A novel analytical solution for constant-head test in a patchy aquifer

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS Int. J. Numer. Anal. Meth. Geomech., 2006; 30: Published online 12 May 2006 in Wiley InterScience ( A novel analytical solution for constant-head test in a atchy aquifer Shaw-Yang Yang 1 Hund-Der Yeh 2, *,y 1 Deartment of Civil Engineering, Vanung University, Chungli, Taiwan 2 Institute of Environmental Engineering, National Chiao Tung University, Hsinchu, Taiwan SUMMARY A mathematical model describing the hydraulic head distribution for a constant-head test erformed in a well situated at the centre of a atchy aquifer is resented. The analytical solution for the mathematical model is derived by the Lalace transforms the Bromwich integral method. The solution for the hydraulic head has been shown to satisfy the governing equations, related boundary conditions, continuity requirements for the hydraulic head flow rate at the interface of the atch outer regions. An efficient numerical aroach is roosed to evaluate the solution, which has an integral covering an integration range from zero to infinity an integr consisting the roduct square of the Bessel functions. This solution can be used to roduce the curves of dimensionless hydraulic head against dimensionless time for investigating the effect of the contrast of formation roerties on the dimensionless hydraulic head distribution. Define the ratio of outer-region transmissivity to atch-region transmissivity as a: The dimensionless hydraulic head for a ¼ 0:1 case is about 2.72 times to that for a ¼ 10 case at dimensionless large time (e.g. t510 6 ) when the dimensionless distance ðrþ equals 10. The results indicate that the hydraulic head distribution highly deends on the hydraulic roerties of two-zone formations. Coyright # 2006 John Wiley & Sons, Ltd. Received 7 August 2004; Revised 15 August 2005; Acceted 2 December 2005 KEY WORDS: constant-head test; atchy aquifer; analytical solution; numerical aroach 1. INTRODUCTION The determination of hydraulic arameters through an aquifer test is commonly used in site characterization. The constant-flux constant-head tests are two different kinds of aquifer tests. The former requires maintaining a constant well discharge (or injection) measure the *Corresondence to: Hund-Der Yeh, Institute of Environmental Engineering, National Chiao Tung University, 75, Po-Ai St., Hsinchu, Taiwan. y hdyeh@mail.nctu.edu.tw Contract/grant sonsor: Taiwan National Science Council; contract/grant number: NSC Z Coyright # 2006 John Wiley & Sons, Ltd.

2 1214 S.-Y. YANG AND H.-D. YEH drawdowns at observation wells. The constant-head test is frequently carried out in a single well for which a constant head must be maintained in order to measure the transient flow rate across the wellbore. The constant-head test is suitable to use in the low-ermeability formations for estimating aquifer arameters. An aquifer having a small region of anomalous hydrogeological roerties may be called a well in a cylindrical inhomogeneity (atchy aquifer). A atchy aquifer may have the radius of heterogeneous cylinder (atchy region) u to 60 m [1] can be considered as a comosite aquifer system. During well construction, a wellbore skin of finite thickness may be develoed due to the invasion of drilling mud into the adjacent formation or the removal of fine articles from the surrounding formation by extensive well develoment; consequently, the original homogeneous aquifer may become a comosite aquifer system also. In any case, the thickness of skin zone may range from a few millimeters to several meters, thus must be considered in uming-test-data analyses [2]. Barker Herbert [1] gave a Lalace-domain solution for the roblem of a well umed with a constant rate situated at the centre of disc of a atchy aquifer. Actually, the aquifer is generally of low transmissivity in the atch region of relatively high transmissivity in the outer region under most field conditions [1, 2]. Concets of the infinitesimal finite-thickness regions with a heterogeneous roerty in contrast to that of the outer region are commonly used to investigate the effect of the atch region on the result of well constant-head tests. Streltsova [3, 4] considered an infinitesimal heterogeneous aquifer used the skin factor to reresent the heterogeneous aquifer effect. Novakowski [5] develoed a Lalace-domain solution for the transient flow rate across the wellbore for comosite aquifers with considering the effects of the finite-thickness heterogeneous aquifer well artial enetration. Curves of dimensionless hydraulic head versus dimensionless time were develoed to investigate the influences of the finite-thickness heterogeneous aquifer well artial enetration on the hydraulic head distribution. Markle et al. [6] develoed a model for comosite aquifer with a artially enetrating well that has a finite-thickness heterogeneous aquifer intersects a single vertical fracture. Their results show that the finite-thickness heterogeneous aquifer the well artial enetration can affect the transient flow rate across the wellbore. For a constant-head test in a radial comosite aquifer system, Chang Chen [7] gave the Lalace-domain solutions for the hydraulic heads flow rate across the wellbore. However, their solution for the head distribution in an undisturbed aquifer is incorrect. In this study, we derive a closed-form solution (i.e. time-domain solution) to describe the hydraulic head distribution for a constanthead test in a atchy aquifer with a fully enetrating well. The solution is exressed in an integral form that has an integration range from zero to infinity. The integral is difficult to evaluate due to the integr not only consisting of the roduct square of the Bessel functions, but also having a singularity at the origin. Therefore, we resent a numerical aroach including a root-search scheme, the Gaussian quadrature, the Shanks method to evaluate this solution. This aroach is in a very efficient way when evaluating the solution. Finally, the effects of hydrogeological roerties thickness of the atch region on dimensionless hydraulic head distribution are exlored. In addition, for dealing with the roblem of the well constant-head test, this new solution can be used as a fundamental tool for testing benchmarking numerical codes, erforming sensitivity analysis for model arameters, even identifying the hydrogeological roerties when couling with an otimization algorithm in the test-data analyses.

3 ANALYTICAL SOLUTION FOR CONSTANT-HEAD TEST PROBLEM STATEMENT 2.1. Groundwater flow equations A atchy aquifer system with a well for uming tests is deicted in Figure 1. The confined aquifer consists of two regions with transmissivities T 1 for r w 4r4r 1 in the atch region T 2 for r5r 1 in the outer region. Assumtions made for the confined-aquifer solution are: (1) the aquifer is homogeneous, isotroic, has infinite-extent with a constant thickness, (2) the well is fully enetrated with a finite radius, (3) the initial hydraulic head is constant uniform throughout the whole aquifer. Using these assumtions, the equations which exress the combination of mass balance Darcy s law through the atch outer regions are, 2 h 2 þ 1 ¼ S 1 T ; r w4r4r 1 2 h 2 þ 1 ¼ S 2 T ; r 14r51 ð2þ where the subscrits 1 2, resectively, denote the atch region around the wellbore the outer region, h is the hydraulic head distribution, T is the transmissivity, S is the storage coefficient, r is the radial distance from the centre of well, r w is the radius of well, r 1 is the outer radius of the atch region (or the cylinder radius of the atch region), t is the time from the start of test. These two equations reresent the hydraulic heads within the atch outer regions, resectively. The hydraulic heads are initially assumed to be zero within both the atch outer regions, that is h 1 ðr; 0Þ ¼h 2 ðr; 0Þ ¼0 for r > r w ð3þ As r aroaches infinity, the derivative of hydraulic head with resect to the radial distance tends to zero. Such a boundary condition for the outer region may be secified 2 ð1; tþ ¼ 0 Wellbore Constant head Imermeable layer Well screen T 1 T 2 Aquifer Patch region Outer region r w Imermeable layer r 1 Figure 1. Schematic diagram of the well atchy aquifer configurations.

4 1216 S.-Y. YANG AND H.-D. YEH The boundary condition for maintaining a constant head at r ¼ r w is given by h 1 ðr w ; tþ ¼h w ð5þ where h w is the constant head around the wellbore at any time. The continuities of hydraulic head flow rate at the interface of the atch outer regions, resectively, require h 1 ðr 1 ; tþ ¼h 2 ðr 1 ; tþ ð6þ T 1 ðr 1 ; ¼ T 2 ðr 1 ; ð7þ 2.2. Derivation of closed-form solutions Alying the Lalace transform on (1) (2) subject to the initial condition (3), the solutions of the transformed governing equations with four unknown constant can be obtained. Similarly, the transformed boundary conditions of (4) (7) can be obtained when taking the Lalace transform. The Lalace-domain solutions in the atch outer regions can then be obtained by substituting the transformed boundary conditions into the solutions of the transformed governing equations subsidiary formulas. The results of %h 1 %h 2 exressed for the atch outer regions are, resectively, %h 1 ¼ h w %h 2 ¼ h w f 1 I 0 ðq 1 rþ f 2 K 0 ðq 1 rþ f 1 I 0 ðq 1 r w Þ f 2 K 0 ðq 1 r w Þ K 0 ðq 2 rþ ½r 1 q 1 Š½f 1 I 0 ðq 1 r w Þ f 2 K 0 ðq 1 r w ÞŠ where q 2 1 ¼ S 1=T 1 ; q 2 2 ¼ S 2=T 2 ; is the Lalace variable [8], I 0 ðþ K 0 ðþ are, resectively, the modified Bessel functions of the first second kinds of order zero, rffiffiffiffiffiffiffiffiffiffi S 2 T 2 f 1 ¼ K 0 ðq 1 r 1 ÞK 1 ðq 2 r 1 Þ K 1 ðq 1 r 1 ÞK 0 ðq 2 r 1 Þ ð10þ S 1 T 1 rffiffiffiffiffiffiffiffiffiffi S 2 T 2 f 2 ¼ I 0 ðq 1 r 1 ÞK 1 ðq 2 r 1 ÞþI 1 ðq 1 r 1 ÞK 0 ðq 2 r 1 Þ ð11þ S 1 T 1 where I 1 ðþ K 1 ðþ are the modified Bessel functions of the first second kinds of order one, resectively. Note that the solutions of (8) (9) were also given by Chang Chen [7]. However, their hydraulic head in the outer region is slightly different from (9). They gave a term r 1 q 2 ; instead of r 1 q 1 ; in the denominator of hydraulic head in the outer region. The time-domain solutions of (8) (9) obtained by emloying the method of the Bromwich integral [9] are h 1 ¼ h w þ 2h w Z 1 0 e ðt 1=S 1 Þu 2 t ½A 1ðu; rþb 2 ðuþ A 2 ðu; rþb 1 ðuþš du ½B 2 1 ðuþþb2 2 ðuþš u ð8þ ð9þ ð12þ

5 ANALYTICAL SOLUTION FOR CONSTANT-HEAD TEST 1217 where h 2 ¼ h w 4h w 2 r 1 Z 1 0 e ðt 1=S 1 Þu 2 t ½J 0ðkruÞB 1 ðuþþy 0 ðkruþb 2 ðuþš du ½B 2 1 ðuþþb2 2 ðuþš u 2 rffiffiffiffiffiffiffiffiffiffi S 2 T 2 A 1 ðu; rþ ¼ ½J 0 ðr 1 uþy 1 ðkr 1 uþy 0 ðruþ Y 0 ðr 1 uþy 1 ðkr 1 uþj 0 ðruþš S 1 T 1 ½J 1 ðr 1 uþy 0 ðkr 1 uþy 0 ðruþ Y 1 ðr 1 uþy 0 ðkr 1 uþj 0 ðruþš rffiffiffiffiffiffiffiffiffiffi S 2 T 2 A 2 ðu; rþ ¼ ½Y 0 ðr 1 uþj 1 ðkr 1 uþj 0 ðruþ J 0 ðr 1 uþj 1 ðkr 1 uþy 0 ðruþš S 1 T 1 ½Y 1 ðr 1 uþj 0 ðkr 1 uþj 0 ðruþ J 1 ðr 1 uþj 0 ðkr 1 uþy 0 ðruþš rffiffiffiffiffiffiffiffiffiffi S 2 T 2 B 1 ðuþ ¼ ½J 0 ðr 1 uþy 1 ðkr 1 uþy 0 ðr w uþ Y 0 ðr 1 uþy 1 ðkr 1 uþj 0 ðr w uþš S 1 T 1 ½J 1 ðr 1 uþy 0 ðkr 1 uþy 0 ðr w uþ Y 1 ðr 1 uþy 0 ðkr 1 uþj 0 ðr w uþš rffiffiffiffiffiffiffiffiffiffi S 2 T 2 B 2 ðuþ ¼ ½Y 0 ðr 1 uþj 1 ðkr 1 uþj 0 ðr w uþ J 0 ðr 1 uþj 1 ðkr 1 uþy 0 ðr w uþš S 1 T 1 ð13þ ð14þ ð15þ ð16þ ½Y 1 ðr 1 uþj 0 ðkr 1 uþj 0 ðr w uþ J 1 ðr 1 uþj 0 ðkr 1 uþy 0 ðr w uþš ð17þ where k ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 1 S 2 =T 2 S 1 ; J 0 ðþ Y 0 ðþ are, resectively, the Bessel functions of the first second kinds of order zero, J 1 ðþ Y 1 ðþ are, resectively, the Bessel functions of the first second kinds of order one. Equations (12) (13) are the closed-form solutions of hydraulic heads within the atch outer regions. Detailed derivation for the solution is given in Aendix A Dimensionless variables As a mean of exressing the solution in dimensionless form, the following dimensionless variables are defined: a ¼ T 2 =T 1 ; b ¼ S 2 =S 1 ; t ¼ T 2 t=s 2 r 2 w ; r ¼ r=r w; r 1 ¼ r 1 =r w ; %h D1 ¼ %h 1 =h w ; %h D2 ¼ %h 2 =h w ; h D1 ¼ h 1 =h w ; h D2 ¼ h 2 =h w where a reresents the ratio of outer-region transmissivity to atch-region transmissivity (ratio of the transmissivity), b reresents the ratio of outer-region storage coefficient to atch-region storage coefficient (ratio of storage coefficient), t reresents the dimensionless time during the test, r reresents the dimensionless distance from the centre of well, r 1 reresents the dimensionless thickness of the atch region, %h D1 reresents the dimensionless hydraulic head in the Lalace domain within the atch region, %h D2 reresents the dimensionless hydraulic head in the Lalace domain within the outer region, h D1 reresents the dimensionless hydraulic head in the time domain within the atch region, h D2 reresents the dimensionless hydraulic head in the time domain within the outer region.

6 1218 S.-Y. YANG AND H.-D. YEH The Lalace-domain solutions of (8) (9) on dimensionless hydraulic head in a atchy aquifer are, resectively, %h D1 ¼ 1 f 1 I 0 ðq 1 rþ f 2 K 0 ðq 1 rþ ð18þ f 1 I 0 ðq 1 Þ f 2 K 0 ðq 1 Þ %h D2 ¼ 1 ½f 1 I 0 ðq 1 r 1 Þ f 2 K 0 ðq 1 r 1 ÞŠK 0 ðq 2 rþ ð19þ ½f 1 I 0 ðq 1 Þ f 2 K 0 ðq 1 ÞŠK 0 ðq 2 r 1 Þ Accordingly, (12) (13) can also be, resectively, exressed in dimensionless form as h D1 ¼ 1 þ 2 Z 1 e btw2 =a ½A D1ðw; rþb D2 ðwþ A D2 ðw; rþb D1 ðwþš dw 0 ½B 2 D1 ðwþþb2 D2 ðwþš ð20þ w h D2 ¼ r 1 where w ¼ r w u; A D1 ðw; rþ ¼ ffiffiffiffiffi ab Z 1 0 e btw2 =a ½J 0ðkrwÞB D1 ðwþþy 0 ðkrwþb D2 ðwþš dw ½B 2 D1 ðwþþb2 D2 ðwþš w 2 ½J0 ðr 1 wþy 1 ðkr 1 wþy 0 ðrwþ Y 0 ðr 1 wþy 1 ðkr 1 wþj 0 ðrwþš ð21þ ½J 1 ðr 1 wþy 0 ðkrwþy 0 ðrwþ Y 1 ðr 1 wþy 0 ðkr 1 wþj 0 ðrwþš ð22þ A D2 ðw; rþ ¼ ffiffiffiffiffi ab ½Y0 ðr 1 wþj 1 ðkr 1 wþj 0 ðrwþ J 0 ðr 1 wþj 1 ðkr 1 wþy 0 ðrwþš ½Y 1 ðr 1 wþj 0 ðkr 1 wþj 0 ðrwþ J 1 ðr 1 wþj 0 ðkr 1 wþy 0 ðrwþš ð23þ B D1 ðwþ ¼ ffiffiffiffiffi ab ½J0 ðr 1 wþy 1 ðkr 1 wþy 0 ðwþ Y 0 ðr 1 wþy 1 ðkr 1 wþj 0 ðwþš ½J 1 ðr 1 wþy 0 ðkr 1 wþy 0 ðwþ Y 1 ðr 1 wþy 0 ðkr 1 wþj 0 ðwþš ð24þ B D2 ðwþ ¼ ffiffiffiffiffi ab ½Y0 ðr 1 wþj 1 ðkr 1 wþj 0 ðwþ J 0 ðr 1 wþj 1 ðkr 1 wþy 0 ðwþš ½Y 1 ðr 1 wþj 0 ðkr 1 wþj 0 ðwþ J 1 ðr 1 wþj 0 ðkr 1 wþy 0 ðwþš ð25þ The aquifer roerties are constant through the whole aquifer if the aquifer is homogeneous. Accordingly, both a b are equal to unity the variables q 1 q 2 in (18) (19), resectively, are set to equal q: The Lalace-domain solutions of (18) (19) reduce to the solution for a uniform formation as %h D ¼ 1 K 0 ðqrþ ð26þ K 0 ðqþ In addition, the solutions of (20) (21) describing the dimensionless hydraulic head distributions can then algebraically reduce to h D ðt; rþ ¼1 2 Z 1 e ½J 0ðwÞY tw2 0 ðrwþ Y 0 ðwþj 0 ðrwþš dw ½J0 2ðwÞþY 0 2ðwÞŠ ð27þ w 0

7 ANALYTICAL SOLUTION FOR CONSTANT-HEAD TEST 1219 Notably, (27) is identical to the solution given in Reference [10,. 335, (6)] for a heat conduction roblem. Equation (27), or in a slightly different form, can also be seen in some engineering alications, e.g. Goldstein [11] in a viscous fluid roblem; Carslaw Jaeger [12], Harvard s roblem reort [13], Jaeger [14] in the heat conduction roblems; Hantush [15] in the groundwater roblem. Peng et al. [16] resented an efficient numerical aroach to evaluate (27) with accuracy to five decimal laces for very wide ranges of dimensionless distances times. 3. NUMERICAL STUDY 3.1. Curves of the integr The atchy aquifer system will reduce to a homogenous (uniform) aquifer system if a ¼ 1 (i.e. T 1 ¼ T 2 ). In contrast, the aquifer has T 1 > T 2 for a51 T 1 5T 2 for a > 1: In the following discussion, the ratio of storage coefficient b dimensionless time t are chosen as one. Figures 2(a) (c) show the curves of integrs in (20) (21) against the argument w for r 1 ¼ 3 ðr 1 ¼ 3r w Þ a ¼ 0:1; 1 10 when t ¼ 1: Figures 2(a) (b) demonstrate the lots of integr in (20) for r ¼ 2 ðr ¼ 2r w Þ 3. Within the atch region (i.e. 14r4r 1 ), the curve contains infinite sinuous waves has the same roots for various values of a: When a41; the curve of integr is smooth since the amlitudes of oscillatory waves after the first zero-root are all smaller than : The amlitude of the wave increases with a but for larger w the amlitude tends toward zero. Note that each wave has two eaks for the case with a51 r43: Figure 2(b) shows that the sinuous waves of integr in (21) for r ¼ 10 when a ¼ 0:1; 1 or 10. As a increases the waves decrease in amlitude but increase in wavelength. Figure 3 is an analysis of the effect of different sizes of atch regions, i.e. r 1 ¼ 35 a ¼ 0:1 10, for r ¼ 10 when t ¼ 1: The roots on the integr for r 1 ¼ 3 are slight smaller than those for r 1 ¼ 5ifa ¼ 0:1: On the contrary, the roots for r 1 ¼ 3 are slight larger than those for r 1 ¼ 5ifa ¼ 10: The atch regions change from 3 to 5; consequently, the roots shift slightly along the dimensionless distance. Obviously, the differences of the roots are not aarent for the different sizes of atch regions. Thus, the thickness of the atchy region has significant effect on the integrs. Figures 2 3 demonstrate the atterns of curves that are helful in obtaining the roots of integrs which will be used when (20) (21) are comuted numerically Numerical integrations The imroer integrals of (20) (21) cannot be directly evaluated due to the comlexity of both the Bessel functions their roducts terms aeared in integrs. Each integral is transformed as a sum of infinite series each term of series reresents an area under the integr between two consecutive roots. The roots of integr can be obtained by a rootsearch scheme along a horizontal axis [16]. The Shanks method [17] is alied to accelerate the convergence when evaluating the infinite series. In a root-search scheme, an initial guess value is required to aroximate the roots of the integr along the w-axis. For r > 1 a reasonable ste size for a uniform solution is D ¼ =ðr 1Þ; [13]. This is also an aroriate choice for a atchy-aquifer solution when r is small, choosing the ste size as D results in a good aroximation for the first few roots of integrs in (20) (21). The first root, w 1 ; is then aroximated by =ðr 1Þ: Therefore, the

8 1220 S.-Y. YANG AND H.-D. YEH Intergr in (20) symbol Intergr in (20) symbol (a) w -0.6 (b) w symbol Intergr in (21) (c) Figure 2. Plots of integrs in (20) (21) versus w for t ¼ 1 a ¼ 0:1; 1 10 when: (a) r ¼ 2; (b) r ¼ 3; (c) r ¼ 10: w ste size D 1 from the origin to the first root aroximately equals =ðr 1Þ: However, using the ste size D 1 to estimate the large roots of the integr will be very oor when r is large. Thus, a different aroach for determining the ste size is needed for large r: A reasonable aroach for the second ste size is chosen D 2 ¼ w 1 ; the second root w 2 is aroximately equal to 2w 1 : Similarly, the remaining ste sizes D i are chosen as w i 1 w i 2; the remaining roots are aroximately equal to w i ¼ w i 1 þ D i where i ¼ 3; 4;... : The integrs in (20) (21) are oscillatory functions as dislayed in Figures 2(a) (c). A curve with double-eak waves can be observed for the case with a > 1 14r4r 1 ; in contrary, a curve with single-eak waves can be seen for the case that r > r 1 : Obviously, the roots for the case with double eaks are difficult to obtain by the conventional root-search aroaches such as Newton s method. The bisection method may be slow in convergence when

9 ANALYTICAL SOLUTION FOR CONSTANT-HEAD TEST Intergr in (21) symbol α ρ w Figure 3. Plots of integr in (21) versus w for t ¼ 1; r ¼ 10; a ¼ 0:1 1 when the atch region r 1 ¼ 3or5. emloying to obtain the roots, yet the advantage of this method is that it is guaranteed to work [18]. The bisection method is thus adoted to obtain the sequent roots of integrs in this study. The initial ste size is chosen as D i =10 for the ith root to ensure that the first guess root ðw 1 i ¼ w i 1 þ D i =10Þ is smaller than the target root ðw i Þ: The integrs in (20) (21), Fðw i Þ; are evaluated in a forward direction (i.e. w k i ¼ w i 1 þ kd i =10; k ¼ 1; 2;...) until the condition Fðw k i ÞFðwkþ1 i Þ50 is met to ensure that a target root lies between w k i w kþ1 i : The bisection rocedure is alied to search each root successively until the criterion Fðw i Þ is met. The values of integrs in (20) (21) become infinity as w close to zero aroach zero for large w as indicated in Figures 2(a) (c). Aarently, (20) (21) are not easy to accurately evaluate esecially when w is very close to the origin (singular oint). Harvard s roblem reort [13] rovided an aroach when a ¼ 1 using an infinite series exansion to remove the singularity of integr at the origin. Their aroach can evaluate the integr with good accuracy for a uniform aquifer system when w contains a singular oint. However, their aroach does not work in removing the singularity when a=1: Thus, the following aroach using small increment is roosed in the evaluations of (20) (21). Let e be a very small value, say : Starting from e; the Gaussian quadrature [18] is emloyed to erform the numerical integration for (20) (21) iecewise along the w-axis. The distance between two adjacent roots of integr is divided into several small ste sizes. Within each ste, both the six-term tenterm formulas of the Gaussian quadrature are used to carry out the integration for the same area under the integr. The integration results after alying the six-term ten-term formulas within a small ste size are, resectively, defined as A 6 A 10 : The absolute difference of these two results is defined as DA ¼jA 10 A 6 j: If DA > 10 7 ; a half-ste size ðdw=2þ will be used the same integration rocedure will be reeatedly alied until DA510 7 : If DA510 7 ; the same ste size ðdwþ is used when DA410 7 ; otherwise, a double ste size ð2dwþ is used (i.e. DA510 8 ) for next ste. This rocedure ensures that each integration

10 1222 S.-Y. YANG AND H.-D. YEH result over a small ste has the accuracy at least to seven decimal laces. The initial ste size, Dw i ; for evaluating (20) (21) is chosen as ðw i w i 1 Þ=2: The roosed integration rocedure is adoted over the w-axis. Note that the last ste size should be chosen such that the end of the ste should be located right at a larger root. In short, the area between any two adjacent roots, which reresents a term of infinite series, is obtained simly by adding the sum of the integration results form those small stes within the roots. Finally, the evaluation of infinite series may have the roblem of slow convergence. Therefore, we emloy the Shanks method to accelerate the convergence when evaluating the summation of alternating series. 4. RESULTS Because of no ublished information on the storage coefficient of atchy aquifer, several investigators (e.g. References [2, 5, 19]) assumed that the storage coefficients of the atch outer regions are the same to simlify the roblems. Accordingly, dimensionless storage, b; is set as unity in this study. Equations (20) (21), the time-domain solutions, are evaluated by the roosed aroach. In addition, all evaluations are conducted in double-recision format the convergence criterion for the Shanks method is chosen as 10 5 : The curves of dimensionless hydraulic head versus dimensionless time are develoed to investigate the influences of the atchy aquifer roerties thickness on the hydraulic head distribution in a atchy aquifer system Dimensionless hydraulic head ρ = 2 ρ = Symbol α E-3 1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 1E+9 1E+10 Dimensionless time Figure 4. The effect of hydrogeological roerties in the atch region on the shae of curves for r 1 ¼ 3 r ¼ 2 10 when a ¼ 0:1; 1 or 10.

11 ANALYTICAL SOLUTION FOR CONSTANT-HEAD TEST 1223 Figure 4 exhibits the curves of dimensionless hydraulic head versus dimensionless time for r 1 ¼ 3r ¼ 2 10 when a ¼ 0:1; 1 or 10. Note that the dimensionless thickness of the atch region is equal to r 1 1: At either r ¼ 2orr ¼ 10; the case of a ¼ 0:1 yields the highest dimensionless hydraulic head, the case of a ¼ 1 the second highest, the case of a ¼ 10 the lowest under the same dimensionless time. Lower transmissivity in the atch region roduces lower flow rate toward the outer region during the well test results in smaller dimensionless hydraulic head in the atch outer regions. On the other h, larger transmissivity in the atch region yields larger flow rate across the wellbore results in larger dimensionless heads in the atch outer regions. Table I deicts the ratio of the dimensionless atch-aquifer hydraulic head to the dimensionless single-aquifer hydraulic head when the dimensionless thickness of the atch region, r 1 ; equals 3. At r ¼ 2; the ratio may reach 1.48 if a ¼ 0: if a ¼ 10 when dimensionless time is very small (say, t ¼ 10); on the other h, the ratio is 1.05 if a ¼ 0: if a ¼ 10 when dimensionless time is very large (say, t ). In addition, at r ¼ 10; the ratio may reach 3.41 if a ¼ 0: if a ¼ 10 when t ¼ 10; in contrast, the ratio is 1.09 for a ¼ 0: for a ¼ 10 when t : Those results demonstrate that the difference of dimensionless hydraulic heads between the atch outer regions for a > 1 is larger than that for a51; esecially the dimensionless time is very small. Figure 5 dislays the lot of dimensionless hydraulic head versus dimensionless distance for r 1 ¼ 3 t ¼ 1; 10 2 ; when a ¼ 0:1 or 10. This figure demonstrates the effects of the hydrogeological roerties of the atch region on the shae of curves of the dimensionless hydraulic head. The head increases with dimensionless time in both the atch outer regions; contrarily, the head decreases with increasing dimensionless radial distance. Note that Figure 5 indicates that the dimensionless hydraulic head for a ¼ 0:1 case is about 2.72 times to that for a ¼ 10 case at dimensionless large time (e.g. t510 6 ) when the dimensionless distance ðrþ equals 10. At the interface of the atch outer regions (i.e. r ¼ r 1 ¼ 3), the sloes of curves Table I. The ratio of dimensionless atch-aquifer hydraulic head to dimensionless single-aquifer hydraulic head for r 1 ¼ 3r¼210whena¼0:1or 10. r ¼ 2 r ¼ 10 t h D;0:1:1 h D;10:1 h D;0:1:1 h D;10:1 1:0E þ :0E þ :0E þ :0E þ :0E þ :0E þ :0E þ :0E þ :0E þ :0E þ Note that h D;0:1:1 denotes the ratio of dimensionless hydraulic head when a ¼ 0:1 to dimensionless aquifer hydraulic head when a ¼ 1; h D;10:1 denotes the ratio of dimensionless hydraulic head when a ¼ 10 to dimensionless aquifer hydraulic head when a ¼ 1:

12 1224 S.-Y. YANG AND H.-D. YEH 1.0 Patch region Outer region Symbol 1.E0 1.E2 1.E4 1.E6 0.8 Dimensionless hydraulic head = 0.1 = Dimensionless distance Figure 5. A lot of dimensionless hydraulic head versus dimensionless distance for r 1 ¼ 3 t ¼ 1; 10 2 ; when a ¼ 0:1 or Symbol ρ α = 0.1 Dimensionless hydraulic head α = 1 α = E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6 Dimensionless time Figure 6. The curves of dimensionless hydraulic head versus dimensionless time for r ¼ 10 r 1 ranges from 1 (uniform aquifer) to 5 when a ¼ 0:1 or 10.

13 ANALYTICAL SOLUTION FOR CONSTANT-HEAD TEST 1225 are markedly different because of the contrast of transmissivity between the atch outer regions. When a51; the sloe of curve within the atch region is obviously smaller than that within the outer region due to the fact of larger transmissivity of the atch region demonstrated in Figure 5. In contrast, the sloe of curve within a atch region is larger than that within the outer region when a > 1: Obviously, the resence of the atch region influences the hydraulic head distribution in a atchy aquifer. The effect of the atch-region thickness on dimensionless hydraulic head distribution is deicted in Figure 6 where the curves are lotted against dimensionless time at r ¼ 10 dimensionless thickness of the atch region ðr 1 Þ ranging from 2 to 5 when a ¼ 0:1; 1 or 10. This figure demonstrates that increasing dimensionless atch-region thickness increases dimensionless hydraulic head for a ¼ 0:1 decreases dimensionless hydraulic head for a ¼ 10: 5. CONCLUSIONS The mathematical model describing the radial groundwater flow in a confined atchy aquifer system is resent for well constant-head tests. The closed-form solution exressed in term of hydraulic head is derived by the Lalace transforms the Bromwich integral method. In addition, we also have roven that the solution satisfies the governing equations, boundary conditions, continuity requirements at the interface of the atch outer regions. An efficient numerical aroach is roosed to evaluate the solution. The numerical aroach includes a root-search scheme for finding the consecutive roots of integr, the Gaussian quadrature for erforming the numerical integration, the Shanks method for accelerating the convergence when evaluating the Bessel functions the alternating infinite series transformed from the integral. Finally, the influences of the roerties thickness of the atch region on dimensionless hydraulic head distribution are exlored. This new time-domain solution can be used for identifying the aquifer arameters, verifying the numerical code, or studying the effect of the contrast of the formation roerties on the hydraulic head distribution. Three conclusions are drawn as follows: 1. The effect of hydraulic head in a atchy aquifer system for T 1 5T 2 is larger than that for T 1 > T 2 : At r ¼ 2; the ratio may reach 1.48 if a ¼ 0: if a ¼ 10 when dimensionless time is very small; on the other h, the ratio is 1.05 if a ¼ 0: if a ¼ 10 when dimensionless time is very large. In addition, at r ¼ 10; the ratio may reach 3.41 if a ¼ 0: if a ¼ 10 when dimensionless time is very small; in contrast, the ratio is 1.09 for a ¼ 0: for a ¼ 10 when dimensionless time is very large. 2. The resence of the finite-thickness atch region around a test well substantially influences the shae of hydraulic head distribution. The curve develoed for the case with a finite thickness of the atch region shows an abrut change in sloe at the interface of the atch outer regions. This indicates that the hydraulic head distribution deends on the hydraulic roerties of both the atch outer regions. 3. The thickness of the atch region significantly affects the distribution of dimensionless hydraulic head in a atchy aquifer system. The dimensionless hydraulic head for a ¼ 0:1 case is about 2.72 times to that for a ¼ 10 case at dimensionless large time (e.g. t510 6 ) when the dimensionless distance ðrþ equals 10. It is to conclude that a thicker atch region give higher hydraulic head when T 1 > T 2 lower hydraulic head when T 1 5T 2 :

14 1226 S.-Y. YANG AND H.-D. YEH APPENDIX A: DERIVATIONS OF (12) AND (13) The inverse Lalace transforms of (8) (9) in the time domain can be obtained by using the Lalace inversion integral [9] as h 1 ¼ 1 Z aþi1 e t %h 1 d ða1þ 2i a i1 h 2 ¼ 1 Z aþi1 e t %h 2 d ða2þ 2i a i1 where is a comlex variable a is a large, real, ositive constant, so much so that all the oles lie to the left of line ða i1; a þ i1þ: The integrs in (8) (9) contains a branch oint at ¼ 0 on the cut lane. Thus, these integrations may require the usage of the Bromwich contour integrals for the Lalace inversion. The closed contour of integr is shown in Figure A1 with a cut along the negative real axis, where d is a small ositive number. A single branch oint with a singularity (ole) at ¼ 0 exists in the integr in (8). Accordingly, this integration can be carried out by the use of contour integral. The integrals taken along BCD GHA tend to zero as R!1: Consequently, (8) can be suerseded by the sum of integrals along DE FG, the small circle EF around the origin as d! 0: In other words, the integral can then be written as Z Z Z 1 h 1 ¼ lim e t %h 1 d þ e t %h 1 d þ e t %h 1 d ða3þ d!0 2i EF DE FG R!1 y C B D R E δ x G F H A Figure A1. A lot of the closed-contour integration of the %h 1 %h 2 as a function of ¼ x þ yi for calculating the inverse Lalace transform.

15 ANALYTICAL SOLUTION FOR CONSTANT-HEAD TEST 1227 Along the small circle EF as d! 0; introducing the limiting form into the first term on the right-h side (RHS) of (A3) obtains h 11 ¼ lim h w f 1 I 0 ðq 1 rþ f 2 K 0 ðq 1 rþ ða4þ!0 f 1 I 0 ðq 1 r w Þ f 2 K 0 ðq 1 r w Þ As! 0; q 1 q 2 aroach to zero. Thus, I 0 ð0þ I 1 ð0þ aroach one zero, resectively, K 0 ð0þ K 1 ð0þ become indefinite. The result of (A4) can be obtained as h 11 ¼ h w ða5þ For the second term on the RHS of (A3) along DE, we introduce the change of variable ¼ u 2 e i T 1 =S 1 ; use the formulas [10,. 490, (25) (26)] K v ðze ð1=2þi Þ¼ 1 2 ieð1=2þvi ½ J v ðzþiy v ðzþš ða6þ I v ðze ð1=2þi Þ¼e ð1=2þvi J v ðzþ where v ¼ 0; 1; 2;... : The second term on the RHS of (A3) leads to h 12 ¼ h w i Z 1 0 e ðt 1=S 1 Þu 2 t A 1ðuÞþiA 2 ðuþ du B 1 ðuþþib 2 ðuþ u Likewise, introducing ¼ u 2 e i T 1 =S 1 ; the integral along EF gives minus the conjugate of (A8) as h 13 ¼ h w i Z 1 0 e ðt 1=S 1 Þu 2 t A 1ðuÞ ia 2 ðuþ du B 1 ðuþ ib 2 ðuþ u The closed-form solution of (12) can then be obtained by combining (A5), (A8), (A9). Also, the closed-form solution of (13) can be obtained in a similar manner. ða7þ ða8þ ða9þ APPENDIX B: PROOF OF THE CLOSED-FORM SOLUTION This section gives the roof that the time-domain solution satisfies the governing equation, boundary conditions, continuity requirements at the interface of the atch outer regions. For the outer boundary, r!1; one has J 1 ð1þ ¼ 0 Y 1 ð1þ ¼ 0: Taking the derivative of (13) with resect to r letting ¼ 4h w r!1 2 r 1 ¼ 0 Z 1 0 e ðt 1=S 1 Þu 2 t ½ kuj 1ð1ÞB 1 ðuþ kuy 1 ð1þb 2 ðuþš du ½B 2 1 ðuþþb2 2 ðuþš u 2 Therefore, the outer boundary condition of (4) is satisfied. When r ¼ r w ; A 1 ðu; r w Þ¼B 1 ðuþ A 2 ðu; r w Þ¼B 2 ðuþ; consequently, the integr in (12) becomes zero. Thus, one obtains the result of (5) which is the boundary condition for maintaining a constant head around the well. Letting r ¼ r 1 using the formula [12] J 1 ðuþy 0 ðuþ J 0 ðuþy 1 ðuþ ¼2=ðuÞ ðb2þ ðb1þ

16 1228 S.-Y. YANG AND H.-D. YEH Equations (14) (15), resectively, become A 1 ðu; r 1 Þ¼ ½J 1 ðr 1 uþy 0 ðr 1 uþ Y 1 ðr 1 uþj 0 ðr 1 uþšy 0 ðkr 1 uþ ¼ 2 r 1 u Y 0ðkr 1 uþ ðb3þ A 2 ðu; r 1 Þ¼½J 1 ðr 1 uþy 0 ðr 1 uþ Y 1 ðr 1 uþj 0 ðr 1 uþšj 0 ðkr 1 uþ ¼ 2 r 1 u J 0ðkr 1 uþ ðb4þ Substituting (B3) (B4) into (12) yields (13). Here we have shown the continuity of hydraulic head between the atch outer regions. Taking the derivatives of A 1 ðu; rþ A 2 ðu; rþ with resect to r letting r ¼ r 1 ; resectively, 1 ðu; ¼ 2 rffiffiffiffiffiffiffiffiffiffi S 2 T 2 Y 1 ðkr 1 uþ ðb5þ r¼r1 r 1 S 1 T 2 ðu; ¼ 2 rffiffiffiffiffiffiffiffiffiffi S 2 T 2 J 1 ðkr 1 uþ ðb6þ r¼r1 r 1 S 1 T 1 Furthermore, letting r ¼ r 1 multilying by T 1 on both sides after taking the derivative of (12) with resect to r 1 T ¼ 4h rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z w S 2 T 1 T 1 2 r¼r1 2 e ðt 1=S 1 Þu 2 t ½J 1ðkr 1 uþb 1 ðuþþy 1 ðkr 1 uþb 2 ðuþš du r 1 S 1 0 ½B 2 1 ðuþþb2 2 ðuþš ðb7þ u Similarly, one can 2 T ¼ 4h rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z w S 2 T 1 T 1 2 r¼r1 2 e ðt 1=S 1 Þu 2 t ½J 1ðkr 1 uþb 1 ðuþþy 1 ðkr 1 uþb 2 ðuþš du r 1 S 1 0 ½B 2 1 ðuþþb2 2 ðuþš ðb8þ u Therefore, we 1 T ¼ T 2 ðb9þ r¼r1 Again, we have shown the continuity of flow rate between the atch outer regions. We can carry similar stes demonstrated above by taking the first second derivatives with resect to r the derivative with resect to t for (12) (13) to rove that the time-domain solutions satisfy the governing equations, (1) (2). McLachlan [20, ] gave the 0 ðruþ ¼ uj 1 1 ¼ u 1 ru J 1ðruÞ J 2 ðruþ ðb11þ

17 ANALYTICAL SOLUTION FOR CONSTANT-HEAD TEST 1229 Similarly, 2 ru J 1ðruÞ ¼J 2 ðruþþj 0 0 ¼ uy 1 ðruþ ðb12þ 1 ¼ u 1 ru Y 1ðruÞ Y 2 ðruþ ðb14þ 2 ru Y 1ðruÞ ¼Y 2 ðruþþy 0 ðruþ ðb15þ Based on formulas (B10) (B12), one can 2 J 0 2 þ 0 ðruþ ¼ u 2 J 0 ðruþ ðb16þ 2 Y 0 2 þ 0 ðruþ ¼ u 2 Y 0 ðruþ ðb17þ After taking the first second derivatives with resect to r for A 1 ðu; rþ using formula (B17), one can 2 A 1 ðu; 2 þ 1 1 ðu; rþ S 2 T 2 ¼ ½J 0 ðr 1 uþy 1 ðkr 1 uþð u 2 Y 0 ðruþþ Y 0 ðr 1 uþy 1 ðkr 1 uþð u 2 J 0 ðruþþš S 1 T 1 ½J 1 ðr 1 uþy 0 ðkr 1 uþð u 2 Y 0 ðruþþ Y 1 ðr 1 uþy 0 ðkr 1 uþð u 2 J 0 ðruþþš ðb18þ Alying (14) 2 A 1 ðu; 2 þ 1 ðu; rþ ¼ u 2 A 1 ðu; rþ ðb19þ Similarly, one can 2 A 2 ðu; 2 þ 2 ðu; rþ ¼ u 2 A 2 ðu; rþ ðb20þ Substituting (12) into (1) using (B19) (B20), the left-h side of (1) 2 h 2 þ 1 ¼ 2h Z 1 w e ðt 1=S 1 Þu 2 t ½ u2 A 1 ðu; rþšb 2 ðuþ ½ u 2 A 2 ðu; rþšb 1 ðuþ du 0 ½B 2 1 ðuþþb2 2 ðuþš ðb21þ u Taking the derivative of (12) with resect to t multilying by S 1 =T 1 ; the RHS of (1) yields S 1 T ¼ 2h w Z 1 0 e ðt 1=S 1 Þu 2 t ½ u2 A 1 ðu; rþšb 2 ðuþ ½ u 2 A 2 ðu; rþšb 1 ðuþ du ½B 2 1 ðuþþb2 2 ðuþš u ðb22þ Since (B21) equals (B22), we have shown that the closed-form solution of (12) satisfies (1). Likewise, one can rove that (13) satisfies (2) also.

18 1230 S.-Y. YANG AND H.-D. YEH ACKNOWLEDGEMENTS This study was artly suorted by the Taiwan National Science Council under grant NSC Z The writers areciate the comments suggested revisions of two anonymous reviewers that hel imrove the clarity of our resentation. REFERENCES 1. Barker JA, Herbert R. Puming tests in atchy aquifers. Ground Water 1982; 20(2): Novakowski KS. A comosite analytical model for analysis of uming tests affected by wellbore storage finite thickness skin. Water Resource Research 1989; 25(9): Streltsova TD, McKinley RM. Effect of flow time duration on buildu attern for reservoirs with heterogeneous roerties. Society of Petroleum Engineers Journal 1984; 24: Streltsova TD. Well Testing in Heterogeneous Formations. Wiley: New York, 1988; Novakowski KS. Interretation of the transient flow rate obtained from constant-head tests conducted in situ in clays. Canadian Geotechnical Journal 1993; 30: Markle JM, Rowe RK, Novakowski KS. A model for the constant-heat uming test conducted in vertically fractured media. International Journal for Numerical Analytical Methods in Geomechanics 1995; 19: Chang CC, Chen CS. Analysis of constant-head for a two-layer radially symmetric nonuniform model. Proceeding of the Third Groundwater Resources Water Quality Protection Conference, National Central University, Chung-Li, Taiwan, 1999; Siegel MR. Lalace Transforms. Schaum Publishing Co.: New York, Hildebr FB. Advanced Calculus for Alications. (2nd edn). Prentice-Hall: New York, Carslaw HS, Jaeger JC. Conduction of Heat in Solids (2nd edn). Clarendon Press: Oxford, U.K., Goldstein S. Some two-dimensional diffusion roblems with circular symmetry. Proceedings of the London Mathematical Society (II) 1932; Carslaw HS, Jaeger JC. Some two-dimensional roblems in conduction of heat with circular symmetry. Some Problems in Conduction of Heat 1939; 46: Harvard Problem Reort. A Function Describing the Conduction of Heat in a Solid Medium Bounded Internally by a Cylindrical Surface. Comutation Laboratory of Harvard University, 1950; 76: Jaeger JC. Numerical values for the temerature in radial heat flow. Journal of Mathematical Physics 1956; 34: Hantush MS. Flow of ground water in ss of nonuniform thickness. Part I: Flow in wedge-shaed aquifer. Journal of Geohysical Research 1962; 67(2): Peng HY, Yeh HD, Yang SY. Imroved numerical evaluation of the radial groundwater flow equation. Advances in Water Resources 2002; 25(6): Shanks D. Non-linear transformations of divergent slowly convergent sequences. Journal of Mathematical Physics 1955; 34: Gerald CF, Wheatley PO. Alied Numerical Analysis (4th edn). Addison-Wesley: California, Butler Jr JJ. Puming tests in nonuniform aquifers}the radially symmetric case. Journal of Hydrology 1988; 101: McLachlan NW. Bessel Functions for Engineers (2nd edn). Clarendon Press: Oxford, U.K., 1955.