Analytic Element Modeling of EmbeddedMultiaquifer Domainsby Mark Bakker

AbstractAn analytic element approach is presented for the modeling of multiaquifer domains embedded in a single-

aquifer model. The inside of each domain may consist of an arbitrary number of aquifers separated by leakylayers. The analytic element solution is obtained through a combination of existing single-aquifer and multiaqui-fer analytic elements and allows for the analytic computation of head and leakage at any point in the aquifer.Along the boundary of an embedded multiaquifer domain, the normal flux is continuous everywhere; continuityof head across the boundary is met exactly at collocations points and approximately, but very accurately, inbetween. The analytic element solution compares well with an existing exact solution. A hypothetical examplewith a river intersecting two embedded domains illustrates the practical application of the proposed approach.

IntroductionOne of the major strengths of the analytic element

method is the modeling of embedded domains with a dif-ferent transmissivity; these domains are called inhomoge-neities in most analytic element models (e.g., Strack andHaitjema 1981; Strack 1989). The contrast between thetransmissivity inside and outside of an inhomogeneity maybe arbitrarily high; efficient and extremely accurate sol-utions may be obtained by using higher order elements andoverspecification (Jankovic and Barnes 1999). Inhomoge-neities have great practical value as the division of an aqui-fer into zones (inhomogeneities) of constant transmissivityor hydraulic conductivity is often a good modeling practice.Using a limited number of zones with constant parametersfacilitates the calibration of ground water models (e.g., Hillet al. 1998), independent of whether the problem is solvedwith analytic elements or a discrete numerical method.

Recently, a new analytic element formulation wasdeveloped for flow in multiaquifer systems (Bakker andStrack 2003); multiaquifer systems are defined as systemsof multiple aquifers separated by leaky resistance layers.

One of the major benefits of the formulation is that it doesnot require an areal discretization of the model area (e.g., intriangles or quadrilaterals) to solve the flow problem, justlike the analytic element method for single-aquifer flow.Multiaquifer analytic element equations were presented forwells, line sinks, and circular infiltration areas in the latterreference; equations for cylindrical inhomogeneities inmultiaquifer systems were presented in Bakker (2003).

The objective of this paper is to model flow in a sin-gle aquifer that contains embedded domains consisting ofmultiple aquifers and leaky layers. Each embedded multi-aquifer domain may consist of an arbitrary number ofaquifers separated by leaky layers; an embedded domainwith two aquifers is shown in Figure 1. The leaky layersinside a domain may represent clay lenses of limitedextent, for example. Multiaquifer domains embedded ina single aquifer cannot be modeled efficiently with manyof the existing ground water models. The main difficultyis caused by the requirement of most models that modellayers exist over the entire model area. This makes it dif-ficult to simulate embedded domains with a differentnumber of aquifers. For example, if a single aquifer con-tains embedded domains of both three and five aquifers,the model must consist of five model layers everywhere,even in areas where there are only one or three aquifers.This puts stringent restrictions on the maximum cell sizein areas where multiple layers together simulate flow inone aquifer (e.g., Bakker 1999). Some numerical modelsallow cells to be ‘‘inactive’’ (e.g., MODFLOW), but thisrequires the arbitrary modeling decision to cut an aquifer

Department of Biological and Agricultural Engineering,University of Georgia, Athens, GA 30602; fax (706) 542-8806;mbakker@engr.uga.edu

Received June 2003, accepted May 2004.Copyright ª 2005 The Author(s)Journal compilationª 2006 National Ground Water Associationdoi: 10.1111/j.1745-6584.2005.00080.x

Vol. 44, No. 1—GROUND WATER—January–February 2006 (pages 81–85) 81

off along some line, rather than allow multiple aquifers toflow freely into one aquifer. More elegant solutions fornumerical models include the use of embedded models(Schaars et al. 2003) or three-dimensional finite elements.

The approach presented here allows for the model-ing of a single aquifer with an arbitrary number ofembedded multiaquifer domains, each consisting of an ar-bitrary number of aquifers and leaky layers. Accuracyand performance of the approach are assessed throughcomparison with an exact solution and the presentation ofa hypothetical example.

TheoryConsider steady-state flow in a single aquifer. Flow is

treated as confined with a piecewise constant trans-missivity. The aquifer contains multiaquifer domains con-sisting of several permeable layers (aquifers) separated byleaky layers of lower permeability (Figure 1). Aquifersand leaky layers are numbered from the top down,with leaky layer n on top of aquifer n. The Dupuit-Forchheimer approximation is adopted for flow in theaquifers and is interpreted to mean that the resistance tovertical flow within an aquifer is neglected. Flow in theleaky layers is approximated as vertical. The vertical leak-age qz [L/T] between aquifers n and n 2 1 is computed as:

qz = ðhn2hn21Þ=cn ð1Þ

where hn [L] is the head in aquifer n and cn [T] is theresistance to vertical flow of leaky layer n. The boundaryof each domain is formed by a polygon of straight line-segments. The boundary conditions along the boundary ofan embedded multiaquifer domain are twofold. First, theheads are continuous across the boundary, and second, thevertically integrated flux normal to the boundary is con-tinuous across the boundary.

Geohydrologic features outside all multiaquifer do-mains are modeled with the analytic elements for single-aquifer flow presented in Strack (1989). Following Strack(1989), analytic element equations are written in termsof a discharge potential F [L3/T], defined for confinedflow as:

F = Th ð2Þ

where T is the transmissivity of the aquifer. In aquiferswith a piecewise constant T, F fulfills Laplace’s

differential equation (or Poisson’s differential equation inthe presence of areal recharge). Analytic elements areanalytic solutions to the differential equation for specificboundary conditions. Analytic elements used in this paperinclude wells, line sinks and line doublets. Line sinks areused to model inflow/outflow along river segments. Linedoublets, which are line elements along which the poten-tial jumps but the flow normal to the element is continu-ous, are used to model boundaries of inhomogeneities(Strack 1989).

Inside a multiaquifer domain, transmissivities arerepresented by a vector ~s, where the components of thevector represent the transmissivities of the individualaquifers. The discharge potential inside a domain is alsoa vector, ~/, defined for a domain with M aquifers as:

~/ = ðs1h1; s2h2;.; sMhMÞ ð3Þ

The potential inside a domain is written as:

~/ = ~/o þ~/i ð4Þ

where ~/o is the potential due to elements outside thedomain and ~/i is the potential due to elements inside thedomain. The potential due to elements outside the domainmay be written as:

~/o = F~s=s ð5Þ

where F is the discharge potential of the elements outsidethe domain and s is the comprehensive transmissivityinside the domain, the sum of the transmissivities of allaquifers. Equation 5 has two nice properties. First, it en-sures that the vertically integrated flux (summed over allaquifers) normal to the boundary of the domain is contin-uous. Second, it creates no leakage between aquifersinside the domain as it creates no head differencebetween aquifers. Heads will be made continuous acrossthe boundary through the addition of line elements alongthe boundary of the domain.

The potential ~/i represents geohydrologic featureson the inside of an embedded domain, which are modeledwith the multiaquifer analytic elements presented byBakker and Strack (2003) and Bakker (2003). In bothreferences, analytic element equations are written in theform of Equation 3, and leakage between aquifers is rep-resented exactly. It is shown in Bakker and Strack (2003)that the comprehensive potential for a multiaquifer ana-lytic element (the sum of the potentials in all aquifers)fulfills Laplace’s equation. Hence, the discharge potentialfor a multiaquifer analytic element outside its multi-aquifer domain may be represented by its comprehensivepotential. Again, this ensures that the vertically integratedflux normal to the boundary of the domain is continuous.

Two additional analytic elements are added along theboundary of a domain to make sure that the heads arecontinuous across the boundary. The aquifer outside thedomain is connected to all aquifers inside the domain.Hence, the heads in all aquifers inside the domain areequal at a point on the boundary of the domain. This isaccomplished by putting multiaquifer line sinks with zeronet discharge along the boundary of the domain. The linesinks redistribute flow between aquifers such that the

Figure 1. An embedded multiaquifer domain with twoaquifers.

82 M. Bakker GROUND WATER 44, no. 1: 81–85

heads in all aquifers are equal on the boundary. As thenet discharge of each line sink, and thus its comprehen-sive potential, is zero, it has no effect on the potentialoutside the domain. Once the heads are equal in all aqui-fers along the boundary of the domain, single-aquifer linedoublets may be used to make sure that the heads arecontinuous across the boundary. This is equivalent to themodeling of a transmissivity inhomogeneity in a singleaquifer where the transmissivities inside and outside theinhomogeneity are s and T, respectively (e.g., Strack1989, sec. 35). Inside the domain, the potential due toa line doublet is computed with Equation 5. The approachpresented here is restricted to cases where the head alongthe boundary of the domain may be approximated as thesame in all aquifers. This approximation may be reason-able for the case where a multiaquifer domain is embed-ded in a single aquifer or when the boundary between twomultiaquifer domains is formed by a vertical, highly con-ductive fault zone.

The combination of single-aquifer analytic elementsand multiaquifer analytic elements described in the fore-going ensures that continuity of the normal component ofthe vertically integrated flux is met exactly everywherealong the boundary of a multiaquifer domain. Continuityof head is met exactly at collocation points on the bound-ary and approximately between them. The leakage betweenaquifers inside a multiaquifer domain fulfills Equation 1exactly. The approach is implemented in the open-sourcemultiaquifer analytic element program TimML (Bakker2004), which is written in Python (www.python.org). Theboundaries of the multiaquifer domains are modeled withconstant-strength multiaquifer line sinks and second-orderline doublets. It will be shown that this gives accurate sol-utions through comparison with an exact solution and thepresentation of a hypothetical example.

Intermezzo: The Role of Leakage FactorsIn the discussion of the comparison and example in

the next two sections, reference will be made to leakagefactors. The theory and practical significance of leakagefactors is well published (Hemker 1984; Maas 1986;Bruggeman 1999; Bakker 1999, 2001; Bakker and Strack2003); it is summarized here, and some conclusions forareal infiltration are drawn.

Leakage factors are characteristic lengths of a multi-aquifer system. As a rule of thumb, and in the absence ofareal infiltration, the leakage between aquifers approacheszero and thus, the heads in all aquifers are nearly equal,at a distance of three to six times the largest leakage fac-tor away from any aquifer feature (such as a well, stream,or discontinuity in aquifer properties). An aquifer with Pleaky layers has P leakage factors; leakage factors may becomputed from the aquifer and leaky layer properties (seeany of the aforementioned references).

In the presence of a constant areal infiltration N[L/T], the leakage between aquifers, and thus the dif-ference in head between aquifers, will be constant at adistance of three to six times the largest leakage factoraway from any aquifer feature. Beyond this distance, the

upward leakage, qzn, from aquifer n to aquifer n 2 1 maybe approximated as:

qzn = 2N

s

XM

i = n

si ð6Þ

where M is the number of aquifers. Since the upwardleakage is also proportional to the difference in headbetween aquifers (Equation 1), the constant head differ-ence between aquifers is:

hn212hn =Ncns

XM

i = n

si ð7Þ

Rearrangement gives an expression for the resistance cn:

cn =sðhn212hnÞ

NPM

i = nsi

ð8Þ

This equation has practical value as it may be used toestimate the resistance of a leaky layer when the head dif-ference across a leaky layer is measured far away fromgeohydrologic boundaries; it also requires estimates ofthe areal recharge N and the ratio of the transmissivities.

Comparison with Exact SolutionThe accuracy of the approach presented in the fore-

going is assessed through comparison with an existingexact solution. Consider a well near an infinitely longstraight boundary between a single-aquifer and a two-aquifer system (Figure 2); the boundary between theone-aquifer and two-aquifer zones runs along the y-axis(dotted line). To the left of the y-axis, there is one aquiferwith a transmissivity of T = 55 m2/d. To the right of they-axis, there is a two-aquifer system; the transmissivities

−1500 −1000 −500 0 500 1000 1500−1500

−1000

−500

0

500

1000

1500

x (m)

y (m

)

λ

Figure 2. Comparison of exact solution (bottom half) andanalytic element solution (top half). One aquifer (left) andtwo aquifers with well in bottom aquifer (right). Head con-tours in single aquifer and upper aquifer (solid) and loweraquifer (dashed).

M. Bakker GROUND WATER 44, no. 1: 81–85 83

are s1 = 15 m2/d and s2 = 40 m2/d, and the resistance ofthe leaky layer is c = 8250 d so that the leakage factor isk = 300 m. A well with discharge Q = 200 m3/d isscreened in the lower aquifer of the two-aquifer zone at(x, y) = (k/2, 0). The exact solution for the case that T ands are equal (as it is here) is given by Hunt and Curtis(1989); a solution for differing T and s is given by Maas(2000); both exact solutions also adopt the Dupuit-Forchheimer approximation within an aquifer.

In the analytic element model, the two-aquiferdomain is represented by a rectangular box of 1625 by3000 m from (x, y) = (0, 21500) to (x, y) = (1625, 1500).Line sinks and line doublets vary in length from 25 mnear the well to 750 m far away from the well. The prob-lem is solved with TimML, and contour plots of the headin the upper aquifer (solid) and lower aquifer (dashed) areshown in Figure 2; in the one-aquifer zone, the head con-tours are solid lines. Note that the head in both aquifers isindeed continuous across the boundary between the one-aquifer and two-aquifer zones (the dotted line). The flowfield is symmetric across the x-axis (the dashed-dottedline). The TimML solution is contoured above the dashed-dotted line and the exact solution below it. The exact andapproximate solutions are virtually identical. The cone ofdepression at the well in the lower aquifer is clearly visi-ble. The largest drawdown in the upper aquifer is muchsmaller and occurs on the boundary of the multiaquiferdomain (in this case at the origin), rather than above thewell. The heads in the two aquifers are nearly equal ata distance of 4k away from the well; the ‘‘3’’ in Figure 2is four leakage factors away from the well.

Example of a Stream Crossing TwoEmbedded Domains

A hypothetical example is presented of a streamcrossing two domains with a different number of aquifersin each. Consider flow to the irregular east-west–runningstream shown in Figure 3 (heavy gray line). For simplic-ity, the head along the stream is equal to 70 m every-where. The stream extends to the left and right of the

figure where it remains straight. Flow is toward thestream, and the head at (x, y) = (0, 2000) is fixed at 80 m;the transmissivity of the aquifer is T = 50 m2/d. Thestream crosses two multiaquifer domains (the heavy solidlines). The domain on the left consists of two aquifers(s1 = s2 = 20 m2/d, c2 = 9000 d, k1 = 300 m), andthe domain on the right consists of three aquifers (s1 = 10m2/d, s2 = s3 = 15 m2/d, c2 = 9000 d, c3 = 4000 d, k1 =160 m, k2 = 282 m).

Especially near the intersections of the stream andthe boundaries of the domains, the heads and flow are ex-pected to vary rapidly over short distances. Hence, thestream and the boundary of the domains must be discre-tized into several sections to obtain an accurate solution.The stream is modeled with 46 line sinks and the domainswith a total of 57 line sinks and 19 line doublets. Con-tours of the heads are shown in Figure 3. The solid linesare contours of heads in the single aquifer and the upperaquifer of each domain. The dashed lines are contours ofheads in the second aquifer of each domain, and the dot-ted lines are contours in the third aquifer of the rightdomain. Continuity of head across the boundaries of theembedded domains may be verified visually and has beencomputed for the multiaquifer domain on the left. The dif-ference in head across the boundary is computed at 10evenly spaced points along each segment of the polygonalboundary. The average difference in head between theinside and the outside of the domain is 4.9 mm in the topaquifer and 4.1 mm in the bottom aquifer; the maximumdifference in head across the boundary is 5.5 cm in the topaquifer and 2.6 cm in the bottom aquifer.

The size of both domains is on the order of a coupleof times the size of the leakage factors. The domains arethus not large enough for the leakage (and thus the headdifference between aquifers) to vanish in the center partof the domains. Inside each domain, the lowest head inthe lower aquifers does not occur directly below thestream but ~150 to 200 m north or south of the stream.

The effect of a well screened in the bottom aquiferof the three-aquifer domain is investigated. A well witha discharge of Q = 200 m3/d is added to the model at

500 1000 1500 2000 2500

−500

0

500

x (m)

y (m

)

72.2 h = 72.6 m

71.8 71.4

7170.6

70.2

70.270.6

71 71.4

71.8 72.2

72.6

Figure 3. A stream (heavy gray line) and two embedded multiaquifer domains (heavy solid lines) containing two aquifers (left)and three aquifers (right). Solid contours are single aquifer and upper aquifer, dashed contours are second aquifer, dottedcontours are third aquifer.

84 M. Bakker GROUND WATER 44, no. 1: 81–85

(x, y) = (1750, 0). Contour lines of the area around thethree-aquifer domain are shown in Figure 4 for the solu-tion with the well. The well creates a large drawdown inthe bottom aquifer (dotted lines); the drawdown in theupper two aquifers is smaller. It may be concluded fromthe head contours that the well draws water from bothsides of the stream.

Conclusion and DiscussionAn analytic element approach was presented for the

modeling of steady-state flow in a single-aquifer modelwith embedded multiaquifer domains. Head and verticalleakage may be computed analytically at any point in theaquifer. Continuity of flow is met exactly along theboundary of each embedded domain; continuity of headis met exactly at collocation points along the boundaryand approximately but very accurately between them.The approach is especially useful for the modeling offlow in aquifers with leaky clay lenses of limited extent.

The approach may be extended to several other set-tings. Flow in the single aquifer may be modeled asunconfined (where the transmissivity varies continuouslywith the saturated thickness) by application of the dis-charge potential for unconfined flow presented by Strack(1989). Inside an embedded domain, the flow in the topaquifer may be perched and separated from the secondaquifer by an unsaturated zone; this application is some-what restricted as it requires the transmissivity in the topaquifer to be modeled as constant, and it requires thewater table in the bottom aquifer to be separated hydrauli-cally from the top aquifer over the entire multiaquiferdomain. The approach was presented for embedded mul-tiaquifer domains in a single-aquifer model, but theinverse may be modeled just as well. Such a model repre-sents embedded single-aquifer domains inside a multi-aquifer model; the embedded single-aquifer domains may

represent, for example, buried bedrock valleys that cutthrough all aquifers and are filled with unconsolidatedmaterial. The presented approach may also be used tomodel multiaquifer domains that are connected througha vertical, highly conductive fault zone. Finally, theembedded multiaquifer domains may be used to modellocal quasi–three dimensional flow in stratified or verti-cally anisotropic aquifers. The different aquifers then rep-resent different model layers that may have differenthydraulic conductivities.

AcknowledgmentThe development of embedded multiaquifer domains

was funded by the U.S. EPA Ecological Research Divi-sion, Athens, Georgia, through contract 1L-1216-NAEX;the project manager was Stephen R. Kraemer.

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version 2.0. http://www.engr.uga.edu/~mbakker/TimML.html(accessed August 2004).

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Bakker, M. 2001. An analytic, approximate method for model-ing steady, three-dimensional flow to partially penetratingwells.Water Resources Research 37, no. 5: 1301–1308.

Bakker, M. 1999. Simulating groundwater flow in multi-aquifersystems with analytical and numerical Dupuit-models.Journal of Hydrology 222, no. 1–4: 55–64.

Bakker, M., and O.D.L. Strack. 2003. Analytic elements formultiaquifer flow. Journal of Hydrology 271, no. 1–4: 119–129.

Bruggeman, G.A. 1999. Analytical Solutions of Geohydro-logical Problems, Developments in Water Science, 46.Amsterdam: Elsevier.

Hemker, C.J. 1984. Steady groundwater flow in leaky multiple-aquifer systems. Journal of Hydrology 72, no. 3–4: 355–374.

Hill, M.C., R.L. Cooley, and D.W. Pollock. 1998. A controlledexperiment in ground-water flow model calibration.Ground Water 36, no. 3: 520–535.

Hunt, B., and T.G. Curtis. 1989. Flow to a well near the bound-ary between a layered and an unlayered aquifer system.Water Resources Research 25, no. 3: 559–563.

Jankovic, I., and R. Barnes. 1999. High-order line elements inmodeling two-dimensional groundwater flow. Journal ofHydrology 226, no. 3–4: 204–210.

Maas, C. 2000. Drawdown pattern due to a well near a geologicfault. Paper presented at the Third International Conferenceon the Analytic Element Method, Brainerd, Minnesota.

Maas, C. 1986. The use of matrix differential calculus in prob-lems of multiple-aquifer flow. Journal of Hydrology 99, no.1–2: 43–67.

Schaars, F., P. Kamps, J. Hoogendoorn, and C. Maas. 2003.MODGRID, simultaneous solving of different groundwaterflow models at various scales; an application to the ground-water model of the Amsterdam dune water area. In Pro-ceedings of MODFLOW and More 2003 Conference,51–53. Golden, Colorado: IGWMC.

Strack, O.D.L. 1989. Groundwater Mechanics. EnglewoodCliffs, New Jersey: Prentice Hall.

Strack, O.D.L., and H.M. Haitjema. 1981. Modeling doubleaquifer flow using a comprehensive potential and distributedsingularities 2. Solution for inhomogeneous permeabilities.Water Resources Research 17, no. 5: 1551–1560.

1200 1400 1600 1800 2000 2200

−400

−200

0

200

400

x (m)

y (m

) 70.2

70.6

71

h = 71.4 m

70.2

70.6

71

71.4

71.8

72.2

Figure 4. Well in bottom aquifer of three-aquifer domain.Stream (heavy gray line) and three-aquifer domain (heavysolid lines). Solid contours are single aquifer and upper aqui-fer, dashed contours are second aquifer, dotted contours arethird aquifer.

M. Bakker GROUND WATER 44, no. 1: 81–85 85