Two-dimensional steady unsaturated flow through embedded elliptical layers

Two-dimensional steady unsaturated flow through

embedded elliptical layers

Mark Bakker

Department of Biological and Agricultural Engineering, University of Georgia, Athens, Georgia, USA

John L. Nieber

Department of Biosystems and Agricultural Engineering, University of Minnesota, St. Paul, Minnesota, USA

Received 22 April 2004; revised 16 August 2004; accepted 29 September 2004; published 14 December 2004.

[1] New analytic element solutions are presented for unsaturated, two-dimensional steadyflow in vertical planes that include nonoverlapping impermeable elliptical layers andelliptical inhomogeneities. The hydraulic conductivity, which is represented by anexponential function of the pressure head, differs between the inside and outside of anelliptical inhomogeneity; both the saturated hydraulic conductivity and water retentionparameters are allowed to differ between the inside and outside. The Richards equation istransformed, through the Kirchhoff transformation and a second standard transformation,into the modified Helmholtz equation. Analytic element solutions are obtained throughseparation of variables in elliptical coordinates. The resulting equations for the Kirchhoffpotential consist of infinite sums of products of exponentials and modified Mathieufunctions. In practical applications the series are truncated but still fulfill the differentialequation exactly; boundary conditions are met approximately but up to machine accuracy,provided that enough terms are used. The pressure head, saturation, and flow may becomputed analytically at any point in the vadose zone. Examples are given of theshadowing effect of an impermeable elliptical layer in a uniform flow field and funnel-type flow between two elliptical inhomogeneities. The presented solutions may be appliedto study transport processes in vadose zones containing many impermeable ellipticallayers or elliptical inhomogeneities. INDEX TERMS: 1875 Hydrology: Unsaturated zone; 1829

Hydrology: Groundwater hydrology; 1866 Hydrology: Soil moisture; KEYWORDS: elliptical inhomogeneity,

vadose zone, analytic element method, Kirchhoff potential

Citation: Bakker, M., and J. L. Nieber (2004), Two-dimensional steady unsaturated flow through embedded elliptical layers, Water

Resour. Res., 40, W12406, doi:10.1029/2004WR003295.

1. Introduction

[2] Groundwater flow through the vadose zone is influ-enced by the presence of embedded soil layers of finiteextent. Under unsaturated conditions, embedded soil layersconsisting of coarser material often have a (much) lowerhydraulic conductivity than the surrounding finer-texturedmaterial. Flow is diverted into preferential flow pathsaround these layers [e.g., Kung, 1990; Ju and Kung,1993, 1997]. Preferential flow paths may have a significantimpact on contaminant transport as they provide a quickpassageway for contaminants, and corresponding shortertravel times affect transport processes such as adsorptionand biodegradation. Contaminant transport may be modeledthrough the explicit inclusion of heterogeneity, or throughthe use of effective hydraulic conductivity and transportproperties for heterogeneous unsaturated soils [e.g., Zamanet al., 2004]. Either way, solutions are dependent on theaccurate representation of the velocity field.[3] Accurate velocity fields may be obtained by applica-

tion of the analytic element method. The analytic element

method was originally developed for the modeling of flowin the saturated zone [Strack, 1989; Haitjema, 1995]. One ofits applications is the modeling of flow through inhomoge-neities, which are defined as bounded domains with differ-ing hydraulic conductivities. Analytic element equationsexist for saturated flow through inhomogeneities of severaldifferent shapes and in several different settings (for anoverview, see Strack [2003]). Example applications ofanalytic element solutions for flow regions containinginhomogeneities include both regional flow models [e.g.,Bakker et al., 1999; Kelson et al., 2002] and the study ofeffective aquifer and transport properties of highly hetero-geneous saturated media using tens of thousands of inho-mogeneities [e.g., Jankovic et al., 2003a, 2003b]. Analyticelement models allow for the analytic computation ofvelocities at any point in the aquifer. Accurate path linesmay be obtained, using the analytic velocity field, throughapplication of, for example, an improved Euler method[e.g., Strack, 1989, p. 315]. Contaminant transport process-es, such as adsorption and biodegradation, may be per-formed along these path lines [e.g., Strack and Fairbrother,1997; Soule, 1997].[4] Numerous exact solutions exist for steady flow in the

vadose zone where the hydraulic conductivity is representedCopyright 2004 by the American Geophysical Union.0043-1397/04/2004WR003295$09.00

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WATER RESOURCES RESEARCH, VOL. 40, W12406, doi:10.1029/2004WR003295, 2004

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by an exponential function of the pressure head [e.g., Philip,1998, and references therein]. Philip [1998] presented anexact solution for unsaturated flow through parabolic inho-mogeneities; this solution constitutes the only exact solu-tion, that the authors are aware of, for unsaturated flowthrough a domain with an inhomogeneity. Philip [1998]considered a parabolic domain of which the interior is filledwith coarser material than outside. For this case, thesubmaximal capillary barrier as he called it, the saturatedhydraulic conductivity and the sorptive number for theporous media inside the inhomogeneity are distinctly dif-ferent from the surrounding porous media. He examinedbackground flow conditions that led either to diversion offlow (shedding of the flow) around the inhomogeneity orconvergence of flow (concentration of the flow) into theinhomogeneity.[5] Warrick and Knight [2002] were the first to apply

analytic element techniques to simulate flow in the vadosezone; they presented a solution for a single circular inho-mogeneity in a uniform flow field. Warrick and Knight[2003] derived solutions for profiles composed of two orthree layers of different saturated hydraulic conductivitiesand Warrick and Knight [2004] presented an analyticelement solution for three-dimensional unsaturated flowaround one spherical inhomogeneity. In the latter threepapers the saturated hydraulic conductivity was the onlyphysical soil property varied between the distinct media.Bakker and Nieber [2004] presented a general analyticelement framework, including cylindrical drains, and cylin-drical inhomogeneities that differ in both saturated hydrau-lic conductivity and water retention parameters inside andoutside the inhomogeneity. They showed that their solutionfor cylindrical inhomogeneities in unsaturated flow has thesame advantages as the corresponding analytical elementsolutions for saturated flow in that the solutions fulfill thegoverning equation exactly and boundary conditions can bemet up to machine accuracy. This allows for the analyticcomputation of the pressure head, saturation, and velocity atany point in the vadose zone.[6] Alternatively, solutions to problems of unsaturated

flow may be obtained by application of boundary discreti-zation methods such as the boundary element method anddomain discretization methods such as the finite element,control volume, or finite difference methods. The boundaryelement method has been applied to the quasilinear form ofthe Richards equation [e.g., Pullan and Collins, 1987], but,to date, has not been applied to unsaturated flow regionswith embedded inhomogeneities. Boundary elements andanalytic elements have certain similarities and certain differ-ences, which are discussed in Bakker and Nieber [2004] [cf.Strack, 2003]. The domain discretization methods are onlyaccurate when the grid is refined near the boundaries ofinhomogeneities, with the amount of refinement increasinggreatly as the material contrast increases [Oldenburg andPruess, 1993]. A major difficulty in using these methods isthat the required grid refinement near these boundaries is afunction of the flow field and is, in general, not known apriori. These conventional numerical methods for the solu-tion of unsaturated flow have more flexibility than theanalytic element method in terms of the choice of hydraulicconductivity functions and the incorporation of transientflow. However, when the hydraulic conductivity function

may be approximated as exponential and the flow as steady,it appears that the analytic element method has the advan-tage over these methods in terms of ease of application andaccuracy.[7] The objective of this paper is to present new

analytic element solutions for the modeling of verticaltwo-dimensional steady unsaturated flow through a vadosezone containing soil layers of elliptical shape. The embed-ded elliptical layers consist of coarser or finer material thanthe surrounding soil and may be modeled either as imper-meable objects or as elliptical inhomogeneities with ahydraulic conductivity function that differs between theinside and outside, including differences in both the satu-rated hydraulic conductivity and the water retention param-eters. Analytic element equations will be derived throughseparation of variables in terms of elliptical coordinates. Thesame approach was applied recently to model flow through10,000 elliptical inhomogeneities in a saturated aquifer[Suribhatla et al., 2004] and to model flow to ellipticallakes and through elliptical inhomogeneities in multiaquifersystems [Bakker, 2004b]. Analytic element equations willbe derived for elliptical inhomogeneities, and it will beshown how these same equations may be used to modelimpermeable elliptical layers. As an example, the solutionsare applied to study the shadowing effect of impermeableellipses and to study funnel-type flow between two ellipticalinhomogeneities.

2. Basic Equations and Problem Statement

[8] Consider steady state, two-dimensional unsaturatedflow in the vertical plane. A right-handed, Cartesian x, zcoordinate system is adopted with the z axis pointingvertically down (Figure 1). Flow is governed by Darcy’slaw:

qx ¼ �K@h

@xqz ¼ �K

@h

@zþ K; ð1Þ

where qx and qz (L T�1) are the components of the specificdischarge, h (L) is the pressure head and K(h) (L T�1) is thehydraulic conductivity. The divergence of the specificdischarge is zero for steady state flow:

@qx@x

þ @qz@z

¼ 0; ð2Þ

such that

@

@xK@h

@x

� �þ @

@zK@h

@z

� �� @K

@z¼ 0: ð3Þ

This equation is linearized with the Kirchhoff transforma-tion [Philip, 1969, 1998]:

H hð Þ ¼Z h

�1K sð Þds h � he; ð4Þ

where H (L2 T�1) is the Kirchhoff potential, and he (L) isthe air entry pressure head of the soil. Differentiation of (4)gives

@H

@x¼ K hð Þ @h

@x

@H

@z¼ K hð Þ @h

@z: ð5Þ

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[9] The exponential hydraulic conductivity function isadopted (also known as the Gardner [1958] model):

K hð Þ ¼ Ks exp a h� heð Þ½ ¼ Ke exp ahð Þ h � he; ð6Þ

where Ks (L T�1) is the saturated hydraulic conductivity, a(L�1) is a parameter dependent on the pore size distributionof the porous medium, and Ke = Ks exp(�ahe) (L T�1).Substitution of (6) for K in (4) gives

H ¼ Ke

aexp ahð Þ ¼ K

a: ð7Þ

Combination of (3), (5), and (7) gives the linear differentialequation

r2H � a@H

@z¼ 0; ð8Þ

where r2 is the two-dimensional Laplacian. As differentialequation (8) is linear, solutions to the differential equationmay be superimposed. A solution in terms of the Kirchhoffpotential H may be converted into the pressure head h withthe inverse of (7):

h ¼ 1

aln

aHKe

� �: ð9Þ

The specific discharge vector (1) in terms of H becomes

qx ¼ � @H

@xqz ¼ � @H

@zþ aH : ð10Þ

[10] An analytic element solution will be derived for anelliptical inhomogeneity in a general, unsaturated flow field.The elliptical inhomogeneity is centered at (x, z) = (xc, zc),has a long semiaxis equal to a, a short semiaxis equal to b,and an inclination of b (Figure 1). Inside the inhomogeneity,the hydraulic conductivity parameters are Ke = K1 and a =a1; outside the inhomogeneity the parameters are Ke = K0

and a = a0. Flow is governed by differential equation (8)both inside and outside the inhomogeneity, but, since thevalues of a and Ke differ between the inside and outside,separate equations must be derived for the inside and

outside of the inhomogeneity. The Kirchhoff potential Hdue to the elliptical inhomogeneity must be finite every-where, be continuous everywhere except across the bound-ary of the inhomogeneity, and vanish at infinity. Theboundary conditions across the boundary of the inhomoge-neity are that the pressure head h is continuous, and that thecomponent of flow normal to the boundary of the ellipse iscontinuous [e.g., Strack, 1989, section 35].

3. Analytic Element Solution

[11] An analytic element solution to the stated problem isobtained through introduction of the transform variable Qdefined as

Q ¼ H exp �a z� zcð Þ=2½ : ð11Þ

Substitution of (11) for H in (8) yields the modifiedHelmholtz equation (or Yakawa equation [Pullan andCollins, 1987])

r2Q ¼ a2

4Q: ð12Þ

An analytic element solution to (12) for domains withelliptical boundaries may be obtained through separation ofvariables in terms of elliptical coordinates.

3.1. Elliptical Coordinates

[12] First, a local X, Z coordinate system is defined asshown in Figure 1. The X, Z coordinates are obtained fromthe x, z coordinates through a rotation:

X ¼ x� xcð Þ cos bþ z� zcð Þ sin b

Y ¼ � x� xcð Þ sin bþ z� zcð Þ cos b:ð13Þ

Figure 1. An elliptical inhomogeneity; circles withcrosses are focal points.

Figure 2. Elliptical coordinates h and y; solid lines are his constant, and dashed lines are y is constant.

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The elliptical coordinates h, y are defined as [Moon andSpencer, 1971]

X ¼ d cosh h cosy Z ¼ d sinh h siny; ð14Þ

where 2d is the focal distance of the ellipse. The coordinateh is constant along confocal ellipses and varies from zeroalong the line connecting the foci to infinity for (x, z)approaching infinity (Figure 2). The y coordinate variesfrom �p to p, jumps from �y to +y across the lineconnecting the foci, and is discontinuous by 2p along the Xaxis for X < �d. The focal distance may be computed fromthe lengths a and b of the semiaxes as

d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � b2

p: ð15Þ

The value of h along the boundary of the ellipse is called h*and may be computed as

h* ¼ arccosh a=dð Þ: ð16Þ

3.2. Mathieu Functions

[13] As already stated, a solution for the transformvariable Q is sought through separation of the modifiedHelmholtz equation (12) in elliptical coordinates. Thisresults in modified Mathieu differential equations in h andy [Moon and Spencer, 1971, p. 20]. The solution to thesedifferential equations consists of modified Mathieu func-tions. The use of Mathieu functions has become practicalsince the development of algorithms and C++ subroutinesfor the computation of Mathieu functions for orders up to200 and values of ad/2 up to 800 by Alhargan [2000a,2000b]. An overview of the work on Mathieu functions isalso given in these references. The notation for Mathieufunctions varies; an overview of notations used before 1965is given in Abramowitz and Stegun [1965]. Both theroutines and notation of Alhargan are used in this paper.Mathieu functions were used recently to model saturatedflow to elliptical lakes and through elliptical inhomogene-ities in multiaquifer systems by Bakker [2004b].

[14] Modified Mathieu functions that are a function of yare called circumferential functions. There are two kinds ofcircumferential functions. Functions of the first kind areperiodic, and functions of the second kind are nonperiodic.As nonperiodic functions create discontinuities in the flowfield (recall that y varies from �p to +p), they cannot beused here. Modified Mathieu functions that are a function ofh are called radial functions. The general solution to (12)may be written as (excluding the nonperiodic circumferen-tial functions)

Qn h;y; da=2ð Þ ¼

Qen da=2;yð Þ AnIen da=2; hð Þ þ BnKen da=2; hð Þ½

þ Qon da=2;yð Þ CnIon da=2; hð Þ þ DnKon da=2; hð Þ½ ; ð17Þ

where Qen and Qon are even and odd modified circumfer-ential Mathieu functions of the first kind, Ien and Ion areeven and odd modified radial Mathieu functions of the firstkind, and Ken and Kon are even and odd modified radialMathieu functions of the second kind, n is the integer orderof the functions (even functions start at n = 0, oddfunctions start at n = 1), and An, Bn, Cn, Dn are constants.The complete solution for Q is obtained through super-position of terms of different order. Using (17) and theinverse of (11) gives the solution for the Kirchhoffpotential:

H ¼ expa z� zcð Þ

2

X1n¼0

Qn: ð18Þ

[15] Modified radial Mathieu functions of order 3 arerepresented graphically in Figure 3. The functions Ien arefinite at h = 0 and have a zero derivative Ie0n = dIen/dh ath = 0, while the functions Ion are zero at h = 0 and have afinite derivative at h = 0. Both Ion and Ien and their firstderivatives approach infinity for h approaching infinity.Conversely, both Kon and Ken and their first derivatives arefinite for h = 0 and approach zero for h approachinginfinity.

Figure 3. (left) Third-order radial modified Mathieu functions and (right) their derivatives for da/2 = 5.

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[16] The functions Ken and Kon cannot be used inside anelliptical inhomogeneity, as they create a jump in either theKirchhoff potential (and thus the pressure head) or the flowacross the line connecting the foci while both of these arerequired to be continuous across this line. Hence theanalytic element solution for the potential inside the inho-mogeneity is obtained through the combination of (17) and(18), by neglecting the radial functions of the second kind,and by using that a = a1, which gives

H ¼ expa1 z� zcð Þ

2

X1n¼0

AnQen da1=2;yð ÞIen da1=2; hð Þ½

þ BnQon da1=2;yð ÞIon da1=2; hð Þ h � h*; ð19Þ

where An and Bn are coefficients that will be computed suchthat the boundary conditions are met along the boundary ofthe ellipse, and B0 = 0 (since no odd functions of order zeroexist). The potential in equation (19) is continuous acrossthe line connecting the foci, which may be seen as follows.Recall that y jumps from �y to +y along this line. Thefunctions Qen are even and thus give the same value for yand �y. The functions Qon are odd and jump across theline connecting the foci, but since all Ion equal zero for h =0, the terms QonIon are again continuous. An expression forthe specific discharge vector inside the inhomogeneity maybe obtained from differentiation as is explained inAppendix A.[17] The functions Ien and Ion cannot be used outside the

elliptical inhomogeneity, as they approach infinity for happroaching infinity while the potential should vanish there.Hence the analytic element solution for the potential outsidethe inhomogeneity is obtained through the combination of(17) and (18), by neglecting the radial functions of the firstkind, and by using that a = a0, which gives

H ¼ expa0 z� zcð Þ

2

X1n¼0

CnQen da0=2;yð ÞKen da0=2; hð Þ½

þ DnQon da0=2;yð ÞKon da0=2; hð Þ h h*; ð20Þ

where Cn and Dn are coefficients that will be computed suchthat the boundary conditions are met along the boundary ofthe ellipse, and D0 = 0. An expression for the specificdischarge vector outside the inhomogeneity may beobtained from differentiation as is explained in Appendix A.[18] Solutions (19) and (20) constitute the analytic ele-

ment solution for an elliptical inhomogeneity. The solutionfulfills the governing differential equation exactly and thepotential is finite everywhere, continuous everywhere insideand outside the inhomogeneity, and vanishes at infinity. Thefinal task is to determine the coefficients An and Bn in (19)and Cn and Dn in (20) such that the boundary conditionsacross the boundary of the inhomogeneity are met.

3.3. Solution of the Coefficients

[19] As stated, analytic element solutions are derived forelliptical inhomogeneities in general flow fields. TheKirchhoff potential in the vadose zone is obtained throughsuperposition of the potentials of all analytic elements. Todate, analytic elements are available for uniform flow,cylindrical drains, and cylindrical inhomogeneities [Bakkerand Nieber, 2004] and the elliptical inhomogeneities

derived here. The coefficients of the different elementsare computed in an iterative manner by solving for thecoefficients of one element at a time while keeping thecoefficients of all other elements fixed. The final solutionis obtained by cycling through all elements repeatedlyuntil an accurate solution is obtained. This procedureconverges well and was first proposed for solving analyticelement models by Barnes and Jankovic [1999].[20] What remains to be shown here is how to solve for

the coefficients of one elliptical inhomogeneity in a general(fixed) flow field. The infinite series in the analytic elementsolution (19) and (20) are truncated at P terms. Thetruncated series fulfill the differential equation (8) exactlyand thus the truncation affects only the accuracy with whichthe boundary conditions can be met. The truncated seriescontain P + 1 unknown coefficients An and Cn and Punknown coefficients Bn and Dn (recall that B0 = D0 = 0)for a total of 4P + 2 unknown coefficients. The boundaryconditions of continuity of head and continuity of normalflow are applied at 2P + 1 collocation points along theboundary of the inhomogeneity. The collocation points areequally spaced in y along the boundary; collocation point pis at (h, y) = (h*, yp). The boundary conditions may bewritten as

hþ h*;yp

� �¼ h� h*;yp

� �p ¼ 1; . . . ; 2P þ 1 ð21Þ

qþh h*;yp

� �¼ q�h h*;yp

� �p ¼ 1; . . . ; 2P þ 1; ð22Þ

where the superscripts ‘+’ and ‘�’ stand for evaluation justinside and just outside the inhomogeneity, respectively.Equations (21) and (22) constitute 4P + 2 equations in the4P + 2 unknown coefficients. Equations (22) are linear inthe coefficients An, Bn, Cn, Dn (expressions for qh are givenin Appendix A), but equations (21) are nonlinear. Manydifferent techniques exist for the solution of nonlinearequations. Here, the equations are linearized and a solutionis obtained iteratively, following the procedure presented byBakker and Nieber [2004]; details of the procedure aregiven in Appendix B.

3.4. Impermeable Ellipses

[21] The derived analytic element equation for the outsideof an inhomogeneity (20) is directly applicable to modelflow around an embedded impermeable elliptical layer. Theonly difference is in the computation of the 2P + 1coefficients Cn and Dn. The boundary condition of zeronormal flow is applied at 2P + 1 control points (h*, yp):

q�h h*;yp

� �¼ 0 p ¼ 1; . . . ; 2P þ 1: ð23Þ

This system of 2P + 1 linear equations may be solved with astandard method (e.g., LU decomposition [Press et al.,1992]).

3.5. Implementation and Performance

[22] The presented analytic element formulation for el-liptical inhomogeneities has been implemented in a smallPython program (http://www.python.org). The time to com-pute the coefficients of one ellipse increases approximatelylinearly with the number of ellipses, and thus the compu-


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tational time to obtain a solution for all ellipses increasesquadratically with the number of ellipses. It is noted,however, that the number of cycles through all elementsto obtain convergence does, in general, not grow beyond15–20, as has been reported for analytic element solutionsof large numbers (thousands or more) of saturated inhomo-geneities [e.g., Barnes and Jankovic, 1999]. Once a solutionhas been obtained, the time to compute a pressure head orvelocity at a point is a linear function of the number ofellipses. Several algorithms have been developed to de-crease computation time for large numbers of inhomogene-ities. Barnes and Jankovic [1999] suggested to use fewerterms in the series expansion farther away from an inho-mogeneity. Strack et al. [1999] proposed a superblockapproach, in which the combined effect of groups of

inhomogeneities is represented by one series expansion.Finally, use can be made of parallel processing; it has beenshown that computational time reduces linearly with anincrease in the number of processors [Jankovic et al.,2003b].

4. Examples

[23] Two examples are presented of elliptical soil layersin an otherwise uniform vertical flow field. The Kirchhoffpotential for a vertical uniform flow with qz = qz0 is [e.g.,Bakker and Nieber, 2004]

H ¼ qz0=a: ð24Þ

Figure 4. Streamlines around an impermeable ellipse for saturated flow and unsaturated flow with threevalues of aa.

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In the absence of any other flow features, a uniform flow qz0results in a uniform pressure head, as may be seen fromcombining (9) and (24):

h ¼ 1

aln

qz0

Ke

� �: ð25Þ

[24] The examples are intended to demonstrate the effectof elliptical layers on flow in the vadose zone and todemonstrate the accuracy of the presented analytic elementsolution; comprehensive studies of flow and transportthrough vadose zones with numerous elliptical layers formsthe topic of future research. The first example concerns flowaround an impermeable elliptical layer. The second exampleconcerns funnel-type flow between two elliptical inhomo-geneities and includes an error analysis.

4.1. Example 1: Flow Around an ImpermeableElliptical Layer

[25] Consider one impermeable elliptical layer, centeredat the origin, in an otherwise homogeneous vadose zone.The long semiaxis is a, the short semiaxis is a/5, and theinclination is 15�. Away from the ellipse, flow is uniform inthe vertical direction at a rate of qz0. For this specificproblem, the pattern of streamlines is independent of thevalue of the uniform flow, but is dependent on the value ofa of the soil surrounding the ellipse, and thus on the poresize distribution of the medium. Streamline patterns areshown in Figure 4 for three different values of the dimen-sionless quantity aa and for the equivalent case of saturatedflow. A large value of aa represents a large ellipse, or acoarse soil, or a combination thereof. Note that streamlinesfor the saturated case are point-symmetric with respect tothe origin, while the streamlines for unsaturated flow arenot.[26] The impermeable ellipse has a shadowing effect on

the unsaturated flow, i.e., there is an area below the ellipsewhere there is relatively little flow and where the soil isrelatively dry. This shadowing effect is larger for largervalues of aa (i.e., for larger ellipses or for coarser soils).The variation of qz with x is evaluated at two different levels

below the ellipse (Figure 5). At z = 2a there is a noticeableeffect on qz for all three quantities of aa, with the largesteffect for aa = 50 where qz varies from 3.10 times theuniform flow to only 0.0032 times the uniform flow. At z =10a there is little variation of qz left for the case with aa = 1,but here is still a significant variation of qz for the two largervalues of aa. The largest variation is for aa = 50, whichresults in a variation of qz/qz0 from 0.38 to 1.66. Thepresence of the impermeable ellipse affects the flow overa large distance below the ellipse and will have a significanteffect on the transport of any dissolved constituents. Theeffect increases for larger values of aa.

4.2. Example 2: Funnel-Type Flow Between TwoElliptical Layers

[27] Consider two elliptical inhomogeneities in an other-wise vertical uniform flow qz0. The two elliptical inhomo-geneities consist of coarser textured sand than thesurrounding soil. For comparison, the soil on the inside ofthe inhomogeneity will be referred to as ‘coarse’ and thesoil on the outside as ‘fine’. The saturated hydraulicconductivity of the coarse soil is approximately three timeslarger than the Ks of fine soil, and the a value of the coarsesoil is almost twice as large as the fine soil (Table 1).[28] To investigate the expected behavior of the two

ellipses of coarse soil embedded in the fine soil, theratio of the two hydraulic conductivity functions iscomputed as

Kcoarse

Kfine

¼ K1

K0

exp a1 � a0ð Þh½ ; ð26Þ

and this ratio is plotted for the coarse and fine soils inFigure 6. The equivalent values of h for three values of the

Table 1. Parameters of Soils Used in Example 2

Porous Media a, m�1 he, m Ks, m d�1 l, m�1

Fine soil 12.9 �0.25 0.36 3.9Coarse soil 21.2 �0.15 1.0 6.2

Figure 5. Variation of qz along z = 2a and z = 10a for example 1.


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uniform flow, computed using (25), are also shown inFigure 6. From this plot it is observed that for any value ofthe pressure head below the air entry value of the fine soil,the hydraulic conductivity of the coarse soil is smaller thanof the fine soil and thus the ellipses are expected to divertthe flow.[29] The saturation S as a function of h is approximated

with an exponential function:

S ¼ exp l h� heð Þ½ : ð27Þ

The values for l are also given in Table 1. A fit between theexponential hydraulic conductivity and saturation functionsand the van Genuchten relationships [van Genuchten, 1980]for these two soils is given by Bakker and Nieber [2004]. Itis shown in this reference that the exponential function fitsboth the van Genuchten hydraulic conductivity andsaturation functions fairly well over the range of pressureheads considered in this example.[30] The two elliptical inhomogeneities are centered at

(x, z) = (0, 0) and (x, z) = (0.4, �0.3) (Figure 7). The longand short semiaxes of both ellipses are 0.25 m and 0.05,respectively (an axes ratio of a/b = 5), and the inclinationsare +30� and �30�, respectively. The problem is solvedwith the described analytic element method for severalvalues of the uniform flow. Contours of the hydraulic head(f = h � z) for the case that qz0 = 0.005 Ks,fine are shown inFigure 7 (dashed lines); the equivalent pressure head for thisuniform flow is h = �0.66 m. Path lines are started at z =�1.2 m at intervals of Dx = 0.6 m, and are traced using theimproved Euler method and a stepsize of 5 mm; the pathlines are also shown in Figure 7 (solid lines).[31] The accuracy of the analytic element solution is

evaluated for the case shown in Figure 7. As statedpreviously, the analytic element solution meets the govern-ing differential equation exactly and the boundary condi-tions approximately. Accurate numerical evaluation of thederivatives was used to verify that the potential solution, as

implemented in the computer program, satisfies differentialequation (8). The problem is solved with P terms in theinfinite series (19) and (20). To assess the accuracy of thesolution between the 2P + 1 collocation points, the averageand maximum absolute difference in head and normal flowacross the boundary of the ellipse centered at the origin arecomputed at 8P + 4 points (equally spaced in y) along theboundary. The iteration procedure for one ellipse, as de-scribed in Appendix B, was continued until the averagedifference in head across the boundary at collocation pointswas smaller than 10�10. The overall solution was obtainedby solving for the two ellipses consecutively until bothellipses fulfilled this condition. The results for differentvalues of P are reported in Table 2. All four error approx-imations decrease with increasing P. Experimentation hasshown that the number of terms in the series required for anaccurate solution increases when the contrast in unsaturated

Figure 6. Ratio of unsaturated hydraulic conductivitiesversus pressure head for soils used in example 2.

Figure 7. Funnel-type flow between two elliptical in-homogeneities. (top) Hydraulic heads (dashed lines) andpath lines (solid lines) for a uniform flow of qz0 =0.005 Ks,fine. (bottom) Variation of qz along z = 0.3.

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hydraulic conductivity across the inhomogeneity boundaryincreases and when the shapes of the ellipses are moreelongated. It is noted that for practical purposes it issufficient to stop the iterative procedure for one ellipse whenthe average difference in pressure head across the boundaryat collocation points is smaller than 10�5 or 10�6 m.[32] As can be seen from Figure 7, the two ellipses divert

much of the flow; this kind of flow has been referred to asfunnel-type flow [Kung, 1990; Ju and Kung, 1993, 1997].

As the hydraulic conductivity inside the two ellipses issignificantly smaller than outside, much of the flow isdiverted through the small gap (‘the funnel’) between thetwo ellipses. It is of special interest for purposes ofcontaminant transport to evaluate the variation of thespecific discharge in the vadose zone. The vertical compo-nent of the specific discharge along z = 0.3 is presented inFigure 7 for three different values of qz0/K0 and for the casewhere the two ellipses are treated as impermeable objects.

Table 2. Approximate Error in Boundary Condition of Ellipse Centered at Origin

P 8 12 16 20 24 28

jDhjavg 5.6 � 10�4 2.8 � 10�5 8.8 � 10�7 6.7 � 10�9 1.4 � 10�9 1.0 � 10�10

jDhjmax 4.4 � 10�3 2.2 � 10�4 7.1 � 10�6 5.0 � 10�8 1.2 � 10�8 8.3 � 10�10

jDqrjavg 7.5 � 10�6 3.1 � 10�7 6.9 � 10�9 5.7 � 10�10 4.9 � 10�11 2.2 � 10�12

jDqrjmax 5.2 � 10�5 2.3 � 10�6 3.0 � 10�8 4.7 � 10�9 4.2 � 10�10 1.8 � 10�11

Figure 8. Funnel-type flow between two longer elliptical inhomogeneities. (top) Hydraulic heads(dashed lines) and path lines (solid lines) for a uniform flow of qz0 = 0.005 Ks,fine. (bottom) Variation of qzalong z = 0.3.


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The maximum value of qz increases from 1.25 qz0 for qz0 =0.05 Ks,fine to 1.59 qz0 for qz0 = 0.0005 Ks,fine, whichapproaches the result obtained with two impermeable ellipses(1.64 qz0).[33] The effect of more elongated ellipses is investigated

next. The long semiaxes of the ellipses is doubled to 0.5 m(a/b = 10), while keeping the size of the gap between theellipses the same. A flow net for this case is presented inFigure 8, which also includes a graph of the variation of qzat z = 0.3. The two long inhomogeneities divert even moreflow toward the gap between them than for the case withshorter ellipses. As a result the maximum value of the flowincreases as well. At an elevation of z = 0.3, the maximumvalue of qz has almost doubled as compared to the case withthe shorter ellipses. For qz0 = 0.0005 Ks,fine the maximumvalue increased from qz = 1.59 qz0 to qz = 3.11 qz0. Furtherfunneling of flow toward the gap is obtained with evenlonger ellipses. For example, for the case of two ellipseswith long semiaxes of 1 m (a/b = 20), the maximum valueof qz at z = 0.3 is q = 5.35 qz0 for qz0 = 0.0005 Ks,fine.[34] The increase in flow in the gap also results in a

saturation buildup around the gap both inside and outsidethe ellipses; contours of the saturation are shown in Figure 9for the case with qz0 = 0.005 Ks,fine. Far away from theellipses, where flow is uniform, the saturation is S = 0.20.The maximum saturation of S = 0.43 is reached at the top ofthe right ellipse. The lowest values of the saturation occurinside the ellipses, near the left tip of the left ellipse and theright tip of the right ellipse; the minimum value is S = 0.05.The increased saturation near the gap results in an increasedamount of flow through the elliptical layers near the gap, ascan be seen from the streamlines in Figure 8.

5. Discussion

[35] For application of the presented solution to an actualfield situation it is necessary to characterize the location andsize of embedded inhomogeneities within the subsurfaceenvironment. This may at first seem to be a daunting task.

However, current developments of technologies for nonin-vasive characterization of the vadose zone provide promis-ing opportunities for such descriptions. These technologiesinclude Ground Penetrating Radar [Boll et al., 1996],Electrical Resistance Tomography [Daily et al., 1992],and Geophysical Diffraction Tomography [King and Witten,1989]. While these methods can be used to identify thegeometric locations of features possessing distinct texturalcharacteristics, point sampling of these features is needed toquantify the hydraulic properties of these features.[36] The solution presented in this manuscript is based on

the use of an exponential form for the hydraulic conductiv-ity, the Gardner soil representation. This function is limitedin application to pressures less than the air entry pressure ofthe soil. Therefore the analytic element solution is not validin regions where the pressure exceeds the air entry pressureof the soil. Certainly it is possible that small perchedsaturated zones (pressures greater than the air entry pres-sure) can occur at the boundary of inhomogeneities, per-meable or impermeable, provided the uniform flow is highenough. The applications shown in this manuscript did notconsider such high flow rates. A possible approach to treatoccurrences of perched saturated zones is to solve theHelmholz equation in the unsaturated domain and the Lap-lace equation in the saturated domain, and to match bound-ary conditions across the interface between the domains.The identification of the a priori unknown interface will bechallenging but probably not insurmountable.[37] The presented solutions are also limited to steady

state flow conditions. Several analytic element formulationsexist for the modeling of transient saturated flow [e.g.,Furman and Neuman, 2003; Bakker, 2004a]. To extendthe capability of the analytic element method to transientunsaturated flows, the transient form of the Richards equa-tion needs to be linearized, for example by the use of anaverage diffusivity term.[38] Finally, the analytic element solutions presented

here, and other solutions referenced in the Introduction,consider geometries with smooth boundaries (circles, ellip-

Figure 9. Saturation distribution for case of Figure 8 with qz0 = 0.005 Ks,fine; contour interval DS = 0.02.

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ses, etc.). However, the method is not limited to suchshapes. Inhomogeneities of any shape may be constructedas a polygon of line segments, where each line segmentrepresents an analytic element (similar to the analyticelement approach for saturated flow). Line elements canalso be used to represent such features as lateral imperme-able boundaries, variable fluxes or pressure heads at theland surface, and the water table boundary of an underlyingsaturated zone.

6. Summary and Conclusions

[39] Analytic element solutions were presented for steadyunsaturated flow around nonoverlapping impermeable el-liptical layers and through nonoverlapping elliptical inho-mogeneities in a vertical plane. The hydraulic conductivitywas represented as an exponential function of the pressurehead and both the saturated hydraulic conductivity and thewater retention properties were allowed to vary betweenthe inside and outside of an inhomogeneity. It is noted thatthe use of the exponential function for the hydraulicconductivity as well as the limitation to steady flow areessential to allow for an analytic solution of the problem, ashas been recognized by many researchers. Solutions wereobtained through separation of variables in elliptical coor-dinates and the use of modified Mathieu functions; theMathieu functions were computed with the algorithms ofAlhargan [2000a, 2000b]. The solutions allow for theanalytic computation of pressure head, saturation, and flowat any point in the vadose zone where the pressure is lessthan the air entry value. Accurate path lines were obtainedwith an improved Euler method. Contaminant transport,including adsorption and biodegradation, may be simulatedalong these path lines. The derived solution method wasapplied to the simulation of unsaturated flow in the vicinityof two ellipses located in a vertically uniform flow field. Itwas found that the ellipses divert a significant amount of theflow and thereby concentrated it into channels of flowsimilar to the funnel flow phenomenon described by Kung[1990]. An increase in the length of the ellipses increasedthe amount of concentrated flow. With the presented ana-lytic element solution the flow through vadose zones con-sisting of many elliptical layers may be simulated accuratelyto determine effective hydraulic conductivity and transportproperties of the unsaturated zone.

Appendix A: Specific Discharge Vector

[40] Equations for the specific discharge vector may beobtained from (10). Substitution of the inverse of (11) for Hin (10) and differentiation gives

qx ¼ � expa z� zcð Þ

2

@Q@x

qz ¼ � expa z� zcð Þ

2

@Q@z

� a2Q

� :

ðA1Þ

The gradient vector of the transform variable Q is called~g =~rQ. The h and y components of~g may be written, with theequations by Moon and Spencer [1971, p. 18], as

~g ¼ gh; gy� �

¼ 1

w@Q@h

;@Q@y

� �; ðA2Þ

where

w ¼ d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosh2 hþ cos2 y

q: ðA3Þ

The h and y derivatives of Q are straightforward, as Q iswritten in separated form; h and y derivatives of themodified Mathieu functions are given by Alhargan [2000a,2000b]. The x and z components of~g may be obtained froma rotation as

gx ¼ @Q=@x ¼ gh cosg� gy sin g

gz ¼ @Q=@z ¼ gh sin gþ gy cosg;ðA4Þ

where g is the inclination of a unit vector in the h direction:

g ¼ arctancoshh sinysinhh cosy

� �þ b: ðA5Þ

The x and z components of the specific discharge vectormay now be obtained from (A1) and (A4) as

qx ¼ � expa z� zcð Þ

2

@Q@h

cos g

w� @Q

@ysin g

w

� �

qz ¼ � expa z� zcð Þ

2

@Q@h

sin g

wþ @Q

@ycosg

w� aQ

2

� �:

ðA6Þ

Similarly, the h and y components of the specific dischargevector may be obtained from (A6) by a rotation over �g,which gives

qh ¼ � expa z� zcð Þ

2

1

w@Q@h

� aQ2

sin g

� �

qy ¼ � expa z� zcð Þ

2

1

w@Q@y

� aQ2

cos g

� �:

ðA7Þ

Continuity of flow across the line connecting the foci is met,as may be seen by evaluating qh and qy(A7) along h = 0.

Appendix B: Solution of Nonlinear Equations

[41] The nonlinear set of equations (21) are solved asfollows. Substitution of (9) for h in (21) gives

1

a0

lna0H

� h*;yp

� �K0

¼ 1

a1

lna1H

þ h*;yp

� �K1

: ðB1Þ

This equation may be written as

lna0

K0

H� h*;yp

� �� ¼ ln

a1

K1

Hþ h*;yp

� ��

þ n lna1

K1

Hþ h*;yp

� �� ; ðB2Þ

where n = (a0 � a1)/a1. Taking the exponential of bothsides of equation (B2) and rearrangement of terms gives

a0

K0

H� h*;yp

� �� a1

K1

Hþ h*;yp

� �� n a1

K1

Hþ h*;yp

� �¼ 0

p ¼ 0; . . . ; 2P þ 1: ðB3Þ


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This equation is linearized and solved iteratively byevaluating the term [a1H

+(h*, yp)/K1]n using the values

of the coefficients at the previous iteration. As such, (B3)represents 2P + 1 linear equations. Combined with the 2P +1 equations for continuity of normal flow, this results in asystem of 4P + 2 linear equations, which is solved for eachiteration. After every iteration, continuity of normal flow ismet exactly (within machine accuracy) at the collocationpoints along the boundary. The boundary condition ofcontinuity of pressure head improves during every iteration.The iterative process is continued until the difference inpressure head across the boundary is smaller than aspecified tolerance. Once this tolerance is reached, the nextanalytic element is solved. Extensive experimentation hasshown that this procedure is very stable and that thedifference in pressure head across the boundary decreasesby an order of magnitude every 2 to 3 iterations. It is notedthat the number of iterations needed to achieve convergencereduces between consecutive cycles as the coefficients ofthe previous cycle are used as a starting point. For extremecontrasts in hydraulic conductivity (a decrease of six ordersof magnitude, for example) the procedure may break downas negative potentials are computed at collocation pointsduring the iteration process. An alternative solution methodshould be developed to solve those extreme cases, or theyshould be modeled as impermeable.

[42] Acknowledgment. This research was funded in part by USDAgrant 2002-34393-12117.

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��M. Bakker, Department of Biological and Agricultural Engineering,

University of Georgia, Athens, GA 30602, USA. ([email protected])J. L. Nieber, Department of Biosystems and Agricultural Engineering,

University of Minnesota, St. Paul, MN 55455, USA. ([email protected])

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Two-dimensional steady unsaturated flow through embedded elliptical layers

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