Multilayer Analytic Element Modeling of RadialCollector Wellsby Mark Bakker1, Victor A. Kelson2, and Kenneth H. Luther3

AbstractA new multilayer approach is presented for the modeling of ground water flow to radial collector wells. The

approach allows for the inclusion of all aspects of the unique boundary condition along the lateral arms of a collec-tor well, including skin effect and internal friction losses due to flow in the arms. The hydraulic conductivity maydiffer between horizontal layers within the aquifer, and vertical anisotropy can be taken into account. Theapproach is based on the multilayer analytic element method, such that regional flow and local three-dimensionaldetail may be simulated simultaneously and accurately within one regional model. Horizontal flow inside a layeris computed analytically, while vertical flow is approximated with a standard finite-difference scheme. Results ob-tained with the proposed approach compare well to results obtained with three-dimensional analytic element solu-tions for flow in unconfined aquifers. The presented approach may be applied to predict the yield of a collectorwell in a regional setting and to compute the origin and residence time, and thus the quality, of water pumped bythe collector well. As an example, the addition of three lateral arms to a collector well that already has three later-als is investigated. The new arms are added at an elevation of 2 m above the existing laterals. The yield increaseof the collector well is computed as a function of the lengths of the three new arms.

IntroductionSince their development by Leo Ranney in the 1930s

(Todd 1959), radial collector wells (Ranney wells) havebecome quite popular as ground water supply wells. Aradial collector well consists of a number of horizontal,lateral arms, which are open to the aquifer, all connectedto a vertical cylindrical caisson. Many new collector wellshave lateral arms composed of wound stainless steelscreens, but in most older collector wells, the arms weremade of steel pipe with slots punched or cut into them.Typically, the lateral arms are installed by driving them

into the aquifer through ports in the caisson, usinga hydraulic jack. Recent advances in drilling technologyhave made it possible to install larger and longer laterals.

During operation, ground water enters the wellthrough the slots in the lateral arms and flows into thecaisson, where one or more pumps are installed, oftenwith variable-speed capability. Compared to verticalwells, collector wells are expensive to build but offermany benefits: drawdowns and thus electrical costs tendto be lower, entrance velocities tend to be smaller, foulingof the well screen occurs at a much slower rate, and wellsneed less frequent cleaning. Collector wells are oftenused to induce recharge from surface water and areinstalled close to major streams, in some cases with lat-eral arms that extend beneath the streams. Computermodels of collector wells may be used to predict yieldsand to investigate water quality effects of inducedrecharge entering the well. In addition, models may beused to evaluate innovative collector well designs, suchas long arms constructed by microtunneling.

Some numerical studies of collector wells appear inthe literature. Ophori and Farvolden (1985) presented oneof the first numerical models of a collector well in

1Corresponding author: Department of Biological and Agri-cultural Engineering, University of Georgia, Athens, GA 30602;fax (706) 542-8806; mbakker@engr.uga.edu

2Wittman Hydro Planning Associates Inc., 320 West 8thStreet, Bloomington, IN 47404; fax (812) 333-3080; vic@wittmanhydro.com

3Department of Mathematics and Computer Science, ValparaisoUniversity, Valparaiso, IN 46383; fax (219) 464-5065; ken.luther@valpo.edu

Received January 2005, accepted May 2005.Copyright ª 2005 National Ground Water Association.doi: 10.1111/j.1745-6584.2005.00116.x

926 Vol. 43, No. 6—GROUND WATER—November–December 2005 (pages 926–934)

Ontario, California; they initially used a one-layer finite-element model and later replaced the collector well bythree point sinks and used the method of images to modelthe nearby river. Eberts and Bair (1990) modeled regionalflow to four collector wells in Columbus, Ohio. Theyused a two-aquifer MODFLOW model and representedeach collector by one grid cell of 1000 by 1000 feet.A more detailed study of this site was completed byCunningham et al. (1995), who developed a three-aquiferMODFLOW model with variable cell sizes. Chen et al.(2003) developed a polygon finite-difference model ofa horizontal well. They approximated the flow inside thepipe by flow through an equivalent porous medium ofwhich the hydraulic conductivity was a function of theflow in the pipe.

Some analytic solutions exist for specific cases. Solu-tions for two-dimensional flow to radial collector wellswere presented by, e.g., Huisman (1972) and Strack(1989). One of the first three-dimensional analytic solu-tions was developed by Hantush and Papadopulos (1962),who developed a model for a confined aquifer with a uni-form inflow along line sinks representing the arms of thecollector well. Bischoff (1981) applied the boundary inte-gral equation method to model three-dimensional flow toa collector well with three lateral arms in a confined aqui-fer subject to uniform flow; the arms of the collector weretreated as constant head and the inflow was allowed to varyin a piecewise linear fashion. Haitjema (1982) presented ananalytic solution for flow to a radial collector well inwhich each lateral was represented by a third-order hori-zontal line sink and a horizontal line doublet. A three-dimensional analytic element solution for a horizontal well(in essence one arm of a collector well) in a confined aqui-fer was presented by Steward and Jin (2001). Semianalyticsolutions (using Laplace transforms) for three-dimensionalflow to horizontal wells in unconfined and leaky aquiferswere presented by Zhan and Zlotnik (2002) and Zhan andPark (2003).

Radial collector wells have large pumping rates andoften have a regional impact on the flow field. Simulationof the interaction of collector wells with other wells andregional surface water requires a regional model. At thesame time, the interaction between the collector well andneighboring streams requires local three-dimensionaldetail. Furthermore, variation of inflow along the lateralarms of the collector, skin effects, and head losses insidethe arms must be taken into account. The combination ofregional flow, local three-dimensional detail, and theunique boundary conditions along collector arms canmake it impractical to use gridded models. These circum-stances are, however, very well suited for application ofthe analytic element method.

The objective of this paper is to present a practicalanalytic element approach for the modeling of flow toradial collector wells. The basic idea behind the approachis to divide the aquifer up into a number of horizontallayers, one of which contains the radial collector well.This is of course a common approach in finite-differenceor finite-element modeling; the novelty here is that flowin the horizontal direction will be modeled analytically.A similar multilayer approach gave accurate results for

the modeling of three-dimensional flow to partially pene-trating wells (Bakker 2001). Flow in the multilayer modelis simulated with existing analytic elements for multi-aquifer flow (Bakker and Strack 2003). The collector armswill be modeled with multiaquifer line sinks. Strings ofline sinks can accurately represent varying extraction rates,and incorporation of skin effects and internal head lossesis relatively straightforward. Through comparison withthree-dimensional analytic element solutions for collectorwells, it will be shown that the proposed multilayer ana-lytic element approach gives accurate results. A more com-prehensive example illustrating many of the capabilitiesof the proposed approach will be presented at the end ofthis paper.

Analytic Multilayer ApproachIt is proposed to model ground water flow to a collec-

tor well with a multilayer approach, whereby the aquiferis divided vertically into a number of horizontal layerswith homogeneous properties, and the resistance to verti-cal flow is neglected within a layer (the Dupuit approxi-mation). The vertical specific discharge qz,i betweenthe center of layer i and the center of overlying layeri 2 1 is approximated with a standard finite-differenceapproximation

qz;i ¼ ðhi 2 hi21Þ=ci ð1Þwhere hi is the head in layer i, and ci is given by

ci ¼Hi21

2kv;i21

1Hi

2kv;ið2Þ

where kv,i is the vertical hydraulic conductivity of layer i,and Hi is the thickness of layer i. Steady flow in layer i isgoverned by the differential equation

r2hi ¼qz;i 2 qz;i1 1

Tið3Þ

where Ti [L2/T] is the transmissivity of layer i, and r2 isthe two-dimensional (horizontal) Laplacian. In a multi-layer system with N layers, flow is governed by N linkeddifferential equations (Equation 3) with qz,i defined as inEquation 1

r2hi ¼hi 2 hi21

ciTi2

hi1 1 2 hici1 1Ti

i ¼ 1;.;N ð4Þ

When the top and bottom of the aquifer are treated asimpermeable, c1 ¼ cN11 ¼ N. In this fashion, the for-mulation for flow in a multilayer system, also referred toas quasi–three-dimensional flow, is mathematically equiva-lent to the formulation for flow in a multiaquifer systemwhere ci is the vertical resistance of a leaky layer betweenaquifers i and i 2 1 (Leake and Mock 1997). Conse-quently, quasi–three-dimensional flow in an aquifer maybe simulated with analytic elements that were developedfor multiaquifer flow.

The analytic element method is based on the super-position of analytic solutions to the differential equationgoverning the flow (here the system 4). Each analyticsolution, termed an analytic element, has one or morefree parameters, or degrees of freedom; examples of free

M. Bakker et al. GROUND WATER 43, no. 6: 926–934 927

parameters are the discharge of a well or the inflow alonga line sink (representing a stream segment, for example).The free parameters of an element may either be speci-fied or may be computed to meet specified boundary con-ditions at control points. As most analytic elements havea linear dependence on their free parameters (includingall analytic elements used in this paper), solution of ananalytic element model requires the solution of a linearsystem of equations (e.g., Strack 1989). Analytic ele-ments for multiaquifer flow include wells, line sinks, cir-cular area sinks (Bakker and Strack 2003), cylindrical andelliptical inhomogeneities (Bakker 2003, 2004a), andembedded multiaquifer domains (Bakker 2005).

Ground water flow to a radial collector well will besimulated with the multilayer approach by discretizing theaquifer vertically in a number of layers. At the level ofthe arms of the collector well, the thickness of the layerwill be equal to the diameter of the arms; the collectorarms will be modeled with strings of line sinks that arescreened only in this layer. Horizontal flow in the modelwill be computed analytically, while vertical flow is com-puted with a standard finite-difference scheme. There isone important restriction in application of the multilayeranalytic element approach, namely that transmissivitiesmust be piecewise constant. This means that in an uncon-fined aquifer, the transmissivity in the top layer must beapproximated by an average transmissivity. This is oftenreasonable for the modeling of radial collector wells, asthe drawdown of the water table will often be relativelysmall. The benefits of the approach far outweigh thisrestriction. First, the solution is analytic and no horizontalgrid is needed to obtain a solution. This gives a largedegree of flexibility in placement of the arms, which isespecially useful during the design phase, when differentnumbers, orientations, and lengths of arms are considered.Second, the hydraulic conductivity may differ betweenlayers; this is useful for the modeling of flow in stratifiedaquifers, which is difficult to do in three-dimensional ana-lytic models (Haitjema 1987). Third, incorporation of ver-tical anisotropy in the aquifer is straightforward, as it onlyaffects the resistance ci (Equation 2) between layers. Andfourth, incorporation of the multilayer model in a regionalone-aquifer (or even multiaquifer) model is straightfor-ward, as analytic elements that fully penetrate all layers ofan aquifer may be superimposed.

Multilayer Analytic Element Formulation forRadial Collector Wells

The inflow of ground water qs [L/T] at a point alongan arm of a collector well is a function of the differencein the hydraulic head h in the aquifer just outside the armand the hydraulic head / inside the arm:

qs ¼ ðh 2 /Þ=cs ð5Þ

where the skin effect is represented by the resistance toinflow cs [T] of the well screen. The discharge per unitlength of arm r [L2/T] is obtained by multiplication of qswith the circumference of the arm

r ¼ ðh 2 /Þ=d ð6Þwhere d ¼ c/(2pR) is called the conductance of the armand R is the radius of the arm. Both h and /, and theinflow r, may vary along the arm.

Consider one arm of a collector well in an aquiferthat is discretized vertically in N piecewise homogeneouslayers; the aquifer is discretized in such a way that atthe level of the arm the thickness of the layer is equal tothe diameter of the arm. Ground water flow to the armis modeled with a string of M multiaquifer line sinks(Bakker and Strack 2003), which are probably moreappropriately referred to as multilayer line sinks in thecurrent context. The line sinks are numbered 1 through Mstarting at the end of the arm and ending at the caisson. Amultilayer line sink is an analytic solution to the systemof differential equations (Equation 4) for the case wherewater is extracted at a constant rate along a line segmentpositioned in one layer; the strength (the free parameter)rm of line sink m is the discharge per unit length of linesink and is constant along the entire length of the linesink. In this fashion, the inflow of ground water along thearm is approximated as piecewise constant. The totalinflow Qarm into the arm may be computed as

Qarm ¼XMm¼1

rmLm ð7Þ

where Lm is the length of line sink m. In the remainder ofthis section, equations will be derived for the computationof the strengths rm of the line sinks representing the arm.

An equation for the strength of line sink m is ob-tained through application of Equation 6 at the center ofthe line sink

rm ¼ ðh�m 2 /mÞ=d ð8Þ

where h�m is the head at the center of the line sink just out-side the arm (in the layer containing the arm) and /m isthe head inside the arm at the center of the line sink. Thedifference in head between the centers of line sinks m andm1 1 inside the arm is called �/m

/m 2 /m1 1 ¼ �/m ð9Þ

and is a function of the discharge inside the arm and thewall friction of the arm. It is proposed to compute thehead loss �/m with the Darcy1-Weisbach friction lossequation (e.g., Munson et al. 1998, equation 8.34), whichmay be written in terms of the discharge Qm between thecenters of line sinks m and m 1 1 as

�/m ¼ flm

2p2R5

Q2m

2gð10Þ

where f is the friction factor, lm is the distance betweenthe centers of line sinks m and m 1 1, and g is the accel-eration of gravity. The friction factor f depends on theroughness d of the wall of the arm and may depend onthe Reynolds number of the flow inside the arm (e.g.,Munson et al. 1998). At present, accurate values of d are

1Yep, the same guy.

928 M. Bakker et al. GROUND WATER 43, no. 6: 926–934

not known for drainpipes. In addition, the effect of thecontinuous inflow along the arm on the head loss insidethe arm is unknown. One of the few lab experimentsmeasuring head losses inside horizontal wells was per-formed by Chen et al. (2003). They measured head lossesinside a slotted PVC pipe positioned inside a sandbox; thepipe was 4.5 m long and had a diameter of 5.4 cm. Theywere able to identify several different flow types insidethe well, varying from laminar to fully turbulent flow.However, not enough information is available to computevariations of f along steel collector arms, and it is pro-posed to approximate f by a constant value until furtherresearch makes it possible to describe the pipe frictionloss more accurately. A reasonable value of f may beestimated, for example, from Moody’s diagram (e.g.,Munson et al. 1998, figure 8.20). The effect of the pipefriction on the well yield is investigated in the exampleapplication at the end of this paper.

The discharge in the arm varies continuously. As thedischarge in the arm is zero at the tip of the arm, the totaldischarge Sm in the arm at the center of line sink m maybe written as

Sm ¼Xm21

j¼1

rjLj 1 rmLm=2 ð11Þ

The average discharge Qm ¼ (Sm 1 Sm11)/2 betweenthe centers of line sinks m and m1 1 may now be written as

Qm ¼Xm21

j¼1

rjLj 13rmLm 1 rm1 1Lm1 1

4ð12Þ

Equation 9 may be applied to all line sinks of thearm, except for the last one. Combining Equations 8 and9 to eliminate /m and /m11 gives

h�m 2 h�m1 1 2 dðrm 2 rm1 1Þ ¼ �/m m ¼ 1;.;M 2 1

ð13Þwhere �/m is given by Equation 10, using Equation 12for Qm.

The difference in head between the center of the lastline sink of the arm (M) and the a priori unknown head /c

in the caisson is/M 2 /c ¼ �/M ð14Þ

The head difference is due to wall friction plus a headloss due to outflow into the caisson (Munson et al. 1998,sec. 8.4.2)

�/M ¼�flM2R

1 1

�Q2

M

p2R42gð15Þ

where QM is the average flow between the center of linesink M and the caisson

QM ¼XM21

j¼1

rjLj 13rMLM

4ð16Þ

Combination of Equations 8 and 14 gives

h�M 2 drM 2 /c ¼ �/M ð17Þ

where �/M is given by Equation 15, using Equation 16for QM.

For a collector well with P arms, with each arm rep-resented by Mp line sinks, the total number of unknownstrengths is

PPp¼1 Mp: Equations 13 and 17 for each arm

represent the same number of equations, although this in-troduces one additional unknown: the head /c in the cais-son. The final equation is that the sum of the totaldischarges of each arm (Equation 7) is equal to the totaldischarge Qcw of the collector well:

XPp¼1

Qarm; p ¼ Qcw ð18Þ

The left-hand side of Equations 13, 17, and 18 islinear in the strengths of the line sinks (and any otheranalytic elements in the model) and can easily beincorporated into the system of linear equations that needsto be solved to obtain an analytic element solution. How-ever, the right-hand side of the former two equations isa nonlinear function of the discharges in the arm andthus the strengths of the line sinks along it. It is proposedto obtain a solution to the stated equations iteratively.Starting with an initial estimate of the inflow along thearms (for example, uniform), the head losses �/m may becomputed and the linear system may be solved to obtainnew values for the strengths of the line sinks. These newvalues are used to compute new values of �/m, and theprocess is repeated until it converges; for certain prob-lems, a relaxation factor is used in the computation of thenew values of �/m. This iterative scheme is relativelyquick as only the right-hand side of the system of equa-tions changes, and thus the matrix has to be inverted onlyonce.

Comparison of Multilayer and Three-DimensionalAnalytic Element Solutions

In this section, results of the multilayer approach asoutlined in the previous section will be compared tothree-dimensional analytic element solutions for flow toradial collector wells in unconfined aquifers. The objec-tive is to assess the accuracy of the multilayer solution torepresent local three-dimensional head and flow varia-tions. The multilayer approach has been implemented ina prototype module for the multilayer analytic elementmodel TimML (Bakker 2004b). The three-dimensionalanalytic element solution will be discussed first.

Three-Dimensional Representation ofa Radial Collector Well

Steady three-dimensional flow in piecewise homoge-neous aquifers may be modeled with analytic elementsthat fulfill the three-dimensional Laplace equation.Three-dimensional analytic elements include line sinksand circular infiltration areas (Haitjema 1985) and ellipti-cal inhomogeneities (Fitts 1991; Jankovic and Barnes1999). Horizontal confining boundaries of an aquifer maybe simulated with the method of images. In case ofunconfined flow, the upper boundary is formed by thephreatic surface. This boundary may be approximatedthrough the method introduced by Luther and Haitjema(1999), which will be used here. Skin effects and friction

M. Bakker et al. GROUND WATER 43, no. 6: 926–934 929

losses inside the arm are neglected, i.e., the head is ap-proximated as constant along all arms.

In the three-dimensional model, the P arms of theradial collector well are represented by three-dimensionalline sinks. The inflow along each line sink is varied line-arly (similar to the approach used by Bischoff [1981] andothers) and is made continuous from one line sink to thenext. Hence, an arm consisting of Mp line sinks will haveMp 1 1 unknown strength parameters, representing theinflow at the end points of the line sinks. Mp 1 1 controlpoints are distributed along arm p at the end points of theline sinks, but a distance equal to the radius R from thecenterline of the arm. Linear equations for the unknownstrengths at the corner points are obtained by requiringthat the difference in head between two adjacent controlpoints equals zero. This results in Mp equations. In addi-tion, the difference in head between the P control pointsat the end points of the arms at the caisson should alsoequal zero, resulting in P 2 1 additional equations. Thefinal equation is that the total flow of all arms equals thetotal flow from the collector (Equation 18). This results ina total of

PPp¼1 ðMp 1 1Þ linear equations for as many

unknown strength parameters.As noted when modeling flow to partially penetrat-

ing wells (Haitjema and Kraemer 1988) and horizontalwells (Luther 1998; Steward and Jin 2001), a piecewiselinear strength distribution alone is sometimes unable tomaintain, near the end points of the arm, the cylindricalshape of the equipotential required along the wellscreen—particularly for arms with a large radius. How-ever, for most radial collector wells the radius is smallrelative to the arm length, and the linear strength distribu-tion gives acceptable accuracy.

In cases of unconfined flow, the upper boundary ofthe aquifer is a nonlinear free boundary whose position isnot known ahead of time. The phreatic surface is identi-fied as the surface along which the fluid pressure pf iszero (atmospheric)

pf ðx; y; zÞ ¼ 0 ð19Þ

and along which the three-dimensional normal flux ~qn iszero (i.e., no recharge)

~qnðx; y; zÞ ¼ 0 ð20Þ

The location of the phreatic surface is found itera-tively through application of the approach described byLuther and Haitjema (1999) for unconfined flow neara partially penetrating well. The essence of the approachis as follows. Many (hundreds) of control points are dis-tributed laterally above the collector well at an elevationchosen to be an initial guess for the location of the phre-atic surface. In addition, many three-dimensional analyticelements (point sinks, line sinks, ring sinks, etc.) are dis-tributed laterally above the anticipated location of thephreatic surface; the strength parameters of these addi-tional elements, which all lie outside the aquifer, are com-puted iteratively such that the required conditions 19 and20 are met at control points along the phreatic surface.First, the zero normal flow condition (Equation 20) isapplied at the phreatic surface control points and thestrengths of the additional analytic elements is computed.But now the second condition of zero pressure (Equation19) is not met, so the phreatic surface control points aremoved vertically to the location where the pressure iszero. At these new locations, the zero normal flow condi-tion (Equation 20) is applied and the process is repeateduntil it converges on the location of the phreatic surface.As reported by Luther and Haitjema (1999), this processconverges generally in five or six iterations. A similarapproach was used by Jankovic and Barnes (2001).

Comparison for a Horizontal WellConsider flow to a horizontal well in an unconfined

aquifer with a hydraulic conductivity of 150 m/d (Figure 1).The elevation of the phreatic surface is fixed to 24 m at(x, y) ¼ (60, 0) m. (These highly conductive aquifer prop-erties are representative for alluvial valleys coincidentwith major rivers in the Midwest of the United States.) A

−60 −30 0 30 60−60

−30

0

30

60

h = 24

−60 −30 0 30 60−60

−30

0

30

60

h = 24

Figure 1. Contours of head at the level of the horizontal well (left) and of the phreatic surface (right); multilayer solution(solid) and three-dimensional solution (dashed); the dotted line in the right figure marks the location of the deeper-lying hori-zontal well; contour interval is 0.2 m.

930 M. Bakker et al. GROUND WATER 43, no. 6: 926–934

60-m-long horizontal well is located along the x-axis,a distance of 3 m above the horizontal bottom of the aqui-fer. The diameter of the horizontal well is 0.3 m, and thedischarge is 12,000 m3/d. For the multilayer solution, theaquifer is divided into 12 layers. The layers are thinnernear the well to represent local three-dimensional effectsmore accurately. The tops of the layers are chosen at thefollowing elevations (m): 24, 16, 11, 7, 5, 4.05, 3.45,3.15, 2.85, 2.55, 1.95, and 1, and a base at 0; the horizon-tal well is located in layer 8 (from 3.15 to 2.85 m) and isrepresented by 10 multilayer line sinks of equal length.For the three-dimensional solution, the horizontal well ismodeled with 10 three-dimensional line sinks of equallength. The skin effect and internal friction of the hori-zontal well are neglected, and thus the head is constantalong the arm.

A comparison of the multilayer and three-dimensionalsolutions is presented in Figures 1 and 2; solid lines repre-sent the multilayer solution and dashed lines the three-dimensional solution. Contours of head at a horizontalplane through the arm are shown in Figure 1. The head inthe arm for the multilayer solution is 22.34 m vs. 22.41 mfor the three-dimensional solution. Contours of the head inthe top layer of the multilayer solution and the phreaticsurface of the three-dimensional solution are also shown inFigure 1; the head in the top layer and the elevation of thephreatic surface directly above the center of the well are23.42 m for both solutions. A comparison of heads in a ver-tical cross section along the well is shown in Figure 2.Here the approximation of the multilayer solution is morevisible: the head does not vary in the vertical directionwithin a layer. In the right half of the figure, a dot is plot-ted on the contour line at the middle of each layer to facili-tate comparison with the three-dimensional contours; theapproximation is still very reasonable.

Path lines in the same cross section were started atx ¼ 260 and x ¼ 60 m at 11 starting points in the verticalwith intervals of 2 m. The path lines were obtained witha standard predictor-corrector method (e.g., Strack 1989,p. 315); the vertical component of flow in the multilayersolution varies linearly in the vertical direction withina layer (Strack 1984). The path lines generated with the

two solutions are again very similar. It is concluded thatfor the presented case, flow to a horizontal well in anunconfined aquifer may be represented accurately witha 12-layer model with a constant transmissivity in the toplayer. The next question is whether the same model maybe used to accurately simulate flow to a radial collectorwell with five arms.

Comparison for a Radial Collector WellConsider flow to a radial collector well with five lat-

eral arms, evenly distributed around the caisson; theradius of the caisson is 3 m. Each arm is 60 m long andhas a diameter of 0.3 m (equivalent to the horizontal wellused in the previous example) and is located at the sameelevation and in the same aquifer as in the previousexample. For the multilayer solution, the vertical discreti-zation of the aquifer is the same as in the previous exam-ple. The center of the coordinate system is chosen at thecenter of the caisson and the head is fixed to 24 m at (x,y) ¼ (100, 0). The total discharge of the radial collectorwell is 60,000 m3/d; skin effects and internal friction inthe lateral arms are again neglected. Each lateral is repre-sented by 10 line sinks of equal length.

A comparison of head contours at the level of the col-lector arms is shown in Figure 3, as well as a comparisonof contours of the phreatic surface and the head in the toplayer of the multilayer model. The contours produced withthe two solutions are again very similar. The head in thecaisson for the multilayer solution is 20.74 vs. 20.82 m forthe three-dimensional solution. The lowest head in the toplayer of the multilayer solution is 21.44 vs. 21.46 m forthe lowest point of the phreatic surface in the three-dimensional solution. As for the horizontal well, the multi-layer solution is an accurate approximation and only slightlyoverpredicts the drawdown, but the differences are small.

Example ApplicationAs an example application, the expansion of a collec-

tor well from three to six arms is studied, and the effect ofthe lengths of the three new arms on the yield of the col-lector well will be computed. Consider a radial collector

−60 −30 0 30 600

3

10

20

24

h = 23.8

−60 −30 0 30 600

3

10

24

Figure 2. Head contours with contour interval of 0.2 m (left) and path lines (right) in a vertical cross section through the horizon-tal well; multilayer solution (solid) and three-dimensional solution (dashed); head contours of multilayer solution are vertical ineach layer, and the dots represent the position of the contours at the center of each layer, as computed by the multilayer solution.

M. Bakker et al. GROUND WATER 43, no. 6: 926–934 931

well with initially three lateral arms of 60 m and a caissonwith a radius of 3 m. The arms are located 3 m above thebase of the aquifer, have a radius of 0.15 m, and areequally spaced around the caisson. The aquifer is approxi-mately 24 m thick. The bottom 10 m of the aquifer has ahydraulic conductivity of 200 m/d, and the material aboveit has a hydraulic conductivity of 100 m/d. The verticalhydraulic conductivity is 60 m/d in the entire aquifer. Thehead in the aquifer is fixed to 24 m at (x, y) ¼ (200, 0).The friction factor of the arms is set to f ¼ 0.02, and theskin effect is neglected.

The yield of the collector well for a head in the cais-son of 20 m may be computed with the described proce-dure when Equation 18 is replaced by /c ¼ 20. Theaquifer is divided into 18 layers of which the tops are atthe following elevations (m): 24, 16, 12, 10, 8, 7, 6, 5.45,5.15, 4.85, 4.55, 4, 3.45, 3.15, 2.85, 2.55, 1.95, 1, and thebase at 0. The collector well arms are in layer 14. Theproblem is solved and the computed yield of the collectorwell is 22,400 m3/d. For this configuration, the lowestlevel of the head in the top layer (an approximation of thephreatic surface) is 22.1 m. The difference in head

between the caisson and the phreatic surface is signifi-cantly larger than in the comparisons of the previous sec-tion because the vertical hydraulic conductivity is smaller.

Next, the collector well is modified to increase theyield. Three laterals are added at an elevation of 5 mabove the base (layer 9), again equally spaced around thecaisson, but out of synch with the lower laterals (Figure 4).For a head in the caisson of 20 m, the addition of threearms with a length of 60 m increases the yield by only57% to 35,200 m3/d; the discharge of the original threearms actually reduces to 14,600 m3/d, while the combineddischarge of the three new arms is 20,600 m3/d. Theeffect of the lengths of the three new arms on the wellyield is shown by plotting the total discharge of the col-lector well as a function of the lengths of the new arms(Figure 5, solid line). It appears from Figure 5 that thetotal discharge of the collector does not increase asquickly with arm length when the arm length is verylarge. This characteristic is investigated further by plot-ting the arm length vs. the additional discharge due to theextension of the new arms by 20 m (Figure 6). This graphshows clearly that for arm lengths beyond ~170 m, the

−100 −50 0 50 100−100

−50

0

50

100

h = 24

−100 −50 0 50 100−100

−50

0

50

100

h = 24

Figure 3. Contours of head at the level of the collector well (left) and of the phreatic surface (right); multilayer solution (solid)and three-dimensional solution (dashed); contour interval is 0.5 m.

−150

0150

−150

0

1500

24

x

y

z

Figure 4. Three-dimensional path lines to collector well withthree arms of 60 m, and three arms of 120 m.

0 50 100 150 200 2502

4

6

8

10

12

14

Lengths of new arms (m)

Dis

char

ge (

x104

m3 /

d)

f = 0.02f = 0

Figure 5. Collector well yield as function of lengths of newarms.

932 M. Bakker et al. GROUND WATER 43, no. 6: 926–934

increase in yield due to an additional section of 20 m de-creases due to the internal wall friction of the arms (thetotal yield of the collector well still increases, of course).

The head inside the arms varies because of the inter-nal wall friction of the arms. For example, for a length of120 m, the head at the tip of the arm is 1.75 m larger thanin the caisson. To justify the chosen value of the frictionfactor, the Reynolds number for flow in the arms is com-puted. For a discharge of 5000 m3/d, the velocity in thearm is 0.8 m/s, which gives Re ¼ 100,000; inspection ofMoody’s diagram shows that f ¼ 0.02 is reasonable forthis Reynolds number. The wall friction has a significanteffect on the yield, as may be seen from Figure 5, wherethe dashed line represents the yield for the case withoutwall friction ( f ¼ 0); the effect of the wall friction be-comes more important for longer arms (and thus higherflow rates). Note that the dashed line in Figure 5 is con-cave, and thus the extension of the new arms by another20 m always increases the discharge by an even largeramount than the previous extension of 20 m.

For illustration purposes, several three-dimensionalpath lines are generated for the case where the new armshave a length of 120 m (Figure 4). Six path lines werestarted from two locations (the vertical dotted lines), withvertical intervals of 4 m; a projection on the verticalplane x ¼ 2150 is shown in Figure 7. Path lines started at

(x, y) ¼ (280, 150) all end at one of the new, longerarms. Path lines started in the bottom part of the aquifer at(x, y) ¼ (40, 2150) flow to a short arm, while path linesstarted at higher elevations flow to a long arm. A smallabrupt change in direction of the path lines is visible atthe horizontal level where the horizontal hydraulic con-ductivity changes from 100 to 200 m/d (z ¼ 10). Originsand travel times of path lines may be computed to esti-mate the water quality in the collector well.

Summary and ConclusionsAn accurate and practical analytic element approach

was presented for the modeling of steady flow to radialcollector wells. The approach allows for vertical stratifi-cation and vertical anisotropy of the aquifer. The uniqueboundary condition along collector arms may be takeninto account fully, including skin effect and internal wallfriction in the collector arms. It was shown that the wallfriction has a significant effect on the well yield. Thewall friction was approximated by a constant friction fac-tor f; detailed experiments are needed to describe headlosses inside horizontal arms more accurately. The inclu-sion of the approach in a regional analytic element modelis straightforward, as two-dimensional analytic elementsthat fully penetrate the aquifer may be superimposed. Theapproach takes full advantage of the fact that the analyticelement method is able to simulate regional flow andlocal detail accurately in the same model. The flexibilityof the analytic element method allows for the inclusion ofother features such as sections of lateral arms that consistof regular pipe rather than drainpipe. The computationaleffort of the approach is manageable: all multilayer exam-ples presented in this paper required only seconds to solveon a regular PC.

AcknowledgmentsThe authors thank Hongbin Zhan, Charlie Fitts,

Randy Hunt, and Henk Haitjema for their constructivereviews.

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0 50 100 150 200 2504

5

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ition

al d

isch

arge

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03 m

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dditi

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