Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue1.1999:

Flowing partially penetrating well: solution to a mixed-type boundary

value problem

G. Cassiani

a,b

, Z.J. Kabala

a,*

, M.A. Medina Jr.

a a_{Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708, USA}

b_{ENI S.p. A ± Agip Division ± Geodynamics and Environment Unit (GEDA), Via Emilia 1, San Donato Milanese I-20097, Italy}

Received 1 October 1998; received in revised form 15 December 1998; accepted 17 December 1998

Abstract

A new semi-analytic solution to the mixed-type boundary value problem for a ¯owing partially penetrating well with in®nitesimal skin situated in an anisotropic aquifer is developed. The solution is suited to aquifers having a semi-in®nite vertical extent or to packer tests with aquifer horizontal boundaries far enough from the tested area. The problem reduces to a system of dual integral equations (DE) and further to a deconvolution problem. Unlike the analogous Dagan's steady-state solution [Water Resour. Res. 1978; 14:929±34], our DE solution does not su€er from numerical oscillations. The new solution is validated by matching the corresponding ®nite-di€erence solution and is computationally much more ecient. An automated (Newton±Raphson) parameter identi®cation algorithm is proposed for ®eld test inversion, utilizing the DE solution for the forward model. The procedure is computationally ecient and converges to correct parameter values. A solution for the partially penetrating ¯owing well with no skin and a drawdown±drawdown discontinuous boundary condition, analogous to that by Novakowski [Can. Geotech. J. 1993; 30:600±6], is compared to the DE solution. The D±D solution leads to physically inconsistent in®nite total ¯ow rate to the well, when no skin e€ect is considered. The DE solution, on the other hand, produces accurate results. Ó _{1999 Elsevier Science Ltd. All} rights reserved.

1. Introduction

The ¯owing well test is a well-established single-borehole technique used in the ®eld by hydrologists [3,8,16,23], geotechnical engineers [32,40,46], and pe-troleum engineers [6,11,25]. Since the ¯ow rate to a well is induced under a constant head, the test is also known as the constant-head or constant-pressure test. The measured quantities are the constant head and the re-sulting ¯ow rate (inbound or outbound). Mathematical models are needed to interpret the ®eld data. A paucity of such models exists for fully penetrating ¯owing wells; even fewer models are available for partially penetrating ¯owing wells, due to the inherent diculty of solving the resulting mixed boundary value problem.

Only hydraulic conductivity (or permeability) can be estimated from the approaches of Dagan [8], who worked out a steady-state solution using Green's func-tions, and of Tavenas et al. [40], who used Hvorslev's quasi-steady-state ¯ow formula for hydraulic

conduc-tivity and studied numerically the e€ects of probe

ge-ometry on the shape factor in this formula.

Transmissivity and storativity can be estimated from the Hantush [16] transient model for a fully penetrating ¯owing well. This model has been extended to account for in®nitesimal thickness skin by Kabala and Cassiani [24].

Novakowski [32] pointed out that models that neglect the skin e€ect [27], i.e. the changed ¯ow ®eld in a dis-turbed zone in the immediate vicinity of the well, pro-duce biased results in interpreting ®eld tests. The disturbed zone can be of ®nite or in®nitesimal thickness. Recently, Novakowski [32] developed a general model, which accounts for the thick skin and partial penetration. However, it contains a large number of parameters and is based on a physically inconsistent well-face boundary condition that causes the model to predict non-physical behaviors for some parameter choices (e.g., when no skin e€ect is considered) as will be shown in this paper.

Although Novakowski [32] uses a point source solu-tion, which he then integrates, his solution is equivalent to that for the initial boundary-value problem with a *_{Corresponding author. Tel.: +1 919 660 5479; fax: +1 919 660 5219;}

e-mail: kabala@copernicus.egr.duke.edu

the Fourier transform. It is not yet clear what kind of error introduces the simpli®ed boundary condition. Some of the many parameters in Novakowski's [32] model, especially the speci®c storativity and hydraulic conductivity of the skin, are not likely to be determined from the ¯owing-well-test ®eld data, or other single borehole tests [27]. It may thus be advantageous to lump the ®nite-thickness skin parameters into a single skin factor that would be representative of an equivalent skin of in®nitesimal thickness as proven by Moench and Hsieh [27].

1.1. Objectives

The purpose of this paper is to derive a new semi-analytic solution for the response of a partially pene-trating ¯owing well, situated in an aquifer of in®nite extent and thickness, and to compare it to the numerical solution obtained via ®nite di€erence approximation. A critical analysis of an existing approximate solution to the same problem [32] will also be given by comparison to the new solution.

2. Mixed-type boundary value problems

Numerous problems of mathematical physics can be described by initial boundary value problems (IBVP) with the mixed-type boundary conditions, i.e. with the ®rst-type boundary condition on one part of the boun-dary (speci®ed dependent variable, e.g. piezometric head) and the second-type boundary condition on an-other part of the boundary (speci®ed derivative of the dependent variable, e.g. ¯ux). Such problems arise in potential theory and its numerous applications to engi-neering [12,26,38], fracture mechanics [10] heat con-duction and, in particular, dip-forming processes in metallurgy [14], surface rewetting [42], contact resistance to heat transfer [36], heat transfer from partially insu-lated pipes [37], and many others. Flow to partially penetrating wells should also be described by IBVPs with the mixed-type boundary condition on the well face [47].

Huang [20] and Wilkinson and Hammond [47] noted that mixed-type boundary value problems cannot be solved by the conventional integral transform methods (or expansion in terms of orthogonal functions). Only a paucity of analytic solutions to such problems have been found by means of more subtle techniques that include the dual integral/series equations [38], Weiner±Hopf technique [31], dual integral equation (DE) [43], and

The solutions to the parabolic mixed-type IBVPs may display some disturbing properties. For example, Bas-sani et al. [1] demonstrated that a steady-state temper-ature ®eld of a wedge-shaped region, where the boundary condition changes abruptly at the vertex of the wedge, su€ers from the heat ¯ux singularity (un-bounded point ¯ux at the vertex) whenever the included angle of the wedge is greater thanp=2. We note that this result has analogous implications for the ¯ux at the edges of the ®lter of a partially penetrating well.

As recognized by Wilkinson and Hammond [47], who applied an approximate perturbation technique, the mixed-type boundary conditions arise naturally in the description of ¯ows to partially penetrating wells with the pressure head speci®ed on the ®lter face and no-¯ux speci®ed on the well casing. With the exception of a few [35,45] and others who solved numerically the mixed-type boundary value problems in well hydraulics, all other researchers who approached such problems ana-lytically, avoided solving them by replacing the mixed-type boundary condition of speci®ed-head/no-¯ux either by a ®rst-type (Dirichlet) boundary condition of speci-®ed-head/zero-head [32] or by a second-type (Neumann) boundary condition of speci®ed-¯ux/no-¯ux [8,9,15,17, 18,22,28±30,33,41,45]. The simpli®ed solutions obtained in this manner are to a certain degree physically incon-sistent, as explicitly acknowledged by Muskat [28] and Hantush [15,17,19], and discussed in detail by Ruud and Kabala [35].

3. The ¯owing partially-penetrating well problem

Cases to which the presented solution applies: (a) partially penetrating ¯owing well in an aquifer of semi-in®nite vertical extent; (b) double packer test geometry in an aquifer of in®nite vertical extent. Consider a ¯owing well drilled in a formation of semi-in®nite ver-tical extent, such as shown in Fig. 1. Note that the bottom of the wellbore is assumed not to collect any ¯ow from the aquifer. The wellbore of radius rw

pene-trates the aquifer down to a depth l. The imposed drawdown in the well is sw. The drawdown at the

dis-tancerfrom the well, the distancezfrom the top of the aquifer, and timet is s…r;z;t†. The aquifer parameters are horizontal hydraulic conductivity Kr, vertical hy-draulic conductivity Kz, and speci®c storativitySs. The governing partial di€erential equation is

Kr o2_s

o_r2 ‡

1

r

o_s

o_r

‡Kz o2_s

o_z2 ˆSs

o_s

A skin of in®nitesimal thickness [27], is assumed characterized by the skin factor g, de®ned by the fol-lowing boundary condition at the well screen (z6l):

ÿgrw

o_s

o_rjrˆrw‡s

_j

rˆrw ˆsw: …2†

We cast the problem in dimensionless terms, by de-®ning the following parameters:

Dimensionless well screen length

dˆl=rw: …3†

Dimensionless drawdown

sˆs=sw: …4†

Dimensionless time

aˆ tKr Ssr2w

: …5†

Anisotropy ratio

a2_ˆKz

Kr …

6_†

and dimensionless spatial coordinates:

q_ˆr=rw; …7†

f_ˆz=rw: …8†

The boundary value problem to be solved follows from the partial di€erential equation (1), that in di-mensionless terms reads:

o2

s

o_q2‡

1 q

o_s

o_q‡a

2o 2

s

o_f2ˆ o_s

o_a; …9†

subject to the following boundary and initial condi-tions:

sq;f;a_ˆ0_{† ˆ}0; …10_†

s…qˆ 1;f;a† ˆ0; …11†

ÿgos

o_qjqˆ1‡s…qˆ1;f;a† ˆ1 for 06f6d

o_s

o_qjqˆ1ˆ0 ford<f;

…12†

o_s

o_fjfˆ0ˆ0; …13†

sq;f_{ˆ 1};a_{† ˆ}os

o_fjfˆ1ˆ0: …14†

Note that Eq. (12) is a mixed boundary condition at the well face, i.e. a third-type (or Robin) boundary condi-tion along the well screen of dimensionless lengthd, and a second-type (Neuman) no-¯ow boundary condition along the casing.

We also note that, due to the symmetry around the

zˆ0 plane, a solution for a partially penetrating ¯ow-ing well in the aquifer of in®nite thickness (or a double packer test with long enough packers to prevent back-¯ow) shown in Fig. 1 can be directly obtained from the solution of Eqs. (9)±(14).

The solution technique employs the DE approach [43] in a manner analogous to that used by Cassiani and Kabala [5]. It is described in Appendix A. The relevant computational details are discussed in Appendix B. sw

z r

(a) (b)

r

packer packer

z

and parameter values.

In Fig. 2, we present an example of point ¯ux at the well face for di€erent times. The curves characteristically exhibit sharp peaks at the extremes of the well screen due to strong contribution of vertical ¯ux in the sur-rounding aquifer. They are analogous to those noted by Dagan [8] in his semi-analytic solutions for steady-state ¯ow to partially penetrating wells in water-table aqui-fers, and by Widdowson et al. [45] in their correspond-ing numerical solutions for partially penetratcorrespond-ing slug tests. Unlike Dagan's results, only tiny oscillations occur in our solution.

For the case of no skin (g_ˆ0) we validate the total dimensionless ¯uxG…†calculated with our methodology by comparing it to a numerical solution obtained via ®nite di€erence algorithm of Ruud and Kabala [34]. As seen in Fig. 3 the two curves show an excellent match. We note that at large times all curves approach quasi-steady state.

The e€ect of the skin factorg is shown in Figs. 4±6, where type curves for d ˆ10, 20 and 50 are shown for gˆ0 through 10.

Similarly to the fully penetrating ¯owing wells [24], the skin e€ect in partially penetrating wells produces an early-time in¯ection point in theG…†curves. For large times the skin factor lowers the curve asymptote.

The DE approach is ideally suited for incorporation into an automated parameter identi®cation algorithm. Letti; Qi; iˆ1;. . .;ndatabe ®eld data of total ¯ow rate

versus time, collected during a constant-head test on a partially penetrating well of known geometry. The for-mation parameter vector b_{ˆ f}b1;b2;b3g, b1ˆKr;

b2ˆSs; b3ˆgcan be obtained fromti; Qiby using an optimization algorithm that minimizes the di€erence between ®eld data and model prediction. If f…t† is the adopted model, let the distance vector and the Jacobian matrix be given respectively by

F ˆ ff…ti† ˆQig iˆ1;. . .;ndata …15†

and

Aˆ _oof bjjtˆti

iˆ1;. . .;ndata; jˆ1;2;3: …16†

An estimate of the parameter vector b is obtained by Newton±Raphson iteration

Fig. 2. Dimensionless point ¯ux for the case of no skin (g_ˆ0), at di€erent time steps, ford_ˆ100.

101 104 109 1014 α=t K_r / (S_s r_w2₎

101 100

G(

α

)

δ

10 20 50 100

Hantush [1959]

Fig. 3. Dimensionless total ¯ux at the wellscreen as a function of di-mensionless time, for the no-skin case (g_ˆ0) and di€erent well screen lengths. The [16] solution for a fully penetrating ¯owing well is shown for reference. Also, a comparison is made with the results obtained using the ®nite di€erence (FD) code by [34].

105 100 105 1010 α=t K_r / (S_s r_w2₎

101 100 101

G(

α

)

η

0

1

3 4 5 6 7 8 9 10 2

δ

=10

bk‡1

ˆbk_ÿAk

‡ Fk; …17†

where k is the iteration step, and A_‡ is the pseudo-in-verse matrix ofA, de®ned as

A_‡ˆATA_†ÿ1AT; …18_†

whereT stands for transpose and _ÿ1 for inverse ([13], p. 243).

In this paper the DE algorithm is used to compute

f…t†. For the repeated computation off…t†required by Eqs. (15) and (16), the fast computation of DE is es-sential.

An example of Newton±Raphson parameter ®tting is shown in Fig. 7. The data points are simulated using the ®nite di€erence code by Ruud and Kabala [34]. Provided

that the ®rst guess for the parameters is not too far from the true values (approximately within an order of mag-nitude) convergence is assured.

5. Comparison with existing models

In Fig. 3 we compare the results produced by our model to the corresponding results for fully penetrating ¯owing wells. At early times the DE solution ap-proaches, as expected, the corresponding solution for a fully penetrating well, i.e. the no-skin solution of Han-tush [16]. In addition, as mentioned earlier, the DE point ¯ux curves in Fig. 2 exhibit sharp increases at the well screen edges, a behavior similar to that pointed out by Dagan [8] in his model for water table aquifers.

The Novakowski [32] model was developed originally for ®nite-thickness aquifers; however, his methodology can be easily extended to aquifers of in®nite or semi-ini®nite vertical extent, such as the ones considered here. In this section we consider the case with no skin, which can be directly compared to our approach. We name this method the drawdown±drawdown (D±D) ap-proach, since it involves a discontinuous drawdown boundary condition at the well face.

Consider the IBVP given by Eqs. (9)±(11) and (14) and the following condition at the well-face, used by Novakowski [32]:

s…qˆ1;f;a† ˆd…fÿf0†; …19_†

where hered_†is the Dirac delta function.

Employing cosine Fourier transform instead of ®nite cosine Fourier transform used by Novakowski [32] we obtain

105 100 105 1010 α=t K_r / (S_s r_w2₎

101 100 101

G(

α

)

η

0

1

3 4 5 6 7 8 9 10 2

δ

=20

Fig. 5. Dimensionless total ¯ux at the wellscreen as a function of di-mensionless time, ford_ˆ20.

105 100 105 1010 α=t K_r / (S_s r_w2₎

101 100 101

G(

α

)

η

0

1

3 4 5 6 7 8 9 10 2

δ

=50

Fig. 6. Dimensionless total ¯ux at the wellscreen as a function of di-mensionless time, fordˆ50.

Fig. 7. Example of Newton±Raphson ®tting to data. The data are produced using the FD model by Ruud and Kabala [34]. The ®tting is obtained using the DE solution as the physical model. The starting point is Krˆ10ÿ7 m/s, Ssˆ10ÿ5 1/m, and convergence to 0.1% is

Integration along the well face leads to the solution for drawdown in the Laplace domain

Note that this solution is identical to that obtained by imposing a unit drawdown on the well screen and zero drawdown on the casing.

The point ¯ux at the well face follows from the above by di€erentiation Integration of Eq. (22) along the well face leads via Eq. (A.22) to the dimensionless total ¯ux G_†, analo-gous to Novakowski's [15], (A.9)

The above integral is divergent, as is the corresponding series solution of Novakowski [15], Eq. (A.9) for the case of no skin. The reason why the total ¯ux is in®nite in this solution can be seen in Figs. 8 and 9. There we present the point ¯ux at dimensionless timeaˆ103_{for a}

well of dimensionless screen lengthdˆ100 and no skin (g_ˆ0) calculated from our DE solution, (A.19), and from the D±D solution (22). The D±D solution shows strong spurious oscillations, known as Gibbs e€ect, caused by truncation in the frequency domain. The curves in Figs. 8 and 9 appear to be blurred because of this e€ect. The DE solution is represented by a thick dot±dashed line. Note that the D±D solution produces spurious negative ¯ux outside the well casing. Along the well screen away from its edges the two solutions are very close to each other. However, they are substantially di€erent around the wellscreen edge. The DE solution takes reasonable values whereas the D±D solution

di-verges to in®nity as 1=…fÿd†, leading to in®nite total ¯ux (the area under the curve), clearly a physical im-possibility.

6. Conclusions

The main results of this paper are:

1. A new semi-analytic solution to the mixed-type boun-dary value problem for a ¯owing partially penetrating

-200 -100 0 100 200

Fig. 8. Comparison between the dual integral equation solution (DE) and the drawdown±drawdown (D±D) solution ford_ˆ100 and time a_ˆ103 _{and no skin (g}

ˆ0). The D±D solution exhibits spurious negative ¯ux on the well casing. For a closer look, see Fig. 9.

well with in®nitesimal skin situated in an anisotropic aquifer is developed via Laplace and Fourier trans-forms. The solution is suited for aquifers of semi-in-®nite vertical extent for packer tests with aquifer horizontal boundaries far enough from the tested area. The problem can be reduced to a system of DE and further to the deconvolution Eq. (A.18), which we solve by discretizing the integral. Inversion of the Laplace and Fourier transforms are handled numerically via Stehfest and fast Fourier transform (FFT) algorithms.

2. The solution is achieved by discretizing a deconvolu-tion equadeconvolu-tion. An ecient discretizadeconvolu-tion needs to sat-isfy two opposite requirements: (i) it has to be ®ne enough to sample adequately the kernel function, and (ii) it must be crude enough to prevent unwanted oscillations. For no skin,gˆ0, the density of 0.6/a2

nodes per unit fproduces accurate results, whereas for g_ˆ10 the corresponding density increases to 1.6/a2 nodes per unitf.

3. The new solution is computationally robust due to the impulse-like nature of the kernel in the deconvo-lution equation. We note that unlike the analogous Dagan's [8] steady-state solution for ®nite-thickness uncon®ned aquifers, our DE solution embedded in Eq. (A.18) does not su€er from numerical oscilla-tions.

4. Our DE solution matches the corresponding ®nite-di€erence solution obtained using the code developed by Ruud and Kabala [34]. The DE solution is tationally much more ecient. Although the compu-tational burden is generally limited, it increases with the well screen lengths.

5. Based on our DE solution (forward model), an auto-mated (Newton±Raphson) parameter identi®cation algorithm is developed for ®eld test interpretation. The procedure is computationally ecient and con-verges to correct parameter values, provided that the initial parameter guess is not too far from the true values (approximately within one order of magni-tude).

6. A solution for the partially penetrating ¯owing well with no skin and D±D discontinuous boundary con-dition, analogous to that of Novakowski [15], Eq. (A.9) was also derived. The D±D solution was compared to the DE solution. The D±D solution leads to physically inconsistent in®nite total ¯ow rate to the well, when no skin e€ect is considered. The DE solution, on the other hand, produces accurate results.

Acknowledgements

The authors are indebted to Nels C. Ruud for the use of numerical code developed in his dissertation. This

research was performed while G. Cassiani was a grad-uate student and Z.J. Kabala and M.A. Medina, Jr. were his faculty advisors in the Department of Civil and Environmental Engineering at Duke University, Dur-ham, NC.

Appendix A. The solution technique

We eliminate the time variableafrom the problem by applying the Laplace transform to Eqs. (9), (11)±(14), and using the initial condition (10). The Laplace-trans-formed variable is denoted by a bar, thus

The transformed boundary-value problem is

o2 Next, we eliminate f by applying the Fourier cosine transform to Eq. (A.2) and Eq. (A.3) and by using Eq. (A.6). In order to apply properly the mixed boun-dary condition (A.4), Tranter ([43], p.51) suggests leav-ing it out and applyleav-ing it only after invertleav-ing back from the Fourier domain. The Fourier-transformed variable is denoted by a hat, thus ^s…q;n;p† ˆF_fs…q;

f;p†;f!ng ˆ2=p R1

0 s…q;f;p†cos…nf†df. We thus

arrive at the Bessel di€erential equation

o2_^s

subject to the boundary condition

^sq_{ˆ 1};n;p_{† ˆ}0: …A:8_†

The general solution is in the form of

^s…q;n;p† ˆA…n;p†K0 q

where K0 is the modi®ed Bessel function of order zero

o_s whereK1 is the modi®ed Bessel function of order one.

Substitution of Eq. (A.10) and Eq. (A.11) into Eq. (A.4) leads to a system of two integral equations, known as DE ([43], p. 111):

Systems (A.12) and (A.13) can be solved in a number of ways [43] we transform it into a deconvolution problem in the space domain. This approach is convenient be-cause the Bessel function kernels in Eqs. (A.12) and (A.13) have closed-form Fourier cosine transforms ([2], vol. I, p. 56): With Eq. (A.14) and Eq. (A.15) and the convolution theorem for Fourier transforms, problem (A.12) and (A.13) can be restated as:

Z ‡1

tically zero outside the well screen interval _jf_j6d, so that Eq. (A.16) and Eq. (A.17) can be conveniently re-duced to the Fredholm integral equation

Z ‡d

be obtained by Fourier transform

o_s Fourier transformations can be conveniently carried out via the fast Fourier transform (FFT) [4]. We use Stehfest [39] algorithm to obtain the point ¯uxo_s†=o_q†j

qˆ1…f;a†

in the dimensionless time domain.

The total dimensionless ¯ux to the partially pene-trating well can be de®ned analogously to the corre-sponding dimensionless ¯ux for a fully penetrating ¯owing well given by Hantush [16]

G…a;d_{† ˆ} Q…t† 2pKrlsw

; …A:20_†

whereQ…t†is the total dimensional ¯ux, computed as

Q…t† ˆ ÿ2prwKr Z l

0

o_s

o_rjrw dz: …A:21†

Consequently, G can be expressed in terms of the di-mensionless point ¯ux as

G…a;d† ˆ ÿ1

The convolution approach is more stable because it is consistent with the slight approximation introduced by limiting the deconvolution from _ÿd to _‡d only (see Eq. (A.18)).

Appendix B. Computational details

Although generally deconvolution problems are ill-posed [7], in some cases highly accurate solutions can be obtained. Dagan [8] notes that ``an accurate solution

‰ Š is expectable if the kernel is strongly peaked at fˆx, whereas the solution will worsen if the kernel is ¯at''. Fortunately, the kernel in Eq. (A.18) is strongly peaked atf_ˆxfor all values ofp, and thus for all values of dimensionless timea.

We note that unlike Dagan's [8] analogous steady-state solution for ®nite-thickness uncon®ned aquifers, our DE solution embedded in Eq. (A.18) does not su€er from severe numerical oscillations.

Solving Eq. (A.18) requires a carefully calibrated discretization along the well screen. An ecient dis-cretization can be found by satisfying as well as possible the two opposite qualitative requirements: (i) it has to be ®ne enough to sample adequately the kernel function, and (ii) it must be crude enough to prevent unwanted oscillations.

Our numerical experiments demonstrate that an e-cient discretization can be determined independently of d. However, it depends ona2 _{andg. For no skin,gˆ0,}

the density of 0.6/a2 nodes per unitfproduces accurate results, whereas for g_ˆ10 the corresponding density increases to 1.6/a2 _{nodes per unit f. For intermediate}

cases proportional increments can be used. Note how the dependence ona2 _{corresponds to a pure scaling of}

the boundary value problem in the vertical direction. Since the optimalDfis independent ofd, the number of points needed is a linear function of d itself. Conse-quently, the computational burden increases with the size of the deconvolution matrix arising from discreti-zation of Eq. (A.18), asdincreases.

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