supplemental materials modeling flow into horizontal wells ... · forchheimer model that utilizes...

Supplemental Materials

Modeling Flow into Horizontal Wells in aDupuit-Forchheimer Model

Henk Haitjema, Sergey Kuzin, Vic Kelson, and Daniel Abrams

August 8, 2011

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Original publication

“Modeling Flow into Horizontal Wells in a Dupuit-Forchheimer Model”, Henk Haitjema,Sergey Kuzin, Vic Kelson, and Daniel Abrams, Ground Water, 2010, in press.

Abstract

Horizontal wells or radial collector wells are used in shallow aquifers to enhance water with-drawal rates. Groundwater flow patterns near these wells are three-dimensional, but difficultto represent in a 3D numerical model because of the high degree of grid refinement needed.However, for the purpose of designing water withdrawal systems it is sufficient to obtainthe correct production rate of these wells for a given drawdown. We developed a Cauchyboundary condition along a horizontal well in a Dupuit-Forchheimer model. Such a steadystate 2D model is not only useful for predicting groundwater withdrawal rates, but also forcapture zone delineation in the context of source water protection. A comparison of ourDupuit-Forchheimer model for a radial collector well with a three-dimensional model yieldsa 5% lower production rate. Particular attention is given to horizontal wells that extendunderneath a river. A comparison of our approach with a 3D solution for this case yieldssatisfactory results, at least for moderate to large river bottom resistances.

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Summary of results

In the next two sections the primary results from the original publication are summarizedwith references to appendices not included in the original publication. A third sectioncontains a validation of the case for a horizontal well (lateral) underneath a river, which wasalso not included in the original publication.

Horizontal well in a confined aquifer

The drawdown (φL − φw) at a infinitely long horizontal well in a confined aquifer, whereresistance to vertical flow is included (2D flow in the vertical plane), is given by,

φL − φw = − σ

2πkln

2 sin(π2rwH

) sin(π h+rw/2H

)

cosh(π LH

)− cos(π hH

)

(1)

where k and H are the hydraulic conductivity and thickness of the aquifer, respectively. Theparameter rw [L] is the radius of the horizontal well, which must be small compared to H,for instance rw < 0.1H. In practice this means that aquifers should not be thinner thanabout 2 to 3 meters. The horizontal well is located at a distance h above the aquifer baseand the head φL occurs at a distance L from the well. The head in the horizontal well isφw. The analysis that leads to (1) is similar to that in Strack (1989), page 375 - 377 andprovided in Appendix A. Comparison of the result in (1) with Dupuit-Forchheimer flow toa stream yields a fictitious stream resistance of

cls = − w

2πkln

4 sin

(π

2

rwH

)sin

(πh+ rw/2

H

)(2)

The width w may be selected arbitrarily. With (2) applied as bottom resistance to a streamin a Dupuit-Forchheimer model, this stream may be interpreted as a horizontal well, yieldingapproximately the same production rate as a horizontal well in a three dimensional model.

Horizontal well underneath a river with bottom resistance

A solution for the case of an infinite long horizontal well (lateral) underneath an infinitelylong (and wide) river is given by Bruggeman (1999, case 356.08 on page 311), which leadsto the following expression for the drawdown (φ0 − φw):

φ0 − φw =σ

k

∞∑n=0

1

αn

cos(αn

hH

)1 + ε

α2n+ε2

cos

(αnh+ rwH

) (3)

where φ0 is the river stage, φw the head in the horizontal well, and where αn are the rootsof:

α tanα = ε (4)

The parameter ε is defined as:

ε =H

k cb(5)

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with cb the resistance to flow in the river bottom sediments, defined as:

cb =δ

ks(6)

where δ [m] and ks are the thickness and the vertical hydraulic conductivity of these sedi-ments, respectively. The roots of (4) are periodic and may be written as:

αn = α + nπ (7)

where α is the principal value of (4). A graph for α as a function of ε is provided in AppendixB.

A resistance to vertical flow in the horizontal plane c2 is obtained as:

c2 =w

k

∞∑n=0

1

αn

cos(αn

hH

)1 + ε

α2n+ε2

cos

(αnh+ rwH

) (8)

The resistance c2 includes the resistance to (a) vertical flow through the river bottom, (b)horizontal flow in the aquifer, and (c) vertical flow in the aquifer. When modeling a horizontalwell underneath a river in a Dupuit-Forchheimer model we are already including the first twotypes of resistances. This resistance, therefore, must be subtracted from c2 when defining theCauchy boundary condition. This leads to the following resistance to be applied as bottomresistance in a Dupuit-Forchheimer model:

cls = c2 −λw

2kH(9)

where λ is a characteristic leakage length defined as:

λ =√kHcb (10)

Some graphs for c2 are provided in Appendix B.

Validation of equation (9)

The use of (9) may be validated by use of a high resolution MODFLOW model with ahorizontal well (lateral) that extends underneath the entire width of a river and with no-flowboundaries coinciding with these river boundaries. In such a setup the flow is forced to betwo-dimensional in the vertical plane, perpendicular to the well, anywhere underneath theriver. This situation is consistent with the conditions for which (9) was developed. The riverstage and head in the horizontal well are φ0 = 30 meters and φw = 20 meters, respectively.The radius of the horizontal well is rw = 0.2553 meters, while the aquifer thickness is H = 24meters. The well is midway the aquifer (h/H = 0.5), which has a hydraulic conductivity ofk = 141.67 meters per day.

Table 1 presents the total flow [m3/day] in the lateral obtained in three differ-ent ways. Firstly, Qexact is obtained by use of (3). Secondly, QMF has been obtained from across-sectional MODFLOW model representing the aquifer underneath a stretch of the river

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cb Qexact QMF error QGF error

[days] [m3/day] [m3/day] [%] [m3/day] [%]

0.01 110,313.6 108,226.5 -1.9 108,362.0 -1.80.1 90,713.0 90,005.5 -0.8 89,533.2 -1.31 50,388.8 50,054.0 -0.7 50,287.8 -0.210 19,841.7 - - 19,915.4 0.37

Table 1: Comparison of total inflows into a lateral underneath a stream for the case of alateral that extends underneath the entire river that is bounded by no-flow boundaries toforce flow perpendicular to the lateral.

of length 480 meters with cells that are 0.1 meters on a side remote from the lateral. Nearthe lateral the cells have been refined down to 0.00625 meters on a side. Finally, QGF hasbeen obtained by use of the Dupuit-Forchheimer model GFLOW using a no-flow boundaryalong a 480 meter long stream stretch for values of cb that are between 0.01 and 1 day anda 1600 meter stream stretch for the case of cb = 10 [days]. In this manner we ensured thatfor all cases the stream on either side of the lateral is longer than 4λ thus capturing streamseepage sufficiently far away, see the original publication. The GFLOW model is a Dupuit-Forchheimer model that utilizes (9) with (8) and (10). No results are presented for the riverbottom resistances of cb = 10, 000 [days] and cb =∞ [days]. For the first case the length ofthe river would have to be about 20,000 meters in each direction from the lateral in order tocapture enough of the river seepage, which is cumbersome to implement. For the latter casethe river is completely separated from the aquifer, hence there would be no flow at all intothe lateral. We also omitted the MODFLOW run for cb = 10 [days], which would require anunduly long river section in the high resolution MODFLOW model. The percent error in thecolumn that follows QMF is (QMF − Qexact)/Qexact ∗ 100%, while the values in the columnfollowing QGF are defined as (QGF −Qexact)/Qexact ∗ 100%. It is seen that the MODFLOWcross-sectional model as well as the GFLOW model using the cls values obtained from (9)are quite accurate.

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Figure 1: A confined aquifer with a lateral at a distance h above its base (z-plane) is mappedonto a reference plane (ζ-plane).

Appendix A: Derivation of equation (1)

The analysis in this appendix is similar to that presented by Strack (1989) page 375 - 377.Figure 1 depicts a vertical section over an aquifer with a small diameter lateral at a distanceh above the aquifer base; the z-plane on the left. The upper half of the ζ-plane on the rightin Figure 1 is a conformal map of this z-plane. The aquifer top and bottom, which formno-flow boundaries in the z-plane, are mapped along the real ξ-axis in the complex ζ-planewith the numbering along these boundaries detailing which points map where. The mappingfunction is given by Strack (1989) equation 32.30:

ζ = eπz/H (11)

where z = x + iy and ζ = ξ + iη, see Figure 1. The lateral with discharge σ [m2/day] islocated at zw = ih and maps onto ζw = exp(iπh/H). Half of the flow toward the lateral inthe z-plane comes from +∞ (between points 2a and 2b), which corresponds to infinity inthe upper half-plane (ζ-plane). The other half of the flow comes from −∞ in the z-plane(between the points 4a and 4b), which corresponds to the origin in the ζ-plane. The latterwater is supplied by a point source with strength σ/2 at ζ = 0 (not shown in Figure 1). Tomaintain the no-flow condition along the horizontal axis in the reference plane, η = 0, animage lateral at ζw = exp(−iπh/H) and an image of the source with strength σ/2 at ζ = 0are added. The resulting complex potential Ω(ζ) is given by:

Ω(ζ) =σ

2πln

(ζ − ζw)(ζ − ζw)

ζ+ Ω0 (12)

where Ω0 is an integration constant.

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The specific discharge potential Φ [m2/day], defined as the product of thehydraulic conductivity k and the head φ,

Φ = kφ (13)

is the real part of Ω (Strack 1989), so that with (11) and (12):

Φ(z) = <σ

2πln

(eπz/H − eiπh/H)(eπz/H − e−iπh/H)

eπz/H

+ Φ0 (14)

where Φ0 is the real part of Ω0. If the potential at some point z = L is given as Φ = ΦL

expression (14) becomes:

Φ(z) = <σ

2πln

(eπz/H − eiπh/H)(eπz/H − e−iπh/H)eπL/H

eπz/H(eπL/H − eiπh/H)(eπL/H − e−iπh/H)

+ ΦL (15)

In (15) constants (independent of z) have been added to the argument of the logarithm suchthat it becomes 1 at z = L, so that Φ = ΦL at z = L.

The potential inside (and just outside) the lateral is denoted by Φw = kφw,where φw is the head inside the lateral. For small diameter laterals the head will be nearlyconstant along the perimeter of the lateral. In writing (12) it was tacitly assumed thatthe lateral could be seen as a point in the z-plane, mapping to a point in the ζ-plane.Since we will apply the analysis to laterals of finite diameter, the analysis presented here isapproximate. The potential at a collocation point zw, selected on top of the lateral,

zw = i(h+ rw) (16)

is set to Φw. Substituting (16) into (15), thus replacing z by zw, results in the followingexpression for the potential Φw:

Φw = <

σ

2πln

(eiπh+rwH − eiπ hH )(eiπ

h+rwH − e−iπ hH )eπ

LH

eiπh+rwH (eπ

LH − eiπ hH )(eπ

LH − e−iπ hH )

+ ΦL (17)

The argument of the logarithm in (17) may be rewritten as follows. Referring to the argumentas A:

A =eiπ

hH ei

π2rwH

(eiπ2rwH − e−iπ2 rwH

) (eiπ( h

H+ rw

2H) − e−iπ( h

H+ rw

2H))eiπ2rwH eπ

LH

eiπ( hH

+ rwH

)eπLH

(1− e−π( L

H−i h

H)) (

1− e−π( LH

+i hH

))eπ

LH

(18)

The last exponential term in the numerator cancels the last exponential term in the denom-inator. The two first exponential terms in the numerator with what is now the last termin the numerator cancel the first exponential term in the denominator. Rearranging termsleads to:

A =


) (eiπ( h

H+ rw

2H) − e−iπ( h

H+ rw

2H))

(eπ2( LH−i h

H) − e−π2 ( L

H−i h

H)) (eπ2( LH

+i hH

) − e−π2 ( LH

+i hH

)) (19)

The denominator may be multiplied out and rearranged to give:

A =


) (eiπ( h

H+ rw

2H) − e−iπ( h

H+ rw

2H))

(eπ

LH + e−π

LH

)−(eiπ

hH + e−iπ

hH

) (20)

7

Dividing the expressions between parentheses in the numerator by 2i and those in the de-nominator by 2 yields:

A =2 sin(π

2rwH

) sin(π h+rw/2H

)eiπ

cosh(π LH

)− cos(π hH

)(21)

Replacing the argument of the logarithm in (17) by (21) and writing the expression as adrawdown using (13) yields:

φ0 − φw = − σ

2πkln

2 sin(π2rwH

) sin(π h+rw/2H

)

cosh(π LH

)− cos(π hH

)

(22)

where use has been made of (13).

8

Appendix B: Graphs for α and c2.

Evaluation of (8) is complicated by the infinite series. For the reader’s convenience someresults are presented in dimensionless form in Figure 3 where k c2/w (kc/w on the graph) isplotted versus ε for various values of rw/H (r/H on the graph) and h/H. The graph willonly provide approximate results, however. For more accurate values of c the value of α maybe read from the graph in Figure 2 and used with (7) in the infinite sum in (8) which may beaborted after sufficient accuracy is obtained. For further accuracy the implicit equation (4)may be evaluated instead of using the graph in Figure 2. For small values of ε it was foundthat several thousand Newton-Raphson iterations were required to obtain accurate valuesfor α.

Additional graphs for the resistance c2, for various values of k, H, and streambottom resistance cb follow in Figure 4 through Figure 10. The parameters in these graphsare presented in dimensionless form as kc∗/w, where c∗ = c2, as kc/H (= 1/ε), where c = cb,and as r/H, where r = rw.

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Figure 2: Principal values for α as a function of ε to be used in equation (7)

.

10

Figure 3: Resistance c2 due to 3D flow underneath a stream with a resistance layer is plottedas the dimensionless parameter k c/w (c = c2) versus ε (= H/kcb) . The dotted lines referto the case of h/H = 0.2, while the solid lines refer to the case h/H = 0.5.

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Figure 4: Resistance c2 due to 3D flow underneath a stream with a resistance layer withresistance c = cb. The parameters are: hydraulic conductivity k, stream bottom resistancec = cb, aquifer thickness H, lateral radius r = rw, lateral elevation h, line-sink width w, andresistance c∗ = c2. The curves are for r/H =0.001, 0.002, 0.004, 0.006, 0.008, 0.01, 0.02,0.04, 0.06, 0.08 and 0.1

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