solver water resources

8/8/2019 Solver Water Resources

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Applications of Microsoft Excel Solver Function in Water Resource Engineering

1aXing Fang,

1bShoudong Jiang,

1bKumar Raut,

1bJinwei Qiu

Abstract

Water Resource Engineers should be competent in hydraulic and hydrologic principals,as well as in the application of principals to the solution of practical problems. Civil engineeringgraduates are faced with hydraulic and hydrology problems too complex to solve by hand.While most commercially available software packages obscure the theoretical background for

program algorithms. The Solver function in Microsoft Excel provides a valuable tool for bridging those gaps. Students and engineering professionals can use appropriate linear ornonlinear mathematical equations to represent a hydraulic and hydrologic system, and then usethe Solver function to solve equations for various combinations of input data. This paperdescribes and demonstrates how effective the Solver function can be used to solve varioushydraulics and hydrology problems, for example, compute normal depth and critical depth forchannel flow, water depths before and after a hydraulic jump, constant rainfall loss, Hortonsinfiltration parameters, and aquifer constant for confined aquifers. The Solver function canreplace traditional methods such as trial-and-error and chart method to solve various problems inthe water resource engineering area. Using the Solver function not only provides more accuratesolution but also saves time and effort to solve the same or similar type of problems since

spreadsheet developed for using Solver can be used repeatedly with different input data,constraints, changing cells (variables), and target cell (parameter).

Introduction

There are many problems in the field of water resource engineering which are difficult tosolve analytically and are normally solved by a trial-and-error method or using monographs orcharts (Roberson et al., 1997). Students and professional practitioners in water resourceengineering often need to determine normal depth, critical depth, and conjugate depths for ahydraulic jump. Mannings equation can be used to compute normal depth, but there is noanalytical solution for non-linear (Mannings) equation even for simple rectangular channelgeometry. Chow (1959) developed monographs to compute normal depth and critical depth for______________________________________________________________________

1aAssociate professor, 1bGraduate students, Department of Civil Engineering, Lamar University,Beaumont, Texas 77710-0024, [email protected]


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rectangular, trapezoidal with equal side slope, and circular channels. Monographs can not beused for triangular channels, trapezoidal channels with different side slopes, and irregular shapedchannels. These monographs are adopted for many textbooks (e.g., Roberson et al, 1997), butuse of monographs often loses accuracy due to the limit of resolution. Alternatively a trial-and-error method is often used. Therefore, civil engineering graduates are faced with dilemma to

deal with such type of problembecause most realistic hydraulics and hydrology problems are toocomplex to solve by hand, while most commercially available software packages are complexand obscure the theoretical background for program algorithms.

As long as unknown parameters and given parameters or characteristics of the problemcan be presented into equations, unknown parameters can be easily solved by using Solverfunction in the Microsoft Excel. The Solver function provides a valuable tool for bridging thegap between understanding of hydraulic and hydrologic principals and solution methods topractical problems in water resource engineering. The Solver function is an attractive tool forstudents and engineers to use for at least two reasons. Firstly, Excel is perhaps the most familiarspreadsheet used both in business and universities and as such is very accessible. Secondly, thespreadsheet offers very convenient data entry and editing features which allows the students and

professionals to gain greater understanding of how to solve linear and non linear equations.Students and engineering professionals can use appropriate linear or nonlinear mathematicalmodels or equations to depict a realistic system, and then use an equation Solver package tosolve models and equations for various combination of input data desired. For example, theSolver function was used to solve reservoir optimization problems for water supply and energygeneration by Dr. Fontane (2001) at the Colorado State University. This paper will describe anddemonstrate how effective the Solver function can be used for the solutions of hydraulics andhydrology problems, for example, to compute normal depth and critical depth for channel flowand water depths before and after a hydraulic jump, to estimate constant rainfall loss, Hortonrainfall loss parameters, and aquifer constants for confined aquifers. Water Resource Engineersshould be competent in hydraulic and hydrologic principals, as well as in the application of

principals to the solution of practical problems.

Solving Non-linear Equation Using the Solver function in Microsoft Excel

Microsoft Excel provides two tools (Goal Seek and Solver) to find roots of non-linearequations (Liengme, 2002), such as equations often used to describe hydraulic and hydrologyprinciples. Some of non-linear equations may be solved analytically, for example, a quadraticequation, but for many other non-linear equations the analytical solution may be very complex ornot exist at all. Therefore, numerical methods, e.g., the Newton-Raphson method, are often usedto find approximate roots or numerical solutions for those non-linear equations. Goal Seek is avery easy tool to solve equations but it has its limitations (Liengme, 2002). The Solver function

in Microsoft Excel is much powerful than Goal Seek, and it was originally designed foroptimization problems but it is useful for root finding of non-linear equations with variousconstraints. Solver is licensed to Microsoft by Frontline Systems, Inc. whose web site(www.Solver.com) has much more information on the product. Several advantages of usingSolver in comparison to using Goal Seek are (Liengme, 2002):

(1). When you have used Solver once on a worksheet, it will retain its settings when it is nextused on that worksheet.


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(2). Whereas Goal Seek allows you to vary or solve one cell (variable), with Solver you canvary 200 cells but using no more than 16 ranges.

(3). Solver permits constraints, for example, you can set that a varied cell always has apositive value or greater than a specific value.

Solver is a useful tool for conducting a what-if analysis when we need to adjust the values inmore than one cell and have multiple constraints for those values. Many problems in waterresources engineering were used to solve by a trial-and-error method, and by using the Solverfunction, these problems can be solved very easily with high accuracy of results devotingminimum amount of time.

Figure 1 shows basic setting to use the Solver in Microsoft Excel, and more details will be illustrated later when the Solver is used to solve various types of problems in the waterresource engineering. When using the Solver, one must identify or set minimum threecomponents of the problem one wants to solve,

A target cell that one wants to set to a certain value (solve equation) or that one wants tomaximize or minimize (optimization problems). This cell must contain a formula whichis ultimately linked with changing cells. The target cell can only be a single cell,therefore, sometimes it is necessary to combine equations and to reduce targeted values(goals) into one single cell.

Equal to, which is used to specify whether one wants the target cell to be maximized,minimized, or set to a specific value. If one wants a specific value, type it in the box.

Changing cells, which is the cell or cells that can be adjusted until the constraints in theproblem are satisfied and the target cell reaches its targeted value. These cells are alsocalled adjustable cells whichare unknowns one wants to solve in nonlinear equations.Again, the adjustable cells must be related directly or indirectly to the target cell.

and Subject to the Constraints which lists the current restrictions on the problem and is oneof the optional components used to solve some of problems, for example, in the Muskingumrouting equation, weighting factor X lies between 0 and 0.5, which is set as a constraint.

There are many problems in the water resource engineering which are difficult to besolved analytically and are often solved by a trial-and-error method or using monographs orcharts (Roberson et al., 1997). As long as unknown parameters and given parameters orcharacteristics of the problem can be presented into equations or formulas, unknown parameterscan be easily solved by using the Solver function in the Microsoft Excel. Several examples ofusing the Solver function in Microsoft Excel to solve trial-and-error type of problems areillustrated below.


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Figure 1. Basic settings for using Solver in Microsoft Excel.

Compute Critical Depth of Open Channel Flows

Determination of critical depth in an open channel flow is important to hydraulic analysisand design. The critical depth (yc) for a rectangular channel is determined using the equation,

3/12 )/( gqyc = (1)

where q is the discharge per unit width of the channel, and g is the acceleration of gravity. Forcritical depth computation in trapezoidal, circular, and natural channels, there is no analyticalequation to give direct solution of yc, and yc is often determined by a trial-and-error method tomake Froude Number [Fr, defined in equation (2)] to be 1.0. To assist engineers to determinethe critical depth, a monograph was developed by Chow (1959) and has limited resolution onsolution accuracy.

1/

/3

2

====c

c

cc

c

c

crc

gA

TQ

TgA

AQ

gD

VF (2)

where Q is the discharge, Vc is the flow velocity, Dc is the mean water depth and equal to Ac/Tc,Ac is the flow cross-sectional area, Tc is the flow top width of the channel (at the free surface),and subscript c stands for under the critical flow condition. Above flow characteristics (Vc,Dc, Ac, Tc) are the function of water depth (yc) and channel geometry. It is typically difficult tosolve yc analytically from equation (2), while a trial-and-error method to make the Froudenumber equal to 1 is time consuming, tedious and may not be able to find the exact solution.Using the equation (2) with equations for channel geometrical parameters (given in varioushydraulics books and engineering handbooks, e.g., Chow, 1959; Simon and Korom, 1997; Saleh,

Choose a cell in which should be maximized, minimized, or equal to a particular quantity

Choose cell(s) thatthe Solver will

manipulate inorder to change ormake the targetcell to designatedvalue

Solver allows userto add constraintsto control theways in which itmanipulateschanging cell(s).


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2002), the example below demonstrates that an Excel spreadsheet can be developed and used tocompute the critical depth with the application of the Solver function in Microsoft Excel.

Figure 2 shows an Excel spreadsheet to compute the critical depth in a trapezoidalchannel. This spreadsheet can be used to determine critical depth for a triangular channel if thechannel bottom width (B) is set to be zero and for a rectangular channel if channel size slopes Z 1

and Z2 are set to be zero (H:V). The initial value of the critical depth is arbitrarily set to be 1 ftfor a trapezoidal channel with side slopes of 2 and 3, bottom width of 20 ft, and discharge of 600cfs, and computed corresponding Froude number is 4.96 which is much greater than the targetedvalue of 1. After the Solver function is activated by selecting Solver under Tools pull-down menu of Excel, Solver Parameters window is shown in Fig 1. User has to select or setone (and only one) target cell and select one or more than one changing cell(s). For criticaldepth computation, the target cell is set as the cell D18 (Fig. 1) which is computed Froudenumber, and the changing cell is D10 which is the (critical) water depth (yc). It is optional to addany constrains to further control the ways in which Solver manipulates the changing cell(s), forexample, user could add D10 > 1 based on the initial value of water depth and correspondingFroude number, but this constrain is not necessary. After applying the Solver function computed

critical depth is given as 2.697 ft.

This spreadsheet can be reused again and again to compute critical depth with othercombinations of channel bottom width, side slopes and discharges. This spreadsheet can also bereused to solve for channel geometrical parameters when the critical depth is known, forexample, lets assume the critical depth to be 2 ft, find corresponding discharge Q. What userhas to do is to set D10 = 2ft and set the changing cell as D9, then click on Solve button,computed discharge is 365.9 cfs. Similar spreadsheet for a circular channel (underground sewerpipes) is also developed (Figure 3). Application of the Solver function to compute critical depthis useful to college students in learning open-channel hydraulics and to engineers for theirprofessional practices, which they dont have to heavily rely on other complex software.

Compute Normal Depth of Open Channel Flows

Normal depth in an open channel flow is another important parameter for hydraulicanalysis and design. Normal depth (yn) is the water depth in an open channel under steadyuniform flow condition. Discharge of a uniform flow in a channel is often computed using theMannings equation,

Q = (/n) ARh2/3

S1/2 (3)

where equals to 1.49 for English units and 1.0 for SI units, n is the Mannings roughnesscoefficient, A is the flow area, Rh is the hydraulic radius and equals to A/P, P is the wetted

perimeter, and S is the bed slope of the channel. In the above equation (3),ARh2/3

= n Q/[S

0.5

]is known as section factor, and /n ARh2/3 = Q/S1/2 is the conveyance of the channel which is a

measure of the carrying capacity of the channel section. For a simple-geometry channel whereAR

2/3 always increases with increasing depth, each discharge has a corresponding unique valueof depth at which uniform flow occurs.

Since an analytical solution of the equation (3) to compute normal depth is difficult, intypical hydraulics textbooks (e.g., Roberson et al., 1997), it is recommended to compute thesection factor first from given discharge, channel slope and roughness, and then to use a trial-


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Figure 3. Spreadsheet for computing critical depth for circular channels.

The spreadsheet in Fig. 4 can be reused to solve other channel geometry and flowparameters under normal flow condition. For example, when discharge (Q), Mannings (n) anddepth of flow are given for the channel section, the equation (3) can be used to determine the bottom width under constrains of maximum permissible velocities and permissible side slopebased on channel material (Fortier and Scobey, 1926; Chow, 1959). Figure 5 shows modifiedspreadsheet for a channel design problem. It needs to design a trapezoidal channel to carry 600cfs flood flow away from the proposed office-residential complex and design an earth linedchannel using sandy loam on the site with channel slope of 0.0002. Suggested channel side slopefor using sandy loam is 3 horizontal versus 1 vertical (Chow, 1959), and maximum permissible

velocity and Mannings n for sandy loam are 1.75 ft/s and 0.02 (Fortier and Scobey, 1926),which haven been input in Fig. 5. Two design parameters are bottom width and channel depth,and were determined to be 146.96 ft and 2.25 ft (Fig. 5) after applying the Solver function.Target cell is still computed discharge to be 600 cfs, and adjustable cells are D3 and D4corresponding channel width and depth, and one constrain is used: D17


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An effort is made to compute normal depth for an irregular shaped channel as shown inFig. 6. It requires a user to input channel geometry in station (x, ft or m) and correspondingelevation (y, ft or m). Elevation can be given as relative to any arbitrary datum. Channelgeometrical parameters for an irregular shaped channel, e.g., flow area, wetted perimeter, and topwidth, has to be computed using coordinate method (Anderson and Mikhail, 1998), which was

implemented by writing a VBA function under Microsoft Excel as partially shown in Fig. 7.After channel parameters can be computed by VBA function, the Solve function can then beused again to determine the normal depth, and also critical depth. Computed normal depth is1.096 ft as initial depth was assumed as 2.0 ft. Above spreadsheets are carefully designed tosolve problems for both SI and FPS units: a cell (D8 in Fig. 6) is used as 1 for SI and 0 for FPSunits, and the cell for computed discharge using Mannings equation does automatically switchcoefficient between 1.49 for FPS and 1.0 for SI units.

Figure 4. An Excel spreadsheet to compute normal depth in a trapezoidal channel.


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Figure 5. An Excel spreadsheet for a channel design problem.

Compute Water Depth Before or After a Hydraulic Jump

The hydraulic jump occurs when the flow changes from supercritical flow in an upstreamsection to subcritical flow in a downstream section. The upstream and downstream depths of thehydraulic jump are determined by applying the momentum equation because the energy loss in ahydraulic jump is not clearly known and the energy equation is not a suitable tool for analysis ofthe velocity-depth relationships for a hydraulic jump. If the resistance of the channel bottom is

negligible, applying the momentum equation for a horizontal channel gives,P1*A1 + QV1 = P2*A2 + QV2 (4)

where P1 and P2 are the upstream and downstream pressure at the centroids of the respectiveareas A1 and A2, V1 and V2 are the upstream and downstream flow velocity, and Q is the flowdischarge in the channel. Experiments also show that equation (4) can be applied to all channelsof moderate slope (So < 0.02) (Roberson et al., 1997). For horizontal rectangular channels, touse the momentum equation (4) with the continuity equation (V1A1= V2A2) leads to,


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Figure 6 An Excel spreadsheet to compute normal depth in an irregular shaped channel.

)181(2

21

12 += rF

yy (5)

wherey1andy2are water depths before and after the hydraulic jump, and Fr1 is Froude number[defined in equation (2) above] at the cross section 1. The experimental relations betweeny1/y2and Froude numberFr1 for hydraulic jumps in rectangular channels with various bottom slopes


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(S from 0 to 0.30) was given by Chow (1959). For trapezoidal channel, analytical formula cant be developed and momentum equation (4) is used to determine depths before and after ahydraulic jump from one to the other, but an iterative or trial-and-error method is typically usedto solve the equation (4) (Roberson et al., 1997). Excel spreadsheets can be developed as shownin Figure 8 to compute conjugate depths before and after a hydraulic jump for trapezoidal and

triangular channels. The Solver function can set one of the conjugate depths as changing celland set the difference of (PA+QV) before and after the jump as the target cell and its value to bezero. Necessary geometrical relationships have to be developed within Excel for trapezoidalchannel (Fig. 8). Froude numbers at the cross sections before and after the jump give usefulinformation to validate the solution. Again, when B or side slopes Z1 and Z2 are set to be zero,the spreadsheet can be used for a triangular and rectangular channel.

Figure 7 Channel geometry function developed using VBA under Microsoft Excel.


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Figure 8. An Excel spreadsheet to compute depth before and after a hydraulic jump for atrapezoidal channel.


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Determine Aquifer Transmissivity and Storativity

Transmissivity and storativity are the hydraulic properties of an aquifer. The capacity ofan aquifer to transmit water is called its transmissivity. Transmissivity is defined as the amountof water that can be transmitted horizontally through a unit width and full saturated thickness ofan aquifer under a unit hydraulic gradient, and equals to hydraulic conductivity (K) times aquifer

thickness (b). Storativity is the volume of water that an aquifer will release or take into storage per unit surface area per unit change in hydraulic head. Storativity is dimensionless, and forconfined aquifers it ranges from 0.005 to 0.00005 (Freeze and Cherry, 1979)

When a new well is first pumped, a large portion of the discharge comes directly from thestorage volume released as the cone depression develops. When steady-state conditions ofgroundwater flow around a well are not encountered, a non-equilibrium equation must be used.There are two approaches, which have been developed to solve the non-equilibrium equation, arather rigorous method of C.V. Theis (1935) and a simplified procedure such as that proposed byCooper and Jacob (1946). Theis (1935) provided a solution to unsteady flow equation (6) in aconfined aquifer of constant thickness b.

th

TSh c

=2 (6)

where h is the piezometer head (ft or m), T is the aquifer transmissivity (ft2/day, m2/day or gpd/ftin American practice) and Sc is the storativity and also known as aquifer storage coefficient.Theis (1935) stated that the drawdown (s) in an observation well located at a distance r (lengthunit) from the pumped well is given by:

T

uWQdu

u

e

T

Qs

u

u

4

)(

4==

(7a)

T

uWQ

s

)(6.114

= (7b)

where Q is the constant pumping rate [ft3/day or m

3/day for Equation (7a), gpm for equation (7b)

with T of gpd/ft], and u is a dimensionless variable defined as:

2

4r

tT

Su c= or 2

87.1r

tT

Su c= with T of gpd/ft (8)

The integral in equation (7a) is commonly called as well function of u and is written as W(u).W(u) can be evaluated from the infinite series when u is small:

....!5*5!4*4!3*3!2*2

ln577216.0)(5432

++++=uuuu

uuuW (9)

In order to solve equations (6) and (7) and to determine aquifer constants (T and S c), two log-logplots should be developed and utilized (Viessman et al., 2003): the first one is a log-log plot of uversus W(u) (known as a type curve) and the second one is a log-log plot of the observed data r2/tversus drawdown (s). W(u) ands are ordinates and u and r

2/tare abscissas. The two curves are

imposed and moved about until segments of two curves coincide. In this operation the axes mustremain parallel. A coincide point is then selected on the matched curves and both plots marked.


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Corresponding values ofu, W(u), s and r2/tare determined from the plots at marked point, thenaquifer constants can be computed from equations (6) and (7).

The graphic method is time-consuming because two log-log plots are needed. The abovemethod also generates some errors because the values of u, W(u), s and r2/t depend on thecoincide point. Different engineers may find different points due to personal experience.

Cooper-Jacob (1946) simplified the Theis method when the pumping time (t) is long enough (orr is small enough) then the u parameter becomes small enough (


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Determination of Aquifer Constants for Confined Aquifers

6.6802E-02

1.9347E+03 1.2051E+04 gpd/ft

129600.0

Computed pumping rate from gpm (ft

3

/day) 693360.0 3000 gpm

Aquifer storativity, S (dimensionless)

Aquifer transmissivity, T (ft2/day or m

2/day)

Pumping rate from the well, Q, (ft3/day or m

3/da

Unsteady-State Anaysis of Confined Aquifer (Theis Equation) - Using Solver Function

Time (t) Distance (r) log[t/r2] Measured u w(u) Computed s2

(min) (ft or m) sm, (ft or m) sc (ft or m) (sm-sc)2

60.0 100.0 -2.222 0.60 2.07167279 0.04462 0.238 1.311E-01

120.0 100.0 -1.921 1.40 1.03583640 0.20726 1.105 8.713E-02

180.0 100.0 -1.745 2.40 0.69055760 0.38125 2.032 1.352E-01

240.0 100.0 -1.620 2.90 0.51791820 0.53936 2.875 6.198E-04

300.0 100.0 -1.523 3.30 0.41433456 0.67972 3.623 1.045E-01

360.0 100.0 -1.444 4.00 0.34527880 0.80460 4.289 8.352E-02480.0 100.0 -1.319 5.20 0.25895910 1.01779 5.425 5.082E-02

600.0 100.0 -1.222 6.20 0.20716728 1.19474 6.369 2.845E-02

720.0 100.0 -1.143 7.50 0.17263940 1.34562 7.173 1.070E-01

1080.0 100.0 -0.967 9.10 0.11509293 1.69749 9.049 2.640E-03

1440.0 100.0 -0.842 10.50 0.08631970 1.95779 10.436 4.077E-03

Sum of sqaure difference of drawdown, s2

7.351E-01

Root mean sqaure of residuals of drawdown (ft or meter) 2.585E-01

Drawdown - Time and Distance Analysis (Cooper-Jacob Method)

7.4592

15.653

3.1800E+03

7.9711E-03

3.9607E-02Estimated initial aquifer storativity, S (dimensionless)

Slope of drawdown curve (s per log cycle)

Intercept of drawdown curve

Estimated initial aquifer transmissivity, T (ft2/day or m

2/day)

(t/r2)o (min/ft

2or min/m

2)

Figure 10a. Parts 1 to 3 of an Excel spreadsheet to estimate aquifer constants using Cooper-Jacob method and Solver function.


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Drawdow n Curve---Cooper-Jocob Me thod

y = 7.4592x + 15.653

R2

= 0.932

0.0

2.0

4.0

6.0

8.0

10.0

12.0

-3.0 -2.0 -1.0 0.0

Log (t/r2)

Drawdown,s(ftorm)

Measured Compu ted L inea r (Measured)

Figure 10b. Part 2 (graphic plot) of an Excel spreadsheet to estimate aquifer constants usingCooper-Jacob method and Solver function.

part 4 is a plot of s versus log )/( 2rt including a linear regression line, regression equation and

computed drawdown; for part 2, column B (time t), C (well location r), and E (measured

drawdown) are given measurements; and column D is computed log )/( 2rt ; columns F and G are

u and W(u) computed from Theis equations 8 and 9; column H is computed drawdown s fromequation 7b; and column I is squared difference of measured and computed drawdown. Sum of

squared difference of drawdown (s2) is used as target cell of the Solver function, and rootmean square of errors (residuals) of drawdown (RMSE) is also computed. In order to apply theSolver function to determine storativity and transmissivity, initial guess values of Sc and T mustbe given and was found to be difficult to specify them since aquifer constants vary in very widerange. Without specifying reasonable initial guess values for Sc and T, the Solver function doesnot give any useful result at all. Reasonable initial guess values are developed by using Copper-

Jacob method in parts 3 and 4.Columns D and E of part 2 are used to generate a plot of s versus log )/( 2rt in order to

develop linear regression equation and trend line. Two constants of linear regression equation forpart 4 are input into part 3, and transmissivity and storativity are determined from Cooper-Jacobequations (10) and (11). Estimated transmissivity and storativity are then input in cells F3 andF4 as initial values, and the Solver function is applied to get final aquifer constants. Thespreadsheet documents a streamlined procedure to determine aquifer constants and was tested byseven problems in three hydrology textbooks (Linsley et al., 1986; Gupta, 2001; Viessman and


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Lewis, 2003). Results of estimated aquifer constants by different methods are summarized inTable 1. Results using Cooper-Jacob method are typically acute based on errors (Table 1) onestimated and measured drawdown, and results using Solver function and initial valuesdeveloped from Cooper-Jacob method are most accurate for all problems tested. Results usingCooper-Jacob method developed by one analyst to another could be different if it is necessary to

remove some data points where u isnt small enough. For example, Figure 10 used all elevenmeasure data points, T and Sc are estimated as 3180 ft2/day and 0.0396 by using Cooper-Jacob

method with error (RMSE) in drawdown estimation of 0.97 ft. If the first three data points withlarger u are removed, T and Sc are estimated as 2329 ft

2/day and 0.0505 with RMSE of 0.72 ft.After applying the Solver function with anyone of the two results developed by Cooper-Jacobmethod, it will have the same final results of T and Sc to be 1935 ft

2/day and 0.0668 with RMSEof 0.26 ft. Therefore, using the Solver function coupled with Jacob method leads to muchaccurate estimation on T and Sc for confined aquifers (Table 1).

Table 1 Comparison of aquifer constants estimated by different methods and associated RMSR(root mean square of residuals) between measured and estimated drawdown.

Testing Storativity, ScProblems Theis Chart Method Cooper-Jacob Method Jacob Plus Solver

A 1.9300E-04 1.7971E-04 1.8310E-04

B 1.2000E-04 9.4238E-05 9.4880E-05

C 6.4000E-02 3.9607E-02 6.6800E-02

D 2.2000E-01 1.5769E-01 2.6363E-01

E 4.8800E-04 3.5171E-04 4.5675E-04

F 1.0640E-02 1.1206E-02 1.2466E-02

G 2.6300E-02 3.7823E-04 4.4526E-04

Testing Transmissivity, T (ft2/day)

Problems Theis Chart Method Cooper-Jacob Method Jacob Plus SolverA 1.3880E+04 1.3696E+04 1.3696E+04

B 1.1940E+03 1.2243E+03 1.2243E+03

C 2.1240E+03 3.1800E+03 1.9347E+03

D 1.4747E+04 1.8442E+04 1.5084E+04

E 9.0817E+02 9.7920E+02 8.9282E+02

F 2.2122E+04 2.2807E+04 2.2014E+04

G 1.2313E+04 1.5197E+04 1.5197E+04

Testing Root mean sqaure of residuals S (ft or meter)

Problems Theis Chart Method Cooper-Jacob Method Jacob Plus Solver

A 6.5466E-02 3.4917E-02 3.3347E-02

B 2.2773E-02 1.3302E-03 2.1100E-04

C 2.9845E-01 9.6502E-01 2.5851E-01

D 1.0909E-01 1.5875E-01 1.6569E-02

E 9.2394E-01 1.1184E+00 7.7471E-01

F 7.9603E-02 4.6188E-02 3.9810E-02

G 1.1137E+00 1.3502E-01 1.0502E-01

Note: A and B are Examples 4.9 and 4.10 from Gupta (2001); C is Example 6.1 from Linsley etal. (1986); and D, E, F, and G are Example 10.7, Problems 10.22, 10.26, and 10.30 fromViessman and Lewis (2003), respectively.


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Estimate Constant Rainfall Loss Rate

In order to develop hydrographs for a watershed, rainfall and runoff model can be used,and the model typically requires to estimate or specify rainfall loss in order to compute rainfallexcess hyetograph from given rainfall distribution. One of rainfall loss model is constant rainfallloss or initial loss plus constant rainfall loss, which is called as phi index method. If the volume

(in inches) of the direct runoff hydrograph (DRH) or total rainfall excess, rainfall hyetograph ordistribution, and initial loss are given, it requires a few iteration to determine the constant rainfall

loss or loss rate (, in/hr) (Chow et al, 1988). Initial loss may distribute in the first or severaltime steps. If the constant rainfall loss is greater than rainfall itself within a time interval, therainfall loss should be set as rainfall itself.

In order to determine the constant rainfall loss, it needs to specify those logicalrelationships in the spreadsheet before the Solver function can be used. For example, the firstcell for initial loss (C8 in Fig. 11) uses a logical formula =IF(F2


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Figure 11 An Excel spreadsheet to compute constant rainfall loss.


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Estimate Hortons Infiltration Parameters

Hortons infiltration model (Horton, 1935) is another common rainfall loss model andstates that

kt

ccp effftf+= )()( 0 (12)

wherefp is the infiltration capacity or potential (in/hr) at time t, kis a constant (1/hr) representingthe rate of decrease in infiltration capacity, fc is the final or equilibrium infiltration capacity(in/hr), and fo is the initial infiltration capacity (in/hr). Actually, the equation (12) only givesinfiltration capacity when ponding occurs immediately after rainfall starts (t= 0), that is whenrainfall intensity i > fo. If constant rainfall intensity i is less than fo, ponding time (tp) andequivalent starting time (to) can be determined as (Chow et al., 1988)

+=

c

cocop

fi

fffif

ikt ln

1(13)

c

copo

fiff

ktt

+=ln1 (14)

If time tis greater than the ponding time tp, the equation (12) above can be used to compute theinfiltration capacity by replacing tto (t - to). Actual infiltration rate is always the minimum ofrainfall intensity and infiltration capacity. Hortons infiltration parameters can be estimated frominfiltration rate measurements (e.g. using infiltrometer) as shown in Figure 12. When the Solverfunction is used to estimate infiltration parameters, the target cell is H54 which is the root meansquare of errors or residuals (RMSE) between measured and predicted infiltration rate, andadjustable cells could be four cells C56 to C59. From equations (13) and (14), one can see that tois not independent offo, fc, k, and i. Therefore, five constraints are used with the Solver function

as shown in Figure 13:fc or C56 0,fo orC58 >= 0.6 (constant rainfall intensity), to or C60


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Figure 12 An Excel spreadsheet to estimate Hortons Infiltration parameters.


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Figure 13 Set up of Solver parameters for estimating Hortons infiltration parameters.

Summary and Conclusion

Water Resource Engineers should be competent in hydraulic and hydrologic principals,as well as in the application of principals to the solution of practical problems. Microsoft Excelhas a Solver function that can be used to solve non-linear equations for various combinations ofinput data desired. This paper has described and demonstrated how effective the Solver functioncan be used to solve various hydraulics and hydrology problems, for example, compute normaldepth and critical depth for channel flow, water depths before and after a hydraulic jump,

constant rainfall loss, Hortons infiltration parameters, and aquifer constants for confinedaquifers. The Solver function is an effective tool to solve one or multiple unknown parametersas long as unknown parameters can be presented as one or several linear or non-linear equationsor formulas. Various constraints can be used with Solver function which greatly increases thepower of the Solver function. The Solver function can be utilized to replace traditional methodssuch as trial-and-error method and chart method to solve various problems in the water resourceengineering area. It not only provides more accurate solution but also saves time and efforts tosolve the same or similar type of problems since spreadsheet developed with using the Solvercan be utilized repeatedly with different input data, constraints, changing cells (variables), andtarget cell (parameter). Excel spreadsheets developed can be useful tools to do necessarycalculations for engineering design and analysis and also to check or understand outputs ofcomplex software packages in hydraulics and hydrology. To develop a spreadsheet using theSolver function may not be easy task because it requires developer has clear understanding ofhydraulic and hydrological principles of the problem, and utilize principles to develop correctgeometrical relationships, apply correct linear or non-linear equations or formulas, and codenecessary logical algorithms in Microsoft Excel. Development of Excel spreadsheets using theSolver function can help students, graduates, and professionals to get in-depth understanding ofbasic principles in hydraulics and hydrology.


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