Water Resources Research

An integral transform approach for a mixed boundary problem involving a flowing partially penetrating well with infinitesimal well skin

Authors


Abstract

[1] A flowing partially penetrating well with infinitesimal well skin is a mixed boundary because a Cauchy condition is prescribed along the screen length and a Neumann condition of no flux is stipulated over the remaining unscreened part. An analytical approach based on the integral transform technique is developed to determine the Laplace domain solution for such a mixed boundary problem in a confined aquifer of finite thickness. First, the mixed boundary is changed into a homogeneous Neumann boundary by substituting the Cauchy condition with a Neumann condition in terms of well bore flux that varies along the screen length and is time dependent. Despite the well bore flux being unknown a priori, the modified model containing this homogeneous Neumann boundary can be solved with the Laplace and the finite Fourier cosine transforms. To determine well bore flux, screen length is discretized into a finite number of segments, to which the Cauchy condition is reinstated. This reinstatement also restores the relation between the original model and the solutions obtained. For a given time, the numerical inversion of the Laplace domain solution yields the drawdown distributions, well bore flux, and the well discharge. This analytical approach provides an alternative for dealing with the mixed boundary problems, especially when aquifer thickness is assumed to be finite.

1. Introduction

[2] The constant drawdown pumping test is particularly useful in low-transmissivity aquifers [Jones et al., 1992; Jones, 1993]. Also, pumping under fixed drawdown is a common means of controlling off-site migration of contaminated groundwater [Hiller and Levy, 1994] or recovering the free product of light nonaqueous phase liquids [Murdoch and Franco, 1994; Abdul, 1992]. A pumping well may fully or partially penetrate the aquifer and may be subject to well skin effects. In well hydraulics a flowing fully penetrating well can be appropriately simulated as a Dirichlet (the first type) boundary of a prescribed potential, and the associated models can be solved with the conventional integral transform technique [e.g., Jacob and Lohman, 1952; van Everdingen and Hurst, 1949; Hantush, 1964; Clegg, 1967; Hurst et al., 1969; Ehlig-Economides and Ramey, 1981; Olarewaju and Lee, 1989; Mishra and Guyonnet, 1992]. On the other hand, a flowing partially penetrating well must be considered as a mixed boundary that involves two different types of boundary conditions. For example, the flowing partially penetrating well with infinitesimal skin shown in Figure 1 is a typical mixed boundary such as

equation image
equation image

where the Cauchy (the third type) condition is prescribed along the well screen extending from ξd to ξl, while the Neumann (the second type) condition of no flux is stipulated over the remaining unscreened part (all variables are defined in the Notation section unless otherwise noted.) If there is no skin (i.e., Sk = 0), equation (1a) becomes a Dirichlet condition, but equations (1a) and (1b) still remain as a mixed boundary. Not directly solvable by the integral transform technique or the method of separation of variables, mathematical models involving a mixed boundary are generally dealt with by special solution techniques [Sneddon, 1966; Noble, 1958; Fabrikant, 1991].

Figure 1.

Schematic diagram of a flowing partially penetrating well with infinitesimal skin in a confined aquifer of finite thickness.

[3] For the well hydraulics model that involves equations (1a) and (1b) in a confined aquifer of infinite thickness, Cassiani et al. [1999] used the dual integral equation method to develop the Laplace domain solution. Neglecting skin effect in equations (1a) and (1b), Selim and Kirkham [1974] used the Gram-Schmidt orthonormalization method to find the steady state solution where the aquifer has a finite horizontal extent. For similar mixed boundary problems in heat conduction, Huang [1985] used the Weiner-Hopf technique to find the solution in a semi-infinite slab, and Huang and Chang [1984] combined the Green's function approach with conformal mapping to determine the solution in an elliptic disk.

[4] There are other groundwater problems associated with a mixed boundary. Dagan [1978] employed the Green's function method to find the solutions for the slug test in a partially penetrating well. For constant rate pumping in a partially penetrating well, Gringarten and Ramey [1975] used the source function approach, Wilkinson and Hammond [1990] used the perturbation method, and Cassiani and Kabala [1998] used the dual-integral equation method to determine the solutions for different conditions. However, a partially penetrating well under a constant rate pumping can also be formulated as a homogeneous Neumann boundary [Hantush, 1961; Javandel, 1982; Dougherty and Babu, 1984; Moench, 1985; Streltsova, 1988]. Ruud and Kabala [1997] noted that the use of the homogeneous Neumann boundary as the substitute for the mixed boundary should be exercised with extreme caution. Nevertheless, whether a mixed boundary or a homogeneous Neumann boundary should be used is subject to debate, for Hantush [1964] pointed out “neither a uniform flux nor a uniform drawdown is really conceived along the face of the well, because of several involved field and operational conditions.” Yet it is very clear that equations (1a) and (1b) are a mixed boundary condition.

[5] Aquifer thickness is finite under field conditions. As stated above, Cassiani et al. [1999] have determined the Laplace domain solution for the problem involving equations (1a) and (1b) by assuming infinite aquifer thickness. In the solution, infinite thickness cannot be reduced to finite thickness. Thus this solution may be appropriate for early times during which pressure change caused by the constant drawdown pumping has not reached the bottom of the aquifer or for the special condition where the screen length is significantly shorter than aquifer thickness. Such limitations can be removed by assuming finite aquifer thickness. Considering that the dual integral equation method as employed by Cassiani et al. [1999] is not suitable for finite aquifer thickness and the integral transform technique is rather prevalent, the purpose of this paper is to develop an analytical approach using appropriate integral transforms to solve the problem involving equations (1a) and (1b) with finite aquifer thickness.

2. Analytical Approach and Solutions

[6] Assuming that the confined aquifer is homogeneous, anisotropic, and of finite thickness, the problem of interest is represented by equations (1a) and (1b) and the following equations:

equation image
equation image
equation image
equation image

[7] The above model becomes identical to that of Cassiani et al. [1999] if β approaches infinity. Under the constant drawdown condition, well bore flux entering through the well screen varies along the screen length and is time dependent such as

equation image

[8] In spite of qw(ξ, τ) being unknown a priori the first step of the analytical approach is to replace equation (1a) with equation (6) for transforming the mixed boundary into an appropriate homogeneous Neumann boundary. Then, equations (2), (3), (4), and (5) subject to equations (1b) and (6) form the modified model that can be solved by the Laplace transform with respect to τ and the finite Fourier cosine transform with respect to ξ. As a result, the Laplace domain solution of the modified model is

equation image

where p is the Laplace transform variable and n is the finite Fourier cosine transform variable.

[9] The determination of qw(ξ, τ) calls for discretizing λ, the dimensionless screen length, into M segments. For I = 1, 2, …M, each segment has a constant length of Δξi, of which the center locates at equation image with ξ0 = ξd and ξM = ξl. If λ is uniformly divided, Δξi equals λ/M; otherwise, Δξi is variable. Gringarten and Ramey [1975] and Dagan [1978] also adopted this well screen discretization approach to deal with mixed boundary problems that are different from the current one.

[10] Now that the water volume inside the well of a fixed drawdown is constant (i.e., no well bore storage), the total well bore flux entering through the entire screen length sustains the well discharge Qw(τ). That is, equation (6) is subject to the following constraint:

equation image

where qi(τ) is well bore flux applied to the ith segment. Rigorously speaking, only when M approaches infinity can the summation term in equation (8) be exactly equal to Qw(τ). Nevertheless, this equality can be accurately approximated by properly choosing M as discussed below. Introduction of equation (8) to equation (7) yields

equation image

where

equation image
equation image
equation image

[11] Since H1 is independent of ξ and the integration of H2 from 0 to β is zero, H1 signifies the vertically averaged drawdown measured by fully penetrating observation wells. Accordingly, H2 represents the partial penetration effect. By direct substitution it can be shown that equation (9) satisfies equations (2), (3), (4), and (5) in the Laplace domain. The satisfaction of the boundary conditions set forth by equations (1b) and (6), however, is to be verified by noting that

equation image

[12] As given by Gradshteyn and Ryzhik [1980, equation (1.442–1)], the following relation is useful for the calculation of the infinite series in equation (13):

equation image

Substitution of equation (14) into equation (13) yields ∂H/∂ρ = equation image (p) for equation image and ∂H/∂ρ = 0 for 0 ≤ ξ ≤ ξd and ξl ≤ ξ ≤ β they are the Laplace transform of equations (6) and (1b), respectively. This completes the verification the equation (9) is the correct Laplace domain solution of the modified model. Now it is time to reinstate equation (1a), of which the Laplace transform for the discretized well screen is

equation image

The application of equation (15)) to equation (9) results in M equations as

equation image

where

equation image

where δij is zero for ij and unity for i = j. By matrix inversion of equation (16), {equation image(p)} can be determined. Finally, introduction of {equation image(p)} to equation (9) completes the reinstatement of equation (1a) in the Laplace domain and restores the relation between the solution obtained and the original model.

3. Numerical Evaluation and Verification of Solutions

[13] In equation (16), [aij] is symmetric if Δξi is constant; otherwise, [aij] is asymmetric. In either case, [aij] can be inverted with appropriate numerical methods. As discussed by Hurst et al. [1969], the skin factor Sk is positive for it is only appropriate for “positive” skin that is less permeable than the aquifer. As indicated by equation (17), Sk only contributes to the diagonal elements of [aij], thereby creating no difficulty in the matrix inversion of [aij]. After [aij]−1 is acquired, {equation image(p)} can be obtained. With this known {equation image(p)}, h(ρ, ξ, τ) is determined by inverting equations (10) and (11) with an appropriate numerical method (e.g., Talbot [1979] method or the Stehfest [1970] method with eight weighting factors). The well discharge Qw(τ) is calculated using equation (8), in which {qi(τ)} is computed by numerically inverting {equation image(p)}. For a given τ, therefore, each time the numerical inversion process determines the vertically averaged drawdown h1(ρ, τ), the partial penetration effect h2(ρ, ξ, τ), well bore flux {qi(τ)}, and the well discharge Qw(τ).

[14] If the screen length is nonuniformly divided, [aij] is asymmetric and involves M2 different elements to be determined. If the screen length is uniformly discretized, [aij] is symmetric and involves M(M + 1)/2 different elements, of which only 2M calculations are required due to constant Δξi. Therefore it is recommended that the screen length be uniformly divided with an appropriate M. The segment number M should not be too small (say, <20), or well bore drawdown will lose accuracy, particularly in the vicinity near the screen ends in contact with the aquifer material. For a large M the accuracy of well bore flux and well bore drawdown is secured while the computational work for the determination of [aij] is increased. However, the calculation of the semi-infinite series in equation (17) can be facilitated with the aid of the Euler transformation [Press et al., 1992] for Δξi > 0.02β and by the Poisson transformation [Marshall, 1998] for Δξi < 0.02β.

[15] Considering equations (1a) and (6) as the crux of the matter, the validity of solutions can be verified by inspecting whether these two conditions are substantiated in calculation. As shown in Figure 2, {qi(τ)} and h(ρ = 1, ξ, τ) are calculated for β = 1000, λ = 500, and ξd = 0 with different Sk values. The screen length is uniformly divided into 100 segments (i.e., M = 100). In Figure 2a, well bore flux is nonuniformly distributed along the screen length with a maximum at the screen end of ξl = 500 due to the strong contribution from vertical groundwater flow induced by the partial penetration effect. Similar well bore flux distributions were displayed by Dagan [1978] and Cassiani et al. [1999]. As Sk gets larger, well bore fluxes decrease, and their distributions become more uniform. The associated drawdown distribution at ρ = 1 is shown in Figure 2b. Drawdown along the screen length is uniformly equal to 1 when there is no skin and is equal to 1 − Skqi(τ) when there is skin, satisfying equations (1a) and (6). Owing to the impedance of skin, therefore well bore drawdown of Sk = 10 is less than that of Sk = 0 by the amount of Skqi(τ), where qi(τ) pertains to Sk = 10. For λ ≤ ξ ≤ β, h(1, ξ, τ) decreases rapidly from the screen end of ξl = 500, while the corresponding horizontal hydraulic gradients are zero such that equation (1b) is satisfied.

Figure 2.

(a) Distribution of well bore flux along the well screen for different Sk. (b) Drawdown distribution at ρ = 1 for different Sk.

[16] The accuracy of the solutions is checked against the well discharges presented by Cassiani et al. [1999] for λ = 10, 20, and 50 with ξd = 0 and β → ∞. For equation (8) the condition of β approaching infinity can be properly handled by setting the penetration ratio ω = λ/β equal to 0.001 and less. As shown in Figure 3, Qw(τ) from equation (8) matches well with the published results. As ω increases to unity, the fully penetrating condition is invoked. For a flowing, fully penetrating well with infinitesimal skin, Hurst et al. [1969] gave the Laplace domain solution for the cumulative production that is the integration of the well discharge with respect to time. From this cumulative production solution [Hurst et al., 1969, equation (12)] the Laplace domain solution of the well discharge can be easily derived as

equation image
Figure 3.

The well discharges determined in the current study coincide with the published results; ω = 10−3 refers to aquifer thickness being infinite; and ω = 1 pertains to the fully penetrating condition.

[17] As shown in Figure 3, Qw(τ) from equation (8) of ω = 1 is also in excellent agreement with Qw(τ) determined by equation (18) with the Stehfest [1970] method. Therefore the numerical technique developed for the solution evaluation is accurate.

4. Discussion

[18] Normally, the analysis of field data from a constant drawdown test is made use of specific drawdown that is defined as drawdown divided by the well discharge. In a low-transmissivity aquifer, sometimes the constant drawdown pumping test is performed at a single well that only hw/Qw(t) is available for estimating the aquifer parameters. In Figure 4, semilog plot of 1/[ωQw(τ)] is shown for three conditions: a fully penetrating well with no skin (ω = 1 and Sk = 0), a fully penetrating well with skin (ω = 1 and Sk = 10), and a partially penetrating well with skin (ω = 0.5, Sk = 10, and different β2/κ). At large τ, all the curves become parallel straight lines of a constant slope 1.151. These straight lines are separated by vertical displacement representative of well skin and restricted flow entry influence denoted by Sp. In general, Sp is manifested by aquifer anisotropy, the penetration ratio, and the location of the screen length. As a result, the straight lines can be expressed as

equation image

of which the dimensional form is

equation image
Figure 4.

Semilog plot of 1/[ωQw(τ)] for three conditions: (1) ω = 1 and Sk = 0; (2) ω = 1 and Sk = 10; and (3) ω = 0.5, Sk = 10, and different β2/κ. At late time, all curves become straight lines parallel to the straight line of ω = 1 and Sk = 0; τp marks the beginning time of the late-time straight line.

[19] If both Sk and Sp are zero, Tr and S can be determined in a similar way as the well-known Cooper and Jacob [1946] method. That is, the horizontal transmissivity Tr can be determined from the slope of the large-time straight line, and the storage coefficient S can be determined from the intercept between the straight line and the abscissa. However, if the well is partially penetrating, whether there is skin or not, Tr can still be determined from the slope, but S cannot be estimated from the intercept because it is shifted by an unknown amount of Sk + Sp. In this event the estimation of S requires drawdown data from fully penetrating observation wells h1(r,t) and the well discharge Qw(t). As illustrated in Figure 5, all the curves of different Sk and ω converge to a single straight line at large τ/ρ2. As noted by Clegg [1967], the approximate Laplace inversion method developed by Schapery [1961] is appropriate for the flowing well problems. By this method the dimensional expression of this straight line is found to be

equation image

which is identical to the large-time approximation solution given by Mishra and Guyonnet [1992] for a fully penetrating flowing well. Since the specific drawdown h1(r, t)/Qw(t) at large times are independent of Sk and ω (and thus Sp), S can be determined from the intercept of the large-time straight line and the abscissa of logarithmic time.

Figure 5.

Semilog plot of h1(ρ, τ)/[ωQw(τ)] for ρ = 1 and ρ = 103 with different Sk and ω. All curves converge to a single straight line at large τ/ρ2.

[20] In Figure 4, τp marks the beginning time of the straight line approximation. Streltsova [1988] discussed the use of τp in determining Kz for the constant rate pumping in a partially penetrating well. For the current study, if the partial penetration screen extends from the top of the aquifer (i.e., ξd = 0), it is also possible to determine Kz from τp, provided S is known. It is found that τp for ξd = 0 is equal to 0.25β2/κ, which is independent of Sk and ω. This relation leads to

equation image

where tp is the dimensional time at which the large-time straight line of field data hw/Qw(t) begins. Therefore, if S is obtained from h1(r, t)/Qw(t), then Kz can be calculated using equation (21).

[21] Finally, it is shown that the analytical approach developed here is not limited to β being finite. By letting β approach infinity, H1 of equation (10) is zero, and H2 of equation (11) forms the solution for aquifer thickness being semi-infinite. In the limit of β approaching infinity the infinite series in equation (11) becomes a semi-infinite integral, and the associated solution is

equation image

[22] The integral transform solution (22) is complementary to the dual-integral solution given by Cassiani et al. [1999]. Moreover, equation (22) can also be obtained by solving the modified model with the Laplace and the Fourier cosine transforms, instead of the “finite” Fourier cosine transform. When equation (16) is used to determine well bore flux, the elements of [aij] must be redefined in accordance with equation (22).

5. Conclusion

[23] The analytical solution approach and the numerical evaluation method developed here allows for the use of the Laplace and the Fourier transforms to determine the Laplace domain solution of the well hydraulics model involving a flowing partially penetrating well with infinitesimal skin. This solution technique complements the special solution techniques for the mixed boundary problems, especially when aquifer thickness is assumed to be finite.

Notation
hw

constant drawdown prescribed at the well [L].

h(ρ, ξ, τ) =

h(r, z, t)/hw.

h(r, z, t)

dimensional drawdown [L].

H(ρ, ξ, p)

dimensionless drawdown in the Laplace domain.

K0(x), K1(x)

modified Bessel function of the second kind with zero order and first order, respectively.

n

finite Fourier cosine transform parameter of ξ.

p

Laplace transform parameter of τ.

qw(ξ, τ) =

qw(z, t)rw/Krhw.

qw(z, t)

dimensional well bore flux [L/T].

equation imagew(ξ, p)

Laplace transform of qw(ξ,τ).

qi(τ)=

qi(t)rw/Krhw.

qi(t)

well bore flux of the ith segment [L/T].

equation image(p)

Laplace transform of qi(τ).

Qw(τ)=

Qw(t)/2πhwKrequation image.

Qw(t)

dimensional well discharge [L3/T].

equation imagew(p)

Laplace transform of Qw(τ).

Sk

dimensionless skin factor.

β =

b/rw.

b

the aquifer thickness [L].

rw

well bore radius [L].

ρ =

r/rw.

r

radial distance [L].

τ =

Krt/Ssrw.

Ss

specific storage.

t

time [T].

κ =

Kz/Kr.

Kr, Kz

horizontal and vertical hydraulic conductivity, respectively [L/T].

χn =

equation image

χn′=

equation image

λ =

l/rw.

l

screen length [L].

ω

partial penetration ratio, equal to λ/β.

ξ =

z/rw.

z

vertical distance [L].

ξd =

dd/rw.

dd

depth to top of the well screen [L].

ξl=

dl/rw.

dl

depth to bottom of the well screen [L].

ξi

depth to the top of the ith well segment.

equation imagei

depth to the center of the ith well segment.

Δξi

length of the ith well segment.

Ancillary