dat Output01 dat Solver01 dat file is used to set the convergence criteria and the value of the over relaxation parameter Open this file using a text editor Accept the flag Y for running the model This flag can be set to N to produce th
48. e set of additional files listed in Table 2 Section 6 1 without running the model Set the allowable error to 0 0001 Set the maximum number of iterations to 100000 The model will perform the calculation of the next time step if the number of iterations undertaken in the current time step exceeds this specified number Accept the value of the successive over relaxation parameter as 1 6 Solver01 dat file should now look like Figure 21 P Solver01 dat Notepad File Edit Format View Help activate solver Y for yes Y Allowable error 0 0001 Maximum number of iterations Figure 21 Modified SolverO1 dat file 44 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 SETTING THE LOCATION OF OBSERVATION BOREHOLES InputO1 dat I Well01 dat Clock01 dat Solver01 dat Output01 dat Output01 dat file is used to specify the output files produced by the model Open this file using a text editor Accept all the flags Y in this input file and accept that the results be produced at the end of each time step Specify the number of observation boreholes There is only one observation borehole in this exercise Set the location of the observation borehole and its depth The observation borehole is assumed to be 10 m away from the centre of the abstraction borehole Figure 15 Adjust the radius value to 10 m There is one slice in this model so set the angle Theta to a value of 0 0 The d
49. e are KrFile01 txt KtFile01 txt and SsFileOl txt These files are used to set the new horizontal hydraulic conductivity values along the radial direction the horizontal hydraulic conductivity values along the circumferential direction and the specific storage values respectively If the vertical hydraulic conductivity values and the specific yield values need alteration the files KvFileO1 txt and SyFile01 txt are used respectively Open KrFile01 txt input file using a text editor Do not allow the text editor to wrap lines This file lists the hydraulic conductivity values at nodes located along one radial direction on one line starting with the node adjacent to the pumped well and ending with the node at the outer boundary A number of input data lines equal to the number of slices specified in Input01 dat input file are listed in this file Thus there must be eight lines of input data in this example A header line that precedes the input data line indicates the slice number and the angle in radians to which the data on this line belong Note that it is possible to know the node radius by using the file NodeRadius01 dat Figure 28 indicates that the radial hydraulic conductivity values at the nodes located at Slice 1 angle 0 785 rd need to be reduced Change the values of the first data line of the KrFile01 dat data file from 10 to 0 01 Figure 29 E KrFile01 txt Notepad TEE File Edit Format View
50. e included in the model by setting the flag on line 14 to Y The well loss factor Section 3 5 is specified on line 16 of this input file Finally the number of abstraction phases and the corresponding abstraction rate are specified in this file The number of abstraction phases is specified on line 18 The abstraction phase durations and the abstraction rates are specified on lines 20 onward To include a recovery phase the abstraction rate of the abstraction phase can be set to zero The well loss factor will have no effects on the drawdown values simulated during this phase even if the well loss flag is set to Y Table 12 Well01 dat input file Line Core of the input file 1 Hydraulic conductivity representing water m day 2 10000 3 Borehole depth m 4 3 0 5 Pump depth m 6 3 7 Casing Please specify S for single value or M for multiple values at the end of this line S 8 0 0 9 Set gradual delivery at the start of each phase 10 N 11 Time after which pump delivers full abstraction rate days 12 0 00208 13 Include Well Losses 14 N 15 well losses factor 16 0 08 17 Number of Abstraction Phases 18 1 19 Q m3 day duration days 20 1000 10 0 31 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 6 11 POSSIBLE ADDITIONAL INPUT FILES Aquifer heterogeneity can be added to the model through the use of a set of files each dedicated to one hydraulic parameter All these files have the txt e
51. emaining part of the simulation time of the 22 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 abstraction phase When a new abstraction phase starts the time step is increased logarithmically again starting from the logarithmic pattern start time defined on line 4 In some cases the time series of the additional abstraction wells abstraction rates recharge rates river stage changes etc start at a time different from the pumping test start time An additional input parameter is included in the model and is used to synchronise the pumping test and other features times The value of this parameter is defined on line 2 of the input file Table 3 Clock01 dat input file Input file name Number of time steps per cycle 1 Reference Start Time days 2 0 3 Start Time days 4 0 00001 5 Time after which the time step becomes constant this is also the constant time step days 6 10000 7 8 10 6 3 INPUT01 DAT INPUT FILE Input01 dat file is used to set the size and geometry of the model grid as well as setting the values of the hydraulic parameters of the different layers of the aquifer This file is divided into two parts The first is used to set the structure and size of the grid Table 4 and the second is made of a set of sections used to set the hydraulic parameters of the layers Table 5 The example given in Table 5 is repeated in this file a number of times equal to the number of layers in the
52. ent of the water table and its effects on the connections between the model MOGES L5 teste decree ette e iret Feo b Pra dec d tede edad e ia DR TE Sete ves 10 Figure 5 Example of a seepage face occurring at a partly penetrating cased well 10 Figure 6 Representation of well losses in the numerical model eeesssss 11 Figure 7 An example of a sparse matrix where the stars illustrate a given number in the matrix and blank areas illustrate numbers equal to zero seen 12 Figure 8 Time stepplng eoe eR Pe RR ER te Reis 13 Figure 9 Control Panel Window seeessseseessseeeeeeeeeeeene eene nnrn enne 18 Figure 11 The Edit System Variable dialog box eene 19 Figure 12 Example of a batch file ssessesseeeeeeeeeeeeeeeenneeen nennen 19 Figure 13 Starting a command line window from the Windows start menu 19 Figure 14 Defining internal boundaries The right figure shows an example where only one side of the rectangle causes internal boundaries eene 24 Figure 15 Illustration of Section 1 conceptual model eeeee 40 Figure 16 the sequence of modifying tutorial 1 input files eee 40 Figure 17 Modified main section of InputO1 dat file see 41 Figure 18 Modified second section of Input01 dat file eee 42 Figure 19
53. epth of the observation borehole must be set to a value equal to the depth of the grid line the observation borehole connects to This is 5 0 m There is only one gridline in this tutorial OutputO1 dat file must now look like Figure 22 P Output01 dat Notepad File Edit Format View Help Printout node type h Printout node Radius printout node Constants Printout node pointers Printout node extra flow set the script files Nor of time steps after which results amp waterTable are recorded Number of observation boreholes pa Radius m Theta radian Depth m 10 0 0 0 5 Figure 22 Modified Output01 dat file RUNNING THE MODEL AND EXAMINING THE OUTPUT FILES Run the model as explained in Section 5 of the manual A summary of the model status is displayed in the DOS box This includes the number of iteration undertaken to reach a solution for each time step Phase number time step number cumulative time and real time Cumulative time plus reference start time Figure 23 Note that in Figure 23 the change of the length of time step when time is less than day and the constant length of the time step when time becomes more than 1 day 45 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 ct C WINDOWS system32 cmd exe Licensing code developed by C R Jackson 2006 Layer Borehole Free Surf Link Done Read DD Read Coef Solver 3xNo w No sx Solving Iterations Phase Time step Time 1705 1 1
54. er from ss 1044 33 water from ss 967 885 water from ss 889 147 water from ss 807 871 water from ss 723 545 water from ss 635 816 water from ss 545 water from ss 452 523 water from ss 361 101 water from ss 274 524 water from ss 197 002 water from SY 66 532 water from sY 137 473 water from sv 211 529 water from SY 288 04 water from sY 366 808 water from sY 448 1 water from sYs 532 435 water from sY 620 171 water from sy 710 991 water from SY 803 471 water from sY 894 895 water from sY 981 473 water from sy 1059 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 from from from from well well well well 000673582 000377924 000216653 00013575 water from ss 132 203 water from SS 82 2819 water from ss 47 2957 water from ss 25 2841 water from sy 1123 8 water from SY 1173 72 water from sy 1208 7 water from sy 1230 71 Total water m3 d 1256 Total water m3 d 1256 Total water m3 d 1256 1256 Total water m3 d Total water m3 i Total water Total water Total water Total water Total water Total water Total water Total water Total water Total water Total water Total water m3 d 1256 Tot
55. f the manual 41 MULTIPLE LAYERS AND MULTIPLE NUMERICAL GRID LINES IN EACH LAYER The radial flow model can incorporate multiple layers of finite difference nodes Each finite difference layer represents one discrete geological layer The nodes of this layer hold its top and base elevations and its different hydraulic characteristics It is also possible to represent one geological layer with multiple finite difference layers The main reason for that is to improve the simulation of groundwater flows in the vertical direction if only one geological layer is studied These finite difference layers will divide the total saturated thickness of the geological layer into equal horizontal slices The nodes of all the finite difference layers will hold the same characteristics of the geological layer but their presence will allow the specification of different hydraulic features at different elevations The groundwater head values calculated at these nodes can be also used to contour the groundwater heads in the corresponding vertical section 4 2 CONFINED AND UNCONFINED CONDITIONS Both confined and unconfined aquifers can be modelled When a confined aquifer system is studied the nodes representing the system are immobile and the layer transmissivity and storage coefficient are independent of the groundwater heads The storage coefficient at all aquifer nodes are calculated in terms of the specific storage When unconfined systems are studied an additi
56. grid nodes the effects of a certain abstraction rate on the simulated groundwater heads will be more pronounced if the well is placed on a node close to the centre of the model 4 13 RIVERS Because of its grid structure the radial flow model cannot represent linear features such as rivers accurately In the current version of the model rivers are represented as leakage nodes but with stage elevations used as heads in the model that vary with time All nodes belonging to one river have the same stage elevation at a given time This simple representation allows the inclusion of the effects of rivers on the simulated groundwater flows but this is by no means a detailed representation of rivers 16 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 5 Running the model 5 1 INSTALLING THE EXECUTABLE To run the radial flow model copy the executable file into a selected directory on a selected drive copy MLWell exe to CAML Well directory for example Set the path of the system so it points to this directory Follow these steps Open the Control Panel Window and double click the command System Figure 9 A dialog box similar to the one shown in Figure 10 will appear Select the Advanced tab of this dialog box and click the button Environment Variables to produce the dialog box shown in Figure 10a In the Environment Variables dialog box scroll down the lower list box until you find the command
57. he simulated drawdown values however this will be at the expense of the runtime of the model Well nodes Water table nodes Broken connection y uem A caused by casing A i p e e a Layer Broken connection caused by the formation of seepage face 46 b e b Layer2 Aquifer node connected to a well node Aquifer node connected to 1 6 P gt P Layer3 opposite aquifer nodes E e e p Layer 4 Figure 5 Example of a seepage face occurring at a partly penetrating cased well 10 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 When a seepage node occurs the abstraction rate must be reduced This reduction is quantified at the end of the time step using the groundwater heads established at the grid nodes The groundwater heads are then recalculated for the same time step using the reduced abstraction rate and applying the seepage face conditions groundwater head at a seepage node elevation at that node See Section 2 3 The seepage flow is then calculated and compared to the previously calculated seepage flow This procedure is repeated until the difference between the newly and previously calculated seepage flows becomes negligible 3 5 REPRESENTATION OF WELL LOSSES Well losses are caused by the turbulent nature of the groundwater flows approaching the pumped well The high velocity of these groundwater flows affec
58. ice Number 10 10 10 Slice Number 10 10 10 Slice Number 10 10 10 Slice Number 10 10 10 Slice Number 8 dangle 0 01 0 01 0 01 0 01 0 01 Figure 30 Example of the KtFileO1 txt input file Open SsFile01 inp input file using a text editor Do not allow the text editor to wrap lines This file has the same format as KrFile01 inp and KtFile01 inp input files Change the values of the second data line from 0 0004 to 0 00004 See Figure 31 P SsFile01 txt Notepad File Edit Format View Help Layer Number 1 slice Number 1 dAngle 0 785398 0 0004 0 0004 0 0004 0 0004 0 0004 0 0004 0 0004 0 0004 Slice Number 2 dAngle 1 5708 0 00004 0 00004 0 00004 0 00004 0 00004 0 00004 dangle 2 35619 0 0004 0 0004 0004 0 0004 0004 0004 dangle 3 14159 0 0004 0 0004 0004 0004 0004 0004 Slice Number 5 dAngle 3 92699 0 0004 0 0004 Q 0004 0 0004 0004 0004 0004 0004 Slice Number 6 dAngle 4 71239 0 0004 0 0004 0 0004 0 0004 0004 0004 0004 0004 Slice Number 7 dAngle 5 49779 0 0004 0 0004 0 0004 0 0004 0004 0004 0004 0004 Slice Number 8 dAngle 6 28319 0 0004 0 0004 0 0004 0 0004 A 0004 oO 0 0 0 0 0 0o o Figure 31 Example of the SsFileO01 txt input file Open Output01 dat input file Set the number of observation boreholes to 2 Set the radius angle and depth of the first borehole to 100 m 0 78 rd 1 4 and 5 0
59. included in the model so that these drawdown values and water table node locations are produced only after a number of time steps have been elapsed The number of elapsed time steps is specified on line 14 of Table 6 Drawdown values at selected observation boreholes can be produced The number of these observation boreholes is specified on line 16 of this input file The locations of these observation boreholes are specified on line 18 and onward 26 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Table 6 Output01 dat input file Core of the input file Printout node type Y Printout node Radius Y Printout node Constants Y Printout node pointers Y 1 2 3 4 5 6 7 8 9 Printout node extra flow Y Set the script files Y Number of iterations after which results amp WaterTable are recorded 1 Number of observation boreholes 1 Radius m Depth m 0 1 6 28 3 0 6 5 PUMPING01 DAT INPUT FILE Additional abstraction wells can be included in the model using the Pumping01 dat input file It must be noted that these wells are not represented in the model as comprehensively as the central well is and only the abstraction rate of these wells are allocated at their representative nodes in the model The number of abstraction wells is specified on line 2 of this input file Table 7 The part starting from line 3 to line 10 must be repeated for each additional abstraction well The length of th
60. internal boundaries recharge etc The aim of this tutorial is to provide a guide for the user to build the numerical model and to become familiar with its functionality The first tutorial gives a step by step sequence to build a simple one dimensional numerical model to run it and to examine its outputs The results obtained from the Theis solution for an observation borehole are used to check the model output In the second tutorial the aquifer heterogeneity is included This requires building a two dimensional model R 0 in this tutorial and updating the additional input files In the third tutorial groundwater flows in an unconfined aquifer are simulated Water table recharge well losses and seepage face processes are included in this tutorial 82 TUTORIALI1 USING A BASIC MODEL TO SIMULATE GROUNDWATER DRAWDOWN VALUES IN A HOMOGENEOUS AQUIFER 8 2 1 Problem Description Theis equation calculates the drawdown in an infinite confined and homogeneous aquifer with the following additional conditions The rest water level is horizontal there is no recharge and the abstraction well is small so no well storage is included The Theis equation takes the following form 2 s ww u IF Where s is drawdown L Q is abstraction rate L T T is transmissivity L T r is radius L S is the storage coefficient Dimensionless t is time T W u is the well function that can be either evaluated numerically or retrieved from tables for
61. ion represents a flow balance which must always be satisfied The radial flow model checks the flow balance during the iteration process and when the residual error of this equation at all nodes is smaller than a specified value i e the convergence criterion the iteration process stops The model run time depends on the values of the convergence criterion and the parameter discussed in Section 3 6 Increased accuracy requires a very small convergence criterion value leading to large number of iterations and consequently an unacceptable increase in the computing time The efficiency of computation can be improved by changing the value of a however numerical difficulties may arise especially when solving the head values inside the pumped well where the vertical hydraulic conductivity value is several orders of magnitude larger than the value of the horizontal hydraulic conductivity 3 8 TIME STEPPING The numerical solution requires that groundwater heads be calculated at discrete time steps The implicit form of the numerical finite difference equations does not impose a restriction on 12 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 the time step size to maintain the stability of the numerical solution see Section 3 6 However analysis of pumping test shows that significant groundwater heads changes happen at the early times of the test and then the rate of groundwater head changes reduces with the progress of the test unless a b
62. is part varies depending on the number of data held in the abstraction rate time series specified on line 7 see below so line 10 may not be necessarily the end of this part On line 5 the location of the additional abstraction well is specified This is specified by the distance between this well and the central well radius Radius the vertical section the well is located on Angle and the depth calculated from the top of the saturated part of the aquifer Depth The number of data point in the abstraction rate time series is specified on line 7 of Table 7 The time series which take the form of time days followed by abstraction rate m day are included from line 9 onward Note that the time are compared to the reference start time specified on line 2 of the ClockO1 dat file Table 3 and the abstraction rates are set accordingly 27 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Table 7 Pumping01 dat input file Core of the input file Number of additional abstraction points 1 K k k k k k k k k kk k New Well 2g os K 2 k ee k k k k k k k k k K Location Radius m Angle Depth m 1 2 3 Number of data in Abstraction time series 2 Time series Real time days m3 day 0 4 12 8 1 2 3 4 5 6 7 8 9 6 6 RECHARGE01 DAT INPUT FILE Recharge can be included in the model through the Recharge01 dat input file Table 8 The recharge cannot be spatially varied over the model area
63. it is uniformly distributed over this area However the recharge can vary with time The recharge has units of m day The number of data point in the recharge rate time series is specified on line 2 of Table 8 The time series data which take the form of time days followed by recharge rate m day are included from line 4 onward Note that times are compared to the reference start time specified on line 2 of the ClockOl dat file Table 3 and the recharge rates are set accordingly Table 8 Recharge01 dat input file Core of the input file Number of data in recharge time series 1 Time series Real time days m day 0 0 00001 6 7 RIVERO1 DAT INPUT FILE The number of rivers is specified on line 2 of River01 dat input file Table 9 For each river the number of nodes forming the river and their locations must be specified For the first river the number of its nodes is specified on line 5 The locations of these nodes are specified in polar coordinates radius and angle on line 7 and onward The hydraulic conductance value controlling the river aquifer interaction is calculated using the river bed thickness and a vertical hydraulic conductivity refer to Section 2 4 The values of these parameters are specified at each river node In the input file these are specified on the same line where the location of the node is specified line 7 in Table 9 28 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 1
64. iver01 dat Rivers and river stage time series Section 6 7 Solver01 dat Numerical solvers Section 6 8 WaterTable01 dat Set criterion to connect water table Section 6 9 nodes to aquifer nodes Well01 dat Borehole depth Section 6 10 Pump depth Casing depth Well losses Abstraction rate time series Stop time Abstraction and recovery phases 21 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Table 2 List of possible additional input files Input file name Model Features Relevant section KrFile01 inp Aquifer characteristics changing the Section 6 11 hydraulic conductivity in the radial direction KtFile01 inp Aquifer characteristics changing the Section 6 11 hydraulic conductivity in the circumferential direction KvFile01 inp Aquifer characteristics changing the hydraulic conductivity in the vertical direction SsFile01 inp Aquifer characteristics changing the Section 6 11 specific storage SyFile01 inp Aquifer characteristics changing the Section 6 11 specific yield ZBaseFile01 inp Layer dimensions changing the Section 6 11 elevations of the base of layers ZTopFile01 inp Layer dimensions changing the Section 6 11 elevations of the top of layers NodelInitDDwn0l inp Initial drawdown values Section 6 11 62 CLOCKO01 DAT INPUT FILE Clock01 dat input file is used to set the time stepping of the numerical solution Table 3 As discussed in Section 3 8 time steps increases from a start time value to
65. l the model and simulate groundwater flows in complicated aquifer systems OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 2 Theoretical background This section describes the mathematical equation used to represent the groundwater flow processes in the model Section 2 1 details the basic flow equation describing the groundwater flow movement in porous medium Section 2 2 details the equation describing the movement of the water table Section 2 3 details the equation describing the occurrence of the seepage face and Section 2 4 details the equation describing the river aquifer interaction 21 BASIC FLOW EQUATION The radial flow model solves the flow equation describing flow through a porous medium which under confined condition is given by i k 2 0 x 2d d x e d x 2 s 2y r or or or ra 00 oz Oz gt where Equation 1 r is the radius measured from the centre of the pumped well L is the angle measured from a reference axis Dimensionless z is the elevation measured from a selected datum respectively L h r Qz t is the groundwater head value at a point r 0 z and time t L k r Qz kar Oz and k r Gz are the hydraulic conductivity values in the radial circumferential and vertical directions respectively L TIVE S is the specific storage IET N is an external source term TH This equation is derived by writing the mass balance equation of the water entering leaving and being stored in
66. lds information about time descritisation the start of simulation time and is the right place to specify the simulation stop time However the duration of pumping is also one of the abstraction well characteristics and can be specified in WellO1 dat input file Indeed it has been found that it is more convenient to specify the duration of pumping and consequently the simulation stop time in WellO1 dat Table 1 shows a list of the main input files required to run the radial flow model If aquifer heterogeneity is included in the model additional files are required These are listed in Table 2 Generally input values are entered in the main input file as one value on one line This is most often preceded by a comment line explaining the use of the input value Exceptions may be found in files where time series input data are entered or where the values of X and Y coordinates are specified 20 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Table 1 List of main minimum input files Input file name Model Features Relevant section Clock01 dat Time discretisation Section 6 2 Start time 2 InputOl dat Well size Section 6 3 Layering Grid structure Groundwater system Boundary conditions Aquifer characteristics Initial conditions Output01 dat Time series at selected observation Section 6 4 boreholes Specify output files Pumping01 dat Additional abstraction points Section 6 5 Recharge01 dat Recharge time series Section 6 6 R
67. le followed by a string holding the full path of the working directory Press enter to run the model 17 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 amp Control Panel File Edit View Favorites Tools Help Q Back z amp I Search Folders la X i Eii Address Control Panel B Name Comments T Game Controllers Add remove and c Be Hummingbird InetD Hummingbird InetD Dinternet Options Configure your Inte akeyboard Customize your key Email Windows Messaging ES Microsoft Mail Postoffice Administers a Micro B Mouse Customize your mo Network Connections Connects to other c a NVIDIA Control Panel Configure your NYI 3NVIDIA nView Desktop Manager Configure your NYI Phone and Modem Options Configure your tele Power Options Configure energy s So Printers and Faxes Shows installed prin L Regional and Language Options Customize settings Sy Scanners and Cameras Add remove and c Scheduled Tasks Schedule computer e Security Center View your current s Dalsoundmax Control Panel For So Sounds and Audio Devices Change the sound s 4 Speech Change settings for Dsymantec LiveUpdate This applet allows y See information abo VW Tablet Properties Tablet Properties EB Taskbar and Start Menu Customize the Start S User Accounts Change user accou eu Windows Firewall Configure the Wind s Wireless Network Setup Wizard Set up or addto
68. le in a text editor o Change the well radius to a value of 0 001 m o Change the aquifer radius to a value of 10000 m o Change the number of radial intervals to 8 intervals per 10 fold increase As stated before a one dimensional numerical model will be built for this section A set of nodes located along one radial direction are therefore required This is located in one circumferential slice of a layer Specifying one circumferential interval and one layer in the Input01 dat file does not need to be changed The next flag is related to the outer boundary condition and is set to the character T indicating that this boundary is impermeable The flag related to the aquifer conditions is set to the character C indicating that the aquifer is under confined conditions The remaining parameters in the first section of this input files are not used here and can be left as defined in the dummy file The first section of the input files must now look like Figure 17 P Input01 dat Notepad File Edit Format View Help well radius m Q01 Aquifer radius m 10000 Number of radial intervals per cycle 8 Number of circumferencial intervals 1 Nbr of Layers He Boundary condition I for Impermeable and F for Fixed Problem type u for unconfined and C for Confined ie Read initial heads from a file Cy for yes and N for no N Read aquifer characteristics from files v for yes and N for no N Set conditional
69. ll nodes for each time step This file is composed of sections the number of which is equal to the number of numerical time steps The values of each section correspond to one time step The format of these sections is the same as the one discussed in Section 7 8 and shown in Table 18 37 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 7 10 WTRELWATERO01 OUT OUTPUT FILE This file holds the amount of water released from each water table node m for each time step Values of each line of this file are those calculated at water table nodes located along one radial direction A number of lines equal to the number of vertical sections in the model are written to the file before the start of a new time step 7 11 WTRESULT01 0UT OUTPUT FILE File that holds the drawdown values at the water table nodes m for each time step The drawdown values of each line of this file are those calculated at water table nodes located along one radial direction A number of lines equal to the number of vertical sections in the model are written to the file before the start of a new time step 38 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 8 Tutorials 81 INTRODUCTION The radial flow model can be used in its simplest one dimensional form to simulate groundwater flows to pumped wells in homogeneous aquifers or in more complicated forms to include the heterogeneity of the aquifers and other processes such as well losses seepage face
70. m and of the second borehole to 100m 3 92rd 57 4 and 5 0m respectively Open InputOl dat input file and change the flag regarding reading the aquifer characteristics from files too Y The ninth entry line of this file Open Solver01 dat input file Set the flag that activate the solver back to Y Open WellO1 dat and set the abstraction phase duration to 100 days 51 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 8 3 3 Running the Model Run the model executable as explained in the previous section or as in Section 5 of the manual Open Timeseries01 out output file using a text editor Note that there are five columns of data in this file This is because two observation boreholes are specified in Output01 dat The last two columns give the drawdown values at these observation boreholes Plot and compare the time drawdown curves produced at the observation boreholes See Figure 32 Time days 0 01 0 10 1 00 10 00 100 00 E c z o 5 s S a e Results at Observation Borehole 1 Low K e Results at Observation Borehole 2 High K Figure 32 Comparison between the numerical results produced at the observation borehole located in the low permeability part of the aquifer Observation Borehole 1 and those produced at the observation borehole located in the high permeability part of the aquifer Observation Boreh
71. me of a program Folder document or Internet resource and Windows will open it For you x Figure 13 Starting a command line window from the Windows start menu 5 3 MODEL RUN TIME The run time of the numerical model will vary with the type of problem to be solved The complexity of the problem determines the level of discretisation required to simulate the groundwater flows accurately and consequently the number of nodes included in the model The more nodes there are in the model the longer is the model run time The high conductance value selected to represent the water inside the central pumped well may create numerical difficulties and increase the time required for the solution to converge The user can experiment with the value of this parameter to optimise the run time of the model however the mass balance must always be checked to make sure that the model has produced an accurate solution 19 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 6 Model input files 6 1 THE PHILOSOPHY OF MODEL INPUT FILES Each file of the radial flow model input files is designed to contain one type of information that is related to one specific feature For example an input file is dedicated to set the pumped well characteristics another input file to set the river characteristics etc However a conflict between the most appropriate places for one specific type of information may occur For example the input file Clock01 dat ho
72. model The first section of this input file starts by specifying the central well radius and the aquifer radius These are specified on lines 2 and 4 respectively Table 4 As discussed in Section 3 1 the space intervals between the grid nodes increase in a logarithmic pattern along the radial direction The number of space intervals in each ten fold increase is specified on line 6 of the input file Table 4 The aquifer can be represented using one grid located within one vertical section It is also possible to divide the aquifer horizontally using multiple grids with each grid located within a separate vertical section to represent the horizontal spatial variations of the aquifer characteristics more accurately The number of vertical sections is specified on line 8 of the input file The number of aquifer layers is specified on line 10 of the input file The user can set a numerical model in its simplest form using one layer model with one vertical grid having one line of node A two dimensional model in a vertical plan can be built by increasing the number of layers the value on line 10 and a two dimensional model can be built in a horizontal plan by increasing the number of vertical sections The value on line 8 A three dimensional model can be built by specifying values on lines 8 and 10 greater than unity 23 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 The outer aquifer boundary can be specified as an impermeable bounda
73. n at B is used to calculate the conductance between B and C Figure 3 The definition of hydraulic parameters at grid nodes 3 3 REPRESENTATION OF THE WATER TABLE Equation 5 shows that water table nodes provide the vertical flow component to the underlying aquifer However these nodes can also transfer water horizontally and circumferentially to adjacent water table nodes The numerical model incorporates a novel grid structure in which the water table and the aquifer nodes can move vertically based on the head values calculated at the end of each time step The positions of the aquifer nodes that are directly connected to the water table nodes are modified so that they always occupy the mid point between the water table and the base of the corresponding aquifer layer Figure 4 This collapsing numerical grid significantly reduces the groundwater head oscillations caused by the creation of internal boundaries These boundaries are created when the free surface falls sufficiently and the horizontal conductance between two nodes is set to zero to disconnect an aquifer node from its adjacent free surface node The process is explained below At time zero the water table is horizontal and there is only a vertical connection between the water table nodes and their underlying aquifer nodes Figure 4 In addition the conductance value between nodes Al and A2 is based on the full layer thickness the hashed area in Figure 4a Later the water table an
74. n the low permeability part of the aquifer Observation Borehole 1 and those produced at the observation borehole located in the high permeability part of the aquifer Observation Borehole 2 nr eit en d e ete UI VERSNS doa eg 52 Figure 33 Tutorial 3 calculated water balance calculated eeeeeses 54 Figure 34 Comparison between the simulated results and those calculated using the Neuman SODIUM 55 Figure 35 Runtime information displayed in a DOS box eee 56 Figure 36 SeepageFlowFileO1 dat showing the seepage flows calculated at the different CIM SLEDS ces 57 Figure 37 Comparison between the simulated results with and without a seepage face 57 TABLES Table 1 List of main minimum input files essen nere 21 Table 2 List of possible additional input files eseeeeeeeeeeeeeeeeneeenennnen 22 Table 3 Cl ck01 dat input file teen das idet entente e detenti edd nenne 23 Table 4 Main section of the InputO1 dat input file eese 25 Table 5 Layer specific section of InputO1 dat input file eeeeeeeeeeeee 26 Table 6 Output01 dat input file eesesesseeeeeeeeeeeeee nennen nnne ether nt enn 27 Table 7 PumpingO1 dat input file 00 0 ee eeseeeeesseeeeceenaeeeeeeeeeceseaaeeesseeeecessaeeeeenaes 28 Table 8 RechargeO1 dat input file
75. nce between the water table and aquifer nodes that initiates this disconnection may affect the numerical solution and must be specified by the user Watertable01 dat input file is created for this purpose This distance is specified on line 2 of this file as shown in Table 11 Table 11 WaterTable01 dat input file Core of the input file Minimum distance between the water table node and the grid node m 0 01 6 10 WELLOI DAT INPUT FILE The Well01 dat input file is dedicated to the central abstraction well of the model Table 12 The characteristics of this well and the features included are specified in this file Borehole depth pump depth and casing depth are specified on lines 4 and 6 respectively Note that these depths are taken from the initial position of the rest water table and that these depths will affect the nearest node located along the well face Casing depth is specified on line 8 Important While line 7 is a comment line the last character of this line is used as a flag to control how well casing is specified on line 8 This last character must be either S indicating that there is a single value on line 8 or M indicating that there are multiple values on line 8 If this flag is set as S well casing depth m measured from the rest water table is entered on line 8 Setting this flag to M allows the possibility of disconnecting selected numerical gridlines from the abstraction well i e
76. of the Input01 dat input file Core of the input file Well radius m Al Aquifer radius m 1000 Number of radial intervals per cycle 6 Number of circumferential intervals 1 O oo y Duna FW Ne Number of Layers 1 Boundary condition T for Impermeable and F for Fixed I Problem type U for Unconfined and C for Confined C Read initial heads from a file Y for yes and N for no N Read aquifer characteristics from files Y for yes and N for no N Set conditional boundary Y for yes and N for no if yes include XLL YLL XUR YUR Y 1000000 0 1000000 0 1000000 0 1000000 0 The second part of InputOl dat input file consists of a set of sections used to set the hydraulic parameters of the numerical layers Table 5 shows an example of one section used to set the hydraulic parameters of one numerical layer This section is repeated a number of times equal to the number of layers included in the model This file starts with a header line included for the user to enter any comment if required This line holds a set of character in Table 5 The layer number must be included on line 3 of Table 5 The upper most layer has a number of 1 The thickness of the layer is set on line 5 of Table 5 The aquifer hydraulic parameters are the horizontal hydraulic conductivity the vertical hydraulic conductivity the specific storage and the specific yield The values of these parameters are set on lines 7 9 11 and 13 of Table 5 I
77. ole 2 52 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 8 4 TUTORIAL 3 USING THE MODEL TO SIMULATE GROUNDWATER FLOW IN AN UNCONFINED AQUIFER In this tutorial the model is used to simulate the groundwater flow in an unconfined aquifer The tutorial is divided into two parts In the first part the model results are compared to the Neuman solution In the second part additional grid lines are introduced in the model so that seepage nodes form at the face of the pumped well during the simulation 8 4 1 Description of Problem In this tutorial the time drawdown curve produced by the numerical model at an observation well located at 50m away from a pumped borehole drilled in an unconfined aquifer is compared to the time drawdown T D curve produced using Neuman analytical solution The comparison here is not perfect because Neuman solution does not take into account the well storage when calculating the drawdown at an observation borehole The effects of well storage on the numerical results cannot be completely eliminated as demonstrated in Tutorial 1 Part 1 8 4 2 Modification of input files Create the dummy input files and modify as in Tutorial 1 MODIFYING INPUTO1 DAT INPUT FILE Change the well radius to 0 004 m Change the aquifer radius to 40000 m e Set the problem type to U the 14 line of this input file Set the aquifer thickness to 50 m Set the horizontal hydraulic conductivity to 10 m da
78. onal finite difference layer is added on top of the aquifer finite difference layers The storage coefficient values of the nodes of this additional layer are calculated based on the specific yield term In addition both the water table nodes and the aquifer nodes are allowed to move with the variation of the groundwater heads leading to a variation in both the transmissivity and storage coefficient values as discussed in Section 3 3 4 3 IMPERMEABLE AND FIXED HEAD BOUNDARIES The boundary condition at the outer side of the model must be defined Two boundaries are allowed in the model These are either impermeable boundary Neumann boundary of M 0 or fixed head boundary Dirichlet boundary of h 0 r 44 HETEROGENEITY A heterogeneous aquifer system can be simulated Different hydraulic parameter values can be specified at each finite difference node and hydraulic conductivity values may be different in the r and 0 directions at one node 45 VARIABLE PUMPING RATE AND RECOVERY PHASES The pumping rate at the main abstraction well at the centre of the aquifer can vary with time The approach followed is to use phases of constant abstraction rates Whenever a new 14 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 abstraction rate is applied a new phase starts The recovery is simulated as an abstraction phase but with an abstraction rate of zero 4 6 GRADUALLY INCREASING PUMPING RATE It is possible that a pump does not reach
79. onfined aquifer simulation Water from Well 901 124 Water from SS 98 6012 Total Water m3 d 999 725 Water from Well 800 312 Water from SS 198 263 Total Water m3 d 998 575 Water from Well 694 79 Water from SS 300 882 Total Water m3 d 995 672 7 8 WELLNODECAO01 OUT WELLNODECRO1 0UT WELLNODECSTORAGEO01 0UT AND WELLNODECZ01 0UT OUTPUT FILES These files hold the values of the hydraulic parameters calculated between or at the central well nodes WellNodeCAOl out WellNodeCROl out and WellNodeCZOl dat files hold the hydraulic conductance values calculated between the central well nodes in the circumferential radial and vertical directions respectively WellNodeCStorage01 out holds the storage coefficient calculated at the central well nodes The values of these parameters are calculated using the hydraulic conductivity value specified in the Well01 dat input file Section 6 10 and a storage coefficient value of one There are two vertical lines of nodes opposite to each vertical section in the aquifer Parameter values are produced in the format shown in Table 18 Table 18 Template of output files of the hydraulic parameters calculated at the central well Line Core of the output file 1 Circumferential Number 1 O oo 1 nun FW WN Circumferential Number 2 Fe e e Re om on Un A UU N KF C 7 9 WELLRESULTO1 OUT OUTPUT FILE This file holds the drawdown values at the we
80. oundary condition comes into effect Numerical models designed to simulate groundwater flows converging to pumped wells require small time steps at early times to represent the groundwater head changes accurately and large time step at later times to maintain the efficiency of the model This is achieved in the current model by increasing the time step size in a logarithmic fashion Figure 8 shows a schematic division of time in a logarithmic pattern between time Tsar and Ttogena The number of time steps in each ten fold can be varied based on the complexity of the problem In addition the time stepping can be set to a constant value when a specified time is reached In Figure 8 for example the time step becomes equal to At after the simulation time becomes greater than Ty ogEna When the abstraction rate changes the time steps are recalculated and increased logarithmically with time from the start time Tstat defined in the input file ClockO01 dat to Troggna and then becomes constant and equal to TLogena until the abstraction phase duration reaches the total abstraction time defined in Well01 dat Tstart TLogEnd At increasing logarithmically Constant At equal to Tr ossa Figure 8 Time stepping 13 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 4 Capabilities of the radial flow model This section discusses the capabilities of the radial flow model Each of the features is discussed in detail in the following sections o
81. ped by the British Geological Survey and the University of Birmingham Mansour 2003 The main application of the radial flow model is to simulate groundwater flows to analyse pumping test results The model simulates groundwater flow in homogeneous or heterogeneous aquifer under confined or unconfined conditions Groundwater flows in aquifers under unconfined conditions are simulated using a moving water table and a collapsing finite difference grid technique The model also simulates important processes taking place in the vicinity of the pumped well such as well losses and the seepage face This user manual introduces the finite difference radial flow model and includes two parts Part 1 describes the theoretical background and the basic flow equations used to build the model and their conversion into numerical form Part 2 consists of three tutorials that are designed to help the user to control the model and simulate groundwater flows in complicated aquifer systems OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Acknowledgements The authors would like to Acknowledge Dr Christopher Jackson of the British Geological Survey for his assistance and for developing the licensing code included in the model OR 07 029 Draft 0 1 Contents Foreword Acknowledgements Contents 1 Introduction 2 Theoretical background 2 1 2 2 2 3 2 4 Basic flow equation Aquifers under unconfinrd conditions The inclusion of the free surface
82. r The first zone has the same characteristics of Tutorial 1 aquifer and occupies three quarters of the aquifer The second zone has horizontal hydraulic conductivity equal to half the value of the first zone hydraulic conductivity and a specific storage value that is one tenth the value of the first zone specific storage coefficient In this exercise the drawdown time series recorded at two observation boreholes located at 100 m away from the pumping well one in the middle of the low permeability zone and the other in the opposite high permeability zone as shown in Figure 28 are compared T 2 T Low permeabilit A SS A S O A T A 4 S 21 Observation well 2 SS SS ss 51 4 lt gt Ler Z mut 30 2 Figure 28 Plan of the two permeability zone aquifer 8 3 2 Modification of input files Create the dummy input files and modify the parameter values exactly as in Tutorial 1 Open InputO1 dat input file Change the number of circumferential intervals to eight Open Solver01 dat input file Set the flag that activate the solver to N The aim at this stage is to produce the additional input files only Run the model executable as explained in the previous section or as in Section 5 of the manual 49 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 The additional input files are produced by the model These have a txt extension The files that need to be modified in this exercis
83. rawdown value at the other observation wells in the order they are listed in Output01 dat 47 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Note The user can plot the drawdown values using either the phase time column or the simulation time column and compare with the drawdown values calculated using the Theis solution This comparison is shown in Figure 27 The discrepancy between the results at the early times is due to well storage P TimeSeries01 out Notepad EIE File Edit Format View Help Observation borehole Radii 10 Observation borehole Angles 6 28319 Observation borehole depths 5 Phase PhaseTime days simulationrime days Drawdown 211528e 003 1 211528e 003 1 524492e 002 467799e 003 1 467799e 003 6 296092e 002 778279e 003 1 778279e 003 1 492199e 001 154435e 003 2 154435e 003 2 695336e 001 610157e 003 2 610157e 003 4 152026e 001 162278e 003 3 162278e 003 5 779750e 001 831187e 003 3 831187e 003 7 516987e 001 641589e 003 4 641589e 003 9 322753e 001 623413e 003 5 623413e 003 1 117111e 000 812921e 003 6 812921e 003 1 304597e 000 254042e 003 8 254042e 003 1 493737e4000 000000e 002 1 000000e 002 1 683913e 000 211528e 002 1 211528e 002 1 874738e 000 467799e 002 1 467799e 002 2 065969e 000 778279e 002 1 778279e 002 2 257452e 000 154435e 002 2 154435e 002 2 449091e 000 610157e 002 2 610157e 002 2 640822e4 000 162278e 002 3 162278e 002 2 832608e 000 831187e 002 3 831187e 002 3 024423e4000 641589e
84. roundwater flows to partially penetrating wells can be simulated if there is more than one layer in the aquifer or if the single layer aquifer is represented using more than one numerical grid The central abstraction well will be in contact with the layers that have a total depth equal to the well depth All remaining aquifer layers will be connected using the values of their corresponding hydraulic parameters See Figure 5 4 11 RECHARGE Recharge can be added to the model and can vary with time However the recharge is applied uniformly over the whole model area The recharge is applied to the upper most nodes of the aquifer in a confined groundwater system and is applied to the water table nodes in an unconfined groundwater system 4 12 ADDITIONAL ABSTRACTION POINTS Additional abstraction wells can be placed at any node within the model domain The abstraction rate at these wells can vary with time These abstraction wells are not represented in the model in the same way as the central pumping well Instead the external source of water term of the basic flow equation is simply used to include these wells The value of this term is calculated by dividing the abstraction rate by the area of the node where the pumped 15 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 well is placed This value is then given a negative sign to represent water leaving the aquifer system It must be noted that because the nodal area is not the same at all
85. ry or as a fixed head boundary This is done by typing the letter T for impermeable boundary or the letter F for fixed head boundary on line 12 of the input file The aquifer can be subjected to confined or unconfined conditions This is specified on line 14 of the input file by typing the letter C for the confined conditions and the letter U for the unconfined conditions At the start of the simulation all drawdown values are assumed to be zero However under confined conditions only the user can specify the initial drawdown values The flag on line 16 must be set to Y and initial drawdown values entered in an input file called NodeInitDDwn01 inp the format of which is discussed in Section 6 11 The initial drawdown values cannot be user defined when simulating groundwater flows in unconfined aquifers because it is difficult to know the exact initial location of the water table While the conductance and storage values at all layer nodes are calculated using the hydraulic parameter values specified in the section related to the layer Table 5 The user can overwrite these values using the files listed in Table 2 The flag on line 18 must be set to Y for the model to use these files While the outer boundary condition at the end of the aquifer is set using the flag on line 12 the user can include internal boundaries using the set of values on line 20 These internal boundaries are assumed to be linear and
86. t file InputO1 dat is not found in the working directory It is assumed that if InputO1 dat is missing none of the other input files exists in the working directory The user has the option of accepting or declining the creation of the dummy files Once these dummy files are created they can be edited using any text editor Note If the working directory contains all the input files except InputO1 dat file the user will be prompted with the option of creating dummy input files If accepted the existing input files will be overwritten 8 2 58 Running the Model Follow the instruction given in Chapter 5 to install and run the model executable 8 2 4 Modification of input files Figure 16 shows the preferable sequence of changing the input files Only five input files are modified in this section These are InputOl dat WellOl dat Clock01 dat SolverO1 dat and OutputO1 dat input files The default values assigned to the remaining dummy files created by the model will be accepted InputO1 dat Well01 dat Clock01 dat Solver01 dat OutputO1 dat Figure 16 the sequence of modifying tutorial 1 input files 40 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 DEFINING THE AQUIFER SIZE AND CHARACTERISTICS InputO1 dat I WellO1 dat ClockO1 dat Solver01 dat OutputO1 dat The aquifer size and hydraulic parameter values are specified in InputO1 dat input file Open this input fi
87. t is possible to divide each layer into more than one horizontal section The benefit of that is to include well casing partly penetrating wells and to be able to plot contour lines along a vertical section The number of divisions is specified on line 15 of Table 5 25 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Table 5 Layer specific section of Input01 dat input file Line Core of the input file Kok seo oe oe oe oe seo oe o oe oe oe o K o oe oe k k oe k K oe K k oe K oe K K K k K oe K K K K K K K KK KK K K Layer Number 1 Layer Thickness m 6 0 Horizontal hydraulic conductivity m day 300 Vertical hydraulic conductivity m day 1 O oo 1 Duan A UU Ne 10 Specific storage m 1 11 0 000001 12 specific yield 13 0 2 14 Number of grid lines 15 1 6 4 OUTPUT01 DAT INPUT FILE Output01 dat output file controls the output files produced by the model A set of files that hold the different parameter values calculated internally by the model are produced however flags in this Output01 dat file must be set to Y for these files to be produced These flags and their corresponding explanations occupy the first 12 lines of this input file as shown in Table 6 The simulated drawdown values and the locations of the water table nodes are produced at the end of each time step However for aquifers discretised using large number of nodes these files can become very large in size An option is
88. the well is not cased a seepage face forms through which groundwater seeps out into the atmosphere At any elevation z along this face the groundwater head h can be described by the following equation h z PE Equation 6 where p is the atmospheric pressure ML T z is the elevation from a datum L p is the density of the fluid M L7 The boundary condition at a seepage face can be derived by acknowledging that the atmospheric pressure is equal to zero p 0 This leaves the head to be equal to the elevation z It must be noted that although the groundwater is leaving the aquifer through the seepage face this groundwater must be accounted for when calculating the mass balance of the system The groundwater volume seeping out through the seepage face forms a part of the pumped water and consequently the pumping rate must be reduced accordingly 2 4 INCLUSION OF RIVERS Rivers are included in the model to study the river aquifer interaction and its effects on the drawdown values Rivers are not represented in details in the current version of the model For example changes in river flows with time are not included river bed elevation is flat and river stage elevations at the nodes constituting a river are also flat The specified head nodes connect to the water table nodes if the aquifer system is under unconfined conditions and they connect to the aquifer nodes directly underneath them if the aquifer system is under confined condi
89. tions The hydraulic conductance value controlling the river aquifer interaction is calculated using the river bed thickness and a riverbed vertical hydraulic conductivity The rate of leakage depends on this hydraulic conductance and the difference between the groundwater head and river stage and is expressed by dst H TH Equation 7 Where is the leakage rate per unit area L T is the riverbed vertical hydraulic conductivity L TT is the riverbed thickness L is the head at the aquifer node L mew As is the head at the river node or the river stage L OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 The above equation lacks an important flow mechanism which is the variation of the vertical conductance value with the variation of the head difference between the river and aquifer nodes This mechanism will be included in future releases of the model OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 3 The numerical solution Numerical solutions require the conversion of the mathematical equations describing the flow processes into numerical forms In this section the conversion of the mathematical equations described in Section 2 is presented Section 3 1 describes the conversion of the basic flow equation Equation 3 Section 3 2 details the concept followed to define the hydraulic parameter values at a node Section 3 3 describes the representation of the movement of the water table numerically Section
90. ts the hydraulic conductance of the aquifer close to the pumped well and consequently affects the efficiency and the yield of the pumped well Well losses are included in the model by introducing a factor that has a value less than unity This factor multiplies the hydraulic conductivity of the nodes that are representing the part of the aquifer adjacent to the pumped well as illustrated in Figure 6 Hydraulic conductivity reduced to include well losses Well face Figure 6 Representation of well losses in the numerical model 3 6 SOLVING THE NUMERICAL EQUATIONS THE SUCCESSIVE OVER RELAXATION METHOD SOR The numerical formulation requires the solution of a system of linear equations resulting from writing Equation 8 at all the grid nodes This system of linear equations can be represented in matrix form by A u f where u denotes the groundwater head values and A is the matrix composed of the conductance values The Successive Over Relaxation SOR iterative method is implemented in the model to obtain a solution for these equations The SOR method can be derived as follows Matrix A is characterised by a sparse structure as illustrated in Figure 7 Expressing this matrix as the sum of its diagonal elements and its upper and lower triangular elements given by the matrices D U and L respectively the above linear system can be written as
91. xtension and take the same format illustrated using Table 13 in which p replaces the parameter value to modify This p parameter may be the radial hydraulic conductivity the circumferential hydraulic conductivity the vertical hydraulic conductivity the specific storage or the specific yield When confined aquifers are studied it is possible to start the numerical simulation with initial drawdown values different than zero An input file that takes the same format of the one shown in Table 13 is used for this purpose and the p values will be the drawdown values The format of these input files is designed to hold a sequence of information related to the layers starting from the uppermost layer down to the base layer The parameter values must be specified for all nodes located in the layer before proceeding to the next lower layer Similarly if more than one vertical section is used in the model hydraulic parameter values at all nodes located in each vertical section must be specified before proceeding to the next vertical section If the parameter p represents the specific storage the specific yield or the initial head this p value represent the aquifer characteristics at the node itself however if p represents the hydraulic conductivity the p value will represent the aquifer characteristics between the current node and the adjacent node see Section 3 2 32 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 Table
92. y e Set the vertical hydraulic conductivity to 1 m day Set the specific storage to 0 000001 m e Set the specific yield to 0 02 Change the number of gridlines in Layer 1 to 3 MODIFYING WELLO1 DAT INPUT FILE Change the borehole depth to 41 6 e Change the pump depth to 41 6 Set the abstraction rate to 1256 0 m day Set the abstraction duration to 10 0 day MODIFYING OUTPUTO1 DAT INPUT FILE Change the observation borehole radius to 40 Change the observation borehole depth to 25 0 53 OR 07 029 Draft 0 1 Last modified 2008 02 08 10 18 8 4 3 Running the model and examining the output files Follow Section 5 instructions to install and run the model executable CHECKING THE WATER BALANCE To check the water balance open Waterbalance01 out output file using a text editor Figure 33 shows part of the water balance file Observe the extra column that gives the water released from the water table The sum of total water released is given in the last column and this must be equal to the abstraction rate WaterBalance01 out Notepad File Edit Format View from well from well from well from well from wel from well from well from wel from well from well from wel from well from well Help 52783 349002 135568 0735153 0444384 0282815 0184526 0121436 00796248 00514855 00325292 00199299 00117841 water from ss 1182 94 water from ss 1118 18 wat

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