Approximate Solution for a Transient Hydraulic HeadDistribution Induced by a Constant-Head Test at a Partially

Penetrating Well in a Two-Zone Confined AquiferShaw-Yang Yang1; Ching-Sheng Huang2; Chih-Hsuan Liu3; and Hund-Der Yeh, Aff.M.ASCE4

Abstract: A mathematical model describing the transient hydraulic head distribution induced by constant-head pumping/injection at apartially penetrating well in a radial two-zone confined aquifer is a mixed-type boundary value problem. The analytical solution of themodel is in terms of an improper integral with an integrand having a singularity at the origin. The solution should rely on numerical methodsto evaluate the integral and handle the problems of convergence and singularity. This study aims at developing a new approximate solutiondescribing the transient hydraulic head distribution for a constant-head test (CHT) at a partially penetrating well in the aquifer. This approxi-mate solution is acquired based on a time-dependent diffusion layer approximation proposed in the field of electrochemistry. The diffusionlayer can be analogous to the radius of influence in the area of well hydraulics. The approximate solution is in terms of modified Besselfunctions for aquifers with a partially penetrating well and can reduce to a simpler form in terms of a natural logarithmic function for the caseof well full penetration. The predicted hydraulic heads from the present approximate solution are compared with those estimated by theLaplace-domain solution of the model. The result shows that the predicted spatial head distributions are accurate in the formation zoneand fairly good in the skin zone. In addition, the present solution gives an accurate temporal head distribution at a specific location whenthe radius of influence is far away from the observation wells. This newly developed approximate solution has advantages of easy computingand good accuracy from practical viewpoint, and thus is a handy tool to evaluate temporal and spatial hydraulic head distributions for theCHT. DOI: 10.1061/(ASCE)HY.1943-7900.0000884. © 2014 American Society of Civil Engineers.

Author keywords: Skin zone; Finite Fourier cosine transform; Radius of influence; Analytical solution.

Introduction

A constant-head test (CHT) is one of the methods used inaquifer site characterization, especially suitable to apply to low-permeability aquifers (e.g., Jones et al. 1992; Jones 1993). The dataobtained from the test is generally analyzed to determine aquiferparameters such as hydraulic conductivity and specific storage.During the test, the well water level is maintained constant, andthe wellbore inflow rate is recorded. The aquifer parameters canthen be estimated by fitting the wellbore flow rates predicted bya solution to the observed test data. Over the last few decades, therehave been a number of research works regarding the developmentof analytical solutions for the CHT executed at a fully penetratingwell (e.g., Carslaw and Jaeger 1940; Van Everdingen and Hurst1949; Jacob and Lohman 1952; Hantush 1964; Peng et al. 2002;Yeh and Chang 2013). Carslaw and Jaeger (1940) developed an

analytical solution describing a temporal and spatial distributionof temperature in an infinite medium, and the solution can be analo-gous to the groundwater flow problem. Van Everdingen and Hurst(1949) derived two analytical solutions for a terminal pressure andflow rate in finite and infinite oil reservoirs. Jacob and Lohman(1952) obtained an analytical solution describing a wellbore inflowrate for a CHT in a confined aquifer and developed a formula toestimate the storage coefficient and transmissivity. Hantush (1964)presented a transient drawdown solution for a CHT in a confinedaquifer system. Peng et al. (2002) derived analytical solutions for ahydraulic head and flow rate subject to the constant-head boundarycondition. However, their solution is in terms of an improper inte-gral with the integrand having a singularity at the origin and shouldrequire numerical methods to evaluate the integral and handle theproblems of convergence and singularity.

The test well may partially penetrate an aquifer when the aquiferthickness is large or the well is installed to monitor the groundwatercontamination at a specific depth interval. The screen portion in thepartially penetrating well for a CHT is considered as a constant-head boundary (or Dirichlet boundary) while the unscreenedportion is treated as a no-flow boundary. The groundwater flowinduced by constant-head pumping in a partially penetratingwell is therefore a mixed boundary value problem. Duffy’s book(Duffy 2008) offers a good coverage for solving transient mixedboundary value problems, which were absent in both Sneddon(1966) and Fabrikant (1991). Chang and Yeh (2009) used the per-turbation method and dual series equation to obtain a semianalyt-ical solution for a CHTwith a screen installed at the upper or lowerportion of the well in a confined aquifer with a finite thickness.Later, Chang and Yeh (2010) further considered a well screen in-stalled at any portion of the well and developed a semianalytical

1Professor, Dept. of Civil Engineering, Vanung Univ., No.1 VannungRd., Chungli, Taoyuan 320, Taiwan. E-mail: shaoyang@vnu.edu.tw

2Postdoctoral Researcher, Institute of Environmental Engineering,National Chiao Tung Univ., No.1001 University Rd., Hsinchu 300, Taiwan.E-mail: cshuang0318@gmail.com

3M.S. Student, Institute of Environmental Engineering, National ChiaoTung Univ., No.1001 University Rd., Hsinchu 300, Taiwan. E-mail:jhihsyuan.liu@gmail.com

4Professor, Institute of Environmental Engineering, National ChiaoTung Univ., No.1001 University Rd., Hsinchu 300, Taiwan (correspondingauthor). E-mail: hdyeh@mail.nctu.edu.tw

Note. This manuscript was submitted on August 11, 2013; approved onFebruary 7, 2014; published online on March 26, 2014. Discussion periodopen until August 26, 2014; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Hydraulic Engineer-ing, © ASCE, ISSN 0733-9429/04014030(9)/$25.00.

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solution for a CHT by means of a triple series equation method.Recently, Yeh and Chang (2013) reviewed the recent researchesabout a CHT for the cases of both fully and partially penetratingwells.

The properties of an aquifer formation near a wellbore may bechanged due to the well construction and/or well development; con-sequently, a homogeneous aquifer may then become a two-zoneaquifer. A negative skin is referred to as a disturbed zone with alarger transmissivity when compared with the undisturbed forma-tion zone. Contrarily, a positive skin is referred to as a zone with asmaller transmissivity as compared with the formation zone. Thewell hydraulics for a CHT with considering the skin zone weresolved analytically by several researches utilizing different assump-tions (e.g., Novakowski 1993; Cassiani and Kabala 1998; Cassianiet al. 1999; Yang and Yeh 2005, 2006). Novakowski (1993) usedthe methods of Laplace transform and finite Fourier cosine trans-form to derive a semianalytical solution for a transient wellboreflow rate in a partially penetrating well with a finite-thickness skin.They drew the type curves accounting for the influences of theskin and well penetration. Cassiani et al. (1999) assumed thatthe skin thickness is infinitesimally small and treated its effectas a skin factor. They derived a Laplace-domain solution usingthe dual integral equation method for a hydraulic head distributiondue to constant-head pumping at a partially penetrating well in aninfinite confined aquifer in both horizontal and vertical directions.They assigned a specific head on the well face and used the no-flowboundary condition along the casing. Their solution is suitable toapply if the screen length is significantly shorter than the aquiferthickness or the packer test with the horizontal boundary being faraway enough from the test well. Chang and Chen (2002) used thesame mathematical model as Cassiani et al. (1999) but assumed theaquifer thickness is finite and tackled the well skin effect as a skinfactor. They portioned the well screen length into many segmentsand considered the wellbore flux entering through the well screenas unknown. The accuracy of their solution may depend on the sizeand number of the segment. Yang and Yeh (2005) provided a semi-analytical solution for a CHT performed at a partially penetratingwell and considered the effect of a finite-thickness skin. They ini-tially assumed an unknown uniform flux along the well screen,treated the flux as a Neumann boundary condition, and then deter-mined the unknown flux using the constant head condition. Thisnew solution has better accuracy as compared with Novakowskisolution (1993). Later, Yang and Yeh (2006) derived an analyticalsolution for describing the hydraulic head distribution induced by aCHT in a patchy confined aquifer with a fully penetrating well.They presented a numerical approach including a root-searchscheme, the Gaussian quadrature, and the Shanks method to evalu-ate the time-domain solution. Barua and Bora (2010) developed asteady/quasi-steady model and derived an analytical solution forbottom flow at a partially penetrating well with a skin zone in aconfined aquifer. Their solution is useful to analyze the flow behav-ior near the pumping well in an artesian aquifer.

The semianalytical and analytical solutions for the CHT men-tioned above are generally laborious in calculation. In the past, sev-eral studies had attempted to develop approximate solutions for theCHT problems in different ways. Haitjema and Kraemer (1988)used several point sinks to represent a partially penetrating welland developed an approximate solution to describe the three-dimensional flow. Wilkinson and Hammond (1990) used a pertur-bation technique to solve an approximate solution for the transientpressure at a partially penetrating well in a confined aquifer witha semi-infinite thickness. Mishra and Guyonnet (1992) employedthe Boltzman transformation technique to solve the problem andobtained an approximate drawdown solution. Bakker (2001)

considered an aquifer divided into several layers for heterogeneityand provided an approximate solution describing steady-statethree-dimensional flow induced by a partially penetrating wellin the aquifer. Perrochet (2005) developed a simple analyticalexpression to approximate the well function GðαÞ of Jacob andLohman (1952) using ln ð1þ ffiffiffiffiffiffi

παp Þ−1 with α defined as dimen-

sionless time and calculated the transient wellbore inflow rate.Based on the work of Perrochet (2005), Renard (2005) providedan approximate solution using a weighted average between theearly- and late-time asymptotes and analyzed the maximum relativedifference between the exact and approximate dimensionlessdischarges in a pumping well.

The solutions mentioned above are summarized in Table 1. Theauthors classified the solutions into a single confined aquifer with-out a skin zone and an aquifer system with skin and formationzones. The solutions are further categorized according to the wellpenetration, solution description, and mathematical technique.

Interestingly, some approximate solutions reported in the areaof electrochemistry (e.g., Fang et al. 2009; Bieniasz 2011; Phillipsand Mahon 2011) had a mathematical model exactly the same asthe CHT problem. The diffusion equation at the microcylinderelectrode is similar to the groundwater flow equation for the CHTin a groundwater problem. Fang et al. (2009) developed an approxi-mate transient solution based on a time-dependent diffusion layerapproximation. They extended the steady-state solution obtainedfrom a steady-state diffusion equation with a finite boundary con-dition for the cylinder electrode to a transient solution by intro-ducing the time-dependent diffusion layer. The predicted resultsfrom their approximate solution match well with experimental data.Recently, Bieniasz (2011) presented a procedure to compute a chro-noamperometric current at cylindrical wire electrodes. The currentis in fact analogous to a wellbore flux in the CHT. His proceduredepends on a local polynomial approximation over the entire timedomain, and his results have accuracy to 14 to 15 decimals. Morerecently, Phillips and Mahon (2011) desingularized the improperintegral in a wellbore flux solution and evaluated the obtained in-tegral using the Laguerre-Gauss quadrature. Their estimated resultsare correct to 10 decimal digits.

The existing analytical solutions describing the hydraulic headdistribution in a single or two-zone confined aquifer system for aCHT are all in complicated forms and difficult to evaluate theirnumerical values. The purpose of this paper is therefore to developa simple approximate solution for the temporal hydraulic head dis-tribution induced by a CHTat a partially penetrating well in a radialtwo-zone confined aquifer system. A steady-state solution is firstderived from the state-steady groundwater flow equation with theremote boundary represented by the radius of influence using thefinite Fourier cosine transform. The approximate transient solutionis then obtained by replacing the radius of influence with a functionintroduced by Fang et al. (2009) for a time-dependent diffusionlayer. This approximate solution can reduce to a simpler form interms of a natural logarithmic function for a two-zone confinedaquifer system under a fully penetrating well condition. The resultsestimated from the present approximate solution are compared withthose predicted by Yang and Yeh solution (2005).

Methodology

Mathematical Model

Fig. 1 shows a schematic diagram of an aquifer configuration for aCHT performed at a partially penetrating well in a radial two-zoneconfined aquifer system. The symbols are defined as follows: r is a

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radial distance from the center of the test well; z is an elevationfrom the lower impermeable layer; rw is the radius of the well;rs is the outer radius of the skin zone (or skin thickness); R isthe radius of influence; a and b are lower and upper elevationsof the well screen, respectively; L is the aquifer thickness; h0 isa constant well water level at any time; the hydraulic head fromthe potentiometric surface in the skin zone is h1 and in the forma-tion zone is h2. Note that the absolute value of the hydraulic headrepresents the drawdown or the change in the hydraulic head sincethe reference datum is set right at the initial potentiometric surface.The notations mentioned above are listed in Notation List. Themathematical model describing the hydraulic head distributionin the two-zone confined aquifer is developed based on followingassumptions: (1) the aquifer is homogeneous and of finite extentwith a constant thickness and (2) the initial piezometric head isconstant and uniform over the whole aquifer.

Define the dimensionless parameters and variables as

h1D ¼ h1h0

; h2D ¼ h2h0

; h0D ¼ 1; rD ¼ rrw

;

zD ¼ zrw

; rSD ¼ rSrw

; RD ¼ Rrw

; κ1 ¼Kz1

Kr1;

κ2 ¼Kz2

Kr2; aD ¼ a

rw; bD ¼ b

rw; LD ¼ L

rw; γ¼Kr2

Kr1ð1Þ

where the subscripts 1 and 2 are the skin and formation zones, re-spectively; the subscript D is a dimensionless symbol; and Kr andKz are the radial and vertical hydraulic conductivities, respectively.The dimensionless notations are also listed in Notation List.

The governing equations describing steady-state dimensionlesshydraulic head distributions in the skin and formation zones forfinite aquifers are expressed, respectively, as (Yang and Yeh 2005)

∂2h1D∂r2D þ 1

rD

∂h1D∂rD þ κ1

∂2h1D∂z2D ¼ 0; 1 ≤ rD ≤ rSD ð2Þ

and

∂2h2D∂r2D þ 1

rD

∂h2D∂rD þ κ2

∂2h2D∂z2D ¼ 0; rSD ≤ rD ≤ RD ð3Þ

The dimensionless hydraulic head in the formation zone at theremote boundary is assumed to maintain zero according to thereference datum and can be expressed as

h2D ¼ 0 at rD ¼ RD ð4Þ

At the interface between the skin and formation zones, thecontinuity requirements for the hydraulic head and mass fluxare, respectively,

h1D ¼ h2D at rD ¼ rSD ð5Þand

∂h1D∂rD ¼ γ

∂h2D∂rD at rD ¼ rSD ð6Þ

where γ, the conductivity ratio, has been defined in Eq. (1).

Table 1. Classification of the Solutions Involving CHT in a Confined Aquifer

ReferencesWell

penetration Solution description Mathematical technique

Single aquifer without skin zoneJacob and Lohman (1952)a Fully Discharge rate Laplace transformHantush (1964)a Fully Drawdown Laplace transformHaitjema and Kraemer (1988)b Partially Piezometric head Equation describing discharge potentialWilkinson and Hammond (1990)b Partially Water pressure Perturbation techniqueMishra and Guyonnet (1992)b Fully Drawdown Boltzman transformationBakker (2001)b Partially Piezometric head in a layered aquifer Equation describing discharge potentialPerrochet (2005)b Fully Discharge rate Straightforward logarithmic functionRenard (2005)b Fully Discharge rate Weighted mean between early and late time asymptotesChang and Yeh (2009)c Partially Drawdown and discharge rate Perturbation method and dual series equationChang and Yeh (2010)c Partially Drawdown and discharge rate Triple series equation

Aquifer system with skin and formation zonesNovakowski (1993)a Partially Wellbore flow rate Laplace transform and finite Fourier cosine transformCassiani et al. (1999)c Partially Drawdown Dual integral equationYang and Yeh (2005)c Partially Hydraulic head and wellbore flow rate Laplace transformYang and Yeh (2006)a Fully Hydraulic head Laplace transform and Bromwich integralBarua and Bora (2010)a Partially Hydraulic head Fourier cosine transform

aTime-domain solution.bApproximate solution.cLaplace-domain solution.

Fig. 1. Schematic diagram of a radial two-zone confined aquifersystem for constant head pumping at a partially penetrating well

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The average of dimensionless hydraulic heads along the wellscreen is equal to the dimensionless well water level, i.e.,

1 ¼ 1

bD − aD

ZbD

aD

h1DdzD at rD ¼ 1 ð7Þ

Note that the unity on the right-hand side (RHS) of Eq. (7) rep-resents the dimensionless water level. At the well, the flux inducedby the CHT is considered as uniformly distributed over the screen.The dimensionless wellbore flow rate is therefore assumed as

−∂h1D∂rD ¼ qD at rD ¼ 1 and aD ≤ zD ≤ bD ð8Þ

where qD ¼ q=2πðb − aÞKr1h0. The no-flow condition over thewell casing is denoted as

∂h1D∂rD ¼ 0 at rD ¼ 1 and zD < aD or zD > bD ð9Þ

Steady-State Head and Wellbore Inflow Rate Solutions

The solutions of the model in Fourier domain can be obtained afterapplying finite Fourier cosine transforms to zD in Eqs. (2)–(6), (8),and (9). Detailed derivation for the solution is given in Appendix.The steady-state solution obtained by taking inverse Fourier trans-forms for the dimensionless hydraulic heads in the skin and forma-tion zones is

h1DðrD; zDÞ ¼ qD

�ϕ1oðrDÞ þ 2

X∞n¼1

ϕ1nðrD; nÞ cosðωzDÞ�

ð10Þ

and

h2DðrD; zDÞ ¼ qD

�ϕ2oðrDÞ þ 2

X∞n¼1

ϕ2nðrD; nÞ cosðωzDÞ�

ð11Þ

with

ϕ1oðrDÞ ¼φγ

�γ ln

�rDrSD

�þ ln

�rSDRD

��ð12Þ

ϕ1nðrD; nÞ ¼δ

nπλ1

A3A1

ffiffiffiffiffiκ1

p þ A4A2

ffiffiffiffiffiκ2

pγ

A5A1

ffiffiffiffiffiκ1

p þ A6A2

ffiffiffiffiffiκ2

pγ

ð13Þ

ϕ2oðrDÞ ¼φγln

�rDRD

�ð14Þ

ϕ2nðrD; nÞ ¼ − δnπω

A7A8

A5A1

ffiffiffiffiffiκ1

p þ A6A2

ffiffiffiffiffiκ2

pγ

ð15Þ

A1 ¼ I0ðrSDλ2ÞK0ðRDλ2Þ − I0ðRDλ2ÞK0ðrSDλ2Þ ð16Þ

A2 ¼ I1ðrSDλ2ÞK0ðRDλ2Þ þ I0ðRDλ2ÞK1ðrSDλ2Þ ð17Þ

A3 ¼ I1ðrSDλ1ÞK0ðrDλ1Þ þ I0ðrDλ1ÞK1ðrSDλ1Þ ð18Þ

A4 ¼ I0ðrDλ1ÞK0ðrSDλ1Þ − I0ðrSDλ1ÞK0ðrDλ1Þ ð19Þ

A5 ¼ I1ðλ1ÞK1ðrSDλ1Þ − I1ðrSDλ1ÞK1ðλ1Þ ð20Þ

A6 ¼ I0ðrSDλ1ÞK1ðλ1Þ þ I1ðλ1ÞK0ðrSDλ1Þ ð21Þ

A7 ¼ I0ðRDλ2ÞK0ðrDλ2Þ − I0ðrDλ2ÞK0ðRDλ2Þ ð22Þand

A8 ¼ I1ðrSDλ1ÞK0ðrSDλ1Þ þ I0ðrSDλ1ÞK1ðrSDλ1Þ ð23Þwhere λi ¼ ω

ffiffiffiffiffiκi

p, i ¼ 1 or 2; ω ¼ nπ=LD, δ ¼ sinðωbDÞ−

sinðωaDÞ; Iμð·Þ and Kμð·Þ are the modified Bessel functions ofthe first and second kinds with order μ, respectively; and φ ¼ðbD − aDÞ=LD is the penetration ratio of the well screen lengthover the aquifer thickness. Notice that the right-hand parts ofEqs. (10) and (11) have two terms; the first term is the solutionconsidering a full penetrating well and the second term is a simpleseries accounting for the effect of a well partial penetration.

The unknown qD in Eqs. (10) and (11) is determined based onthe boundary condition of Eq. (7). Substituting Eq. (10) into Eq. (7)and integrating the result with respect to zD yields

qD ¼�ϕ1oð1Þ þ

2

bD − aD

X∞n¼1

δωϕ1nð1; nÞ

�−1ð24Þ

where the definitions of ω and δ are given in Notation List, andϕ1oð1Þ and ϕ1nð1; nÞ are the results of substituting rD ¼ 1 intoEqs. (12) and (13), respectively. Finally, the steady-state solutionfor dimensionless hydraulic heads in the skin and formation zonescan be obtained after substituting Eq. (24) into Eqs. (10) and (11).

Approximate Transient Head and Wellbore Inflow RateSolutions

Fang et al. (2009) proposed a time-dependent diffusion layer toachieve a transient diffusion for a chronoamperometric current.The diffusion layer is assumed dependent on both time t and dif-fusion coefficientD0, and expressed as

ffiffiffiffiffiffiffiffiffiffiπD0t

p. The diffusion equa-

tion describing the current and its current solution have the samemathematical forms as the groundwater flow equation and its well-bore solution for the CHT. By the analogy of the groundwater flowinduced by the CHT to the chronoamperometric current in electro-chemistry, the radius of influence R for the finite aquifer is replacedwith rw þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

πðKr2=Ss2Þtp

where Ss2 is the specific storage of theformation zone and the rw is because of considering the skin zone.The dimensionless radius of influence is written as

RD ¼ 1þ ffiffiffiffiffiffiffiffiπtD

p ð25Þwhere the dimensionless time tD is defined as Kr2t=ðSs2r2wÞ.Substituting Eq. (25) into Eqs. (12) and (14) results in, respectively

ϕ1oðrD; tDÞ ¼φγ

�γ ln

�rDrSD

�þ ln

�rSD

1þ ffiffiffiffiffiffiffiffiπtD

p��

ð26Þ

and

ϕ2oðrD; tDÞ ¼φγln

�rD

1þ ffiffiffiffiffiffiffiffiπtD

p�

ð27Þ

Note that A1, A2, and A7 also contain the parameter RD. Replac-ing RD by 1þ ffiffiffiffiffiffiffiffi

πtDp

, Eqs. (13) and (15) respectively, become

ϕ1nðrD; tD; nÞ ¼δ

nπλ1

A3A1ðtDÞ ffiffiffiffiffiκ1

p þ A4A2ðtDÞ ffiffiffiffiffiκ2

pγ

A5A1ðtDÞ ffiffiffiffiffiκ1

p þ A6A2ðtDÞ ffiffiffiffiffiκ2

pγ

ð28Þ

and

ϕ2nðrD; tD; nÞ ¼ − δnπω

A7ðtDÞA8

A5A1ðtDÞ ffiffiffiffiffiκ1

p þ A6A2ðtDÞ ffiffiffiffiffiκ2

pγ

ð29Þ

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Accordingly, Eqs. (10) and (11), the steady-state solution, be-come functions of time and may be written, respectively, as

h1DðrD; zD; tDÞ ¼ qDðtDÞ�ϕ1oðrD; tDÞ

þ 2X∞n¼1

ϕ1nðrD; tD; nÞ cosðωzDÞ�

ð30Þ

and

h2DðrD; zD; tDÞ ¼ qDðtDÞ�ϕ2oðrD; tDÞ

þ 2X∞n¼1

ϕ2nðrD; tD; nÞ cosðωzDÞ�

ð31Þ

where qDðtDÞ, the function of time, can be obtained by replacingthe variables ϕ1oð1Þ and ϕ1nð1; nÞ in Eq. (24) with ϕ1oð1; tDÞ andϕ1nð1; tD; nÞ, respectively. These two equations describing the tran-sient head distribution for the CHT in a two-zone aquifer systemwith a partially penetrating well will be compared with the semi-analytical solution presented in Yang and Yeh (2005).

Special Case 1: Fully Penetrating Well in Two-ZoneAquifers

Letting aD ¼ 0 and bD ¼ LD, one can obtain φ ¼ 1 and δ ¼ 0,which makes Eqs. (28) and (29) all equal to zero. Under thiscircumstance, the test well fully penetrates the confined aquifer.The dimensionless wellbore flow rate becomes

qDðtDÞ ¼γ

lnðr−1SDÞ þ ln½rSD=ð1þffiffiffiffiffiffiffiffiπtD

p Þ� ð32Þ

For a fully penetrating well, Eq. (30) for the dimensionlesshydraulic head in the skin zone reduces to

h1DðrD; tDÞ ¼ 1 − γ ln rDγ lnðrSDÞ þ ln½ð1þ ffiffiffiffiffiffiffiffi

πtDp Þ=rSD�

ð33Þ

Similarly, Eq. (31) for the dimensionless hydraulic head in theformation zone becomes

h2DðrD; tDÞ ¼ 1 − γ ln rSD þ lnðrD=rSDÞγ lnðrSDÞ þ ln½ð1þ ffiffiffiffiffiffiffiffi

πtDp Þ=rSD�

ð34Þ

Special Case 2: Fully Penetrating Well inHomogeneous Aquifers

When the skin zone is absent, the aquifer is homogeneous with theratio of γ equaling unity. Both Eqs. (33) and (34) can then reduce to

hDðrD; tDÞ ¼lnð1þ ffiffiffiffiffiffiffiffi

πtDp

=rDÞlnð1þ ffiffiffiffiffiffiffiffi

πtDp Þ ð35Þ

and Eq. (32) becomes

qDðtDÞ ¼ 1= lnð1þ ffiffiffiffiffiffiffiffiπtD

p Þ ð36Þ

It is interesting to note that the wellbore inflow rate representedby Eq. (36) is exactly the same as the one given in Perrochet (2005);yet, the procedure or methodology to derive the solution is totallydifferent.

Results and Discussion

The present approximate solution, Eqs. (30) and (31), is verifiedthrough the comparisons of the predicted temporal and spatial hy-draulic head distributions with the Yang and Yeh solution (2005)for a partially penetrating well. In the following analysis, the useof Eq. (25) to predict the transient flow behavior is examined.The validity of the present solution to the aquifer system subject tothe effect of the skin zone is investigated. Moreover, the temporaldistribution of the wellbore flow rate concerned in a CHT is ana-lyzed. In the analysis, the default values used in calculationsare rD ¼ 20, zD ¼ 100; tD ¼ 104; LD ¼ 200; rSD ¼ 5; φ ¼ 0.1;γ ¼ 0.1; κ1 ¼ 1;and κ2 ¼ 1.

The Situation of Using Eq. (25)

The spatial distributions of the hydraulic heads predicted by thepresent approximate solution and Yang and Yeh solution (2005)at tD ¼ 104, 106, and 108 are shown in Fig. 2. The radius of influ-ence RD, defined by Eq. (25), at various times is also indicated inthe figure. The results from both solutions agree well in the forma-tion zone but slightly deviate in the skin zone and near rD ¼ RD.Such discrepancies are due to the boundary effect. The present sol-ution considers an outer zero-head boundary which moves outwardwith time described by Eq. (25). The head distributions at rD ¼ 10,20, and 30 are underestimated because the zero-head boundary,specified at the RD, is not far away in early time as shown in Fig. 3.The figure shows the temporal distribution curve of RD and indi-cates that the differences in the predicted heads are very small andnegligible after tD ¼ 103 (i.e., RD > 57).

Effect of Considering Skin Zone on Solution Validity

The effect of aquifer heterogeneity on the validity of the presentapproximate solution is examined because the radius of influencedenoted as Eq. (25) was developed based on a homogeneousmedium (Fang et al. 2009). Positive and negative skins account

Fig. 2. Spatial distributions of hydraulic head predicted by the presentapproximate solution and Yang and Yeh solution (2005) at tD ¼ 104,106 and 108

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for the heterogeneity. The hydraulic heads predicted by the presentapproximate solution and Yang and Yeh solution (2005) under neg-ative skin (γ ¼ 0.1), no skin (γ ¼ 1), and positive skin (γ ¼ 10)conditions are displayed in Fig. 4(a) for the spatial distributionsand Fig. 4(b) for the temporal distributions. The figure shows thatthe values of γ seldom affects the validity of the present solution.In addition, the negative skin case produces the higher head thanthe other two cases at the same tD.

The effect of the skin width rSD on the validity of the presentsolution is also examined via the spatial head distributions underthe negative skin condition when rSD ¼ 2, 5, and 10 as demon-strated in Fig. 5. The rSD ranges from 2 to 10 in engineering prac-tice. The figure shows that the curve slopes at the interfacesbetween the skin and formation zones are discontinuous. It seems

reasonable to reveal that the predicted head in the formation zoneis valid even for the largest rSD.

Temporal Distribution of Wellbore Flow Rate

The temporal distributions of the wellbore flow rate qD defined byEq. (24) with (25) γ ¼ 1 (no skin) and LD ¼ 200 when φ ¼ 0.1,0.4, 0.8 and 1.0 (full penetration case) are shown in Fig. 6.The prediction from Yang and Yeh solution (2005) is also takento examine the validity of the present solution. The figure showsboth solutions agree very well to each φ over the entire period.

Fig. 3. Temporal distributions of hydraulic head predicted by thepresent approximate solution and Yang and Yeh solution (2005) atrD ¼ 10, 20 and 30

Fig. 4. (a) Spatial distributions; (b) temporal distributions of the hydraulic head predicted by the present approximate solution and Yang and Yehsolution (2005) under negative skin, no skin, and positive skin conditions

Fig. 5. Spatial distributions of hydraulic head predicted by thepresent approximate solution and Yang and Yeh solution (2005) forskin widths rSD ¼ 2, 5 and 10

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In addition, the flow rate decreases with increasing time andstabilizes as time is very large. The larger φ produces the smallerflow rate at a fixed tD.

Concluding Remarks

An approximate solution describing the transient head distributionin a radial confined aquifer has been developed for a CHT in apartially penetrating well with the presence of a skin zone. Thistransient approximate solution is derived based on a steady-statesolution and a formula regarding to a time-dependent diffusionlayer proposed by Fang et al. (2009). The solution can reduceto a very simple form in terms of a natural logarithmic functionif the well fully penetrates the aquifer. The authors have investi-gated the validity of the present solution by the comparison withYang and Yeh solution (2005). Following conclusions can bedrawn in regard to the applicability and/or accuracy of the presentsolution:1. The predicted head will be underestimated if the observation

well is located near the radius of influence to the boundaryeffect; and

2. The present solution generally gives accurate and fairly goodspatial head distributions in the formation and skin zones,respectively, when the radius of influence is far away fromthe observation well.

Based on the above conditions, the present approximate solutionhas the advantages of easy calculation and good accuracy fromengineering viewpoint, and can be used as a convenient tool toevaluate the temporal and spatial head distributions in the aquifersystem and wellbore flow rate for a CHT.

Appendix. Derivation of Steady-State Solution

The solutions for the dimensionless hydraulic head within the skinand formation zones are developed by applying finite Fourier co-sine transform with respect to the spatial variable zD. The definitionof the transform is given below (Kreyszig 1993)

Fc½hDðzDÞ� ¼ h̄DðnÞ ¼Z

LD

0

hDðzDÞ cos�nπLD

zD

�dzD ð37Þ

where n is an integer from 0; 1; 2; : : :∞. The transform has theproperty of

F

�d2hDðzDÞ

dz2D

�¼ ð−1Þn dhDðzDÞ

dzD

����zD¼LD

− dhDðzDÞdzD

����zD¼0

− n2π2

L2D

h̄DðnÞ ð38Þ

The formula for the inverse transform is defined as

hDðzDÞ ¼1

LDh̄Dð0Þ þ

2

LD

X∞n¼1

h̄DðnÞ cos�nπLD

zD

�ð39Þ

Applying the transform to Eqs. (2)–(6) and (8) results in follow-ing equations:

∂2h̄1D∂r2D þ 1

rD

∂h̄1D∂rD − λ2

1h̄1D ¼ 0 ð40Þ

∂2h̄2D∂r2D þ 1

rD

∂h̄2D∂rD − λ2

2h̄2D ¼ 0 ð41Þ

h̄2D ¼ 0 at rD ¼ RD ð42Þ

h̄1D ¼ h̄2D at rD ¼ rSD ð43Þ

∂h̄1D∂rD ¼ γ

∂h̄2D∂rD at rD ¼ rSD ð44Þ

and

−∂h̄1D∂rD ¼ qDδ

ωat rD ¼ 1 ð45Þ

where the definitions of the dimensionless variables and parametersare listed in Notation List.

The general Fourier-domain solution of Eqs. (40) and (41) forthe skin and formation zones are, respectively,

h̄1D ¼ C1I0ðλ1rDÞ þ C2K0ðλ1rDÞ ð46Þand

h̄2D ¼ C3I0ðλ2rDÞ þ C4K0ðλ2rDÞ ð47Þwhere I0ð·Þ and K0ð·Þ are the modified Bessel functions of the firstand second kinds of order zero, respectively, and C1, C2, C3, andC4 are undetermined coefficients.

Substituting Eqs. (46) and (47) into Eqs. (42)–(45), respectively,yields

C3I0ðλ2RDÞ þ C4K0ðλ2RDÞ ¼ 0 ð48Þ

C1I0ðλ1rSDÞ þ C2K0ðλ1rSDÞ ¼ C3I0ðλ2rSDÞ þ C4K0ðλ2rSDÞð49Þ

C1λ1I1ðλ1rSDÞ − C2λ1K1ðλ1rSDÞ¼ γ½C3λ2I1ðλ2rSDÞ − C4λ2K1ðλ2rSDÞ� ð50Þ

and

Fig. 6. Temporal distributions of wellbore flow rate predicted by thepresent approximate solution and Yang and Yeh solution (2005) forpenetration ratios φ ¼ 0.1, 0.4, 0.8 or 1.0

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C1λ1I1ðλ1Þ − C2λ1K1ðλ1Þ ¼ − qDδω

ð51Þ

The coefficients can be determined by simultaneously solvingEqs. (48)–(51) as

C1 ¼q̄Dλ1

A1

ffiffiffiffiffiκ1

pK1ðrSDλ1Þ þ A2

ffiffiffiffiffiκ2

pγK0ðrSDλ1Þ

A5A1

ffiffiffiffiffiκ1

p þ A6A2

ffiffiffiffiffiκ2

pγ

ð52Þ

C2 ¼q̄Dλ1

A1

ffiffiffiffiffiκ1

pI1ðrSDλ1Þ − A2

ffiffiffiffiffiκ2

pγI0ðrSDλ1Þ

A5A1

ffiffiffiffiffiκ1

p þ A6A2

ffiffiffiffiffiκ2

pγ

ð53Þ

C3 ¼q̄Dω

A8K0ðRDλ2ÞA5A1

ffiffiffiffiffiκ1

p þ A6A2ffiffiffiffiffiκ2

pγ

ð54Þ

and

C4 ¼q̄Dω

A8I0ðRDλ2ÞA5A1

ffiffiffiffiffiκ1

p þ A6A2

ffiffiffiffiffiκ2

pγ

ð55Þ

where the A1–A8 are defined in Eqs. (16)–(23).Consequently, the solution for the dimensionless hydraulic

heads within the skin and formation zones can be obtained by sub-stituting Eqs. (52)–(55) into Eqs. (46) and (47), respectively, as

h̄1D ¼ qDδωλ1

A3A1

ffiffiffiffiffiκ1

p þ A4A2

ffiffiffiffiffiκ2

pγ

A5A1ffiffiffiffiffiκ1

p þ A6A2ffiffiffiffiffiκ2

pγ

ð56Þ

and

h̄2D ¼ − qDδω2

A7A8

A5A1

ffiffiffiffiffiκ1

p þ A6A2

ffiffiffiffiffiκ2

pγ

ð57Þ

Applying the inverse transform defined by Eq. (39) to Eqs. (56)and (57) leads to Eqs. (10) and (11), respectively.

Notation

The following symbols are used in this paper:(a, b) = lower and upper elevation of the well screen,

respectively (L);(aD, bD) = (a=rw, b=rw);

h0 = constant head imposed inside the well (L);(h1, h2) = hydraulic head in the skin and formation zones,

respectively (L);(h1D, h2D) = (h1=rw, h2=rw);(Kr1, Kz1) = radial and vertical hydraulic conductivity of the

skin zone, respectively (L=T);(Kr2, Kz2) = radial and vertical hydraulic conductivity of the

formation zone, respectively (L=T);L = aquifer thickness (L);q = uniform flow rate on the screen (L3=T);

qD = q=½2πðb − aÞKr1h0�;R = radius of influence (L);r = radial distance from the well center (L);

(rD, zD, tD) = [r=rw, z=rw, Kr2t=ðSs2r2wÞ];rS = radial distance from the well center to the

skin-formation intersection (L);(rSD, RD, LD) = (rS=rw, R=rw, L=rw);

rw = radius of the pumping well (L);Ss = specific storage of the aquifer (L−1);t = time since the beginning of the CHT (T);z = elevation from the aquifer bottom (L);

γ = Kr2=Kr1;(κi, ω, λi) = (Kzi=Kri, nπ=LD, ω

ffiffiffiffiffiκi

p) where i ¼ 1

or 2; and(φ, δ) = [ðbD − aDÞ=LD, sinðωbDÞ − sinðωaDÞ].

Acknowledgments

Research leading to this paper has been partially supported by thegrants from Taiwan National Science Council under the contractnumbers NSC 101-2221-E-009-105-MY2 and 102-2221-E-009-072-MY2.

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