Published online 28 February 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.607
A new solution for a partially penetrating constant-rate pumping
well with a finite-thickness skin
Pin-Yuan Chiu
1, Hund-Der Yeh
1,∗,†and Shaw-Yang Yang
2 1Institute of Environmental Engineering, National Chiao Tung University, Hsinchu 300, Taiwan2Department of Civil Engineering, Vanung University, Chungli 320, Taiwan
SUMMARY
A mathematical model describing the constant pumping is developed for a partially penetrating well in a heterogeneous aquifer system. The Laplace-domain solution for the model is derived by applying the Laplace transforms with respect to time and the finite Fourier cosine transforms with respect to vertical co-ordinates. This solution is used to produce the curves of dimensionless drawdown versus dimensionless time to investigate the influences of the patch zone and well partial penetration on the drawdown distributions. The results show that the dimensionless drawdown depends on the hydraulic properties of the patch and formation zones. The effect of a partially penetrating well on the drawdown with a negative patch zone is larger than that with a positive patch zone. For a single-zone aquifer case, neglecting the effect of a well radius will give significant error in estimating dimensionless drawdown, especially when dimensionless distance is small. The dimensionless drawdown curves for cases with and without considering the well radius approach the Hantush equation (Advances in Hydroscience. Academic Press: New York, 1964) at large time and/or large distance away from a test well. Copyright q 2007 John Wiley & Sons, Ltd.
Received 16 August 2006; Revised 28 December 2006; Accepted 3 January 2007
KEY WORDS: groundwater; analytical solution; constant pumping test; heterogeneous aquifer; finite Fourier cosine transforms; Laplace transforms; partially penetrating well
1. INTRODUCTION
The pumping test with a constant pumping rate is a popular technique for high-transmissivity aquifers. The aquifer parameters such as transmissivity and storage coefficient can be determined from a test-data analysis with the measured drawdowns. These aquifer parameters have been widely
∗Correspondence to: Hund-Der Yeh, Institute of Environmental Engineering, National Chiao Tung University, Hsinchu
300, Taiwan.
†E-mail: [email protected]
utilized for estimating the water resources in the ground-water hydraulics. Most of the data analysis methods used in the ground-water field are developed for a fully penetrating well and the well diameter is considered as infinitesimally small. In addition, the aquifer is assumed as homogenous, isotropic, infinite-extent and with a constant thickness. For a small-diameter well, the aquifer parameters are usually estimated according to the Theis formula[1] in which the pumping well was considered as a line source. On the other hand, the aquifer parameters for a large-diameter well may be founded using an approach suggested by Papadopulos and Cooper[2] in which the wellbore storage was also considered. Several researchers had provided the analytical models or results for the various types of problems in engineering applications, e.g. [3–5]. Hantush [3] developed an analytical model of a constant pumping test in a partially penetrating well. His solution in the Laplace domain was obtained via the Laplace transform and the finite Fourier sine transform, and the time-domain solution was derived using the inverse Laplace transforms.
An aquifer having a small region of anomalous hydrogeological properties may be called a cylindrical inhomogeneity (patchy aquifer). A patchy aquifer may have the radius of heterogeneous cylinder (patch region) up to 60 m[6] and can be considered as a composite aquifer system. The patch zone may affect the pumping drawdown; consequently, the aquifer-drawdown distribution depends on the thickness and properties of the patch and formation zones.
In the well constructions, the well drilling induces the invasion of drilling mud into aquifer and may produce a positive patch zone that has a lower permeability than that of the original formation. In contrast, the extensive well development and substantial spalling and fracturing of borehole wall may increase the permeability of the adjacent formation around the wellbore and form a negative patch zone. In any case, the thickness of a patch zone may range from a few millimetres to several meters and thus must be considered in the pumping-test data analyses[7]. For wells in a heterogeneous aquifer, Novakowski[7] presented a composite analytical solution by using the Laplace transforms. He provided some type curves generated from the Laplace-domain solution and used them to explore the effects of the wellbore storage and patch zone on the head distributions. Using the Laplace transforms and Bromwich integral method, Yeh et al.[8] obtained the time-domain solution for a radial two-layer drawdown equation for an aquifer under constant-flux pumping in a finite-radius well. They also proposed a numerical method to efficiently evaluate the solution with accuracy to five decimal places. The existing solutions, addressing the problems of the partially penetrating well in a heterogeneous aquifer system, had been developed mostly under some simplified conditions. Examples for the solutions in a petroleum industry are Bixel and van Poolen [9] and Jargon [10] and in a groundwater hydraulic are Barker and Herbert[6] and Butler [11]. Cassiani and Kabala [12] mentioned that those articles simplified the problems by assuming a uniform point flux along a screened portion of the wellbore. Considering the wellbore storage and skin effect, Park and Zhan[13] provided a solution of groundwater flow for a finite-diameter horizontal well in an anisotropic leaky aquifer. Their solution was derived based on the separation of the source and geometric functions. Zhan and Bian [14] had derived the closed-form solutions of the steady-state leakage rates and volumes for both the constant-rate and constant-drawdown pumping wells. With the scale-invariant relationship, those solutions of the total leakage rate and volume can be generalized to finite size aquifers with impermeable boundaries.
Markle et al.[15] developed an analytical model for a constant-head test conducted in a vertically fractured media. Their solution was presented in the Laplace domain and numerically inverted to obtain the values of the time-domain solution. They provided small- and large-time approximations to analyse the test data. More recently, Moench[16] presented a Laplace-domain solution for flow
to a finite-diameter well in a homogeneous, anisotropic unconfined aquifer. His solution accounts for the effects of the wellbore storage and patch zone and allows for the non-instantaneous release of water from an unsaturated zone. Cassiani et al. [17] developed a semi-analytical solution for a pumping test on the partially penetrating wells in a confined aquifer that accounting not only for the wellbore storage, infinitesimal skin, and anisotropic aquifer, but also for a mixed-type boundary condition at a well face. Their solution was obtained via the method of the dual integral equation. Inversion of the Laplace transform and Fourier transform were handled numerically via the Stehfest algorithm and the fast Fourier transform.
The purpose of this article is to present a new mathematical model describing a constant pumping test in a partially penetrating well that has a patch zone in a radial confined aquifer system. The Laplace-domain solution is derived by applying the Laplace transforms with respect to time and the finite Fourier cosine transforms with respect to vertical co-ordinates. Then the time-domain results are evaluated when applying the modified Crump algorithm[18, 19] to invert the Laplace-domain solution. Simplified solutions obtained with the flux–flux discontinuous boundary conditions (e.g. [3, 8]) are compared with our solution.
2. MATHEMATICAL MODEL
A partially penetrating well in a heterogeneous aquifer system is illustrated in Figure 1. Several assumptions made for the solutions in terms of drawdowns are:
(1) The aquifer is anisotropic, infinite-extent and with a constant thickness. (2) The well is partially penetrated with a finite radius.
(3) The pumping flow rate is maintained at a constant value throughout the whole test period. (4) The patch zone has uniform thickness in the z-direction within the top and bottom
imper-meable layers.
A term representing the vertical flow is included in the equations of a radial confined aquifer system to account for the effect of well partial penetration. According to the assumptions mentioned above, the governing equations of drawdowns, s(r, z, t), can be expressed within the patch and formation zones, respectively, as
Kr1* 2 s1(r, z, t) *r2 + Kr1 r *s1(r, z, t) *r + Kz1 *2 s1(r, z, t) *z2 = Ss1 *s1(r, z, t) *t , rwrr1 (1) and Kr2* 2 s2(r, z, t) *r2 + Kr2 r *s2(r, z, t) *r + Kz2 *2 s2(r, z, t) *z2 = Ss2 *s2(r, z, t) *t , r1r<∞ (2) where the subscript 1 denotes the patch zone, the subscript 2 denotes the formation zone, Kr is
the hydraulic conductivity in the radial direction, Kz is the hydraulic conductivity in the vertical
direction, Ssis the specific storage, r is the radial distance from the centreline of pumping well,
rw is the radius of pumping well, r1 is the outer radius of patch zone and t is the time from the
Q Impermeable layer Impermeable layer = b2 = b1 = b’2 = b’1 Formation zone Confined aquifer Well screen
Pumping well Observation well
= L = 0 rw r Patch zone r1
Figure 1. Schematic diagram of the well and aquifer configurations.
The drawdowns are initially assumed to be zero within both the patch and formation zones, these are
s1(r, z, 0) = s2(r, z, 0) = 0 (3)
The drawdown tends to be zero when r approaches infinity. Therefore, the outer boundary condition for the formation zone is given by
s2(∞, z, t) = 0 (4)
The continuities of drawdown and flux between the patch and formation zones, respectively, require s1(r1, z, t) = s2(r1, z, t) (5) and Kr1*s1(r1, z, t) *r = Kr2 *s2(r1, z, t) *r (6)
The lower and upper boundary conditions in a z-direction are, respectively,
*s1(r, 0, t)
*z =
*s2(r, 0, t)
and
*s1(r, L, t)
*z =
*s2(r, L, t)
*z = 0 (8)
where L is the thickness of the confined aquifer.
According to Darcy’s law, the boundary condition for maintaining a constant flux across the screen is assumed as *s1(rw, z, t) *r = − Q 2rwKr1(b2− b1)[U(z − b1) − U(z − b2)], 0zL (9)
where Q is the pumping rate, b1and b2are, respectively, the lower and upper vertical co-ordinates
of well screen, and U(Q) is a unit step function defining that U(z − bi) equals one when biz but
equals zero otherwise for i= 1 or 2. Equation (9) assumes that the flow rate along the well screen is uniform. Such an assumption is similar to that made in Reference[20, p. 304, (15) and (16)].
2.1. Analytical solutions
To solve the boundary value problem, the Laplace transform and finite Fourier cosine transform are applied to the governing equations and boundary conditions. The Laplace transform is taken with respect to time and the finite Fourier cosine transform is taken with respect to the z-co-ordinate. The inverse finite Fourier transform is analytically performed to obtain the solutions for drawdowns within the patch and formation zones. Detailed derivations for the Laplace-domain solutions are given in Appendix A and the results are
s1(r, z, p) = Q 4T2 1 p 2T2 rwT1 3I0(q3r) + 4K0(q3r) ∞ + Q 4T2 1 p 4T2 (b2− b1)rwT1 × ∞ n=1 1 0 I0(q1r) +2 0 K0(q1r) F(b1, b2) cos(wnz) (10) and s2(r, z, p) = Q 4T2 1 p 2T2 rwT1 5K0(q4r) ∞K0(q4r1) + Q 4T2 1 p 4T2 (b2− b1)rwT1 ∞ n=1 0 K0(q2r) K0(q2r1) F(b1, b2) cos(wnz) (11)
where p is the Laplace variable, F(b1, b2) = [sin(wnb2) − sin(wnb1)]/wn, I0(Q) and K0(Q) are the
modified Bessel functions of the first and second kinds of order zero, and I1(Q) and K1(Q) are the
modified Bessel functions of the first and second kinds of order one. Notice that the right-hand side of (10) and (11) have two terms; the first term represents the solution for a confined radial flow, and the second term contains a summation term accounting for the effect of a partially penetrating well.
2.2. Average drawdown to the observation well
The water level in an observation well as shown in Figure 1 represents the average drawdown in an aquifer that is in contact with the well screen (or perforated section) of an observation well. The average drawdown in an observation well that is screened between the depths of b1 and b2 can be obtained by integrating the drawdown equations with respect to z between the limits of
b1 and b2, and then dividing the result by(b2− b1). Thus, the average drawdown within the patch and formation zones can be expressed as
s1(r, p) = Q 4T2 1 p 2T2 rwT1 3I0(q3r) + 4K0(q3r) ∞ + Q 4T2 1 p 4T2 (b2− b1)(b2− b1)rwT1 × ∞ n=1 1 0 I0(q1r) + 2 0 K0(q1r) F(b1, b2)F(b1, b2) (12) and s2(r, p) = Q 4T2 1 p 2T2 rwT1 5K0(q4r) ∞K0(q4r1) + Q 4T2 1 p 4T2 (b2− b1)(b2− b1)rwT1 × ∞ n=1 K 0(q2r) 0K0(q2r1) F(b1, b2)F(b1, b2) (13)
2.3. Full penetration with a patch zone
For the case of a fully penetrating well with a patch zone, b1 equals zero and b2 equals the
thickness of confined aquifer, L. The Laplace-domain solutions of drawdown distributions within the patch and formation zones can, respectively, reduce to
s1(r, p) = Q 4T2 1 p 2T2 rwT1 3I0(q3r) + 4K0(q3r) ∞ (14) and s2(r, p) = Q 4T2 1 p 2T2 rwT1 5K0(q4r) ∞K0(q4r1) (15)
These two solutions can also be obtained by solving (1)–(6) when neglecting the second derivative of the drawdown with respect to z in (1) and (2)[8]. Thus, (9), the average flow rate across the wellbore, turns out to be
*s1(rw, t)
*r = −
Q
2rwKr1L
(16)
2.4. Solution of radial single-zone flow
If the patch zone is absent, the formation becomes a single-zone aquifer system. For the case of the isotropic aquifer and partially penetrating well, the Laplace-domain solutions of (10) and (11)
reduce to s(r, z, p) = Q 4T 2 rw K0(q6r) pq6K1(q6rw)+ Q 4T 4 (b2− b1)rw × ∞ n=1 K0(q5r) pq5K1(q5rw) F(b1, b2) cos(wnz) (17)
The limit of (17) as rw approaches zero can be written as
s(r, z, p) = Q 4T rwlim→0 2 rw K0(q6r) pq6K1(q6rw)+ Q 4T rwlim→0 4 (b2− b1)rw × ∞ n=1 K0(q5r) pq5K1(q5rw) F(b1, b2) cos(wnz) (18)
Carslaw and Jaeger[21] gave a formula
I0(x)K1(x) + K0(x)I1(x) =
1
x (19)
According to I0(0) = 1 and I1(0) = 0, the limit of (19) as x → 0 gets
lim x→0[x K1(x)] = 1 (20) Accordingly, (18) reduces to s(r, z, p) = Q 4T 2K0(q6r) p + Q 4T 4 (b2− b1)× ∞ n=1 K0(q5r) p F(b1, b2) cos(wnz) (21)
which is the Laplace-domain solution for a line source case presented in Hantush[3].
2.5. Dimensionless solutions
The dimensionless variables are defined as = Kz/Kr, = Ss/Kr, = Kr2/Kr1, = Krt/Ssrw2,
= r/rw, LD= L/rw, B1= b1/rw, B2= b2/rw,wnD= n/LD, = s(4T )/Q and = s(4T )/
Q where represents the ratio of vertical hydraulic conductivity to horizontal hydraulic
conduc-tivity, represents the ratio of specific storage to horizontal hydraulic conductivity, represents the ratio of formation horizontal hydraulic conductivity to patch horizontal hydraulic conductivity,
represents the dimensionless time during the pumping, represents the dimensionless distance
from the centreline of well, LD represents the dimensionless thickness of confined aquifer,
represents the dimensionless drawdown in the Laplace domain and represents the dimensionless drawdown in the time domain.
The Laplace-domain solutions for dimensionless average drawdowns of (12) and (13) are
1(, p) = 2 p 3DI0(q3D) + 4DK0(q3D) ∞D + 1 p 4 (B2− B1)(B2 − B1) × ∞ n=1 1D 0D I0(q1D) +2D 0D K0(q1D) F(B1, B2)F(B1, B2) (22)
and 2(, p) = 2 p 5DK0(q4D) ∞DK0(q4D1) + 1 p 4 (B2− B1)(B2 − B1) × ∞ n=1 DK0(q2D) 0DK0(q2D1) F(B1, B2)F(B1, B2) (23) 3. NUMERICAL IMPLEMENTATION
The Laplace transforms are commonly used to solve the differential and integral equations. In many engineering problems, the Laplace-domain solutions for the mathematical models are tractable, yet the corresponding solutions in the time domain may not be easily solved. Under such circumstances, the methods of numerical Laplace inversion such as the Stehfest method[22], Crump method [23], or Talbot method[24] may be used. The Laplace inversion transform of (22) and (23) are performed to three decimal places using the routine INLAP of IMSL[19], developed according to the work of de Hoog et al.[18]. 0.1 1 10 100 1000 0 4 8 12 Di m e n s io nle s s d ra w dow n Our solution Hantush's solution ρ = 1 ρ = 5 τ/ρ2
Figure 2. Comparison of the results between our solution and Hantush’s solution for = 0.1 when = 1 or 5.
4. DISCUSSION OF RESULTS
4.1. Effect of well radius
The effect of well radius on the drawdown due to a constant-flux pumping can be clearly explored by comparing the present solution with the Hantush solution[3]. The dimensionless drawdown curves according to our solution and the Hantush solution[3] is plotted in Figure 2 for = 0.1 when
= 1 or 5. For comparison purpose, the axis of dimensionless time is chosen as /2(T t/Sr2)
which was also used in Hantush[3]. Figure 2 indicates that neglecting the effect of a well radius may make significant error in estimated dimensionless drawdowns, especially is small. For = 1 (at a well), the differences of dimensionless drawdowns between two solutions are large when
/2<1 and very small when /2100. For = 5, the dimensionless drawdowns for the solutions
with and without considering the well radius are almost identical. In fact, both the dimensionless drawdown curves approach Hantush’s equation at very large dimensionless time.
4.2. Effect of anisotropy
This section investigates the effect of the anisotropy for = 0.1, = 1, 1= 4 and LD= 200
when = 0.01, 0.1 or 1. The vertical flow occurs near a pumping well when the well is partially
1 1E+001 1E+002 1E+003 1E+004 1E+005 1E+006 1E+007 1E+008 0 20 40 60 80 Dimensionless drawdown 0.01 0.1 1 Dimensionless time ( τ )
Figure 3. Dimensionless drawdown versus dimensionless time() for = 0.1, = 1, 1= 4, and LD= 200 when = 0.01, 0.1 or 1.
1 1E+001 1E+002 1E+003 1E+004 1E+005 1E+006 1E+007 1E+008 0 20 40 60 80 100 Dimensionless drawdown Dimensionless time ( τ ) ρ1 25 12 8 4 2 1 (no patch)
Figure 4. The effect of dimensionless patch thickness on dimensionless drawdown dis-tribution for = 10, = 1, = 0.1, B1= 20, B2= 180 and LD= 200. The dimensionless
patch thickness(1) ranges from 1 (no patch) to 25.
penetrated. Therefore, the vertical hydraulic conductivity is an important parameter to be consid-ered. For field problems, the ratio of vertical hydraulic conductivity to radial hydraulic conductivity
() ranges from 0.01 to 1. Figure 3 shows that the difference of dimensionless drawdowns is
ap-parent when is large; in contrast, the difference of dimensionless drawdowns is small when is small, i.e.<100. On the other hand, the effect of on dimensionless drawdown is significant at large dimensionless time. In addition, the slope of three curves is approximately identical when
105. This result indicates that the vertical flow effect is noticeable when is small. Obviously,
the erroneous results for a pumping-test data analysis performed in an anisotropic aquifer will be made if is assumed to be one.
4.3. Effect of patch thickness
The effect of dimensionless patch thickness on dimensionless drawdown distribution is displayed in Figure 4. The curves are plotted for = 10, = 1, = 0.1, B1= 20, B2= 180, and LD= 200 when
1ranges from 1 to 25. Note that the dimensionless patch thickness is equal to1− 1 and 1= 1
represents no patch case. The permeability of the patch for this case is one order of magnitude lower than that of a formation ( = 10). With a thicker patch zone, the presence of a patch zone is
identifiable for a lower permeability. The dimensionless drawdowns for1= 2–25 are greater than that of an uniform medium (1= 1) at early dimensionless time, reflecting the effect of a patch zone. Obviously, without considering the presence of a patch for a two-zone aquifer system, the predicted drawdown will be under-estimated, for a negative patch and over-estimated for a positive patch. Notice that the slope of these dimensionless drawdown curves tends to equal that of an uniform medium when 104, implying that the patch effect diminishes at large dimensionless time.
4.4. Effect of well partial penetration
When = 0.1 or 10, Figure 5 depicts the relationship of dimensionless drawdown versus
dimen-sionless time for = 1, = 1, 1= 4, LD= 200, = 0.1, 0.4, 0.8 and 1.0. Note that = 1
repre-sents a fully penetrating well case. The dimensionless drawdown tends to increase rapidly with
10 10 10 10 10 10 10 10 10 0.0 100.0 200.0 Dimensionless drawdown 0.1 0.4 0.8 1.0 Symbol Dimensionless time ( τ ) φ
Figure 5. The plot of dimensionless drawdown versus dimensionless time for = 1, = 1, 1= 4, LD= 200, = 0.1, 0.4, 0.8 and 1 when = 0.1 or 10. The dot-ted line represents the dimensionless time-drawdown curve for = 0.1 and the solid
dimensionless time and stabilize when dimensionless time is very large (104 for = 0.1 and
102 for = 10). The dimensionless drawdown with a partially penetrating well significantly
differs from that with a fully penetrating well. Obviously, the dimensionless drawdown increases with decreasing at the same dimensionless time. The effect of a well partial penetration increases with dimensionless time. In addition, the effect of a well partial penetration on the drawdown with a negative patch( = 0.1) is larger than that with a positive patch ( = 10).
5. CONCLUSIONS
New Laplace-domain solutions had been developed for a constant pumping at a partially penetrating well in a heterogeneous aquifer. The solutions were derived to account for the effects of the patch thickness and well partial penetration on the drawdown distributions. The derived solution considers the effects of well radius and provides appropriate mathematical models for the analyses of pumping test data. An efficient numerical inversion approach is used for evaluating this Laplace-domain solution. The results show that this solution can be used to investigate the effects of the patch thickness, well radius and well partial penetration on the drawdown distributions. The solution for the case with a well radius has shown to reduce to that presented by Hantush[3] if the well radius is neglected. This study demonstrates that Hantush’s solution gives significant errors in the drawdown when the observation well is close to a pumping well and/or the pumping time is very small.
APPENDIX A: DERIVATIONS OF (10) AND (11)
The solutions of drawdown within the patch and formation zones are derived via Laplace transform with respect to time variable t and the finite Fourier transform with respect to spatial variable z. The appropriate finite Fourier transform is given by[25]
F[s(z)] =s(wn) =
L
0
s(z) cos(wnz) dz, 0zL (A1)
wherewn= n/L, n = 0, 1, 2, . . . . The transform has following operational property:
F d2s(z) dz2 = (−1)n ds(z) dz z=L − ds(z) dz z=0 − w2 ns(wn) (A2)
Applying Laplace transform and the finite Fourier cosine transform, (1) and (2) give the following subsidiary equations: d2s1(r, wn, p) dr2 + 1 r ds1(r, wn, p) dr = q 2 1s1(r, wn, p), rwrr1 (A3) and d2s2(r, wn, p) dr2 + 1 r ds2(r, wn, p) dr = q 2 2s2(r, wn, p), r1r<∞ (A4)
where p is the Laplace transform variable of time variable t,s is the transformed drawdown,
q1=
1w2n+ 1p, and q2=
The transformed boundary conditions are s2(∞, wn, p) = 0 (A5) and ds1(rw, wn, p) dr = − 1 p Q 2rw(b2− b1)Kr1 F(b1, b2) (A6)
The continuity conditions required at the interface between the patch and formation zones are
s1(r1, wn, p) =s2(r1, wn, p) (A7) and ds1(r1, wn, p) dr = ds2(r1, wn, p) dr (A8)
The general solutions of (A3) and (A4) are
s1(r, wn, p) = C1I0(q1r) + C2K0(q1r) (A9)
and
s2(r, wn, p) = C3I0(q2r) + C4K0(q2r) (A10)
where C1, C2, C3and C4are the undetermined constants.
Substituting (A9) and (A10) into (A5)–(A8), one obtains
C1= − 1 p Q 2rw(b2− b1)Kr1 1 0 (A11) C2= − 1 p Q 2rw(b2− b1)Kr1 2 0 (A12) C3= 0 (A13) and C4= − 1 p Q 2rw(b2− b1)Kr1 0K0(q2r1) (A14)
Consequently, the solutions of the drawdowns within the patch and formation zones can be obtained by substituting the constants of (A11)–(A14) into (A9) and (A10) as
s1(r, wn, p) = − 1 p Q 2rw(b2− b1)Kr1 F(b1, b2) 1 0 I0(q1r) + 2 0 K0(q1r) (A15) and s2(r, wn, p) = − 1 p Q 2rw(b2− b1)Kr1 F(b1, b2) K0(q2 r) 0K0(q2r1) (A16)
Applying the inverse finite Fourier transform gives s1(r, z, p) = Q 4T2 1 p 2T2 rwT1 3I0(q3r) + 4K0(q3r) ∞ + Q 4T2 1 p 4T2 (b2− b1)rwT1 × ∞ n=1 1 0 I0(q1r) +2 0 K0(q1r) F(b1, b2) cos(wnz) (A17) and s2(r, z, p) = Q 4T2 1 p 2T2 rwT1 5K0(q4r) ∞K0(q1r1) + Q 4T2 1 p 4T2 (b2− b1)rwT1 × ∞ n=1 K 0(q2r) 0K0(q2r1) F(b1, b2) cos(wnz) (A18)
Equations (A17) and (A18) are, respectively, the Laplace-domain solutions for the drawdowns within the patch and formation zones.
NOMENCLATURE
b1 lower z co-ordinate of well screen
b2 upper z co-ordinate of well screen
B b/rw
B b/rw
Kr hydraulic conductivity in a radial direction
Kz hydraulic conductivity in a vertical direction
L thickness of confined aquifer
LD L/rw
S storage coefficient
Ss specific storage
s drawdown distribution
T transmissivity
t time from the star of pumping
p Laplace variable
Q constant flow rate into or out the wellbore
q1 1w2n+ 1p q2 2w2n+ 2p q3 1p q4 2p q5 w2 n+ p
q6
p
qnD qn× rw, n = 1, 2, . . . , 6
wnD n/LD, n = 1, 2, . . .
r radial distance from the centreline of well
r1 outer radius of patch region
rw radius of the pumping well
wn n/L, n = 0, 1, 2, . . .
z vertical distance from a lower impermeable layer
I0(u), K0(u) modified Bessel functions of the first and second kinds of order zero
I1(u), K1(u) modified Bessel functions of the first and second kinds of order one
1I0(q1r1) + 2K0(q1r1) 0 q1[2K1(q1rw) − 1I1(q1rw)] 1 q1K0(q2r1)K1(q1r1) − q2K0(q1r1)K1(q2r1) 2 q1I1(q1r1)K0(q2r1) + q2I0(q1r1)K1(q2r1) 3 q3K0(q4r1)K1(q3r1) − q4K0(q3r1)K1(q4r1) 4 q3I1(q3r1)K0(q4r1) + q4I0(q3r1)K1(q4r1) 5 3I0(q3r1) + 4K0(q3r1) ∞ q3[4K1(q3rw) − 3I1(q3rw)] D 1D(q1D1) + 2DK0(q1D1) 0D q1D[2DK1(q1D) − 1DI1(q1D)] 1D q1DK0(q2D1)K1(q1D1) − q2DK0(q1D1)K1(q2D1) 2D q1DI1(q1D1)K0(q2D1) + q2DI0(q1D1)K1(q2D1) 3D q3DK0(q4D1)K1(q3D1) − q4DK0(q3D1)K1(q4D1) 4D q3DI1(q3D1)K0(q4D1) + q4DI0(q3D1)K1(q4D1) 5D 3DI0(q3D1) + 4DK0(q3D1) ∞D q3D[4DK1(q3D) − 3DI1(q3D)] n Kzn/Krn, n = 1, 2 n SSn/Krn, n = 1, 2 Kr2/Kr1 r/rw s(4T )/Q Krt/Ssrw2 (B2− B1)/LD F(b1, b2) [sin(wnb2) − sin(wnb1)]/wn F(b1, b2) [sin(wnb2) − sin(wnb1)]/wn Subscripts D dimensionless 1 patch zone 2 formation zone w pumping well
ACKNOWLEDGEMENTS
The authors appreciate anonymous reviewers and editor for their constructive comments and suggested revisions that help improve the clarity of our presentation. This study was partly supported by the Taiwan National Science Council under the grant NSC 93-2218-E-009-056.
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